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The document outlines the International Astronomy and Astrophysics Competition Qualification Round 2019, consisting of various problems related to astronomy and astrophysics, including topics like planetary distances, gravitational forces, and phenomena like polar lights. Participants must solve problems to qualify for the pre-final round, with specific scoring criteria based on correct solutions and steps taken. The document also provides submission guidelines and deadlines for participants.

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International Astronomy and

Astrophysics Competition
Qualification Round 2019

Problem A : Planets and Stars (5 Points)

Fill in the blank spaces with the correct information:

The earth has a distance of (1) light minutes to our sun. When the moon covers the sun we call
this event a (2). There are eight planets in the solar system and (3) is the heaviest of them all. The
smallest planet is (4) and it circles the sun in just (5) days. Besides the planets, there are thousands
of stars visible in the night sky. The brightest star is called (6) and it is just one of about (7) billion
stars in our Milky Way. The (8) galaxy is the closest spiral galaxy to our Mily Way.

(1) (2) (3) (4)

(5) (6) (7) (8)

Problem B : The Size of Jupiter (5 Points)

The Earth has a radius of RE ≈ 6371 km and an average density of ρE ≈ 5.514 cmg 3 . Jupiter is
kg
much bigger and heavier with a radius of RJ ≈ 70000 km and an average density of ρJ ≈ 1326 m3.

Use these values to answer following questions (write down your steps):

(a). Approximately how many Earths fit into Jupiter (by volume)?
(b). How many times heavier is Jupiter compared to Earth?

www.iaac.space 1/3
Problem C : Space Race to the Moon (5 Points)
Alice and Bob are doing a space race from Earth to the Moon which is d ≈ 384000 km far away.
Alice’s spaceship flies with a constant speed of v = 500 km
h
. Bob’s spaceship starts slowly but
km
accelerates constantly with a = 1.4 h2 .

Who wins this space race? (Write down your steps.)

Problem D : Forces between Earth and Moon (5 Points)

Alice and Bob will encounter on their space race the so called Lagrange point L1 at which the
forces from Earth and Moon cancel out. The gravitational force on the spaceships is given by

ME MM
F (r) = mG −
r2 (d − r)2

with the constant G, the mass of the spaceship m, the masses ME , M of Earth and Moon, the dis-
tance d between Earth and Moon, and the distance r of the spacecraft to the center of the earth.
For this problem we assume that the Earth-Moon system is at rest.

(a). Use F (r) to find a formula that calculates the distance to the Lagrange point L1 .
(b). Explain missing aspects in this calculation due to the assumption that the ’Earth-Moon sys-
tem is at rest’.

www.iaac.space 2/3
Problem E : Polar Lights (5 Points)
Since the existence of humans we were fascinated by the natural phenomena of polar lights (au-
rora). The various colors in the skies have inspired many stories and are a symbol for the beauty
of nature. Today, we understand the underlying scientific reasons for this phenomena.

Explain the causes and scientific reasons that explain polar lights.

General Information and Submission

You can write the solution by hand on this paper or type it on a computer. To qualify for the pre-final
round you have to get at least 15 points (Junior, under 18 years) or 20 points (Youth, over 18 years). Make
sure to submit your solution until 14. April 2019 23:59 UTC+0 online at www.iaac.space/submission. In case
of questions or comments you can contact the team of IAAC at info@iaac.space. Good luck!

www.iaac.space 3/3
International Astronomy and
Astrophysics Competition
Qualification Round 2019

Problem A : Planets and Stars (5 Points)

(1) ≈ 8 (2) solar eclipse (3) Jupiter (4) Mercury
(5) ≈ 88 (6) Sirius (7) 200 ± 150 (8) Andromeda

Problem B : The Size of Jupiter (5 Points)

(a). 1326 Earths
(b). 319 times

Problem C : Space Race to the Moon (5 Points)

Bob wins. (Time Alice: 768 hours, Time Bob: 741 hours)

Problem D : Forces between Earth and Moon (5 Points)

√ q −1
M√ MM
(a). F (r1 ) = 0 =⇒ r1 = d √ME + E
MM
= d 1 + ME

(b). Missing aspect: angular velocity of objects (causes additional L-points).

Problem E : Polar Lights (5 Points)

sun activity (e.g. storms) → charged particles (electrons, protons) → interacting with earth’s
magnetic field → follow trajectory of magnetic field to poles → ionization and excitement of
atoms in atmosphere (higher state) → atmosphere atoms emit light (return to ground state) →
colors depending on gas (oxygen: yellow, green; nitrogen: reg, violet, blue)

www.iaac.space Plain Solutions

International Astronomy and
Astrophysics Competition
Pre-Final Round 2019

Important: Read all the information on this page carefully!

General Information
• We recommend to print out this problem sheet. Use another paper to draft the solutions to the
problems and write your final solution (with steps) on the provided space below the problems.

• You may use extra paper if necessary, however, the space under the problems is usually enough.

• Typing the solution on a computer is allowed but not recommended (no extra points).

• The 10 problems are separated into three categories: 4x basic problems (A; four points), 4x advanced
problems (B; six points), 2x research problems (C; ten points). The research problems require you
to read a short scientific article to answer the questions. There is a link to the PDF article.

• You receive points for the correct solution and for the performed steps. Example: You will not get
all points for a correct value if the calculations are missing.

• Make sure to clearly mark your final solution values (e.g. underlining, red color, box).

• You can reach up to 60 points in total. You qualify for the final round if you reach at least 25 points
(junior, under 18 years) or 35 points (youth, over 18 years).

• It is not allowed to work in groups on the problems. Help from teachers, friends, family, or the inter-
net is prohibited. Cheating will result in disqualification! (Textbooks and calculators are allowed.)

Uploading Your Solution

• Please upload a file/pictures of (this sheet with) your written solutions: https://iaac.space/status

• Only upload one single PDF file! If you have multiple pictures, please compress them into one
single file. Do not upload your pictures in a different format (e.g, no Word and Zip files).

• The deadline for uploading your solution is Sunday 12. May 2019, 23:59 UTC+0.

• The results of the pre-final round will be announced on Monday 20. May 2019.

Good luck!

www.iaac.space 1/11
Problem A.1 : Journey to Proxima Centauri (4 Points)
The diameter of the Sun is 1.39 million kilometres and the Earth is 8.3 light minutes far away.
Proxima Centauri is the nearest star - it has a distance of 4.24 light years to our Sun.

(a) How long does it take to travel to Proxima Centauri with

(i) an airplane (920 km/h) or
(ii) with the Voyager 1 space probe (17 km/s).

(b) Let the Sun have the size of a tennis ball (diameter: 6.7 cm): How far away is the Earth and
how far away is Proxima Centauri on this scale?

www.iaac.space 2/11
Problem A.2 : Orbit of the Solar System (4 Points)
The Milky Way has a diameter of about 150,000 light years. Our solar system is located 27,000
light years from the center of the Milky Way and orbits the center with a speed of 220 km/s.

(a) How long does it take for the solar system to circle the center of the Milky Way?
(b) The earth has formed about 4.5 billion years ago. How often has the earth circled the center?

www.iaac.space 3/11
Problem A.3 : Distance to Arcturus (4 Points)
The stellar parallax of the star Arcturus in the constellation Boötes was measures with 0.0900 .

(a) Calculate the distance (in parsec) between Arcturus and the Earth.
(b) How long does it take to send a light message from Earth to Arcturus?

www.iaac.space 4/11
Problem A.4 : From Earth to Mars (4 Points)
For a special mission to Mars you need to know the smallest distance between Earth and Mars.
However, you have lost your astronomy book and you could only find these values:

Distance Earth to Sun: 149.6 million km

Orbital period Earth: 1.00 years
Orbital period Mars: 1.88 years

By using these values and assuming that Mars and Earth move an circular orbits, calculate the
smallest possible distance between Earth and Mars.

www.iaac.space 5/11
Problem B.1 : New Star (6 Points)
You have discovered a new star in the Milky Way: Your new star is red and has 3/5 the tempera-
ture of our Sun. The new star emits a total power that is 100,000 times greater than the power
emitted by our Sun.

(a) Determine the spectral type (i.e. spectral classification) of the new star.
(b) How many times bigger is the radius of the new star compared to the radius of our Sun?

www.iaac.space 6/11
Problem B.2 : Moon Satellite (6 Points)
The Moon has a mass of M = 7.3 · 1022 kg, a radius of R = 1.7 · 106 m and a rotation period of
T = 27.3 days. Scientists are planning to place a satellite around the Moon that always remains
above the same position (geostationary).

(a) Calculate the distance from the Moon’s surface to this satellite.
(b) Explain if such a Moon satellite is possible in reality.

www.iaac.space 7/11
Problem B.3 : Binary Star System (6 Points)
You are the captain of a spaceship that is circling through a binary star system. Due to the gravi-
tational forces and the rocket engines, the orbit of your spaceship looks like that:

The position of your spaceship (in AU) at the time t (in days) is given by:

x = 5 sin(t) y = sin(2t) z=0

(a) How long does it take your spaceship to circle the orbit once?
(b) Find an equation that calculates the velocity v(t) of your spaceship at a given time t.
(c) The two stars are positioned at the points (4, 0, 0) and (−4, 0, 0): What is the distance of your
spaceship to the stars at the time t = π2 ?

www.iaac.space 8/11
Problem B.4 : Asteroid Collision (6 Points)
A warning system has calculated that two asteroids will collide not far from Earth any time soon.
The smaller asteroid has the mass m and moves with the velocity vm . The bigger asteroid has the
mass M = 3m and the velocity of vM = 21 vm . They collide at an angle of α = 60◦ and turn into a
single heavy asteroid (inelastic collision):

β
M
α

(a) Calculate the velocity of the single object after the collision.
(b) Determine the angle β after the collision.

www.iaac.space 9/11
Problem C.1 : The Sunburst Arc (10 Points)
This problem requires you to read following recently published scientific article:

The Sunburst Arc: Direct Lyman α escape observed in the brightest known lensed galaxy.
T.E. Rivera-Thorsen, H. Dahle, M. Gronke, M. Bayliss, J.R. Rigby, R. Simcoe, R. Bordoloi, M. Turner, and G. Furesz,
Astronomy & Astrophysics 608, (2017). Link: https://www.aanda.org/articles/aa/pdf/2017/12/aa32173-17.pdf

Answer following questions related to this article:

(a) Why is it difficulty for LyC radiation to escape galaxies with high star formation rates?

(b) What is the difference between the density-bounded medium and the picket fence model?

(d) What is the Sunburst Arc and how was it discovered?

(e) When did the scientist observe the object and which instruments did they use?

(f) What is the redshift of the Sunburst Arc and how was it determined from the data?

(g) Explain the difference between the right and the left diagram in figure 4 (see article).

www.iaac.space 10/11
Problem C.2 : Dark Matter (10 Points)
This problem requires you to read following recently published scientific article:

Probing Dark Matter Using Precision Measurements of Stellar Accelerations.

A. Ravi, N. Langellier, D.F. Phillips, M. Buschmann, B.R. Safdi, R.L. Walsworth,
arXiv:1812.07578, (2018). Link: https://arxiv.org/pdf/1812.07578.pdf

Answer following questions related to this article:

(a) What are the current methods to determine the dark matter density / radial velocity and what
are the disadvantages of these methods?

(b) Explain the new method for measuring dark matter density that is proposed in the article.

(c) Let ar (r) be the dark matter

contribution to the
acceleration: Show that the dark matter den-
1 ∂a r
sity is given by ρDM ≈ 2(A − B)2 − with Oort constants 2(A − B)2 = 2GM (r)/r3 .
4πG ∂r

(d) How much does the velocity of the stars change during the lifetime of a human (80 years)?

(e) What is important for measuring stellar accelerations and which instruments can be used?

(f) Explain the curves of the four diagrams in figure 2 (see article).

www.iaac.space 11/11
International Astronomy and
Astrophysics Competition
Pre-Final Round 2019

Solutions to the Pre-Final Round 2019

Please note that there are many ways to reach the final solutions.
Not all detailed steps are elaborated in this solution document.

(a) How long does it take to travel to Proxima Centauri with

(i) an airplane (920 km/h) or
(ii) with the Voyager 1 space probe (17 km/s).

(b) Let the Sun have the size of a tennis ball (diameter: 6.7 cm): How far away is the Earth and
how far away is Proxima Centauri on this scale?

Solution a.i:
c
tair = 4.24y · ≈ 5 million years
vair
Solution a.ii:
c
tvoy = 4.24y · ≈ 75 thousand years
vvoy
Solution b:
L: scaled distance, dS : diameter Sun, D: diameter of tennis ball
8.3 min · c
Learth = D · ≈ 7.2 m
dS
4.24 y · c
Lstar = D · ≈ 1930 km
dS

(a) How long does it take for the solar system to circle the center of the Milky Way?
(b) The earth has formed about 4.5 billion years ago. How often has the earth circled the center?

Solution a:
2πrsun 2πtsun c
T = = ≈ 231 million years
vsun vsun
Solution b:
4.5 · 109 y
≈ 19.5 rotations
231 · 106 y

www.iaac.space 3/11
Problem A.3 : Distance to Arcturus (4 Points)
The stellar parallax of the star Arcturus in the constellation Boötes was measures with 0.0900 .

(a) Calculate the distance (in parsec) between Arcturus and the Earth.
(b) How long does it take to send a light message from Earth to Arcturus?

Solution a:
Stellar parallax p = 0.0900 :
1pc · 100
d= = 11.11 pc
p
Solution b:

11.11 pc = 32.22 ly ⇒ 36.22 years

Distance Earth to Sun: 149.6 million km

Orbital period Earth: 1.00 years
Orbital period Mars: 1.88 years

By using these values and assuming that Mars and Earth move an circular orbits, calculate the
smallest possible distance between Earth and Mars.

Solution:
Kepler’s third law → T 2 /R3 = const.
" 2/3 #
TE2 2
TM 2
TM TM
3
= 3
= ⇒ d= − 1 RE ≈ 78.28 million km
RE RM (RE + d)3 TE

(a) Determine the spectral type (i.e. spectral classification) of the new star.
(b) How many times bigger is the radius of the new star compared to the radius of our Sun?

Solution a:
Red color, 3/5 cooler than sun → Class M star

Solution b:
Stefan-Boltzmann law: (total power) L = σAT 4 = 4πσ · R2 T 4
Properties of Sun: T0 , R0 , T0 :
2 4 r 2
L 4πσ · R2 T 4 R T R L T0
= 2 4
= ⇒ = ≈ 878 times bigger
L0 4πσ · R0 T0 R0 T0 R0 L0 T

(a) Calculate the distance from the Moon’s surface to this satellite.
(b) Explain if such a Moon satellite is possible in reality.

Solution a:
For a stable orbit (using Newton’s law of gravitation):
2
M 2 2π
agrav = aradial ⇒ G 2
= (h + R)ω = (h + R)
(h + R) T
" #1/3
2
T
⇒ h= GM − R ≈ 86, 500 km
2π

Solution b:
Distance Earth-Moon: 384,000 km → gravitational interference from earth → not possible

The position of your spaceship (in AU) at the time t (in days) is given by:

x = 5 sin(t) y = sin(2t) z=0

Solution a:
Superposition movement:

2π 2π
x = 5 sin(t) = 5 sin t ⇒ t= t ⇒ T = 2π ≈ 6.28 days
T T

Solution b:
   
5 sin(t) 5 cos(t)
~x(t) =  sin(2t)  ⇒ ~v (t) = ~x˙ (t) = 2 cos(2t) ⇒ v(t) = 25 cos2 (t) + 4 cos2 (2t)
    p

0 0

Solution c:
 
4 q
~
d± (t) = ~x(t) ± 0 ⇒ d± (t) = (5 sin(t) ± 4)2 + sin2 (2t)
 

0
π π
⇒ d+ = 9 AU, d− = 1 AU
2 2

β
M
α

(a) Calculate the velocity of the single object after the collision.
(b) Determine the angle β after the collision.

Solution a:
Inelastic collision → p~ = const. (conservation of momentum):
3
p~ = M~vM + m~vm = M vM êM + mvm êm = vm êM + mvm êM
2
= (M + m)~v = 4m~v
" ! !# !
3 1 1 3 1 cos α 1 3/2 + cos α
⇒ ~v = vm êM + vm êm = vm + = vm
8 4 4 2 0 sin α 4 sin α
√
1 19 vm
q
⇒ v = |~v | = vm (3/2 + cos α)2 + sin2 α = vm ≈
4 8 2
Solution b:
! !
cos β 1 3/2 + cos α 1
~v = vêv = v = vm ⇒ v sin β = vm sin α
sin β 4 sin α 4

1 vm
⇒ β = arcsin sin α ≈ 23.41◦
4 v

www.iaac.space 9/11
Problem C.1 : The Sunburst Arc (10 Points)
This problem requires you to read following recently published scientific article:

Answer following questions related to this article:

(a) Why is it difficulty for LyC radiation to escape galaxies with high star formation rates?
→ containing large amounts of neutral hydrogen (opaque to LyC at column densities)

(b) What is the difference between the density-bounded medium and the picket fence model?
→ density-bounded medium: region is surrounded by gas with sufficiently low column density
to not completely weaken the LyC radiation
→ picket fence model: region is surrounded by optically thick gas, but does not completely cover
the source (radiation can pass through holes)

(c) Explain the spectral shape of the perforated shell model (see article: figure 1, right box).
→ narrow central peak due to escaping radiation (through holes)
→ overlaid by characteristic double-peak profile that emerges from optically thick gas

(d) What is the Sunburst Arc and how was it discovered?

→ a (lensed) galaxy (PSZ1-ARC G311.6602-18.4624)
→ discovered due to Sunyaev-Ze’dovich effect in the Planck data

(e) When did the scientist observe the object and which instruments did they use?
→ observations: UT 24 May 2017, beginning at 03:31, and UT 30 March 2016, beginning at 09:06
→ instruments: Magellan Echellette (MagE) spectro., Folded-port InfraRed Echelle (FIRE) spectro.

(f) What is the redshift of the Sunburst Arc and how was it determined from the data?
→ z = 2.37094 ± 0.00001, by fitting a single Gaussian profile to the strong emission lines

(g) Explain the difference between the right and the left diagram in figure 4 (see article).
→ right diagram: plot of the observation data
→ fitting and subtracting the central peak (of right diagram) to a Gaussian profile → left diagram

www.iaac.space 10/11
Problem C.2 : Dark Matter (10 Points)
This problem requires you to read following recently published scientific article:

Probing Dark Matter Using Precision Measurements of Stellar Accelerations.

A. Ravi, N. Langellier, D.F. Phillips, M. Buschmann, B.R. Safdi, R.L. Walsworth,
arXiv:1812.07578 [astro-ph.GA], (2018). Link: https://arxiv.org/pdf/1812.07578.pdf

Answer following questions related to this article:

(a) What are the current methods to determine the dark matter density or radial velocity and
what are the disadvantages of these methods?
→ current methods: Doppler shifts, dispersion of local stellar velocities (vertical direction)
→ disadvantages: indirect and subject to large systematic uncertainties

(b) Explain the new method for measuring dark matter density that is proposed in the article.
→ direct measurement of stellar accelerations → gravitational potential → dark matter density

(c) Let ar (r) be the dark matter

contribution to the
acceleration: Show that the dark matter den-
1 ∂a r
sity is given by ρDM ≈ 2(A − B)2 − with Oort constants 2(A − B)2 = 2GM (r)/r3 .
4πG ∂r

→ contribution to the acceleration: ar (r) = −G Mr(r)

2
M 0 (r) M 0 (r)
→ dark matter density: ρDM = V (r) = 4πr2 ⇒ M 0 (r) = 4πr2 ρDM
M 0 (r)

∂ar M (r) 2 1 2 ∂ar
→ = 2G 3 − G 2 = 2(A − B) − 4πGρDM ⇒ ρDM ≈ 2(A − B) −
∂r r r 4πG ∂r

(d) How much does the velocity of the stars change during the lifetime of a human (80 years)?
→ rate: 0.5 cm/s/year → velocity change in 80 years: 40 cm/s

(e) What is important for measuring stellar accelerations and which instruments can be used?
→ requirement: extremely stable calibration of the spectrograph
→ spectrograph → possible instruments for calibration: laser frequency comb (astro-combs)

(f) Explain the curves of the four diagrams in figure 2 (see article).
→ (a) vaccel : acceleration due to gravitation Milky Way, constant rate
→ (b) vcomp : stellar companions (e.g. star cluster), periodic change
→ (c) vplan : planets around star, (many) overlapping periodic oscillations
→ (d) vnoise : noise (e.g. magnetic activities, instruments), random with some recurring noise

www.iaac.space 11/11
International Astronomy and
Astrophysics Competition
Final Round 2019

Important: Read all the information on this page carefully!

• Please read all questions carefully!

• This exam consists of 40 multiple-choice questions.

• To every question, there are four possible answers: A, B, C and D.

• Only one of the four answers is correct!

• Every correct answer gives you one point.

• There are no negative points for wrong answers.

• You have strictly 60 minutes to solve as many problems as possible.

• Write your answers on the Your-Answers-page only (see next page)!

• After the 60 minutes, return the exam (including all question pages) back to your teacher.

• You are allowed to...

– use a pencil/pen for writing.

– use extra blank papers for personal notes.

• You are not allowed to...

– work more than 60 minutes on this exam.

– use electronic devices (e.g. internet, calculators).
– use any source of information (e.g. notes, books).
– receive help from your supervisor or other students.
– take this exam, your answers, or the questions with you (after the exam).

• Cheating Policy: In addition to the presence and supervision of your teacher during the examina-
tion we have various additional methods to detect cheating: This includes methods to detect time
violations as well as to detect the usage of tools (e.g. internet, textbooks) for cheating.
Cheating will result in immediate disqualification!
Good luck!

IAAC Final Exam 2019 1/8

Your Answers

Name:

Please write your answers on this page!

No. 1 No. 2 No. 3 No. 4 No. 5 No. 6 No. 7 No. 8 No. 9 No. 10

No. 11 No. 12 No. 13 No. 14 No. 15 No. 16 No. 17 No. 18 No. 19 No. 20

No. 21 No. 22 No. 23 No. 24 No. 25 No. 26 No. 27 No. 28 No. 29 No. 30

No. 31 No. 32 No. 33 No. 34 No. 35 No. 36 No. 37 No. 38 No. 39 No. 40

Please write A, B, C or D into the boxes to give your answers.

IAAC Final Exam 2019 2/8

Question 1 : Most people us C°(degree Celsius) to measure everyday temperatures. Astronomers
prefer to use K (Kelvin) to measure temperatures. Approximately how many Kelvin are 25 C°?

(A) 200 Kelvin (B) 250 Kelvin (C) 300 Kelvin (D) 350 Kelvin

Question 2 : Mars might be a place for future human explorations. However, humans can not
breathe on the surface of Mars because the atmosphere consists mostly of ...

(A) Nitrogen (B) Argon (C) Methane (D) CO2

Question 3 : The two moons of Mars are called ...

(A) Tritos and Desmos (B) Tritos and Deimos

Question 4 : On which planet in our solar system can you find the Great Red Spot?

(A) Venus (B) Mars (C) Jupiter (D) Saturn

Question 5 : The axis of the Earth is tilted at an angle of approximately ... relative to the orbital
plane around the Sun.

(A) 20.3 degrees (B) 21.4 degrees (C) 22.7 degrees (D) 23.5 degrees

Question 6 : The Pleiades is an open star cluster that plays a role in many ancient stories and is
well-known for containing ... bright stars.

(A) 5 (B) 7 (C) 9 (D) 12

Question 7 : Previous IAAC rounds featured Proxima/Alpha Centauri as closes star(system) to the
Earth. Which one is the second closest star(system)?

(A) Wolf 359 (B) Sirius (C) 61 Cygni (D) Barnard’s Star

Question 8 : The visible part of the electromagnetic spectrum is between ...

(A) 240 to 680 nm. (B) 360 to 620 nm. (C) 380 to 740 nm. (D) 420 to 810 nm.

Question 9 : The constellation ... is a bright W-shaped constellation in the northern sky.

(A) Centaurus (B) Cygnus (C) Cassiopeia (D) Cepheus

IAAC Final Exam 2019 3/8

Question 10 : What is the so-called bolometric luminosity in astronomy?

(A) The luminosity, integrated over vertically polarized wavelengths.

(B) The luminosity, integrated over horizontally wavelengths.
(C) The luminosity, integrated over visible wavelengths.
(D) The luminosity, integrated over all wavelengths.

Question 11 : The second cosmic velocity (or escape velocity) is the speed required by an object
to escape the gravitational field of a celestial body with mass M and radius R. Which formula
correctly calculates this velocity? (G: gravitational constant)
q q q q
GM
(A) v = R
(B) v = 2 GM
R
(C) v = 2GM
R
(D) v = GM
2R

Question 12 : The International Space Station (ISS) circles the Earth approximately 410 km above
the ground. Find the best estimate for the orbital speed of the ISS:

(A) 19,000 km/h (B) 21,000 km/h (C) 28,000 km/h (D) 32,000 km/h

Question 13 : Find the best approximation for the surface temperature of the Sun:

(A) 6000 K (B) 7000 K (C) 9000 K (D) 13000 K

Question 14 : The four big moons of Jupiter are Callisto, Europa, Ganymede, and Io. Which one
of them has the smallest distance to Jupiter?

(A) Callisto (B) Europa (C) Ganymede (D) Io

Question 15 : The name of the black hole in the center of our Milky Way is ...

(A) Altair A* (B) Alsephina A* (C) Fomalhaut A* (D) Sagittarius A*

Question 16 : Approximately how far away is the Andromeda Galaxy?

(A) 1.7 million light years (B) 2.1 million light years
(C) 2.5 million light years (D) 3.2 million light years

Question 17 : When was the telescope invented by Galileo?

(A) 1409 (B) 1509 (C) 1609 (D) 1709

IAAC Final Exam 2019 4/8

Question 18 : The astronomical unit parsec (pc) plays a crucial role in astronomy. One parsec is
equal to about 3.26 light-years. How is one parsec defined in astronomy?

(A) Distance at which one astronomical unit measures one arcsecond from Earth.
(B) Orbital distance of the solar system around the center of the Milky Way in one year.
(C) Effective distance of the solar wind (i.e. the radius of the heliosphere).
(D) Historical distance to the brightest star Sirius.

Question 19 : The picture below shows a very famous nebula: What is the name of this nebula?

Figure 1: Nebula. Credit: NASA

(A) Crab Nebula (B) Orion Nebula (C) Ring Nebula (D) Carina Nebula

Question 20 : The star Betelgeuse is one of the brightest stars in the night sky with remarkable
red color. In which constellation is Betelgeuse located?

(A) Cassiopeia (B) Cygnus (C) Ursa Major (D) Orion

Question 21 : What does the astronomical term ecliptic describe?

(A) The path of the Sun in the sky throughout a year.

(B) The axial tilt of the Earth throughout a year.
(C) The movement of the stars due to Earth’s rotation.
(D) The central line through the axis of rotation.

Question 22 : Approximately how long does it take Pluto to orbit the Sun once?

(A) 150 years (B) 200 years (C) 250 years (D) 300 years

IAAC Final Exam 2019 5/8

Question 23 : Sunspots are black regions that temporarily appear on the Sun. Their number
highly increases every ... years. This period is also called the solar cycle.

(A) 9 (B) 11 (C) 13 (D) 15

Question 24 : The Large Magellanic Cloud is ...

(A) a dwarf galaxy orbiting the Milky Way.

(B) the closest planetary nebula to the Earth.
(C) a bright star cluster discovered by Magellan.
(D) the outer arm of the Milky Way named after Magellan.

Question 25 : The Milky Way is part of a giant supercluster with a diameter of 160 Mpc. What is
the name of this supercluster?

(A) Virgo (B) Laniakea (C) Sculptor (D) Boötes

Question 26 : Why are Cepheid stars relevant for astronomers?

(A) To measure interstellar mass. (B) To measure galactic distances.

Question 27 : What type of radiation causes a black hole to evaporate over time?

(A) Schwarzschild radiation (B) Planck radiation

Question 28 : In astronomy, the concept of black bodies is very important to better calculate the
radiation of stars. Which one is the correct definition of a black body?

(A) An idealized physical object that reflects all electromagnetic radiation.

(B) An idealized physical object that absorbs all electromagnetic radiation.
(C) An idealized physical object that reflects all polarized radiation.
(D) An idealized physical object that absorbs all polarized radiation.

Question 29 : One astronomical unit (AU) is equal to approximately ...

(A) 130 million km (B) 150 million km (C) 170 million km (D) 190 million km

IAAC Final Exam 2019 6/8

Question 30 : Radio telescopes are crucial for astronomical observations. Which one of these
well-known radio telescopes has the largest parabolic antenna?

(A) Green Bank Telescope (B) Arecibo Telescope

Question 31 : Which one of these constellations is not located along the Milky Way in the sky?

(A) Perseus (B) Cygnus (C) Scorpius (D) Leo

Question 32 : A comet’s tail points in the following direction:

(A) away from the Sun (B) towards the Sun

Question 33 : As the life of a star progresses, heavy elements are produced. The elements form
layers around the star in this order (starting from the outer layer):

(A) H → He → Li → N → O → Si → Fe (B) H → He → C → O → Ne → Si → Fe
(C) H → He → Li → O → Ne → Si → Fe (D) H → He → C → N → O → Si → Fe

Question 34 : The so-called dark energy is a model to explain ...

(A) the radiation of black holes.

(B) the mass distribution of galaxies.
(C) the acceleration of the universe.
(D) the microwave background of the universe.

Question 35 : As a star collapses at the end of its life, the triple-alpha reaction takes place. Which
one of these equations describes this reaction correctly?

(A) 21 H +21 H +21 H → 63 Li + γ

(B) 42 He +42 He +42 He → 12
6C +γ
(C) 21 H +21 H +42 He +42 He → 12
6C +γ
(D) 21 H +42 He → 63 Li + γ

Question 36 : Besides large astronomical objects, astrophysicists are also interested in small par-
ticles from space. Which one of these particles has the weakest interactions with other particles?

(A) µ Muons (B) Antiparticles (C) ν Neutrinos (D) π Pions

IAAC Final Exam 2019 7/8

Question 37 : The term Schwarzschild radius usually describes properties of ...

(A) red dwarfs. (B) pulsars. (C) black holes. (D) galaxies.

Question 38 : What is the correct numerical value and unit of the Boltzmann constant?

(A) 1.38 × 10−21 m3 · kg · s−2 · K −1 (B) 1.38 × 10−22 m2 · kg · s−3 · K −1

Question 39 : What is true for a type-Ia ("type one-a") supernova?

(A) This type occurs in binary systems.

(B) This type occurs often in young galaxies.
(C) This type produces gamma-ray bursts.
(D) This type produces high amounts of X-rays.

Question 40 : The famous Drake equation attempts to answer the following question:

(A) Will the Sun become a black hole? (B) Is the universe infinitely large?
(C) How old is the visible universe? (D) Are we alone in the universe?

IAAC Final Exam 2019 8/8

International Astronomy and
Astrophysics Competition
Final Round 2019

Important: Read all the information on this page carefully!

• Please read all questions carefully!

• This exam consists of 40 multiple-choice questions.

• To every question, there are four possible answers: A, B, C and D.

• Only one of the four answers is correct!

• Every correct answer gives you one point.

• There are no negative points for wrong answers.

• You have strictly 60 minutes to solve as many problems as possible.

• Write your answers on the Your-Answers-page only (see next page)!

• After the 60 minutes, return the exam (including all question pages) back to your teacher.

• You are allowed to...

– use a pencil/pen for writing.

– use extra blank papers for personal notes.

• You are not allowed to...

– work more than 60 minutes on this exam.

IAAC Final Exam 2019 1/2

Your Answers

Name: Solution

Please write your answers on this page!

No. 1 No. 2 No. 3 No. 4 No. 5 No. 6 No. 7 No. 8 No. 9 No. 10

C D D C D B D C C D

No. 11 No. 12 No. 13 No. 14 No. 15 No. 16 No. 17 No. 18 No. 19 No. 20

C C A D D C C A A D

No. 21 No. 22 No. 23 No. 24 No. 25 No. 26 No. 27 No. 28 No. 29 No. 30

A C B A B B D B B B

No. 31 No. 32 No. 33 No. 34 No. 35 No. 36 No. 37 No. 38 No. 39 No. 40

D A B C B C C C A D

Please write A, B, C or D into the boxes to give your answers.

IAAC Final Exam 2019 2/2

International Astronomy and
Astrophysics Competition
Qualification Round 2020

Problem A : The Solar System (5 Points)

Fill in the blank spaces with the correct information:

The Sun is in the center of Solar System and is composed mainly of the elements (1) and (2). The
distance from the Earth to the Sun is also called (3). Many people dream about building a colony on
Mars, but the atmosphere is primarily made of (4). We have discovered the (5) between Mars and
Jupiter, which contains millions of small objects. Jupiter has a total of (6) moons: The four largest
moons are easily visible with a telescope and (7) is the closest and most active one. Uranus and Nep-
tune are the outermost planets and it takes Neptune (8) years to complete one orbit around the Sun.

(1) (2) (3) (4)

(5) (6) (7) (8)

Problem B : Cosmic Scales (5 Points)

Assume the diameter of the Earth (12,700 km) is scaled down to 1 cm and answer the following:

(a) How large is the Sun (diameter: 1.4×106 km) on this scale?
(b) How far away is the nearest star (distance: 4.24 light-years) on this scale?

www.iaac.space 1/3
Problem C : Distance to the Moon (5 Points)
During the daylight, you hold a ruler in a distance of 60 cm away from your eyes, and you find
the size of the Moon to be 0.55 cm (try it yourself!). At night, you use a telescope to observe rock
formations and craters on the Moon to estimate the diameter of the Moon to be about 3500 km.

Find the distance to the Moon by using only the information from this experiment.

Problem D : Energy of Satellites (5 Points)

A satellite of mass mS orbits the Earth (with mass mE and radius RE ) with a velocity v and an
altitude h. The gravitational force FG and the centripetal force FC are given by:

mS · mE mS · v 2
FG = G , FC = , G = const.
(RE + h)2 RE + h

(a) Find an equation for the kinetic energy Ekin (h) of a satellite with an altitude h.
(b) Based on the kinetic energy, how much liquid hydrogen (energy density: 106 J/Litre) is at least
needed to bring a small 1 kg satellite in an orbit of 400 km. (Use literature to find mE , RE , G.)

www.iaac.space 2/3
Problem E : Nuclear Fusion (5 Points)
The light from the Sun is essential for all life on Earth. For a long time, we did not understand
where all of this energy is coming from and how the sunlight is generated. Today, we know that
the process of nuclear fusion is responsible for the energy production in the Sun and other stars.

Explain how nuclear fusion in the Sun produces the sunlight we see on Earth.

General Information and Submission

You can write the solution by hand on this paper or type it on a computer. To qualify for the pre-final round,
you have to get at least 15 points (Junior, under 18 years) or 20 points (Youth, over 18 years). Make sure to
submit your solution until Friday 15. May 2020 23:59 UTC+0 online at www.iaac.space/submission. In case
of questions or comments, please feel free to contact us via email: info@iaac.space. Good luck!

www.iaac.space 3/3
International Astronomy and
Astrophysics Competition
Qualification Round 2020

Problem A : The Solar System (5 Points)

(1) hydrogen (2) helium (3) astronomical unit (4) carbon dioxide
(5) asteroid belt (6) 79 (7) Io (8) ≈ 165 years

Problem B : Cosmic Scales (5 Points)

(a) ≈ 110 cm
(b) ≈ 31,580 km

Problem C : Distance to the Moon (5 Points)

sM sR sM
dM
= dR
=⇒ sM = dM · dM
≈ 382, 000 km

Problem D : Energy of Satellites (5 Points)

(a) Ekin (h) = 12 mS v 2 = 21 Gm S mE
RE +h
(b) V = Ekin (h)/ρ ≈ 29.4 Litre

Problem E : Nuclear Fusion (5 Points)

two protons fuse → transformation of one proton into a neutron: deuterium → deuterium col-
lides with another proton → release of gamma rays → the gamma rays travels to the surface of
the Sun → sunlight

www.iaac.space Plain Solutions

International Astronomy and
Astrophysics Competition
Pre-Final Round 2020

Important: Read all the information on this page carefully!

• You may use extra paper if necessary, however, the space under the problems is usually enough.

• Typing the solution on a computer is allowed but not recommended (no extra points).

• The 10 problems are separated into three categories: 4x basic problems (A; four points), 4x advanced
problems (B; six points), 2x research problems (C; ten points). The research problems require you
to read a short scientific article each to answer the questions. There is a link to the PDF article.

• You receive points for the correct solution and for the performed steps. Example: You will not get
all points for a correct value if the calculations are missing.

• Make sure to clearly mark your final solution values (e.g. underlining, red color, box).

• You can reach up to 60 points in total. You qualify for the final round if you reach at least 25 points
(junior, under 18 years) or 35 points (youth, over 18 years).

Uploading Your Solution

• Please upload a file/pictures of (this sheet with) your written solutions: https://iaac.space/login

• Only upload one single PDF file! If you have multiple pictures, please compress them into one
single file. Do not upload your pictures in a different format (e.g, no Word and Zip files).

• The deadline for uploading your solution is Sunday 21. June 2020, 23:59 UTC+0.

• The results of the pre-final round will be announced on Monday 29. June 2020.

Good luck!

www.iaac.space 1/12
Problem A.1: Interstellar Mission (4 Points)
You are on an interstellar mission from the Earth to the 8.7 light-years distant star Sirius. Your
spaceship can travel with 70% the speed of light and has a cylindrical shape with a diameter of
6 m at the front surface and a length of 25 m. You have to cross the interstellar medium with an
approximated density of 1 hydrogen atom/m3 .

(a) Calculate the time it takes your spaceship to reach Sirius.

(b) Determine the mass of interstellar gas that collides with your spaceship during the mission.

Note: Use 1.673 × 10−27 kg as proton mass.

www.iaac.space 2/12
Problem A.2: Time Dilation (4 Points)
Because you are moving with an enormous speed, your mission from the previous problem A.1
will be influenced by the effects of time dilation described by special relativity: Your spaceship
launches in June 2020 and returns back to Earth directly after arriving at Sirius.

(a) How many years will have passed from your perspective?
(b) At which Earth date (year and month) will you arrive back to Earth?

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Problem A.3: Magnitude of Stars (4 Points)
The star Sirius has an apparent magnitude of -1.46 and appears 95-times brighter compared to
the more distant star Tau Ceti, which has an absolute magnitude of 5.69.

(a) Explain the terms apparent magnitude, absolute magnitude and bolometric magnitude.
(b) Calculate the apparent magnitude of the star Tau Ceti.
(c) Find the distance between the Earth and Tau Ceti.

www.iaac.space 4/12
Problem A.4: Emergency Landing (4 Points)
Because your spaceship has an engine failure, you crash-land with an emergency capsule at the
equator of a nearby planet. The planet is very small and the surface is a desert with some stones
and small rocks laying around. You need water to survive. However, water is only available at the
poles of the planet. You find the following items in your emergency capsule:

• Stopwatch

• Electronic scale

• 2m yardstick

• 1 Litre oil

• Measuring cup

Describe an experiment to determine your distance to the poles by using the available items.

Hint: As the planet is very small, you can assume the same density everywhere.

www.iaac.space 5/12
Problem B.1: Temperature of Earth (6 Points)
Our Sun shines bright with a luminosity of 3.828 x 1026 Watt. Her energy is responsible for many
processes and the habitable temperatures on the Earth that make our life possible.

(a) Calculate the amount of energy arriving on the Earth in a single day.
(b) To how many litres of heating oil (energy density: 37.3 x 106 J/litre) is this equivalent?
(c) The Earth reflects 30% of this energy: Determine the temperature on Earth’s surface.
(d) What other factors should be considered to get an even more precise temperature estimate?

Note: The Earth’s radius is 6370 km; the Sun’s radius is 696 x 103 km; 1 AU is 1.495 x 108 km.

www.iaac.space 6/12
Problem B.2: Distance of the Planets (6 Points)
The table below lists the average distance R to the Sun and orbital period T of the first planets:

Distance Orbital Period

Mercury 0.39 AU 88 days
Venus 0.72 AU 225 days
Earth 1.00 AU 365 days
Mars 1.52 AU 687 days

(a) Calculate the average distance of Mercury, Venus and Mars to the Earth.
Which one of these planets is the closest to Earth on average?
(b) Calculate the average distance of Mercury, Venus and Earth to Mars.
Which one of these planets is the closest to Mars on average?
(c) What do you expect for the other planets?

Hint: Assume circular orbits and use symmetries to make the distance calculation easier. You can
approximate the average distance by using four well-chosen points on the planet’s orbit.

www.iaac.space 7/12
Problem B.3: Mysterious Object (6 Points)
Your research team analysis the light of a mysterious object in space. By using a spectrometer,
you can observe the following spectrum of the object. The Hα line peak is clearly visible:

(a) Mark the first four spectral lines of hydrogen (Hα, Hβ, Hγ, Hδ) in the spectrum.
(b) Determine the radial velocity and the direction of the object’s movement.
(c) Calculate the distance to the observed object.
(d) What possible type of object is your team observing?

www.iaac.space 8/12
Problem B.4: Distribution of Dark Matter (6 Points)
The most mass of our Milky Way is contained in an inner region close to the core with radius R0 .
Because the mass outside this inner region is almost constant, the density distribution can be
written as following (assume a flat Milky Way with height z0 ):
(
ρ0 , r ≤ R0
ρ(r) =
0, r > R0

(a) Derive an expression for the mass M (r) enclosed within the radius r.
(b) Derive the expected rotational velocity of the Milky Way v(r) at a radius r.

(c) Astronomical observations indicate that the rotational velocity follows a different behaviour:

p 5/2 5
vobs (r) = Gπρ0 z0 R0 −
1 + e−4r/R0 4

Draw the expected and observed rotational velocity into the plot below:

(d) Scientists believe the reasons for the difference to be dark matter: Determine the rotational
velocity due to dark matter vDM (r) from R0 and draw it into the plot above.
(e) Derive the dark matter mass MDM (r) enclosed in r and explain its distributed.
(f) Explain briefly three theories that provide explanations for dark matter.

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(extra page for problem B.4: Distribution of Dark Matter)

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Problem C.1 : Detection of Gravitational Waves (10 Points)
This problem requires you to read the following recently published scientific article:

Observation of Gravitational Waves from a Binary Black Hole Merger.

B. P. Abbott et al., LIGO Scientific Collaboration and Virgo Collaboration
arXiv:1602.03837, (2016). Link: https://arxiv.org/pdf/1602.03837.pdf

Answer following questions related to this article:

(a) How was the existence of gravitational waves first shown?

(b) Which detectors exist around the world? Why did only LIGO detect GW150914?

(c) Explain the components of the LIGO detectors.

(d) Describe the different sources of noise. How was their impact reduced?

(e) What indicates that the gravitational wave originated from the merger of a black hole?

(f) Which are the methods used to search for gravitational wave signals in the detector data?

(g) How were the source parameters (mass, distance, etc.) determined from the data?

www.iaac.space 11/12
Problem C.2 : First Image of a Black Hole (10 Points)
This problem requires you to read the following recently published scientific article:

First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole.
The Event Horizon Telescope Collaboration, arXiv:1906.11238, (2019). Link: https://arxiv.org/pdf/1906.11238.pdf

Answer following questions related to this article:

(a) Calculate the photon capture radius and the Schwarzschild radius of M87∗ (in AU).

(b) Why was it not possible for previous telescopes to take such a picture of the black hole?

(d) Explain the two algorithms used to reconstruct the image from the telescope data.

(e) What parameters were required for the GRMHD simulations to generate an image?

(f) Explain the physical origins of the features in Figure 3 (central dark region, ring, shadow).

(g) How can the image resolution be increased in future observations?

www.iaac.space 12/12
International Astronomy and
Astrophysics Competition
Pre-Final Round 2020

Solutions to the Pre-Final Round 2020

Please note that there are many ways to reach the final solutions.
Not all detailed steps are elaborated in this solution document.

www.iaac.space 1/16
Problem A.1: Interstellar Mission (4 Points)
You are on an interstellar mission from the Earth to the 8.7 light-years distant star Sirius. Your
spaceship can travel with 70% the speed of light and has a cylindrical shape with a diameter of
6 m at the front surface and a length of 25 m. You have to cross the interstellar medium with an
approximated density of 1 hydrogen atom/m3 .

(a) Calculate the time it takes your spaceship to reach Sirius.

(b) Determine the mass of interstellar gas that collides with your spaceship during the mission.

Note: Use 1.673 × 10−27 kg as proton mass.

Solution a:

8.7 ly 8.7 c · yr
t= = = 12.4 yr
0.7c 0.7c

Solution b:
Number of collisions with atoms:
2
d
N =ρ·V =ρ·A·s=ρ·π ·s
2

Total mass of interstellar gas:

2
d
M = mH2 · N = 2mp · N = 2mp · ρ · π · s = 7.8 · 10−9 kg
2

www.iaac.space 2/16
Problem A.2: Time Dilation (4 Points)
Because you are moving with an enormous speed, your mission from the previous problem A.1
will be influenced by the effects of time dilation described by special relativity: Your spaceship
launches in June 2020 and returns back to Earth directly after arriving at Sirius.

(a) How many years will have passed from your perspective?
(b) At which Earth date (year and month) will you arrive back to Earth?

Solution a:
→ No time dilation: 24.8 years

Solution b:
With time dilation:
tSpaceship tSpaceship
tEarth = q =√ = 34.7 yr
1 − vc2
2 1 − 0.72

→ Date of arrival: January 2055

www.iaac.space 3/16
Problem A.3: Magnitude of Stars (4 Points)
The star Sirius has an apparent magnitude of -1.46 and appears 95-times brighter compared to
the more distant star Tau Ceti, which has an absolute magnitude of 5.69.

(a) Explain the terms apparent magnitude, absolute magnitude and bolometric magnitude.
(b) Calculate the apparent magnitude of the star Tau Ceti.
(c) Find the distance between the Earth and Tau Ceti.

Solution a:
→ Apparent magnitude: brightness observed from Earth (relative scale, historical background)
→ Absolute magnitude: apparent magnitude in 10 parsecs distance
→ Bolometric magnitude: including all wavelengths (not only visible light)

Solution b:
The apparent magnitudes are denoted by mS , mτ and it is IS /Iτ = 95:

mτ = mS + 2.5 · log(IS /Iτ ) = 3.49

Solution c:
The absolute magnitude is denoted by M :
m−M +5
m − M = 5 log(r) − 5 =⇒ r = 10 5 = 3.63 pc

www.iaac.space 4/16
Problem A.4: Emergency Landing (4 Points)
Because your spaceship has an engine failure, you crash-land with an emergency capsule at the
equator of a nearby planet. The planet is very small and the surface is a desert with some stones
and small rocks laying around. You need water to survive. However, water is only available at the
poles of the planet. You find the following items in your emergency capsule:

• Stopwatch

• Electronic scale

• 2m yardstick

• 1 Litre oil

• Measuring cup

Describe an experiment to determine your distance to the poles by using the available items.

Hint: As the planet is very small, you can assume the same density everywhere.

Solution:

1. You collect a small rock from the surface.

2. You measure the mass m0 of the rock (on this planet) with the electronic scale. The Earth
mass m can be determined with the acceleration g (see 5): m = 1N · m0 /g

3. By using the measuring cup and the oil, you determine the volume V of the rock.

4. This gives you the density of the rock ρ = m/V . As the planet is small, you assume this
density for the planet. The formula for the mass of the planet is M = ρ · V = ρ · 34 πR3 .

5. You place the yardstick vertically into the air and drop a small rock from a height h down to
the ground. You measure the falling time t with your stopwatch. This let’s you determine
the acceleration of the stone: g = 2h/t2

6. From Newton’s law of universal gravitation it follows that the gravitational acceleration at
the surface is g = G RM2 with the gravitational constant G. By using the formula for the
3g
planet’s mass, you determine the radius of the planet: R = 4πGρ

7. You know that you have landed at the equator of the planet. Using basic geometry, you
then determine the distance from the equator to the poles: d = 2πR
4
= π2 R

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Problem B.1: Temperature of Earth (6 Points)
Our Sun shines bright with a luminosity of 3.828 x 1026 Watt. Her energy is responsible for many
processes and the habitable temperatures on the Earth that make our life possible.

Note: The Earth’s radius is 6370 km; the Sun’s radius is 696 x 103 km; 1 AU is 1.495 x 108 km.

Solution a:
2
The energy is distributed on a sphere with radius 1 AU. The surface pointing to the Sun is πRE :
2
AE πRE
Eday = L · ·t=L · ·t
A1AU 4π · (1AU )2

→ Result: 1.7 x 1017 W, which is 1.5 x 1022 J/day

Solution b:
From V = Eday /ρE it follows 4.0 x 1014 Litres.

Solution c:
We use the Stefan-Boltzmann law with σ = 5.67 · 10−8 mW
2 K 4 and an emissivity ε of 0.7:

s
2 LE · ε
LE · ε = 4πRE εσTE4 =⇒ TE = 4
2
4πRE σ

→ Result: 254.7 K, which is -18.5 ◦ C

Solution d:
→ the greenhouse effect, layered structure of the atmosphere, etc.

www.iaac.space 6/16
Problem B.2: Distance of the Planets (6 Points)
The table below lists the average distance R to the Sun and orbital period T of the first planets:

Distance Orbital Period

Mercury 0.39 AU 88 days
Venus 0.72 AU 225 days
Earth 1.00 AU 365 days
Mars 1.52 AU 687 days

Hint: Assume circular orbits and use symmetries to make the distance calculation easier. You can
approximate the average distance by using four well-chosen points on the planet’s orbit.

Solution:
Assuming circular orbits, the position ~ri (t) of a planet i at a given time t is
!
cos(ωi t)
~ri (t) = Ri
sin(ωi t)

with the angular velocity ω = 2π/T . The distance of two planets at the time t is given by
q
∆r(t) = |~r1 (t) − ~r2 (t)| = [R1 cos(ω1 t) − R2 cos(ω2 t)]2 + [R1 sin(ω1 t) − R2 sin(ω2 t)]2 .

The circular symmetries allow us to fix a single point of planet 2 for averaging over time:
q
∆r(t) = [R1 cos(ω1 t) − R2 ]2 + [R1 sin(ω1 t)]2

Instead of averaging over all t, an approximation with four equally distributed points is sufficient:
4
1 X T1
h∆ri = · ∆r k
4 k=1 4

Solution a:
Average Distance (without approximation)
Mercury-Earth 1.04 AU 1.04 AU
Venus-Earth 1.12 AU 1.13 AU
Mars-Earth 1.67 AU 1.69 AU

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→ Mercury is the closest to Earth.

Solution b:
Average Distance (without approximation)
Mercury-Mars 1.54 AU 1.55 AU
Venus-Mars 1.60 AU 1.61 AU
Earth-Mars 1.67 AU 1.69 AU

→ Mercury is the closest to Mars.

Solution c:
Mercury is on average the closest planet to all planets of the solar system.

www.iaac.space 8/16
Problem B.3: Mysterious Object (6 Points)
Your research team analysis the light of a mysterious object in space. By using a spectrometer,
you can observe the following spectrum of the object. The Hα line peak is clearly visible:

Solution a:
The Hα peak is located at 800 nm (without red shift: 656 nm). This yields the red shift z:
λobs
z= − 1 = 0.22
λexp

From λobs = (z + 1) · λexp it follows:

Hα Hβ Hγ Hδ
At rest 656 nm 486 nm 434 nm 410 nm
With red shift 800 nm 593 nm 529 nm 500 nm

Solution b:
The red shift corresponds to the radial velocity of an object:
s
1 + v/c (z + 1)2 − 1
z= −1 ⇒ v = c = 58881 km/s
1 − v/c (z + 1)2 + 1

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→ positive red shift, i.e. the object is moving away

Solution c:
According to Hubble’s law, v = H0 · d with the distance d and the constant H0 = 70 km/s/Mpc:
v
d= = 841 M pc
H0
Solution d:
→ possible objects for this large distance: galaxy or quasar

www.iaac.space 10/16
Problem B.4: Distribution of Dark Matter (6 Points)
The most mass of our Milky Way is contained in an inner region close to the core with radius R0 .
Because the mass outside this inner region is almost constant, the density distribution can be
written as following (assume a flat Milky Way with height z0 ):
(
ρ0 , r ≤ R0
ρ(r) =
0, r > R0

(a) Derive an expression for the mass M (r) enclosed within the radius r.
(b) Derive the expected rotational velocity of the Milky Way v(r) at a radius r.

(c) Astronomical observations indicate that the rotational velocity follows a different behaviour:

p 5/2 5
vobs (r) = Gπρ0 z0 R0 −
1 + e−4r/R0 4

Draw the expected and observed rotational velocity into the plot below:

www.iaac.space 11/16
Solution a:
It is M (r) = V (r) · ρ0 with a volume of V (r) = πr2 z0 :
(
ρ0 · πr2 z0 , r ≤ R0
M (r) =
ρ0 · πR02 z0 , r > R0

Solution b:
Using basic mechanics we get:
r
v 2 (r) M (r) M (r)
ar = ag ⇒ =G 2 ⇒ v(r) = G
r r r
Using M (r) we can write the final answer as:
( √ √
Gπρ0 z0 · r, r ≤ R0
v(r) = √ √
Gπρ0 z0 · R0 / r, r > R0

Solution c:

Solution d:
The difference between observed and expected rotational velocity:
r !
p 5/2 5 R0
vDM (r) = vobs (r) − v(r) = Gπρ0 z0 R0 −4r/R
− −
1+e 0 4 r

Solution e:
2
→ It is MDM (r) = vDM (r)r/G; dark matter seems to increase with increasing r (halo).

Solution f:
→ Examples: WIMPS, Axions, Sterile Neutrinos, Black Holes, MACHOS, MOND, etc.

www.iaac.space 12/16
Problem C.1 : Detection of Gravitational Waves (10 Points)
This problem requires you to read the following recently published scientific article:

Observation of Gravitational Waves from a Binary Black Hole Merger.

B. P. Abbott et al., LIGO Scientific Collaboration and Virgo Collaboration
arXiv:1602.03837, (2016). Link: https://arxiv.org/pdf/1602.03837.pdf

Answer following questions related to this article:

(a) How was the existence of gravitational waves first shown?

→ by the discovery of the binary pulsar system PSR B1913+16 and observations of its energy loss

(b) Which detectors exist around the world? Why did only LIGO detect GW150914?

→ TAMA 300 (Japan), GEO 600 (Germany), LIGO (United States), Virgo (Italy)
→ only LIGO detectors were observing at the time of GW150914

(c) Explain the components of the LIGO detectors.

→ two detectors in Hanford and Livingston (10ms light distance)

→ Michelson interferometer (1064-nm Nd:YAG laser, 4km arm length)
→ resonant optical cavity (multiplies the effect of a gravitational wave on the light)
→ transmissive power-recycling mirror at the input (additional resonant buildup of the laser)
→ transmissive signal-recycling mirror at the output (broadening of bandwidth)

(d) Describe the different sources of noise. How was their impact reduced?

→ seismic noise: quadruple-pendulum system, active seismic isolation platform

→ thermal noise: low-mechanical-loss materials
→ optical phase fluctuations: very low pressure in tubes
→ environmental disturbances: seismometers, accelerometers, microphones, magnetometers,
radio receivers, weather sensors, ac-power line monitors, cosmic-ray detector

(e) What indicates that the gravitational wave originated from the merger of a black hole?

→ the objects must be very close and very compact; neutron star pair: insufficient mass; black
hole and neutron star pair: merge at much lower frequency
→ decay of the waveform consistent with the oscillations of a relaxing black hole

www.iaac.space 13/16
→ consistency checks: mass/spin of final black hole, waveform power series, graviton properties

(f) Which are the methods used to search for gravitational wave signals in the detector data?

→ generic transient search: minimal assumptions about waveforms

→ binary coalescence search: using waveforms predicted by general relativity

(g) How were the source parameters (mass, distance, etc.) determined from the data?
→ estimate: with matched-filter search; refinement: general relativity-based models

www.iaac.space 14/16
Problem C.2 : First Image of a Black Hole (10 Points)
This problem requires you to read the following recently published scientific article:

Answer following questions related to this article:

(a) Calculate the photon capture radius and the Schwarzschild radius of M87∗ (in AU).

√ √
→ photon capture radius: Rc = 27rg = 27GM/c2 = 333 AU
→ Schwarzschild radius: RS = 2rg = 2GM/c2 = 128 AU

(b) Why was it not possible for previous telescopes to take such a picture of the black hole?

→ due to limited baseline coverage

→ very-long-baseline interferometry; measures visibility of radio brightness directly

→ observation at 1.3 mm with eight stations over six geographical locations
→ the separate telescopes simultaneously sample and record the radiation field from the source
→ synchronization with GPS; equipped with hydrogen maser frequency standard

(d) Explain the two algorithms used to reconstruct the image from the telescope data.

→ CLEAN: inverse-modeling approach, deconvolves the interferometer point-spread function

from the Fourier-transformed visibilities
→ RML (regularized maximum likelihood): forward-modeling approach, searches for an image
that is consistent with the observed data and favours specified image properties

(e) What parameters were required for the GRMHD simulations to generate an image?

→ properties of the fluid (magnetic field, velocity field, and rest-mass density), the emission and
absorption coefficients, the inclination, the position angle, the black hole mass and distance

(f) Explain the physical origins of the features in Figure 3 (central dark region, ring, shadow).

www.iaac.space 15/16
→ emission ring and shadow: combination of an event horizon and light bending
→ north-south asymmetry: produced by strong gravitational lensing and relativistic beaming
→ central flux depression: observational signature of the black hole shadow

(g) How can the image resolution be increased in future observations?

→ shorter wavelength, by adding more telescopes and space-based interferometry

www.iaac.space 16/16
International Astronomy and
Astrophysics Competition
Final Round

Final Round Exam 2020

The final round exam was given in the form of an online exam.
Each participant was given a subset of 40 questions in random order.
This paper version is only available for training purposes.

IAAC Final Exam 2020 1/9

Question 1 : How many parsecs are approximately 23 light years?

(A) 6 (B) 7 (C) 8 (D) 9

Question 2 : Which one is the biggest moon of Jupiter?

(A) Io (B) Europa (C) Ganymede (D) Callisto

Question 3 : Which object was investigated by the Cassini research mission?

(A) A comet (B) Jupiter (C) Saturn (D) Pluto

Question 4 : What is the distinctive color of the star Betelgeuse?

(A) Red (B) Orange (C) Blue (D) White

Question 5 : The International Space Station circles the Earth approximately every ...

(A) 60 minutes (B) 90 minutes (C) 2 hours (D) 5 hours

Question 6 : The well-known Hyades star cluster is located in the constellation ...

(A) Sagittarius (B) Virgo (C) Cancer (D) Taurus

Question 7 : What is the diameter of the Sun approximately?

(A) 0.6 million km (B) 0.9 million km (C) 1.4 million km (D) 1.9 million km

Question 8 : The planet Jupiter orbits the Sun in ...

(A) 12 years (B) 15 years (C) 18 years (D) 23 years

Question 9 : The value of the Hubble constant is approximately ...

(A) 35 (km/s)/Mpc (B) 70 (km/s)/Mpc

(C) 120 (km/s)/Mpc (D) 180 (km/s)/Mpc

IAAC Final Exam 2020 2/9

Question 10 : The first gravitational wave GW150914 was detected by using a Michelson interfer-
ometer with an arm length of ...

(A) 4 km (B) 6 km (C) 8 km (D) 10 km

Question 11 : The Event Horizon Telescope is a very long baseline interferometry array observing
at a wavelength of ...

(A) 0.9 mm (B) 1.3 mm (C) 1.7 mm (D) 1.9 mm

Question 12 : Light with a wavelength of 100 micrometres is called ...

(A) ultraviolet light (B) infrared light

(C) visible light (D) radio waves

Question 13 : How high above the Earth’s equator is the geostationary orbit located?

(A) 26000 km (B) 36000 km (C) 46000 km (D) 56000 km

Question 14 : Voyager 1 is the most distant man-made object. How fast is Voyager 1 moving away
from the Sun at the moment?

(A) 17 km/s (B) 19 km/s (C) 23 km/s (D) 28 km/s

Question 15 : When was the dwarf planet Pluto discovered?

(A) 1630 (B) 1730 (C) 1830 (D) 1930

Question 16 : How many times more mass has the Earth compared to the Moon?

(A) 15 times

(B) 30 times

(D) 80 times

IAAC Final Exam 2020 3/9

Question 17 : What does the astronomical term barycentre describe?

(A) The centre of zero gravitational potential of multiple objects.

(B) The centre of mass of multiple each other orbiting objects.

(C) The central axis of rotation of a spinning stellar object.

(D) The central axis of the precession of a rotating stellar object.

Question 18 : How many Lagrange points exist in a two-body system?

(A) 3 (B) 5 (C) 8 (D) 16

Question 19 : The astronomical term axial precession describes the changing ...

(A) angular velocity of the rotational axis.

(B) angular momentum of the rotational axis.

(C) orientation of the rotational axis.

(D) torque of the rotational axis.

Question 20 : Which astronomical term describes the alignment of three celestial objects?

(A) Conjunction (B) Eclipse (C) Transit (D) Syzygy

Question 21 : The farthest point of a planet to the Sun is called ...

(A) Aphelion (B) Perihelion (C) Apogee (D) Perigee

Question 22 : What is a synodic month?

(A) The time between two full moons.

(B) The time between two Venus appearances.

(D) The time for one Earth rotation in respect to the Sun.

IAAC Final Exam 2020 4/9

Question 23 : The astronomical object Hale-Bopp is a ...

(A) Dwarf Planet (B) Meteor (C) Asteroid (D) Comet

Question 24 : The Olbers’ paradox raises the following question:

(A) Is the universe infinitely large?

(B) Is the universe infinitely old?

(C) Are we alone in the universe?

(D) Why is the night sky dark?

Question 25 : The picture below shows a very close and well-known galaxy:

What is the name of this galaxy?

(A) Pinwheel Galaxy (B) Whirlpool Galaxy

(C) Bode Galaxy (D) Triangulum Galaxy

Question 26 : The correct astronomical term for shooting stars (or falling star) is ...

(A) Asteroid (B) Meteoroid (C) Meteor (D) Comet

Question 27 : The calculated age of the universe is ...

(A) 11.8 billion years (B) 12.8 billion years

(C) 13.8 billion years (D) 14.8 billion years

IAAC Final Exam 2020 5/9

Question 28 : What is expected to happen at the end of the Sun’s lifetime?

(A) Become a white dwarf (B) Become a brown dwarf

(C) Explode as a supernova (D) Become a neutron star or black hole

Question 29 : Approximately how far away is the Sun from the centre of the Milky Way?

(A) 12,000 light-years (B) 17,000 light-years

(C) 21,000 light-years (D) 27,000 light-years

Question 30 : Which one is the correct order of spectral classes according to the star’s surface
temperature (from high to low)?

(A) O, A, B, F, G, K, M (B) O, B, A, F, G, K, M

Question 31 : A rocket that has accelerated to the second cosmic velocity can...

(A) circle around the Earth in a stable orbit.

(B) circle around the Earth in an elliptic orbit.

(C) escape the gravitational field of the Earth.

(D) escape the gravitational field of the Sun.

Question 32 : What is the diameter of the observable universe in parsec?

(A) 13 billion pc (B) 29 billion pc (C) 36 billion pc (D) 55 billion pc

Question 33 : The well-known and very large volcano named Olympus Mons is located on ...

(A) Mars (B) Venus

(C) Io (Jupiter moon) (D) Europa (Jupiter moon)

IAAC Final Exam 2020 6/9

Question 34 : About how many Earth days are one Mars day?

(A) 1 day (B) 1.5 days (C) 2 days (D) 2.5 days

Question 35 : The fascinating aurorae (or polar lights) are located in the ...

(A) Ionosphere (B) Stratosphere (C) Troposphere (D) Exosphere

Question 36 : In which year did humans first land and walk on the Moon?

(A) 1969 (B) 1986 (C) 1968 (D) 1976

Question 37 : The term heliocentric means ...

(A) centred around the Earth (B) centred around the Venus

Question 38 : Saturn’s moon Titan has many lakes which are made of liquid ...

(A) Oxygen and Nitrogen (B) Nitrogen and Ethane

(C) Ethane and Methane (D) Methane and Nitrogen

Question 39 : Copernicus is located on our moon and refers to a well-known ...

(A) Mountain (B) Sea (C) Crater (D) Apollo landing

site

Question 40 : What type of nebula is the famous Orion Nebula?

(A) Planetary Nebula (B) Dark Nebula

(C) Absorption Nebula (D) Emission Nebula

Question 41 : How is the path in the sky called that the Sun appears to cross within a year?

(A) Equinox (B) Ecliptic

(C) Epicycle (D) Celestial Equator

IAAC Final Exam 2020 7/9

Question 42 : Spectral line splitting caused by magnetic fields is called ...

(A) Planck Effect (B) Russell Effect (C) Raylight Effect (D) Zeeman Effect

Question 43 : In which wavelengths is the cosmic background radiation concentrated?

(A) Infrared Wavelengths (B) Gamma Wavelengths

(C) Ultraviolet Wavelengths (D) Microwave Wavelengths

Question 44 : Kepler’s Laws state that the cube of a planet’s semi-major axis is proportional to
the ...

(A) square of the average distance.

(B) covered area by the planet.

(C) cube of the average distance.

(D) square of the orbital period.

Question 45 : What is the astronomical term for the outer boundary of the Sun’s magnetic field?

(A) Heliopause (B) Heliosheath

(C) Heliotail (D) Termination Shock

Question 46 : The angle between an inferior planet and the Sun when observed from Earth is
called ...

(A) Inclination (B) Declination (C) Ecliptic (D) Elongation

Question 47 : The summer solstice in the northern/southern hemisphere occurs in ...

(A) June / November (B) July / November

(C) June / December (D) July / December

Question 48 : Absolute magnitude is a star’s apparent magnitude in a distance of ...

(A) 5 pc (B) 10 pc (C) 15 pc (D) 20 pc

IAAC Final Exam 2020 8/9

Question 49 : What is the visible part of the Sun called?

(A) Radiative Zone (B) Heliosphere (C) Photosphere (D) Tachocline

Question 50 : The average albedo of the Earth (planetary albedo) is between ...

(A) 20-25 % (B) 25-30 % (C) 30-35 % (D) 35-40 %

IAAC Final Exam 2020 9/9

International Astronomy and
Astrophysics Competition
Final Round

Final Round Exam 2020 - Solutions

No. 1 No. 2 No. 3 No. 4 No. 5 No. 6 No. 7 No. 8 No. 9 No. 10

B C C A B D C A B A

No. 11 No. 12 No. 13 No. 14 No. 15 No. 16 No. 17 No. 18 No. 19 No. 20

B B B A D D B B C D

No. 21 No. 22 No. 23 No. 24 No. 25 No. 26 No. 27 No. 28 No. 29 No. 30

A A D D D C C A D B

No. 31 No. 32 No. 33 No. 34 No. 35 No. 36 No. 37 No. 38 No. 39 No. 40

C B A A A A C C C D

No. 41 No. 42 No. 43 No. 44 No. 45 No. 46 No. 47 No. 48 No. 49 No. 50

B D D D A D C B C C

IAAC Final Exam 2020 1/1

International Astronomy and
Astrophysics Competition
Qualification Round 2021

Problem A : Observing the Night Sky (5 Points)

Fill in the blank spaces with the correct answers:

Approximately how many stars are visible with the naked eye in the night sky?
(1)

Where in the night sky can you observe the famous double star system Mizar and Alcor?
(2)

What kind of celestial object is Neowise C/2020 F3 and what makes it special?
(3)

Which very intense meteor shower is taking place annually in December?

(4)

What are the names of the following three well-known constellations?

(5) (6) (7)

www.iaac.space 1/3
Problem B : Shock Wave Escape (5 Points)
The star of a distant solar system explodes as a supernova. At the moment of the explosion, an
resting exploration spaceship is 15 AU away from the shock wave. The shock wave of the explo-
sion travels with 25000 km/s towards the spaceship. To save the crew, the spacecraft makes use
of a special booster that uniformly accelerates at 150 m/s2 in the opposite direction.

Determine if the crew manages to escape from the shock wave. (Neglect relativistic effects.)

Problem C : Mysterious Planet (5 Points)

A research team has discovered that a moon is circling a planet of our solar system: The moon
orbits the planet once every 7 hours on a nearly circular orbit in a distance R of 48000 km from
the centre of the planet. Unfortunately, the mass m of the moon is not known. Use Newton’s law
of gravitation with G = 6.67 · 10−11 m3 /(kg·s2 ) to approach the following questions:
mM
F =G· (1)
R2
(a) Based on the observations, determine the total mass M of the planet.
(b) Which moon and planet of our solar system is the team observing? (Use literature.)

Problem D : Gravitational Constant (5 Points)

An astronaut working on the Moon tries to determine the gravitational constant G by throwing
a Moon rock of mass m with a velocity of v vertically into the sky. The astronaut knows that the
Moon has a density ρ of 3340 kg/m3 and a radius R of 1740 km.

(a) Show with (1) that the potential energy of the rock at height h above the surface is given by:

4πG R3
E=− mρ · (2)
3 R+h
(b) Next, show that the gravitational constant can be determined by:
−1
3 v2

R
G= 1− (3)
8π ρR2 R+h

www.iaac.space 2/3
Problem E : Pulsars (5 Points)
Radio telescopes are an essential tool for modern astrophysics. They played a crucial role in
discovering a fascinating astronomical object: Pulsars - highly compact objects that periodically
emit radiation. Pulsars are still an active part of astrophysical research.

Explain how pulsars are formed and the causes for their pulsating behaviour.

General Information and Submission

You can write the solution by hand or type it on a computer. To qualify for the pre-final round, you have to
get at least 15 points (Junior, under 18 years) or 20 points (Youth, over 18 years). Make sure to submit your
solution by Friday 30. April 2021 23:59 UTC+0 online at www.iaac.space/submission. In case of questions or
comments, please feel free to contact us via email: info@iaac.space. Good luck!

www.iaac.space 3/3
International Astronomy and
Astrophysics Competition
Qualification Round 2021

Problem A : Observing the Night Sky (5 Points)

(1) 2k - 4.5k (one half) (2) Ursa Major (3) Comet, eye visibility (4) Geminids
(5) Cassiopeia (6) Cygnus (7) Andromeda

Problem B : Shock Wave Escape (5 Points)

v2
vt = 12 at2 + dAU =⇒ a > 2dAU
≈ 139 m/s2 =⇒ The crew manages to escape!

Problem C : Mysterious Planet (5 Points)

R 2πR 2 R 4π 2 R3

(a) GmM/R2 = F = mv 2 /R =⇒ M = v 2 G = T G
= T 2G
≈ 1.03 · 1026 kg
(b) (Planet) Neptune, (Moon) Naiad

Problem D : Gravitational Constant (5 Points)

(a)
∞
4πR3 R3
Z
1 1 4πG
E= F ds = −GmM =− ρ · Gm =− mρ ·
R+h R+h 3 R+h 3 R+h
(b)
−1
mv 2 mv 2 3 v2

4πG 1 1 R
+E(0) = E(h) ⇒ =− mρR3 − ⇒ G= 1−
2 2 3 R+h R 8π ρR2 R+h

(c) G ≈ 6.63 · 10−11

Problem E : Pulsars (5 Points)

(individual written solution)

www.iaac.space Basic Solution

International Astronomy and
Astrophysics Competition
Pre-Final Round 2021

Important: Read all the information on this page carefully!

• You may use extra paper if necessary, however, the space under the problems is usually enough.

• Typing the solution on a computer is allowed but not recommended (no extra points).

• The 10 problems are separated into three categories: 4x basic problems (A; four points), 4x advanced
problems (B; six points), 2x research problems (C; ten points). The research problems require you
to read a short scientific article each to answer the questions. There is a link to the PDF article.

• You receive points for the correct solution and for the performed steps. Example: You will not get
all points for a correct value if the calculations are missing.

• Make sure to clearly mark your final solution values (e.g. underlining, red color, box).

• You can reach up to 60 points in total. You qualify for the final round if you reach at least 25 points
(junior, under 18 years) or 35 points (youth, over 18 years).

Uploading Your Solution

• Please upload a file/pictures of (this sheet with) your written solutions: https://iaac.space/login

• Only upload one single PDF file! If you have multiple pictures, please compress them into one
single file. Do not upload your pictures in a different format (e.g, no Word and Zip files).

• The deadline for uploading your solution is Sunday 6. June 2021, 23:59 UTC+0.

• The results of the pre-final round will be announced on Monday 14. June 2021.

Good luck!

www.iaac.space 1/14
Problem A.1: Equatorial Coordinate System (4 Points)
Astronomers need to identify the position of objects in the sky with very high precision. For that,
it is essential to have coordinate systems that specify the position of an object at a given time.
One of them is the equatorial coordinate system that is widely used in astronomy.

(a) Explain how the equatorial coordinate system works.

(b) What is the meaning of J2000 that often occurs together with equatorial coordinates?

The object NGC 4440 is a galaxy located in the Virgo Cluster at the following equatorial coordi-
nates (J2000): 12h 27m 53.6s (right ascension), 12◦ 170 3600 (declination). The Calar Alto Observatory
is located in Spain at the geographical coordinates 37.23◦ N and 2.55◦ W.

www.iaac.space 2/14
Problem A.2: Resolution of Telescopes (4 Points)
Telescopes are an essential tool for astronomers to study the universe. You plan to build your
own telescope that can resolve the Great Red Spot on the surface of Jupiter at a wavelength of
600 nm. The farthest distance between the Earth and Jupiter is 968 × 106 km and the Great Red
Spot has currently a diameter of 16,500 km.

(a) Use the Rayleigh criterion to determine the diameter of the lens’ aperture of your telescope
that is needed to resolve the Great Red Spot on Jupiter.

Impacts have formed many craters on the Moon’s surface. You would like to study some of the
craters with your new telescope. The distance between Moon and Earth is 384,400 km.

(b) What is the smallest possible size of the craters that your telescope can resolve?

www.iaac.space 3/14
Problem A.3: Total Solar Eclipse (4 Points)
A total solar eclipse occurs when the Moon moves between the Earth and the Sun and completely
blocks out the Sun. This phenomenon is very spectacular and attracts people from all cultures.
However, total solar eclipses can also take place on other planets of the Solar System.

Determine for each of the following moons if they can create a total solar eclipse on their planet.

Moon Radius Distance to Planet Planet Distance to the Sun

Phobos 11 km 9376 km Mars 228 × 106 km
Callisto 2410 km 1.883 × 106 km Jupiter 779 × 106 km
Titan 2574 km 1.222 × 106 km Saturn 1433 × 106 km
Oberon 761 km 0.584 × 106 km Uranus 2875 × 106 km

Note: The radius of the Sun is 696 × 103 km.

www.iaac.space 4/14
Problem A.4: Special Relativity - Part I (4 Points)
Special relativity has become a fundamental theory in the 20th century and is crucial for ex-
plaining many astrophysical phenomena. A central aspect of special relativity is the transfor-
mation from one reference frame to another. The following Lorentz transformation matrix gives
the transformation from a frame at rest to a moving frame with velocity v along the z-axis:
 
γ 0 0 γβ

 0 1 0 0 
 
 0 0 1 0 
γβ 0 0 γ

where β = v/c with c being the speed of light in a vacuum, and γ is the Lorentz factor:
1
γ=p
1 − β2

(a) State and explain the two traditional postulates from which special relativity originates.
(b) Draw a plot of the Lorentz factor for 0 ≤ β ≤ 0.9 to see how its value changes.

One of the many exciting phenomena of special relativity is time dilation. Imagine astronauts
in a spaceship that is passing by the Earth with a high velocity.

(c) Are clocks ticking slower for the people on Earth or for the astronauts on the spaceship?
(d) How fast must the spaceship travel such that the clocks go twice as slow?

www.iaac.space 5/14
Problem B.1: Space Cannon (6 Points)
Scientists are developing a new space cannon to shoot objects from the surface of the Earth di-
rectly into a low orbit around the Earth. For testing purposes, a projectile is fired with an initial
velocity of 2.8 km/s vertically into the sky.

Calculate the height that the projectile reaches, ...

(a) assuming a constant gravitational deceleration of 9.81 m/s2 .
(b) considering the change of the gravitational force with height.

Note: Neglect the air resistance for this problem. Use 6.67×10−11 m3 kg−1 s−2 for the gravitational
constant, 6371 km for the Earth’s radius, and 5.97 × 1024 kg for the Earth’s mass.

www.iaac.space 6/14
Problem B.2: Shock Wave (6 Points)
This year’s qualification round featured a spaceship escaping from a shock wave (Problem B).
The crew survived and wants to study the shock wave in more detail. It can be assumed that
the shock wave travels through a stationary flow of an ideal polytropic gas which is adiabatic on
both sides of the shock. Properties in front and behind a shock are related through the three
Rankine-Hugoniot jump conditions (mass, momentum, energy conservation):

v12 v2
ρ1 v1 = ρ2 v2 ρ1 v12 + p1 = ρ2 v22 + p2 + h1 = 2 + h2
2 2
where ρ, v, p, and h are the density, shock velocity, pressure, and specific enthalpy in front (1 )
and behind (2 ) the shock respectively.
Shock front

v2 , ρ2 , p2 , h2 v1 , ρ1 , p1 , h1

(a) Explain briefly the following terms used in the text above:
(i) stationary flow
(ii) polytropic gas
(iii) specific enthalpy

(b) Show with the Rankine-Hugoniot conditions that the change in specific enthalpy is given by:

p2 − p1 1 1
∆h = · +
2 ρ1 ρ2

The general form of Bernoulli’s law is fulfilled on both sides of the shock separately:

v2
+Φ+h=b
2
where Φ is the gravitational potential and b a constant.

(c) Assuming that the gravitational potential is the same on both sides, determine how the con-
stant b changes at the shock front.

(d) Explain whether Bernoulli’s law can be applied across shock fronts.

www.iaac.space 7/14
(extra page for problem B.2: Shock Wave)

www.iaac.space 8/14
Problem B.3: Interplanetary Journey (6 Points)
A space probe is about to launch with the objective to explore the planets Mars and Jupiter. To
use the lowest amount of energy, the rocket starts from the Earth’s orbit (A) and flies in an ellip-
tical orbit to Mars (B), such that the ellipse has its perihelion at Earth’s orbit and its aphelion at
Mars’ orbit. The space probe explores Mars for some time until Mars has completed 1/4 of its orbit
(C). After that, the space probe uses the same ellipse to get from Mars (C) to Jupiter (D). There the
mission is completed, and the space probe will stay around Jupiter.

The drawing below shows the trajectory of the space probe (not drawn to scale):

D C
Sun

Earth
Mars

Jupiter

Below you find the obrital period and the semi-major axis of the three planets:

Orbital period Semi-major axis

Earth 365 days 1.00 AU
Mars 687 days 1.52 AU
Jupiter 4333 days 5.20 AU

How many years after its launch from the Earth (A) will the space probe arrive at Jupiter (D)?

www.iaac.space 9/14
(extra page for problem B.3: Interplanetary Journey)

www.iaac.space 10/14
Problem B.4: Special Relativity - Part II (6 Points)
Space and time are interconnected according to special relativity. Because of that, coordinates
have four components (three position coordinates x, y, z, one time coordinate t ) and can be ex-
pressed as a vector with four rows as such:
 
ct

 x 

 
 y 
z

The spaceship from problem A.4 (Special Relativity - Part I) travels away from the Earth into the
deep space outside of our Milky Way. The Milky Way has a very circular shape and can be ex-
pressed as all vectors of the following form (for all 0 ≤ ϕ < 2π):
 
ct

 0 

 
 sin ϕ 
cos ϕ

(a) How does the shape of the Milky Way look like for the astronauts in the fast-moving space-
ship? To answer this question, apply the Lorentz transformation matrix (see A.4) on the circular
shape to get the vectors (ct0 , x0 , y 0 , z 0 ) of the shape from the perspective of the moving spaceship.

(b) Draw the shape of the Milky Way for a spaceship with a velocity of 20%, 50%, and 90% of the
speed of light in the figure below (Note: The ring shape for a resting spaceship is already drawn.):

www.iaac.space 11/14
(extra page for problem B.4: Special Relativity - Part II)

www.iaac.space 12/14
Problem C.1 : Earliest Galaxy Group (10 Points)
This problem requires you to read the following recently published scientific article:

Onset of Cosmic Reionization: Evidence of an Ionized Bubble Merely 680 Myr after the Big Bang.
V. Tilvi et al 2020 ApJL 891 L10. Link: https://iopscience.iop.org/article/10.3847/2041-8213/ab75ec

Answer the following questions related to this article:

(a) What is the so called cosmic reionization process?

(b) What are Lyα lines and why did the researches want to observe them?

(d) How is confirmed that the peaks seen in Figure 3 are actually from Lyα emissions?

(e) How are the bubble sizes of the galaxies estimated?

(f) What is special about the findings in the article and what are the scientific implications?

www.iaac.space 13/14
Problem C.2 : Massive Protostar Jet (10 Points)
This problem requires you to read the following recently published scientific article:

Measuring the ionisation fraction in a jet from a massive protostar.

Fedriani, R., Caratti o Garatti, A., Purser, S.J.D. et al. Nat Commun 10, 3630 (2019).
Link: https://www.nature.com/articles/s41467-019-11595-x.pdf

Answer the following questions related to this article:

(a) Why are massive stars important for the development of the universe?

(b) How can the ionised part of jets be observed?

(c) What kind of region is G35.2N? Describe how it is structured.

(d) What is the ionisation fraction χe and how do the authors calculate its value?

(e) How is the mass-loss rate being determined for knots K3 and K4? Why not for K1 and K2?

(f) Why is the ionisation fraction so small for G35.2N?

www.iaac.space 14/14
International Astronomy and
Astrophysics Competition
Pre-Final Round 2021

Solutions to the Pre-Final Round 2021

Please note that there are many ways to reach the final solutions.
Not all detailed steps are elaborated in this solution document.

www.iaac.space 1/15
Problem A.1: Equatorial Coordinate System (4 Points)
Astronomers need to identify the position of objects in the sky with very high precision. For that,
it is essential to have coordinate systems that express the position of an object at a given time.
One of them is the equatorial coordinate systems that is widely used in astronomy.

(a) Explain how the equatorial coordinate system works.

(b) What is the meaning of J2000 that often occurs together with equatorial coordinates?

Solution a:
→ Origin: Earth’s center, 0◦ Latitude: Celestial equator , 0◦ Longitude: Vernal equinox (intersec-
tion between celestial equator and ecliptic), Coordinates: Declination (the angle between object
and celestial equator, ±90◦ ) and right ascension (the angle between vernal equinox and object,
360◦ or 24h)

Solution b:
→ Orientation of reference frame is not fixed due to precession, nutation, and other movements;
reference point in time for the equinox is necessary, the epoch; J2000 refers to midday on the 1.
January 2000

Solution c:
It is 37.23◦ - 12◦ 170 3600 < 90◦ , thus NGC 4440 is observable.

www.iaac.space 2/15
Problem A.2: Resolution of Telescopes (4 Points)
Telescopes are an essential tool for astronomers to study the universe. You plan to build your
own telescope that can resolve the Great Red Spot on the surface of Jupiter at a wavelength of
600 nm. The farthest distance between the Earth and Jupiter is 968 × 106 km and the Great Red
Spot has a current diameter of 16,500 km.

(a) Use the Rayleigh criterion to determine the diameter of the lens’ aperture of your telescope
that is needed to resolve the Great Red Spot on Jupiter.

Impacts have formed many craters on the Moon’s surface. You would like to study some of the
craters with your new telescope. The distance between Moon and Earth is 384,400 km.

(b) What is the smallest possible size of the craters that your telescope can resolve?

Solution a:
For the required angular resolution θ we get (z: distance to Jupiter, d: diameter of object):

θ d d
sin = =⇒ θJ = 2 arcsin = 1.705 × 10−5
2 2z 2z

With the Rayleigh criterion θJ = 1.22λ/D we get:

1.22λ
D=
θ
→ Diameter of the lens’ aperture: 4.3 cm

Solution b:
For the minimum crater diameter d we get:

θ
d = 2z · sin
2

→ Smallest possible diameter: 6.6 km

www.iaac.space 3/15
Problem A.3: Total Solar Eclipse (4 Points)
A total solar eclipse occurs when the Moon moves between the Earth and the Sun and completely
blocks out the Sun. This phenomena is very spectacular and attracts people from all cultures.
However, total solar eclipses can also take place on other planets of the Solar System.

Determine for each of the following moons if they can create a total solar eclipse on their planet.

Moon Radius Distance to Planet Planet Distance to the Sun

Phobos 11 km 9376 km Mars 228 × 106 km
Callisto 2410 km 1.883 × 106 km Jupiter 779 × 106 km
Titan 2574 km 1.222 × 106 km Saturn 1433 × 106 km
Oberon 761 km 0.584 × 106 km Uranus 2875 × 106 km

Note: The radius of the Sun is 696 × 103 km.

Solution:
For a total eclipse, the planet must be within the shadow of the moon (umbra). The distance be-
tween the moon and the planet d must be smaller than the distance of the shadow dU . Geometry
gives us (D: distance planet–Sun, RS : radius of the Sun, R: radius of the moon):
D dU R
= =⇒ dU = D ·
RS R RS
With the condition d ≤ dU we get:

Moon dU
Phobos 3603 km total eclipse not possible
6
Callisto 2.697 × 10 km total eclipse possible
6
Titan 5.300 × 10 km total eclipse possible
6
Oberon 3.143 × 10 km total eclipse possible

www.iaac.space 4/15
Problem A.4: Special Relativity - Part I (6 Points)
Special relativity has become a fundamental theory in the 20th century and is crucial for ex-
plaining many astrophysical phenomena. A central aspect of special relativity is the transfor-
mation from one reference frame to another. The following Lorentz transformation matrix gives
the transformation from a frame at rest to a moving frame with velocity v along the z-axis:
 
γ 0 0 γβ

 0 1 0 0 
 
 0 0 1 0 
γβ 0 0 γ

where β = v/c with c being the speed of light in a vacuum, and γ is the Lorentz factor:
1
γ=p
1 − β2

(a) State and explain the two traditional postulates from which special relativity originates.
(b) Draw a plot of the Lorentz factor for 0 ≤ β ≤ 0.9 to see how its value changes.

One of the many exciting phenomena of special relativity is time dilation. Imagine astronauts
in a spaceship that is passing by the Earth with a high velocity.

(c) Are the clocks ticking slower for the people on Earth or for the astronauts on the spaceship?
(d) How fast must the spaceship travel such that the clocks go twice as slow?

Solution a:
(1) The laws of physics are invariant in all inertial frames of reference.
(2) The speed of light in vacuum is the same for all observers.

Solution b:

www.iaac.space 5/15
Solution c:
The clocks are ticking slower for the spaceship.

Solution d:
From the Lorentz factor we get:
r
1 1
γ=p =⇒ β = 1− 2
1 − β2 γ

With γ = 2 we have: β = 0.87, thus 87% of the speed of light

www.iaac.space 6/15
Problem B.1: Space Cannon (4 Points)
Scientists are developing a new space cannon to shoot objects from the surface of the Earth di-
rectly into a low orbit around the Earth. For testing purpose, a projectile is fired with an initial
velocity of 2.8 km/s vertically into the sky.

Calculate the height that the projectile reaches, ...

(a) assuming a constant gravitational deceleration of 9.81 m/s2 .
(b) considering the change of the gravitational force with height.

Note: Neglect the air resistance for this problem. Use 6.67×10−11 m3 kg−1 s−2 for the gravitational
constant, 6371 km for the Earth’s radius, and 5.97 × 1024 kg for the Earth’s mass.

Solution a:
Kinetic energy of the projectile:
1
Ekin = mv02
2
Potential energy at height h due to the force of gravitation F1 = mg:
Z h
Epot (h) = F1 dy = mgh
0

The maximum is reached as soon all kinetic energy is converted into potential energy:

v02
Ekin = Epot (h) ⇒ h =
2g
→ Height of the projectile: 400 km

Solution b:
Considering the gravitational force (R Earth’s radius, M Earth’s mass, G gravitational constant):
mM
F2 (y) = G
(R + y)2
This yields the potential energy:
Z h Z h Z R+h
1 1 1 1
Epot (h) = F2 (y) dy = GmM dy = GmM dr = GmM −
0 0 (R + y)2 R r2 R R+h

With energy conservation it follows:

1
h= v02
−R
1
R
− 2GM

→ Height of the projectile: 426 km

www.iaac.space 7/15
Problem B.2: Shock Wave (6 Points)
This year’s qualification round featured a spaceship escaping from a shock wave (Problem B).
The crew survives and wants to study the shock wave in more detail. It can be assumed that
the shock wave travels through a stationary flow of an ideal polytropic gas which is adiabatic on
both sides of the shock. Properties in front and behind a shock are related through the three
Rankine-Hugoniot jump conditions (mass, momentum, energy conservation):

v2 , ρ2 , p2 , h2 v1 , ρ1 , p1 , h1

(a) Explain briefly the following terms used in the text above:
(i) stationary flow
(ii) polytropic gas
(iii) specific enthalpy

(b) Show with the Rankine-Hugoniot conditions that the change in specific enthalpy is given by:

p2 − p1 1 1
∆h = · +
2 ρ1 ρ2

The general form of Bernoulli’s law is fulfilled on both sides of the shock separately:

v2
+Φ+h=b
2
where Φ is the gravitational potential and b a constant.

(c) Assuming that the gravitational potential is the same on both sides, determine how the con-
stant b changes at the shock front.

(d) Explain whether Bernoulli’s law can be applied across shock fronts.

www.iaac.space 8/15
(extra page for problem B.1: Shock Wave)

Solution a:
(i) stationary flow: fluid properties f (e.g. v, ρ, p) do not change with time, i.e. ∂t f = 0
(ii) polytropic gas: it applies that p · V n is constant for some index n
(iii) specific enthalpy: enthalpy is the sum of a system’s internal energy U and p · V , thus H =
U + pV ; specific enthalpy is enthalpy per unit of mass, this h = H/m

Solution b:
From momentum conservation we obtain:
ρ2 v22 + p2 − p1 ρ1 v12 + p1 − p2
v12 = v22 =
ρ1 ρ2
With energy conservation and mass conservation it then follows:

v12 − v22 1 ρ2 v22 + p2 − p1 ρ1 v12 + p1 − p2

p2 − p1 1 1
∆h = h2 − h1 = = − = · +
2 2 ρ1 ρ2 2 ρ1 ρ2

Solution c:
From Bernoulli’s law we get:

v22 v2 v 2 − v22
b2 − b1 = + Φ + h2 − 1 − Φ − h1 = 1 + ∆h = 0
2 2 2

Solution c:
As the constant b does not change, Bernoulli’s law can be applied across shock fronts.

www.iaac.space 9/15
Problem B.3: Interplanetary Journey (6 Points)
A space probe is about to launch with the objective to explore the planets Mars and Jupiter. To
use the lowest amount of energy, the rocket starts from the Earth’s orbit (A) and flies in an ellip-
tical orbit to Mars (B), such that the ellipse has its perihelion at Earth’s orbit and its aphelion at
Mars’ orbit. The space probe explores Mars for some time until Mars has completed 1/4 of its orbit
(C). After that, the space probe uses the same ellipse to get from Mars (C) to Jupiter (D). There the
mission is completed, and the space probe will stay around Jupiter.

The drawing below shows the trajectory of the space probe (not drawn to scale):

D C
Sun

Earth
Mars

Jupiter

Below you find the obrital period and the semi-major axis of the three planets:

Orbital period Semi-major axis

Earth 365 days 1.00 AU
Mars 687 days 1.52 AU
Jupiter 4333 days 5.20 AU

How many years after its launch from the Earth (A) will the space probe arrive at Jupiter (D)?

www.iaac.space 10/15
(extra page for problem B.2: Interplanetary Journey)

Solution:
This problem requires the use of the third Kepler’s laws. The ellipses have a semi-major axis of:
R1 + R2
R12 =
2
As T 2 /R3 is constant for all planets and ellipses orbiting the Sun, we get (M for Mars):
2 2
3/2 3/2
TM T12 R12 R1 + R2
3
= 3
=⇒ T12 = TM = TM
RM R12 RM 2RM

The time for the manoeuvre τ is half of the ellipse, thus τ12 = T12 /2. This gives for A to B:
3/2
TM RE + RM
τAB =
2 2RM

which is 518 days. For B to C we have TM /4, thus 172 days. And for C to D:
3/2
TM RM + RJ
τCD =
2 2RM

which is 1129 days. The total time is then given by:

TM
τ = τAB + + τCD
4
→ Total time for the mission: 1560 days or 4.2 years

www.iaac.space 11/15
Problem B.4: Special Relativity - Part II (6 Points)
Space and time are interconnected according to special relativity. Because of that, coordinates
have four components (three position coordinates x, y, z, one time coordinate t ) and can be ex-
pressed as a vector with four rows as such:
 
ct

 x 

 
 y 
z

The spaceship from problem A.4 (Special Relativity - Part I) travels away from the Earth into the
deep space outside of our Milky Way. The Milky Way has a very circular shape and can be express
as all vectors of the following form (for all 0 ≤ ϕ < 2π):
 
ct

 0 

 
 sin ϕ 
cos ϕ

(b) Draw the shape of the Milky Way for a spaceship with a velocity of 20%, 50%, and 90% of the
speed of light in the figure below (Note: The ring shape for a resting spaceship is already drawn.):

www.iaac.space 12/15
(extra page for problem B.4: Special Relativity - Part II)

Solution a:
Applying the Lorentz transformation matrix to get the new shape:

ct0
      
γ 0 0 γβ ct γct + γβ cos ϕ
 x0   0 1 0 0  0
    0 
= =
     
y0
  
   0 0 1 0   sin ϕ   sin ϕ 
z0 γβ 0 0 γ cos ϕ γβct + γ cos ϕ

With ct0 = γct + γβ cos ϕ, we can convert into the time of the moving frame:

ct0
 

 0 

 
 sin ϕ 
0
βct + γ(1 − β 2 ) cos ϕ

With γ(1 − β 2 ) = 1/γ, we then get:

ct0
 

 0 

 
 sin ϕ 
0
βct + cos ϕ/γ

Solution b:
We use t0 = 0 and get due to the length contraction:

www.iaac.space 13/15
Problem C.1 : Earliest Galaxy Group (10 Points)
This problem requires you to read the following recently published scientific article:

Onset of Cosmic Reionization: Evidence of an Ionized Bubble Merely 680 Myr after the Big Bang.
V. Tilvi et al 2020 ApJL 891 L10. Link: https://iopscience.iop.org/article/10.3847/2041-8213/ab75ec/pdf

Answer the following questions related to this article:

(a) What is the so called cosmic reionization process?

→ epoch in universe development: neutral transparent universe from galaxies etc.
→ start: UV radiation from galaxies/groups ionized local surroundings
→ end: entire intergalactic medium is ionized

(b) What are Lyα lines and why did the researches aim to observe them?
→ transition lines of hydrogen, from 2nd to 1st orbit, emission at 121.6 nm (UV)
→ indication for ionized intergalactic medium

(c) What do the authors intend to point out with Figure 1 (see article)?
→ significant fluxes in redder wavelengths
→ no fluxes in visible wavelengths
→ indication for being at high redshifts

(d) How is confirmed that the peaks seen in Figure 3 are actually from Lyα emissions?
→ by considering the line asymmetry and calculating the Skewness

(e) How are the bubble sizes of the galaxies estimated?

→ theoretical model, relation between Lyα luminosity and bubble size predicted via simulations

(f) What is special about the findings and what are the scientific implications?
→ most distant galaxy group found yet
→ supports for inhomogeneous reionization through ionized bubbles

www.iaac.space 14/15
Problem C.2 : Massive Protostar Jet (10 Points)
This problem requires you to read the following recently published scientific article:

Measuring the ionisation fraction in a jet from a massive protostar.

Fedriani, R., Caratti o Garatti, A., Purser, S.J.D. et al. Nat Commun 10, 3630 (2019).
Link: https://www.nature.com/articles/s41467-019-11595-x.pdf

Answer the following questions related to this article:

(a) Why are massive stars important for the development of the universe?
→ synthesising most of the chemical elements
→ major feedback into the molecular clouds where stars are born

(b) How can the ionised part of jets be observed?

→ radio continuum emission from protostellar jets interpreted as thermal bremsstrahlung
→ (radio emission does not suffer from extinction)

(c) What kind of region is G35.2N? Describe how the region is structured.
→ high-mass star-forming region
→ two main cores: core A and core B
→ Core B: binary system, B-type stars

(d) What is the ionisation fraction χe and how is it being calculated?

→ total number density over electron number density: χe = ne /ntot
→ ntot : from properties of Fe II emission lines
→ ne : from ratio of Fe II lines or properties of radio emission

(e) How is the mass-loss rate being determined for knots K3 and K4? Why not for K1 and K2?
→ mass-loss rate via Ṁ = M v⊥ /k⊥
→ mass of knots not directly, but via M = µmH ntot V
→ K1: no velocity information available
→ K2: only upper limit on ne given

(f) Why is the ionisation fraction so small for G35.2N?

→ Explanation 1: stage of high ionisation not yet reached
→ Explanation 2: still accreting, thus protostar is swollen and too cool to photo-ionise

www.iaac.space 15/15
International Astronomy and
Astrophysics Competition
Final Round

Final Round Exam 2021

The final round exam was given in the form of an online exam.
Each participant was given a subset of 30 questions in random order.
This paper version is only available for training purposes.

IAAC Final Exam 2021 1/8

Question 1 : Which intense meteor shower peaks every year in August?

(A) Lyrids (B) Leonids (C) Perseids (D) Geminids

Question 2 : The distance between two objects with equal mass is doubled. By how much does
the gravitational force between the objects decrease?
√
(A) 1/ 2 (B) 1/2 (C) 1/4 (D) 1/8

Question 3 : A spaceship accelerates at 150 m/s2 from rest. How long does it take to reach a 1.2
km distant asteroid?

(A) 2 Seconds (B) 3 Seconds (C) 4 Seconds (D) 5 Seconds

Question 4 : What kind of radiation do pulsars emit?

(A) Compton radiation (B) Lyα radiation

(C) Planck radiation (D) Synchrotron radiation

Question 5 : The gravitational potential energy E between two objects with a distance R to each
other can be expressed as ...

(A) E = −G MR·m (B) E = −G MR·m

(C) E = G MR·m (D) E = G MR·m

Question 6 : What instrument is commonly used to observe and study pulsars?

(A) Laser interferometer (B) Radio telescopes

(C) Magnetic spectrometer (D) Radar telescopes

Question 7 : What type of astronomical object are pulsars most commonly?

(A) White dwarfs (B) Neutron stars (C) Quasars (D) Black dwarfs

IAAC Final Exam 2021 2/8

Question 8 : The two coordinate axis of the Equatorial Coordinate System are called ...

(A) Elevation and azimuth (B) Declination and right ascension

(C) Azimuth and declination (D) Ecliptic latitude and longitude

Question 9 : The Vernal equinox is the intersection point between ...

(A) the Earth’s rotational axis and the horizon

(B) the celestial equator and the horizon

(C) the Earth’s rotational axis and the ecliptic

(D) the celestial equator and the ecliptic

Question 10 : The closest star system to the Earth is called Alpha Centauri. How many stars be-
long to the Alpha Centauri system?

(A) 2 (B) 3 (C) 4 (D) 5

Question 11 : The picture below shows a satellite galaxy of the Milky Way: What is the name of
this satellite galaxy?

(A) Sagittarius Dwarf Spheroidal (B) Triangulum Galaxy

(C) Andromeda Dwarf (D) Large Magellanic Cloud

Question 12 : Stars from which spectral type have the lowest temperature?

(A) Class O (B) Class F (C) Class K (D) Class M

IAAC Final Exam 2021 3/8

Question 13 : Which one of the following expressions is the Rankine-Hugoniot jump condition
for the conservation of momentum through a shock wave?

(A) ρ1 v1 = ρ2 v2 (B) ρ1 v12 + p1 = ρ2 v22 + p2

v12 v22
(C) 2
+ h1 = 2
+ h2 (D) ρ1 v1 + h1 = ρ1 v1 + h2

Question 14 : Which one of the following expressions is correct for the Lorentz factor γ and β =
v/c?

(A) γ12 = 1 + β 2 (B) γ12 = βγ 2 (C) 1

γ2
= 1 − β2 (D) γ12 = βγ 2

Question 15 : The intergalactic medium is permeated mainly by ...

(A) neutral heavy elements (B) ionized heavy elements

(C) neutral hydrogen (D) ionized hydrogen

Question 16 : In a far-away solar system, a planet circles the main star with an orbital period of
100 days and a semi-major axis of 2 AU. Another planet with a semi-major axis of 8 AU is observed
in the same solar system. What is the orbital period of the new planet?

(A) 200 days (B) 400 days (C) 800 days (D) 1000 days

Question 17 : How is the part of a moon’s shadow called in which a total solar eclipse is possible?

(A) Umbra (B) Penumbra (C) Antumbra (D) Apex

Question 18 : The diameter of the lens’ aperture of your telescope is 5 m and you observe at a
wavelength of 1000 nm. What is the resolution of this telescope?

(A) 1.22 · 109 (B) 2.44 · 109 (C) 122 · 109 (D) 244 · 109

Question 19 : A fast travelling spaceship has a Lorentz factor of γ = 3. This corresponds to a

velocity v of ...
√ √ √ √
(A) 8/3c (B) 3/ 8c (C) 8/9c (D) 9/ 8c

IAAC Final Exam 2021 4/8

Question 20 : Imagine that a total solar eclipse happens on the Earth today. What is the phase
of the Moon on this day?

(A) New moon (B) Waxing crescent

(C) Waning gibbous (D) Full moon

Question 21 : The negative effect of incoming wavefront distortions on telescope images is re-
duced by using a system called ...

(A) Adaptive spectroscopy (B) Active spectroscopy

(C) Adaptive optics (D) Active optics

Question 22 : You are given a list of stars and their apparent magnitude. Which one is the faintest
star?

(A) Star A, 5 mag (B) Star B, 0.3 mag

(C) Star C, -0.2 mag (D) Star D, -5 mag

Question 23 : The right ascension is measured in units of ...

(A) kelvin (B) meters (C) hours (D) declinations

Question 24 : The Lyman-alpha line can be found at a wavelength of around ...

(A) 120 nm (B) 330 nm (C) 860 nm (D) 1100 nm

Question 25 : Which transition is called Lyα lines?

(A) Hydrogen, 1st to 2nd orbit (B) Helium, 1st to 2nd orbit

IAAC Final Exam 2021 5/8

Question 26 : Below, you can see the rough sketch of a telescope. What type of telescope is
drawn?

(A) Newtonian telescope (B) Refractor telescope

(C) Cassegrain telescope (D) Catadioptric telescope

Question 27 : An Analemma is a diagram showing the position of from a fixed location and
the same mean solar time over the course of a year.

(A) the Sun (B) the Moon

(C) the Vernal equinox (D) Jupiter

Question 28 : Which astronomical coordinate system uses the observer on Earth as a reference
point?

(A) Horizontal Coordinate System (B) Equatorial Coordinate System

(C) Ecliptic Coordinate System (D) Galactic Coordinate System

Question 29 : Which type of star is only stable due to the pressure of an electron gas?

(A) Neutron stars (B) Supergiant stars

(C) Blue dwarfs (D) White dwarfs

Question 30 : The colour of a star depends on the ...

(A) mass (B) radius

(C) temperature (D) orbital velocity

IAAC Final Exam 2021 6/8

Question 31 : An object is moving with a velocity of 0.6c in your line of sight. What is the redshift
z of this object?
√
(A) z = 1 (B) z = 2 (C) z = 1.5 (D) z = 2.5

Question 32 : In which one of the following time frames after the big bang did the cosmic reion-
ization process begin?

(A) 0 - 999 hundred years (B) 1 - 999 thousand years

(C) 1 - 999 million years (D) 1 - 14 billion years

Question 33 : Massive stars are essential for the universe because they ...

(A) neutralize the intergalactic medium. (B) emit synchrotron radiation.

Question 34 : The drawing below shows a comet in four positions around the Sun. For which
position does the comet’s tail point in the correct direction?

(A) Position A (B) Position B (C) Position C (D) Position D

Question 35 : What spectral region can be observed from the Earth even when there are clouds
in the sky?

(A) Gamma wavelengths (B) Radio wavelengths

(C) Ultraviolet wavelengths (D) X-ray wavelengths

IAAC Final Exam 2021 7/8

Question 36 : What was the last planet visited by Voyager 1?

(A) Jupiter (B) Saturn (C) Uranus (D) Neptune

Question 37 : The time between two full moons is approximately ...

(A) 28 days (B) 28.5 days (C) 29 days (D) 29.5 days

Question 38 : Which one of the following objects is called a black dwarf in astronomy?

(A) A star that will become a black hole (B) A cooled white dwarf

(C) A low radiation-emitting star (D) A high radiation-absorbing star

Question 39 : Which planet has an axial tilt of 98 degrees?

(A) Saturn (B) Uranus (C) Neptune (D) Pluto

Question 40 : The smallest distance between Pluto and the Sun is about ...

(A) 20 AU (B) 30 AU (C) 40 AU (D) 50 AU

IAAC Final Exam 2021 8/8

International Astronomy and
Astrophysics Competition
Final Round

Final Round Exam 2021 - Solutions

No. 1 No. 2 No. 3 No. 4 No. 5 No. 6 No. 7 No. 8 No. 9 No. 10

C C C D A B B B D B

No. 11 No. 12 No. 13 No. 14 No. 15 No. 16 No. 17 No. 18 No. 19 No. 20

D D B C D C A D A A

No. 21 No. 22 No. 23 No. 24 No. 25 No. 26 No. 27 No. 28 No. 29 No. 30

C A C A C A A A D C

IAAC Final Exam 2021 1/1

International Astronomy and
Astrophysics Competition
Qualification Round 2022

Problem A : The James Webb Space Telescope (5 Points)

The James Webb Space Telescope (JWSP) was finally launched into space on December 25, 2021,
and is about to become one of the most important scientific instruments of our time. Name the
major components of the JWSP shown in the figure below:

(A) (B) (C)

(D) (E) (F)
(G) (H) (I)
(J)

www.iaac.space 1/3
Problem B : Very Dense Earth (5 Points)
Neutron stars are some of the densest objects in the universe. They form during supernova ex-
plosions and are very small compared to other astronomical objects. You can assume 5 x 1017
kg/m3 as the average density of a neutron star. The Earth has a total mass of about 5.97 x 1024 kg.

What would be the diameter of the Earth if it had the density of a neutron star?

Problem C : Asteroid Field (5 Points)

Assume that a gravitational anomaly in the solar system has shifted a field of asteroids into Earth’s
orbit, and the field is now moving directly towards Earth. The asteroid field has a density of ρ
(asteroids/volume), and each asteroid has an average mass of m. The asteroid field stretches
over a distance d in Earth’s path. Assume that the Earth moves with a velocity of v, the asteroids
with u, and that v u. Let R be the radius and M the mass of the Earth. The Earth collides with
the asteroid field:

v u

Show that the slow-down ∆v of the Earth due to the asteroid collisions is given by:

1
∆v = v 1 − m
1 + πR2 dρ M

Problem D : Position of the JWST (5 Points)

The James Webb Space Telescope is positioned around 1.5 million kilometres from the Earth on
the side facing away from the Sun. The telescope remains at this distance and orbits around the
Sun with the Earth’s orbital velocity.

(a) Why is it important to position the JWST behind the Earth?

(b) Determine the angular velocity ω of the telescope as it orbits around the Sun.

The centrifugal Fω and gravitational force FG are acting on objects orbiting the Sun: F = Fω −FG

(c) Based on this, how much should the telescope accelerate towards or away from the Sun?
(d) Why is the orbit of the telescope stable nonetheless? What other forces need to be considered?

www.iaac.space 2/3
Problem E : Infrared Radiation (5 Points)
The electromagnetic spectrum contains various types of radiation with different properties. The
James Webb Space Telescope is not like an optical telescope at home and does not capture visible
light; instead, it observes the sky in the infrared spectrum.

Explain what infrared radiation is and answer the following questions: How is infrared radiation
different from visible light? Why does the JWST observe infrared light, and what are the scientific
advantages for astronomers?

General Information and Submission

You can write the solution by hand or type it on a computer. To qualify for the pre-final round, you have to
get at least 15 points (Junior, under 18 years) or 20 points (Youth, over 18 years). Make sure to submit your
solution by Friday 20. May 2022 23:59 UTC+0 online at www.iaac.space/submission. In case of questions or
comments, please feel free to contact us via email: info@iaac.space. Good luck!

www.iaac.space 3/3
International Astronomy and
Astrophysics Competition
Qualification Round 2022

Problem A : James Webb Space Telescope (5 Points)

(A) Primary mirror (B) Science instrument module (C) Optics subsystem
(D) Secondary mirror (E) Sunshield (F) Star trackers
(G) Spacecraft bus (H) Antenna (I) Solar array
(J) Stabilization flap

Problem B : Very Dense Earth (5 Points)

1/3 1/3
M 4 3M 6M
V = = πR3 ⇒ D = 2R = 2 = = 283.6 m
ρ 3 4πρ πρ

Problem C : Asteroid Field (5 Points)

Number of asteroid collisions: N = ρV = ρ · πR2 · d; momentum conservation:

Mv 1
M v + N mu ≈ M v = (M + N m)w ⇒ w = ⇒ ∆v = v − w = v 1 − m
M + Nm 1+NM

Problem D : Position of the JWST (5 Points)

(a) To block the radiation of the Sun
(b) ω = 2π/365d = 0.017/d
(c) a = mF
= ω 2 (dE + d) − G (dEM+d)2 = 2.57 · 10−4 m/s = 22.2 m/d (away from the Earth)
(d) The gravitational force of the Earth FG,E → enables the stable Lagrange point L2

Problem E : Infrared Radiation (5 Points)

→ wavelength 1 mm to 700 nm; lower energy than red; → infrared mostly blocked by Earth’s
atmosphere; → cosmological redshift: capable of observing more distant objects
→ observing through dust and clouds possible

www.iaac.space Basic Solution

Problem A

A) Paimay muwo
n t Module
B In+egrated Science Instnu me
COptical Telescope Element
muay
b) Secon dayy
E Sunshield
F) S t t a c k s
) Spacecnaft bu*
H) E a t h pointing antenna

I) Solan atna
J) Momentum laap

Problem B
&neutron stan =
5 X1o kg /m
AVHage density
Mass o Eath 5.97 Xio kgF
oF Hadius R,
Eath to
be a sphee
the
Assuming th = 4TR$
the Ea
Yoome of 3

Nou Mass= Densi

Volume

MasS
So Vowme
Densi

R 3 =5.47X1o *
5 X lo

m°
2850
50465.
03I
7R
m.
on R 141.78
2
28833.5
.5 m
m
is 2R a

diamete

So, neutnon sta

densi f a
Ea
F ax
nt h
had t h e the actval
valwe
the m a8ainst
Theefone , if be 28 3.
5t
aoould only times Jes),
44, 934
diamete
ts about
km-
12,¥42
of
Problem C

Eanth

Ma ss M

Radios R

asteoid that is
going to col li de
Let us
u+ ind o u t the mass of
oith Eanth. encounte a
eiuculax croSS-section
ield will
The asteoid.
F Eath ie.
astenoids volumne

of asteroid R d x m
tokal mass
So aY ma ss of
Vowme no./vo. each aStexoid

Rdem
the Eaxth and
Collision,
shalI consid e
af teH t h e
that
Also, w e
moving with velo c v
m e a single s r e m
asteoids beco

E aH th

collision to be elasic,
Assuming the ConsUvekX
will be
Total
kineic en eg
- c
(Rdpm )u* ( M +7Rd pm )vi2
+
Mv 2

be eons e ve d
momentum
will
OTotal
v/
=(M + TR*pdm )
-C2)

Mv +CTR Pdm u
Re-aLanging CD
M (v2- v ) =
R pdm. (v/ - u a) -(3)

Re-asanging C2)
-(4
M CV-v') TR2 pdm(v-u
=

Solving (4);
TR*pdmv
-
TR2e dmu
MV Mv
t TR* e dm u = v?
MV
M+TRpdm

V+ TR m ( dividing each tepm by M)

or v M

I+7TR pd m
M

+TR2p dividin each tom by

+TRpd m.
v>>u
RP M = o in e e
v'
It TRpd m
M

V
V-
Av= V -v/
1+7RPdm
M
1
oH Av= v ( 1 -
1+R ed
asteHoid
due to the
Av of the Eax th
slou-douwn
The

is thus ob t a i n e d .
eollis io ns
PROBLE MD
Fayth.. The TWST shall
the JwST behind the
imponbant to position
fou ils hea ing eppec+).
aIt s
which is Knouwn
pximaily odseve infraned light
fainb infnane & signals,
it
need s to be
Since it will be obsexving vouy
emitt of infnaned nadiaHons.
Sun,uwhich is maja
a
Shielded from the

HOT SIDE
85°e COLD SIDE ( 233° e

Solo
Panet
Science instoment S
and
shield

SUN'S
Miw o
RAYS

TWST

ib behind the
Eanth is the avai la biily
Anothe eason for pla.cing to stable
L&. Placing a body at L2 Leads
a

o Lagrang
the point
bodies obiting each o t h e Het staying-
Config wation fo thuee oth . each
aelaive to
in the same posihion

L3
Sun L E oth

PosiHon
TwST

L5
third laa of peiodS iis knouon that
b From Keples
T2 = KA3
T is time peiod
the onbit
is adius of
2m-3
is a constant K = 2.9 x 1o
K 4T the Sun
hene, Ms
is mas of
GMs

4o 3TWST is
+1.5 milion km
I A U.
dis ranee betoeen
distance between eath and TwST
Sun and eaHth

million km-
6
S =151 miltion km +
So
I52.5 X 1o m

So, T (k)

-2.97X10-/9 x (152.5 xio)*J

324 5 5lo 2.88

= 1. 1 3 5 X lo-7 Had s-!

velocityw 27T
.Angulax

Fne Fw- F

mV -
m ms G

2
Hadus 52.5x/ m-mass o
JWST
mg- mass oF
Sun
Acceleati on a = Fnet

Sun m

(-m
x 66.67 X10-
[sox1o3]'-2xiox
I52. 5Xio
152.5 X1ol
here
the Sun
=
1. 655 x 10 m/s away fpom opbital veloa, F
TwST is same aas
hat of Eax th
30 kms-
d) The orbit of the tele scope is sta ble nonetheless. Th is is because
hexe, in the paevious calculaHon , we had not considered the gnavitaHonal

pul Eantm.
Eax th shall abbract TwST wi th an acceleuaH on a

E GMe =(6.6 XIo-")x C6xio 24) = .44 X io-4 m/s

C1.5x10)
This balances the outwand accelenai on of 1.656 10- m/s
dua ls the Eaxth needs 5
The foce of gxavi taton a atbachon

be eonsideed.

Pmoblen E

Eleetromagne ic cwaves having wavelength in the ange oF

8000 A to 1oA C1 mm) axe called infrar ed radiations.
J
william Hexsheli st detected it in 1800 as a paxt of the

spectrum which is invisible' but has a

"Stmo ng- heating
effeet'onlike the visi ble spectrum(A>4o0 L5 800 nm)
TwST to Study gala, stax and plane+ formation-
aims
in the oniveHSe So it will back in time. Tbe univeXae
have to look
the fa h coe look , the m o r e xedshifte&
is expanding and thus
which coas inital
This implies t h a t lighE
the tight is. to infraned
emi Hed visible
in the
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on
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time they ane obseve&.
spectnom by the
takes place
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clouds. The dust obscores
in the c e n t e n s of dense, dusty
visible wavelengths
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the
ets o aboot+ Eanths
e a n peneiate
the d u s t . Maneove, obje

emit most of theix uadiahon a mid- infrane

temp e a t x e

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TWST coill a l l o u t h e seientists td withess
Hence, the
the distan t xeaches of pace and an
epoch o tHme n e v e

Combined asith the

data om Chanda
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wSTill
and Hobbls pace Telescope,
X~Hay obsexval5 and b e t e ondaxstan du'ng of owe
present a complete pictue
&.
Cosmos l5 the sc'enHs
Problem A : The James Webb Space Telescope
Solution:
(A) Primary Mirror
(B) Integrated Science Instrument Module (ISIM)
(C) Fine Steering Mirror
(D) Secondary Mirror
(E) Sunshield
(F) Star Trackers
(G) Spacecraft Bus
(H) Earth-pointing Antenna
(I) Solar Array
(J) Momentum Flap

1
Problem B: Very Dense Earth
Solution:
Let,
Mass of the Earth = Me
Mass of the Earth with new density = Me ′
Radius of the New Earth = Rn
Diameter of the new Earth = Dn
Density of the new Earth = ρn
Volume of the new Earth = V ′
From the question,

M e = Me ′
⇒ Me = V ′ ρn
4
⇒ Me = πRn 3 ρn
3
Me 4π 3
⇒ = Rn
ρn 3
3Me
⇒ = Rn 3
4πρn
s
3Me
⇒ Rn = 3
4πρn
s
3 3 × 5.97 × 1024
⇒ Rn =
4 × 3.1416 × 5 × 1017
⇒ Rn = 141.788 m
⇒ Rn ≈ 142 m
⇒ Dn = (142 × 2) m
∴ Dn = 284 m

∴ Diameter of the New Earth, Dn = 284 m (Answer)

2
Problem C : Asteroid Field
Solution:
Given that,
The number density of asteroid field = ρ
Average mass of asteroids = m
∴ Total mass of asteroids per unit volume = mρ
Assuming the asteroid field as sphere,
Diameter of the field = d
Mass of Earth = M
Radius of Earth = R
∴ Cross section of Earth, A = πR2

Figure 1: Earth and Its Motion Through the Asteroid Field

During the collision, Earth will go through the asteroid field while making a cylindrical path
with volume V , and the mass of the asteroids in that path will be added to Earth assuming they
won’t burn in the atmosphere.

∴ V = Ad
⇒ V = πR2 d

So, the mass added to Earth after collision is equal to V mρ.

Let,
Mass of Earth after collision, M ′ = M + V mρ
Velocity of Earth after collision = v ′

3
From momentum conservation,

M v = M ′ v ′ [∵ v >> u]
Mv
⇒ v′ =
M′
Mv
⇒ v′ =
M + V mρ
Mv
⇒ v′ =
M + (πR2 d)ρm
Mv
⇒ v′ =
M + πR2 dρm
Mv
⇒ v′ = m
M 1 + πR2 dρ
M
′ v
∴v = m
1 + πR2 dρ
M

Now the slowdown ∆v can be written as:

∆v = v − v ′
v
⇒ ∆v = v − m
1 + πR2 dρ
 M 
1
∴ ∆v = v 1 − m

1 + πR2 dρ
M
 
1
∴ The slow-down ∆v of the Earth due to the asteroid collisions is: ∆v = v 1 − m
 (Showed)
1 + πR2 dρ
M

4
Problem D: Positions of the JWST
Solution:
a) It is important to place JWST behind the Earth to protect it from sun’s radiation in different
wavelengths; which would disturb JWST’s working procedure. Earth works as a shield for JWST,
protecting it from various radiations from sun. This way, JWST is protected from incoming pow-
erful radiation from the sun.

b) We know,

Distance of Earth from the Sun, R = 1.5 × 1011 m

Orbital period of Earth, T = 365.25 days

= (365.25 × 24 × 60 × 60) s
= 31557600 s
Given that,
Distance of JWST from Earth, r = 1.5 × 106 km
Let,
Angular velocity of JWST = ω
Orbital velocity of Earth around the Sun = v
Distance of JWST from Sun = R′

∴ R′ = R + r
⇒ R′ = (1.5 × 106 + 1.5 × 1011 ) m
= 1.500015 × 1011 m

We know,
2πR
v=
T
2 × 3.1416 × 1.5 × 1011

⇒v=
31557600
∴ v = 29865.389 ms−1

Since JWST moves with the same orbital velocity,

v
ω= ′
R

29865.389
⇒ω= rad/s
1.500015 × 1011
⇒ ω = 1.99 × 10−7 rad/s
∴ ω ≈ 2 × 10−7 rad/s

∴ The angular velocity of JWST is approximately 2 × 10−7 rad/s. (Answer)

5
c) The centrifugal force Fω and gravitational force FG are acting on objects orbiting the Sun
according to the following equation:

F = Fω − FG
mv 2 GM m
⇒ ma = −
r r2
2
mv r − GM m
⇒ ma =
r2
2
m(v r − GM )
⇒ ma =
r2
v 2 r − GM
⇒a=
r2
2
v r GM
⇒a= 2 − 2
r r
v2 GM
⇒a= − 2
r r
2 GM
⇒a=ω r− 2
r
h 2 i 6.673 × 10−11 × 2 × 1030
⇒ a = 2 × 10−4 × 1.5 × 106 × 1 × 103 − 2
(1.5 × 106 × 1 × 103 )
∴ a = 0.6844 m/s2

Since the value of the acceleration is positive, we can say that the telescope will be accelerating away

from the sun at a value of 0.6844m/s2 . (Answer)

d) From (c), we can see that the telescope will accelerate away. It seems to match with the real
situation, since JWST orbits in L2 (Lagrange Point 2), which is an unstable equilibrium. However,
to stay stable, JWST must employ Station Keeping, which indicates the process of a spacecraft
keeping its distance to another spacecraft/celestial body constant to stay stable in orbit. For that,
JWST has to apply periodic thrusts to keep itself stable.

6
Problem E: Infrared Radiation
Solution:
Infrared Radiation: Definition
Infrared radiation is a part of the electromagnetic spectrum which has a longer wavelength than
visible light and the range for this radiation usually lies in 1 mm to 700 nm. We can see infrared
radiation using various detectors. The most common source for infrared radiation is the Blackbody
radiation of the objects within room temperature. It is often referred to as IR and has 3 major
regions:
NIR: Near Infrared;

MIR: Middle Infrared;

FIR: Far Infrared.

Difference from Visible Light

The main difference of infrared radiation from visible light is the wavelength. It is longer than that
of the visible ones, so it is invisible to Human eyes. However, detectors allow us to see and detect
infrared radiation. The detectors detect the detected radiations and those are then converted to
visible light for us to see. Contrary to visible light, infrared radiation has no color and most objects
radiate more energy in infrared radiations than that of the visible light.

Reasons for JWST Working in Infrared Radiation

JWST’s primary emphasis is on infrared astronomy. Observing in the infrared spectrum is a must
for observing much older galaxies and stars. The main reasons why Infrared radiation is needed for
JWST are:

Cosmological Redshift: Usually light from farther objects reaches to us later than light from
the closer objects. Because of the universe’s expansion, light becomes redshifted as it keeps
travelling, and objects at extreme distances are therefore easier to see in the Infrared region
of the electromagnetic spectrum. JWST’s infrared capabilities are expected to let it see back
in time to the first galaxies forming just a few hundred million years after the Big Bang. For
this reason, infrared radiation is the way to go.
Better Interstellar Visibility: The interstellar space is filled with dust, gas and other objects.
These gases and dust obscure visible light, therefore limiting visibility and observation range
of the Optical telescopes. However, infrared radiations better penetrates obscuring dust and
gas and can pass more freely through regions of cosmic dust that scatter visible light. This
is why infrared radiation-based observations allow the study of objects and regions of space
which would be obscured by gas and dust in the visible spectrum, such as the star-forming
molecular clouds, the planet-bearing circumstellar disks, and active galaxy cores. Relatively
cool objects (temperatures less than several thousand degrees) emit their radiation primarily
in the infrared, since IR has a longer wavelength and therefore, contains lesser energy per
quanta compared to visible light. As a result, most objects that are cooler than stars are

7
better studied in the infrared. This includes the clouds of the interstellar medium, brown
dwarfs, planets both in our own and other solar systems, comets etc.
Detecting Exoplanetary Atmospheres: In Earth, the atmosphere prevents most of the infrared
radiation coming outside from space. This characteristic can be seen in most of the atmo-
spheres depending on the contents of the atmosphere. So JWST can detect infrared radiations
near exoplanets and see if it allows IR to go through its nearby areas or not. This can prove
atmosphere’s existence and therefore make a great contribution in the field of studying exo-
planets.

Advantages For Astronomers

Earth allows only a sliver of the whole range of the Infrared radiation into the atmosphere. This is
why ground-based infrared based studies are very limited and so, a space-based Infrared telescope
helps in a lot of ways. The first effort like this is the Spitzer Space Telescope, which observed in the
infrared region. However, JWST is better than its predecessors in that it has better technology and
methods to take images and analyze observed regions of space. Being placed at L2, JWST will be
protected from Sun’s intense radiation and therefore can function properly. In this way, JWST will
hopefully produce more sophisticated images than ever before. Potentially being one of the most
important scientific instruments of our time, JWST will look at a time way beyond Hubble Space
Telescope has seen. It will hopefully provide us with images showing states of the early Universe
and how galaxies formed. This can boost astronomical researches and is therefore a very important
factor for astronomers. Thus, JWST can help astronomers analyze the growth of the universe and
learn about our surroundings a lot.

Sources
• Wikipedia
• NASA JWST Webpage
• JWST YouTube Channel

8
International Astronomy and
Astrophysics Competition
Pre-Final Round 2022

Important: Read all the information on this page carefully!

• You may use extra paper if necessary, however, the space under the problems is usually enough.

• Typing the solution on a computer is allowed but not recommended (no extra points).

• The six problems are separated into three categories: 2x basic problems (A; four points), 2x ad-
vanced problems (B; six points), 2x research problems (C; eight points). The research problems re-
quire you to read a short scientific article to answer the questions. There is a link to the PDF article.

• You receive points for the correct solution and for the performed steps. Example: You will not get
all points for a correct value if the calculations are missing.

• Make sure to clearly mark your final solution values (e.g. underlining, red color, box).

• You can reach up to 36 points in total. You qualify for the final round if you reach at least 18 points
(junior, under 18 years) or 24 points (youth, over 18 years).

Uploading Your Solution

• Please upload a file/pictures of (this sheet with) your written solutions: https://iaac.space/login

• Only upload one single PDF file! If you have multiple pictures, please compress them into one
single file. Do not upload your pictures in a different format (e.g, no Word and Zip files).

• The deadline for uploading your solution is Sunday 26. June 2022, 23:59 UTC+0.

• The results of the pre-final round will be announced on Monday 4. July 2022.

Good luck!

www.iaac.space 1/9
Problem A.1: Looking back with the JWST (4 Points)
The James Webb Space Telescope (JWST) will allow us to look back in time and observe the early
universe. You are a scientist trying to observe an object that emitted its light a long time ago.

(a) Explain why the light you receive from the object is red-shifted.

The object has a redshift of 7.6 and the JWST observes the object at a wavelength of 2 microme-
tres (mid-infrared light).

(b) What is the wavelength of the light emitted by the object?

www.iaac.space 2/9
Problem A.2: Counting Asteroids (4 Points)
An extraterrestrial civilisation lives on a planet with a very elliptical orbit. Additionally, thousands
of large asteroids orbit their solar system. The civilisation uses the light from their home star to
count the number of asteroids in the direct line between the star and their planet.

For a first measurement, they count the asteroids for 60 days and detect 1000 objects. Several
months later, they start a second measurement: This time, they count for 80 days.

How many asteroids will they detect during the second measurement? Explain why.
(Note: Assume that the asteroids are homogeneously distributed in their solar system.)

www.iaac.space 3/9
Problem B.1: Rotating the JWST (6 Points)
The JWST has a propulsion system to adjust the orbit and orientation of the telescope.

For this problem, we assume that the JWST only consists of the 18 primary mirror segments (with
a weight of 40 kg each, m1 ) forming a cylinder with a radius of 3.3 m (R), the Aft optical subsystem
with a weight of 120 kg (m2 ) forming a cone with a radius of 65 cm (r), and the science instrument
module with a weight of 1400 kg (m3 ) forming a cuboid with a side length of 5.3 m (a):

Aft Optical Subsystem

(rotational axis)
Primary Mirror

Science Instrument Module

(a) Derive a general expression for the moment of inertia I of the telescope’s shape with respect
to the dimensions R, r, a and the masses m1 , m2 , m3 . (Hint: Derive the moment of inertia for the
individual components first. The rotational axis is the axis of symmetry.)

(b) Calculate the numerical value of I for the JWST. (Use only the values from the text above.)

To perform calibration measurements, the researchers need to rotate the telescope by 90 de-
grees. For that, they fire the MRE-1 thrusters at the bottom edge of the primary mirror (see figure)
for 0.5 seconds with a thrust of 2.5 newtons.

www.iaac.space 4/9
(extra page for problem B.1: Rotating the JWST)

www.iaac.space 5/9
Problem B.2: Changing Temperature (6 Points)
The energy of our Sun is responsible for life on Earth. We are very lucky that the Sun has the right
conditions and that the Earth is at the exact right position to create habitable temperatures.

(a) Find an equation for the surface temperature of the Earth TE (R, T ) with respect to the ra-
dius R and the surface temperature T of the Sun.
(Note: Approach the Earth and the Sun as black bodies; then, account for the Earth’s albedo of 30%
and add an atmosphere correction factor of 1.13 to the surface temperature of the Earth.)

The radius of the Sun is 696 x 103 km, and the surface temperature is 5772 K:

(b) Confirm with your equation that Earth’s current surface temperature is 15 ◦ C.

The two axes of the diagram below display a relative change in the surface temperature (x-axis)
and radius (y-axis) of the Sun.

(c) Draw a black line in the diagram for all pairs (R, T ) that still result in a temperature of 15 ◦ C
on the Earth. If the Sun’s temperature increases by 10%, how much needs the radius to decrease
to maintain 15 ◦ C on Earth?

(d) Draw a grey area in the diagram for all (R, T ) that result in a temperature ± 10◦ from 15 ◦ C.

www.iaac.space 6/9
(extra page for problem B.2: Changing Temperature)

www.iaac.space 7/9
Problem C.1 : The Surface of Planets (8 Points)
This problem requires you to read the following recently published scientific article:

Inferring Shallow Surfaces on Sub-Neptune Exoplanets with JWST.

Shang-Min Tsai et al 2021 ApJL 922 L27. Link: https://iopscience.iop.org/article/10.3847/2041-8213/ac399a/pdf

Answer the following questions related to this article:

(a) What is a proxy? What proxy is this study trying to find, and what are they doing differently
compared to previous studies?

(b) Explain the meaning and use of the following acronyms: HELIOS, Exo-FMS, HAZMAT, NIRSpec.

(c) Make a sketch of the components used to model the planet (including the pressure-longitude
grid and the equatorial regions):

(d) Explain the components of Figure 1. Why was it included in the paper?

(e) Why is CH4 not a suitable proxy for the surface pressure?

(f) You detect CH3 OH but non NH3 in the atmosphere of a sub-Neptune planet. What type of
surface does this planet have?

www.iaac.space 8/9
Problem C.2 : Black Holes and the JWST (8 Points)
This problem requires you to read the following recently published scientific article:

The Age of Discovery with the James Webb:

Excavating the Spectral Signatures of the First Massive Black Holes.
Inayoshi, K. et al. arXiv:2204.09692 [astro-ph.GA] (2022). Link: https://arxiv.org/pdf/2204.09692.pdf

Answer the following questions related to this article:

(a) What are massive black holes (BH)? Why is the observation of young massive BHs important?

(b) What is the spectral energy distribution (SED)?

(d) What are broad-band filters, and what is their use in astronomy?

(e) Explain the increase of all lines for high z in Figure 3, top-left panel.

(f) Explain the meaning and use of the magenta rectangle in Figure 4.

www.iaac.space 9/9
International Astronomy and
Astrophysics Competition
Pre-Final Round 2022

Solutions to the Pre-Final Round 2022

Please note that there are many ways to reach the final solutions.
Not all detailed steps are elaborated in this solution document.

www.iaac.space 1/??
Problem A.1: Looking back with the JWST (4 Points)
The James Webb Space Telescope (JWST) will allow us to look back in time and observe the early
universe. You are a scientist trying to observe an object that emitted its light a long time ago.

(a) Explain why the light you receive from the object is red-shifted.

The object has a redshift of 7.6 and the JWST observes your object at a wavelength of 2 microme-
tres (mid-infrared light).

(b) How long was the wavelength of the light emitted by the object?
(c) What type of radiation was originally emitted by the object?

Solution a:
General relativity → expansion of universe → space between objects stretches → wavelength
stretches → redshift

Solution b:
λ2 λ2
z= − 1 ⇒ λ1 =
λ1 z+1
→ Result: 233 nm

Solution c:
→ UV radiation

www.iaac.space 2/??
Problem A.2: Counting Asteroids (4 Points)
An extraterrestrial civilisation lives on a planet with a very elliptical orbit. Additionally, thousands
of large asteroids orbit in their solar system. They use the light from their home star to count the
number of asteroids in the direct line between the star and their planet.

For a first measurement, they count the asteroids for 60 days and detect 1000 objects. Several
months later, they start a second measurement: This time, they count for 80 days.

How many asteroids will they detect during the second measurement? Explain why.
(Note: Assume that the asteroids are homogeneously distributed in their solar system.)

Solution:
— Homogeneous distribution: constant density ∆N/∆A
— Kepler’s 2nd law: equal area covered in equal time, i.e. constant ∆A/∆t

∆N2 ∆N1 ∆A2 ∆t2

= =⇒ ∆N2 = ∆N1 · = ∆N1 ·
∆A2 ∆A1 ∆A1 ∆t1

→ Result: 1333 asteroids during the 2nd measurement

www.iaac.space 3/??
Problem B.1: Rotating the JWST (6 Points)
The JWST has a propulsion system to adjust the orbit and orientation of the telescope.

Aft Optical Subsystem

(rotational axis)
Primary Mirror

Science Instrument Module

(b) Calculate the numerical value of I for the JWST. (Use only the values from the text above.)

www.iaac.space 4/??
(extra page for problem B.1: Rotating the JWST)

Solution a:
Primary mirror (with V = πR2 h):
Z R
2 R4 ρV R2 nm1 R2
I= 2πr⊥ dr⊥ h · r⊥ ρ = 2πρh · = =
0 4 2 2

Aft optical system (with V = πr2 h/3):

h ry/h h
r4 y 4 r4 h5 πρr4 h 3m2 r2
Z Z Z
2
I= dy 2πr⊥ dr⊥ · r⊥ ρ = 2πρ
= 2πρ · dy ·
= =
0 0 0 4h4 20h4 10 10
p
Science instrument module (with V = a2 h and r⊥ = x2 + y 2 ):
a/2 a/2 a/2
a3 a4 m3 a2
Z Z Z
2 2 2
I= dx dy · h(x + y )ρ = hρ dx x a + = hρ · =
−a/2 −a/2 −a/2 12 6 6

Total moment of inertia:

nm1 R2 3m2 r2 m3 a2
I= + +
2 10 6
Solution b:
→ Moment of inertia: 10490 kg·m2

Solution c:
The initial time for the acceleration is negligible; the torque due to thrust is:
RF
τ = R · F = I · ω̇ ⇒ ω̇ =
I
This yields the final angular velocity (with t0 = 0.5 seconds):
ω RF t0
ω̇ = ⇒ ω = ω̇t0 =
t0 I
Finally, we have:
∆ϕ I
ω= ⇒ ∆t = ∆ϕ ·
∆t RF t0
→ Result (with ∆ϕ = π/2): 3994 seconds or 66.6 minutes or 1h7min

www.iaac.space 5/??
Problem B.2: Changing Temperature (6 Points)
The energy of our Sun is responsible for life on Earth. We are very lucky that the Sun has the right
conditions and that the Earth is at the exact right position to create habitable temperatures.

The radius of the Sun is 696 x 103 km, and the surface temperature is 5772 K:

(b) Confirm with your equation that Earth’s current surface temperature is 15 ◦ C.

The two axes of the diagram below display a relative change in the surface temperature (x-axis)
and radius (y-axis) of the Sun.

(d) Draw a grey area in the diagram for all (R, T ) that result in a temperature ± 10◦ from 15 ◦ C.

www.iaac.space 6/??
(extra page for problem B.2: Changing Temperature)

Solution a:
From the Stefan-Boltzmann law we get for the radiation of the Sun:

L = 4πR2 σT 4

The radiation arriving on the Earth is (d equals 1 AU):

2
πRE rE2
LE = L · = L ·
4πd2 4d2
This gives us the temperature for the Earth (with α = 0.7 and correction β = 1.13):
1/4 1/4 α 1/4 √
R2 α

2 LE α
LE α = 4πRE σTE4 ⇒ TE = β 2
=β T =β RT
4πRE σ 4d2 4d2

Solution b:
→ Result of TE : 288 K, that is 15 ◦ C

Solution c:
→ The radius needs to decrease to about 0.83, that is by 27%.

Solution c+d:

www.iaac.space 7/??
Problem C.1 : The Surface of Planets (8 Points)
This problem requires you to read the following recently published scientific article:
Inferring Shallow Surfaces on Sub-Neptune Exoplanets with JWST.
Shang-Min Tsai et al 2021 ApJL 922 L27. Link: https://iopscience.iop.org/article/10.3847/2041-8213/ac399a/pdf

Answer the following questions related to this article:

(a) What is a proxy? What proxy is this study trying to find, and what are they doing differently
compared to previous studies?

→ To determine a variable (not directly measurable) with another correlating variable (directly
measurable); proxy for the presence of surfaces; interaction between day and night with a 2D
model

(b) Explain the meaning and use of the following acronyms: HELIOS, Exo-FMS, HAZMAT, NIRSpec

→ HELIOS: a radiative transfer model; Exo-FMS: 3D global circulation model; HAZMAT: a program
for studying certain planets; NIRSpec: an instrument of JWST

(c) Make a sketch of the components used to model the planet (including the pressure-longitude
grid and the equatorial regions):

→ (Pressure: z-direction; longitudes: x-direction; equatorial regions: 45-135, 135-225, 225-325,

325-45 degrees)

(d) Explain the components of Figure 1. Why was it included in the paper?

→ Temperature-pressure profile of planet K2-18b for different surface positions, i.e. pressure
levels; lines overlap continuously, i.e. surface has minimal effect on the profile;

(e) Why is CH4 not a suitable proxy for the surface pressure?

→ Still evolving after Myr for quite M star (see Figure 3d)

(f) You detect CH3 OH but non NH3 in the atmosphere of a sub-Neptune planet. What type of
surface does this planet have?

→ Shallow surface

www.iaac.space 8/??
Problem C.2 : Black Holes and the JWST (8 Points)
This problem requires you to read the following recently published scientific article:

The Age of Discovery with the James Webb:

Excavating the Spectral Signatures of the First Massive Black Holes.
Inayoshi, K. et al. arXiv:2204.09692 [astro-ph.GA] (2022). Link: https://arxiv.org/pdf/2204.09692.pdf

Answer the following questions related to this article:

(a) What are massive black holes (BH)? Why is the observation of young massive BHs important?

→ M > 109 MO ; their origin and formation pathway are constrained

(b) What is the spectral energy distribution (SED)?

→ received energy received per wavelength (of an object)

→ excited in the dense gaseous disk; from Lyβ fluorescence

(d) What are broad-band filters, and what is their use in astronomy?

→ block part of spectrum; improve signal ratio, focus on specific wavelengths

(e) Explain the increase of all lines for high z in Figure 3, top-left panel.

→ absorption due to the intergalactic medium

(f) Explain the meaning and use of the magenta rectangle in Figure 4.

→ color-cut condition visualized, objects in that region are potentially seed BHs

www.iaac.space 9/??
IAAC 2022 Pre-Final Round Solution
Otávio Casagrande Ferrari - Brazil

1. Problem A.1: Looking back with the JWST

a) The light is red-shifted because of the expansion of the Universe. As the Universe expands, the
distance between two consecutive pulses of light is “stretched”, making the wavelength of the light
to increase. This increase in the wavelength led the light to be red-shifted.
b) The redshift z is defined as follows:

λ − λ0
z=
λ0
in which λ is the wavelength of the light observed and λ0 is the wavelength of the light emitted by
the source. Therefore:

λ 2 · 10−6
λ0 = = ⇒ λ0 = 230 nm
1+z 1 + 7.6
c) Light with wavelength within the range 200-300 nm, which is the case of the light emitted by
the object, is characterized as middle ultraviolet .

2. Problem A.2: Counting Asteroids

As the distribution of asteroids is assumed to be homogeneous, the number N of asteroids detected
is directly proportional to the area ∆A swept by a segment of line connecting the planet to the
star, that is:
N ∝ ∆A
Now, recalling Kepler’s second law we know that the rate that this area is swept is constant:
∆A
= constant ⇒ ∆A ∝ ∆t
∆t
Finally, because N ∝ ∆A and ∆A ∝ ∆t, we can conclude that N ∝ ∆t. Then:
N N 1000 asteroids
= constant ⇒ = ⇒ N = 1300 asteroids
∆t 80 days 60 days
Note that the value obtained by the calculations is approximately 1333 asteroids, but the final
answer above is given with a more appropriate quantity of significant figures.

Page 1
3. Problem B.1: Rotating the JWST
a) The moment of inertia is given by:
Z
I= r2 dm (1)

integrating over the entire volume of the object. Firstly, we are going to deduce the moment of
inertia of a generic cylinder, cuboid and cone.

• Cylinder with mass M and radius R:

The element of mass is dm = ρ r dθ dr dh. So, by eq. (1):

H 2π R
2πρR4 H
Z Z Z
I=ρ r3 dr dθ dh ⇒ I =
0 0 0 4
But M = ρπR2 H, then:

1
I= M R2
2
• Cone with mass M and base radius R:
ρH
The element of mass is dm = ρ r dθ dr dh, where h = H R
r ⇒ dh = H
R
dr. So dm = R
r2 dθ dr2 .
Therefore, from eq. (1) the moment of inertia is given by:

2π R
R5 πρR4 H
Z ZZ
ρH ρH
I= r3 dr2 dθ = 2π ⇒I=
R 0 0 R 20 10
But M = 31 ρπR2 H, then:

3
I= M R2
10
• Cuboid with mass M , two sides equals to ℓ and the side parallel to the axis equals to h:

The element of mass is dm = ρ dx dy dz and the distance r from the axis is simply r2 = x2 + y 2 .
So, by eq. (1):
Z ℓ/2 Z ℓ/2 Z h/2 Z ℓ/2 Z ℓ/2
2 2
I=ρ (x + y ) dx dy dz = ρ h (x2 + y 2 ) dx dy
x=−ℓ/2 y=−ℓ/2 z=−h/2 x=−ℓ/2 y=−ℓ/2
3
ℓ3 ℓ ℓ3 ℓ3 ρ h ℓ ℓ3 ℓ3 ρhℓ4

ℓ ℓ
I = ρh + + + = + =
3 23 23 3 23 23 3 22 22 6

But M = ρhℓ2 , then:

1
I = M ℓ2
6

Page 2
• JWST:

Now we can finally calculate the moment of inertia of the JWST as just adding the separated
moment of inertia of the primary mirror (cylinder with a radius of R and mass m1 ), Aft optical
subsystem (cone with a radius r and mass m2 ) and the science instrument module (cuboid with a
side length of a and mass m3 ). Therefore, we have:

1 3 1
I = m1 R2 + m2 r2 + m3 a2
2 10 6

b) Substituting the values of the variables above (m1 = 18 · 40 kg = 720 kg, m2 = 120 kg, m3 =
1400 kg, R = 3.3 m, r = 0.65 m and a = 5.3 m), we obtain:

I = 10489.9 kg m2 ⇒ I = 10500 kg m2
in which the final answer was given with a more appropriate number of significant figures.
c) The torque τ applied is given by:

τ = F · R = Iα
where F is the force and α is the angular acceleration provided. The angular acceleration is
therefore given by:

FR 2.5 · 3.3
α= = = 7.865 · 10−4 rad/s2
I 10489.9
By the end of the thrust (after δt = 0.5 s), the JWST have rotated by an angle ∆θ:

1
∆θ = α δt2 = 9.831 · 10−5 rad
2
which is a negligible angle. The angular velocity ω after the thrust is given by ω = α δt =
3.932 · 10−4 rad/s. Therefore the time ∆t needed to the JWST rotate by 90◦ = π/2 rad is:

π/2 − ∆θ
∆t = = 3994 s
ω
So our final answer is:

∆t = 4.00 · 103 s = 1.11 hours

4. Problem B.2: Changing Temperature

a) First, the luminosity LE emitted by Earth is given by Stefan-Boltzmann’s law:

2 4
LE = 4πσRE TE
in which RE is Earth’s radius and TE its temperature.

Page 3
The luminosity L⊙,E of the Sun that is absorbed by Earth is given by:

2
πRE
L⊙,E = (1 − α) L⊙
4πd2
where L⊙ is the luminosity emitted by the Sun, d is the Sun-Earth distance and α is the albedo.
But, the luminosity of the Sun is L⊙ = 4πσR2 T 4 , then:

2
πRE
L⊙,E = (1 − α) σR2 T 4
d2
Now, to discover the Earth’s temperature, we assume that Earth is in thermal equilibrium, so that
the energy emitted (LE ) is equal to the energy absorbed (L⊙,E ):
r
πR2 R
LE = L⊙,E ⇒ 4πσRE TE = (1 − α) 2E σR2 T 4 ⇒ TE =
2 4
T (1 − α)1/4
d 2d
Finally, adding an atmosphere correction factor of f = 1.13 to the surface temperature of the
Earth, we get:

r
R
TE = f T (1 − α)1/4
2d

b) Substituting the numerical values f = 1.13, d = 1.496·1011 m, R = 6.96·108 m and T = 5772 K,

we get:

TE = 287.7 K ≈ 15◦ C

c) Radius R and temperatures T that still result in the same temperature TE are given by the
equation:

2d TE2 1
R = 2√
f 1 − α T2

Note that the factor multiplying 1/T 2 is a constant, which we are going to call k. We have:

k = RT 2 = R⊙ T⊙2

Therefore:
−2
R T
=
R⊙ T⊙

Tracing this curve (basically of the type y = 1/x2 ) in the graph given, we have:

Page 4
If the the Sun’s temperature increases by 10% (T = 1,1T⊙ ), to maintain n 15◦ C on Earth, we need
the radius to be:

R = (1,1)−2 R⊙ ≈ 0.826 R⊙
R⊙ −R
Therefore, the radius needs to decrease by R⊙
≈ 0.174 = 17.4%. Final answer:

The radius needs to decreases by 17%

5. Problem C.1: The Surface of Planets

a) A proxy is a variable that is not directly relevant to determine the variable in study, but that
is easier to measure and that have a certain correlation to the variable being studied, so it can be
used in place of other immeasurable variables. In this study, they are trying to track several trace
gases as a proxy to discover if the planet has a surface, since rocky surface is likely not accessible
to observations. A key addition to previous work is that this study applies a 2D model including
day–night transport to reevaluate the viability of utilizing atmospheric chemistry as a proxy for
the presence of surfaces.
b) They are all computer codes that are used to model and study exoplanetary atmospheres.
c)
d) Figure 1 is a graph of the pressure-temperature profiles of K2-18b with various surface pressure
levels (Pb). For each surface pressure level, a line is drawn in the graph, showing how the tempe-
rature of the atmosphere changes as the pressure changes. Figure 1 was included in the paper to
show that the presence of a surface turns out to have negligible thermal effects.
e) CH4 is still continuously evolving after millions of years with a quiet M star, making it ambi-
guous to compare with the deep-atmosphere abundance as a proxy for surface pressure.
f) Detection of CH3 OH but without NH3 indicates the presence of a surface.

Page 5
6. Problem C.2: Black Holes and the JWST
a) Massive black holes are massive astronomical objects that have undergone gravitational collapse,
leaving behind spheroidal regions of space from which nothing can escape, not even light. It’s
believed that almost every large galaxy has a supermassive black hole at its center. The observation
of young massive black holes strongly constrains their origin and formation pathway, increasing
our knowledge of these exotic astronomical objects, specially their formation.
b) It is a frequently used plot/graph in astronomy. It’s basically a plot of the flux density (a
quantity that measures the energy in the form of light) emitted by an object as a function of its
wavelength/frequency.
c) The production of the three OI lines is a result of Lyβ fluorescence that occurs when a popu-
lation in n = 3 of hydrogen is built up by collisional excitation and thus tightly correlates to the
enhancement of Balmer lines.
d) Broad-band filters are optical filters that block the light pollution in the sky and transmit the
H-alpha, H-beta, and O-III spectral lines which makes observing nebulae from the city and light
polluted skies possible. Broadband filters are particularly designed for nebulae observing. The
term ‘broadband’ refers to the width of the frequency/wavelength spectrum over which a given
observation takes place.
e) The increase of all lines for z > 14 in the top-left panel in Figure 3 is due to intergalactic
medium (IGM) absorption.
f) The magenta rectangle in Figure 4 separates the accreting seed BHs to the other three cases,
because the former tend to be redder at the same redshift range and thus can be distinguishable
by the color cuts that are denoted by the magenta rectangle in Figure 4.

Page 6
International Astronomy and
Astrophysics Competition
Final Round

Final Round Exam 2022

The final round exam was given in the form of an online exam.
Each participant was given a subset of 20 questions in random order.
This paper version is only available for training purposes.

IAAC Final Exam 2022 1/6

Question 1 : What is the name of the JWST component highlighted below?

(A) Primary mirror (B) Secondary mirror

(C) Optics subsystem (D) Antenna

Question 2 : What is the name of the JWST component highlighted below?

(A) Stabilization flap (B) Spacecraft bus

(C) Antenna (D) Star tracker

Question 3 : What is the name of the JWST component highlighted below?

(A) Antenna (B) Sunshield

(C) Optics subsystem (D) Stabilization flap

IAAC Final Exam 2022 2/6

Question 4 : When a neutron star rotates, it becomes a ...

(A) Neutron dwarf (B) Rotar (C) Quasar (D) Pulsar

Question 5 : If the Earth had the density of a neutron star, what would be the diameter of the
Earth?

(A) between 1 - 100 m (B) between 100 - 500 m

(C) between 500 - 1000 m (D) between 1000 - 5000 m

Question 6 : Where in space is the JWST located?

(A) Geostationary orbit (B) In the Moon’s shadow

(C) Lagrange point (D) Between Earth and Moon

Question 7 : Two asteroids with masses m1 , m2 and velocities v1 , v2 collide horizontally and
merge into a single object. What is the velocity of the new asteroid?

(A) m1 v1 +m
2
2 v2
(B) m1mv11 +m
+m2 v2
2

m1 v1 −m2 v2 m1 v1 −m2 v2
(C) 2
(D) m1 +m2

Question 8 : Which one of these wavelengths is considered ultraviolet radiation?

(A) 150 meters (B) 150 millimeters

(C) 150 micrometers (D) 150 nanometers

Question 9 : Which one of these wavelengths is considered infrared radiation?

(A) 150 meters (B) 150 millimeters

(C) 150 micrometers (D) 150 nanometers

IAAC Final Exam 2022 3/6

Question 10 : The planet’s albedo is the fraction of incident light ...

(A) reflected by the planet’s surface. (B) absorbed by the planet’s surface.

Question 11 : Earth’s current average surface temperature is around ...

(A) 5 ◦ C (B) 10 ◦ C (C) 15 ◦ C (D) 20 ◦ C

Question 12 : Which of the following acronyms refers to an instrument of the JWST?

(A) NIRSpec (B) HELIOS (C) Exo-FMS (D) HAZMAT

Question 13 : A star has the luminosity L0 . The temperature T of the star doubles. How does the
luminosity change?

(A) 2 × L0 (B) 4 × L0 (C) 8 × L0 (D) 16 × L0

Question 14 : An object emits light at a wavelength of 200 nanometers. You receive the light at
1000 nanometers. What is the redshift z?

(A) z = 1.2 (B) z = 4 (C) z = 5 (D) z = 6

Question 15 : Why is it hard to observe the universe with infrared radiation from the Earth’s
surface?

(A) Harmful to humans (B) Blocked by the atmosphere

(C) Reflected by the atmosphere (D) Distorted by the atmosphere

Question 16 : The moment of inertia of a solid cylinder with radius R and mass M is given by ...

(A) M R2 /2 (B) 2M R2 /5 (C) 3M R2 /10 (D) M R2 /3

Question 17 : Keplers 2nd law states that ...

(A) the orbit of a planet is an ellipse. (B) dA/dt is constant.

(C) a3 /T 2 is constant. (D) a2 /T 3 is constant.

IAAC Final Exam 2022 4/6

Question 18 : Scientists detect no CH3 OH and no NH3 in the atmosphere of a sub-Neptune planet.
What type of surface does this planet probably have?

(A) Shallow surface (B) Water oceans

(C) Dry surface (D) Methane oceans

Question 19 : Sub-Neptune planets are ...

(A) smaller than Neptune. (B) bigger than Neptune.

(C) farther away than Neptune. (D) closer than Neptune.

Question 20 : The surface temperature (in ◦ C) of the Sun is close to ...

(A) 5200 ◦ C (B) 5500 ◦ C (C) 5800 ◦ C (D) 6000 ◦ C

Question 21 : An object’s spectral energy distribution (SED) is formally given by ...

(A) dE/dλ (B) dE/dt (C) dz/dλ (D) dz/dt

Question 22 : Galileo discovered with his telescope that Venus ...

(A) has a diameter similar to the Earth (B) has mountains on the surface.

Question 23 : The Jovian planets are ...

(A) Mercury, Venus, Earth, Mars (B) Mercury, Venus, Earth, Mars, Jupiter, Sat-
urn

(C) Jupiter, Saturn, Uranus, Neptune (D) Uranus, Neptune

Question 24 : Neptune’s diameter is similar to ...

(A) Venus (B) Jupiter (C) Saturn (D) Uranus

IAAC Final Exam 2022 5/6

Question 25 : Compared to the Sun’s surface temperature, sunspots are ...

(A) cooler (B) hotter

(C) same temperature (D) sometimes hotter and sometimes cooler

Question 26 : The brightest star in the night sky is called ...

(A) Polaris / North star (B) Betelgeuse

(C) Alpha Centauri (D) Sirius

Question 27 : Alpha Centauri is the closest star system to the Earth. It is located on the ...

(A) northern hemisphere, 1.3 light-years away

(B) northern hemisphere, 4.3 light-years away

(C) southern hemisphere, 1.3 light-years away

(D) southern hemisphere, 4.3 light-years away

Question 28 : The atmosphere of Venus contains high amounts of ...

(A) Methane (B) Sulfur (C) Phosphorus (D) Iron

Question 29 : What is the name of Jupiter’s moon shown in the figure below?

(A) Io (B) Europa (C) Callisto (D) Ganymede

Question 30 : The moon Titan orbits around ...

(A) Mars (B) Jupiter (C) Saturn (D) Neptune

IAAC Final Exam 2022 6/6

International Astronomy and
Astrophysics Competition
Final Round

Final Round Exam 2022 - Solutions

No. 1 No. 2 No. 3 No. 4 No. 5 No. 6 No. 7 No. 8 No. 9 No. 10

B C D D B C D D C A

No. 11 No. 12 No. 13 No. 14 No. 15 No. 16 No. 17 No. 18 No. 19 No. 20

C A D B B A B B A B

No. 21 No. 22 No. 23 No. 24 No. 25 No. 26 No. 27 No. 28 No. 29 No. 30

A C C D A D D B A C

IAAC Final Exam 2022 1/1

International Astronomy and
Astrophysics Competition
Qualification Round 2023

Problem A : The Classification of Galaxies (5 Points)

Galaxies are some of the most beautiful objects in the universe and are observable in many differ-
ent shapes, colours and sizes. Astronomers have classified galaxies into different groups: spiral
(SA), intermediate spiral (SAB), barred spiral (SB), lenticular (S0), elliptical (E), and irregular (Irr).

Which galaxy classes are illustrated by the shapes below (A1-A4)? Find the correct class (B1-B4)
and name (C1-C4) of each galaxy shown in the images: NGC 2337, NGC 300, NGC 1365, Messier 110

(A1) (A2) (A3) (A4)

(B1) (B2) (B3) (B4)

(C1) (C2) (C3) (C4)

www.iaac.space 1/3
Problem B : The Speed of Light (5 Points)
Light travels extremely fast through the universe. However, the speed of light is limited to about
300,000 km/s. Because of that, it takes sunlight 8.3 minutes to reach the Earth.

How long does it take light from the Sun’s surface to reach Mars (223 million km distance to the
Sun), Jupiter (777 million km) and Pluto (5,906 million km), respectively?

Problem C : Elliptical Orbit (5 Points)

Objects go around the Sun in elliptical orbits. Especially comets can have orbits with a high ec-
centricity. The newly found comet P/2023 IAAC has a semi-major axis of 16.5 AU and a semi-minor
axis of 8.3 AU. The comet’s mass is negligible compared to the Sun (1.9 x 1030 kg).

P1 P2

The vis-viva equation gives the orbital speed of an object travelling along the ellipse:
s
2 1
v(x) = µ − , µ = G(m1 + m2 )
x a

Here, a is the semi-major axis, m1 and m2 are the masses of the orbiting bodies, x is the distance
between the comet and the centre of mass, and G is the gravitational constant.

(a) Calculate the eccentricity of P/2023 IAAC’s orbit around the Sun.
(b) Which one of the points P1 , P2 , P3 is the aphelion and which one the perihelion?
(c) Determine the comet’s speed at the three points P1 , P2 , P3 .

Problem D : Distance between Stars (5 Points)

Determining the distance to stars can be challenging. The parallax method is one way of finding
the distance to many stars around us. Your research team measures the parallax of two stars that
have a distance of 5 degrees from each other in the night sky: The first star has a parallax of 0.11
arcsec, and the second has a parallax of 0.13 arcsec.

How far apart are the two stars from each other? Express your answer in light-years.

www.iaac.space 2/3
Problem E : Dark Energy (5 Points)
Cosmology studies the dynamics of the universe on its largest scales. Its research reveals how the
universe evolves over time and, in particular, how it expands. The term dark energy frequently
appears in cosmology.

What does the term dark energy describe? What are evidences for the existence of dark energy?

General Information and Submission

You can write the solution by hand or type it on a computer. To qualify for the pre-final round, you
have to get at least 15 points (Junior, under 18 years) or 20 points (Youth, over 18 years). Make sure to
submit your solution by Friday 21. April 2023 23:59 UTC+0 online at www.iaac.space/submission. In case
of questions or comments, please feel free to contact us via email: info@iaac.space. Good luck!

www.iaac.space 3/3
International Astronomy and
Astrophysics Competition
Qualification Round 2023

Problem A : The Classification of Galaxies (5 Points)

(A1) irregular (A2) elliptical (A3) spiral (A4) barred spiral
(B1) spiral (B2) irregular (B3) barred spiral (B4) elliptical
(C1) NGC 300 (C2) NGC 2337 (C3) NGC 1365 (C4) Messier 110

Problem B : The Speed of Light (5 Points)

t = d/c: Mars: 12.4 minutes, Jupiter: 43.2 minutes, Pluto: 328.1 minutes

Problem C : Elliptical Orbit (5 Points)

p
(a) ε = 1 − (b/a)2 = 0.86
(b) P1 : perihelion, P2 : aphelion
p
(c) x1 = a(1 − ε) (2.3 AU), x2 = a(1 + ε) (30.7 AU), x3 = b2 + (aε)2 (16.4 AU)
=⇒ v1 : 26231 m/s, v2 : 1965 m/s, v3 : 7223 m/s

Problem D : Distance between Stars (5 Points)

p
d = 1/p: d1 : 9.09 pc (29.6 ly), d2 : 7.69 pc (25.1 ly) =⇒ d12 = (d2 sin(ϕ))2 + (d1 − d2 cos(ϕ))2
Result: 5.1 light-years

Problem E : Dark Energy (5 Points)

phenomena in the large-scale universe causing accelerated expansion; evidences: redshift change
with distance, missing global curvature (from cosmic microwave background), etc.

www.iaac.space Basic Solution

International Astronomy and
Astrophysics Competition
Pre-Final Round 2023

Important: Read all the information on this page carefully!

• You may use extra paper if necessary, however, the space under the problems is usually enough.

• Typing the solution on a computer is allowed but not recommended (no extra points).

• You receive points for the correct solution and for the performed steps. Example: You will not get
all points for a correct value if the calculations are missing.

• Make sure to clearly mark your final solution values (e.g. underlining, red color, box).

• You can reach up to 36 points in total. You qualify for the final round if you reach at least 18 points
(junior, under 18 years) or 24 points (youth, over 18 years).

Uploading Your Solution

• Please upload a file/pictures of (this sheet with) your written solutions: https://iaac.space/login

• Only upload one single PDF file! If you have multiple pictures, please compress them into one
single file. Do not upload your pictures in a different format (e.g, no Word and Zip files).

• The deadline for uploading your solution is Sunday 28. May 2023, 23:59 UTC+0.

• The results of the pre-final round will be announced on Monday 5. June 2023.

Good luck!

www.iaac.space 1/9
Problem A.1: Parabolic Trajectory (4 Points)
As presented in the qualification round, the comet P/2023 IAAC circles the Sun in an elliptical or-
bit. There are other comets with a parabolic trajectory, for example the comet C/2023 IAAC.

(a) Explain the meaning of the letters P and C in the names of the two comets.

The vis-viva-equation can be extended for different types of trajectories as follows:

s a > 0 for ellipses
2 1
v(x) = µ − with a = ∞ for parabolas
x a
a < 0 for hyperbolas

and µ = G(m1 + m2 ). Here, m1 is the comet’s mass, m2 the Sun’s mass (1.9 x 1030 kg), x the dis-
tance between the comet and the Sun, and G is the gravitational constant (6.67 x 10-11 m3 kg-1 s-2 ).

(b) Determine the velocity (in km/s) of C/2023 IAAC for a distance of 0.8 AU to the Sun.

www.iaac.space 2/9
Problem A.2: Brightness of a Binary Star (4 Points)
A binary star system consists of two stars very close to one another. The two stars have apparent
magnitudes of m1 = 2 and m2 = 3. The apparent magnitude m is defined with a stars’ flux
density F , compared to a reference star with m0 and F0 :

F
m − m0 = −2.5 log10
F0

Calculate the total apparent magnitude of the binary star system.

www.iaac.space 3/9
Problem B.1: Temperature of the Sun (6 Points)
Assume a constant density ρ̄ of 1.4 x 103 kg·m-3 for the entire Sun. The ideal gas law states that

pV = N kT

with the pressure p, the volume V , the number of particles N , the Boltzmann constant k (1.38 x
10-23 m2 kg s-2 K-1 ) and the temperature T .

(a) Show that the temperature T at a certain pressure p is given by

pm̄
T (p) =
ρk

with the average particle mass m̄ within the Sun (1.02 x 10-27 kg).

(b) Explain why the Sun must be in a state of hydrostatic equilibrium:

dp
= −g(r)ρ̄
dr

(d) Use the condition of hydrostatic equilibrium to show that the pressure p inside the Sun at
a radius of R/4 from the centre is about 1.26 x 1014 Pa, where R is the Sun’s radius of 0.7 x 109 m.

(e) Determine the Sun’s temperature at a radius of R/4. Why is this result only a broad estimate?

www.iaac.space 4/9
(extra page for problem B.1: Temperature of the Sun)

www.iaac.space 5/9
Problem B.2: Escaping a Star (6 Points)
It takes many years for a photon produced in a star’s centre to reach its surface and escape into
space. This is due to its constant interaction with other particles. To estimate the time it takes for
a photon to escape a star’s interior, we assume that the photon is deflected in equal time intervals
into a random direction in a two-dimensional space (i.e., a random walk):

At each step i, the photon moves a constant distance ε in an angle ϕi , thus changing its position:
!
cos(ϕi )
∆~xi = ε
sin(ϕi )

(a) Determine the distance R(n) from the centre (0,0) after n steps.

Assume that the step-distance ε is about 1.0 x 10−4 m for a photon moving inside the Sun:
(b) How many steps does the photon need to reach the Sun’s surface?

www.iaac.space 6/9
(extra page for problem B.2: Escaping a Star)

www.iaac.space 7/9
Problem C.1 : Zhurong Rover Mars Landing (8 Points)
This problem requires you to read the following recently published scientific article:

Geomorphic contexts and science focus of the Zhurong landing site on Mars.
Liu, J., Li, C., Zhang, R. et al. Nat Astron 6, 65–71 (2022).
Link: https://www.nature.com/articles/s41550-021-01519-5.pdf

Answer the following questions related to this article:

(a) What was done to find a suitable landing site and which factors were considered?

(b) Describe three geomorphic features present near the landing site.

(c) Which features described in (b) are likely related to volcanism?

(d) Name all instruments on board the Zhurong rover and explain the purpose of MarSCoDe.

(e) Which instrument measures the magnetic field and for which scientific reason?

(f) What is possible evidence for ancient oceans that the Zhurong rover may find?

www.iaac.space 8/9
Problem C.2 : Young Stars in the Galactic Centre (8 Points)
This problem requires you to read the following recently published scientific article:

Detection of an excess of young stars in the Galactic Centre Sagittarius B1 region.

Nogueras-Lara, F., Schödel, R. & Neumayer, N. Nat Astron 6, 1178–1184 (2022).
Link: https://www.nature.com/articles/s41550-022-01755-3.pdf

Answer the following questions related to this article:

(a) How high is the star formation rate in the Galactic Center and how was it estimated?

(b) What is the missing clusters problem?

(c) What is the Ks luminosity function?

(d) How do the regions in Figure 4 compare to each other?

(e) Why is it hard to determine the location of where the young stars formed?

(f) What do the findings tell us about the evolution of the young stars?

www.iaac.space 9/9
International Astronomy and
Astrophysics Competition
Pre-Final Round 2023

Important: Read all the information on this page carefully!

• You may use extra paper if necessary, however, the space under the problems is usually enough.

• Typing the solution on a computer is allowed but not recommended (no extra points).

• You receive points for the correct solution and for the performed steps. Example: You will not get
all points for a correct value if the calculations are missing.

• Make sure to clearly mark your final solution values (e.g. underlining, red color, box).

• You can reach up to 36 points in total. You qualify for the final round if you reach at least 18 points
(junior, under 18 years) or 24 points (youth, over 18 years).

Uploading Your Solution

• Please upload a file/pictures of (this sheet with) your written solutions: https://iaac.space/login

• Only upload one single PDF file! If you have multiple pictures, please compress them into one
single file. Do not upload your pictures in a different format (e.g, no Word and Zip files).

• The deadline for uploading your solution is Sunday 28. May 2023, 23:59 UTC+0.

• The results of the pre-final round will be announced on Monday 5. June 2023.

Good luck!

www.iaac.space 1/10
Problem A.1: Parabolic Trajectory (4 Points)
As presented in the qualification round, the comet P/2023 IAAC circles the Sun in an elliptical or-
bit. There are other comets with a parabolic trajectory, for example the comet C/2023 IAAC.

(a) Explain the meaning of the letters P and C in the names of the two comets.

The vis-viva-equation can be extended for different types of trajectories as follows:

s a > 0 for ellipses
2 1
v(x) = µ − with a = ∞ for parabolas
x a
a < 0 for hyperbolas

(b) Determine the velocity (in km/s) of C/2023 IAAC for a distance of 0.8 AU to the Sun.

Solution a:
Comet classification; P for periodic, C for non-periodic; (additionally: X for no orbit, D for lost)

Solution b:
Because m2 m1 , we have µ = Gm2 . With a = ∞ for C/2023 IAAC we get 1/a = 0 and
r
2Gm2
v(x) =
x
which yields a velocity of 46 km/s.

www.iaac.space 2/10
Problem A.2: Brightness of a Binary Star (4 Points)
A binary star system consists of two stars very close to one another. The two stars have apparent
magnitudes of m1 = 2 and m2 = 3. The apparent magnitude m is defined with a stars’ flux
density F , compared to a reference star with m0 and F0 :

F
m − m0 = −2.5 log10
F0

Calculate the total apparent magnitude of the binary star system.

Solution:
The flux densities must be added together (not the magnitudes). From the definition we get

F = F0 · 10−0.4·(m−m0 )

and thus F1 = F0 · 10−0.8 and F2 = F0 · 10−1.2 . This gives us

F = F1 + F2 = F0 · (10−0.8 + 10−1.2 )

and
m = −2.5 log10 10−0.8 + 10−1.2

and we get 1.64 for the total magnitude of the binary star system.

www.iaac.space 3/10
Problem B.1: Temperature of the Sun (6 Points)
Assume a constant density ρ̄ of 1.4 x 103 kg·m-3 for the entire Sun. The ideal gas law states that

pV = N kT

with the pressure p, the volume V , the number of particles N , the Boltzmann constant k (1.38 x
10-23 m2 kg s-2 K-1 ) and the temperature T .

(a) Show that the temperature T at a certain pressure p is given by

pm̄
T (p) =
ρk

with the average particle mass m̄ within the Sun (1.02 x 10-27 kg).

(b) Explain why the Sun must be in a state of hydrostatic equilibrium:

dp
= −g(r)ρ̄
dr

(d) Use the condition of hydrostatic equilibrium to show that the pressure p inside the Sun at
a radius of R/4 from the centre is about 1.26 x 1014 Pa, where R is the Sun’s radius of 0.7 x 109 m.

(e) Determine the Sun’s temperature at a radius of R/4. Why is this result only a broad estimate?

www.iaac.space 4/10
Solution a:
With the ideal gas law, N = m/m̄ and ρ = m/V we obtain the answer:

1 m ρkT pm̄
p= · · kT = =⇒ T =
V m̄ m̄ ρk

Solution b:
Sun does not collapse; gravity (inward) in balance with pressure-gradient force (outward).

Solution c:
The mass M (r) enclosed within the radius r is
4
M (r) = V (r) · ρ̄ = πr3 ρ̄
3
and thus we get for the gravitational acceleration:

M (r) 4
g(r) = G 2
= πGρ̄r
r 3
Solution d:
The condition of hydrostatic equilibrium gives us
dp 4
= − πGρ̄2 r
dr 3
and by integrating from R/4 to the surface we get:
Z 0 Z R
4 5
dp = − πGρ̄2 rdr =⇒ P = πGρ̄2 R2
P 3 R/4 8

This gives the answer of 1.26 x 1014 Pa.

Solution e:
Using the equation from (a) yields 6.65 x 106 K; problem: assumption of constant density.

www.iaac.space 5/10
Problem B.2: Escaping a Star (6 Points)
It takes many years for a photon produced in a star’s centre to reach its surface and escape into
space. This is due to its constant interaction with other particles. To estimate the time it takes for
a photon to escape a star’s interior, we assume that the photon is deflected in equal time intervals
into a random direction in a two-dimensional space (i.e., a random walk):

At each step i, the photon moves a constant distance ε in an angle ϕi , thus changing its position:
!
cos(ϕi )
∆~xi = ε
sin(ϕi )

(a) Determine the distance R(n) from the centre (0,0) after n steps.

Assume that the step-distance ε is about 1.0 x 10−4 m for a photon moving inside the Sun:
(b) How many steps does the photon need to reach the Sun’s surface?

www.iaac.space 6/10
Solution a:
The position ~x after n steps is given by
n n
!
X X cos(ϕi )
~x = ∆~xi = ε
i=1 i=1
sin(ϕi )

and thus we get for the distance:

v !2 !2
u n n
u X X
R(n) = |~x| = ε · t cos(ϕi ) + sin(ϕi )
i=1 i=1
v
u n n
uX X X X
2 2
=ε·u
t cos (ϕi ) + cos(ϕi ) cos(ϕ j ) + sin (ϕ i ) + sin(ϕi ) sin(ϕj )
i=1 1≤i,j≤n i=1 1≤i,j≤n
i6=j i6=j
v
u n
uX
cos2 (ϕi ) + sin2 (ϕi )

=ε·t
i=1
v
u n
uX √
=ε·t 1=ε· n
i=1

(Because the ϕi are randomly distributed, the sums with i 6= j vanish.)

Solution b:
From (a) we get n = (R/ε)2 , which are 1026 steps.

Solution c:
The time can be determined by considering the total distance traveled (c: speed of light):
n·ε
t=
c
This gives a final answer of about 1,060,000 years.

www.iaac.space 7/10
Problem C.1 : Zhurong Rover Mars Landing (8 Points)
This problem requires you to read the following recently published scientific article:

Geomorphic contexts and science focus of the Zhurong landing site on Mars.
Liu, J., Li, C., Zhang, R. et al. Nat Astron 6, 65–71 (2022).
Link: https://www.nature.com/articles/s41550-021-01519-5.pdf

Answer the following questions related to this article:

(a) What was done to find a suitable landing site and which factors were considered?

→ Tianwen-1 orbiter: three months remote sensing survey

→ local elevation, slope, rock distribution, thermal environment, communicate with Earth

(b) Describe three geomorphic features present near the landing site.

→ possible choices: Rampart craters, Cones, Ridges, Troughs, Transverse aeolian ridges (TARs)

(c) Which features described in (b) are likely related to volcanism?

→ correct answers: Cones, Ridges, Troughs

(d) Name all instruments on board the Zhurong rover and explain the purpose of MarSCoDe.

→ M. Rover Penetrating Radar (RoPeR), M. Rover Magnetometers (RoMAG), M. Surface Composi-

tion Detector (MarSCoDe), Multispectral Camera (MSCam), NaTeCams, M. Climate Station (MCS)
→ MarSCoDe: laser-induced breakdown spectroscopy to reveal rock compositions

(e) Which instrument measures the magnetic field and for which scientific reason?

→ Mars Rover Magnetometers (RoMAG)

→ to learn about the remanent magnetization and possible intrinsic magnetic field

(f) What is possible evidence for ancient oceans that the Zhurong rover may find?

→ low dielectric constant compared with typical volcanic materials

→ hydrated minerals, water-related sedimentary rocks and textures, and alteration minerals
→ anomalous chemistry

www.iaac.space 8/10
Problem C.2 : Young Stars in the Galactic Centre (8 Points)
This problem requires you to read the following recently published scientific article:

Detection of an excess of young stars in the Galactic Centre Sagittarius B1 region.

Nogueras-Lara, F., Schödel, R. & Neumayer, N. Nat Astron 6, 1178–1184 (2022).
s Link: https://www.nature.com/articles/s41550-022-01755-3.pdf

Answer the following questions related to this article:

(a) How high is the star formation rate in the Galactic Center and how was it estimated?

→ 0.1 solar mass per year in the past 10-100 Myr

→ radio to high-energy emission, finding of massive young stars, detection of classical Cepheids,
luminosity functions

(b) What is the missing clusters problem?

→ high star formation rate, but only two young massive clusters known (Arches and Quintuplet)

(c) What is the Ks luminosity function?

→ number of stars per luminosity interval

→ contains information about its formation history

(d) How do the regions in Figure 4 compare to each other?

→ control and inner Galactic Centre fields agree: most older than 7 Ga
→ Sgr B1 region significantly different: younger on average, high contribution from 2-7 Ga

(e) Why is it hard to determine the location of where the young stars formed?

→ due to age uncertainty and unknown distance along the line of sight

(f) What do the findings tell us about the evolution of the young stars?

→ stars form in massive stellar associations that can contain clusters and later disperse while
orbiting through the nuclear stellar disk

www.iaac.space 9/10
International Astronomy and
Astrophysics Competition
Final Round

Final Round Exam 2023

The final round exam was given in the form of an online exam.
Each participant was given a subset of 20 questions in random order.
This paper version is only available for training purposes.

IAAC Final Exam 2023 1/6

Question 1 : Which one of these shapes illustrates an elliptical galaxy?:

(A) A (B) B (C) C (D) D

Question 2 : What is the class of this galaxy?:

(A) Spiral (B) Barred Spiral (C) Lenticular (D) Elliptical

Question 3 : Which class of galaxies does this shape illustrate?:

(A) Barred Spiral (B) Lenticular (C) Elliptical (D) Irregular

Question 4 : The speed of light is approximately ...

(A) 3 · 103 km/s (B) 3 · 105 km/s

(C) 3 · 106 km/s (D) 3 · 109 km/s

IAAC Final Exam 2023 2/6

Question 5 : What is the most common type of galaxy in the universe?

(A) Spiral galaxy (B) Lenticular galaxy

(C) Irregular galaxy (D) Elliptical galaxy

Question 6 : How far away from Earth is a star with an observed parallax of 0.1 arcsecs?

(A) 1 pc (B) 3 pc (C) 10 pc (D) 30 pc

Question 7 : Which one of these points is the aphelion of the ellipse?

(A) A (B) B (C) C (D) D

Question 8 : What is true about the orbit of a comet called C/2023 A3?

(A) non-periodic orbit (B) periodic orbit

(C) elliptical orbit (D) no meaningful orbit

Question 9 : The term dark energy is related to ...

(A) the gravitation of black holes. (B) the matter in the universe.

(C) the rotation of galaxies. (D) the expansion of the universe.

Question 10 : Which landforms near the Zhurong landing site on Mars are likely related to vol-
canism?

(A) Rampart craters, Ridges, Troughs (B) Ridges, Rampart craters, Transverse aeo-
lian ridges

(C) Ridges, Cones, Transverse aeolian (D) Cones, Ridges, Troughs

ridges

IAAC Final Exam 2023 3/6

Question 11 : The gravitational acceleration at the surface of a star with average density ρ and
radius R is given by ...

(A) 34 πGρR (B) 43 πGρR2

(C) 43 πGρR3 (D) 34 πGρ/R

Question 12 : The Zhurong Mars rover landed at an elevation of around ...

(A) −8, 000 metres (B) −4, 000 metres

(C) 4, 000 metres (D) 8, 000 metres

Question 13 : Which instrument onboard the Zhurong Mars rover measures the magnetic field?

(A) RoMAG (B) RoPeR (C) MarSCoDe (D) NaTeCams

Question 14 : Which object was historically the reference value m0 for the apparent magnitude?

(A) The Sun (B) North/Pole Star

(C) Sirius (star) (D) Vega (star)

Question 15 : About how much brighter is a star with an appearent magnitude of m = 1 com-
pared to a star with an apparent magnitude of m = 2?

(A) 2.5-times brighter (B) 5.0-times brighter

(C) 10-times brighter (D) 25-times brighter

Question 16 : Hydrostatic equilibrium can best be described as the equilibrium between ...

(A) temperature and pressure. (B) friction and pressure.

(C) density and pressure. (D) gravity and pressure.

Question 17 : The ideal gas law states that ... <br><br>(T : temperature, p: pressure, V : volume,
N : number of particles, k: Boltzmann constant)

(A) pV = N k/T (B) pT = N kV (C) pT = N k/V (D) pV = N kT

IAAC Final Exam 2023 4/6

Question 18 : How long is a year on Mars?

(A) ≈ 300 Earth-days (B) ≈ 500 Earth-days

(C) ≈ 700 Earth-days (D) ≈ 1,000 Earth-days

Question 19 : Where on Mars did the Zhurong rover of the Tianwen-1 mission land?

(A) Site A (B) Site B (C) Site C (D) Site D

Question 20 : The colour of light with a wavelength of 700 nanometers is ...

(A) Green (B) Blue (C) Yellow (D) Red

Question 21 : How high was the star formation rate in the Galactic Center in the past 10-100 billion
years?

(A) 0.01 solar masses per year (B) 0.1 solar masses per year

Question 22 : The two young massive clusters near the Galactic Centre are called ...

(A) Sagittarius and Parsec (B) Sagittarius and Quintuplet

(C) Arches and Parsec (D) Arches and Quintuplet

Question 23 : How far away is the centre of the Milky Way from Earth?

(A) ≈ 10,000 light-years (B) ≈ 25,000 light-years

(C) ≈ 50,000 light-years (D) ≈ 100,000 light-years

IAAC Final Exam 2023 5/6

Question 24 : The atmosphere of Mars consists primarily of ...

(A) Nitrogen (B) Oxygen

(C) Carbon dioxide (D) Carbon monoxide

Question 25 : Approximately how many stars are there in the Milky Way?

(A) 20 billion (B) 50 billion (C) 200 billion (D) 500 billion

Question 26 : The bending of starlight when it passes through a gravitational field is called ...

(A) stellar parallax (B) gravitational redshift

(C) gravitational lensing (D) stellar occultation

Question 27 : The outermost layer of the Sun that is visible during a total solar eclipse is called
the ...

(A) Corona (B) Photosphere (C) Chromosphere (D) Heliosphere

Question 28 : The fusion process that generates the Sun’s energy is called ...

(A) Carbon-nitrogen-oxygen cycle (B) Triple-alpha process

(C) Proton-proton chain (D) Bethe-Weise process

Question 29 : How does the temperature of a star affect the distribution of photon energies it
emits?

(A) Higher temperature leads to higher- (B) Lower temperature leads to higher-energy
energy photons photons

Question 30 : The asteroid belt of the solar system is between the orbits of ...

(A) Earth and Mars (B) Mars and Jupiter

(C) Jupiter and Saturn (D) Saturn and Uranus

IAAC Final Exam 2023 6/6

International Astronomy and
Astrophysics Competition
Final Round

Final Round Exam 2023 - Solutions

No. 1 No. 2 No. 3 No. 4 No. 5 No. 6 No. 7 No. 8 No. 9 No. 10

B A A B D C C A D D

No. 11 No. 12 No. 13 No. 14 No. 15 No. 16 No. 17 No. 18 No. 19 No. 20

A B A D A D D C B D

No. 21 No. 22 No. 23 No. 24 No. 25 No. 26 No. 27 No. 28 No. 29 No. 30

B D B C C C A C A B

IAAC Final Exam 2023 1/1

International Astronomy and
Astrophysics Competition
Qualification Round 2024

Problem A : The Southern Hemisphere (5 Points)

Most people live in the northern hemisphere of the Earth and observe the northern night sky.
However, the night sky in the southern hemisphere is different and features many interesting &
important astronomical objects that cannot be seen in the northern hemisphere.

Below, you see a sketch of a particular section of the southern hemisphere night sky. Find the
correct names of the objects tagged with the letters A to H.

Note: A,B,H refer to stars; F refers to a constellation; C,D,E,G refer to other objects.

B
C

D
E

(A) (B) (C) (D)

(E) (F) (G) (H)

www.iaac.space 1/3
Problem B : Stars in the Milky Way (5 Points)
Our Sun is one star among billions of stars in the Milky Way. However, it is difficult to determine
the exact number of stars. By observing the nearby stars around us, you find that the local den-
sity of stars is 0.05 stars/(light-year)3 . For simplicity, assume that the Milky Way has a cylindrical
shape with a diameter of 100,000 light-years and a thickness of 1,000 light-years.

Use this information to estimate the total number of stars in the Milky Way.

Problem C : Gravity at the ISS (5 Points)

The International Space Station (ISS) orbits the Earth at an altitude of around 410 km. The ISS
does not fall towards the Earth, and the astronauts & objects are floating weightlessly. On Earth’s
surface, objects fall with a gravitational acceleration of approximately 9.81 m/s2 .

(a) Show that the following equation gives the percentage P of how much the grav. acceleration
decreases between Earth’s surface (R: Earth’s radius, 6371 km) and an object at altitude z:
2
1
P (z) = 1 − z
1+ R
(b) What is P (z) for the altitude of the ISS, and why are objects weightless nevertheless?
(c) At what distance does the grav. acceleration become only 1% of that on Earth’s surface?

Problem D : Field of View of the ISS (5 Points)

The astronauts living on the ISS have a spectacular view of Earth. However, they see only a cer-
tain part of the Earth at a given time, and the planet’s curvature limits their view.

Use geometry to determine (a) the field of view angle θ, (b) the total distance S visible, and (c)
the percentage of Earth’s surface that astronauts can see.

www.iaac.space 2/3
Problem E : Microwave Background (5 Points)
Looking into the universe around us reveals billions of galaxies across all directions. However,
when looking past them into the empty space, we detect something called the cosmic microwave
background (CMB). It was discovered by accident and is immensely important for cosmology.

Explain what the CMB is and how it was discovered.

General Information and Submission

You can write the solution by hand or type it on a computer. To qualify for the pre-final round, you must get at least
15 points (Junior, under 18 years) or 20 points (Youth, over 18 years). Make sure to submit your solution by Friday
26. April 2024 23:59 UTC+0 online at www.iaac.space/submission. If you have questions or comments, please feel
free to contact us via email: info@iaac.space. Good luck!

www.iaac.space 3/3
International Astronomy and
Astrophysics Competition
Qualification Round 2024

Problem A : The Southern Hemisphere (5 Points)

(A) Canopus (B) Achernar (C) LMC (D) SMC
(E) Eta Carinae Nebula (F) Crux (G) Omega Centauri (H) Alpha Centauri

Problem B : Stars in the Milky Way (5 Points)

N = V · ρ = π · R2 · h · ρ = π · (50, 000 ly)2 · 1, 000 ly · 0.05/ly 3 = 393 · 109 (i.e., 393 billion stars)

Problem C : Gravity at the ISS (5 Points)

2 2
M R 1
(a) g(z) = γ (R+z)2 =⇒ P (z) = 1 − g(z)/g(0) = 1 − R+z
= 1 − z
1+ R
(b) P (410 km) = 11.7 %; ISS orbits around Earth → centrifugal force
2
1
(c) z(P ) = R · √1−P −1 → z(99 %) = 57, 339 km

Problem D : Field of View of the ISS (5 Points)

R
= 139.9◦

(a) θ = 2 · arcsin R+z
◦ −θ
(b) S = 2πR · 180360◦
= 4459 km
◦
(b) 2πR (1 − cos(90 − θ/2))/(4πR2 ) = 3 %
2

Problem E : Microwave Background (5 Points)

(What?) Hot, dense primordial state → cosmic expansion → temperature of the universe gradu-
ally decreased → state of increased transparency → photons can escape → released photons are
CMB (How?) 1965, Arno Penzias and Robert Wilson → troubleshooting a persistent background
noise in their radio telescope → faint, uniform signal coming from all directions in the sky → CMB

www.iaac.space Basic Solution

International Astronomy and
Astrophysics Competition
Pre-Final Round 2024

Important: Read all the information on this page carefully!

• You may use extra paper if necessary, however, the space under the problems is usually enough.

• Typing the solution on a computer is allowed but not recommended (no extra points).

• You receive points for the correct solution and for the performed steps. Example: You will not get
all points for a correct value if the calculations are missing.

• Make sure to clearly mark your final solution values (e.g. underlining, red color, box).

• You can reach up to 36 points in total. You qualify for the final round if you reach at least 18 points
(junior, under 18 years) or 24 points (youth, over 18 years).

Uploading Your Solution

• Please upload a file/pictures of (this sheet with) your written solutions: https://iaac.space/login

• Only upload one single PDF file! If you have multiple pictures, please compress them into one
single file. Do not upload your pictures in a different format (e.g, no Word and Zip files).

• The deadline for uploading your solution is Sunday 2. June 2024, 23:59 UTC+0.

• The results of the pre-final round will be announced on Monday 10. June 2024.

Good luck!

www.iaac.space 1/9
Problem A.1: Rotation of the Earth (4 Points)
Rockets allow us to launch spacecraft and satellites into space, which are an essential part of our
modern world. However, rocket launches need careful planning, and some places on Earth pro-
vide better launch conditions than others.

(a) Explain why most rocket launches take place close to the equator.
(b) Find an equation v(ϕ) that calculates the rotational speed v of the Earth at latitude ϕ.
(c) Calculate the rotational speed at 5◦ S (near the equator) and 80◦ N (near the pole).

Note: The Earth’s radius is 6371 km.

www.iaac.space 2/9
Problem A.2: Altitude of the ISS – Part 1 (4 Points)
The International Space Station (ISS) does not orbit the Earth at a perfectly constant altitude. In-
stead, the ISS changes its altitude over time due to collisions with atmospheric particles (down-
wards) and boosters that are used to adjust the orbit (upwards). The diagram below displays the
ISS’s altitude above the ground for the last months (December 2023 to April 2024).1

(a) How much did the ISS descend and ascend between December 2023 and April 2024?
(b) Determine the average rate of descent of the ISS.

Space is often considered to start at an altitude of 100 km above ground (the edge of space).

1
A high-resolution version of the diagram is available online: https://iaac.space/A2-AltitudeISS.png

www.iaac.space 3/9
Problem B.1: Altitude of the ISS – Part 2 (6 Points)
As mentioned in Problem A.2, the ISS loses altitude due to the collision with atmospheric parti-
cles. This causes the ISS to experience a drag force Fd according to the drag equation
1
Fd = · ρ · Cd · A · v 2
2
with the atmospheric density ρ, the dimensionless drag coefficient Cd , the ISS’s cross-sectional
area A, and the ISS’s speed relative to the particles v.

Use the rate of descent from Problem A.2 to estimate

(a) the atmospheric density ρ at the current position of the ISS and
(b) the total amount of matter (in kg) which collides with the ISS every day.

Note: The following constants may be helpful: the gravitational constant: 6.67 x 10-11 m3 kg-1 s-2 ,
the mass of Earth: 5.97 x 1024 kg, the drag coefficient of the ISS: 1.3, the cross-sectional area of
the ISS: 4800 m2 , the total mass of the ISS: 450 tons.

www.iaac.space 4/9
(extra page for problem B.1: Altitude of the ISS – Part 2)

www.iaac.space 5/9
Problem B.2: Stars in the Milky Way (6 Points)
In Problem B of the Qualification Round, you estimated the number of stars in the Milky Way by
assuming a constant density of stars throughout the galaxy. However, the density of stars is not
constant and varies significantly across different regions.

(a) Name the three regions A, B, C marked in the horizontal Milky Way drawing below.

A B C

Scientists have developed a basic model for the Milky Way to describe the density distribution of
stars ρ(r) at distance r from the center by evaluating the three regions A, B, C:

r r r
ρ(r) = Ψ · exp ΩA − + exp ΩB − + exp ΩC −
RA RB RC

The model parameters have the values below:

Ψ = 10−4 stars/(light-year)3
RA = 20 light-years
RB = 12 · 103 light-years
RC = 5 · 104 light-years
ΩA = 21, ΩB = −3, ΩC = −8

(b) Create a double logarithmic plot of the density distribution ρ(r) with respect to r.

Note: Assume that the Milky Way has a constant thickness of 1,000 light-years.

www.iaac.space 6/9
(extra page for problem B.2: Stars in the Milky Way)

www.iaac.space 7/9
Problem C.1 : Einstein Ring around Galaxy (8 Points)
This problem requires you to read the following recently published scientific article:

A massive compact quiescent galaxy at z = 2 with a complete Einstein ring in JWST imaging.
van Dokkum, P., Brammer, G., Wang, B. et al. Nat Astron 8, 119-125 (2024).
Link: https://www.nature.com/articles/s41550-023-02103-9.pdf

Answer the following questions related to this article:

(a) What kind of object is the article about, and how was it discovered?

(b) Describe all elements visible in Figure 1a, including the names JWST-ER1g and JWST-ER1r.

(d) According to the study, what is and what is not likely to cause the lensing mass?

(e) Explain what the IMF is by using Figure 3a.

(f) Why is the ring in Figure 1a a gravitational lens and not a ring galaxy?

www.iaac.space 8/9
Problem C.2 : Volcanic Activity on Io (8 Points)
This problem requires you to read the following recently published scientific article:

Io’s polar volcanic thermal emission indicative of magma ocean and shallow tidal heating.
Davies, A.G., Perry, J.E., Williams, D.A. et al. Nat Astron 8, 94-100 (2024).
Link: https://www.nature.com/articles/s41550-023-02123-5.pdf

Answer the following questions related to this article:

(a) What is different about the presented observations compared to previous studies?

(b) Explain how the north pole, south pole and the lower latitudes differ in volcanic activity.

(c) What is Loki Patera and Tvashtar Paterae?

(d) How many hot spots did the researchers find in the north and south polar caps?

(e) According to the study’s findings, which mechanism likely causes Io’s volcanic activity?

(f) Describe how the scientists identified hot spots from the observation pixel data.

www.iaac.space 9/9
International Astronomy and
Astrophysics Competition
Pre-Final Round 2024

Solutions to the Pre-Final Round 2024

Please note that there are many ways to reach the final solutions.
Not all detailed steps are elaborated in this solution document.

Note: The Earth’s radius is 6371 km.

Solution a: The rocket gains additional speed due to the rotational speed of the Earth closer to
the equator; thus, providing additional free energy (i.e., less fuel or more payload).

Solution b: (T = 24 h; R: Earth’s radius)

2πRϕ 2πR cos ϕ
v(ϕ) = =
T T
Solution c: 5◦ S: 1661.6 km/h (i.e., 461.5 m/s); 80◦ N: 289.6 km/h (80.5 m/s)

(a) How much did the ISS descend and ascend between December 2023 and April 2024?
(b) Determine the average rate of descent of the ISS.

Space is often considered to start at an altitude of 100 km above ground (the edge of space).

Solution a: Descent: ≈14.3 km; Ascent: ≈14.7 km

Solution b: Rate of descent: 0.119 km/d (i.e., 0.0014 m/s)

Solution c: From 417 km to 100km: 2664 days or 7.3 years

1
A high-resolution version of the diagram is available online: https://iaac.space/A2-AltitudeISS.png

Use the rate of descent from Problem A.2 to estimate

(a) the atmospheric density ρ at the current position of the ISS and
(b) the total amount of matter (in kg) which collides with the ISS every day.

www.iaac.space 4/9
(extra page for problem B.1: Altitude of the ISS – Part 2)

p
Solution a: Because v = γM/(R + h), the energy of the ISS at altitude h is

Mm mv 2 γM m
E = −γ + =− .
R+h 2 2(R + h)

This energy is dissipated away due to the drag force:

dE 1 γM m ḣ
Fd · v = =⇒ ρcd Av 3 =
dt 2 2 (R + h)2
p
This gives with v = γM/(R + h):

m ḣ
ρ= √ ·√
cd A γM R+h

By using the rate of descent ḣ from A.2, we get the result: 1.94 x 10-12 kg/m3

Solution b: The volume crossed per orbit is V̂ = A · 2π(R + h). The orbital period is
s
U 2π(R + h) (R + h)3
T = =p = 2π
v γM/(R + h) γM

Thus, the volume per time is given by:

r
t γM
V (t) = V̂ · = A ·t
T R+h
With m = ρ · V , we get for 24 hours: 6.16 kg

(a) Name the three regions A, B, C marked in the horizontal Milky Way drawing below.

A B C

The model parameters have the values below:

Ψ = 10−4 stars/(light-year)3
RA = 20 light-years
RB = 12 · 103 light-years
RC = 5 · 104 light-years
ΩA = 21, ΩB = −3, ΩC = −8

(b) Create a double logarithmic plot of the density distribution ρ(r) with respect to r.

Note: Assume that the Milky Way has a constant thickness of 1,000 light-years.

www.iaac.space 6/9
(extra page for problem B.2: Stars in the Milky Way)

Solution a: A: center or bulge; B: disk; C: halo

Solution b:

Solution c: Let i ∈ {A, B, C}. We integrate over a cylindrical volume with radius R and height h
to get the total number of stars N :
Z R Z R
N= ρ(r)dV = ρ(r) · 2πhr · dr
0 0
Z R " #
X r
= Ψ· exp Ωi − · 2πhr · dr
0 i
Ri
XZ R
r

= Ψ · 2πh exp Ωi − · rdr
i 0 R i

X Z R/Ri
= Ψ · 2πh Ωi 2
e Ri e−x · xdx
i 0

X
Ωi 2 R −R/Ri
= Ψ · 2πh e Ri 1 − +1 e
i
Ri

The result is 331 · 109 stars (i.e., 331 billion stars).

www.iaac.space 7/9
Problem C.1 : Einstein Ring around Galaxy (8 Points)
This problem requires you to read the following recently published scientific article:

Answer the following questions related to this article:

(a) What kind of object is the article about, and how was it discovered?

→ massive compact galaxy at redshift z = 2

→ discovered with the JWST NIRCam, COSMOS-Web project

(b) Describe all elements visible in Figure 1a, including the names JWST-ER1g and JWST-ER1r.

→ center: JWST-ER1g, compact early-type galaxy (red center, blue disk)

→ ring: JWST-ER1r, Einstein ring due to gravitational lensing, including two red concentrations

(c) Find the galaxy’s age, mass, and radius and compare them to our Milky Way.
→ age: 1.9 Gyr; Milky way: 13.6 Gyr, in much older stage
→ mass: 1.3 x 1011 M ; Milky way: 8.9-15.4 x 1011 M , more massive
→ effective radius: 1.9 kpc; Milky way: around 26 kpc, much larger

(d) According to the study, what is and what is not likely to cause the lensing mass?

→ unlikely: gas (otherwise high star formation rate), extra dark matter in the Einstein ring
→ likely: low mass stars (dominate mass but low light contribution)

(e) Explain what the IMF is by using Figure 3a.

→ initial mass function: distribution of masses for a stellar population

→ Figure 3a: different models, x: masses, y: number of stars

(f) Why is the ring in Figure 1a a gravitational lens and not a ring galaxy?

→ photometric redshift of ring is higher than central galaxy

→ morphology of red knots (mirrored!)

www.iaac.space 8/9
Problem C.2 : Volcanic Activity on Io (8 Points)
This problem requires you to read the following recently published scientific article:

Answer the following questions related to this article:

(a) What is different about the presented observations compared to previous studies?

→ global mapping not possible before, poles not visible

→ now observations of Io’s poles due to Juno spacecraft

(b) Explain how the north pole, south pole and the lower latitudes differ in volcanic activity.
→ polar volcanoes less energetic but same density
→ fewer active volcanoes in south polar region compared to north (also less radiance/area)
→ polar volcanoes generate less energy than lower latitude volcanoes

(c) What is Loki Patera and Tvashtar Paterae?

→ Loki Patera: power thermal source at 310W 12N

→ Tvashtar Paterae: site of episodic, vigorous volcanic activity at 124W 62N

(d) How many hot spots did the researchers find in the north and south polar caps?

→ north: 20; south: 12; total: 22

(e) According to the study’s findings, which mechanism likely causes Io’s volcanic activity?

→ general reason: tidally induced internal heating

→ specific mechanism: not tidal heating in deep mantel; instead shallow asthenospheric heating

(f) Describe how the scientists identified hot spots from the observation pixel data.

→ nightside observations: pixel brightness threshold 50% greater than background

→ dayside observations: lower threshold, up to 1.2 x background, further criteria

www.iaac.space 9/9
International Astronomy and
Astrophysics Competition
Final Round

Final Round Exam 2024

The final round exam was given in the form of an online exam.
Each participant was given a subset of 20 questions in random order.
This paper version is only available for training purposes.

IAAC Final Exam 2024 1/5

Question 1 : Which one of these stars can be seen only in the Southern Hemisphere?

(A) Antares (B) Arcturus

(C) Alpha Centauri (D) Altair

Question 2 : Which one of these constellations can be seen only in the Southern Hemisphere?

(A) Cepheus (B) Cassiopeia (C) Cygnus (D) Crux

Question 3 : What kind of astronomical object is the Large Magellanic Cloud?

(A) Constellation (B) Dwarf Galaxy (C) Nebula (D) Star Cluster

Question 4 : How many stars are there in the Milky Way?

(A) 10 - 100 billion (B) 100 - 400 billion

(C) 400 - 900 billion (D) 900 - 1200 billion

Question 5 : On Earth’s surface, objects fall with a gravitational acceleration of approximately ...

(A) 9.8 m/s2 (B) 11.9 m/s2 (C) 19.5 m/s2 (D) 32.2 m/s2

Question 6 : The cosmic microwave background can be detected using a ...

(A) x-ray telescope (B) optical telescope

(C) infrared telescope (D) radio telescope

Question 7 : The rotational speed at Earth’s equator is about 1,600 km/h. What is the rotational
speed at Earth’s poles?

(A) 0 km/h (B) 800 km/h (C) 1,131 km/h (D) 1,600 km/h

Question 8 : Which latitude is best suited for a rocket launch?

(A) 5◦ N (B) 15◦ S (C) 45◦ N (D) 90◦ S

IAAC Final Exam 2024 2/5

Question 9 : The altitude of the International Space Station (ISS) is around ...

(A) 250 km (B) 380 km (C) 410 km (D) 520 km

Question 10 : The ISS descends over time primarily due to ...

(A) changes in Earth’s rotational speed. (B) pressure from the solar wind.

Question 11 : Space is often considered to start at an altitude of ...

(A) 50 km above ground (B) 100 km above ground

(C) 150 km above ground (D) 200 km above ground

Question 12 : What is the name of the region marked in the horizontal Milky Way drawing below?

(A) galactic bulge (B) galactic disk

(C) galactic halo (D) galactic continuum

Question 13 : What is the name of the region marked in the horizontal Milky Way drawing below?

(A) galactic bulge (B) galactic disk

(C) galactic halo (D) galactic continuum

IAAC Final Exam 2024 3/5

Question 14 : Einstein rings are caused by ...

(A) interference of light. (B) magnetic field distortion.

(C) gravitational lensing. (D) light scattering from interstellar dust.

Question 15 : What is the initial mass function (IMF) in astronomy?

(A) frequency of star formation events in (B) spatial distribution of stars in a star cluster
a galaxy

Question 16 : What is a spectral energy distribution (SED) plot in astronomy?

(A) plot of energy versus wavelength (B) plot of energy versus distance

Question 17 : Io is a moon of the planet ...

(A) Venus (B) Jupiter (C) Saturn (D) Uranus

Question 18 : Which spacecraft orbits currently around Jupiter?

(A) Voyager (B) Cassini (C) New Horizons (D) Juno

Question 19 : How do volcanoes at the poles of the moon Io differ from volcanoes at the equator?

(A) polar volcanoes are less energetic (B) polar volcanoes are less densely dis-
tributed

Question 20 : How many volcanic hot spots were observable on the surface of the moon Io be-
tween 2017 and 2022?

(A) 26 (B) 56 (C) 136 (D) 266

IAAC Final Exam 2024 4/5

Question 21 : Which mechanism causes volcanic activity on the moon Io?

(A) solar radiation heat absorption (B) impact from meteor showers

Question 22 : How many people have stepped on the moon as of 2024?

(A) 8 (B) 12 (C) 22 (D) 32

Question 23 : Which one of these planets is never visible at midnight?

(A) Venus (B) Mars (C) Jupiter (D) Uranus

Question 24 : Schmidt, Cassegrain, and Galilean are names for famous ...

(A) spacecraft types (B) comet types

(C) telescope types (D) wavelength types

Question 25 : Most stars are cooler than the Sun and emit most of their energy as ...

(A) ultraviolet light (B) visible light

(C) radio waves (D) infrared light

Question 26 : Astronomers use Cepheid stars as a measurement for ...

(A) velocity (B) distance

(C) diameter (D) spectral composition

IAAC Final Exam 2024 5/5

International Astronomy and
Astrophysics Competition
Final Round

Final Round Exam 2024 - Solutions

No. 1 No. 2 No. 3 No. 4 No. 5 No. 6 No. 7 No. 8 No. 9 No. 10

C D B B A D A A C D

No. 11 No. 12 No. 13 No. 14 No. 15 No. 16 No. 17 No. 18 No. 19 No. 20

B C A C C A B D A D

No. 21 No. 22 No. 23 No. 24 No. 25 No. 26

C B A C D B

IAAC Final Exam 2024 1/1

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