Singapore Astronomy Olympiad 2022

Cover Page

1. Ensure that all materials (answer sheets, graphing paper, practical question paper) you
intend to submit for grading have been combined to one PDF file. Amendments to
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 Your participant code. Question Marks Question Marks
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 Page number – in increasing order.
1.2 3.2
 DO NOT WRITE YOUR NAME
1.3 3.3
ON YOUR ANSWER SCRIPT.
1.4 Total
1.5 4.1
3. For all components of this Olympiad, order Total 4.2
your answer scripts as follows: 2.1 Total
 The corresponding Summary 2.2 P(A)
Answer Sheet
2.3 P(B)
 Answer sheets with your workings
i. Within each section, sort by 2.4 P(C)
question number. (EG. 2.1 2.5 Total
before 2.3 before 2.4) 2.6 OVERALL
ii. For the practical round, Total Theory
include your star chart
DA
responses here
Practical
GRAND TOTAL

FOR OFFICIAL USE ONLY

Marker’s Initials Signature

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of SAO Answer Sheets
 Singapore Astronomy Olympiad 2022
Theory

Instructions
1. The theoretical portion of this Olympiad lasts for 105 minutes and is worth a total of 102
marks.
2. Fill in these details on the Theory Summary Answer Sheet (Attached as a MS word document)
and each side of an SAO answer sheet:
• Year of competition
• Your participant code
• The page number – which should be continuous from 1 to N
• The part of the paper, and the question number
3. Cross out all workings or answers you do not wish to be evaluated.

4. If you require assistance (enquire about an ambiguity or possible errata, etc.), please get the
attention of the invigilators.
5. Once the Theory Round is over, you are to combine all documents into one singular PDF file to
be uploaded. More details on the order can be found in the Cover Page word document.

Competition Rules and Regulation

1. Only the use of scientific calculators is permitted. No graphing or programmable calculators are
allowed.
2. Disruptive behaviour, cheating, collusion to cheat or any integrity-related offences are grounds for
immediate disqualification.

3. You may opt to retain the question paper, constants sheet and answer script for personal use.
4. Toilet breaks are prohibited for the duration of the paper owing to its decentralized nature. Par-
ticipants are encouraged to use the washroom before the paper commences.

2
Contents

1 Section A [12] 4
1.1 Exoplanet Exploration [4] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Galactic Collision [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Space Elevators [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Lunar Evaporation [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5 Artificial Illumination [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Section B [54] 7
2.1 Eruption [9] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Kerr [9] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Magnetic Sails [9] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 Standard Candle [9] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.5 Radio Array [9] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.6 Strömgren Spheres [18] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Section C: Long Question [36] 12

3.1 Part 1: Lagrange Points [8] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2 Part 2: Roche Lobes [13] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.3 Part 3: Evolution [15] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1 Section A [12]
Write all your responses in the summary answer sheet provided.

1.1 Exoplanet Exploration [4]

Consider the following hypothetical exoplanets:

Radius Mass Mean Density Average Parent Star

Atmosphere
(Earths) (Earths) (kg/m3 ) Temperature (°C) Classification
SAO001 0.000012 6.27 × 10−19 2000 125 None B2 III (Blue Giant)
SAO002 1.57 7.3 11026 -59 None Wolf-Rayet Star

Based on humanity’s current technology, which of the following modes of transport is most practical for
exploration of each of the exoplanet’s surface: [4]

(A) Rover with wheels

(B) Helicopter

(C) Ion Thruster

(D) Aeroplane

1.2 Galactic Collision [2]

In five billion years, the Milky Way is projected to collide with its closest neighbour, the Andromeda
Galaxy.

Number of Stars Galactic Disk Diameter (kly) Central Black Hole Mass (M⊙ )
Milky Way 3 × 1011 185 3.6 × 106
Andromeda Galaxy 1012 220 1.5 × 108

Based on your background knowledge and the estimates in the table above, which of the following
statements are false?
I. A large number of stellar collisions will occur
II. Some stars will be ejected from the combined galaxy

III. Many supernovae will be triggered by the merger

IV. Central black holes from both galaxies are likely to merge

(A) I and III

(B) I and IV
(C) I and II

(D) II and IV
1.3 Space Elevators [2]
Space elevators are a planet-to-space transportation system involving a cable structure extending per-
pendicularly from the planet surface. Modern space elevator concepts rely on a tensile design, with a
counterweight in space using centrifugal force to negate the cable’s gravitational weight.

Which of the following statements are true?

I. The counterweight must always be above the altitude for geostationary orbit
II. The total mass of climber and payload being lifted should be less than the mass of the counterweight
III. To minimize counterweight mass, space elevators with tensile designs should be deployed at the
equator

IV. Space elevators will never work on a tidally locked planet

(A) I and III

(B) I and II
(C) II and III

(D) III and IV

1.4 Lunar Evaporation [2]
Hawking radiation refers to the thermal radiation emitted by a black hole due to relativistic quantum
effects. The radiation power and temperature of a black hole are, respectively, related to mass by:
1 1
P ∝ T ∝
M2 M
Assume the moon has collapsed into a moon-mass black hole. From Earth, what would be observed in
the final moments before its evaporation?

I. The black hole’s radiant flux increases rapidly

II. Rate of DNA ionization increases rapidly

III. Spaghettification occurs to Earth due to extreme tidal forces

IV. Earth is irradiated primarily by infrared radiation

(A) III and IV

(B) I and III

(C) I and II
(D) II and IV

1.5 Artificial Illumination [2]

For two decades, the Proton-M rocket has been a cornerstone of Russia’s heavy-lift rocket launches. The
rocket has a burn time of 130s and a first stage carrying 458,900 kg of fuel (8.11kJ/kg).

Calculate the apparent magnitude of the rocket as seen by a observer 1500km away, given that solar
flux at Earth is 1361 W/m2 and the Sun’s apparent magnitude is -26.74. Assume 0.5% of energy from
combustion is converted into isotropic light.

(A) 16.1
(B) 1.83
(C) 11.7
(D) -3.49
2 Section B [54]
Section B(1): Short Questions [36]
This section contains a total of 5 short-length questions worth 9 marks each.

Answer any FOUR (4) questions.

2.1 Eruption [9]

In 2009, the islands of Hunga Tonga and Hunga Ha’apai (20.57° S, 175.38° W) merged due to the eruption
of its submarine volcano. At 1726 Tongan civil time (UTC+13), January 15, 2022, the newly-combined
island was torn back apart due to another volcanic eruption, thought to be one of the strongest in
three decades. This eruption released a pressure wave that spread radially outward, resulting in several
tsunami advisories in Hawaii and Japan.

Residents of Anchorage, Alaska (61.22° N, 149.90° W) reported hearing a sonic boom at 0300 AKST
(UTC-9).

(a) Calculate the average velocity of this pressure wave, stating any assumptions made. [4]
(b) How long after the eruption is sunset? Take the effect of atmospheric distortion at the horizon to
be θatm = 34′ . [5]

2.2 Kerr [9]

The optical Kerr effect is the change in refractive index of a medium induced by the electric field of
light. For a beam of light, this creates a refractive index gradient analogous to a gradient index lens and
results in self-focusing. It has been conjectured that the optical Kerr effect also occurs in a vacuum.

The equation for critical power is given by:

λ2
Pcr = α
4πn0 n2
For relativistic momentum:
E 2 = (pc)2 + (m0 c2 )2
Take the constants to be α = 1.862, n2 = 1.555 × 10−37 m2 /W (for a vacuum), and n0 to be the refractive
index of the medium.

(a) Find the critical power for a 820nm laser in a vacuum [1]
(b) Currently, the most powerful laser is the ZEUS three-petawatt (peta = 1015 ). Assuming the
maximum power of lasers increase by an order of magnitude every eight years, estimate how long
it takes before a laser will attain the power found in part (a). [2]

Breakthrough Starshot is a project aiming to launch miniature interstellar probes to Alpha Centauri
using light sails. Assume the probes are fully reflective and propelled by aforementioned self-focusing
laser on Earth with wavelength and power found in part (a) with a pulse length of ten picoseconds.

(c) For a probe with mass 5g, how long would would it take to reach Alpha Centauri, which is 4.37
light years away? The time taken for acceleration can be taken as negligible. [6]
2.3 Magnetic Sails [9]
Magnetic sails are a proposed method of propulsion using a static magnetic field to deflect charged
particles radiated by the Sun, thus imparting momentum. Technological limitations on magnetic sails
include identifying suitable high-temperature superconductors, with the highest currently known only
being superconducting below 133 K.

A typical design generates the magnetic field using a large loop of superconducting wire positioned per-
pendicularly to the direction of charged particle flow (ignore orientation in diagram). For simplicity, take
the loop to be a flat 2D disc with a radially symmetric hole.

Assume the solar wind emanates radially from the Sun with a constant velocity of 500km/s. At a distance
of 1AU, the proton flow is 5 × 1014 protons/m2 s, and is inversely proportional to the square of distance
from the Sun. The magnetic field resulting from the superconductor has an effective proton deflection
radius of 20km, within which it deflects solar wind particles perpendicularly from their initial direction.
(a) Assuming the sail reaches thermal equilibrium with no cooling system, find the minimum distance
from the Sun for the magnetic sail to function (with currently available superconductors). [3]

(b) Find the force by solar wind on the sail at the distance in part (a) [4]
(c) Find the mass of the magnetic sail such that it remains stationary (not orbiting) at the distance
found in part (a) [2]
2.4 Standard Candle [9]

Consider a type 1A supernova in a distant galaxy with a peak luminosity of 5.8 × 109 L⊙ . Suppose you
observe this supernova, and find it to have an apparent luminosity 1.7 × 10−8 that of Vega’s. Subsequent
measurements of its host galaxy finds that the 21cm line has been Doppler shifted to a wavelength of
22.5cm.

For such distant objects, the intensity of measured light is reduced due to the expansion of the universe,
and the luminosity distance is no longer equal to the comoving distance. The luminosity distance is given
by:
dl = (1 + z)dc
Where dl and dc are the luminosity and comoving distances respectively, and z is the redshift of the
object in question.

(a) Calculate the redshift to the host galaxy. [2]

(b) Calculate the comoving distance to the type 1A supernova. [4]
(c) Hence, calculate the Hubble constant, and subsequently the Hubble time. [3]
2.5 Radio Array [9]
The SKA (Square Kilometre Array) is a large international radio telescope being built in South Africa
and Australia. The SKA1-mid telescope in South Africa (30° S) consists of 133 dish antennas of diameter
12 m observing in the frequency range of 350 MHz to 15.3 GHz. Using interferometry, these dishes will
function as a single telescope with a baseline length of 150 km.

An astronomer points one of the dish’s antennas towards a radio source with a known incident flux of
1.3 × 10−20 W/m2 .

(a) Assume that the collected photons are evenly distributed across the frequencies 0.95 GHz to 1.76
GHz. What is the average number of photons reaching the detector of each radio dish every second?
[2]

(b) If instead observing using a frequency of 7.825 GHz, what is the angular resolution of the entire
interferometer array in arcseconds? [2]

To detect a faraway point source using the telescope, the source’s signal must be sufficiently strong
compared to the noise level of the telescope. The noise level, σ is defined as follows:
2kB Tsys
σ= √
A ti ∆v
In terms of the system temperature Tsys , aperture area of the telescope A, integration time ti and band-
width ∆v.

A radio galaxy has a known spectral flux density of 2.5 × 10−3 Jy at an observing frequency of 0.4
GHz. At this frequency, the SKA1-mid array has Tsys = 150 K and bandwidth 12.5 MHz. The Jansky
is a unit frequently used to describe the spectral flux density of radio sources with conversion 1 Jy =
10−26 W/m2 Hz. The entire radio array is used to observe the galaxy.

(c) Calculate the minimum integration time ti,min to achieve a 40σ detection (the signal is 40× the
noise level σ). [2]
(d) What is the minimum spectral flux density of a source located on the celestial equator, such that
it may still be observed in the SKA1-mid array for a 30σ detection? Consider the maximum
integration time due to the rotation of the Earth. [3]
Section B(2): Medium Questions [18]
Write all your responses in the summary answer sheet provided.

2.6 Strömgren Spheres [18]

Strömgren spheres are spheres of ionised hydrogen formed around hot stars of spectral types O and B,
where neutral hydrogen (HI) is ionised by the Lyman-continuum photons from the star (hv > 13.6 eV)
to form a region of ionised hydrogen (HII-region).

Consider a luminous blue star located within a uniform spherical hydrogen cloud with a density of 108
molecules per cubic metre and some temperature THII . The star emits 1049 photons per second, with all
such photons assumed to be ionizing.

Suppose the hydrogen within the Strömgren sphere is fully ionized. Within the Strömgren sphere, the
rate of recombination and ionisation is balanced. The rate of recombination is proportional to the
number density of protons (np ) and of electrons (ne ), as well as a temperature-dependent constant of
recombination α. At THII , α(THII ) = 10−19 m3 s−1 . For simplicity, ignore the effect of any photons
created via recombination.

(a) Derive an algebraic expression for the radius of a Strömgren Sphere and calculate its value using
the given parameters. Express your answer in light years. [5]
(b) Estimate the timescale (in years) it takes for such a sphere to form, if recombination does not take
place. [2]

In reality, 40% of recombination events actually emit line photons that will further ionize the hydrogen
cloud.

(c) Hence, derive a new expression for the radius of a Strömgren Sphere taking this percentage into
account. [2]

P∞
Hint: k=0 ark = a
1−r for |r| < 1

The Bubble Nebula is a famous example of a Strömgren sphere located approximately 3400 pc from
Earth. It is approximately spherical with observed angular diameter 3′ and surface brightness of
14.94 mag/arcmin2 . Ignore the contribution of the luminous blue star in the brightness of the nebula
and assume the sphere is of uniform density.

(d) Calculate the apparent magnitude of the Bubble Nebula as observed from Earth. [2]
(e) Find the difference in apparent magnitude of the Bubble Nebula as observed from the centre of the
nebula as compared to your answer in (d). [7]
3 Section C: Long Question [36]
Write all responses on the answer sheets provided. The marks are stated in the brackets [ ] at the end
of each sub-part.

3.1 Part 1: Lagrange Points [8]

The Lagrange points of a system are points of gravitational equilibrium in a two-body system. The
L4 and L5 points are naturally stable and are known to harbour objects such as the Greek and Trojan
asteroids in the Sun-Jupiter system.

On 25 Dec 2021, the JWST was launched towards the L2 point which is a favourable location as it
offers a view into deep space unobstructed by neither the Sun nor Earth. Although the point is not
intrinsically stable, station keeping manoeuvres will allow the JWST to remain in orbit for the duration
of its mission.

(a) For a two-body system with masses M1 ≫ M2 , find the distance between L2 and the secondary
body with mass M2 in terms of the orbital separation a.

Hence or otherwise, calculate the distance between L2 and Earth for the Sun-Earth system. [8]

3.2 Part 2: Roche Lobes [13]

When studying the gravitational interactions of a binary system, it is also useful to look at the Roche
lobe of each body. This is the region around each component where matter is gravitationally bound to
and will orbit it.

The shape of the donor’s Roche lobe is non-spherical with shape dependent on the mass ratio of the
binary, defined as q ≡ M1/M2 . In a theoretical treatment, it is convenient to approximate the Roche lobe
by a spherical volume with radius given by Eggleton (1983):

0.44q 0.33
RL ≈ a for 0.1 < q < 10
(1 + q)0.2

(b) Briefly describe how L1 is related to the boundary of a Roche lobe [2]

(c) Show that the ratio of Roche lobe radii for 0.1 < q < 10 is given by [4]

RL,1
≈ q 0.46
RL,2

In a binary system, Roche lobe overflow occurs when the stellar envelope of a component expands past its
Roche lobe, and mass is siphoned onto the accreting object through L2. With mass-transferring binary
systems, strong tidal forces tend to circularize the orbit relatively fast.

Conventionally, the mass ratio for such a system is defined q ≡ Md/Ma , where Md and Ma are the
masses of the donor and accretor respectively. In a semi-detached binary, only one component overflows
its Roche lobe. Effectively, the donor’s radius is equivalent to the radius of its Roche lobe.
(d) Express the orbital period T of a semi-detached binary system in terms of the mean density of the
donor star ρ̄ and other relevant quantities. [7]
3.3 Part 3: Evolution [15]
The Roche lobe index ζL relates the response of the donor star’s Roche lobe to the donor star’s mass:

RL ∝ MdζL

In studying the dynamics and evolution of these systems, it is convenient to express the change in an
observed quantity over time as a fraction of the original quantity. For example, if a quantity Q can be
expressed as
X aY b
Qn = k
Zc
For some constants a, b, c, k and n, and quantities X, Y and Z, then the rate at which Q changes is given
by
Q̇ Ẋ Ẏ Ż
n =a +b −c
Q X Y Z
The notation Q̇ denotes the change in quantity Q over time. For instance, the change in masses of the
stellar components are related by
Ṁa = −η Ṁd
This models mass transfer with 0 ≤ η ≤ 1 being the fraction of mass outflow from the donor star that is
captured by the accretor. In reality, usually η < 1 since processes such as stellar wind carry mass out of
the system.

The angular momentum of a binary system is given by

M12 M22
J2 = G a(1 − e2 )
M1 + M2
(e) For a circularized semi-detached binary undergoing conservative mass transfer:

(i) Determine the value of J/J

˙ and ė/e [2]
(ii) Hence or otherwise, express the change in orbital separation ȧ/a in terms of other relevant
quantities of the system [6]
(f ) Derive ζL (q) for the case of conservative mass transfer. [4]

Hint: The expression has the form ζL (q) = a + bq for constants a and b

(g) A study of a semi-detached binary system determines the masses of the donor and accretor to be
1.49M⊙ and 3.22M⊙ respectively, with a period of 0.901 days. Calculate the density of the donor.
Hence or otherwise, suggest (with explanation), which of the following the donor is likely to be: [3]

• Main Sequence star

• Giant star
• White dwarf
• Neutron star
End of Theory Paper