Name
______________________________
Hour ___________
Back Ground: Back in 1609, Johannes Kepler did not have a computerized
planetarium that he could use to visualize the solar system at hatever instant he chose!
"hat he did have as an enormous #uantity o$ data inherited $rom %ycho Brahe, an
o&servational astronomer so care$ul and meticulous that his data can still &e used today
to locate astronomical o&'ects! %ycho had hoped that Kepler ould &e a&le to use his
data to prove mathematically the correctness o$ his on (eocentric cosmolo(y )a
complicated variation o$ *tolemy+s model,! "hat happened instead as that Kepler ent
here the data took him, and it asn+t in %ycho+s direction!
Kepler+s -irst .a states that planets or&it the sun, and the or&its are ellipses! "e
are (oin( to take a very small portion o$ data similar to hat as availa&le to Kepler, and
determine the or&it o$ /ars usin( the
method that he used!
Kepler as a&le to determine the
or&ital period o$ /ars )ith respect to the
stars,! He determined that /ars or&ited
the 0un once every appro1imately 623
days! %his is important since this tells us
that /ars ill &e at the same location in its
or&it every 623 days! "hy is this
important4 %ycho+s accurate data $or /ars
in the sky only provided direction &ut not
its distance! But one can trian(ulate the
position i$ one o&serves /ars in the same
position )623 days apart,!
Objectives:
5pply (eometric methods o$
trian(ulation
6se o&servational data to construct the
or&it o$
the planet /ars
7emonstrate Kepler+s -irst .a o$
*lanetary
/otion
.a&el and de8ne the properties o$ an
ellipse 5pply the properties o$ ellipses to
the or&it to
determine the eccentricity e o$ /ars+
or&it 9alculate the percent error in your
e1perimental
value o$ e
Procedure:
1) :et the lon(itudes $or mars o&served
positions
Fram
e
Date Longitude
A
/ar! ;1,19<1
B
-e&! =, 19<<
C
5pril ;0,
19<<
D
/ar! 2, 19<=
E
/ay ;6,
19<=
F
5pril 1;,
19<3
G
0ept! 16,
19<9
5u(! >, 19>1
!
Nov! ;;,
19>1
"
?ct! 11,
19><
#
Jan! ;1,
19>>
L
7ec! 9, 19>=
$
/ar! 19,
19>6
%
-e&! <, 19>2
O
5pril >, 19>2
P
-e&! ;1,
19=0
/ars ?r&it .a&
$rom the %ransparencies! @ound to the nearest
hal$A de(ree, and enter them in ta&le 1!
&) %ape $our sheets o$ paper to(ether at their ed(es to make one lar(e s#uare )or i$
availa&le (ra& one
lar(e s#uare sheet,! -ind the center and la&el it ith a dot to represent the sun!
') 9onstruct the or&it o$ the BarthC 0tart &y assumin( that the Barth+s or&it is circular(
use a compass to
dra a circle D a radius o$ 10cm to represent the or&it o$ earth around the sun!
)) Be(in on the ri(ht horizontal end o$ your circle and la&el that position 0E! /ark the
other $our points
o$ the a1is+ )these represent the e#uino1es and solstices, ith 90E, 120E, and ;30E
respectively (oin(
counter clockise!
*) *lot the si1teen Barth positions &y usin( their dates! 0ince there are almost the
same num&er o$ days
as there are de(rees, make one day one de(ree! Fou ill need to 8(ure out ho
many days there are
&eteen certain dates! )@emem&er -e&! G ;2 days! 5pr!, June, 0ept!, Nov! are all
<0 days! Jan!, /ar!,
/ay, July, 5u(!, ?ct!, 7ec! are all <1days!,
Bach data pair represents an o&servation o$ /ars, taken $rom here on Barth! Hn
each case, the o&servations are separated in time &y e1actly 623 days, or the
len(th o$ one /artian year! %hus, each pair o$ o&servations puts /ars in the same
place on its or&it relative to the sun! Hoever, that does not put it in the same
place relative to the Barth! %his is &ecause the Barth is also movin( ith respect to
the sun, and a$ter 623 days, the Barth is de8nitely not &ack in the same place it
started!
0o, to 8nd here /ars is ith respect to the
sun, e need to kno here the Barth is ith
respect to the sun, and then here /ars is ith
respect to the Barth! %he 8(ure illustrates thisC
%he heliocentric lon(itude o$ Barth is the
position o$ the Barth relative to the sun, and
the (eocentric lon(itude o$ /ars is the position
o$ /ars relative to the Barth!
+) -rom each pair o$ Barth positions there ill
&e
one /ars position! 5lays measure $rom
the east
)0E,! %he point o$ intersection o$ the to
lines
ill &e the position o$ /ars!
,) %o 8nd the areas that /ars seeps out &eteen positions, you need to 8nd the
avera(e radius o$ the
positions! %hen use that in the $olloin( $ormula to (et the amount o$ area per day
the or&it seeps
outC I J r
;
<60
Conc-usion .uestions:
1) 7oes your plot a(ree ith Kepler+s conclusion that the or&it is an ellipse4
&) "hat is the avera(e sunAtoA/ars distance in astronomical units )56,4
') 5s seen $rom the sun, hat is the direction )lon(itude, o$ /ars+ nearest and
$arthest positions4
)) 7urin( hat month is the Barth closest to the or&it o$ /ars4 "hat ould &e the
minimum separation &eteen the Barth and /ars4
*) "hat is the eccentricity o$ the or&it o$ /ars4
+) 7oes your plot o$ /ars+ or&it a(ree ith Kepler+s la o$ areas, hich states that a
line dran $rom the sun to the planet seeps out areas proportional to the time
intervals4 -rom tour or&it, you see that /ars as at point B+ on -e&ruary =, 19<<,
and point 9+ on 5pril ;0, 19<<! %here are ei(ht such pairs o$ dates in your data!
%he time intervals are diKerent $or each pair!
,) Ho much )&y hat percenta(e, do they vary4
/) "hat is the uncertainty in your area measurements4
0) Hs the uncertainty the same $or lar(e areas as small4
11) 7o your results &ear out Kepler+s la o$ areas4
ints 2or 3uestions:
1!
a, "hen determinin( the ma'or a1is, hich is the (reatest distance across the or&it,
the sun ill &e
at one o$ the $oci! %he second $oci is symetrically placed on the a1is! 9onstruct
the ellipse usin(
a piece o$ strin( e#ual to the len(th o$ the ma'or a1is!
&, 5lso anser #uestions 1 &y
determinin(
i$ *-
1
L *-
;
G constant! %ry this $or
to
diKerent points on the plot and
determine
the M diKerence!
;! @ememe&er that 156 on your plot is
10cm!
also @
ave
G aD;!
<! 0E is to the ri(ht!
>! :ive minimum separations in 56!
=! eGcDa
6A10!
a! %hese deal ith the .a o$ 5reas! 7etermine the areaDday values $or three
se(ments o$ the
plot $or #uestion 6 )See PPT).
&! M variation G .ar(est N smallest O
100M
5v(!
c! -or #uestion 10, (ive a 'usti8cation
$or
your anser!
