The Orbit of Mars C
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Towson University *
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Subject
Astronomy
Date
Apr 3, 2024
Type
docx
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7
Uploaded by ProfCloverYak6
The Orbit of Mars
Big Idea
Tycho Brahe made a number of observations of the positions of Mars during the latter part of the 16
th
century. Despite not having a telescope, Brahe was able to obtain the most accurate measurements of the positions of Mars of his time. His assistant, a young mathematician named Johannes Kepler, devised a method of triangulation to determine the orbit of Mars around the Sun.
In this experiment, you will:
Recreate Kepler’s measurements using Tycho Brahe’s data;
Analyze the properties of Mars’ orbit, and;
Investigate a modern claim about Mars’ appearance in the night sky.
Setup
You will need:
Ruler
Protractor
Compass
Pencil (tip: if you can use a colored pencil in addition to a regular pencil, that would be great)
Calculator (optional)
Part I: Brahe’s data
Kepler knew from Brahe’s observations that the sidereal period of Mars is 687 days
, so every 687 days, Mars would return to the same position among the fixed stars. He also knew that the Earth completes two orbits around the Sun in 730 days
. That means by the time Mars completes one full orbit around the Sun, Earth will not quite have completed two full orbits. Below is a table of Brahe’s data, grouped into five pairs of dates, each 687 days apart:
Pair
Date Pair (687 days apart)
Heliocentric longitude of Earth
Geocentric longitude of Mars
1
February 17, 1585
January 5, 1587
159
23'
115
21'
135
12'
182
08'
2
September 19, 1591
August 6, 1593
5
47'
323
26'
284
18'
346
56'
3
December 7, 1593
October 25, 1595
85
53'
41
42'
3
04'
49
42'
4
March 28, 1587
February 12, 1589
196
50'
153
42'
168
12'
218
48'
5
March 10, 1585
January 26, 1587
179
41'
136
06'
131
48'
184
42'
Every Martian year (687 days) Mars returns to the same point in its orbit around the Sun, thus if we view Mars at these intervals we can, by triangulation, determine that point. You should follow the procedure below to get the first point, then repeat four more times to get the orbit.
1
Name: ________________________
________________________
Date:
________________________
Part 2: Plot the orbit
Attached is a diagram of the Sun with the orbit of Earth drawn around it (the orbits of Mercury and Venus
are drawn in as well, to help show their relative distances.) The dashed horizontal line indicates where the
Sun would appear to an observer from Earth (on the opposite side of the Sun) on the March equinox (March 21). This position represents 0 degrees of heliocentric longitude.
1.
With the protractor and Sun as the center, plot the heliocentric longitude of the Earth as a point on
the Earth's orbit as given in the table (159 degrees).
2.
Now with the protractor and using the Earth as the center plot the geocentric position of Mars (135 degrees). You can use the horizontal lines to help make sure your protractor is lined up at 0 degrees longitude. Your drawing should be similar to Figure 1:
Figure 1
3.
Now repeat for the Jan. 5
th
1587 date. First mark the position of Earth from its heliocentric longitude, and from that point draw a line to the geocentric longitude of Mars. The point of intersection is the position that Mars had on these two dates. Draw a dot there to represent Mars. Label this as position P1
. Your drawing should be similar to Figure 2:
Figure 2
4.
Repeat the above steps for the remaining four pairs of dates in the data table. Label the positions of Mars as P2
, P3
, P4
, and P5
.
Kepler chose the first two sets of data to represent aphelion
and perihelion
, respectively for Mars. 5.
Draw a line from the first position for Mars P1
to the second position for Mars P2
. This line should pass close to the Sun (if your line passes nowhere near the Sun, your measurements for the
2
Earth and/or Mars were off and you’ll need to try again). This line is called the major axis
. 6.
Measure the major axis in centimeters to the nearest millimeter (tenth of a cm)____________cm.
7.
Find the middle of the major axis by dividing the length of the major axis by 2. Mark the center of the major axis and label it “midpoint”.
8.
Measure the distance from the midpoint of the major axis to either end of it in centimeters. This length is defined as the semimajor axis
.___________cm. Label this length a
.
Part 3: Kepler’s third law
Let’s calculate the value of Mars’ semimajor axis in Astronomical Units (AU). An Astronomical Unit
is defined as the distance from the Sun to the orbit of the Earth.
9.
Find the scale for astronomical units on your graph by measuring the distance from the Sun to the
Earth in centimeters to the nearest millimeter (tenth of a cm). Scale: 1 AU = ______________ cm
10.
Using your scale, calculate the semimajor axis of Mars in AU: _______________________ AU
11.
Now calculate the semimajor axis of Mars in km. 1AU is 149 million km (
1.49
×
10
8
km
), so multiply your answer from step 10 by 149: _________________million km.
12.
Express your answer in scientific notation: _________________ km
Now that we know Mars’ semimajor axis in AU (
a)
, we can use Kepler’s third law to calculate its orbital period around the Sun in years (
P)
. Recall:
P
2
=
a
3
…which means we can solve for the period P
like this:
P
=
√
a
3
13.
Using the above formula calculate the orbital period of Mars in years, using “a” from question 10:
P
Mars
=
√
¿
¿
3
=
¿
years
3
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