Lab-3 The-Orbit-of-Mars
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Mountain View College *
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2301
Subject
Astronomy
Date
Apr 3, 2024
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4
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The Orbit of Mars
Materials: protractor, compass, centimeter ruler, paper, scientific calculator, Textbook.
Object: To reconstruct the approximate orbit of Mars by using some of Tycho's data.
Tycho Brahe collected a number of observations of the positions of Mars as seen from Earth during the latter part of the
16th century. In order to plot the orbit of Mars it is necessary not only to know where Mars was in relation to the earth
but also to know where the earth was in relation to the Sun. Mars’s period of revolution about the Sun is 687 days. So
any two observations of the position of Mars separated by exactly 687 days would view Mars at the same position in its
orbit. Where those two lines of sight (from Earth to Mars) crossed would be Mars’s position in space. The table below
lists pairs of Mars positions observed by Tycho and arranged by Kepler to be 687 days apart. These were recorded in
Keplers book of 1609, Astronomia Nova (“The New Astronomy”).
Date
Heliocentric Longitude of
Earth
Geocentric Longitude of
Mars
1585 Feb 17
159°
135°
1587 Jan 05
115°
182°
1591 Sep 19
6°
284°
1593 Aug 06
323°
347°
1593 Dec 07
86°
3°
1595 Oct. 25
42°
50°
1587 Mar 28
197°
168°
1589 Feb 12
154°
219°
1585 Mar 10
180°
132°
1587 Jan 26
136°
185°
Procedure: You may work in groups. However, EACH student will turn in the results of the following procedures and your
own sketch!
1. Place the graph paper in front of you with the long side horizontal. Place a small dot, representing the Sun, near the
center of your graph paper. Label the dot "Sun". Draw a straight line from the center (the Sun) to the right-hand edge.
Label this line the “autumnal equinox.” This line represents the 0° direction in space. All angles should be measured
counter-clockwise from this direction
2. Draw a circle of radius 5.0 centimeters centered on the Sun. This circle represents the orbit of Earth. In reality, the
Earth’s orbital eccentricity is 0.016, or 1.6% deviation from perfect circularity.
Note: This scale means that 5.0 cm = 1 A.U. Where 1 A.U. is the average distance between the Earth and Sun (93 million
miles); Label this circle "Earth’s Orbit".
3. Using the protractor centered on the Sun with 0° toward the autumnal equinox, and using the heliocentric longitude of
the Earth (as given in the table above) plot the positions of the Earth with dots on the Earth’s orbit. You should label the
date next to each of the 10 dots!
4. The observations of Mars are paired. Go to the first entry (Feb 17, 1885). Move the protractor so that the Earth is at
the protractor’s center, but the 0° direction is still parallel to the autumnal equinox line. Find the geocentric longitude of
Mars observed for that date and mark it. Draw a line in this direction starting from Earth and proceeding nearly to the
edge of the page (as in figure 1).
5. Repeat step-4 for the second entry of the first observation pair (see figure-2). Continue this process for all data
entries in table.
6. For each observation pair Mars is located at the intersection of the two lines. Put a conspicuous dot on each of the
five intersections (as in figure 2). When completed you should have 5 paired lines that meet outside earth’s orbit. STOP
if not and review steps 3 to 5!
Figure 1
Figure 2
Figure 3
7. Kepler chose the first two sets of data to represent aphelion and perihelion respectively for
Mars, mark these in drawing. Then, draw a line from the aphelion to the perihelion for Mars. This line should go through
(or pass close to) the Sun. If not, something has gone wrong. This line is called the major axis of the orbit. (Today, half of
this is known as the mean distance Sun-Earth).
Measure the major axis in centimeters to the nearest millimeter (tenth of a centimeter)
Major axis = 15.5 cm.
8. Find the middle of the major axis by dividing the length of the major axis by 2. This length is defined as the
semi-major axis = 7.75 cm
Mark the center of the major axis and label it “midpoint”.
9. Using the compass draw a circle representing Mars' orbit by placing the point of the compass on the midpoint and
the pencil part either on the perihelion or aphelion points and making a circle. If you found the midpoint correctly your
orbit should pass through both perihelion and aphelion points. Your diagram should look somewhat like the one in figure
3. The other three points of the orbit should pass quite close to the circle that you drew. No wonder Kepler and others
initially thought that planetary orbits were circular!
DO NOT continue if orbit does not look like figure 3, review for errors from step 1.
10. Calculate the length of the semi-major axis of the orbit of Mars in A.U. and in miles. Remember that on our drawing
that 5 cm = 1 A.U. = 93 000 000 miles. Show your work!
Semi-major axis = 1.496 A.U.
Semi-major axis =139,061,888 miles
11. Look up the accepted value for Mars’s semi-major axis length in AU in the appendix of your textbook. Compare your
calculated value against the accepted one by calculating the percent deviation, using the following formula. Show your
work clearly. Note that to multiply by 100% means you multiply by 100 and attach the percent sign to the answer.
%
?????
=|
𝐴??????? ?𝑎𝑙???
−
𝑦??? ?𝑎𝑙???𝑎??????? ?𝑎𝑙??
| × 100%
(the bars in formula means the absolute value)
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