Lab-3 The-Orbit-of-Mars
pdf
keyboard_arrow_up
School
Mountain View College *
*We aren’t endorsed by this school
Course
2301
Subject
Astronomy
Date
Apr 3, 2024
Type
Pages
4
Uploaded by ConstableResolve12083
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)
Show your work here:
1.524-1.496/1.524*100%=1.837% or about 1.84%
12. What is the very closest that Mars can get to Earth in miles? Hint: Measure the closest distant that the two orbits get
on your diagram in centimeters, and then convert to AU, then to miles. Show your work.
Closest distance =33,922,500 miles
Closest distance≈0.3645AU×93 million miles/AU Closest distance ≈ 3 ,922,500 miles
13. What is the very farthest apart that Mars and Earth can be in miles, not necessarily at opposition but anytime? Hint:
Earth and Mars are not always on the same side of the Sun! Show your work.
Farthest distance in miles≈(1.5237+1.0)×93 million miles Farthest distance in miles ≈ 2.5237 × 93=234,279,000miles
14. Finally we will calculate the eccentricity of the orbit of Mars. The eccentricity is a number that tells us how oval an
ellipse is, for example a perfectly circular orbit would have an eccentricity of zero and a flattened out oval would have an
eccentricity of 0.9. Eccentricities of all ellipses lie between 0 and up to, but not including 1. To find the eccentricity
follow this simple formula: The eccentricity equals the distance from the Sun to the midpoint divided by the length of the
semi-major axis. Both numbers should have the same units (for example, centimeters) so that after the division, the
eccentricity has no units.
Show your work.
Your value of eccentricity =0.2398.
1 AU is approximately 1.496 × 1 0 13 1.496×10 13 centimeters.
Closest distance in centimeters ≈ 0.3645 AU × 1.496 × 1 0 13 cm/AU Closest distance in
centimeters≈0.3645AU×1.496×10 13 cm/AU Semi-Major Axis in centimeters ≈ 1.5237 AU × 1.496 × 1 0 13 cm/AU
Semi-Major Axis in centimeters≈1.5237AU×1.496×10 13 cm/AU Plug the values into the eccentricity formula:
Eccentricity = Closest distance in centimeters Semi-Major Axis in centimeters Eccentricity= Semi-Major Axis in
centimeters Closest distance in centimeters
Eccentricity ≈ 0.3645 × 1.496 × 1 0 13 1.5237 × 1.496 × 1 0 13
Eccentricity≈ 1.5237×1.496×10 13 0.3645×1.496×10 13
Eccentricity ≈ 5.4492 × 1 0 12 2.2721 × 1 0 13 Eccentricity≈ 2.2721×10 13 5.4492×10 12
Eccentricity ≈ 0.2398
Eccentricity≈0.2398 Therefore, the eccentricity of the orbit of Mars is approximately 0.2398 ]
15. In the textbook, look up the accepted value for the eccentricity of Mars’s orbit.
Accepted value of eccentricity = .0935
Final Questions:
16. Conceptual question: A certain planet (B) has an eccentricity of 0.005; another planet (C) has an eccentricity of
0.034; which planet B or C has the most circular orbit? Explain: Planet C is more circular. This is because a perfect
circle will have an eccentricity of zero.
Use the diagram that you constructed in parts 1-9 of the lab to answer the following questions:
17. “Opposition” is when a planet-Earth-Sun line-up occurs, with the Earth between the planet and the Sun.
Oppositions that occur during which month provide the best oppositions to view Mars, because Mars is closer to the
Earth than at other oppositions. See your chart for the answer to this. The month that provides the best oppositions to
Your preview ends here
Eager to read complete document? Join bartleby learn and gain access to the full version
- Access to all documents
- Unlimited textbook solutions
- 24/7 expert homework help
view Mars is Augest.
18. Make an "E" on your graph paper at the position that the Earth would occupy about Jan 1st every year.
19. Make an "O" at the position Mars would occupy if it were in opposition on Jan 1st.
20. Make a "C" at the position that Mars would occupy if it were in conjunction on Jan 1st. “Conjuction” is when a
planet-Sun-Earth line-up occurs, with the Sun between the planet and the Earth.
21. If you lived on Mars, Earth would be an inferior planet. Place an "I" at the two places that Earth could be located to
be at greatest elongation if viewed from Mars when Mars was located at position 1 (aphelion). Greatest elongation
occurs when an inferior planet makes the largest possible angle with respect to the Sun from the observer’s viewpoint.
NOTE: Attach Drawing and submit for grading!