2 687 in scientific notation

Astronomy Lab

The Orbit of Mars

Big Idea Tycho Brahe made a number of observations of the positions of Mars during the latter part of the 16th 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'

Name: ______________________ Collaborator(s): ______________________

Date: ______________________ 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.

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. 5th 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

Name: ______________________ Collaborator(s): ______________________

Date: ______________________

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 Earth and/or Mars were off and you’ll need to try again). This line is called the major axis of the orbit.

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 miles. 1AU is 93 million miles, so multiply your answer from step 10 by 93: _________________million miles

12. Express your answer in scientific notation: _________________ miles

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:

𝑎! = 𝑃!

…which means we can solve for the period P like this:

𝑃 = 𝑎!

13. Using the above formula, calculate the orbital period of Mars in years:

___________! = ________𝑦𝑒𝑎𝑟𝑠

Name: ______________________ Collaborator(s): ______________________

Date: ______________________ Part 4: Eccentricity Kepler believed that the orbits of the planets were divinely constructed, and therefore must be perfectly circular. Using your compass, draw a circle centered on the Sun with a radius equal to your measurement of Mars’ semimajor axis in step 8.

14. Do the positions of Mars (P1, P2, etc.,) line up on your circle? ________________

Despite what Kepler wanted to believe, his data showed him that the orbits of Mars and the other planets were not circular, but elliptical with the Sun at one focus.

Eccentricity, e, is a number that tells us how elliptical an ellipse is. For example a perfectly circular orbit would have an eccentricity of zero and a flattened out orbit 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 (marked in step 7) divided by the length of the semimajor axis. You do not need to convert from centimeters to AU or miles before dividing.

15. 𝑒!"#!!!"#$% = _______________ / _____________ = ______________

Determine the accuracy of your measurement: The known eccentricity of Mars’ orbit is e = 0.09. How close is your value? Calculate the percent error in your result:

16. % 𝑒𝑟𝑟𝑜𝑟 = !"#$" !!!"#$ ! !"#$" !

𝑥 100 = !.!"! ___________ !.!"

𝑥 100 = _____________%

Part 5: Mythbusting Mars A friend sends you the following email:

On August 27 at 00:30, lift up your eyes and look up at the night sky. On this night, the planet Mars will pass just 34.65 million miles from the earth. To the naked eye it will be twice the size of the full Moon! The next time Mars will be so close to the Earth as much as in 2287. Share the news with your friends, because no one living on this earth has ever seen this!

Using your plot of Mars’ orbit, what is the closest Mars and Earth could possibly get to one another (for this question, let’s assume that Earth’s orbit is perfectly circular) in miles? Express your answer in scientific notation.

17. Minimum distance between Earth and Mars _______________________ miles. Show your work:

18. Does Mars ever get to within 34.65 𝑥 10! miles of Earth? _____________

Name: ______________________ Collaborator(s): ______________________

Date: ______________________ Let’s investigate the claim that Mars will be twice the size of the full Moon at its closest approach.

On August 27, 2003, Mars made the closest approach to Earth in recorded history due to a near synchronization of Earth being at aphelion (furthest orbital point from the sun) and Mars being at perihelion (closest orbital point from the sun). The distance between the planets that day was a mere 55.8 million km.

Mars has a known diameter of 6790 km, which is about ½ the diameter of Earth, but still much larger than the Moon (3,475 km).

19. Let’s use the small angle formula to calculate the angular diameter, ∅ of Mars on this date. Express your answer in scientific notation:

∅!"#$%#&'($ = 206265 𝑑!"#$%&%' 𝐷!"#$%&'(

= 206265 _____________________

_____________________

= ______________________𝑎𝑟𝑐𝑠𝑒𝑐𝑜𝑛𝑑𝑠

Convert your answer in arcseconds to degrees. Recall that:

1 𝑑𝑒𝑔𝑟𝑒𝑒 = 60 𝑎𝑟𝑐 𝑚𝑖𝑛𝑢𝑡𝑒𝑠 = 3600 𝑠𝑒𝑐𝑜𝑛𝑑𝑠

20. Mars’ maximum angular diameter: ___________________________ degrees (use scientific notation)

21. The angular diameter of the full Moon is about ½ a degree in the sky. Does this value appear to

be larger or smaller than the full Moon? _________________

22. Given the email above, do you agree or disagree with the claim? Explain why or why not:

Name: ______________________ Collaborator(s): ______________________

Date: ______________________