What are the perihelion and aphelion speeds of Mercury?.
The following assignment covers Chapters One to Three. Please answer the following questions and submit your work
using the assignment link. Show all your work in a clear and concise fashion, and include each step you take to reach
the answer. The way in which you obtain your answer is as important as the final answer, itself, and marks will be
given for showing all the relevant steps. Please refer to the appendix in the textbook (pages 392–395) to obtain any
needed planetary values, and to Table 2-2 on page 41 for any needed constants.
Submit your answers to your tutor for grading and feedback using the assignment link. You may submit your
assignment (a) typed as a word-processed document, or (b) hand-written and scanned as a PDF. If you opt to handwrite
your assignment, be sure to write legibly. Always keep a backup copy of your assignment.
Note: Assignments must be submitted as .doc, .docx, or .pdf files.
Each question has its point value marked in bold, for a total of 85 marks on this assignment.
1. [12 marks total]
(a) [4] Use Keplers first law to derive explicit expressions for the perihelion and aphelion distances for a planet
in terms of the semi-major axis a, and the eccentricity e, for an elliptical orbit. Calculate these distances for
Mercury.
(b) [4] What are the perihelion and aphelion speeds of Mercury?
(c) [4] Calculate the maximum angular separation of Mercury as seen from Earth.
(d) [4] What is the angular size of Mercury in arcseconds when it is at this location?
2. [9 marks total] Problem 19 in Chapter Three of the text (page 72):
(a) [4] Consider two particles in circular orbits in Saturns rings 108 m from Saturns center. One is located 1
m farther from Saturn than the other. By Keplers laws, they have different periods and must occasionally
pass each other. How fast do they pass by each other? (Hint: You could compute Vcirc for each particle and
subtract, but the difference in orbits is only one part in 108
, so you would have to maintain accuracy to at least
eight decimal places, which is unrealistic. Instead, use calculus and differentiate Vcirc = (GM)
1/2
(r
-1/2
)
with respect to getting
dVcirc = difference in velocity
dVcirc = 1/2 (GM)
1/2
r
-3/2dr
where dr is the difference in distance.
Note that the text has chosen to ignore the negative sign for this question, which indicates the direction of the
change in velocity.
(b) [3] What if the particles are about 200 km apart, as in the example of Figure 3-17a? Compare your result
with the 60 m/s figure given there (based on a more detailed calculation including gravitational attraction of
the particles for each other).
(c) [2] How fast would a shuttle in a 200-km-high circular orbit approach a space lab in a 201-km-high circular
orbit? (Answer: about 0.6 m/s.)
What are the perihelion and aphelion speeds of Mercury?
