HW#36 Ch 13: Gravitation
Instructions: Be sure to write formulas and show substitutions to earn full credit.
1. The following table shows a list of the orbital radius (semi-major axis) and orbital period of several Neptunian moons.
a) Fill in the table with the quantities indicated.
| Moon | a (km) | T (days) | a3 (m3) | T2 (s2) |
|---|---|---|---|---|
| Naiad | 48,227 | 0.294 | ||
| Thalassa | 50,074 | 0.311 | ||
| Despina | 52,526 | 0.335 | ||
| Galatea | 61,953 | 0.429 | ||
| Larissa | 73,548 | 0.555 |
b) Derive Kepler’s third law relating orbital period to orbital radius.
c) Using Excel, plot the quantities indicated above and fit a trendline or best-fit line. Make sure you label the X and Y axes, including the appropriate units.
d) From the slope of the line, determine the mass of Neptune. Make sure to format the trendline label to show 3 decimal places to avoid rounding errors. Show all work.
2. A satellite of mass m has an elliptical orbit around a planet of mass M. The orbit is characterized by an eccentricity e = 0.5 and semi-major axis a. What is the speed of the satellite in terms of G, m, M and a when it is at the coordinate \(\left(\dfrac{2a}{3},\sqrt{\dfrac{5}{12}}a\right)\)? Use conservation of energy to solve this problem, and assume that the planet is located at \(\left(ae,0\right)\).
3. The center of our galaxy is located in the Sagittarius constellation, a distance of 26,000 light years from Earth. At the center, there is a strong radio source named Sagittarius A*, which is thought to be a supermassive black hole. A star S2 of 15 solar masses orbits Sagittarius A* with a period of 15.78 years, a semi-major axis of 1004 astronomical units (1 A.U. = 1.496 x 1011 m) and an eccentricity e = 0.87. a) Estimate the mass of Sagittarius A* in kg, neglecting the mass of S2. b) Determine the equivalent mass in terms of solar masses by dividing by the sun’s mass. c) If the speed at the perihelion (closest approach) is represented by \({v_p} = \sqrt {\dfrac{{GM\left( {1 + e} \right)}}{{a\left( {1 - e} \right)}}} \), where M is the mass of the black hole, a is the semi-major axis and e is the eccentricity, determine S2’s perihelion speed as a percentage of the speed of light.