The rules: Collaborative work is encouraged. This homework can be done in consultation with your fellow classmates, the AI, or the professor. However, everyone must submit their own solutions to get credit, and all help should be acknowledged (a single sentence mentioning the others in your study group is sufficient). Show your work.
1. Just the facts [5 pts]
(a) [2 pts] At opposition, the angle between the Sun-Earth-Planet (elongation) is _____ degrees.
(b) [3 pts] Sketch the orientation of the Sun-Moon-Earth system for a solar eclipse and for a lunar eclipse.
2. 'Round and 'round we go [29 pts]
(a) [4 pts] Imagine that you are observing the Earth from Jupiter. What
would you observe the Earth's synodic orbital period to be? What would
it be from Venus?
(b) [4 pts] A satellite which always remains over the exact same
spot on earth is in a geosynchronous orbit. Calculated
from the center of the earth, what is the radius of such a
satellite's orbit?
(c) [4 pts] The international space station is currently located at an average height of 392.3 km above the Earth's surface. Calculate
its orbital period and its mean speed (velocity) assuming a circular orbit.
(d) [9 pts] (i) What is the semi-major axis of the least energy elliptical orbit of a space probe from Earth to Venus? (ii) Relative to
the Earth, what is the velocity of such a probe at the Earth's orbit? (iii) When the probe reaches Venus, what is its velocity relative to
that planet?
(e) [8 pts] What would be the correct launch speed for a least-energy orbit to Mars? How long would it take a probe to reach Mars
from the Earth?
3. Weight-y matters [16 pts]
(a) [4 pts] Suppose you travelled to a planet whose mass is the
same as Earth's but whose diameter is twice as large.
Would you weigh more or less on that planet than on
Earth? By what factor?
(b) [4 pts] Titan orbits Saturn at a radius of 1.22186 x 106 km
with an orbital period of 15.945 days. It has a bulk density
of 1880 kg/m3 and a radius of 2575 km. Using this information,
derive the mass of Saturn (show your work!).
(c) [4 pts] Using the orbital data for the Earth found in the back of the book,
estimate the mass of the Sun. Does the mass of the Earth matter
significantly in this calculation?
(d) [4 pts] Compute the mutual gravitational force between you and
(i) another person with mass 100 kg located 1 m from you;
(ii) Mars at opposition. Which is larger?