Homework 8 due Friday, March 5

1. A general rule for superior planets is that the greater the average distance from the planet to the Sun, the more frequently that planet will be at opposition. Explain how this rule comes about.

   Look at the diagram below. Note that a planet that is very far away from the sun moves very little in one year.Therefore, the time between oppositions is close to a year. The planet that is close to the earth has moved significantly in a year. The earth has to go further to catch up to the planet



2. A certain asteroid is 2 AU from the Sun at perihelion and 6 AU from the Sun at aphelion.

a). Find the semimajor axis of the asteroid's orbit.

The semi-major axis is half the major axis of 6 AU + 2 AU. So the semimajor axis is 4 AU.

b). Find the sidereal period of the orbit.

P²= (1 yr²/1 AU³) * (4 AU)³

P = 8 years

3. Suppose that you traveled to a planet with 3 times the mass and 3 times the diameter of the Earth. Would you weigh more of less on that planet than on Earth? By what factor? Show your work.

Your weight is the gravitional force of the planet on you.

F_{planet on you} = G M_planetm_you/(R_{you planet})²

Now the distance between you and the planet is essentially the radius of the planet. So for this case

F_{planet on you} = G (3M_Earthm_you)/((3 M_earth)²=(1/3) F_{earth on you}

Your weight on this planet is one third of what your weight is on earth.

4. How much greater is the Earth's gravitational force on the Moon than the Moon's gravitational force on the Earth?

   They are equal, by Newton's third law.

5. A satellite in earth orbit experiences a small drag force as it starts to enter the earth's atmosphere What happens to its speed? Explain your answer.

   v = 2*p*D/P

   where D is the distance from the satellite to the center of the earth, and P is the period of the satellite. Now P²=k'D³, so

   v = 2*p*D/(k'D)^3/2=

   2*p*D/(k^3/2*D^1/2.

   Note that as D decreases, v increases. So the satellite speeds up as it spirals down to earth.