Question

Suppose the Earth had two Moons instead of one, with the second moon (let’s call it Althea) orbiting in a 2:1 resonance inside of the Moon’s orbit. This means Althea orbits twice for each lunar sidereal period of 27.3 days. The mass and radius of Althea are Ma=5.34*10 ˆ 21 kg and Ra= 869 km.

(a) (1 pt) Draw and label a diagram showing ‘from above’, the possible configuration(s) of Earth and its two moons when tides on Earth would be the strongest.

(b) (2 pts) Find the orbital distance of Althea from Earth, in km.

(c) (2 pts) Which appears larger in Earth’s sky, Moon or Althea? (If you didn’t get an answer for part (b), assume r = 2.40*10 ˆ 5 km.)

(d) (3 pts) Tidal forces of a body (1) on another body (2) have a magnitude equal to Ftidal =4GM₁M₂R₂/r^3 where M₁ and M₂ are the masses of the two bodies, R₂ is radius of body (2), and r is their physical separation. What is the relative strength of Althea’s tides on Earth, compared to the Moon’s tides on Earth?

(e) (3 pts) If Althea is in a retrograde orbit (clockwise from above), tidal friction would cause it to slowly spiral inwards. Standing on Earth’s surface, what would Althea’s apparent angular size be just before it is ripped apart by tidal forces? (Recall that the Roche limit is rc = R_A(2M_E/M_A )^1/3. How does this compare to the angular size of the Sun?

Answer 1