m and 2m respectively, orbiting the same star in circular orbits, the more massive planet is 9.3 times as far from the star as the less massive one.
What is the ratio v1:v2 of the speeds of the 2 planets?
What is the ratio T1:T2 of the orbital speeds of the 2 planets?
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f/F = 2m/(9.3r)^2//m/r^2 = 2(r/9.3r)^2 = 2/9.3^2 = 0.023 = g/G where f = force from 2m at R = 9.3r and F = force from m at R = r. ANS.
We get this by noting the gravitational fields G ~ m/r^2 and g ~ 2m/(9.3r)^2 are proportional to the forces exerted by the planets. A fact we use next.
g/G = .023 = V^2/9.3r//U^2/r = a/A where A and a are the radial accelerations A = U^2/r and a = V^2/(9.3r) So we can write V/U = sqrt(.023*9.3) = 0.462 where V is for 2m at R = 9.3r and U is for m at R = r. ANS
Finally V/U = .462 = w 9.3r/W r = (2pi/T) 9.3r//(2pi/t) r = (t/T) 9.3; so that t/T = .462/9.3 = 0.0497 where w is the angular velocity and T is the period for 2m at 9.3r and W is the angular velocity and t is the period for m at r. ANS.
Note, as you failed to specify which variables went with which planet, I explicitly specified my ratio variables so you can identify them with the planets m and 2m. My ratios might be upside down from what you were looking for; in which case just invert my answers.
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