Two moons, Luma and Nox

PHYSICS · Multi-part · A Level · Physics · Fields and their consequences · Fields and their consequences

Problem

Two moons, Luma and Nox, orbit different planets in nearly circular paths. The available observational data are • Luma: orbital radius RL=1.5×108 mR_L = 1.5\times10^{8}\,\text{m}, orbital period TL=1.20×106 sT_L = 1.20\times10^{6}\,\text{s} • Nox: orbital radius RN=2.8×108 mR_N = 2.8\times10^{8}\,\text{m}, orbital period TN=2.10×106 sT_N = 2.10\times10^{6}\,\text{s} Assume each moon’s mass is negligible compared with the mass of its planet. (a) Derive an expression for the ratio MNML\dfrac{M_N}{M_L} of the masses of the two planets in terms of the measured quantities RLR_L, TLT_L, RNR_N, and TNT_N. (b) Using the expression from part (a), calculate the numerical value of MNML\dfrac{M_N}{M_L}. Show your work clearly. (c) If the radius of Planet N is known to be twice the radius of Planet L, determine the ratio of their surface gravitational fields gNgL\dfrac{g_N}{g_L}. Assume both planets are spherical and have uniform density.

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