Callisto orbits Jupiter at an average distance of 1.88 times 10^6 km with an orb

Question

Callisto orbits Jupiter at an average distance of 1.88 times 10^6 km with an orbital period of 0.0457 yr. Neptune, which is one of the satellites of the Sun orbits its parent at an average distance of 4.50 times 10^9 km with an orbital period of 164.8 yr. (a) Use the above information to find the orbital speeds of Callisto around Jupiter and of Neptune around the Sun. M = (b) What is the expression for the mass M of the parent in terms of the orbital speed v of the satellite, the orbital radius R of the satellite and the gravitational constant G? (Do not substitute numerical values; use variables only.) M = (c) Now use your answers from parts (a) and (b) to find the ratio of the mass of the Sun to that of Jupiter. M_S/M_J =

Explanation / Answer

Let R1=1.88×106 km

       R2=4.50×109 km

Part a.

Orbital speed of callisto around Jupiter.

Vcallisto=2(pi)R1/T=(2×3.14×1.88×109)/(0.0457×365.25×24×3600)=8186.48 m/s

Orbital speed of Neptune around Sun.

VNeptune=2(pi)R2/T=(2×3.14×4.50×1012)/(164.8×365.25×24×3600)=5433.89 m/s

Part b.

Centripetal force =gravitational force

mv2/R=GMm/R2

M=Rv2/G

Part c.

For callisto around jupitor.

MJ=R1 Vcallisto2/G

For Neptune around Sun

Ms=R2 VNeptune2/G

So Ms/ MJ= R2 VNeptune2/ R1 Vcallisto2=1054.59

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