Two planets orbit a much more massive star, as shown in the figure. One of the p

Question

Two planets orbit a much more massive star, as shown in the figure.  One of the planets has a mass m1 = 3.29

Two planets orbit a much more massive star, as shown in the figure. One of the planets has a mass m1 = 3.29 times 1024 kg and it moves in a circular orbit of radius r1 = 6.6930 times 1010 m with a period T1 = 1.0 yr. The other planet has a mass m2 = 2.77 times 1026 kg and it moves in an elliptical orbit, with its closest distance r1 = 6.6930 times 1010 m and its farthest distance r2 = 1.47 times 1011 m. Find the period T2 of planet m2's orbit.

Explanation / Answer

m1 = 3.29*10^24 kg, T1 = 1 year = 365*24*60*60 = 31536000 s

r1 = 6.693*10^10 m

let M is the mass of the star

for planet 1

a) T1 = 2*pi*r1^(3/2)/sqrt(G*M)

==> M = 4*pi^2*r^3/(G*T^2)

==> M = 1.808*10^28 kg

2) v1 = sqrt(G*M/r1) = 4.245*10^3 m/s

3) same as pnanet 1, v1 = sqrt(G*M/r1) = 4.245*10^3 m/s

4) let v2 is the speed at point A

angular momentum of the planet is conserved

m2*v2*r2 = m2*v1*r1

v2 = (r1/r2)*v1 = 1.933*10^3 m/s

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