A planet revolves around a star in an elliptical orbit. At perihelion it has dis

Question

A planet revolves around a star in an elliptical orbit. At perihelion it has distance 7.8 times 10^10 m and speed 5.7 times 10^4 m/s and at aphelion it has distance 7.8 times 10^11 m and speed 5.7 times 10^3 m/s. (a) What is the mass of the star? (b) What is the planet's speed when it is a distance of 2.5 times 10^11 m from the star? What is the angle between the radial vector and the velocity vector at this point? Sketch the orbit and indicate the approximate position and radial and velocity vectors at this point.

Explanation / Answer

(10) VP / Va = 1 + e / (1 - e)

5.7 x 10^4 / 5.7 x 10^3 = 1 + e / (1 - e)

10 - 10e = 1 + e

e = 0.818

ra = a(1 + e) => a = ra / 1 + e = ra / 1.818

v^2 = G M [ 2/r - 1/a]

5.7 x 10^3 )^2 = (6.67 x 10^-11) (M) [ 2 / (7.8 x 10^11) - 1.818 / (7.8 x 10^11)]

M = 2.09 x 10^30 kg

(b)

now at r = 2.5 x 10^11

v^2 = (6.67 x 10^11) (2.09 x 10^30) [ 2 / 2.5 x 10^11 - 1.818 /(7.8 x 10^11)]

v = 2.81 x 10^4 m/s

angular momentum is conserved.

m v r = constant

v r = constant

(7.8 x 10^10 x 5.7 x 10^4) (sin90) = (2.5 x 10^11) (2.81 x 10^4) sin(theta)

sin(theta ) = 0.633

theta = 39.2 deg

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