COULD STEP BY STEPS IN DETAIL AND NAME OF EQUATIONS BEING USED PLEASE BE ADDED I

Question

COULD STEP BY STEPS IN DETAIL AND NAME OF EQUATIONS BEING USED PLEASE BE ADDED

IT WOULD BE AWESOME IF IT CAN BE TYPED FOR BETTER UNDERSTANDING AS WELL,

THANK YOU!

P12: A planet with the mass M= 2 x 1026 kg orbits a massive star with mass 10 times the solar mass. At perihelion, it has a speed of 1.4 x 103 m/s and is 1.0 x 1015 m from the star. The orbital radius increases to 2.77 X 1015 m at aphelion. Note: Msz 2 X 1030 kg a) What is the planet's speed at aphelion? b) What is the period of the orbit? c) What is the energy of the orbit?

Explanation / Answer

(A) L = m v1 r1 = m v2 r2

(1.4 x 10^3) (1 x 10^15) = v (2.77 x 10^15)

v = 505 m/s

(b) a = ra + rp /2 = (1 x 10^15) + (2.77 x 10^15) / 2

a = 1.385 x 10^15 m

and T = 2pi sqrt[ a^3 / G M ]

G M = 6.67 x 10^-11 x 10 x 1.99 x 10^30

putting in,

T = 8.9 x 10^12 sec

(C) energy = - G M Mp / (1 x 10^15) + Mp (1.4 x10^3)^2 /2

= = - 6.95 x 10^31 J

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