Answered: Planet X with a radius of 4.0 Rearth⊕…

Physics for Scientists and Engineers: Foundations and Connections

1st Edition

ISBN:9781133939146

Author:Katz, Debora M.

Publisher:Katz, Debora M.

Chapter7: Gravity

Section: Chapter Questions

Problem 67PQ

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Question

Planet X with a radius of 4.0 Rearth⊕ is found orbiting a star of mass 0.80M⊙ . The eccentricity of its orbit is 0.05, and at its farthest approach to the star, it is 0.63 AU away, while at its closest approach to the star, it is 0.57 AU away.

a. How long does it take Planet X to make a complete orbit around its star in years?

b.The semi major axis of the planet is changed such that its closest approach to the star is now 2.9 AU. Would the gravitational force exerted by the star on Planet X increase or decrease? By what factor? Would the gravitational force exerted by Planet X on the star increase or decrease? By what factor?

c. The mass of the star was increased to 2.4M⊙ . Would the gravitational force exerted by it on Planet X increase or decrease? By what factor? Would the gravitational force exerted by Planet X on the star increase or decrease? By what factor?

Expert Solution

Step 1

$G i v e n : R a d i u s o f t h e p l a n e t, R_{x} = 4 R_{e} (R_{e} = R a d i u s o f e a r t h) m a s s o f t h e c e n t r a l s t a r, M_{s} = 0.80 M_{s u n} e c c e n t r i c i t y o f t h e o r b i t, e = 0.05 c l o s e s t a p p r o a c h o f p l a n e t t o t h e s t a r, r_{1} = 0.57 A U f a r t h e s t a p p r o a c h o f p l a n e t o t h e s t a r, r_{2} = 0.63 A U A s k e d : a . O r b i t a l p e r i o d, T = ? b . i f s e m i - m a j o r a x i s i s c h a n g e d s u c h t h a t, r_{1}' = 2.9 A U, t h e n b y w h a t f a c t o r s w i l l p l a n e t a n d s t a r e x e r t g r a v i t a t i o n a l f o r c e o n e a c h o t h e r . c . I f m a s s o f s t a r i s i n c r e a s e d, M_{s}' = 2.4 M_{s u n}, t h e n b y w h a t f a c t o r s w i l l p l a n e t a n d s t a r e x e r t g r a v i t a t i o n a l f o r c e o n e a c h o t h e r .$

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