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this is gonna be my first post here so I'm not sure how it all works, so just tell me if I do something out of order please. Anyway I have been given this homework assignment and part of it was the question stated below.

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The speed (v) of a planet in orbit around a star of mass M is related to its orbital distance (r) and orbital semimajor axis (a) by

v^{2}=GM((2/r)-(1/a))

Use this equation to show that if the Sun instantly lost a fraction f of its mass, reducing its mass from M to M(1 − f), then the Earth’s (originally circular) orbit would have a period of

T =((1 − f)/(1 − 2f))^{3/2}years

[You may assume Kepler’s Third Law: orbital period is proportional to orbital semimajor

axis to the power 3/2.]

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The Equations that I am supposed to use are stated in the problem, however I also considered the equation for the angular momentum and energy relating the kinetic energy and the gravitational potential energy:

L=mvr and E=1/2mv^{2}-GMm/r

I started off by trying to find an expression for a and the using that in Keplars 3rd rule, however rather than simplifying the whole expression to what they asked it made it more of a mess. I then attempted to solve it for the point at which r=a (initially) and then the same for the new orbit. By then considering the conservation of angular momentum I found an expression on the new a (calling it c) as follows: c= a/(1-f) ( by saying that L^{2}=m^{2}u^{2}a^{2}= m^{2}v^{2}c^{2}(where u is the initial velocity and v is the final velocity) and then putting the expression from the problem in as the v^{2}and then simplifying it down using the initial velocity of u as squrt.(GM/a)) I then put this in keplars 3rd rule but with no success.

I could have made some mistake here or forgot to consider something but I just can not get my head around it.

It would be great if someone could just hint me where I could have gone wrong or what I should/can consider to stay the same during the mass loss.

Thanks a lot in advance already!

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# Homework Help: Orbital Period of the Earth where the Sun's Mass Changes

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