Orbital Period of the Earth where the Sun's Mass Changes

[Maik]

Hey,
this is going to 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

v²=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=1/2mv² -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 L²=m² u² a² = m² v² c² (where u is the initial velocity and v is the final velocity) and then putting the expression from the problem in as the v² 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!

 

[mfb]

I don't see how the angular momentum would help here. Consider v²=GM((2/r)-(1/a)). We know v and r of the Earth today satisfy this equation, and we know r=a as the orbit is supposed to be circular.
How does the equation look like directly after the Sun lost some mass? v and r will still be the same, but two other things change. You can write that as new equation. Set them equal to get rid of v and you should get the new semimajor axis a function of the old one and the mass loss.

 

[TSny]

Hello. Welcome to PF.

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 L²=m² u² a² = m² v² c² (where u is the initial velocity and v is the final velocity) and then putting the expression from the problem in as the v² and then simplifying it down using the initial velocity of u as squrt.(GM/a))

Your result c = a/(1-f) is not correct. I believe your mistake might be due to using the symbol "a" for two different distances. You use "a" to denote the distance of the planet from the star at the moment the star's mass changes. But you also use "a" do denote the semi-major axis in the formula v² = GM[(2/r)-(1/a)]. But the radius of the initial circular orbit does not equal the semi-major axis of the new elliptical orbit.

 

[Maik]

v and r will still be the same, but two other things change. You can write that as new equation. Set them equal to get rid of v and you should get the new semimajor axis a function of the old one and the mass loss.

Thank you so much, by considering that v and r stay the same I was able to solve it.
Just out of curiosity, how do I know that the v and r will stay the same even when the sun losses mass?

 

[Maik]

Your result c = a/(1-f) is not correct. I believe your mistake might be due to using the symbol "a" for two different distances. You use "a" to denote the distance of the planet from the star at the moment the star's mass changes. But you also use "a" do denote the semi-major axis in the formula v² = GM[(2/r)-(1/a)]. But the radius of the initial circular orbit does not equal the semi-major axis of the new elliptical orbit.

That's right, I just saw that in my working now, thank you for pointing it out to me.
I should really be more careful when it comes to things like this.
Thank you for your help!

 

[Janus]

Thank you so much, by considering that v and r stay the same I was able to solve it.
Just out of curiosity, how do I know that the v and r will stay the same even when the sun losses mass?

At the moment the Sun loses mass, the Earth is a certain distance from the Sun (r) and has a given velocity(v). The Sun losing mass will not change these properties at that moment, just the future trajectory of the Earth from that moment on.

 

[Maik]

At the moment the Sun loses mass, the Earth is a certain distance from the Sun (r) and has a given velocity(v). The Sun losing mass will not change these properties at that moment, just the future trajectory of the Earth from that moment on.

Ah I see that makes sense.
Thank you!