How Would a Halved Sun Mass Alter Earth's Orbit?

[AdkinsJr]

Homework Statement

If the mass of the sun were 1/2 it's current mass suddenly show the orbit of the Earth would be a parabola...

The Attempt at a Solution

I'm not sure what kind of approach to apply here, if the total energy equals zero then the orbit is a parabolic trajectory, if it is less than zero the the orbit is circular or elliptical, if I plug in numbers for the mass of the earth, sun, the radius and the average speed I can find escape velocity,

$$E=\frac{1}{2}mv^2 -\frac{GMm}{r}$$

and solve for the velocity $$v=\sqrt\frac{GM}{r}$$

and show that the Earth's average speed exceeds this value, but since these numbers are going to be approximations it's difficult to tell whether it should be parabola or bound orbit...E will not be exactly zero...

 

[mfb]

and show that the Earth's average speed exceeds this value, but since these numbers are going to be approximations it's difficult to tell whether it should be parabola or bound orbit...E will not be exactly zero...

You can do it analytically, assuming Earth has a perfectly circular orbit* you can express the radius as function of the velocity (or vice versa) and calculate the total energy afterwards.

*otherwise the whole statement is not true anyway

 

[AdkinsJr]

That makes sense, by radius I presume you mean the orbital equations solns. Yeah I think that will simplify things down a bit. $$r(\theta)=\frac{\alpha}{1+e cos(\theta)}$$...for a circle e=o so $$r_c=\frac{ml^2}{k}$$ where $$l=L/m$$ and $$k=GMm$$

$$r_c=\frac{m(v_cr_c)^2}{GMm}$$ if I solve for $$V_c$$ I get $$v_c^2=\frac{GM}{r}$$

If I plug this into energy equation it does actually work when I plug in 1/2 M for the mass of the sun in the potential energy term.