Derive the equation for the shape of an orbit

[Cocoleia]

Homework Statement

I am given this equation:

where m and M are masses.
I have to obtain the following as a final result:

Homework Equations

The Attempt at a Solution

so far, I have done this (This may not be the most efficient or the easiest way, but it is how my professor wants it to be done)

I am stuck at what to do for the part where I wrote {steps}

 

[TSny]

There are several places that don't look right.

You wrote

. The right-hand side looks like a constant instead of being a function of $r$.

You wrote

followed by

. These don't agree.

In the expression

it is not clear if the -1 is an exponent on $r$. Note that $r - 1$ in the denominator cannot be correct since $r$ has dimensions but 1 does not. This might be causing the problems in filling in the {steps} to get the final expression for $r$.

You did not define what $p$ stands for. Is it just an arbitrary constant?

 

[Cocoleia]

There are several places that don't look right.

You wrote View attachment 108813. The right-hand side looks like a constant instead of being a function of $r$.

You wrote View attachment 108814 followed by View attachment 108815. These don't agree.

In the expression View attachment 108816 it is not clear if the -1 is an exponent on $r$. Note that $r - 1$ in the denominator cannot be correct since $r$ has dimensions but 1 does not. This might be causing the problems in filling in the {steps} to get the final expression for $r$.

You did not define what $p$ stands for. Is it just an arbitrary constant?

These are the exact steps that my prof wrote on the board, I have no idea, it's possible that it is all wrong. I believe that p is a constant. Usually when stuff isn't defined it's just a constant.

 

[TSny]

The right hand side of

doesn't look right.

To see what it should be, take the given answer

and solve for $\cos(\theta - \theta_0)$.

 

[mpresic]

Your notes are incorrect. You may have copied them incorrectly. You should probably consult the professor, or other students in the class to see if their notes agree with yours. If the professor really did right the equation for effective potential U (eff), (s)he is clearly wrong. U eff should be a function of r, in order to get the equation for the ellipse at the end.

You can also consult several textbooks, such as Symon, Mechanics, or Marion and Thornton, Mechanics of Particles and Systems. There are many steps that are skipped here. The actual argument runs several pages, and involves at least two changes of variable. The notes you have written are sparse. It could be your professor wants you to supply the missing steps using this outline as a guideline. I do think this would be a tough assignment.