Stumped on an orbital mechanics problem...

[GoSlash27]

I'm trying to work out a formula for deriving the eccentricity of an orbit from mu, an object's velocity, and the object's altitude. I'm not having any luck so far...

As an example:

I have a craft in low Kerbin orbit. r= 670 km and mu= 3.532x10^12. Vorb= 2,295.8 m/sec. If I add 10% of Vorb so that V=2525.4 and r= 670 km, What is the resultant apoapsis and what is the velocity at apoapsis?

This seems like it should be a very easy problem, but it's giving me fits.

How do I go about solving this?

Best,
-Slashy

[Dman979]

I saw that you had posted in a topic called "Stumped on an orbital mechanics problem" and figured that you had posted an insightful answer. I'm scared, now, because you're asking the question.

I'd suggest emailing a physics professor at a local college. They like to help people out.

[Padishar]

Assuming that you are starting in a circular orbit and you are basically asking what is the Ap speed of an orbit with Pe of 670Km and speed at Pe of 2525.4 then all you need is the Vis-Viva equation.  GM is the same as mu so you should be able to rearrange that to give a = ... and then plug in the Pe radius and Pe speed.  The a is the semi-major axis of the resulting orbit so the Ap satisfies a = (Pe + Ap) / 2

To work out the speed at Ap, just plug in the value for a and the Ap height for r into the original equation...

Edited April 5, 2016 by Padishar

[GoSlash27]

Padishar,

 That's how I would normally do it, but unfortunately the vis-viva is insufficient for this problem. Neither the apoapsis nor the eccentricity of the orbit are given in this problem.

Best,
-Slashy

[Steel]

Using vis-viva you can work out the semi-major axis of the orbit. Assuming r is the periapsis, you can rearrange the equation r_min = a(1-e) for e. you can then use r_max = a(1+e) to get apoapsis and vis-viva again for velocity.

*This is an over-compicated solution, see my next post*

Edited April 5, 2016 by Steel

[GoSlash27]

I'll try that. Thanks!

-Slashy

[Steel]

24 minutes ago, Steel said:

Using vis-viva you can work out the semi-major axis of the orbit. Assuming r is the periapsis, you can rearrange the equation r_min = a(1-e) for e. you can then use r_max = a(1+e) to get apoapsis and vis-viva again for velocity.

Just realised there's a little over-complication in my answer. Instead of using the equations for r_min and r_max you can just do as @Padishar said and use a = (Pe + Ap) / 2 to get the apoapsis. You only need the r_min and r_max equations if you want a number for the eccentricity.

Edited April 5, 2016 by Steel

[GoSlash27]

Steel, That got it for me. ru/(2u-v^2r)= SMA. That gives me something to hang my hat on!

Thanks!

-Slashy

[GoSlash27]

Yeah, that definitely sorted it out. Given v,r, and u, the velocity at apoapsis is:

Vap= (2vru-v^3r^2)/v^2r^2

simplified;;;

2u/vr-v= Vap

Thanks again!

-Slashy

Edited April 5, 2016 by GoSlash27

[PB666]

1 hour ago, GoSlash27 said:

Yeah, that definitely sorted it out. Given v,r, and u, the velocity at apoapsis is:

Vap= (2vru-v^3r^2)/v^2r^2

simplified;;;

2u/vr-v= Vap

Thanks again!

-Slashy

e = (r_apo - r_pe) / 2a    v = sqrt(u)*sqrt(2/r - 1/a)  vis visa equation.

e = 0.11728

a = 759018 r_ap= 848036.2967

solved using the 2525.4 = SQRT(3.53E12)*SQRT(2/670000-1/a)

2525.4 = 1879255.172 * SQRT(1/335000 - 1/a) ::Insert here because GUI screwed up on cut and paste::::: 1.80588E-06 = 2.98507E-06 - 1/a ::::::::::: 1/a = 1.17919E-06