Orbit phase/True anomaly adjustment

[CaptainKipard]

Wow. I've spent half an hour searching google for a simple comprehensive answer to this question.

I have a satellite in a stationary orbit, but for reasons of science I need to change it's "position in the sky" so that it's sitting right above the KSC.

Now I'm positive I can do it with two burns by increasing my apoapsis, orbiting once, then circularising again at the periapsis, I just need a simple formula that takes the desired phase angle and gives me a delta-v for the first burn.

Thanks for helping.

Edited June 24, 2013 by Cpt. Kipard

[Mr Shifty]

If you, for instance, want to move your longitude 60 degrees westward, then you want to orbit in 7 hours instead of 6, correct?

T₂=T₁*((360 degrees - degrees change eastward)/360)

Since your orbital period is proportional to the 3/2 power of your semi-major axis, you want to increase your sma by (7/6)^2/3=1.108. If your orbit was circular at 2868.75 km altitude (=sma of 3468.75km because of Kerbin's radius), you'd want to increase your sma to 3844.18km, which gives you an altitude at apoapsis of 3019.6 km (=2*sma-Kerbin diameter-starting altitude of 2868.75 km). You can figure out the delta-V change by using the Vis-Viva equation:

Original velocity = (GM*(1/r))^1/2= (3.5316e12*(1/(2868750+600000))^1/2 = 1009 m/s

New velocity = (GM*(2/r-1/a))^1/2=(3.5316e12)*(1/(2868750+600000)-1/3844180)^1/2 = 1057 m/s

Delta-V = 48 m/s

[CaptainKipard]

That's great. I just have 3 questions:

1. What do you call the relationship between orbital period and the semi-major axis.

2. You first said 3/2 but then 2/3. Which is it?

3. I've look at the vis-viva equation and it looks like the one for new velocity. What do you call the equation above it?

[Mr Shifty]

1. What do you call the relationship between orbital period and the semi-major axis.

Kepler's Third Law

2. You first said 3/2 but then 2/3. Which is it?

The orbital period is the 3/2 power of the sma. I was calculating the ratio of the periods, so you have to invert it.

TâˆÂa^3/2 therefore aâˆÂT^2/3

3. I've look at the vis-viva equation and it looks like the one for new velocity. What do you call the equation above it?

It's Vis-Viva. If you have a circular orbit, the radius is equal to the semi-major axis, so 2/r-1/a is 2/r-1/r which is simply 1/r.

Edited June 24, 2013 by Mr Shifty

[Crown]

Very useful in this case are Wikipedia's articles about Kepler's laws of planetary motion and Orbital period. In case you want to do some maths. The rest is all about geometry. ;-)

[CaptainKipard]

Ok thanks, You both get a cookie.