Calculating perigee on changing SOI

[Voxels]

Hi all,

I'm currently planning a Voyager II style flyby of Duna and Jool (similar orbital inclination), and was wondering how you would calculate the perigee of the spacecraft once it has transferred SOI.

Thanks all,

Voxels

[AmpsterMan]

I wouldn't know how to calculate it to be honest. However, if you play with the Maneuver Nodes you should be able to see where the next SOI path will be; the Perihelion should show up there.

[Voxels]

Thanks for the reply- the thing is that I want to calculate the mission before launch, because I enjoy calculating stuff before I launch, as well as allowing me to work out the correct launch window for a dual flyby.

[Smidge204]

Since all the planets are "on rails" you can use Kepler orbit equations and they should be spot-on. All the information you need to define the planet's orbits and locations is available on the wiki and in the game.

Then it's just a matter of solving a bunch of 2-body problems, with SOI determining which planetary body you use.

Might want to write some software - or at least a spreadsheet - to do the math for you... 'cause there's gonna be a lot of it.

=Smidge=

[Yasmy]

I will assume you mean after transferring from Kerbol SOI to outer planetary SOI, since going from Kerbin to Kerbol SOI,

you are pretty much already at your Kerbol periapsis (if you exited Kerbin's SOI reasonably prograde.)

The standard answer to your question is that you don't. The game does it for you.

Changing your periapsis at the target planet takes a tiny bit of delta-v when you are far away,

if you already have an encounter. It won't be part of typical delta-v calculations

or mission planning, other than "now I should adjust my encounter periapsis."

But I enjoy all things nerdy, and if you wish to calculate everything, read my next reply...

Edited January 2, 2014 by Yasmy
Reducing negativity.

[Yasmy]

Honestly, I don't think that the question you asked is really one you need to ask for mission planning.

Like I said in the previous post, periapsis adjustment at the target, while you are far from the target SOI, is cheap.

Generally all you need to do any orbital calculations is the vis-viva equation.

I won't go over how you use it to calculate delta-v for going from one body to another here. There are tutorials around for that.

Or check out en.wikipedia.org/wiki/Hohmann_transfer_orbit.

But I will go ahead and answer your original question anyway.

First you need to know the velocity (both speed and direction) of your spacecraft and the target planet at the SOI boundary.

Next, when entering a planetary body's SOI, you have to subtract the body's orbital velocity from the spacecraft velocity,

to get the spacecraft velocity relative to the planet. Using the planet's average orbital speed will not work unless the planet

is in a perfectly circular, zero inclination orbit. You will have to calculate the planet's trajectory from its orbital elements.

Now the question becomes: Given a velocity v and radius r of an orbit, what is the periapsis p?

The standard way to approach such a calculation is to use conservation of energy and conservation of angular momentum.

Instead I'll use the vis-viva equation (which has conservation of energy implicit in it) and conservation of angular momentum.

Use the vis-viva equation at the SOI boundary to find your semi-major axis: 1/a = 2/r - v²/mu.

Note that r is just the SOI radius.

Use the vis-viva equation again at periapsis: v_p² = mu(2/p - 1/a) = mu(2/p - 2/r) + v²

Now use conservation of angular momentum to eliminate the periapsis velocity, v_p:

At periapsis, the magnitude of the angular momentum is just L = m v_pp,

since at periapsis, the velocity is perpendicular to the radius.

At the SOI boundary, the magnitude of the angular momentum is L = m v r sin(theta),

where theta is the angle between the line from the planet to the spacecraft and the velocity vector.

Conservation of angular momentum says these two values are equal: m v_pp = m v r sin(theta).

Substitute v_p = v r sin(theta) / p into the previous equation: v_p² = mu(2/p - 2/r) + v²

(v r sin(theta) / p)² = mu(2/p - 2/r) + v².

Now multiply both sides by p² and you get a quadratic equation for the periapsis p.

Solve the quadratic equation. There are two roots: the periapsis and the apoapsis.

Since you are on a hyperbolic orbit, the apoapsis will be meaningless. It should be obvious which root is the periasis.

(Hint: It's the positive root.)

Now the important thing to note in all of this, ie, the reason why I said this is pretty much useless to almost everyone,

is that tiny changes in velocity while you are far form the target planet's SOI will have a huge effect on the velocity

at the SOI. Specifically, it will have a huge effect on the sin(theta) term, the angle between the velocity and the line

from the planet to the spacecraft.

Edited January 2, 2014 by Yasmy
Typos