Can't get Jool flyby velocity right

[Kulebron]

I try to calculate the velocity at Jool periapsis from velocity when I enter SOI, and periapsis height.

Here are the equaitons I use:

v² = μ(2/r - 1/a)

from which I get

a = μr/(2μ - rv²)

put the numbers in: μ=2,83E+014, r=2,46E+009, v=1600.

Then, periapsis, q=6140000 (6Mm radius + 140km).

eccentricity, e=1-a/q=20,749

v_q = sqrt[(-μ/a)(1+e)/(e-1)]= ... I get 1602 m/s. (current velocity is 1600.)

What am I doing wrong?

Source of formulae.

Edited December 16, 2013 by Kulebron

[Moar Boosters]

This works pretty well.

Don't ask me how he did the calculations.

http://forum.kerbalspaceprogram.com/threads/30433-A-handy-chart-for-Aerocapture-at-Jool

[Kulebron]

Don't ask me how he did the calculations.

Good material. But I'm interested in the calculation itself.

[Smidge204]

So you enter Jool's SOI at some velocity, which is now expressed as relative to Jool. You then travel to some minimal distance to Jool, and you want to know the velocity at that point?

We can apply conservation of energy. Most problems, truthfully, can be solved through energy if you do all the book keeping. Any other method is really just a shortcut

E = -G(m1*m2/r) + 0.5*m1*v^2 = Constant (assuming you don't spend any fuel)

m1 = Mass of your ship... doesn't matter, it cancels out

G*m2 = 2.8252E14 m^3/s^2

r1 (Entering) = 2.455E9 + 6E6 = 2.461E9 m

r2 (Lowest) = 6.14E6 m

v1 (Entering) = 1600 m/s in your case

v2 (Lowest) = What we're trying to solve

-2.8252E14/r1 + 0.5*v1^2 = -2.8252E14/r2 + 0.5*v2^2

v2 = 9446 m/s

Which certainly sounds reasonable to me.

=Smidge=

[Kulebron]

Yes, I checked my screenshots, the velocity was close to this.

Oh, I see, so you take -G(m1*m2/r) + 0.5*m1*v^2, and equate this for two states, that looks very smart.

μ/r₁ + v₁²/2 = μ/r₂ + v₂²/2

That's cool! Thanks a lot!

A used stage falling vertically.

Edited December 16, 2013 by Kulebron

[Smidge204]

No problem.

Because I'm feeling bored;

-G(m1*m2/r) is potential energy. The term is negative because it's considered "stored" in the fact that you are some distance r from the planet.

0.5*m1*v^2 is the kinetic energy of your ship. Note that normally the mass needs to be included, but in this case mass factors out of all the terms.

So the math is saying your total energy at one point is equal to your total energy at some other point. We consider only potential and kinetic energies here, but if you use your engines then you have to include the energy they contribute as well. Want to include atmospheric braking? Sure, you can express energy converted to heat as a function of various parameters (speed, air density, ship drag etc) and integrate all that as well... though in reality that math can get quite ugly.

=Smidge=

[Libertyacclimated]

Aint been to Jool yet... Perhaps that's my next go...

[Yasmy]

The vis-viva equation is all you need. You applied it correctly at one spot to find the semi-major axis: a = μr/(2μ - rv²)

Now apply it again at periapsis, by plugging the periapsis and semi-major axis in:

v_p² = μ(2/r_p - 1/a) = μ(2/r_p - (2μ - rv²)/μr) = v² + μ(2/r_p - 2/r)

Note that this is the same equation in your reply (post #5). You've shown that the vis-viva equation implies conservation of energy.

In the future you can calculate just about any orbital operation by applying the vis-viva equation.

Plugging in your numbers, I get v_p = 9722 m/s.

Edited December 17, 2013 by Yasmy