How to calculate DV for closing a hyper/para-bolic orbit?

[CaptainKipard]

My kOS circularisation script right now only works for circularising from a smaller orbit, and I'd like to be able to e.g. circularise after a Mun capture.

Thanks.

Edited May 13, 2014 by Cpt. Kipard

[Claw]

If you're dealing with the orbital energy, then you might be able to back calculate the dV required using the Oberth equations. You should be able to calculate the escape velocity, then figure out how much excess energy you have. The Oberth equations should allow you to figure out the dV required at PE burn to cancel that out.

[CaptainKipard]

I don't know how to do that. I've only ever used the vis-viva equation.

Edited May 11, 2014 by Cpt. Kipard

[Claw]

Are you trying to calculate it for an arbitrary point along the orbit? Or at the PE?

[CaptainKipard]

PE, yes. That should make it simpler I think.

[Claw]

Hmm. I will have to think about this a bit. I'm certain there's a simpler equation you can use. It might be as simple as V = Vescape + dV for all speed in excess of Vescape, but there's a bit more to it than that.

[Starstrider42]

I'm pretty sure v_peri = sqrt(μ/a) × sqrt((1+e)/(1-e)) applies even to a hyperbolic orbit. So then the delta-V needed is sqrt(μ/a) × [sqrt((1+e)/(1-e)) - sqrt(1/(1-e))].

Edited May 11, 2014 by Starstrider42
Stupid math error, final orbit at a(1-e)

[maccollo]

Somebody tell me if I'm getting this wrong.

The velocity you want to end up with is that of a circular orbit: sqrt(μ/r)

Your specific kinetic energy in a circular orbit will then be: μ/r/2

Let's say this is Ke1

You can then subtract this from the specific kinetic energy ((Ke=v^2)/2) you will have at PE.

Let's say this is Ke2

So the deltaV for your burn will then be sqrt 2Ke2 - sqrt 2Ke1

Or simply velocity-velocity circular. I like to use specific kinetic energy because it's easier to work with energy than velocity =P.

If you don't know what the velocity will have at periapsis, but you know your current velocity, your PE radius and your current radius, you can derive it from the potential gravitational energy, μ/r, and your current specific kinetic energy.

potential gravitational energy is the energy you would gain in falling to the specified radius from →∞

The specific kinetic energy you would have at periapsis is then μ/r1-μ/r2+Ke

Where r1 is your Pe radius, r2 is your current radius and Ke is your current specific kinetic energy.

Edited May 11, 2014 by maccollo

[Yasmy]

Basically just echoing maccollo here.

It looks like kOS can tell you your current altitude and velocity and your periapsis altitude.

Your periapsis velocity, given your current velocity and orbital radius is sqrt(v^2 + mu (2/rp - 2/r)).

Subtract off circular orbit velocity sqrt(mu/r).

If kOS works in altitude, which appears to be the case, don't forget to add the body radius to the altitudes to get orbital radius at current location and periapsis.

[CaptainKipard]

It would be a lot more helpful if I knew what all of that meant. An equation by itself isn't very helpful unless you know the name of those laws, theories, etc.

E.g. I know vis-viva, so I can look up related equations, or derive my own depending on my needs.

Thanks

Edited May 11, 2014 by Cpt. Kipard

[maccollo]

Well, to sum it up it means is that when you fall a certain height through a gravity well you will always gain the same amount of kinetic energy, which you can get with the gravitational potential.

Therefore, if you know your current altitude, your current speed and the gravitational parameter of the planet you can calculate how fast you will be moving at any given altitude.

Just take your current specific kinetic energy

Add the delta in gravitational potential energy

Revert it back to velocity

Subtract the velocity of a circular orbit

Edited May 11, 2014 by maccollo

[Claw]

Yeah, I tend to agree. You'll have to look at the orbital energies. I have been thinking of a simplistic equation, but you probably have to start with the energy equations and simplify to what you want for your script. What maccollo said in that last post.

[Yasmy]

It would be a lot more helpful if I knew what all of that meant. An equation by itself isn't very helpful unless you know the name of those laws, theories, etc.

E.g. I know vis-viva, so I can look up related equations, or derive my own depending on my needs.

Thanks

maccollo and I are basically both just using the vis-viva equation: v^2 = mu (2/r - 1/a),

though he stated in conservation of energy form: KE + PE = v^2/2 - mu/r = constant.

1) Use the vis-viva equation at any point inside the Mun's SOI to find your semi-major axis: 1/a = 2/r - v^2 / mu

r = current distance from center of Mun (or whatever) = current altitude + body radius (assuming altitude is measured from surface)

v = current velocity

a = semi-major axis

mu = gravitational parameter = G * M = Newton's constant times the mass of the body.

2) Substitute the semi-major axis from step 1 into the vis-viva equation at periapsis (rp) to find your periapsis velocity vp: vp = sqrt(mu (2/rp - 1/a)) = sqrt(mu (2/rp - 2/r) + v^2)

Where ever you are in orbit, this calculation tells you how fast you will be going at periapsis.

r = current distance from the center of the Mun

v = current velocity

rp = periapsis distance from center of the Mun

vp = velocity at periapsis

3) Then to get your delta-v for circularization at periapsis, just subtract off the velocity of a circular orbit from your periapsis velocity:

Again, from the vis-viva equation, the velocity of a circular orbit at radius rp is v = sqrt(mu(2/rp - 1/rp)) = sqrt(mu/rp)

4) dv = sqrt(mu (2/rp - 2/r) + v^2) - sqrt(mu/rp)

kOS can tell you v, r, rp and M. I think you have to supply G. Just make sure that r and rp are relative to the center of the Mun, rather than to the surface.

Clear?

Edited May 12, 2014 by Yasmy

[CaptainKipard]

clear?

Sir, Yes Sir!!!