Help with some math: delta-v to actual velocity relationship

[Rosco P. Coltrane]

Hello. So I'm missing something here and don't what what it is, but I'm sure is something stupid.

What I'd like to do is calculate the dV needed to go from one orbit to another around the same body. Using the Vis-Viva equation I know the speeds I'll have at different altitudes in my (circular) orbit. So my guts tell me there must be some way of calculating the dV that would take me from one velocity to the other, hence from one orbit to the other... but how exactly does one go about calculating that?

So for instance I'm in circular orbit some 2.6 Gm above this planet Sarnus, (yes, Gm), both the Vis-Viva and the navball say I'm traveling at 176 m/s. If I where to lower my orbit to 315 km, then per the equation the speed would be some 3800 m/s. So how much dv would I need to do it? Performing the maneuver myself gives me some 162 at Ap and 1564 at Pe burns to completely circularize at 315 km, so 1726 m/s of dV spent. I just don't know how to get to that number mathematically.

Any help?

Thanks.

[andrewas]

You don't go directly from one circular orbit to another, you go into an intermediate elliptical orbit first. So, calculate the difference in velocity between your starting orbit and the transfer orbit, and the same at the other end, sum the two, and you have your overall deltaV budget.

[Rosco P. Coltrane]

Ha, I told you it was going to be something stupid.

Thanks a lot, andrewas.

[K^2]

You need vis-viva and angular momentum conservation to do transfer calculations. If you apply these to apsides of the elliptical orbit, you get the equation v_p = sqrt(GM/a) * sqrt(r_a/r_p). The formula for v_a is the same, with r_a and r_p switched around.

Between two co-planar circular orbits the simplest and usually (but not always) most efficient transfer is the Hohmann, which simply connects the two orbits with an elliptical one. The above formula is all you need to compute the dV cost of such transfer. In general, the most efficient transfer between two circular orbits is the bi-elliptic transfer, which requires some numerical work to compute on case-by-case basis.

[PB666]

Use the formula K-2 provided with a couple of amendments.

Hohman transfers consist of 2 burns, the first i abbreviate H1 and second H2

It is slightly better if possible to choose a H1 that is radial to the apogee on the target orbit so that the intercept is on the periapsis of the target orbit (H2 is generally 180' from optimal H1). This may require several passes of H1 theta to approximate _or_ simply set H1 at next pass and wait until the opportune moment(s) to collapse the ellipse (H2) successively (over many orbits, takes practice). H2 does not have to be a single burn, it can be broken up into several burns and these successive burns can be accomplished by simply firing at near exactly H2. With proper planning partial H2 burns can collapse the orbit within a km of the target. If you are doing transfers from altitudes that have huge (yearly or more) orbital periods to small orbital periods (monthly), then its best to let the closer body approximate but follow future position on H2 (meaning you might deviate slight from optimal H2), this means that the first partial H2 will bring orbital period very close to the target orbital period so the wait for the intercept burn will be minimal.

The other amendment is that you may also need additional dV to correct inclination, and the amount of fuel varies depending on whether it AN/DN nodes are close to or far from the optimal transfer point. if the AN/DN have a theta ~90' from H2 then you might want to shift transfer points to AN/DN alter both in one vector burn and then once you reach perigee begin collapsing the orbit to match the Theta of your target.

BTW don't forget to also include RCS for precision intercepting. Use RCS fuel ® and (N or H) at the end of burns to precisely adjust your altitude.

Some useful info:

Ax = (Pex + Rc) + (Apx + Rc) where Rc = radius of central body if central body is SOI center.

ax = Ax/2

There are three a to think of

a1 = 0.5 * A1= semimajor axis of starting orbit

a2 = " " of orbit between H1 and H2

a3 = " " of target orbit.

If optimal transfer is used then

two pe

pe1 = pe of starting orbit

pe2 = pe3= Pe of target orbit

three apo

apo1 = starting orbits apo

apo2 = close to point of H2

apo3 = target orbit apo

This give you all the information then to use the formula K-2 provided except apo2.

e1 = (Apo1-Pe1)/A1

(apo2 + Rc) ~ a x (1 - e1^2)/(1 + e1 cosine theta) where theta is the angle of H2 relative to theta of pe1

a2 ~ ((apo2 + Rc) + (Pe3 + Rc))/2

with this you can calculate all the dV. If SOI center is different form central bodies center, then you are going then it will not be precise.

Edited March 1, 2015 by PB666

[PB666]

I should probably also add.

your can derive H2 r because at the point of the burn you would know v (at least in KSP) mu = Central bodies gravitational coefficient

v1® = (mu * (2/r - 1/a1))^0.5

v2(Apo2) = (mu * (2/(Apo2+ Rc) - 1/a2))^0.5

v2(Pe2/3) = (mu * (2/(Pe2 + Rc) - 1/a2))^0.5

v3(pe3) = (mu * (2/(Pe3 + Rc) - 1/a3))^0.5

dV(H1) = v1® - V2(Apo2)

dV(H2) = v2(Pe2) - V3(pe3)

dV = dV(H1) + dV(H2)

[Rosco P. Coltrane]

I actually have pretty much everything covered. The only thing I don't really have is the dV for the inclination change, precisely because the point where you perform it will vary on a case-by-case basis. So my "theoretical situation" approach goes out the window. I can tell you how much dV you'll need to go from X altitude to Y altitude with a decent approximation, but inclination... nope. I could go the route of "the best/worst place to do it is A, you'll need B m/s at that point"... but still.

[Kibble]

I actually have pretty much everything covered. The only thing I don't really have is the dV for the inclination change, precisely because the point where you perform it will vary on a case-by-case basis. So my "theoretical situation" approach goes out the window. I can tell you how much dV you'll need to go from X altitude to Y altitude with a decent approximation, but inclination... nope. I could go the route of "the best/worst place to do it is A, you'll need B m/s at that point"... but still.

Delta vee for an inclination change is related to your orbital speed, and can be imagined as a triangle with your current vector and your desired vector. The third line is the change in velocity. For example a change of 60 degrees in inclination would create an equilateral triangle, and so the required delta vee is equal to your current orbital speed.

[K^2]

I'm pretty sure Rosco knows how to compute dV for inclination change at a given altitude. He seems to be interested in finding best altitude to do so.

I can't help you much. Every time I need optimal bi-elliptic, I end up writing a script in Mathematica to compute dV given altitude of intermediate node, and then optimize that value numerically. Given the monstrosity that is the general form of that dV equation, I don't think there is an analytic solution. If you don't have access to Mathematica, this works equally well in Matlab or its free cousin Octave.

I'm starting to think that we might just need to set up a community pool of various KSP-related Octave scripts.

[Bill Phil]

If it's related to velocity, then the highest altitude possible would be the best, right?

[Kibble]

If it's related to velocity, then the highest altitude possible would be the best, right?

Yes. I guess the question is balancing the Oberth effect with the potential savings from high orbital altitude and inclination. Which is why EML-2 is such a great place, you get loads of both!

[K^2]

If it's related to velocity, then the highest altitude possible would be the best, right?

No, because if your final altitude is low, you are wasting a ton of dV on that transfer to high altitude, which might actually be greater than savings on inclination change. For very small inclination changes it's often most efficient to do inclination change at your initial or final orbit (whichever is higher) and do a standard Hohmann in between. For very large inclination changes, it's best to go to the edge of SOI or Hill Sphere, do the inclination change there, and then return to your target altitude. For everything in between, there's the sweet spot somewhere above the higher of the two orbits and you need to do an honest bi-elliptic optimization.

Also, the limits of how high is "very high" and how low is "very low" is entirely up to the parameters of initial and final orbits.

[PB666]

I'm starting to think that we might just need to set up a community pool of various KSP-related Octave scripts.

There needs to be a repository of useful orbital algorithms. I have an excel sheet using VB scripts where I can enter coordinate and SOI center based velocity vectors and derive the ellipse, but VB is on the wane because of the virus issues. Wikipedia is not the place to look for formula, they are so scattered about willy-nilly, and the formula generally cannot be cut and pasted into anything.

I have a whole bunch of stuff in Excel (nicely tabled) that I can DL, but it assumes they users have access to spreadsheets. And of couse any scripting has to VB, and I haven't migrated to 2010 yet so its its only going to be compatible version<=2007.

[Rosco P. Coltrane]

Well, my exact problem right now is like this:

I'm working on a dV map for the Outer Planets Mod, which adds new planets to KSP. Now listed on that map, along with the dV you need to go there, dV to capture, circularize, take off, etc. there's the dV you need to do the inclination change for the transfer from Kerbin to the other planet.

Now, you (or at least I) would do the inc. change at the An/Dn, but where exactly that An/Dn is, differers from window to window. Because you don't always launch at the very best window possible, you launch at the next one. Planets in this mod also are very far away so waiting for a window takes a long time, so the first acceptable phase angle you get, you launch. As a result, different launches will end up with An/Dn in very different places, sometimes they are close to Kerbin, sometimes they are farther away.

What I've done is use dV = 2*v*sin(ang/2) to calculate said dV. v here being the velocity I have when I do the change and "ang" the angle I'm aiming for. The problem is of course picking a place to do the maneuver, hence v. Also I'm not too what value should I pick for the angle.

But in any case, my head hurts.