How to calculate DeltaV to get to orbit or to get from orbit to land

[tavert]

Maybe it is all about keeping gravity perpendicular to the velocity vector.

Ooh, I like that interpretation, I wasn't thinking quite in those terms but that makes sense. If you apply the right amount of vertical thrust, then any magnitude of horizontal velocity can be equivalent to a circular orbit around an effectively-lower-gravity body (or higher, if you're going too fast and thrusting down instead of up).

As far as starting the burn a bit before periapsis, it might make sense to go from transfer orbit to a circular with a pure retrograde burn, and then switch to constant altitude. One more thing to try, I guess. But damn, these thrust equations are starting to get complicated.

That's about what I had in mind, since in my test case the transfer-to-circular burn was tiny.

I can see that asking for a given non-trivial trajectory could result in a mess. That's why I was thinking it could make sense to just go straight to an optimization formulation, where dynamics are implicit equality constraints. Simulation of a given trajectory is then just a special case of optimization with more constraints (and not worrying about the cost function). More computationally expensive, but maybe less coding effort in the end.

Edited July 10, 2013 by tavert

[K^2]

You can't use standard variation techniques to optimize the trajectory. Any trajectory where full thrust is applied is locally optimal. (Locally, I can go to coordinate system that's free-falling and x axis is aligned with direction of thrust. Then T = m x'', and Euler-Lagrange equation is the trivial 0 = 0.)

Global variation is very difficult, because it's not clear what the boundary condition should be, and how to optimize the overall trajectory. The only thing that even comes to mind is just doing numerical optimization. I was having same difficulties trying to find optimal gravity turn for the ascent through atmosphere.

Speaking of which, I was thinking of take-off instead of landing. Neglecting mass changes, they are time-reversals of each other. If we simply assume that we should be diverting as much thrust as possible to horizontal acceleration, we get constant-altitude right until the point that the circular orbit at zero altitude is achieved. Thereafter, it's just a matter of going to a transfer orbit. So it really does sound like optimal landing will include a switch to constant altitude only after you circularize.

Of course, this only opens up the question of whether a direct pro/retrograde burn is the optimal way to transfer/circularize.

[tavert]

Yeah variation techniques aren't too useful in practical engineering situations, HJB equations are hard. I meant numerical. Non-convexity means we only get local optima unless you also do spatial branch-and-bound, which takes forever. But good initial seeds should spit out better trajectories. And if you're doing forward simulation as a special case of optimization, then you don't have any degrees of freedom to optimize, it's just an implicit feasibility problem.

Yes, this applies to takeoff as well just by switching the initial and final conditions.

Finite-thrust transfer problem could also be an interesting one...

Edited July 10, 2013 by tavert

[K^2]

Heh. I wanted to see if I can at least get an analytic solution to constant altitude takeoff/landing. No dice. The solution can be written in terms of elliptic integrals, which means an infinite series is the closest you can get to a closed form.

[tavert]

Is that what it ends up being? Last month I got to HSpeed'(t) = -sqrt(Thrust^2 - Mass(t)^2 * (Mu - (Altitude + BodyRadius) * HSpeed(t)^2)^2 / (Altitude + BodyRadius)^4) / Mass(t), where Mass(t) = InitialMass - Thrust * t / (g0 * Isp), but didn't try change of variables or anything fancy.

[K^2]

I was actually assuming a constant mass, but since m'(t) is a constant under constant thrust, I doubt it'd make that much difference. And yeah, after giving up trying to solve it by hand, I ran it through Mathematica.

[kalor]

So I know this is an old thread... but I'm curious, in more simple terms (Calc is a bit over my head ) as to whether or not a straight retro burn, or the pitch-up varying thrust method would be more efficient? Or, as another option, the answer of "situation dependent", for atmosphere vs. non-atmospheric, orbital path, etc, etc, changes that answer? I'm currently working on a slightly unconventional Mun lander design where I REALLY need to conserve all the delta-v I can, and this may help significantly.

Of course, if anyone has a more recent thread that I didn't find...

[tavert]

If there's an atmosphere, trajectory optimization is tricky and craft-dependent.

If there's no atmosphere, the constant-altitude landing method (stay at full throttle until the very end, just control vertical speed using pitch) is more efficient than a retrograde suicide burn. The difference depends on TWR, it's more noticeable at lower TWR.

[NASAHireMe]

Sorry to re-open an old thread, but the math in here fascinates me. I can follow half of it, but the other half is either over my head, or I've forgotten how to do it. Just curious, tavert and K^2, how much math background do you have? What courses (i.e. Multivariable, Linear Algebra, etc.) taught you all the stuff to derive these equations? Where'd you learn, and in what course did you learn, how to use Mathematica and Matlab and LaTeX?

[rockbloodystar]

Hi,

I wish to dredge this up.

I am trying to calculate lift to particualr orbit delta-v requirements for most bodies. To test whether i was using the formulae correctly i checked the given orbits and delta-v requirements against this chart: http://wiki.kerbalspaceprogram.com/w/images/7/73/KerbinDeltaVMap.png

For the Moho, Dres and Eeloo, (where ther is not atmosphere) my calcuations require considerably less delta-v for the launch.

For Moho i calculate 891, the chart says 870.

For Dres I calculate 385, the chart says 430.

For Eeloo I calculate 541, the chart says 620.

For Moho (as an example) I am using the following method:

Radius is 250 000m

Surface Velocity is 1.2982

GM (Mu) is 1.6860938*10^11

Target radius is 50 000m (as in map)

v1 = orbital velocity for altitude of 0m = SQRT(GM/Moho's Radius)-surface velocity

= -1.2982+SQRT(1.69e11/250000) = 820

Dv1 = Hohmann calculation = SQRT(1.69e11/250000)*((SQRT((2*300000)/(250000+300000)))-1)=37

Dv2 = Circularisation butn = SQRT(1.69e11/250000)*(1-(SQRT((2*250000)/(250000+300000))))=35

Total = v1 + Dv1 + Dv2 = 891

Why are these so far off?

Am i using thhe incorrect equations, or out of date raw data?

Cheers,

RBS.

[bigcalm]

I'd like to reawaken this thread - as I'm not sure I understand the full range of maths I need to be able to do these calculations and I'm hoping someone can help. Ultimately the goal is be able to calculate the kind of lander I need to take for any body (which can detach itself from an orbiting space station, land, take off, and redock -- and possibly be carrying some additional weight in the form of ore).

I've created a google spreadsheet here: https://docs.google.com/spreadsheets/d/1WaJEZKcyvqzXa0oAx1tEmOYDgXKqiiwEDDSF4xuW-3k/edit?usp=sharing which attempts to allow calculations for Hohmann transfers, change of orbit height, landing and taking off each body. The numbers I get normally when compared with testing are normally within 10% or so. However, occasionally I've been quite out (my Mun landing was so out it's hard to blame MechJeb).

My calculation for "fighting gravity" is simply wrong - can anyone help?

[r3_141592654]

On 7/7/2013 at 8:13 PM, K^2 said:

Finally, the atmosphere. The simple formula to estimate additional delta-V to leave atmosphere is 4gH/v_t. Here, g is surface gravity, H is scale height of the atmosphere, and v_t is the surface terminal velocity. For Kerbin that works out to 1,832m/s. This assumes TWR of roughly 2 of your rocket, which optimizes fuel consumption on ascent.

What exactly is surface terminal velocity? Working this statement backwards: v_t=4gH/dv = 4(9.81)(70,000)/1832 = 1500. I have not been able to match up the 1500 m/s with any data about Kerbin. I've been able to get all other values for dv except for the atmospheric drag portion here. Without atmosphere, Kerbin would have dv of 2398 m/s to 80km circular orbit. Per all of the charts and diagrams the actual value is 3400 m/s so I am missing about 1000 m/s of dv according to the charts. Lastly, I have been unable to find the above equation in any of the documentation thus far and so I am wondering where the dv=4gH/v_t equation comes from. Sorry for reviving an old thread.

[K^2]

14 hours ago, r3_141592654 said:

What exactly is surface terminal velocity? Working this statement backwards: v_t=4gH/dv = 4(9.81)(70,000)/1832 = 1500. I have not been able to match up the 1500 m/s with any data about Kerbin. I've been able to get all other values for dv except for the atmospheric drag portion here. Without atmosphere, Kerbin would have dv of 2398 m/s to 80km circular orbit. Per all of the charts and diagrams the actual value is 3400 m/s so I am missing about 1000 m/s of dv according to the charts. Lastly, I have been unable to find the above equation in any of the documentation thus far and so I am wondering where the dv=4gH/v_t equation comes from. Sorry for reviving an old thread.

H is the scale height, which for Kerbin used to be exactly 5km until the aerodynamics overhaul. Likewise, terminal velocity at sea level was pretty much fixed at just over 100m/s unless you used a very large quantity of certain parts that would alter drag. With new aerodynamics, scale height varies with temperature, averaging about 5.6km, and terminal velocity will depend on your rocket's construction. But this can still be used for a ball park estimate.

Edit: As for the origin, I've derived this equation by integrating vertical ascent at optimal velocity in constant gravity and exponentially thinning atmosphere. That model was surprisingly accurate for the way aerodynamics in the game used to work and is good enough for a ballpark guess with more realistic atmosphere.

Edited January 14, 2022 by K^2

[Prashnat]

How to calculate delta V for Mars 5 sol orbit to the martian surface landing ?