dV Calculations

[0111narwhalz]

This has a fair bit of overlap with Gameplay; however, I think it is a general enough problem to place here.

I have a couple of problems I'd like to solve. They are as follows:
1. What is the minimum average acceleration needed to complete a maneuver as described?
2. Given a one-to-one function representing time with respect to dV (or the other way around, whichever is more convenient), the starting orbital characteristics, and the parent body's physical characteristics, and assuming that the optimal landing site is the desired one, what is the dV required for a soft landing?
3. See ②, only launching this time.

For clarification:
All problems should take place in a vacuum.

1. "As described" here meaning "with a minimum of divergence from the intended result orbit." In the most blunt case, if a maneuver which would result in a capture is executed in a way which leads to an escape, it was not completed as described.

2. I understand that the suicide burn is a Hard Problem. If this problem is similar enough to the suicide burn to be hard as well, simplifying assumptions such as a low orbit where the gravity well is essentially flat or even a constant acceleration are permitted.

Edited November 5, 2018 by 0111narwhalz
spherical cows in a vacuum

[K^2]

1. Infinite. Otherwise, you have to specify tolerances. And then the math gets really hard. But the gist of it is that "high enough to complete transfer before your anomaly changes drastically."

2. Optimal descent is not the suicide burn, except for the actual touchdown, which is almost trivial fraction of dV under right conditions. For an airless body, you first need to transfer to the orbit that has periapsis just above surface at the target sight. I'm not giving exact formula here, because this might involve inclination changes and bi-elliptic transfer. You can look up various transfer formulae. From there, the remaining dV is your periapsis velocity - surface velocity + g * T, where g is surface gravity, and T is time until landing. The later will depend on TWR, and is also not a trivial quantity. But T = v * m / F is a first order estimate, where v is velocity you need to kill, m is mass, and F is thrust. If your TWR is high, this will be fairly precise. If your TWR is only slightly above 1, this will be badly off.

If there is atmosphere, descent is either free or very hard math.

3. For an airless body, identical to descent. For a body with atmosphere, very hard math. A rough estimate penalty for ascent in atmosphere is 4gH/v_t, where v_t is terminal velocity at launch sight, and H is scale height. But this is a VERY rough estimate.

 

In general, simple formulas exist only to a tiny subset of problems. In majority of real scenarios, there are numerical methods that can give you desired solutions. But it's something you write a program for. Not something you just punch into a calculator.

[0111narwhalz]

1 minute ago, K^2 said:

If there is atmosphere, descent is either free or very hard math.

Apologies; I should've noted that all scenarios are to be completed in vacuum. I will edit the OP to clarify this.

2 minutes ago, K^2 said:

1. Infinite. Otherwise, you have to specify tolerances. And then the math gets really hard. But the gist of it is that "high enough to complete transfer before your anomaly changes drastically."

Is there a robust way to formalize your tolerances? And, once you have tolerances, just how hard does the math become?

3 minutes ago, K^2 said:

Optimal descent is not the suicide burn, except for the actual touchdown, which is almost trivial fraction of dV under right conditions.

Can losses from this be ignored, then?

4 minutes ago, K^2 said:

For an airless body, you first need to transfer to the orbit that has periapsis just above surface at the target sight. I'm not giving exact formula here, because this might involve inclination changes and bi-elliptic transfer. You can look up various transfer formulae.

I assume for a simple circular orbit relatively near the surface (a SMA within a factor of… eleven, was it? to the destination) already this is equal to just dropping your periapsis? That is, as I understand it, a relatively simple operation.

6 minutes ago, K^2 said:

From there, the remaining dV is your periapsis velocity - surface velocity + g * T, where g is surface gravity, and T is time until landing.

Of all the equations, I should've been able to figure this one out on my own.

6 minutes ago, K^2 said:

The later will depend on TWR, and is also not a trivial quantity. But T = v * m / F is a first order estimate, where v is velocity you need to kill, m is mass, and F is thrust. If your TWR is high, this will be fairly precise. If your TWR is only slightly above 1, this will be badly off.

Is there a relatively simple way to approximate how off it is, even if only for a small patch of TWR values? Is there a minimum TWR for an approximation within, say, 1% of actuality?

[K^2]

2 hours ago, 0111narwhalz said:

Is there a robust way to formalize your tolerances? And, once you have tolerances, just how hard does the math become?

In principle? Yes. You can work out the relevant cross-section of the target, convert it into a volume in the space of orbital elements, and then see where different burn times put you in that volume. There's probably a first-order formula in there somewhere, but it's still not going to be pretty, and I wouldn't want to try and work it out by hand.

In contrast, consider throwing a computer at it. Just simulate transfers with different burn times, and see whether you still hit the target. Way better precision and much easier to implement.

2 hours ago, 0111narwhalz said:

Can losses from this be ignored, then?

If you aren't planning to hover, yeah. If you're going in the way Apollo 11 did, without knowing if your LZ is safe, and having to traverse the surface looking for alternative, then it becomes significant. The key is that once you are well bellow orbital speed, the amount of fuel you need is just proportional to amount of time you're staying off the ground. It doesn't matter if you're in level hover, or slowly descending, or even going up and down. Gravity transfers momentum to your ship at a constant rate, and it has to be dumped as engine exhaust or you won't like the landing. An extra second of operation is probably negligible compared to your total. If you plan to extend your descent by a minute, you'll have to have a minute-worth of hover of fuel. Or precisely g * T in dV budget.

2 hours ago, 0111narwhalz said:

I assume for a simple circular orbit relatively near the surface (a SMA within a factor of… eleven, was it? to the destination) already this is equal to just dropping your periapsis? That is, as I understand it, a relatively simple operation.

Yeah, if you're well-positioned, you just drop periapsis over LZ. The dV is just the difference between orbital velocity of your original orbit - sqrt(μ/r), and the speed at apoapsis of the eliptical orbit - sqrt(2μ/(R+r)) * sqrt(R/r). Here r is your orbit's radius, and R is body's radius.

2 hours ago, 0111narwhalz said:

Is there a relatively simple way to approximate how off it is, even if only for a small patch of TWR values? Is there a minimum TWR for an approximation within, say, 1% of actuality?

I'm sure there's a number, but I don't see any way of deriving it algebraically. The integral is quite ugly. But again, it's no trouble integrating numerically for any given body. The math for actually computing dV from low circular orbit to landing can be done in an Excel sheet, if you don't want to take time writing actual code for some reason.

[wumpus]

I'm confused.  KSP uses fairly basic calculations to determine each of these burns.  Unfortunately, to keep things simple they are based on some "delta* function thrust" which has infinite thrust for an infinitely short period of time (that integrates to some unit delta-v/kg) and you have to burn some factor of these delta-thrusts.

I'd hate to try to calculate the error involved thanks to realistic  thrust.  Note that if you can use a Mangalyaan manuever (typically called "pe kicking" in KSP) you can get arbitrarily close to minimally wasted delta-v, assuming you can keep your pulses arbitrarily short.  Unfortunately this doesn't help at all for kicks above escape velocity (and obviously the original Mangalyaan plan had to make the last kick hard enough to go from a high Earth orbit to Mars Transfer).

Note that any additional mass required for the last delta-v burn will propagate back through all earlier burns requiring more delta-v and more mass.  Expect some really ugly equations (if they are even linear).  This is the type of thing that really gets done on a computer simulation.

* sorry, but the "delta function" has nothing to do with "delta-v" but was created by Paul Dirac to simply some things.  It is also critical in transformations such as Fourier and Laplace transformations.  It is a function that is zero everywhere but 0, infinite at 1, and integrates to 1 for any integral containing 0.

[0111narwhalz]

8 hours ago, K^2 said:

consider throwing a computer at it.

I do intend to throw a computer at it, yes.

8 hours ago, K^2 said:

You can work out the relevant cross-section of the target, convert it into a volume in the space of orbital elements, and then see where different burn times put you in that volume.

I think this is probably enough information for me to do what I want to do. Thanks.

8 hours ago, K^2 said:

I'm sure there's a number, but I don't see any way of deriving it algebraically. The integral is quite ugly. But again, it's no trouble integrating numerically for any given body.

Could I use

8 hours ago, K^2 said:

Gravity transfers momentum to your ship at a constant rate

to iterate by first finding the minimum dV, then the time to burn it, then the extra dV from gravity losses, and loop that? I'd think it would converge eventually, and I'll probably add 1% or so on top anyways.

I might break the instantaneous maneuver into a series of e.g. ten-second burns and see how well that approximates the described maneuver. Maybe I could even optimize such a sequence to match the orbital parameters after the burn with the ones after the instantaneous maneuver, though that's probably out of scope for now.

[K^2]

4 hours ago, 0111narwhalz said:

I do intend to throw a computer at it, yes.

I've been vague about numerical methods, because there are just a whole lot of methods and techniques that can be relevant, depending on approach, so it'd take me hours to do a proper write up. But if you'll have specific questions, I can probably refer you to specific algorithms that might be helpful.