The "ideal" launch vehicle

[OhioBob]

I recently created a computer simulation in which I can test the performance of various rocket designs. It is based on the stock KSP game, i.e. no mods. I decided to see if I could find the optimum design for a simple two-stage launch vehicle. I looked at several key parameters: thrust-to-weight ratio, propellant distribution, and attitude control. My goal was to maximize payload fraction. I performed dozens of iterations, tweaking these parameters until I could make no further improvements. I thought I'd share my findings. Below is what I found to be the optimum design:

Thrust-to-weight ratio: Stage 1 = 1.64, Stage 2 = 1.31.

Propellant distribution: Stage 1 = 67.7%, Stage 2 = 32.3%.

Attitude control: Pitch over to +87^O at altitude 5300 m; gradually increase pitch staying 3^O ahead of the surface velocity vector; pitch should be 0^O at Stage 2 first burn cutoff, with velocity vector at +3^O.

Payload fraction: 0.162

Orbit: 75,000 m

Of course this is all theoretical. The simulation isn't based on using real in-game components, though typical mass ratios were used. For instance, I assumed the tank dry mass is 1/8^th the mass of the propellant, which is consistent with most tanks found in the game. Similarly, engine mass was estimated as a function of thrust.

Due to the finite number of parts in the game, it is unlikely that these hypothetical ideal numbers can be matched exactly. However, these values can be used as a guideline when designing a launch vehicle. Being off a little bit shouldn't dramatically hurt performance. Conveniently the ideal propellant distribution is very close to a 2:1 ratio, which works nicely with the stock tank sizes. I've found that attitude control is particularly sensitive to TWR. (For instance, with higher TWR you may need to lead the velocity vector by a greater amount to increase the pitch rate.)

Obviously a simple two-stage design isn't going to work in many cases. Nonetheless, some of these basic guidelines might still apply to more complex configurations.

I'm interested to see how these numbers compare to what others have found. I've already seen others write that the optimum liftoff TWR is 1.5-1.7, so clearly my results agree with that part.

(edit to add) One important thing that I forgot to mention is that these numbers are based on insertion into a 75 km orbit.

Edited November 13, 2014 by OhioBob

[boosters++]

Nice! Can we see your code, or pseudo-code, to maybe better show us how you got these numbers?

[Francesco]

pitch should be 0° at Stage 2 first burn cutoff, with velocity vector at +3°.

(were those degrees?)

stage 2 first burn cutoff = coasting to apoapsis, right?

a velocity vector that low means you're very close to orbital speed at thrust cutoff;

a 1.3 stage 2 TWR, in my mind, correlates with 2000-2500 m/s of delta-v in the second stage,

which in turn means throttling back towards the end of the ascent to prevent reaching apoapsis too quickly.

the 2:1 fuel distribution seems a bit low, meaning the first stage should be bigger.

but that's just speculation - off to the VAB!

[Northstar1989]

(were those degrees?)

stage 2 first burn cutoff = coasting to apoapsis, right?

a velocity vector that low means you're very close to orbital speed at thrust cutoff;

a 1.3 stage 2 TWR, in my mind, correlates with 2000-2500 m/s of delta-v in the second stage,

which in turn means throttling back towards the end of the ascent to prevent reaching apoapsis too quickly.

the 2:1 fuel distribution seems a bit low, meaning the first stage should be bigger.

but that's just speculation - off to the VAB!

Uhhh....

No. You don't need to already have apoapsis at the desired height before pointing horizontal to the horizon.

Have you ever *tried* seeing what happens if you burn horizontal? Your apoapsis continues to increase, as long as you still have some upward momentum. Therefore, you do indeed want to be pointing completely horizontal quite a bit before you cut thrust and coast to apoapsis... (in practice, maybe a couple degrees above horizontal, since you DO NOT want your engines to burn *even slightly* downwards at any point during ascent)

Also, you *want* to reach apoapsis quickly. That reduces total gravity-drag during your ascent. I think what you *meant*, is that you don't want to OVERSHOOT your target apoapsis, which is true. The low TWR, and horizontal facing of the upper stage help prevent that though...

Regards,

Northstar

[Red Iron Crown]

I can't speak to your equations used, but 0.162 is a very respectable payload fraction for a two stage rocket lifter.

You might be interested in tavert's Mass Optimal Engine charts (a bit out of date now, but fascinating info if you're into optimizing).

tavert, please come back and update your charts.

[OhioBob]

(were those degrees?)

Yes

stage 2 first burn cutoff = coasting to apoapsis, right?

Correct. A second burn would have to be made at apoapsis to circularize the orbit. I forgot to mention in my original post that I'm assuming a 75 km orbit.

(a velocity vector that low means you're very close to orbital speed at thrust cutoff

Correct again. In practice I'm rarely able to get it to work out as perfectly as the theory. I'm usually going slower than optimal at first burn cutoff and then have to compensate with a fairly high delta-v burn at apoapsis. However, I find that, in theory, performance is optimized by nearly reaching orbital velocity at first burn cutoff and then making a low delta-v second burn (about 60 m/s).

a 1.3 stage 2 TWR, in my mind, correlates with 2000-2500 m/s of delta-v in the second stage,

which in turn means throttling back towards the end of the ascent to prevent reaching apoapsis too quickly.

If performed correctly, I don't think there is any need to throttle back. For the particular scenario that I simulated, cutoff would occur at an altitude of about 50 km and, with such a high horizontal velocity, you'd in a long arcing trajectory a good distance away from apoapsis. According to the simulation, first burn cutoff would occur 6.5 minutes before reaching apoapsis.

the 2:1 fuel distribution seems a bit low, meaning the first stage should be bigger.

but that's just speculation - off to the VAB!

I agree that's low for the real world. I was expecting it to be higher, like about 3:1. In the KSP universe, however, it didn't work out that way.

[OhioBob]

Nice! Can we see your code, or pseudo-code, to maybe better show us how you got these numbers?

Providing the code wouldn't be very practical, but I do provide a description of the basic method in an article I wrote about a my Saturn V simulation. If you're interested you can read it here: Saturn V Launch Simulation.

For the KSP simulation I simply had to modify everything to conform to the Kerbal universe. As I mentioned previously, the mass of all the rocket parts are based on typical ratios found within game. I primarily based it on a "large" sized rocket (Rockomax).

There are five parameters that I attempted to optimize: Stage 1 TWR, Stage 2 TWR, propellant mass distribution, pitch-to-velocity vector separation angle, and flight path angle at engine cutoff.

The pitch-to-velocity separation angle determines how rapidly the rocket pitches over. It is assumed constant throughout the pitch over phase. This angle, along with the initial pitch over altitude, determines the rocket attitude and flight path angle at cutoff.

I started out by assuming all five parameters and then determined the pitch over altitude and cutoff time needed to attain the correct cutoff conditions to reach my target 75 km orbit. It would take several iterations to find the payload mass that exactly exhausted all of the propellant. The next step would be to change one of the assumed parameters and find out in what direction I had to go to increase the payload fraction. I would continue to change the parameter until I reached a maximum on the payload fraction. I would then move to the next parameter and adjust it until I again reached a maximum. Then the next parameter and so. I would then return to the first parameter and go through another iteration of adjustments. I eventually reached a point where changing the numbers resulted in no further increase in payload fraction. The resulting numbers are those that I give in the opening post.

Edited November 13, 2014 by OhioBob

[OhioBob]

I can't speak to your equations used, but 0.162 is a very respectable payload fraction for a two stage rocket lifter.

That's what I thought. With the design philosophy I've used in the past I though I was doing pretty good to get a 0.12 payload fraction. I'm curious to see in practice how close I can get to 0.162. It's surely unrealistic to expect to reach the theoretical limit, but it should be possible to get much closer by implementing some of my findings.

[Rune]

Well, everything here looks very reasonable, and I really have little to add to the discussion since I haven't gone into any mathematical analysis to optimize my rockets. But I can tell you right now, I usually make bigger first stages, mostly so I can make do with lower T/W on the upper stage. It might not be efficient for the climb to orbit since you increase gravity losses slightly due to the increased ascent time, but the engine weight you save is very useful once you are in space and TWR is something you don't care about. Also note that those gravity losses are much lower than if you had a low first stage TWR, since by the time you light these smaller upper stages the effective gravity is much less.

Also, lately and due to the budgets and cost, I am getting hooked on 1.5 SRB booster staging. First, the SRBs are very cheap, and the big first stage engines are expensive, so that's good on that front, you can make do with a much smaller first stage engine that wouldn't be able to lift you off the ground, even. That usually works out to a slightly bigger T/W at liftoff, 1.7, going down after SRB separation to something like 1.4. With that I usually have between 3 and 3.5 km/s in the first stage+boosters, and then my upper stage can perfectly do its thing with T/W 0.8, plus it can be refuelled on orbit for greater in-space delta-v.

So yeah, the numbers you present are probably just perfect for a 75km orbit, if all you care about is minimum takeoff weight and maximum payload lifted. But if what you are minimizing cost... well, fuel is cheap, engines aren't. And if you are going to higher orbits things also change.

Rune. But I like the scientific approach!