Optimal launch trajectory + rocket staging

[Kethas]

Hi all. I've been poking around in KSP on and off for a couple weeks now, and two things have started to bug me. The following seem like they really impact the delta-V of your rocket (or, rather, the portion of your rocket that reaches orbit) and deserve rigorous answers, but I don't know how to address them.

1) Launch trajectory: for a given rocket, what's the flight path (all the way from launch to circularized orbit) that requires the least delta-V?

I strongly suspect that the optimal trajectory keeps the direction of thrust exactly prograde (to minimize steering losses), starts at a small offset from vertical at launch (so that we actually end up in orbit, despite point #1), and burns continuously until orbit is reached (to minimize time spent in freefall pre-orbit, and thus gravitational losses). I haven't a clue where to start designing the relevant equations, though, so I can't produce a checklist like "at 10km altitude be oriented at 80 degrees, at 20 be oriented at 65 degrees," etc.

Correcting for air resistance presumably makes the path "curvier" (i.e. you aim closer to vertical early on, to get out of the atmosphere earlier, and then lean over on your side more quickly).

I did find this Naval Research Laboratory "primer" from the late 70s that's talking about the same stuff, but the equations are illegible.

2) Staging: given fixed total and payload masses and a fixed number of stages (or a fixed set of KSP parts to make your rocket with), what's the optimal rocket staging?

Devote too much fuel (and delta-V, and mass) to your early stages, and you waste fuel accelerating partially-empty tanks and other infrastructure that you can't jettison; make the opposite mistake and you waste fuel accelerating other fuel.

Currently I'm just staging my rockets in a roughly geometric progression (e.g. two tanks + one thruster, then six tanks + three thrusters, then 18 + 9), but I have no evidence that that's the efficient way to do it.

I found this more general description of the problem (great site, by the way) and this paper on mathematically modeling it, but the former only briefly touches on the relevant factors instead of solving them (and eventually devolves into guess-and-check) and the latter (which, I admit, isn't exactly light reading for a Monday morning) only seems to describe how total required rocket mass for a fixed payload mass decreases as the number of stages increases, not how to lay out the actual stages.

Has anyone tackled these problems? If not, how would I start?

I'm also having worlds of trouble keeping aircraft from crashing without a joystick, but that's for another thread. Thanks!

[EndlessWaves]

I'm also having worlds of trouble keeping aircraft from crashing without a joystick, but that's for another thread. Thanks!

It's likely an unstable design rather than a lack of joystick causing that.

I'd imagine it's been discussed before but if you want to build a model yourself you'll need to collect up the relevant equations (atmospheric drag, fuel use rates, gravity etc.) as well as working out the equation or quantity you want to optimise (just fuel use, or are other factors like number of parts important?) and then mash it all together and work out the algebra.

The wiki has some information: http://kspwiki.nexisonline.net/wiki/Kerbin

[Kethas]

From the Atomic Rocket site linked in the OP:

"In the special case where each stage has the same inert-mass fractions and the same specific impulse, they will also have the same ÃŽâ€v fraction. The fraction will be 1 / nstage. For example, if there are four stages, each stage will contribute 1/4 = 0.25 = 25% of the total ÃŽâ€v."

That's a good approximation for a launch vehicle with several homogenous rockets, but can't account for e.g. higher-specific-impulse SRBs or smaller, more efficient rockets for orbital maneuvers in later stages.

[Zephram Kerman]

These two challenge threads thoroughly dwell on efficiency subjects. (There are two more I want to point out, but can't find them now.)

1) the Goddard Problem

2) the Optimal Ascent Challenge

The links go to specific posts which have the (relatively) short answers. Closette's table of terminal velocities helped perfect the ascent speed profile. My post in the ascent challenge has a graph image attached, which reveals a lot about how to shape your gravity turn. Darn, but I wish I could find the other two challenges, about approaches and landings.

[Kethas]

stuff

Thanks, Zephram, that's some great reading. It's nice to know I'm not the first KSP player to ask these questions. I'll definitely take a look, and if you find links to the other two challenges, I'd love those too.