Gravity turn efficiency score

[Rodhern]

How to measure the efficiency of a gravity turn

To make my gravity turn challenge a bit more interesting the scores will be given as percentage scores. I have seen percentage scores used in game reviews, school tests and what not - so it must be a cool score measure right?

I post the score description to this separate post in case anyone wants to discuss and/or refine the idea.

Gravity turn efficiency score

Imagine that we can send our vessel to orbit by doing two short powerful explosion-like burns. First a burn that will shoot us all the way to space, a burn of strength "v" (for vertical) say. Then, just above Kerbin's atmosphere our positive vertical speed runs out, and we do another burn to quickly accelerate to orbital speed, a burn of strength "h" (for horizontal) say. I imagine this 'burn model' as depicted in this (first) figure.

Now imagine another way to send the vessel to orbit. Maybe we don't need to shoot the vessel all the way to space with the first burn. Instead we only clear the thicker part of the atmosphere with the first burn, and then we angle the second burn to take care of both the needed "h" (horizontal component) burn and the missing part of the "v" (vertical component) burn. I imagine that this improved 'burn model' looks like this (second) figure.

The idea of the score is to make a guess for how much total burn strength it would take to first burn and fly vertical and then burn and fly horizontal. We could call this our worst case budget. Once we know how much burn we actually used in total in some instance, we expect to see that we did better than our worst case budget. Compare the actual total burn (i.e. fuel used) with the burn from the second figure to assess if the saving is large (i.e. a large alpha) or small (i.e. an alpha just barely above zero).

Interpretation of alpha

We pick some "worst case budget" as a reference and can then assign an alpha to the actual total burn that we record. The interpretation is that a negative alpha value means we used more fuel than our "worst case budget". We should, in theory, avoid this, as it implies we could save fuel simply by flying straight up and then accelerate along the horizon. In practice though, we might fly an inefficient path for non-fuel related reasons. An alpha of around zero percent means that our fuel usage turned out to match our "worst case budget". An alpha of 100 % is interpreted as the theoretically ideal composite of the vertical and horizontal burns. If the gravity turn efficiency is even better than that, then alpha rises to above 100 %; maybe that is achievable for space planes - time will tell. Technically alpha, as calculated by the formula below, can surpass infinity, but that requires a ridiculously cheap launch trajectory to orbit.

Formula

Label the actual burn strength total as "t", and the vertical and horizontal components of the worst case budget as "v" and "h" respectively. Assuming that "t" is larger than "v", alpha is equal to (h^2-(t-v)^2)/(2*v*(t-v)) .

The Kerbal 1-5 case [edit: for KSP version 1.2.1]

Let us take a "Kerbal 1-5" budget as an example. I have uploaded a KSP version 1.2.1 budget, that may be used to score the entries in this challenge.

Budget and scoring spreadsheet

The budgeted vertical and horizontal components are 2239 and 2220 m/s of vacuum delta-V. The delta-V potential of the stock configured Kerbal 1-5 is 4398 m/s, assuming the main throttle is kept closed until the boosters are done. The budget indicates that, if you fly straight up to 70 km before turning, then the rocket cannot make it to orbit. Notice that in practice, even if launching vertically, we probably want to turn the rocket towards the horizon before we reach 70 km, so the budget is really for a quite hypothetical worst case scenario.

The threshold for an alpha-score of 100 percent is to get to LKO (70 km) using only 3153 m/s of delta-V. To me that goal seems unattainable. On the other hand, perhaps 3153 m/s is not an unreasonable indication of the upper bound for the gravity turn efficiency of the stock Kerbal 1-5.

By the way, the reason the spreadsheet says "NO SCORE" is because the challenge is for an 80 km orbit, so 70 km is too low.

Edited November 3, 2016 by Rodhern
updated scoring spreadsheet for KSP ver. 1.2.1