# Is there an optimal accelaration for first stage going in LEO?

I know every engineering works is based on compromises, and rocket design is no exception. A rocket should go as quickly as possible to high altitude to avoid flying for a long time and at high speed into the atmosphere.

Moreover, the rocket may accelerate more and more as it is lighter while burning fuel and the engines remain the same (almost the same thrust).

Climbing faster may increase max Q (high speed at lower altitude).

Accelerating too quickly may exceed payload acceleration tolerances.

Accelerating quickly requires powerful engines (and perhaps few fuel or smaller tanks to reduce weight).

Not knowing all the constraints and parameter to optimize for rocket launch, I wonder if there is an optimal acceleration profile for launcher take off. As it seems broad, the question is restricted to the first minutes of flights (more or less from take off to 150 km high) and to flights putting payload into LEO.

• Is there any rockets first stage reaching LEO?
– Uwe
Jul 6, 2018 at 15:43
• Optimal for what metric? Cost? Time to orbit? Airframe luminosity from skin heating? Jul 6, 2018 at 15:57
• Gotta be the latter. Jul 6, 2018 at 23:58
• @RussellBorogove I think the OP is asking in good faith. If you have some insight, why not share it with the OP rather than crack a joke at the OP's expense? When someone is trying to understand something they don't already know the answer. In this case the existence of any reasonable optimum would be a helpful contribution, but look how clever I am's can be confusing to many users, therefore not helpful.
– uhoh
Jul 7, 2018 at 0:37
• @RussellBorogove why not go all the way and vote to delete the answer that's already been posted adding the comment "this can not be an answer to the question, as the question is clearly unclear"? I think it's an absolutely reasonable question and i think it would be great if people felt welcome here to ask this kind of question without the sense that they are being berated. The Math Overflow site exists for those too enlightened for Math SE. Perhaps we need a Rocket Overflow for the Rocket Scientists among us?
– uhoh
Jul 7, 2018 at 0:59

## 1 Answer

As you seem to surmise in your question, the answer is "no."

There's no single optimal acceleration profile. In general, you want to be at maximum vertical acceleration to reduce gravity losses and to rise above thick atmosphere; the maximum acceleration is bounded by things like

• stress on the payload
• stress on the airframe
• aerodynamic heating
• cost/availability of engines with sufficient thrust and throttling capability

As Russell's comment astutely points out, the optimum involves even more factors than that and is essentially system/situation dependent.

• Let's add one important metric: cost per kg to orbit. Since large fuel tanks and fuels are cheaper than powerful engines, the usual acceleration profile will be way below optimal at least in the initial phase of flight, resulting in higher losses to gravity than strictly necessary, but lower first stage price, which more than makes it up for the extra fuel expense.
– SF.
Jul 8, 2018 at 1:41
• @SF “way below optimal” by what measure? Jul 8, 2018 at 4:16
• @RussellBorogove: TWR -> time of flight staying at maxQ -> time to orbit -> Delta-V.
– SF.
Jul 8, 2018 at 10:17