How to select appropriate orbital rocket stage thrust?

I’ve been working on a preliminary design of a small-lift launch vehicle capable of sending a small satellite into LEO. Given the required $$\Delta v$$ (~9 km/s for LEO), the mass distribution, including the required propellant mass for each stage and the total, can be calculated by using the well-known optimal rocket staging method. However, the method does not tell the size of the engine of each stage, i.e. the thrust and the burn time. Many sources just assume some thrust-to weight ratio value that is greater than 1 to get the rocket out of gravity well.

My question therefore is: how do I select the appropriate required thrust for each stage of the launch vehicle?

What I think might lead to the solution involves trajectory design of the rocket. For instance, in a gravity turn trajectory, higher thrust leads to lower gravity loss but higher aerodynamic drag loss and the opposite applies for lower thrust. The optimal amount of thrust is therefore the point where the total loss is minimum. Furthermore, thrust is also limited by the maximum dynamic pressure the rocket can handle and the maximum acceleration the payload can withstand. I am not quite sure if determining optimal thrust through trajectory design is the right direction, and trajectory optimization is a large and complex subject by its own. Plus, $$\Delta v$$ loss might not be the only factor for thrust optimization.

I don't know if there's an analytical solution here; you probably have to iteratively simulate the launch trajectory to find good thrust values.

For the first stage, ignition TWR is typically in the 1.2 to 1.4 range, though there are some outliers -- Saturn V at about 1.16, STS closer to 1.5 if I remember rightly.

I had previously thought best performance would come from taking as much fuel as possible, i.e. having a very low TWR at launch, but this turns out not to be the case.

For upper stages, it's not unusual to ignite at a TWR less than 1.0 -- Saturn V second stage starts at about 0.8, and third stage at about 0.6, for example. The stack is turned to some degree towards the horizontal before staging occurs, so there's nothing magical about a 1.0 TWR for upper stages.

• Is there any launch trajectory suitable for early launch vehicle design stage? So far I’ve seen literatures on gravity turn in the atmosphere and a different trajectory when the Knudsen number is high. There are many trajectory design techniques but exploring them seems a daunting task for early vehicle design stage. Also, what should I look at when performing parametric study with trajectory simulation apart from 1) total delta-v loss 2) maximum dynamic pressure and 3) maximum acceleration? Feb 14 '21 at 2:54
• You could just copy an existing launcher's trajectory for a preliminary study. Feb 14 '21 at 4:34
• @CharlesStaats A quick simulation seems to suggest you're right, even though (holding the total stage mass constant) the dry mass is actually decreased in the minimum TWR case -- the engine mass savings outweigh the tankage mass increase. Everything I know is wrong! Feb 14 '21 at 17:41