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I understand how to use lagrange multipliers to obtain the optimal stage masses given the payload mass, the burnout velocity, the structural ratios of each stage and the specific impulses of each stage. However, the "optimal staging" formulation seems to be independent of the mass flow rate $\dot{m}$, which is an essential ingredient into computing the overall thrust T that a stage needs to produce:

$$T=I_{sp}g_0\dot{m}$$

As I understand, the classical optimal staging problem can tell you how much fuel you need in each stage to give the payload a given velocity, but it doesn't tell you how quickly it needs to burn to achieve this. How can I find the (optimal) mass flow rate needed to send the optimally staged rocket's payload to its desired velocity? Is there another formulation of the optimization problem that can help me yield the optimal mass flow rates of each stage in addition to the stage masses? Or is there a way that I can calculate the necessary (optimal) thrusts of each stage first so that I can then derive the optimal mass flow rates?

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    $\begingroup$ As far as I know, this is always solved pragmatically. Your choice of available engines quantizes (or at least constrains, if throttleable) the thrust (and thus mass flow). Add engines until your full-throttle T/W at liftoff is > 1.15 or so. More engines reduces gravity loss but increases dry mass (and drag loss, unless deeply throttleable). In the absence of atmosphere, engine mass, and structural strength limits, you'd want obscenely high thrust for minimum g-loss. $\endgroup$ – Russell Borogove Sep 28 '17 at 20:50
  • $\begingroup$ @RussellBorogove: Why 1.15? Is there something significant about this ratio, specifically? Is there a way to derive an optimal thrust-to-weight ratio for each stage in order to achieve the burnout velocity given the optimal staging? $\endgroup$ – Paul Sep 28 '17 at 20:58
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    $\begingroup$ Initial liftoff T/W must be greater than 1 or you're not going anywhere. At 1.0 exactly you'll spend a lot of time moving slow next to the tower or strongback; it's safer to clear it quickly. 1.15 is about the lowest I've seen in practice. Like I said, in the absence of real world engineering issues the optimal T/W ratio is arbitrarily high. The practical optimum is going to involve structural engineering, aerodynamics, and engine availability. $\endgroup$ – Russell Borogove Sep 28 '17 at 21:06
  • $\begingroup$ (Highest T/W off the pad I've seen was about 1.6: Hayabusa 2 on the H-IIA launcher.) $\endgroup$ – Russell Borogove Sep 28 '17 at 21:19
  • $\begingroup$ So hey, regarding stage ratios, I've read that, all other things being equal, you want similar fuel mass ratio / ∆v contribution for stages with similar Isp, but a greater ratio and thus larger ∆v contribution for stages with lower Isp, which seems a little counterintuitive -- is that the correct relationship? $\endgroup$ – Russell Borogove Sep 28 '17 at 21:58
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This question inherently can't be answered as the problem is incompletely defined.

You have three very different problems based on where you are:

1) In space: Here it's simple: You want the tiniest engine you can get that doesn't sacrifice ISP. A bigger engine will complete the burn faster but so what? Using the same mass as fuel will almost certainly get you there faster by cutting travel time. Look at some of the burn times NASA lists for maneuvers by it's deep space probes and you'll see their engines have to be tiny.

2) On an airless body: Now it gets a lot more complex and there is no optimal answer without considering the whole mission. The higher the thrust of your engine the less delta-v you'll use but you'll be carrying more engine weight--and you'll also have had to get that engine weight there in the first place. Hauling that bigger engine might cost you more fuel than you save by reducing your gravity loss.

3) In atmosphere: Now it's even more complex as you have to consider air drag--now the bigger engine also carries an increased drag penalty as well as an increased weight penalty. Not only must you consider the whole mission but you must consider the shape of your rocket also.

Since cases #2 and #3 are optimizations between competing variables you have to optimize for each mission, there are no simple equations that give you an overall best answer.

Furthermore, unless you want to design an engine for the mission you're limited to what's on the shelf--and that may not be optimum for what you want to do. I play Kerbal Space Program and I've sent out plenty of overpowered spacecraft because the available parts don't match up perfectly with my objectives and because engines might be called upon to operate in multiple realms. Consider a simple mission: Land a probe on the Mun (the closest moon). The first part of the mission is atmospheric flight. The second part of the mission is space flight (transfer orbit, orbital insertion burn). The third part is airless body flight.

Now, the boosters used to get off Kerbin will generally fall back. Space flight is in my category #1--which would say to use the smallest engine. However, I'm going to land which means I must carry enough thrust to land. Why not simply use the landing engine for the space flight also? Thus I end up doing the space mission on engines capable of landing on the Mun, not merely getting there.

Now, consider a more expensive mission: Land on Ike. It's the moon of the Mars-analogue, a bit weaker gravity than the Mun. However, you have to do the interplanetary voyage. If it's a simple lander I would use the same system, just with a bit more fuel. However, if I'm taking something big (say, a manned rover capable of exploring the moon) the economics change: You get a smaller, cheaper rocket by doing all the space maneuvering with a nuclear engine. It doesn't come in a really small size, though, so it doesn't get used for the small stuff.

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  • $\begingroup$ I disagree. The optimal staging problem is well-defined, even if it is not a perfect representation of all of the physics involved in liftoff (any text on orbital mechanics will confirm this assertion). The solution to the problem estimates how much propellant is needed in each stage to achieve a desired burnout velocity (based on idealizations of the rocket equation). For the mass proposed, there must also be a minimum thrust requirement to reach the desired outcome. $\endgroup$ – Paul Sep 30 '17 at 4:36
  • $\begingroup$ @Paul The performance of a particular rocket is well defined by a fairly simple equation. That's not the same thing as finding the optimal values for the whole mission, though. $\endgroup$ – Loren Pechtel Sep 30 '17 at 21:43
  • $\begingroup$ For now, I'm only concerned with analyzing liftoff and reaching orbit at a given speed using staging. The classic staging problem clearly ignores aerodynamic effects. For simplicity, I'm also ignoring aerodynamic effects. $\endgroup$ – Paul Oct 1 '17 at 15:51

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