The maths of staging

[GoSlash27]

Is there some way to mathematically determine whether or not to stage or how many stages to use for a given engine, payload, DV budget, etc?

Best,

-Slashy

[Starhawk]

You could use Meithan's Engine Charts to quickly get some parameters for single-staged vs. non-staged vs. multi-staged solutions.

That might be simplest. Other than that, I'm not sure. In the past, I've tended to try building things different ways in the VAB and see what the results looked like. Meithan's tool has removed much of that for me.

Happy landings!

[GoSlash27]

You could use Meithan's Engine Charts to quickly get some parameters for single-staged vs. non-staged vs. multi-staged solutions.

That might be simplest. Other than that, I'm not sure. In the past, I've tended to try building things different ways in the VAB and see what the results looked like. Meithan's tool has removed much of that for me.

Happy landings!

Starhawk,

I have the ability to generate the data in his charts. I'm just wondering if there's a formula that will predict the crossover point where staging saves mass.

Best,

-Slashy

[Starhawk]

It seems to me that this is more algorithmic than formulaic. Maybe that's due to me being more programmer than mathematician.

It gets tricky right away when I start thinking about it. In general, how do you find the mass optimal two-stage solution for a given dv, payload mass, and starting twr?

There will be any number of solutions depending on how you split the total dv budget between the stages. With multiple stages, one would also have to specify a minimum twr as well as a starting value. The only way that I see is to iterate through the list of engines, run the numbers, do the same thing for the second stage, and total the mass each time and compare.

OK, so then we have to try the same thing for three stages...

How many stages should there be? In the ideal case, with no staging equipment needed, and the stages being any size we like, we would have an infinite number of stages. This would allow us to constantly throw away mass we don't need and keep the flight truly optimal. The reality of the possible sizes of the stages is very different from this ideal case. And decouplers have mass as well.

Hmmm... Maybe I'm overcomplicating it.

Happy landings!

[GoSlash27]

It seems to me that this is more algorithmic than formulaic. Maybe that's due to me being more programmer than mathematician.

It gets tricky right away when I start thinking about it. In general, how do you find the mass optimal two-stage solution for a given dv, payload mass, and starting twr?

There will be any number of solutions depending on how you split the total dv budget between the stages. With multiple stages, one would also have to specify a minimum twr as well as a starting value. The only way that I see is to iterate through the list of engines, run the numbers, do the same thing for the second stage, and total the mass each time and compare.

OK, so then we have to try the same thing for three stages...

How many stages should there be? In the ideal case, with no staging equipment needed, and the stages being any size we like, we would have an infinite number of stages. This would allow us to constantly throw away mass we don't need and keep the flight truly optimal. The reality of the possible sizes of the stages is very different from this ideal case. And decouplers have mass as well.

Hmmm... Maybe I'm overcomplicating it.

Happy landings!

Actually, you're thinking the same way I am about it. In a finite case, the added engine mass and decoupler are balanced by the dead tank mass. Maybe it's just that simple; the mass of the empty tank is twice the mass of the second stage engine and decoupler?

*scratchin' mah head...*

-Slashy

[Orbital Vagabond]

It seems to me that this is more algorithmic than formulaic. Maybe that's due to me being more programmer than mathematician.

It is algorithmic, which, in my view, is primarily due to the step-wise method of dV calculation for staged rockets. It also depends on what value you're trying to optimize: launch mass, dV, cost, etc.

I recall seeing a webpage (which I frustratingly can't find right nowon project rho here) that discussed having a hapless graduate student plug in various values to determine the optimal ration of fuel between two rocket stages to minimize total launch mass. In a two stage system the solution is relatively easy to find, but in a three stage system the solution becomes much more difficult to locate.

If you're optimizing dV, in theory there wouldn't be a solution because larger stage attached below any described rocket, which can have another booster stage attached beneath it, etc etc.

In my experience, it's simply easier and equally (if not more) effective to design rockets with stages for specific jobs, e.g. 1 or 2 stages to launch to LKO, a separate stage for a transfer, etc. You can optimize for each leg of the journey.

Additionally, Wikipedia has some interesting information on optimizing rocket staging here and cites Curtis' Orbital Mechanics for Engineering Students. If there's an answer to you question, it'll probably be in there.

Edited June 4, 2015 by Orbital Vagabond

[Xannari Ferrows]

It is an exponential process. This is the tyranny of the rocket equation kicking in.

[Starhawk]

In a finite case, the added engine mass and decoupler are balanced by the dead tank mass. Maybe it's just that simple; the mass of the empty tank is twice the mass of the second stage engine and decoupler?

Interesting. That's the general direction my intuition was pointing before I wrote the long post above.

It is an interesting problem.

Other, more mundane, tasks are calling on my brain at the moment. I think I'll try and look at a couple of test cases when I have a bit more time.

Happy landings!

[Prepper-Jack]

Don't forget that you can also get better efficiency through staging engines that are more correct for whatever environment you are in. For an early example, you could have a Swivel or Reliant launch to 30k, and have a Terrier take over where the atmosphere becomes super thin. As long as TWR is adequate, you save on fuel efficiency by both dumping the dead weight and switching to a more efficient engine (which can be done before many people, I believe, assume).

There are also other considerations of course. Say you're making an expedition to Mun. Your lander has about 3k of dV with reserves of 360 fuel. You could build a relatively cheap lifting and transfer stage, land once (and move once or twice) and be able to make it home. Alternatively, you could overbuild your lifting and transfer stages, have some junior docking ports on a massive transfer stage, and use that to refuel from Munar orbit - leading to 5 or 6 times the number of landings for perhaps only 2x the cost. Here, the utility of creating an extra 15-20k dV can trump early atmospheric staging concerns.

[Northlight]

Only way to determine wheter to stage or not, (And how many stages you should have) is to do the math for each variant.

Given a target delta-v, and payload.

* How big (in tonnes) is the rocket if only using one stage? (easy calulation)

* How big is the smallest (in tonnes) rocket with two stages? (still easy, but there is a lot of iterations)

* How big is the smallest (in tonnes) rocket with three stages? (easy, but a LOT of work)

Then you can see witch is the most efficient. (in terms of weight)

But it would be cool if there was some "shortcut".

[DancesWithSquirrels]

I don't think it's been updated to 1.0.2, just to 1.0, but the Kerbal Optimal Rocket Calculator does things algorithmically - try all combinations and look at the best results. It's a good starting point, at any rate.

[capi3101]

I don't think it's been updated to 1.0.2, just to 1.0, but the Kerbal Optimal Rocket Calculator does things algorithmically - try all combinations and look at the best results. It's a good starting point, at any rate.

That's a nice tool right there...shouldn't be any major differences between 1.0.0 and 1.0.2 that would affect its operation. Going to have to add it to my own little toolbox set.

[GoSlash27]

Interesting...

My testing so far says that (in a very generic situation with identical payload) staging never reduces total mass except where a single stage is incapable of doing the job.

Example: LV-T45, vacuum Isp, 1.0G minimum accel

DV Mp M1 M2

1000 12 19.2 22.8

2000 8.0 19.0 23.5

3000 4.7 17.3 22.7

5400 0.0 9.22 13.4

I'll have to think on why this would be..

[Jouni]

The good news is that the internet is full of research papers and lecture notes about optimal rocket staging. The bad news is that the solutions are complicated and ugly, even in simplified special cases.

In a typical simplified scenario, the goal is to maximize the payload fraction for a given delta-v target. Rocket stages are defined by two parameters, Isp and dry mass fraction, and you can scale them up and down arbitrarily. There are no TWR requirements, Isp does not change during flight, and only vertical staging is allowed. If all stages are have the same parameter values, each of them should provide the same amount of delta-v. If one stage has a strictly better (Isp, dry mass fraction) combination than another, it should provide more delta-v.

The optimal number of stages depends on the delta-v target and stage parameters. In KSP, the availability of suitable engines also has an important role. One stage per 2000 m/s should be about right for Kerbin launches, with something like 2500 m/s from low-TWR stages with efficient conventional engines. This means two stages to LKO or for an encounter with the Mun, and three stages for Mun landing or Jool encounter.

[ajburges]

First off, there won't be a formula. I'm also sure this problem is also NP complete.

There are two big ways staging gives mass advantage in KSP. Engine specialization and dead weight removal.

Engine specialization is simply using the right engine for the right situation. You won't use a 909 for takeoff just like you would avoid a mainsail for interplanetary travel. The ISPs create relative advantages between atmospheric and vacuum operations.

Dead weight removal is about not carrying mass that has no use. 1+ TWR with 4km/s dV requires a lot of engine. Towards the end of that tank you really don't need all that engine (or tank). If you could lose some of it you would need less fuel for that same dV.

The easiest place to see staging offering a mass advantage is drop tanks. The dry mass of the largest tanks is more that a comparable stack seperator. For a long train of drop tanks, the lack of trailing empty tanks will create a dV advantage which means you need a shorter train for the same total dV.

[Starhawk]

I don't think it's been updated to 1.0.2, just to 1.0, but the Kerbal Optimal Rocket Calculator does things algorithmically - try all combinations and look at the best results. It's a good starting point, at any rate.

This tool appears to do exactly what I was describing earlier. Iterate through all possible given solutions for a specific set of starting parameters and order the list based on the selected criteria. And, yes, it's a very nice tool.

My testing so far says that (in a very generic situation with identical payload) staging never reduces total mass except where a single stage is incapable of doing the job.

I do not understand. From the ideal situation, I would expect more staging to be better in general, until the losses from decoupler mass/increased engine mass outweigh the gains from dropping unused mass.

For my simple test case my goal is to move 10 t to Kerbin orbit. I'm allowing a total dv budget of 3800 m/s.

Meithan's tool says my best single-stage solution is a Mainsail with 70 t of fuel tanks for a total of 85.14 t.

I tried two different splits of the delta-v budget.

1. Upper stage 2000 m/s

Skipper + 14.6 t of fuel tanks + payload = 27.6 t

Lower stage 1800 m/s

Mainsail + 36.4 t of fuel tanks + payload = 70 t

2. Upper stage 1500 m/s

Poodle + 7.8 t of fuel tanks + payload = 19.6 t

Lower stage 2300

Mainsail + 43.8 t of fuel tanks + payload = 69.4 t

I would generally expect more stages to be better up to some point.

Happy landings!

[GoSlash27]

This tool appears to do exactly what I was describing earlier. Iterate through all possible given solutions for a specific set of starting parameters and order the list based on the selected criteria. And, yes, it's a very nice tool.

I do not understand. From the ideal situation, I would expect more staging to be better in general, until the losses from decoupler mass/increased engine mass outweigh the gains from dropping unused mass.

For my simple test case my goal is to move 10 t to Kerbin orbit. I'm allowing a total dv budget of 3800 m/s.

Meithan's tool says my best single-stage solution is a Mainsail with 70 t of fuel tanks for a total of 85.14 t.

I tried two different splits of the delta-v budget.

1. Upper stage 2000 m/s

Skipper + 14.6 t of fuel tanks + payload = 27.6 t

Lower stage 1800 m/s

Mainsail + 36.4 t of fuel tanks + payload = 70 t

2. Upper stage 1500 m/s

Poodle + 7.8 t of fuel tanks + payload = 19.6 t

Lower stage 2300

Mainsail + 43.8 t of fuel tanks + payload = 69.4 t

I would generally expect more stages to be better up to some point.

Happy landings!

Starhawk,

According to my spreadsheet at home, I can do a single stage using a single Skipper at 61.7 tonnes, which is lighter than either of the 2 stage solutions. My solution for the Mainsail is 80.9 tonnes.

I'll have to verify this result against my work spreadsheet. What minimum t/w did you assume?

Scratchin' mah head,

-Slashy

Edited June 5, 2015 by GoSlash27

[Starhawk]

Starhawk,

According to my spreadsheet at home, I can do a single stage using a single Skipper at 61.7 tonnes, which is lighter than either of the 2 stage solutions. My solution for the Mainsail is 80.9 tonnes.

I'll have to verify this result against my work spreadsheet. What minimum t/w did you assume?

Scratchin' mah head,

-Slashy

My numbers for a single stage solution came from using Meithan's tool and setting the atmo to 0.3 as a guesstimate to cover the small portion of the flight done at higher pressure, and using 1.4 as the minimum twr.

That gave the number of 85.14 t for a single stage solution with 3800 m/s and a 10 t payload.

For the multi-stage solutions, I set the lower stage twr at 1.4, and the upper stage at 1.1.

I haven't had the time to verify the numbers and may not get a chance until Sunday.

But something seems odd, that's for sure.

Happy landings!

[GoSlash27]

My numbers for a single stage solution came from using Meithan's tool and setting the atmo to 0.3 as a guesstimate to cover the small portion of the flight done at higher pressure, and using 1.4 as the minimum twr.

That gave the number of 85.14 t for a single stage solution with 3800 m/s and a 10 t payload.

For the multi-stage solutions, I set the lower stage twr at 1.4, and the upper stage at 1.1.

I haven't had the time to verify the numbers and may not get a chance until Sunday.

But something seems odd, that's for sure.

Happy landings!

Ah, I bet that's what it is; you're not keeping the requirement and engine type uniform throughout the budget.

Yeah, I understand that staging is worth it AFA changing requirements for t/w and changes in Isp with pressure, but that can't be solved with an equation.

What my results are suggesting is that for a uniform payload, engine and constant profile demand, a single stage will be lightest so long as it is capable of doing the job.

Bumping the requirements to match 1.4G, the skipper single stage can still do 64.7 tonnes, which is still lighter than the 2 stage solutions.

As I said, I'll have to verify this against my work spreadsheet, which scales the number of engines automatically. It's the model I used to corroborate Meithan's work.

Best,

-Slashy

[Requia]

Starhawk,

According to my spreadsheet at home, I can do a single stage using a single Skipper at 61.7 tonnes, which is lighter than either of the 2 stage solutions. My solution for the Mainsail is 80.9 tonnes.

I'll have to verify this result against my work spreadsheet. What minimum t/w did you assume?

Scratchin' mah head,

-Slashy

A TWR of greater than 1 is generally necessary, no matter how inefficient it is, you need to correct for sea level vs vacuum thrust.

[GoSlash27]

A TWR of greater than 1 is generally necessary, no matter how inefficient it is, you need to correct for sea level vs vacuum thrust.

Requia,

Mathematically you can't do that. Or more specifically you can't reduce that concept down to an equation that'll tell you when to stage a lifter.

And in most cases in KSP 1G acceleration is overkill. That is certainly true for the early going in a Kerbin launch, but not necessary for most of it.

Best,

-Slashy

[Requia]

While the specific case of when is staging right is a good question, in the general case there's a best payload mass % thread somewhere on the forum. It'll give you the general case of what the lightest possible rocket looks like.

- - - Updated - - -

Requia,

Mathematically you can't do that. Or more specifically you can't reduce that concept down to an equation that'll tell you when to stage a lifter.

And in most cases in KSP 1G acceleration is overkill. That is certainly true for the early going in a Kerbin launch, but not necessary for most of it.

Best,

-Slashy

I didn't say acceleration, I said TWR, that skipper SSTO you talk about isn't getting off the launchpad at all until it burns most of your fuel.

[FancyMouse]

you're not keeping the requirement and engine type uniform throughout the budget.

OK this needs some clarification. So your question is specifically about when you only have 1 exact engine to choose from, but you can stack arbitrary number of them in arbitrary stages, and then find the optimal solution?

[GoSlash27]

OK this needs some clarification. So your question is specifically about when you only have 1 exact engine to choose from, but you can stack arbitrary number of them in arbitrary stages, and then find the optimal solution?

Pretty much.

The question is "for an arbitrary engine and uniform profile, at what point will staging yield a reduction in launch vehicle mass?".

What would an equation to solve this problem look like?

- - - Updated - - -

While the specific case of when is staging right is a good question, in the general case there's a best payload mass % thread somewhere on the forum. It'll give you the general case of what the lightest possible rocket looks like.

- - - Updated - - -

I didn't say acceleration, I said TWR, that skipper SSTO you talk about isn't getting off the launchpad at all until it burns most of your fuel.

As a practical matter yes. As a mathematical problem, no.

I am aware that the single stage "solution" will not *actually* function on Kerbin due to changes in pressure affecting Isp and thrust, but that question is not solvable mathematically.

The problem is more generic than that.

If given an arbitrary engine and mission requirement, at what point does staging show a reduction in stage mass?

I would have thought that it would happen when the mass of the empty tanks in the single stage match the mass of twice the added engine in the two stage design, but the model isn't supporting that assumption.

It seems that the single stage design is working out lighter until the DV requirement exceeds the capability of the single stage design.

[FancyMouse]

Without TWR restriction, the answer is never - two simple proposition should kill it. One is for each given stage, more than 1 engine is never better than using just 1 using the same amount of fuel (this easy to prove, since Isp doesn't change, and the ln term will get bigger by reducing engine mass). The other is staging two 1 engines is never better than combining the two stages with the same amount of fuel (to prove this, notice that if fuel mass z is given, then ln((x+z)/x) is a decreasing function, so putting second stage engine and decoupler is always worse).

With TWR restriction, things become interesting, but we need to rigorously define the restriction before we dig into it.

EDIT: screw it - I know I often produce bad proofs. Apparently I forgot empty tank mass - let me recalculate and update later

Edited June 5, 2015 by FancyMouse