Theoretical maximum deltaV and how to get beyond it.

[Mr. Boncrete]

Sorry, but I have to disagree: delta-v is proportional to the logarithm of your initial fuel, and the logarithm doesn't converge - it grows very slowly, but it never reaches a limit.

Isn't that the definition of limits? It would be based on the properties of an individual ship or configuration, but if the delta-v achieved from increasing the quantity of fuel decreases at (an increasing rate), is that not the precise definition of an asymptotic limit?

It would vary between ships, but I am inclined to think that there is most certainly an upper limit unless you use some fancy configuration.

[Xavven]

Isn't that the definition of limits? It would be based on the properties of an individual ship or configuration, but if the delta-v achieved from increasing the quantity of fuel decreases at (an increasing rate), is that not the precise definition of an asymptotic limit?

It would vary between ships, but I am inclined to think that there is most certainly an upper limit unless you use some fancy configuration.

The ln function looks like this: http://en.wikipedia.org/wiki/File:Log.svg

But it isn't asymptotic. It will increase infinitely. I think what jebbe meant was "it has no limit," not "it can't reach the limit".

[UmbralRaptor]

*grumble* necromancers...

But it isn't asymptotic. It will increase infinitely. I think what jebbe meant was "it has no limit," not "it can't reach the limit".

If your mass ratio can go on to infinity (as Jebbe was assuming), then yes, so can your ÃŽâ€Ã¯Â¼Â¶. You'll need × stages, though...

As long as we're rambling about this, how do gravity assists play into the equations? If I understand correctly, a fortuitous gravity assist equals free delta-V (effective delta-V anyway, I know your ship doesn't literally get more thrust). But how far can this go? Do gravity assists always contribute the same amount no matter how many times you do it, or is there also a falloff therÃŽâ€e? Can I get unlimited travel range out of a ship by surfing up and down gravity wells a bunch of times?

The effectiveness of a gravity assist is roughly inversely proportional to your hyperbolic excess speed (relative to the body's SOI). That would put an upper limit, though ÃŽâ€Ã¯Â¼Â¶ from gravity assists is unrelated to the ÃŽâ€Ã¯Â¼Â¶ discussed back in 2013. (It's from the raw rocket design, and is ignoring gravity wells, etc)

[travis575757]

Per stage there doesn't seem to be a boundary in delta v.

(g*Isp) * log mass start/ mass end at a base of e or simply

Ve*ln(m0/m1)

Now there could be limits but theoretically they are very large, probably something you would never get to so decided to test it out.

I graphed it using the isp of a nuclear engine in vacuum with a end mass of 1 and a starting mass of x this wouldn't work this way in the game but it is theoretical

so (9.8 * 800) * ln(x/1) (yeah delta v equation) and their appears to start get significantly less increase at ~70000 delta v

Now if we set the end mass to 0.1 we get ~90000 delta v

In both though adding larger numbers never reaches a limit if there were one then eventually doubling the starting mass would do almost nothing. At no point it does this.

So from what i see there is no horizontal asymptote marking maximum yield, it seems as if though it is diagonal which basically means, no there is no theoretical maximum to delta v (from what im seeing).

If someone could check this out to it would be great, put it into google and see what happens. If you keep zooming out the the increase never comes close to stopping.

[Red Iron Crown]

Per stage there doesn't seem to be a boundary in delta v.

(g*Isp) * log mass start/ mass end at a base of e or simply

Ve*ln(m0/m1)

Now there could be limits but theoretically they are very large, probably something you would never get to so decided to test it out.

I graphed it using the isp of a nuclear engine in vacuum with a end mass of 1 and a starting mass of x this wouldn't work this way in the game but it is theoretical

so (9.8 * 800) * ln(x/1) (yeah delta v equation) and their appears to start get significantly less increase at ~70000 delta v

Now if we set the end mass to 0.1 we get ~90000 delta v

In both though adding larger numbers never reaches a limit if there were one then eventually doubling the starting mass would do almost nothing. At no point it does this.

So from what i see there is no horizontal asymptote marking maximum yield, it seems as if though it is diagonal which basically means, no there is no theoretical maximum to delta v (from what im seeing).

If someone could check this out to it would be great, put it into google and see what happens. If you keep zooming out the the increase never comes close to stopping.

Your error is in assuming that you can increase fueled mass without a corresponding increase in dry mass.

As you add more and more tanks, the mass of the engines and other equipment becomes less and less significant, so the theoretical limit is based on the rocket equation using a single fuel tank's wet mass over the dry mass of a single tank for the mass fraction. This gives an asymptotal quantity that can be approached but never reached, as the other dry mass never completely disappears.

[travis575757]

Re did that then with orange tanks in the equation. Now the theoretical max is 17225 m/s delta v. Sounds strange, was expecting more. BTW this is with 800 isp. I tried this again with 370 Isp and got around 9000m/s delta v. when doing this though to show what i mean i used (800*9.8) * ln(36x/4x+1) the plus one being something like a probe core.

[mhoram]

Per stage there doesn't seem to be a boundary in delta v.

Some time ago there was a similar discussion with te result that there is a dV limit for a single stage that depends on the engine.

http://forum.kerbalspaceprogram.com/threads/60025-Maximum-possible-dV-for-a-single-stage

[Red Iron Crown]

Re did that then with orange tanks in the equation. Now the theoretical max is 17225 m/s delta v. Sounds strange, was expecting more. BTW this is with 800 isp. I tried this again with 370 Isp and got around 9000m/s delta v. when doing this though to show what i mean i used (800*9.8) * ln(36x/4x+1) the plus one being something like a probe core.

Now you're on the right track. As x gets larger and larger, the 1 representing non-tank mass gets less and less significant, so the value without the 1 at all is the asymptote representing an infinite number of tanks.

Doesn't matter which liquid fuel tank you use either, I think the all have the same wet/dry fraction except the Oscar-B and Round-8 which are worse.

[TimMartland]

Theoretically it is infinite.

The tyranny of the rocket equation is the main obstacle in KSP however, as eventually you will have too many parts for the game to be playable. The largest single launch dV I ever had was about 17k, but that almost killed my computer (This is excluding my ION probes, which had an insane dV which I could never really use all of due to their long burn time)

[EtherDragon]

Technically, theoretically, the ln function is infinite, but the amount of fuel (without any extra non-fuel mass) must grow exponentially for more Dv.

But don't get confused with just the ln function and what occurs in the real world (or even in KSP). In KSP, the fact remains - if you take the most efficient engine, and attach it to such a huge rocket that the weight of the engine is a non-factor, you are still stuck with the dry mass of the fuel tanks. For a Rockomax Jumbo-64 that's 4t dry mass 32t fuel.

Just the fuel (one tank) and (an LVN) engine gives you 6.25 empty, 38.25 full = 1449 Dv

1,000 tanks and one lvn, 4,002.25 empty, 36,002.25 full = 1757 Dv

1,000,000 tanks and one lvn, 4,000,002.25 empty 36,000,002.25 full = 1757 Dv

That 4t empty mass per tank sets a hard (mathematical) limit to the Delta V from a single stage.

[Red Iron Crown]

EtherDragon, I think you forgot to multiply by g when computing the dV, seems like you're an order of magnitude too low.

[EtherDragon]

EtherDragon, I think you forgot to multiply by g when computing the dV, seems like you're an order of magnitude too low.

Depends on your units of measure. But yea, the usual thing people do is take Isp and replace it with Ve * g.

Now, one interesting question that the OP asks about is how one can theoretically "get past" this limit. Simple fact - if you carry all your fuel and oxidizer on board, you cant; the only way is to have a more efficient engine. In RL the efficiency of liquid fuel / oxidizer has a well known theoretical max, and it's actually pretty low.

But, we already see examples where the Delta-V can be much greater - Air breathing air-craft don't bring oxidizer, they pull it out of the atmosphere. Air breathers also don't create all the thrust with just fuel; turbo-fans use the air to provide some of the thrust.

Other theoretical space-craft might do something similar with the solar winds. Rather than carry all the fuel with them, they have an energy source that creates a powerful magnetic field accelerating hydrogen atoms (from space) around the ship causing a change in velocity. Think Ion propulsion, but without having to take xenon with you. Possibly, fusion power may be sufficient to take hydrogen atoms abundant in space around our solar system and use fusion propulsion to both power systems and provide lots thrust. Any time you can get fuel or thrust from another source other than what you brought with you your actual Delta-V goes up.

[Xavven]

Technically, theoretically, the ln function is infinite, but the amount of fuel (without any extra non-fuel mass) must grow exponentially for more Dv.

But don't get confused with just the ln function and what occurs in the real world (or even in KSP). In KSP, the fact remains - if you take the most efficient engine, and attach it to such a huge rocket that the weight of the engine is a non-factor, you are still stuck with the dry mass of the fuel tanks. For a Rockomax Jumbo-64 that's 4t dry mass 32t fuel.

Just the fuel (one tank) and (an LVN) engine gives you 6.25 empty, 38.25 full = 1449 Dv

1,000 tanks and one lvn, 4,002.25 empty, 36,002.25 full = 1757 Dv

1,000,000 tanks and one lvn, 4,000,002.25 empty 36,000,002.25 full = 1757 Dv

That 4t empty mass per tank sets a hard (mathematical) limit to the Delta V from a single stage.

It looks like the discussion is split into two threads-- one talking about theoretical max delta v in a single stage, and one talking about theoretical max delta v period.

With no staging allowed, those who have commented on the limits based on the mass ratio of the fuel tanks themselves are correct.

With an infinite amount of staging, there would appear to be no theoretical max just looking at the rocket equation itself, but other things start to become a factor when your rocket gets infinitely large. For one, material strength would keep you from building a very large rocket that wouldn't collapse under its own weight in Kerbin's gravity, or from begin spaghettified from gravitational tidal forces as your rocket gets so tall that the top is in space while the bottom is on the ground. Getting larger still, your rocket would collapse under its own gravity into a neutron star or a black hole. So, while the ln function increases to infinity, even with infinite staging, you still have a limit. And the tyranny of the rocket equation means that at some point, you have to add a LOT of fuel just to get another 1 m/s of delta v such that you run into the materials strength limit really fast!

[PUNiSH3R]

Single stage limit:

Delta-V as a function of Isp and fuel fraction.

Easier to visualize if we take a "slice" at a particular Isp value:

You grok?

[Red Iron Crown]

But, we already see examples where the Delta-V can be much greater - Air breathing air-craft don't bring oxidizer, they pull it out of the atmosphere. Air breathers also don't create all the thrust with just fuel; turbo-fans use the air to provide some of the thrust.

Other theoretical space-craft might do something similar with the solar winds. Rather than carry all the fuel with them, they have an energy source that creates a powerful magnetic field accelerating hydrogen atoms (from space) around the ship causing a change in velocity. Think Ion propulsion, but without having to take xenon with you. Possibly, fusion power may be sufficient to take hydrogen atoms abundant in space around our solar system and use fusion propulsion to both power systems and provide lots thrust. Any time you can get fuel or thrust from another source other than what you brought with you your actual Delta-V goes up.

Delta-V is only a useful calculation for reaction engines that carry all their reaction mass with them, otherwise it's not a fixed quantity.