Ratio of first- to second-stage engine thrust?

[Exosphere]

While looking at the specs of various rockets, I noticed something -- some rockets, such as the Falcon 9, R-7, and Proton, have a relatively low ratio between the thrust of the first stage and that of the second stage. For example, the Proton's "stage thrust ratio" is 4.36:1. On the other hand, rockets such as the Delta II, Delta IV, and Atlas V have a very high ratio -- the Atlas V's stage thrust ratio is 41.85:1.

I know that there may be some factors at play other than squeezing the best performance out of each rocket (for example, the Falcon 9 might have been designed with such a low ratio so the production line could be simplified by using a variant of the Merlin 1 as the second stage engine), but is there any performance (in terms of efficiency at getting the largest payload to orbit) impetus behind either a very high of a very low ratio between the thrust of the first stage engine(s) and that of the second stage? It would seem like both designs would have some advantage.

With a "high thrust" second stage, the rocket could be designed so that it would rely more on its second stage to get into orbit than a rocket with a smaller second stage. This could be an advantage, because the second stage usually has a very high area ratio nozzle, which would improve its specific impulse in a vacuum. Instead of continuing to use the relatively inefficient first stage (with a nozzle designed to perform well at sea level, not in a vacuum) for a significant portion of the rocket's thrust in the upper atmosphere, the rocket could utilize its more efficient vacuum nozzles earlier.

However, if the second stage was made too large in relation to the overall size of the rocket, it would somewhat defeat the purpose of staging. By using a larger second stage and a smaller first stage, the rocket would not get rid of as much mass by staging, which could reduce its overall mass fraction by some measure.

So, is there any way to determine (in general) whether it's more efficient to have a high or a low ratio between the thrust of the first stage vs. that of the second stage?

[Iskierka]

In pretty much all two-stage or two-plus-boost-stage rockets, by the time the second stage is burning they're well beyond the need of overcoming gravity. The thrust ratio comes down to economics - in the Atlas, politics requires them to have lots of separate manufacturing, so two very different engines will get ordered. As such, with no restrictions, the second stage engine is very low thrust, intended to build up velocity slowly rather than fight gravity, and save fuel by increasing mass ratio. In Falcon, and Russian rockets, the engine manufacturers are generally the same. As such, since they already have to make a kind of engine, they'd like to use a similar engine on the next stage, so that parts can be duplicated to save costs.

In rocket launches, what you find is that fuel cost is negligible, consistently less than about half a percent. When you have to use separate manufacturing anyway so you can't save cost by sharing engines, it's preferable to make the rest of the rocket slightly smaller and save a little fuel, as with the Atlas. When you don't, what you find is that most of the cost is in the rocket, and a large fraction of that is in the engines. So, rather than save on fuel or rocket size, it turns out more profitable to save money by duplicating engine components, as SpaceX and the Russians have been doing. If you look at the stage stats, you'll notice that despite the very different thrust ratios, the two stages will cover about the same ranges in terms of dv ratio, and thus contribute about the same getting into orbit regardless of second stage size.

[Camacha]

What about delta-v? How do those compare?

[cantab]

I believe - but am not sure - that it's generally best for each stage in a serially-staged rocket to contribute about the same amount of delta-V. So for a two stage to orbit launcher it'd be about 4500 m/s a stage.

[GoSlash27]

I don't look at the ratio of thrust from one stage to the next, but rather the thrust that I need from each stage to do it's mission.

For a Kerbin payload lift to orbit, I break it down into 3 phases (terminology may be wrong) in reverse order:

3) Injection. Stage + payload to establish uniform orbit, intercept a target, rendezvous and dock. 1,500 M/sec DV at no less than .75 G.

2) Transstage. Stage + injection stage + payload from gravity turn (roughly 7kM) to establishing apoapsis at 80 Km. 2,000 DV at no less than 1G.

1) Boost stage. Stage + subsequent stages + payload from the pad to gravity turn. 1,500 DV at no less than 2G.

I may (and usually do) break these stages down into sub- stages to lighten the upper stages.

Upshot is: The thrust ratio ends up being whatever it happens to be.

Best,

-Slashy

[Camacha]

Yeah, that is what I meant. I feel that the thrust factor is not a driving number.

[sgt_flyer]

For the disparity between proton and other rockets, keep in mind at which point the proton second stage is ignited - waaay earlier during ascent, because proton is a full 3 stage serial rocket + an additional transfer stage most of the time. (block D for proton K, briz-M for proton M)

[Armchair Rocket Scientist]

While looking at the specs of various rockets, I noticed something -- some rockets, such as the Falcon 9, R-7, and Proton, have a relatively low ratio between the thrust of the first stage and that of the second stage. For example, the Proton's "stage thrust ratio" is 4.36:1. On the other hand, rockets such as the Delta II, Delta IV, and Atlas V have a very high ratio -- the Atlas V's stage thrust ratio is 41.85:1.

I know that there may be some factors at play other than squeezing the best performance out of each rocket (for example, the Falcon 9 might have been designed with such a low ratio so the production line could be simplified by using a variant of the Merlin 1 as the second stage engine), but is there any performance (in terms of efficiency at getting the largest payload to orbit) impetus behind either a very high of a very low ratio between the thrust of the first stage engine(s) and that of the second stage? It would seem like both designs would have some advantage.

With a "high thrust" second stage, the rocket could be designed so that it would rely more on its second stage to get into orbit than a rocket with a smaller second stage. This could be an advantage, because the second stage usually has a very high area ratio nozzle, which would improve its specific impulse in a vacuum. Instead of continuing to use the relatively inefficient first stage (with a nozzle designed to perform well at sea level, not in a vacuum) for a significant portion of the rocket's thrust in the upper atmosphere, the rocket could utilize its more efficient vacuum nozzles earlier.

However, if the second stage was made too large in relation to the overall size of the rocket, it would somewhat defeat the purpose of staging. By using a larger second stage and a smaller first stage, the rocket would not get rid of as much mass by staging, which could reduce its overall mass fraction by some measure.

So, is there any way to determine (in general) whether it's more efficient to have a high or a low ratio between the thrust of the first stage vs. that of the second stage?

A major factor is how fast the first stage is moving when the second stage ignites. For example, according to my calculations based on this, the Proton-M's first stage only provides 2750 m/s of dV, while the Atlas V and Delta IV CCBs can provide 6000 m/s of dV (in the lightest configuration). According to

, the Ariane 5 is moving 6.91 km/s at first-stage cutoff.

Now, a higher dV first stage is a higher fraction of the vehicle's launch mass, so the ratio between vehicle mass at first stage ignition and vehicle mass at second stage ignition is higher. For example, the Proton second stage + everything above it is 37% of the vehicle's launch mass. For the Delta IV or Atlas V, it's more like 10%. If the second stage needs to accelerate less mass at a certain minimum TWR, it needs less thrust.

In addition, because the second stage is contributes less of the dV required to reach a stable orbit, it can safely use a lower TWR and still have time to reach orbital velocity without falling back into the atmosphere. The second stage of the Proton-M has a TWR of about 0.9, while the DCSS and Centaur get more like 0.3 depending on the payload.

Falcon 9's second stage probably has a bit more thrust than it needs to because (a) SpaceX wants to use the same engine for both stages to reduce cost ( F9 will probably fly a steeper flight profile than most rockets to reduce return fuel requirements for the first stage.

[Iskierka]

I believe - but am not sure - that it's generally best for each stage in a serially-staged rocket to contribute about the same amount of delta-V. So for a two stage to orbit launcher it'd be about 4500 m/s a stage.

This is often taught but not entirely accurate. Two kerosene stages of 4500 m/s requires a mass ratio of ~3.7 each, with a little spare fuel, or 13.5 overall. If you adjust the split slightly, say 5000/4000, you find that the overall ratio is still 13.5. This doesn't account for certain factors, which when included, results in the upper stage being given more dv, due to factors like a lower minimum TWR requirement, resulting in a lighter engine. Thus making it preferable to have the smaller stage ratio on the lower stage, to avoid burning more fuel to lift the upper stage's. Further upper stages will be optimised for greater expansion, giving a higher Isp and making it preferable to burn with this engine.

As a result splits more like 6000/3000 are common. Higher is seen regularly when cryogenic upper stages are involved, as the kerosene then becomes a boost stage to get the efficient hydrogen stage moving, and they want to use as little weight as possible on the less efficient fuel. Kerosene lower stages with cryogenic upper stages are commonly only 2000-2500 m/s, leaving the rest to the upper stage.

[Armchair Rocket Scientist]

This is often taught but not entirely accurate. Two kerosene stages of 4500 m/s requires a mass ratio of ~3.7 each, with a little spare fuel, or 13.5 overall. If you adjust the split slightly, say 5000/4000, you find that the overall ratio is still 13.5. This doesn't account for certain factors, which when included, results in the upper stage being given more dv, due to factors like a lower minimum TWR requirement, resulting in a lighter engine. Thus making it preferable to have the smaller stage ratio on the lower stage, to avoid burning more fuel to lift the upper stage's. Further upper stages will be optimised for greater expansion, giving a higher Isp and making it preferable to burn with this engine.

This isn't accurate.

Let's check the math. With an ISP of 400 (about midway between kerosene and hydrogen) and assuming a stage's mass is 10% the mass of its fuel, I'll consider various configurations for a rocket with a total of 10,000 m/s of dV and estimate the payload fraction.

0/10000 m/s (SSTO case): Payload fraction -1.4% (not possible).

1000/9000 m/s: Stage 1 payload fraction 75%, stage 2 payload fraction 1.1%. Total payload fraction: 0.8%.

2000/8000 m/s: Stage 1 payload fraction 56%, stage 2 payload fraction 4.3%. Total payload fraction: 2.5%.

3000/7000 m/s: Stage 1 payload fraction 41%, stage 2 payload fraction 8.5%. Total payload fraction: 3.5%.

4000/6000 m/s: Stage 1 payload fraction 30%, stage 2 payload fraction 14%. Total payload fraction: 4.1%.

5000/5000 m/s: Stage 1 payload fraction 21%, stage 2 payload fraction 21%. Total payload fraction 4.3%.

As you can see, payload fraction is maximized when the stages contribute equal dV. However, as you mentioned, this is affected by upper stages having higher ISP, which tips the balance in favor of upper stages.

As a result splits more like 6000/3000 are common. Higher is seen regularly when cryogenic upper stages are involved, as the kerosene then becomes a boost stage to get the efficient hydrogen stage moving, and they want to use as little weight as possible on the less efficient fuel. Kerosene lower stages with cryogenic upper stages are commonly only 2000-2500 m/s, leaving the rest to the upper stage.

This is outright wrong. The lowest dV for a liquid-fueled stage that I've heard of is 2750 m/s on the Proton-M, and it has 3-4 stages. The Saturn V's lower stage gets about 3800 m/s (using the vacuum ISP stats) and as I said, Delta IV and Atlas V both get over 6 km/s from their lower stages. The Ariane 5 probably gets more like 8.5, but its huge SRBs do a lot of the work compared to the GEM-60s on a Delta or Atlas.

The reason for this, incidentally, is that many rockets are optimized for launches to GTO, which requires an extra 2.5 km/s. In reality, an Atlas or Delta's dV is split about evenly, with each stage providing 6000 m/s. On the other hand, the Falcon 9's stages each provide about half the dV for an LEO launch.

[Exosphere]

Well, I found something that kind of answers the question I was asking (although all of your answers were definitely helpful and informative!).

On the Atomic Rockets section of Project Rho (a great site; I'd recommend it to anyone that hasn't heard of it), there's a section that details how to determine how much dV should be allocated to each stage to maximize efficiency. Basically, the process is pretty long and laborious (especially for 3+ stage rockets), and is known as the "iterative process." An example is provided at the end of the page here:

http://www.projectrho.com/public_html/rocket/multistage.php

As far as thrust goes, I was using that metric because I would assume that each upper stage should have a TWR > 1.0. Obviously, the first stage has to have that level of thrust (else the rocket will just sit on the pad and vibrate), but do upper stages also need to have a thrust to weight ratio greater than one? I would assume the only requirement on the upper-most stage is that its TWR is high enough to accelerate the payload to orbital velocity before it falls back into the atmosphere, and I think I remember reading that the Saturn V's S-IVB stage had a TWR<1, so is it safe for an orbital rocket to have such a low thrust to weight ratio? I know the answer probably varies (a rocket launching a payload to GTO has a longer time to accelerate it than one taking a satellite to LEO), but is there any "absolute minimum TWR" that the upper stage of a standard commercial rocket would have to have to accelerate its payload to orbital velocity prior to falling into the atmosphere?

[Kryten]

For the disparity between proton and other rockets, keep in mind at which point the proton second stage is ignited - waaay earlier during ascent, because proton is a full 3 stage serial rocket + an additional transfer stage most of the time. (block D for proton K, briz-M for proton M)

This is backwards-Proton was initially a two-stage rocket, they added more stages because it had the extra capability.