# Why does $I_\text{sp}$ on an upper stage matter so dramatically?

When you compare the upper stage of an Atlas 5/Delta 4, which use an RL-10 upper stage engine vs a Falcon 9 upper stage that uses a Merlin 1D Vac version, you see really big performance differences.

The RL-10 is a LOX/LH2 engine with great specific impulse ($I_\text{sp}$ of 430-450 s depending on version) but crappy thrust of only 20-25 Klbs.

The Merlin 1D vac is an LOX/RP1 engine with so-so $I_\text{sp}$ (good for RP-1 though) at 304 s, but much better thrust at 180 Klbs.

Yet, the RL-10 is considered a much better upper stage than a Merlin 1D.

Sure, that is a pretty big $I_\text{sp}$ gap (25% more for RL-10), but with so much more thrust (8-9 times more for Merlin), it seems like it should not be such a big deal.

I understand that $I_\text{sp}$ is a measure of efficiency, in terms of how many seconds of 'use' you get per unit of fuel. Thus an RL-10 is 25% more efficient than a Merlin 1D. But as long as you are carrying sufficient fuel/oxidizer and have the thrust, why would the 8-9 times different thrust values matter more?

Ultimately this is probably a question of why $I_\text{sp}$ matters so much in this context?

• This is just another example of the tyranny of the rocket equation. You've already burned a lot of fuel to get that pound of propellant up there. – Loren Pechtel Jan 3 '14 at 0:58
• Yep. The upper stage and its propellant are costing you because you need a first stage with enough propellant to get them high & fast enough. Once you're out of significant atmospheric drag and are fast enough to avoid immediate re-entry, the time it takes to enter the desired orbit doesn't cost that much. The mass of the upper stage is still costing you payload capacity, though; a more efficient rocket means more payload for the same combined upper-stage+payload mass (i.e. the stuff the first stage must boost). – CBHacking Jan 17 '16 at 11:11

The mass required to deliver the required $\Delta\mathrm{V}$ depends on the Isp and the dry mass ratio of the stage. It does not depend on the thrust, except to the extent that a high thrust would increase the mass required due to a large engine and feeds. For a given Isp, the thrust just determines how long it takes to use all the propellant. For an upper stage, you're not in as much of a rush as for lower stages. You are already on a trajectory that gives you some time in vacuum to complete your orbit insertion. Once in orbit you have all the time in the world for the next burn of the upper stage, if there is one, to go to a different orbit or escape.

You just need enough thrust to get the burn done before you start falling back, if it's completing an orbit insertion. For the next burn, you need enough thrust so that the burn can be completed before a lot of altitude is gained, to maximize the Oberth effect (more $\Delta$energy per unit $\Delta\mathrm{V}$). More thrust than that doesn't help an upper stage. It just hurts with more mass that isn't needed.

• High trust does have a slight advantage; You get a more efficient burn if you can consentrate it in a single instant. (But this effect is of course very slight compared to a 25% $I_{sp}$) – Taemyr Jan 13 '15 at 11:59
• "For the next burn, you need enough thrust so that the burn can be completed before a lot of altitude is gained, to maximize the Oberth effect" - and even that is just convenience, as the transfer burn can be split into multiple smaller burns near apoapsis, or with extreme ISp engines you just neglect the losses and spiral away. – SF. Jan 15 '16 at 5:04
• You mean periapsis, but yes. – Mark Adler Jan 15 '16 at 6:13
• @MarkAdler: Yes, periapsis. Me bad. – SF. Jan 15 '16 at 11:39

The first part of the ascent profile is nearly vertical so as to get out of the thick atmosphere as soon as possible.

When accelerating vertically, the ship suffers gravity loss:

Every 102 seconds spent in vertical ascent gives you a gravity loss hit of 1 km/s.

It's desirable to reduce time of vertical ascent so as to minimize gravity loss. How fast a ship gets above the atmosphere is related to the thrust to weight ratio, a.k.a. T/W.

If the rocket's thrust exactly equals gravity, 9.8 meters/sec^2, T/W=1 and the rocket just hovers. A big T/W helps the ship ascend faster.

Generally, better ISP means less thrust. Ion engines have fantastic ISP but miniscule thrust.

So it's not really a question of the upper stage needing more ISP. We want good ISP for all the stages. But the booster stage needs more thrust.

• "Every 102 seconds spent in vertical ascent gives you a gravity loss hit of 1 km/s." - now that is an evocative figure that shows how costly gravity drag really is! – SF. Jan 15 '16 at 5:06
• Why did you have to use Comic Sans for that diagram? – Philipp Jan 16 '16 at 11:54
• @Phillip Two reasons: I used Comic Sans because 1) It's a legible, informal typeface. Often I use Myriad or Avante Garde but sometimes feel like I'm getting too stuffy. 2) To flip off the hipster fashion police trying to dictate what fonts to use. Yeah, Comic Sans is used a lot. Because it's a nice typeface. The Oatmeal can eat me. – HopDavid Jan 16 '16 at 14:41