When does $I_{sp}$ become more of a concern than thrust?

Question

In an orbital launch from Earth, the priority in the initial phase of the launch is high thrust, so as to actually get off the ground, and minimise losses due to "gravity drag" and air resistance. Later in the launch, and certainly if you want to go beyond LEO, the most important consideration is $I_{sp}$ — essentially exhaust velocity, to attain maximum delta-V from a given mass ratio.

I'm interested in trying to understand when the transition between these two considerations happens — how high and fast do you need to be before switching to a low-thrust (not very low — we're still talking chemical rockets, not ion or lightsail) but high $I_{sp}$ second stage makes sense. I appreciate that there won't be a sharp answer, but it would be interesting to understand the range of options.

Answer 1

Ideally, thrust and specific impulse would gradually trade off during flight, but that isn’t generally achievable. Instead, large changes in thrust and ISP are done during staging.

It's pretty common for the second stage (or core stage, for a solid booster/liquid sustainer configuration) to start at less than 1:1 thrust-to-weight ratio. That seems like a pretty good proxy for "Isp is more important than thrust": you're literally giving up vertical speed that you've already achieved for gains in fuel efficiency. Because the rocket is rapidly throwing away propellant mass, the thrust-to-weight ratio recovers before long, and staging would be advantageous even if the Isp weren't different between stages, but it's still a significant and easy-to-study marking point.

Case: Saturn V

For the Saturn V, the first staging switches over from the massive kerosene-LOX F-1 engines with a vacuum Isp of 304 seconds to hydrogen-LOX J-2 engines with specific impulse around 421-423 seconds. First stage cutoff takes place at about 162 seconds into the flight, at 70km altitude. Dynamic atmospheric pressure is down to about 450 Pa from the max Q peak of almost 34 kPa -- there's almost no air left. The rocket's velocity is around 2.65 km/s, nearly mach 8. The vertical velocity component is about 900 m/s. When the second stage starts, it's firing at about 0.8 thrust-to-weight, and the vertical velocity starts to decrease significantly, but it's already 38% of the way to orbital altitude; the vertical speed actually never increases from that point on.

Specific impulse is so much more important than thrust from here out that the engines switch to a more fuel-rich mix towards the end of the second stage run, giving up 25% of the thrust for a small increase in specific impulse to 427 seconds.

Case: Space Shuttle

Two minutes into flight, the high-thrust, low Isp solid rocket boosters burn out. This is at a lower altitude than Saturn V, around 50km instead of 70km -- atmospheric resistance is still somewhere around 2 kPa. The main engines at this point are producing about 0.9:1 thrust-to-weight with a specific impulse of 452 seconds. Later in the main engine burn, the engines are deliberately throttled down to maintain a ~3g limit for crew comfort and payload safety. Unlike the Saturn V, where the vertical speed never goes negative, the space shuttle does give up some altitude late in the burn.

Case: Ariane 5

Ariane 5 is another solid booster / hydrogen sustainer rocket. At booster cutoff, the sustainer is delivering about 0.8:1 thrust-to-weight at 432 s Isp. That finishes off at about 3g, and the upper stage takes over, starting at only 0.25:1 thrust-to-weight at 446s Isp. With that small engine, the rocket takes a good long time getting into orbit, and loses substantial altitude before it does so.

Answer 2

It would be very hard to pinpoint a specific unique condition, but there's a bunch of considerations coming into play that decide it.

• switching to vacuum-optimized bell nozzle. This alone gives you a significant ISp boost - which is always welcome, but comes at a cost - the nozzle is big, you can't fit a whole lot of engines or combustion chambers with these, and it would create excessive air drag anyway. So, first thing, the pressure drop. About 10km is the altitude where it starts making sense, but is usually delayed more to get more speed out of 1st stage.
• keeping TWR in check. Too much thrust can be a problem. Engine operates optimally near its peak thrust; throttling down is usually wasteful (lower chamber pressure - lower exhaust speed - worse ISp!) And with amount of fuel dropping, you no longer need all this massive initial thrust to keep the acceleration. Dump the excess of engines (along with the tanks and the rest...) and keep thrust proportional to weight remaining - happily utilizing the freed up rear end space for a huge vacuum nozzle.
• Upper stages suffer from tyranny of rocket equation more. First stage doesn't need to be light, if it can be cheap and strong instead. The "big dumb tube" philosophy - pack it with fuel to the brim, keeping launch TWR barely above 1, don't worry if it's not as efficient if it can make up in raw thrust+delta-V. Bigger isn't always more expensive. But this isn't true with higher stages. Here the tyranny of rocket equation hurts not only the stage in question but all the stages below. Therefore "big, heavy, cheap, strong" ceases to be a viable approach.

And now for a point which isn't really taken into account for the stage drop:

• gravitational drag dropping. It might seem like with reduction of gravitational drag your need for rapid acceleration drops, as the losses drop. Unfortunately, gravitational drag grows with square of the difference between craft velocity and orbital velocity. That means it achieves "low enough" values only quite shortly before circularization. Most of rockets have switched to higher ISp stages long ago at that point.