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Is there a way to quantify how much you would gain by air-launching a rocket compared to a ground launch?

Hobbes
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I came across a fairly detailed comparison made by the team that worked on Interim HOTOL.

Delta-v required for a vertical launch SSTO (e.g. Delta Clipper) to LEO: 9361 m/s

For Interim Hotol:

  • speed supplied by launch aircraft (An-225), launching at Mach 0.8 at 9 km altitude: - 235 m/s
  • drag loss: + 67 m/s
  • gravity losses: - 670 m/s
  • Isp underexpansion losses at low altitude: - 180 m/s
  • Thrust vectoring demands: + 10 m/s
  • Improved engine Isp due to altitude start: - 214 m/s

for a total reduction in delta-V of 1222 m/s, or 13%, translating into a 24% reduction in propellant requirement.

From the book 'Spaceflight in the era of aero-space planes' (R. Hannigan, Krieger publishing, 1994).

Hobbes
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  • Are the +67m/s drag loss related to the attitude change, from horizontal to suborbital trajectory? And are drag gains (from not having to go through 9km of dense atmosphere) already summed up into this result of +67? – user721108 Apr 22 '19 at 14:34
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    "Isp underexpansion losses at low altitude" and "improved engine Isp due to altitude start" seem like they should be the same thing, so I'm a little suspicious of this accounting. – Russell Borogove Apr 22 '19 at 14:39
  • The book doesn't go into more detail than what I've supplied. The calculation was originally made by BAe and TsAGI for a presentation to ESA. – Hobbes Apr 22 '19 at 15:03
  • Could this be the difference? 'Isp underexpansion losses' are the losses you avoid by not running the rocket at 0-9 km altitude, and 'improved engine Isp' is the gain you get by tuning the nozzle for the 9km+ regime. – Hobbes Apr 22 '19 at 15:48
  • What's interesting about this is while Elon Musk has said it's only around a 5% savings, this comparison shows substantially more. – TemporalWolf Apr 22 '19 at 20:18
  • Curious about the gravity loses. This implies it would have taken over a minute to get to 9km up if I'm understanding things right. – GremlinWranger Apr 23 '19 at 07:47
  • @GremlinWranger a linear increasing speed from 0 at ground to 235 m/s at 9 km height would take 76 seconds, a little over a minute. – Uwe Apr 23 '19 at 08:03