# Is this a plot of Isp vs propellant mass fraction for a SSTO vehicle?

Whilst trying to find some information on SSTO designs, I came across this chart:

It was used in an ancient forum thread by as part of an SSTO discussion, but the poster didn't cite the paper they'd taken it from. I got as far as identifying that RBCC meant rocket-based combined cycle, but that was about it.

So. Does this really show Isp vs propellant fraction required for an SSTO, or was the original poster misinformed? (and so what is it showing?)

It would also be nice to know if that line should be straight, or at least curved in only one direction, and if anyone any idea what paper the plot was taken from I'd like to know.

• If you are asking about the thick black line that starts near 1.0 and drops to 0.55, It appears to be a plot of the Tsiolkovsky rocket equation where $\Delta v$ is fixed (possibly 10 km/s) and the parameter $1 - \frac{m_0}{m_f}$ is plotted against ISP. See also (en.wikipedia.org/wiki/Propellant_mass_fraction)
– AJN
Mar 27, 2022 at 11:39
• @AJN I'm guessing more like 7.6 km/s as the fixed $\Delta v$, as this nearly replicates the curve. I guessed 480 seconds as the Isp that requires a propellant mass fraction of 0.8 and solved for $\Delta v$. Mar 27, 2022 at 11:58
• ... and not only $\Delta v$ is fixed, but also the thrust is fixed (a.k.a. gravitational drag is neglected) Mar 27, 2022 at 12:13
• I've found that the open-square-bracket description close-square-bracket open-parenthesis link close-parenthesis is sometimes less problematic. Close parentheses are still problematic. Mar 27, 2022 at 12:39
• The curvature is simple: It has to be S-shaped: For low ISP it has to be tangent to 1 (a lot of fuel and nothing else), for very high ISP it is asymptotic to 0 (almost no fuel). wolframalpha.com/… Mar 27, 2022 at 12:49

A typical rocket needs ~10 km/s of delta-v to reach orbit. An air-breathing vehicle with its slow climb and long path through the atmosphere will need more. Realistically, you'll have different curves for different propulsion technologies and vehicle types, each accounting for the differences in gravity and aerodynamic losses. Here's an attempt based on this derivation from Henry Spencer and Bob Zubrin and the RBCC performance shown in that plot. With L/D of 5 and average acceleration of 0.5g, taking the "equivalent effective specific impulse" from your plot as the specific impulse at $$V_{final}/2$$ and a somewhat wider range for chemical rocket performance: