Payload Fraction Calculation for Reusable Vehicles

When I was learning how to use mass-estimating relationships (MERs) at Georgia Tech, our focus was on reusable launch vehicles, and most of our MERs came from NASA Langley, where my professor had once worked. When it came to much of the reusability aspects of the spacecraft, the MER tended to depend on the entry or landing mass of the spacecraft rather than the initial mass or the propellant mass, as I have previously defined.

Systems like thermal protection, wings, aerosurfaces, and landing gear all tended to scale with entry or landing mass. So to capture these effects in order to help calculate performance of reusable vehicles, I introduce another non-dimensional term. I use the letter epsilon to describe the entry-mass-sensitive mass items divided by the entry mass.

     \begin{displaymath} \epsilon \equiv \frac{m_{entry-mass-sensitive}}{m_{entry}} \end{displaymath}

The difference between the entry mass and the dry mass depends mostly on whether or not the payload is intended to return with the spacecraft, so I include a jettison factor (fjett) whose value is one if all the payload is jettisoned and zero if none of the payload is jettisoned before entry.

The initial mass of the vehicle is simply the mass of all the propellant, all the payload, and the dry mass of the vehicle. If the vehicle is a multi-stage vehicle, then we consider each stage individually, and all of the upper stages are payload for the first stage.

     \begin{displaymath} m_{initial} = m_{payload} + m_{propellant} + m_{vehicle-dry} \end{displaymath}

Using our mass-sensitive terms (initial, propellant, and entry) we can also express the dry weight of the vehicle in terms of these sensitivities. The engines and thrust structure will be accommodated by the initial-mass-sensitive term, the tankage and feedlines will be accommodated by the propellant-sensitive term, and the wings, thermal-protection, landing gear, and other structures will be accommodated by the entry-mass-sensitive term.

     \begin{displaymath} m_{vehicle-dry} = \lambda (m_{propellant}) + \phi (m_{initial}) + \epsilon (m_{entry}) \end{displaymath}

The mass of the spacecraft at atmospheric entry is not the same as the dry mass. It might be returning with some fraction of its payload. Therefore we account for this fact by a simple process of elimination. The entry mass is the initial mass, less the propellant, less the payload multiplied by a fraction that is jettisoned before entry. If we mean to design the vehicle to be able to return with the payload (like the Space Shuttle) then the fraction of payload to be jettisoned would be zero.

     \begin{displaymath} m_{entry} = m_{initial} - m_{propellant} - f_{jett} (m_{payload}) \end{displaymath} \label{entry-mass-eq}

By substituting this definition for the entry mass of the vehicle into the definition of the vehicle mass, we obtain:

     \begin{displaymath} m_{vehicle-dry} = (\lambda - \epsilon) m_{propellant} + (\phi + \epsilon) m_{initial} - (f_{jett} \epsilon) m_{payload} \end{displaymath}

     \begin{displaymath} m_{initial} (1 - \phi - \epsilon) = m_{propellant} (1 + \lambda - \epsilon) + m_{payload} (1 - f_{jett} \epsilon) \end{displaymath}

Let’s use the Greek letter “eta” to define the mass ratio, which we obtain from the rocket equation:

     \begin{displaymath} \eta = \frac{m_{propellant}}{m_{initial}} = \exp\left(\frac{\Delta v}{v_e}\right) \end{displaymath}

I employ the now-familiar substitution for propellant mass:

     \begin{displaymath} m_{propellant} = m_{initial} \left(\frac{\eta - 1}{\eta}\right) = m_{initial} (PMF) \end{displaymath}

     \begin{displaymath} m_{initial} (1 - \phi - \epsilon) = m_{initial} (PMF) (1 + \lambda - \epsilon) + m_{payload} (1 - f_{jett} \epsilon) \end{displaymath}

     \begin{displaymath} m_{initial} (1 - \phi - \epsilon - PMF (1 + \lambda - \epsilon)) = m_{payload} (1 - f_{jett} \epsilon) \end{displaymath}

     \begin{displaymath} \frac{m_{payload}}{m_{initial}} = \frac{1 - \phi - \epsilon - PMF - \lambda (PMF) + \epsilon (PMF)}{1 - f_{jett} \epsilon} \end{displaymath}

     \begin{displaymath} \frac{m_{payload}}{m_{initial}} = \frac{(1 - PMF) - \lambda (PMF) - \phi - \epsilon (1 - PMF)}{1 - f_{jett} \epsilon} \end{displaymath}

     \begin{displaymath} \frac{m_{payload}}{m_{initial}} = \frac{(1 - \epsilon)(1 - PMF) - \lambda (PMF) - \phi}{1 - f_{jett} \epsilon} \end{displaymath}

     \begin{displaymath} FMF = 1 - PMF \end{displaymath}

And simplify:

     \begin{displaymath} \frac{m_{payload}}{m_{initial}} = \frac{(1 - \epsilon) FMF - \lambda (PMF) - \phi}{1 - f_{jett} \epsilon} \end{displaymath}

Which as you can see is the same as the previous expressions

     \begin{displaymath} \frac{m_{payload}}{m_{initial}} = FMF - \lambda (PMF) - \phi \end{displaymath}

if epsilon is zero.

A simple way to understand this expression is to think about it in terms of one. One is the most payload fraction you could have–if all of your spacecraft is payload. But some fraction has to be propellant, according to the rocket equation. So you start out with your final mass fraction (FMF). From that point, which will always be less than one, you subtract lambda times the propellant mass fraction. Then you subtract your phi term, which depends mostly on your engines. Finally you subtract epsilon times your final mass fraction. If you have anything left over, you have a payload fraction. If you were going to jettison your payload before reentry, then the denominator gets a little smaller than one and your payload fraction improves a bit. But it can never improve a payload fraction that is less than zero.

If your payload fraction is less than zero, then you had better go change something to clean things up. You better use a better Isp to improve final mass fraction, or better tankage or propellants to improve lambda, or better engines to improve phi, or better TPS or wings or landing gear to improve epsilon. Because if the numerator of the payload fraction is less than zero, you’ve got no reason to build your rocket.

Now, interestingly, there are a surprising number of occasions when you want to run this equation in reverse. You may already know the payload fraction and you want to know the mass ratio. This might occur, for instance, if you are designing a two-stage rocket and you know (or have specified) the initial masses of both stages. This is often encountered when one is trying to enforce an initial thrust-to-weight of the stage. Then, knowing the initial masses of the two stages, and recognizing that the second stage is the “payload” of the first stage, you already know the payload fraction of the first stage. What you really want to know is the mass ratio of the first stage.

Never fear, it turns out to be just about as easy to isolate mass ratio as one might have hoped. We expand our previous expression, replacing final mass ratio and propellant mass ratio with their definitions in terms of mass ratio.

     \begin{displaymath} PF = \frac{\dfrac{(1 - \epsilon)}{\eta} - \lambda\left(1 - \dfrac{1}{\eta}\right) - \phi}{1 - f_{jett} \epsilon} \end{displaymath}

and then we undertake all the usual algebraic expansion and collection,

     \begin{displaymath} PF(1 - f_{jett} \epsilon) = \frac{(1 - \epsilon)}{\eta} - \lambda + \frac{\lambda}{\eta} - \phi \end{displaymath}

     \begin{displaymath} PF(1 - f_{jett} \epsilon) = \frac{(1 - \epsilon + \lambda)}{\eta} - \lambda - \phi \end{displaymath}

     \begin{displaymath} PF(1 - f_{jett} \epsilon) + \lambda + \phi = \frac{(1 - \epsilon + \lambda)}{\eta} \end{displaymath}

in order to arrive at an expression for mass ratio in terms of payload fraction and the non-dimensional parameters I have previously defined.

     \begin{displaymath} \eta = \frac{1 - \epsilon + \lambda}{PF(1 - f_{jett} \epsilon) + \lambda + \phi } \end{displaymath}

with the mass ratio in hand, we can then use the rocket equation to find the change-in-velocity, which is probably what we were seeking.

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MS, nuclear engineering, University of Tennessee, 2014, Flibe Energy, president, 2011-present, Teledyne Brown Engineering, chief nuclear technologist, 2010-2011, NASA Marshall Space Flight Center, aerospace engineer, 2000-2010, MS, aerospace engineering, Georgia Tech, 1999

About Kirk Sorensen

MS, nuclear engineering, University of Tennessee, 2014, Flibe Energy, president, 2011-present, Teledyne Brown Engineering, chief nuclear technologist, 2010-2011, NASA Marshall Space Flight Center, aerospace engineer, 2000-2010, MS, aerospace engineering, Georgia Tech, 1999
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7 Responses to Payload Fraction Calculation for Reusable Vehicles

  1. Grif Ingram says:

    Dear Mr Sorenson,
    Thank-you for this very informative series! I’ve passed it on to a friend of mine who is doing some studies on various propellant combinations for a given mission, and I think I’ll pass it on to scott Lowther of and Winchell Chung of the “Atomic Rockets” website!
    A small problem is that the public-access computers here at Bexleyheath Library can’t show your equations, and I have to use a local cyber-cafe…which is closed as the two prpprietors take a holiday, meaning I can’t really appreciate the last episode…GRRR!
    I’d imagine that most people reading this will have this data already, but how about a table of exhaust velocities, structural factors, etc, just to spoon-feed me? 😉
    Thanks Again,
    Grif Ingram

  2. Robert Clark says:

    Grif, that Scott Lowther web site is

    Kirk, could you put some numbers in so we can see a comparison between feasible and non-feasible SSTO designs.

    Bob Clark

  3. Fascinating reading! Most of it is way above my pay grade, but I can appreciate the equations.

    Grif Ingram, thanks for offering to clue me in to this blog. However I already stumbled over it.

  4. I agree with Winch above – excellent stuff. It tends to confirm my impression that absent major materials advances, SSTO is a nonstarter, requiring extreme designs that are marginal to work at all, let alone with the robustness for operational use.

    My guess is that the first fully reusable orbital craft will be three stages to orbit, on extended SpaceShipTwo lines – airbreathing first stage, spaceplane second stage, and the orbiter, allowing each stage to be more conservative in requirements and engineering margins.

  5. John Bucknell says:

    This is a little late – but what does this audience think of Paul March’s LA-NTR concept?

  6. Jonathan Goff Jonathan Goff says:

    It’s an interesting concept–IIRC that’s part of what led Aerojet to its Thrust Augmented Nozzles idea. Not sure if it saves the NTR as a worthwhile engine, but the general concept of taking a normally fuel-rich engine, and getting better thrust and propellant density by running LOX-rich augmentation in the nozzle isn’t crazy. I know Gary Hudson has talked about doing that to make SSTO RLVs more feasible. Basically use a LOX/LH2 engine, and then have the TAN be a very lean LOX/LH2 injection. Keeps you at two propellant tanks, but biases the LV’s propellant mixture ratio more towards the LOX (better bulk density), while giving you the benefit of the high Isp of the LOX/LH2 combo once you’re out of the atmosphere.

    Haven’t run numbers, but if I had to do an SSTO, that’s the first approach I’d be looking at.


  7. John Bucknell says:

    Thanks Jon,

    Mostly I was interested in expert commentary on the LANTR concept to see if March had done his analysis correctly. I have not found any such commentary – but if he has valid ideas there has been a lot of concurrent work on Nuclear Electric Thruster reactor designs that may further enhance the nuclear launch field (as they are focused on power to weight as well).


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