A Simple Modification of the Rocket Equation

When I was an undergrad, I spent two summers interning on the X-33 program at the Lockheed Martin Skunk Works in Palmdale, California. It was a fantastic experience and I got to meet with and work with some wonderful people on a very exciting program. Plus I got to live in the Mojave for two whole summers!

But the X-33 program, as we know, failed. Then I went to grad school at Georgia Tech and studied under Dr. John Olds in his Space Systems Design Lab. We worked on highly-reusable hypersonic airbreathing space transportation systems. Mostly we studied them for NASA headquarters. But I couldn’t help but wonder where had X-33 gone so wrong?

After graduating from Georgia Tech, I got a job at NASA Marshall Space Flight Center in March 2000, and I worked for Dr. George Schmidt in the Propulsion Research Center. It was another wonderful place to work, and I was surrounded by people who were studying antimatter, or plasma rockets, or nuclear reactors–all kinds of interesting subjects.

At that time there was a bit of a “fad” making the rounds at NASA HQ, and it went something like this: we needed to figure out how to do a roundtrip Mars mission in a year or less. Someone had decided that a year was all the human body could take, or the public would keep interest it, or something like that.

I protested, pointing out the astrodynamics of the situation would make such a mission nigh under (very nigh unto) impossible. But there was another “fad” making the rounds at NASA HQ that was lulling them into a false sense that it could be done.

They called it “abundant chemical”, and it was based on the notion that if you could wave your arms and imagine that all the chemical propellant you could ever want was somehow waiting for you in low Earth orbit, then you could just build a big honkin’ rocket and go to Mars and come back just as fast as you wanted to. No one was super clear about how all this propellant got to LEO, and the most popular approach seemed to involve huge, poorly-defined guns or sling-a-trons of some sort that would somehow make it happen, but my boss George had a different idea.

He tried to derive a simple variation of the rocket equation that would show the folks pushing this “abundant chemical” idea that it really didn’t matter if they could assume unlimited propellant, that if the delta-V of the mission was too high (and the one-year Mars mission certainly fell in that category) that all the propellant in the world couldn’t do it.

Now, for a quick review, the rocket equation is derived rather quickly by integrating the differential change in velocity (dv) on an object with mass (m) by the expulsion of a differential amount of mass (dm) at an exhaust velocity (ve).

     \begin{displaymath} m\:dv -v_e\:dm = 0 \end{displaymath}

     \begin{displaymath} \int_{v_0}^{v_1}dv = v_e\:\int_{m_1}^{m_0}\frac{dm}{m} \end{displaymath}

     \begin{displaymath} v_1 - v_0 = \Delta v = v_e\:(\ln(m_0) - \ln(m_1)) \end{displaymath}

     \begin{displaymath} \Delta v = v_e\:\left(\ln\left(\frac{m_0}{m_1}\right)\right) \end{displaymath}

The result is the amount of change in velocity (delta-V) that can be expected from the expulsion of some fraction of mass at the given exhaust velocity. The rocket equation can also be rewritten so as to tell you for a given delta-velocity and exhaust velocity, what the mass ratio of the rocket ($\eta$) will be in that situation.

     \begin{displaymath} \eta \equiv \frac{m_0}{m_1} = \exp\left(\frac{\Delta v}{v_e}\right) \end{displaymath}

     \begin{displaymath} \frac{m_0}{m_1} = \exp\left(\frac{7300 \text{m/s}}{4500 \text{m/s}}\right) = 5.06 \end{displaymath}

For instance, if you had a delta-V of 7300 m/s and an exhaust velocity of 4500 m/s, the mass ratio predicted from the rocket equation would be 5.06, in other words, your vehicle would be 5.06 times more massive at the beginning of the delta-V maneuver than at the end.

So let’s imagine a rather simple type of rocket. We’ll say it consists of only three things: propellant, structure, and payload. We’ll assume that the initial mass of the rocket is all three of these together, and that the final mass of the rocket is just the structure and the payload–that all of the propellant was used up in the maneuver. We’ll also assume that we know the delta-V and Isp of the rocket, so that we can calculate the mass ratio.

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

     \begin{displaymath} \eta \approx \frac{m_{payload} + m_{structure} + m_{propellant}}{m_{payload} + m_{structure}} \end{displaymath}

Now here’s the part where George did something that got my creative juices flowing, many years ago. He proposed that we imagine that the structure is some fraction of the mass of the propellant. He called this fraction “lambda”, which is a pretty common Greek letter people use when they’re talking about some structural fraction in the rocket equation. George did up a spreadsheet showing how for some practical value of lambda, you would be limited on how much delta-V you could deliver, even if you had lots of propellant.

I started playing around with the equations and was curious if a closed-form solution might be possible that would relate the payload mass (what we’re after) to the gross mass of the vehicle in the first place. With lambda defined, you can proceed to go and use it to replace the structural mass in this extension of the rocket equation.

     \begin{displaymath} \lambda = \frac{m_{structure}}{m_{propellant}} \end{displaymath}

     \begin{displaymath} m_{structure} = \lambda m_{propellant} \end{displaymath}

     \begin{displaymath} \eta = \frac{m_{payload} + m_{propellant}(\lambda + 1)}{m_{payload} + \lambda m_{propellant}} \end{displaymath}

With a little more algebraic mastication, you can simplify things down to just payload mass and propellant mass.

     \begin{displaymath} m_{payload} (\eta - 1) = m_{propellant} (1 + \lambda - \lambda \eta) \end{displaymath}

     \begin{displaymath} m_{payload} = m_{propellant} \left(\frac{1 + \lambda - \lambda \eta}{\eta - 1}\right) \end{displaymath}

To go further, you need to be able to define propellant mass some other way, and then substitute that definition into this expression. Fortunately, the original rocket equation (assuming you know mass ratio) can be solved another way to give you propellant mass.

     \begin{displaymath} \eta = \frac{m_{initial}}{m_{final}} = \frac{m_{initial}}{m_{initial} - m_{propellant}} \end{displaymath}

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

Then this definition can be substituted into the equation for propellant mass, giving you the expression in terms of only payload mass and initial mass, which is what I was after in the first place. Things simplify quite nicely.

     \begin{displaymath} m_{payload} = m_{initial} \left(\frac{\eta - 1}{\eta}\right) \left(\frac{1 + \lambda - \lambda \eta}{\eta - 1}\right) \end{displaymath}

     \begin{displaymath} m_{payload} = m_{initial} \left(\frac{1 + \lambda - \lambda \eta}{\eta}\right) \end{displaymath}

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

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

And finally there it is. An expression for payload fraction of a rocket, defined as the payload mass divided by the initial mass, with expressions for lambda and mass ratio embedded into the equation. One can simplify it even further by recognizing that the inverse of the mass ratio is simply the final mass fraction (FMF) of the rocket:

     \begin{displaymath} FMF \equiv \frac{1}{\eta} \end{displaymath}

and that one minus this quantity is the propellant mass fraction (PMF) of the rocket:

     \begin{displaymath} PMF \equiv 1 - \frac{1}{\eta} \end{displaymath}

then the expression becomes even more simple and obvious.

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

This equation can be very useful, because if you look at the numerator, you can imagine that there is some value of lambda that makes it zero for any given value of mass ratio. That would be the structural lambda at which point there would be no mass for payload. Finding it is very easy by setting the numerator to zero and solving for lambda.

     \begin{displaymath} \lambda_{max} = \frac{1}{\eta - 1} \end{displaymath}

At last I was beginning to get insight into my original question, which is “why didn’t the X-33 work?” For a single-stage-to-orbit vehicle, burning LOX/LH2 propellant at about 450 sec Isp, you can calculate the mass ratio from the rocket equation. Then you can throw the mass ratio and lambda into the expression I derived to get an idea of what kind of payload fraction you could expect.

In the ten years or so since I first did this work, I’ve taken these derivations much further, and I plan to share with you ever-extended derivations of this sort in upcoming posts.

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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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12 Responses to A Simple Modification of the Rocket Equation

  1. BTW, Sutton calls lambda the “propellant mass fraction” of a rocket stage on page 30, specifically leaving out the payload by defining it as the final stage of the vehicle.. and by that definition only payloadless vehicles are SSTO.. one of the great reasons I love to hate that chapter.

    I think the more fun way of thinking about the equations you presented is to wonder what kind of technology one would need to cause the value of lambda to approach one. ๐Ÿ™‚

  2. gravityloss says:

    Thanks Kirk!

    What’s interesting is the dm_payload/dlambda. That kind of tends to describe the “hardness” or “payload sensitivity to mass growth” of the problem.

    For an SSTO this, if I recall correctly, can be about 10x that of the one for 2sto because of
    1) The much higher dry mass compared to payload, when you compare an SSTO and the second stage of a 2STO (about 10x)
    2) The much higher sensitivity to dry mass when you compare an SSTO and the first stage of a 2STO (about 10x again). In a 2STO, if the first stage underperforms at a certain design point, the second stage can be downsized as a total to restore the delta vee, not just the payload like you would have to do in an SSTO.

    Anybody willing to see that stuff typed up formally? Or are you Kirk going to dive into exactly that? You’re probably going to do something cleverer anyway… ๐Ÿ™‚

    As we know, almost all aerospace programs (except perhaps a few by Ed Heinemann?) have suffered from mass growth and hence generous margins are applied in the industry. Just ask Jim on NSF.

    Hence I think the first high flight rate routinely flying reusables will have at least two stages. This is just one of the reasons, though it is a big one.

  3. Charlie says:

    Your equation and observations are useful.

    Note, a conventional (NERVA, Timberwind) hydrogen propellant nuclear rocket has a lambda which is mostly limited by the bulky and deep-cryo nature of the LH2 propellant, so even the dry mass of the hydrogen tank is likely to exceed 15% of the fully-fueled mass.

    When you are designing an ‘abundant chemical’ rocket, it makes a good deal of sense to use a propellant which both dense and reasonably storable. FLOX/methane is an interesting choice. You get hypergolic ignition, the ability to use a RL-10 variant instead of developing a completely new engine, ISP approaching that of LH2/LOX, a fairly high density, plus “space storability”. It is not too hard to envision a large scale FLOX/methane rocket with a structural fraction under 1%.

    Another alternative to consider for ‘abundant chemical’ is frozen brick propulsion units. In theory, frozen H2O/aluminum hydride bricks could be chucked into a combustion chamber and ignited (detonated?) with a microwave pulse. You’d get a bumpy ride, but it is not too absurd to get a structural fraction under 0.1% or even 0.01%

  4. anon says:

    What Kirk is saying is what a lot of people said. SSTO Cannot be done
    at the current state of technology. That said, some clever improvements
    might get us there. Better cycles, lighter nozzles, clever structure,
    but the trick is getting there with a Reusable.

    I’m sure in the 1920’s the rane equations showed you couldn’t fly across
    the ocean. But we got better.

    The trick is investing in the paths that will get us to better.

  5. Charlie, you read my mind–you’ll like my next few posts.

  6. Charlie says:

    Oops. I need to proofread more carefully.

    I meant H2O2 (peroxide) , not H2O(water).

    Peroxide + AlH3 (aluminum hydride) has very high theoretical ISP (over 400sec with a high expansion engine) and very high density (over 1.4g/cc). Actual ISP will be less due to two-phase losses, but could still exceed 380sec. At simple home-freezer temperatures of -10C the propellant bricks would be highly inert, but a small spark would transform it into alumina dust and 3900C hydrogen gas. BOOM!

    Cooling the chamber/throat/nozzle of an engine like this would be challenging. However, if you don’t need a high average thrust, you could just spread out the pulses over time and let it cool radiatively.

  7. Charlie says:

    Oops again.

    The chamber temperature of H2O2/AlH3 is about 3930K , not 3900C.

    Mea culpa.

  8. nick says:

    Charlie has the inklings of a very interesting approach. Many moons ago I proposed an ice rocket (for in-orbit, not launch). Since IMHO the abundance assumption is silly unless we are talking about ISRU, it’s designed around ISRU: propellant that is simple to extract from naturally occurring ice. It’s a low continuous thrust solar- or nuclear-thermal rocket, or alternatively an arcjet. The propellant is water or ammonia stored as ice and we don’t melt it until we need it.

    As Charlie suggests, getting rid of the tank and storing the propellant as ice can be a huge structural win. But we can get far below 1% structural mass. Lambda for an ice rocket made and used in microgravity could conceivably be 10^-3 or smaller. All we need to keep the ice frozen in deep space at Earth’s heliocentric orbit or beyond is a shade and a coat of sealant. It’s easier than long-term storage of liquid cryogenics and we don’t need active cooling or a tank.

    Since most of the rest of the structural mass is power plant (solar panels, mirrors, or a nuclear reactor) and rocket engines, it works best as a low-thrust continuous rocket to minimize peak power. Higher thrust regimes, for example launching from the moon, would be trickier: this method works best for ice from Ceres, asteroids, Jupiter-family comets, Phobos/Demos, or the like.

    It’s true that we would consume over 98% of our spacecraft mass as propellant on the high delta-v trip back from the Jupiter-family comets to Earth orbit if we used a low specific impulse water thermal rocket. Arcjet and ammonia propellant would work better on that score. But if water ice can be cheaply extracted and given the low lambda this could still be very economical despite the seemingly preposterous waste. If the propellant is cheap and the main structural mass is the powerplant, this turns rocket economics on its head: to a certain extent lower specific impulse is better, because it minimizes peak power. So thermal engines may end up for most purposes better than higher-specific-impulse electric methods.

    Closer to home and to the present, if we want an orbital transfer vehicle based on propellant launched from earth, the abundance assumption doesn’t apply, but there still may be a significant win from the very low lambda made possible by storing the propellant as ice rather than as liquid cryogenic. But the trade-off of giving up higher specific impulse propellants that can’t be economically frozen may not be worth it on-orbit as long as propellant there is expensive.

  9. Ann Onymous says:

    /// Hence I think the first high flight rate routinely flying reusables will have at least two stages. This is just one of the reasons, though it is a big one.////

    The Kistler K-1 came very close from that.

    ///As we know, almost all aerospace programs (except perhaps a few by Ed Heinemann?) have suffered from mass growth and hence generous margins are applied in the industry. Just ask Jim on NSF.///

    ” – Fearing that the Navy would probably lose the contest and that the class of Navy supercarriers would never be built, Ed Heinemann decided that it was absolutely necessary that the weight be kept below 70,000 pounds so that the aircraft would be capable of operating from Midway-class carriers. In mid 1948, Douglas submitted a proposal for a 68,000 pound aircraft. The Curtiss design weighed nearly 100,000 pounds. North American dropped out of the competition, because they did not believe that it was possible to build an aircraft weighing less than 100,000 pounds that met the requirements.

    Although the Navy was skeptical that Douglas really could build a 68,000-pound aircraft that met the requirements, both Douglas and Curtiss were given a preliminary 3-month contract so that they could refine their proposals. On March 31, 1949, Douglas was declared the winner of the contest and was awarded a contract for two XA3D-1s and a single static test airframe – “

  10. Tom D says:

    Nice ideas here. I’m currently most intrigued with rockets that use beamed power (laser or maser) to heat and/or power a reaction mass for near-earth propulsion, but this does look to be going somewhere interesting.

    Speaking of “abundant chemical” propulsion, I’ve got a book that calls for 26 shuttle launches to fill up the tanks on a manned missions to Mars. I had to smile when I ran across it in a used book store. That mission would certainly take a long time to get ready for!

    I’m still mourning the effective moratorium on nuclear propulsion in space, though I expect that to be lifted sooner or later.

  11. Pete says:

    I had thought that TSTO would ultimately be favored over SSTO as the total payload fraction over dry mass and propellant would be significantly less. However, I have just realized that this may not be the case. For SSTO one needs something like 12% dry mass (including payload) with LH2/LOX and 7% with RP-1/LOX. Currently engine and tank mass for LH2/LOX is something like 5% (shuttle ET and SSME), and say 2% for RP-1/LOX. However there is significant room for reduction in these numbers (1 GPa bidirectional composite at 1500 kg/m^3, 2 bar tank pressure and 2xSF gives ~0.2% tank mass). Interestingly such dry mass reductions will benefit LH2/LOX vehicles more, though absolute dry mass cost is perhaps a more important cost metric and LH2 vehicles are notoriously more expensive on a dry mass basis.

    So if we assume tank and engine mass of say 1% for a RP-1 SSTO RLV (the primary dry mass components that mostly scale with propellant mass). Then we are left with 6% for payload, guidance, reusability, etc. Guidance systems will scale somewhat independently of propellant mass, but reusability (deorbiting systems, reentry shields, landing systems, etc.), will scale mostly with payload. Hence total dry mass will actually scale mostly with payload, independent of whether it is a TSTO or SSTO vehicle. In such a scenario the SSTO will have an only slightly reduced payload as compared to the TSTO vehicle (less than 1% GLOW), a disadvantage that is probably out weighed by the operational advantages of SSTO.

    In reality I probably favor assisted SSTO, basically subsonic air launch such that tank insulation, atmosphere compensating engines, low altitude abort, and what not are not required. I might also favor structurally self-sufficient external tanks, leaving them in orbit, simplifying abort and reentry design รขโ‚ฌโ€œ they are a low mass/cost item that is perhaps worth more in LEO than on Earth.

  12. Charlie says:

    Another option is to use a lot of propellant tanks, and construct your tanks and support structure of propellant (not ‘ideal’ propellant, but adequate propellant) and ‘recycle’ your empty fuel tanks into your exhaust stream.

    E.g. – For the FLOX/methane concept I mentioned above, construct your propellant tanks and structure mostly of Aluminum/Lithium alloy (the FLOX tank would require some kind of liner). Design your engine to function in two modes – as a standard expander-cycle biprop, and as a tri-prop hybrid, with solid Aluminum/Lithium fuel getting pushed into the combustion chamber and extra FLOX introduced as well. Designing an efficient way to disassemble your fuel tank into hybrid propellant strips is left as an exercise for the alert reader.

    The limit here becomes the g-loads and throttling. Even if you start out at 0.01g of thrust (in order to minimize engine mass), if 99.9% of your starting mass is fuel, you’ll eventually have to throttle down to 30% just to limit g-loads at burnout to 3g.

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