Selenian Boondocks

As I prepared for this post tonight, I realized that I wasn’t really modifying the rocket equation at all–I have been using the rocket equation and a summation of mass terms to find the payload fraction, which I consider an especially useful value to know.

Furthermore, if you read my previous post, you probably figured out pretty quickly that all of the dry mass in the rocket doesn’t correlate to the propellant mass. That’s a pretty good guess for very large rockets with large delta-V’s, but as you get smaller, the assumption really starts to fall apart.

One of the big masses in a rocket outside of the propellant tanks are the engines themselves, and they really don’t correlate to propellant mass at all. They correlate to the gross mass, because the engines are typically sized to give you some particular value of thrust-to-weight when you light them up. So I went ahead and extended the previous derivation of payload fraction to include this important feature.

So let’s define some terms. Lambda is the relationship between the vehicle dry mass that correlates to propellant, and phi is the relationship between vehicle dry mass that correlates to gross mass.

We plug these definitions into an equation for gross mass, and I simplified things by just going ahead and combining the propellant-sensitive mass into (1+lambda)*propellant_mass. Then I went ahead and replaced propellant mass with the expression from the rocket equation that related propellant mass to gross mass, and now I had the equation entirely in terms of payload mass and gross mass, which is exactly what I was after.

Then it was a matter of doing the algebra to simplify things down until I had an expression for payload fraction like I did before.

Take a look at the final equation. You’ll note a few things. The first one is that if phi is zero, then the expression is just the same as the one I derived before. The second one is how lambda and phi impact the payload fraction differently. You can see that the effect of lambda is reduced somewhat by the mass ratio having one subtracted from it, but that phi has no such reduction.

This is because in the case where you had a vanishing small impulse, MR would be very close to one. And one minus one would be zero and the impact of lambda would be almost eliminated. But because phi is multiplied by one, its effect is still present even if the impulse was very small. This is because any rocket whose engines were sized to deliver a particular thrust-to-weight at ignition would still pay a mass penalty for those engines, even if it just briefly turned them on and then turned them off.

In an upcoming post I will show how to calculate lambda and phi from other vehicle design parameters, like mixture ratio, tank mass per unit volume, propellant densities, thrust-to-weight ratios of engines and vehicles, and so forth.

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).

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 (MR) will be in that situation.

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.

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 modified rocket equation.

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

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.

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

And finally there it is. An expression for payload fraction of a rocket, defined as the payload mass divided by the gross mass, with expressions for lambda and mass ratio embedded into the equation.

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.

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.

While using electromagnetic effects for atmospheric reentry and thermal protection is interesting, it’s only one of several promising options that have been proposed over the years.  There is another application though, where exploiting magnet-hydrodynamic effects could be a much bigger “game changer” — aerobraking and aerocapture for reusable in-space vehicles.

Traditional Aerobraking and Aerocapture
One of the challenges of orbital mechanics is that it takes just as much energy to descend into a gravity well as it does to ascend out of it. One technique that has been used for lowering the propellant cost of descent into the gravity well of a planet with an atmosphere is aerobraking. Aerobraking is the process of taking a spacecraft in an ellpitical orbit around a planet with an atmosphere, and using atmospheric drag at the lowest altitude portion of its trajectory to slowly decrease the altitude of the high end of the elliptical orbit. This process has been used now on about a half-dozen planetary missions, in some cases reducing the propulsion requirements by 1km/s or more, over the course of a couple hundred passes. Aerobraking has been traditionally been done by satellites that aren’t explicitly shaped like a reentry vehicle–in fact most of the drag for typical aerobraking vehicles is produced by using the spacecraft’s solar panels as massive drag brakes!

Artist's Impression of MRO Aerobraking (credit JPL and Wikipedia)

A more aggressive maneuver called aerocapture takes a spacecraft in a hyperbolic (interplanetary) orbit and in a single pass decelerates that vehicle into an elliptical orbit around a planetary body.  Typically the term refers to maneuvers where the ending orbit has an apoasis near the altitude of a circular orbit, though it could also be used to describe a maneuver that uses a single pass through the atmosphere to replace the “capture braking burn” that would normally be used. Aerocapture is a lot more challenging, since the deceleration has to take place a lot lower in the atmosphere in order to provide the required deceleration in such a short distance. This implies much higher forces and heat-fluxes, which require some sort of aeroshield/TPS system.

Here are a few of the main challenges of aerobraking and aerocapture:

1. Dynamic Pressure Loads: Dynamic pressure is the pressure felt on the vehicle by the impingement of the atmospheric molecules.  The equation for dynamic pressure is q = 1/2 * rho * V^2, where lower case q is the dynamic pressure, rho is the instantaneous atmospheric density, and V is the instantaneous relative velocity.  For MRO, the dynamic pressure limits were set at 0.35 Pascals, which correlates to moving at about .76m/s at sea level (ie a slow walking pace).  To give you an idea of how this compares with orbital reentrythe peak dynamic pressure of say a Soyuz in its emergency ballistic reentry mode, is over 40,000 Pa of dynamic pressure, and even a low-G lifting reentry is still in the 10kPa+ range.  Direct entry into the Venusian atmosphere from a hyperbolic interplanetary orbit gets you into the 1MPa range!  Another fun comparison is that the max-Q Xombie or Xoie have seen in flight was around 250Pa.Most of the very low allowable dynamic pressure load for past aerobraking efforts has been driven by the fact that most aerobraking craft to-date have used large flimsy solar panels as their main drag structure.
2. Peak Heat Flux:  The shockwave caused by slamming into gas particles at hypersonic velocities compresses and heats the gas particles to substantial temperatures.    Heat from this shock wave is convected and radiated into the aerobraking spacecraft.  The equation for heat flux is Q = 1/2 * rho * Ch * V^3.  Capital Q is the heat flux (in W/m^2), rho and V are the same as before, and Ch is the heat transfer coefficient.  The heat transfer coefficient, I think, represents what portion of that heating goes into the vehicle itself instead of being carried off by the now quite ruffled atmospheric gas molecules who didn’t see you coming.  Yes it is confusing that dynamic pressure is lower-case q, and heat flux is capital Q.Once again, to give you some scale, the worst case pass for Odyssey had an estimated heat flux of about 500 W/m^2,  which is about 40% of the heat you get in LEO from the solar radiation. For that Soyuz reentry case mentioned earlier, the total heat generated at max-q is in the 240 MW/m^2 range–several times higher than the heat flux at the throat of the SSME or RD-180.  The Venusian direct entry example according to one source would actually be in the 4000MW/m^2 range! Fortunately, I think that for atmospheric reentry the Ch term is relatively low–most of that heat gets carried away by the atmosphere.As with dynamic pressure loads, the reason why peak heating rates are kept so low for most aerobraking missions is that you’re using the large solar panels as most of the drag surface, and they can only take so much heating before their temperatures rise to levels that could permanently degrade their performance.
3. Atmospheric Density Variations: If atmospheric density was nice, constant, and well-known, aerobraking could proceed a lot faster and in a lot fewer passes.  The problem is that at the altitudes where aerobraking takes place (100+km), the density can vary significantly over length scales as small as 20km.  This can be driven by many processes including variations in the solar wind and solar radiation due to sun cycles, weather effects like dust storms for Mars aerobraking, and other effects.  Going off of some data from the Odyssey mission, variations as big as 2-3x were seen in density from pass to pass.   A second-order effect of density variations is that both the drag coefficient and the heat transfer coefficient will vary with atmospheric conditions by noticeable amounts.  Unfortunately,  in many cases you don’t know the density along a given trajectory in advance, so you have to plan for not the average density, but the worst case pass density.   Which means that most of the time you’re getting less deceleration and heating than you could actually withstand, but some of the times you might actually find yourself pushing your limits more than you would like.   This drives you to taking more passes than you’d really like to take in an ideal situation.  These variations get more and more pronounced at higher aerobraking altitudes, where atmospheric density is measured in kilograms per cubic kilometer.Once again, this is an area where using large, sensitive solar panels as your drag devices really hurts. Because you can’t stand high dynamic pressures or heat fluxes, you have to do your passes higher up in the atmosphere. But due to variability in density at those higher altitudes, you end up getting driven even further up to deal with worst case variations. That said, even aerocapture trajectories are high enough altitude that atmospheric variations can be important challenges to deal with.
4. Aerobraking Duration: For most previous Mars and Venus aerobraking missions, velocity changes in the 1-1.2km/s range have taken between 70-150 days, over several hundred passes.  While this is fine for unmanned missions, it’s harder to do for manned missions, where radiation concerns make you want to minimize your time spent in-transit.  The large number of cycles is also a difficulty for missions aerobraking at earth, where each pass will take you through the Van Allen belts.  Lastly, for reusable in-space transports, the total turn-time is an important economic parameter–the more missions you can fly in the same period of time, the fewer vehicles you need to support a given mass throughput.

A couple more quick observations before we jump into using MHD forces to enhance aerobraking:

• For typical aerobraking, the parameter you can control easiest is the periapsis altitude, and thus indirectly the average density.  In other words, if you want to double the drag on a pass, you lower your periapsis to an altitude that has about double the average density.  This also means that to a first order approximation (ie ignoring the relation between density and the heat transfer coefficient) heat flux for traditional aerobraking is going to scale fairly linearly with drag.
• Ballistic coefficient ends up being really important for aerobraking as well–this is the whole reason why the solar panels are used unstowed for aerobraking.  Higher ballistic coefficients mean that you have to dip lower into the atmosphere (and thus get a higher heat flux) to get the same amount of deceleration per pass.
• In spite of the disadvantages of using solar panels as your drag brakes, there are some real advantages to being able to use a aerobraking scheme that doesn’t require your vehicle to be explicitly crammed into a typically reentry-vehicle shape behind a massive heat shield.  It would be nice for instance to be able to get tanker vehicles or orbital tugs back from lunar trajectories or martian trajectories without them having to carry a big aerobraking shield like you see in all the old literature.

Anyhow, that was a quick introduction to aerobraking by a complete non-expert.

Some Backstory on Why I’m Interested in Aerobraking
I started looking into this a few months ago as an alternative to propulsive retrobraking for Centaur-derived cislunar tanker vehicles.  While a Centaur stage actually can do a lunar round trip fully propulsively, with at least some payload delivered to the Moon, the “gearing ratio” (initial mass in LEO compared to payload delivered to LUNO or the Lunar Surface) was pretty pathetic.  Just to use some ballpark numbers, without digging up my more precise calculations, I’m getting around 8000lbs payload to LUNO if you drop it off in orbit and the Centaur only returns to earth, dropping to only 2500lb if the Centaur has to haul the payload all the way there and all the way back propulsively.  However, if you could do 3km/s worth of aerobraking (assuming about 1200m/s worth of burns between the Trans-Earth Injection burn and any periapsis raising maneuvers, including the final circularization), all of the sudden you’re talking about almost 20,000lb of payload on the dropoff mission, and about 13000lb on the round-trip maneuver.  Depending on how massive and expensive the aerobraking system weighs, it makes a massive difference in the performance of a reusable cis-lunar architecture.  For a long time though, I had sort of dismissed aerobraking, because any aeroshield big enough to allow single-pass aerobraking (or few enough passes to be interesting) also ended up looking like it would either be very heavy, or very bulky, or require lots of orbital assembly or some sort of new deployable technology.  Not that any of those other than being too heavy was a total show-stopper, but it definitely made it less attractive for a near-term commercial operation.

Another line of thought I had been wondering about recently was manned cislunar transportation, especially in light of the Augustine Committee report.  One of the big suggestions they made that rubbed a lot of HLV-advocates wrong was the idea of launching the crew on commercial LEO taxi vehicles, and flying Orion up to LEO unmanned.  A lot of people said this was just silly–if you’re launching Orion may as well launch it manned, even though this would require adding launch escape and emergency detection capabilities to the HLV.  I started thinking down the lines of what Orion could look like if it was designed from the start not to carry astronauts until they got to space.  The LAS would go away, as would all the structural requirements for taking those sorts of loads, being able to rapidly drop the service module, etc.  The whole thing could fit inside a fairing, thus simplifying aerodynamics and loads on the front end of Orion.  Heck, it could even be attached to the rest of the stack in whatever orientation made the most sense for mission ops–it wouldn’t be constrained by needing to be on the top in an orientation where the capsule could “get out of Dodge” in a hurry if something “went south” with the HLV.  The more I thought about it, the more I realized that Orion could end up looking like a drastically different vehicle if it was optimized for in-space use and reentry instead of needing to also handle manned ascent to orbit as well.  Then I made an interesting leap of logic.  What if Orion was only meant to be used in space?  I originally sort of dismissed this, since most single-pass aerobraking schemes I knew of would require the thing to be designed like a reentry capsule anyway.

Jumping back to the Centaur-based tug idea, I toyed around with the idea of doing a blog series, seeing if I could make an aerobraking simulator to figure out if a Centaur could without any sort of fancy aerobraking shield actually do a multi-pass aerobraking mission that would get it back to LEO within a reasonable amount of time (say three weeks or less).  However, I stumbled on the papers about magnetic aerobraking right about this point in my thought process, which may possibly provide a solution to both of these problems.

While I don’t have anywhere near the analytical chops to know for sure how far you can push this technology, if it could enable single-pass or at least small number of pass aerobraking without requiring a huge traditional aerobraking shield, interesting things might become possible. Magnetic aerobraking could potentially revolutionize cislunar transportation, enabling low-cost reusable manned and unmanned deliveries based on modified versions of existing LOX/LH2 upper stages, and could allow fully-reusable in-space only manned vehicles that weren’t just an overglorified 1960s-style reentry capsules.  But more on that later.

For now let’s get back to how we can use magneto-hydrodynamic interactions to enhance traditional aerobraking, and see if we can figure out if this idea has merit at all.

Magnetic Aerobraking
Going back to our previous two discussions, one of the key takeaways was that the enhanced braking and thermal protection provided by strong magnetic fields was strongest at high altitudes where atmospheric density was lowest. At high altitudes, the ambient atmospheric density is low, but Joule heating caused by the interactions between ions in the shock layer and the superconducting magnet keeps the electrical conductivity of the plasma in the shock layer high. Also, for aerobraking or aerocapture short of reentry, by definition you are both always at a speed and altitude high enough that you don’t have to worry about the shock layer losing sufficient conductivity for MHD effects to dominate aerodynamic drag effects. The magnetic interaction parameter (Qmhd) introduced in my first post in this series can easily be in the 250-1000+ range at high altitudes compared to down in the 5-50 range you might see during atmospheric reentry. For example, the paper I cited in my first article (Otsu et al) showed that for a vehicle coming back from a GTO-like orbit, you could cut the return time by 70% with a 0.1T magnet, which is about 5x weaker than the magnet assumed for most of the reentry magnetic TPS studies.   While magnetic effects may be helpful for reentry, they truly come into their own for aerobraking and aerocapture.

A few other thoughts:

• While the total drag for a magnetic aerobraking concept can actually be several times the drag of a similar non-magnetic vehicle, the gas-dynamic portion of the total drag actually decreases substantially in the case of magnetic aerobraking.  This is due to a much lower velocity behind the shock layer in the magnetic case.  Figure 9 from the Fujino et al paper I used in the last post (”Numerical Analysis of Reentry Trajectory Coupled with Magnetohydrodynamics Flow Control”, JS&R Vol 45 No 5, pg 911-920) illustrates this beautifully:
• For a vehicle using magnetic braking, most of the total drag force is actually reacted electromagnetically through the magnet itself, not through the surface of the vehicle.  The dynamic pressure that the vehicle surface itself sees is greatly reduced compared to what you would expect at that altitude and entry velocity.
• While in the above case, the dynamic pressure reduction was about 4x at ~75km, this effect is likely to be even more pronounced at the altitudes used for aerobraking (90-120km) where the electromagnetic interaction parameter is substantially higher (40-160x higher) than it is in the case shown above for atmospheric reentry.
• The heat flux seen by the aerobraking vehicle will also be greatly reduced compared to a non-magnetic aerobraking system at a similar altitude and velocity.  This is due to the much thicker shock layer standoff distance and the lower velocity of the particles behind the shock layer.  The Fujino et al paper estimated that the heat flux would roughly be cut in half at 75km with a 0.5T magnet (due to a boundary layer between the bow shock that is twice as thick at that magnetic interaction parameter).
• For higher parameters in the 100-1000 range that you would likely see for aerobraking, this effect should be even more pronounced.  The trend in shocklayer thickness vs. Qmhd shown in Fig 3 of Fujino et al  was linear over the Qmhd range of 0-6.  If it continued out linearly up into the Qmhd 100-1000 range, the shock layer standoff distance would be in the range of 100-125x thicker than without MHD effects, implying a drastically reduced heat flux at aerobraking altitudes.  Unfortunately without having them run the actual analysis, it would be hard to know precisely how well this would work.
• All these factors mean that the same vehicle could use a lower periapsis with a magnetic braking system than without.  The dynamic pressure and heat flux that the vehicle sees at a given periapsis altitude is going to be at least 2-4x and possibly more than an order of magnitude less than it would be without the magnetic field.  Even in the most conservative case (ie assuming that the effect at 100km and aerobraking speeds is no better than at 75km in spite of having a Q 40-160x higher) this would allow you to go to an altitude with at least double the density while keeping the heat flux and dynamic pressure loads within tolerances.  With an effective total drag 4x higher at a given altitude combined with being able to go to a lower periapsis, you get bare minimum a 8x reduction in total aerobraking time compared to the non-magnetic case.
• For the aggressive, “I don’t know if I’m extrapolating way too far” case, you could get even larger reductions in aerobraking time.  Going back to my linear extrapolation on shock layer standoff vs. Qmhd (and thus heat flux vs Qmhd), at Qmhd=250 this would put the shock layer standoff at about 25-30x thicker than the non-MHD case.  The example in Otsu et al gave a Qmhd of 250 using a 0.1T magnet and a 100km periapsis.  Since Qmhd is proportional to B^2 and inversely proportional to rho.  If you increased the magnetic field from 0.1 to 0.5T (similar to what was being suggested for the reentry studies done by Fujino et al and some of the others), you could maintain a Qmhd of 250 even if you increased the local density by a factor of 25.  At Qmhd of 250, the effective drag coefficient is about 3x higher than the non magnetic version.  That would give up to a 75x reduction in aerobraking time compared to the non-magnetic case.
• One other advantage of magnetic aerobraking is that you can drastically vary your effective drag coefficient electrically.  Also, the heating and dynamic pressure are far more driven by the magnetic field strength than by the atmospheric density for the MHD aerobraking case.  These mean that you can afford to take deeper passes without having to worry as much about variability.  If the density is higher than expected, and you have some head-room on your magnet, you can increase the MHD field strength a bit to keep the shock layer back and the dynamic pressure down.  This also could cut trip times in half just by allowing you to base your planning off of the average atmospheric density instead of having to take the mean + 3 standard deviations as your predicted atmospheric density.

I’m rapidly coming up to the point where I’m pretty sure I no longer know what I’m talking about.  At least from here, it looks like there’s a good chance that MHD aerobraking could allow for aerocapture (at least into a high eccentricity elliptical orbit), and very rapid aerobraking down to a circular orbit compared to the non-magnetic case. I think you can extrapolate the conclusions of these papers in these ways, but without having the people with the analysis tools actually verify these claims, I’d still take them with the appropriate sized grain of salt.   Also, my intuition on how a MHD aerobraking vehicle would compensate for density variations is not very good.  That alone could be a paper or a thesis.

So, whether this ends up being a mild curiosity that ends up only being useful in niche applications, or a game-change remains to be seen, but the potential for this being a game-change is real.

In my last post in this series, I’ll go more into some of the implications of what this could do if it works, and some thoughts on how to actually flight-demonstrate MHD aerobraking.

[Edit: It turns out I had misspelled Fujino's name in the original post. Fixed that and added the title of the paper in case people want to get a copy--it's free if you have a JS&R subscription, $15 if you're an AIAA member without a JS&R subscription, and something like $30 if you're not an AIAA member--highly recommended if you're interested in this topic]

Computing the mass ratio for a tapered tether (tether mass/tip mass) was first done (to the best of my knowledge) by Hans Moravec in an appendix to his unpublished 1978 paper, “Non-Synchronous Orbital Skyhooks for the Moon and Mars with Conventional Materials.”

The expression uses the Gaussian error function, erf(x), which is not typically available in a spreadsheet or scientific calculator. erf(x) also cannot be calculated in closed-form–typically the expressions used to calculate it are iterative. Since the tapered tether mass ratio is such a useful design tool to have, I derived a recursive algorithm that computes the ratio in a simple loop, given only the velocity ratio of the tether (tip velocity/characteristic velocity).

Here is the recursion:

and here is psuedocode for the recursion:

subroutine getRatio(double VR, int k) {
double VR2 = VR*VR;
double sum = 0.0;
for (int i = k; i >= 1; i--) {
sum = (VR2/(double)i)*(1.0/(double)(2*i+1) - sum);
}
return 2.0*exp(VR2)*VR2*(1.0 - sum);
}

With about 8 recursions, the results are extremely accurate. The recursion is unstable when the velocity ratio is greater than about 3, but no one should be building tethers with velocity ratios greater than 3! and you can just use Moravec’s expression with erf(x) = 1.0, which is a pretty safe assumption for x > 2.0 or so.

The history of momentum-exchange tethers goes back many, many years but is bound by a common thread that, until recently, limited the realization of this technology. That common thread is the need for high specific tensile strength.

The first idea of concept of a tether dates back to the imagination of Konstantin Tsiolkovsky, the Russian schoolteacher who first developed our modern concepts of rocketry and first derived the rocket equation. In the late 1800s, Tsiolkovsky visited Paris and saw the Eiffel Tower. He was so impressed by the sight that he imagined a tower reaching up far into space. He calculated the height at which such a tower would have to be before the centrifugal force from the Earth’s rotation balanced the pull of gravity (inadvertently calculating the altitude of geosynchronous orbit).

Tsiolkovsky, of course, could not conceive of any material that could withstand the compressive forces of such a structure, but sixty years later, a Russian engineer named Yuri Artsutanov picked up the thread of Tsiolkovsky’s work and first worked out the engineering principles of what is now called a “space elevator”, a long tether hanging all the way from geosynchronous orbit to the surface of the Earth. The space elevator required materials with specific tensile strength far in excess of any known material, and still does. Further conceptual engineering work on the space elevator concept was done in the early 1970’s by American engineer Jerome Pearson.

The space elevator was a hanging tether, and payloads were required to traverse its length in order to achieve orbit. The beginnings of rotating momentum-exchange tethers date to the late 1970s, when Hans Moravec, a robotics researcher at Stanford University (now at Carnegie-Mellon) was intrigued by a suggestion of his friend John McCarthy of a satellite that “rolled like a wheel” around the Earth. Moravec began a scientific investigation of the concept, which he first called a “non-synchronous orbital skyhook” and later a “Rotovator”. Like the space elevator, it reached all the way to the surface of the Earth, but unlike the elevator, it rotated about its axis a number of times per orbit. A payload would be picked up by the tip at the surface of the Earth and then thrown half a rotation later into an interplanetary trajectory. The Rotovator was a good deal shorter than the space elevator (~4200 km vs. 40, 000 – 100,000 km) but was not much better in terms of materials required. Moravec published a paper on the subject in the Journal of Astronautical Sciences where he speculated on advanced forms of matter that might have the strength needed to build the Rotovator.

About a year after the JAS paper was published, Dupont’s development of Kevlar excited Moravec to the possibilities of Rotovators built with conventional materials. He wrote a short paper called on the subject which was never published. The paper showed that Kevlar skyhooks were not feasible around the Earth but could be reasonably built around the Moon and Mars. In an appendix to this unpublished paper, Moravec speculated on the possibility of skyhooks built in interplanetary space that would assist spacecraft traveling between the Earth and Mars. To the great benefit of future tether researchers, his equations for the cross-section of a tether, in the absence of a gravitational field, could be integrated in closed-form. Thus, the Moravec “tether equation” was first derived.



Moravec was able to derive analytical expressions for the area of the tether as a function of its distance from the rotational center. He then numerically integrated the area expression along the length of the tether to calculate volume and mass. As an aside, in an appendix, he considered the case of a tether spinning in free space. When the tension on the tether was only due to centrifugal forces, the area expression could be analytically integrated to a closed-form solution. Thus the Moravec mass ratio was derived.

The equation could be simplified by realizing that fundamentally, the mass ratio is a function only of the velocity ratio, which itself is the ratio of the tip velocity of the tether and the characteristic velocity of the tether material.


Further insight into the value of the equation was gained by comparing it to the rocket equation and noting the similarities and differences.

Moravec wrote a few articles on the subject for space-themed publications, but basically returned to his robotics work. Nevertheless, Moravec’s equation still serves as a foundation to all momentum-exchange tether work to this day.

Rotating momentum-exchange tethers are a very exciting technology, but one of my first thoughts after being exposed to the technology was the tricky rendezvous. The space industry has spent all kinds of money and time on satellite rendezvous, and these are typically slow, long, drawn-out affairs with two satellites in almost precisely identical orbits, slowly closing the distance between each other and finally making a solid connection.

The rendezvous required for a rotating tether and its payload is far more dramatic. The whole point of the operation is to have the tether and the payload in different orbits, so that the rendezvous can lead to an exchange in angular momentum and orbital energy between the two, resulting in a payload boosted to a higher energy orbit (or dropped to a lower energy one).

Thus, you can’t match orbits like you do in conventional rendezvous. The best that you can do is to instantaneously match position and velocity (but not acceleration). So you need an approach to rendezvous that is pretty tolerant of error.

So we threw out the book when it came to trying to think of how to do rendezvous, and came up with something totally different and designed to meet the specific needs of the mission. And I was pretty proud of the result, and still am. Because, you see, this is a bit of an anniversary for tether rendezvous technology. It was five years ago (February 2005) that we successfully demonstrated that the rendezvous technology we had postulated could work, at least at the lab scale.

We took advantage of the fact that the tether was under rotation and experiencing centrifugal acceleration, and that the payload was in free-fall. We simulated this (quite accurately) by hanging the tether’s “catch mechanism” from the ceiling of a racquetball court at Tennessee Tech, and then we “shot” our simulated payload up to the catch mechanism, with its boom positioned to be captured by the catch mechanism when it penetrated the aperture of the catch mechanism. Then the catch mechanism would release and close around the boom, quite quickly, allowing the simulated payload to be caught.

It all worked out a lot better than I thought it would–take a look at our results:

First Catch Mechanism Test

Second Catch Mechanism Test

More Testing with Animation

And here was the press release that came out months later announcing the accomplishment. Our video footage of successful testing got on NASA TV…once.

NASA Engineers, Tennessee College Students Successfully Demonstrate Catch Mechanism for Future Space Tether

In this installment, I want to dig a lot deeper into the mechanics of how one might maximize the utility of MHD effects for orbital reentry. But first, I wanted to spend a few seconds discussing what is important for an RLV TPS system.

RLV Thermal Protection Systems
Protection from the harsh heating environment caused by atmospheric reentry is one of the biggest challenges for reusable vehicles–far more difficult than the often harped-on rocket equation or the “inefficiency of chemical propulsion”. The problem isn’t even the weight of the thermal protection system as much as it is the maintenance requirements. Ideally you’d like a TPS solution that requires very little maintenance, and can be “tested” easily and quickly on the ground before flight, even if it cost you a little extra weight. You’d also prefer something that was relatively simple operationally to use, with a minimum number of failure modes. MHD thermal protection seems like an interesting match for these requirements. I should note however that there are other promising ideas out there such as transpiration cooling that might also work on their own or in combination with MHD thermal protection, but they are beyond the scope of this blog post.

Some Take-Aways from the Literature on MHD Reentry TPS
There have been several interesting papers on this topic, including the JS&R article “Experiment on Drag Enhancement for a Blunt Body with Electrodynamic Heat Shield” that got me thinking about this more seriously, a second JS&R article that goes into experimental proof of the heat flux reduction “Experimental Verification of Heat-Flux Mitigation by Electromagnetic Fields in Partially-Ionized-Argon Flows”, and another JS&R article from a year and a half ago “Numerical Analysis of Reentry Trajectory Coupled with Magnetohydrodynamics Flow Control” that I’ll be leaning on pretty heavily for this discussion. You can purchase the articles from AIAA, or if you already have a subscription to the Journal of Spacecraft and Rockets, you can read them for free.

I’ll briefly summarize some of my takeaways before going into my thoughts on how to move things forward from there:

1. Both analytically and experimentally, magnetic reentry TPS appears to provide large reductions in both peak heating and in total heat load.  The third paper above suggested a 30% reduction in peak heat load and a 40% reduction in total heat load for ballistic reentries.  Under the conditions tested in the second paper, heat reductions up to 85% were shown.
2. The magnetic braking effects dominate aerodynamic braking effects at high altitudes.  This is mostly due to lower density meaning that atmospheric drag is fairly low, while also lower density means that Joule heating caused by the currents (the loop marked “J” in the previous post) induced by the magnetic fields increases the electrical conductivity more effectively than at lower altitudes.
3. The more deceleration that can be done high up in the atmosphere, the lower the peak heating and the lower the total heat load.  The heat flux is roughly proportional to the cube of the velocity.
4. The heat flux reduction from this scheme is dominated by the increased shock layer thickness at high altitudes, and at lower altitudes is dominated by the much lower velocity by the time you get there by getting extra braking high up.
5. Conductivity of the plasma is one of the keys to making this work.  The conductivity in these cases was entirely due to the temperature in the plasma–higher velocities lead to higher temperatures, and Joule heating also leads to higher temperatures.  As velocities slow down, conductivity drops, as does the effectiveness of the braking system.  Below about Mach 12, the only way to keep the flow ionized enough to control magnetically is to add energy via some mechanism.
6. Because of the large induced currents, this idea only works if the heat shield is an electrical insulator.  If it is a conductor, you’ll just generate hall currents in the heat shield which will null out a lot of the benefit of the approach.

Thoughts on Maximizing the Effectiveness of MHD Reentry TPS
Based on these takeaways, and the discussion in the last post, I’ve come up with a few ideas for how to maximize the effectiveness of an MHD heat shield.

1. Use a lifting reentry.  Just as it is possible to offset the CG of a reentry body to generate some aerodynamic lift, it may also be possible to locate and orient the magnet in a way to create both lift and drag.  If you do a force balance on a body in a circular orbit, the downward gravitational force is exactly balanced out by a fictitious centrifugal force due to your forward velocity.  As you decelerate though, that centrifugal force component decreases, but by using lift, you can counteract some of that gravitational force.  This allows you to stay up at a higher altitude longer, which allows you to do more of your deceleration in the lower density air.  This is already used by all manned space capsules as well as the shuttle in order to keep reentry decelerations to a reasonably low level, and also to reduce the peak heating.  This is even more beneficial for magnetic braking concepts, because you can do more of your deceleration at a point where the magnetic effects dominate, electrical conductivities are high, and heat fluxes are low.
2. Use as strong of a magnet as you can reasonably work with.  While there are diminishing returns according to all of these papers, a stronger magnet does help provide more deceleration and shoves the boundary layer away further.
3. Use an alkali seed.  As velocities decrease, it gets harder and harder to maintain the electrical conductivity in the plasma at a high enough value to maintain useful levels of Lorentz interaction.  This is similar to the challenge with MHD electric generators.  In order to keep the conductivity high, injecting an alkali metal into the stream can help.  Alkali metals, particularly Potassium and Cesium have very low ionization energies compared to air.  In a weakly ionized plasma, most of the atoms are actually not atomized–almost all of the conductivity is provided by the small number of atoms that are.  So, a little bit of seeding can go a long way.  This helps you keep your magnetic deceleration forces high even as altitude and velocity drop.  The other nice thing about seeding, is that depending on what the fluid is, it might also cut down on the radiative heat transfer from the hot shock layer back to the heat shield.
4. Heat the plasma.  This may sound counterintuitive, but you might actually get better thermal protection if you start heating the plasma once you get to a certain point.  Below Mach 12, even with seeding, there just isn’t enough heat rise caused by the shock layer to keep the plasma  sufficiently ionized.  But, it is actually possible via several different means to dump a bit of energy back into the shock layer to push the gas back into an ionized state.  It’s unclear at this point if this is worth doing, but if the system is light and simple enough it might be worth considering.  As it is, you’ll have a lot of stored energy in the superconducting magnet, and you probably want to dump that somehow before landing–using it to keep the incoming air ionized a bit longer to get a little more deceleration before you hit the thick air might be worth it.

All told, you’re still going to need some sort of thermal protection for the last bit of deceleration, but the heat loads and max temperatures are so much lower if you can dump say half the reentry velocity while you’re still high up, that the problem becomes a lot easier to deal with.  If you could only get down to Mach 12 with this system, that would cut the peak and total heat loads by at least a factor of 8x.  The heat fluxes at this point would be low enough that you wouldn’t need ablative materials, and could probably use a ceramic tough enough that it was low maintenance.

Anyhow, the key questions I have at this point are: a) what sort of effective “L/D” ratio can you get by varying the location and orientation of the magnet, b) how much does seeding help, c) how long can you stay up in the high altitudes, d) what is the maximum amount of velocity decrease you can provide via this method, e) how strong of a magnet could you reasonably hold on an RLV, f) how does the strong magnetic field interact with the operation of the RLV itself–what does it do to solenoid valves, electric actuators, etc. and is there a way to shield against these issues?

In the next segments, I’m going to talk about another, possibly even more interesting application of this concept, as well as some thoughts on how we can reduce this technology to practice.

Last year, my family went out to the coast to spend the holiday with a good friend who does finance and project management work for a large aerospace company out there. We’ve been brainstorming various space business opportunities for some time to see if there were any interesting areas that we could both make money and make a difference in the utilization of space. While we were out there for Thanksgiving, Colin pitched the concept of making the equivalence of a Travel Agency for unmanned space experiments to fly on suborbital vehicles. I thought it was an intriguing idea at the time, but have been too busy to write anything about it (I’m also somewhat reticent to go too much into details that relate to the business of my day job without getting approval from Dave and the others). Anyhow, Colin started a Space Business blog this past month, and one of his first posts is discussing this very idea. I’d suggest reading the whole thing.

Michael Mealling once quipped that it was far easier to take a business guy and get him interested in space, than it was to take an aerospace engineer, and somehow get him to understand business. I’ll admit to being firmly in the “aerospace engineer that’s trying to understand business” category myself, so I think having blogs like Colin’s out there is a trend I hope to see increasing over the years.

Oh, and welcome to the blogroll, Colin!

I’ve been meaning to write for a while about a rather fascinating, but not very well known, area of research that I think might have significant implications for several areas of space transportation. The research I am referring to is focused on exploiting Magneto-hydrodynamic forces to manipulate weakly-ionized plasmas caused by hypersonic flight in rarefied flows–ie using magnets to shove around the hot flamey stuff caused by slamming into the thin air above us at crazy-high speeds. I’m going to be a tease, and not go into some of the ramifications until later posts in this series, but for now I want to give a bit more of an explanation than I’ve found available in the popular press so far.

Oh, and one small caveat before I jump in–while I think there’s some real potential here, electromagnetics is a topic that I’m truly awful at. I’ve never had another class, including a PhD level turbulent fluid dynamics class that made me feel like such a brow-dragging neanderthal as my Physics 122 class on Electromagnetism. This may be yet another niche technology that while somewhat interesting, ends up not being all that useful. But it looks at least possible that this may become a game changing technology in many space transportation fields. Without further ado, let’s go over some of the basics.

Some Background on MHD Aerobraking and Thermal Protection
The basic concept behind MHD Thermal Protection is that during hypersonic flight, above about Mach 12, the shockwave formed in front of a blunt-bodied vehicle reaches a high enough temperature to form a weakly ionized plasma that is conductive enough to be manipulated by strong magnetic fields. A powerful magnet near the leading part of the vehicle interacts with charged particles in the plasma via the Lorentz force. This force bends the trajectory of charged particles, creates large hall currents, which if I’m understanding correctly repel the magnetic field. These charged particles also impact with the uncharged gas particles nearby (the plasma is only “weakly ionized”) transmitting these forces to them as well. Here’s an interesting diagram I’ll reference from one of the papers I’ll talk about more later (”Trajectory Analysis of Electromagnetic Aerobraking Flight Based on Rarefied Flow Analysis” by Otsu, Katsurayama, and Abe–well worth the $28):

Figure 1 (from Otsu et al): Schematic View of the Flow Around a Vehicle With Applied Magnetic Field and Induced Current

If the magnet is strong enough, this leads to two interesting effects–first, the distance from the vehicle to the bow shock increases (I think the plasma density between the bow shock and the vehicle also decreases, but I’m less sure about that). This can significantly reduce the heat transferred into the vehicle for a given velocity and altitude. The other big effect is that the Lorentz forces create forces that can produce drag or lift. At high altitudes these Lorentz forces can greatly augment the aerodynamic drag forces, effectively making it as though the vehicle had a much lower ballistic coefficient. It should be noted that this force is electrically controllable. In fact, depending on the sophistication of the magnetic apparatus and its location within and orientation with respect to the vehicle, it can possibly also produce lift as well as control torques without the need for aero control surfaces.

Both of these help from a reentry thermal standpoint, because by the time you hit the denser air, where the heating is the highest, you’re going a lot slower than you would’ve been otherwise, and a lot of that earlier braking is done at much lower heating loads than would have been the case without the electromagnetic effects.

Several of the papers I’ve read introduce an interaction parameter term, Q, that relates the relative strength of the Lorentz forces to drag forces. The relationship takes the form:

Equation 1 (from Otsu et al)

Sigma is the conductivity of the weakly ionized plasma, B is the magnetic field strength, L is a reference length (I think related to the magnet configuration), rho is atmospheric density, and V is velocity. As you can see, for a given magnet, the drag forces start dominating as the conductivity drops and as the atmospheric density increases. Atmospheric density increases dramatically as you descend from orbit, so for a reentry application, you get most of your benefit from the first little bit of descent.

We’ll go more into some of these ramifications starting in my next installment.

I’ve had a few more ideas on the Lunar One-Way-To-Stay concept that I figured it would be worth posting now before I forget them.  I still think this is pretty much the only way that there will be a human foot on the Moon this decade.  More importantly, this is the only cost-effective way short of an architecture using both cryogenic depots and RLVs of doing the actual development on the Moon that would be necessary to lay the groundwork for affordable settlement and economic development.

Horse-Trading on Even Earlier Markets
A good point that was made on the same day by Wes Johnson in comments, and my boss Dave on the carpool down to Mojave, was that the “horse-trading” trick at the center of the business concept I gave could work even before manned landings. One of the big challenges with any lunar surface robotic exploration is the lack of a suitable lander. The big up-front development cost of a lander (especially one done the traditional way, without leveraging the capabilities of us VTVL developers) usually makes it harder to get these projects funded. If you could do a deal where the PI for a proposal only had to come up with the launch costs plus the marginal cost of the science payload such as rover(s), ISRU technique demonstrators etc., it might make it easier to close their proposal. More importantly, as Wes pointed out, PI’s on science missions have a lot more leeway on negotiating details of how to get the payload to the destination. You’d give them the same deal as the others–in exchange for covering the launch cost, you give them free delivery to the lunar surface, and get to sell the other half of the payload.

Robotic Precursor Missions
An interesting development in the NASA budget proposal that has gotten almost no real discussion in the blogosphere, was the funding for a series of robotic lander missions on the Moon and possibly other destinations. These could be a very interesting potential market for the initial lander work. I could imagine the private entity trying to build up to the manned one-way missions could set up a Space Act agreement with some of the groups at NASA to facilitate sharing of information on lander systems, then possibly using a combination of more traditional aerospace and newere entrepreneurial space entities (”OldSpace” entities since they tend to have a wider range of specialized knowledge, and “NewSpace” entities since they tend to have ways to flight test hardware cheaper, and to do cheaper rapid prototyping), could develop the lander in support of these missions. The money for the lander development could be mostly made back by selling the remaining hardware space to one or more up-and coming space countries that wants to get a leg-up on their competition (say either India, China, Japan, or South Korea). Groups that aren’t actively planning lunar landers in the near-term, or which might be a bit behind their competitor might be the most natural targets. Imagine South Korea being able to beat Japan to the lunar surface by partnering with a private space company? Or India beating China. South Korea has already demonstrated its interest and willingness to partner with commercial space companies to get a leg-up in regional technical rivalries. Just food for thought.

Also, this might tie into stuff like Project M, a youtube of which has been floating around the intertubes for a week or so. JSC has been working in the background on trying to put together a plan to do a quick robotic lunar lander, “within 1000 days of go-ahead”. If they don’t get the money to do such a project entirely themselves as-planned, teaming with a private entity might still allow them to pull such a feat off.

Lunar Surface Systems
After thinking this over and talking with some of the commenters, I think this is one area that I was being overly optimistic on. There is going to be a fair deal of expense for lunar rovers, life support systems, habitats, ISRU experiments (including stuff like systems to try out regolith fusing), power sources, etc. Some of these could be supplied as “demo units” by companies interested in selling future versions to other private or public expeditions, some could be supplied by governments wanting to pretest systems before sending their own people, but ultimately some of these systems would likely need to be developed by the private developer running the project. The good news is that if you can get initial revenue from selling some robotic flights on the lander, it might be possible to raise enough money to invest in the lunar surface systems.

Anyhow, just some thoughts. I just think it would be ironic if due to the lunar precursor lander funding, Obama’s “Evil Exploration Eradicating NASA Budget Proposal” somehow enabled the US to beat the rest of the world back to the Moon and ultimately cemented its lead in lunar exploration. All without having to blow tens of billions on new launchers.