I’m writing today about a very dubious claim made by the Ad Astra Rocket Corporation about their electric propulsion system VASIMR. VASIMR is a type of electric propulsion, and is probably one of the better kinds under development at this time.
If you don’t know much about electric propulsion or VASIMR, Wikipedia is an excellent reference for both and I recommend it highly before reading on.
The claim that I find so objectionable is that VASIMR enables transportation to Mars within 39 days. Electric propulsion is great for what it’s good for – missions with long travel times where mass is at a premium and power is not. It is not good for manned missions for exactly these reasons.
Dr. Robert Zubrin, true to form, has responded with a bombastic but very much correct rebuttal to this claim. I would like to expand upon his intuition here using some numbers to demonstrate why this claim is so fundamentally and probably intentionally deceptive.
What follows is my methodology. If you don’t care, by all means skip down to the results and the graphics below the second divider line.
In order to do so, I’m going to assume a couple things:
- The change in gravitational potential energy of the Earth and Sun are small in comparison to the kinetic energy of the spacecraft. This is justifiable: The minimum distance between Earth and Mars is about 75 million km. Traversing this distance in 39 days implies a mean velocity of 22 km/s, so 44 km/s to accelerate and decelerate, although the actual peak velocity and therefore total delta-V budget will be much higher. For comparison, the delta-V from Low Earth Orbit to Low Mars Orbit on a minimum energy trajectory is about 6 km/s. Therefore I will treat this as a straight kinematics problem.
- The spacecraft accelerates, turns around, and decelerates with no time in between. This minimizes the power requirements but not the delta-V requirements. While this is not strictly the optimum result (depending in large part on what you’re optimizing for) it’s justified by the fact that electric propulsion systems, VASIMR included, have very low thrust and thus accelerating to acceptable velocities in shorter time spans is even less reasonable. Furthermore, because of the high exhaust velocity the relative cost of higher exhaust velocities is low.
- The transit speeds will be too high to make aerobraking at Mars or Earth a reasonable proposal. In some senses, the possibility of aerobraking cancels out the change in gravitational potential energy which I am neglecting.
- The engines will produce a constant force at all times, but because the mass will vary with time the acceleration will change.
I will cite the sources for my numbers if possible or justify them if not.
Newton’s Second Law states that:
Where F is Force, a is acceleration, and m is mass. Of these, only Mass is a function of time:
So we have:
Where m0 is the initial mass, r is the rate of change of mass (always negative) and t is the time since the engine began firing. Keeping in mind that acceleration is a function of time, I integrated and got the following (Checked with Wolfram Alpha):
Where ΔV is the change in velocity from time 0 to time 1, but not the ΔV of the mission taken as a whole. Basically, we need to solve for when the ship needs to change from speeding up to slowing down by calculating the ΔV from 0 s to t1 and t1 to 39 days, setting them equal to each other, and solving for t1. The result is as follows:
Finally, we need to solve for the amount of force that’s required to do this maneuver. This is a function of total distance. But rather than integrate again and try to solve a nasty and possibly un-solvable algebraic formula (we’re not savages, after all!), I wrote a Matlab code to do the integration for me and allowed me to guess various levels of force until I found one that was right. I realize there are better ways to do this and don’t care very much because this one worked fine.
In order to give real mass breakdowns, payload fractions, etc., I also have to give some numbers to the thrust-to-weight ratio of engines, power sources, and fuel tanks. Therefore, I will use the following numbers for the mass of system components:
- 830 W/kg for the VASIMR engine, as given in this paper. Please note that this is actually an estimate for an engine that hasn’t been built, meaning it is very much open to manipulation, since the author of the paper is also the owner of Ad Astra Rocket Corp. The paper also suggests a pathetic electric-to-kinetic efficiency of 4% for presently existing engines. I will use Mr. Chang Diaz’s projections that future engines can reach 50% efficiency. This means that the kinetic energy of the exhaust will be 415 W/kg of engine.
- I will assume that a solar power system with a specific power of 300 W/kg will be used. This is higher than currently existing designs, which as of 2004 were getting less than 100 W/kg. This is also higher than nuclear systems. Even the SAFE-400 (Go to Wiki) is a very modern nuclear design and doesn’t include any systems to convert thermal energy to electrical energy, its specific power is under 200 W/kg.
- I will assume that tankage requirements constitute 5% of the mass of whatever is in the tank. This is actually really optimistic because VASIMR uses a very light Hydrogen fuel. For example, the Space Shuttle External Tank massed 29,930 kg, and contained 721,045 kg of fuel. However, most of this weight is Liquid Oxygen, which is much more dense than Liquid Hydrogen. Pure Liquid Hydrogen is about 5 times less dense (70 kg/m^3 as compared to 360 kg/m^3) than the Hydrogen/Oxygen mixture used in the shuttle; If the tank contained the same volume of only liquid Hydrogen, the tank’s mass would be more than 20% of the mass of the stuff in the tank. So this is a really generous assumption.
Here are the two MATLAB scripts used for this calculation, linked to on Pastebin. It’s important that the two scripts retain their names, so save VASIMR.m as VASIMR.m and rocket.m as rocket.m. Capitalization matters!
They need to be saved into the same folder in order to work. If you don’t have MATLAB on your computer and don’t want to pay for it, FreeMat should be able to run these programs just as well and doesn’t cost anything. The way the script works is that once you’ve chosen your parameters (I believe I’ve mentioned all the important ones in this post, but please note that the script uses exhaust velocity, which is a factor of 9.8 times higher than Isp) you do guess-and-check by changing the force value until it outputs a “Distance” (This is the ratio of the distance travelled to the minimum distance from Earth to Mars) equal to 1. As I said, there are better ways to do this and I didn’t feel like doing any of them because this works well enough.
Here are my results:
I chose to give a large number of significant digits for the initial acceleration, because it’s very sensitive to slight changes in this value. All numbers are used in their typical way. Normal mass ratios for Marsbound vehicles using chemical fuel are around 3, and anything above about 10 is very high; Above 20 is probably impossible.
The most important number in this chart is the Necessary Reduction Factor (NRF). It describes the ratio of necessary solid mass to the amount of allowable solid mass. For example, if you choose an exhaust velocity such that your rocket has a mass ratio of 4, and its initial mass is 80 tonnes, you can have up to 20 tonnes of solid mass. But let’s say your tanks mass 3 tonnes, your engines mass 17 tonnes, and your power source masses 20 tonnes. That would mean you would need 40 tonnes of solid mass to complete your mission, and you would have a NRF of 40/20=2. Basically, it describes how much you need to shrink down your components to make the mission feasible. For NRFs below 1, you have some amount of payload carrying capability too.
As you can see, there is no value of the exhaust velocity for which the NRF of this system is below 1, or even anywhere close. By picking an Isp value between 5,000 s and 10,000 s, it’s possible to get a value slightly below 12, but not one below 11.
For Dr. Chang Diaz’s claims to be true, VASIMR and all related technologies would have to be at least twelve times lighter than they actually are.
But its even worse than that: The engines that Ad Astra Rocket Corporation has actually tested have Isps of about 2,000 seconds. For rockets with exhaust speeds that low, the mass ratios get so high that it’s nearly impossible to get a value for how much mass is actually left.
Basically, Ad Astra Rocket Corporation is about as close to being able to do this as Ford is to building a car that gets 200 miles to the gallon at 1,500 mph.
The image below shows just how much this technology blows past the mass limits available to it.
The maximum allowable mass is normalized to 1, with the mass of the engines and total system masses expressed relative to this. As you can see, they’re much, much higher.
For anyone who’s interested, here’s a plot of the Position-Time and Velocity-Time profiles for a typical scenario (I used Isp=5,000 s):
If you took high school physics, these graphs should be familiar to you. Notice that the velocity graph’s peak corresponds to the turnaround point, which happens towards the end of the transit because the acceleration increases at the mass decreases.
So, there you have it: Claims debunked. Spread the word.