Electric Propulsion, VASIMR, and 39 Days to Mars

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:

  1. 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.
  2. 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.
  3. 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.
  4. 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:

  1. 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.
  2. 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.
  3. 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:

Results for Various Isp values.

Results for Various Isp values.

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.

Engine Mass and Total System Mass relative to maximum allowable mass

Engine Mass and Total System Mass relative to maximum allowable mass

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):

Position-Time and Velocity-Time graphs for typical transit

Position-Time and Velocity-Time graphs for typical transit

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.

The American Democracy Act

Here’s something for all of you, that should come as no surprise:

Congress is broken.

Congress actually has two jobs:  Writing legislation, and passing legislation.  It’s doing an abysmal job of both.

Writing legislation is the process of determining the problems that need to be solved, researching the issue, and coming up with good policy solutions.  It involves foreseeing issues in implementation and addressing them.  It involves writing the legislation in a concise, legible way.  It involves writing legislation without undue influence from lobbying groups and special interests.  It involves writing legislation that actually works for real Americans.

Passing legislation is the politics required to get to 218 votes (A majority) in the House and 60 votes (A filibuster-proof majority) in the Senate.  It involves the ability to work through the political process in such a way as to get legislation passed, both through choosing passable legislation to fight for and through taking the right strategies to get your legislation signed into law.

Congress’ approval rating hovers around 10%.  By almost any measure, Congress is doing a bad job, and nobody seems to know how to fix it.

Here’s my proposal: American Democracy Act (Link to .doc download of the full text, 8 pages).

Here’s how it works:

The US government will create a website, which will take the form of a wiki, where people can propose and collaborate on legislation.  We’ve seen from Wikipedia, among other things, that with a proper set of guidelines and culture people interacting on the Internet can make some things that are truly great.  There’s a vote page, where people can Support or Oppose legislation, and the legislation with the most support and least opposition is regularly sent to Congress.

Congress has to vote on the legislation, and it will pass if it gets 218 votes in the House and 51 in the Senate (No filibusters permitted!), when it will go to the President to sign.  It’s simple, it doesn’t require a Constitutional amendment, and it gives the American people real say into what their government does in the process of governing them.

If this sounds interesting to you, I encourage you to read the bill, or ask questions, or comment, and spread the word!

Unilateral Metrication

As you may already be aware, I have a particular dislike for the system of units used in the United States, and a great love for the International System (SI, for Système International) Units. It is a great pain to me that my home country is stubbornly holding out against its inevitable adoption of units that simply make sense.

There are plenty of benefits to doing this, but these have all been argued before. And, more importantly, the argument has been settled: 95.6% of humanity lives in countries where SI units, or some variant thereof, are or shortly will become the standard and the norm. This is perhaps the only social construct which we as a species have ever really agreed on and used in exactly the same way.

The meter and the gram, two of seven base units in the metric system, were defined in a law passed by the First Republic of France on April 7, 1795, nearly 220 years ago. Since then the metric system’s progress towards universal human adoption has been direct and linear. The United States is the only country that is both large enough and stupid enough to hold on to an archaic and poorly designed system of units whose obsolescence has long been clear. This has left it with an unfortunate and often confusing mix of archaic US “Imperial” units and metric units. Basically, it boils down to this: Any time measurement accuracy is important, measurement will be done in metric units. This holds for the fields of science, engineering, construction, and medicine. Whenever it is necessary that people who may or may not have graduated high school understand what you’re doing, US units are used in preference to a quick explanation of the metric system.

As on many other issues, American politicians and cultural leaders are united against the tide of logic. This goes beyond the two-party system. While both Democrats and Republicans are opposed to metrication, so is the Constitution party, the Libertarian party, and the Green party. Actually, so far as I can tell, there is no political party in the United States, no matter how minor, that supports metrication as an element of their platform. This is a disgrace.

But I digress. What can we do about this? Well, there are two answers. The first is that we can and should start a metrication party, and we should lobby all parties, major and minor, to add metrication to their platform. I suspect the Greens would be most amenable, but more on this later.

What I propose is that we begin metrication with ourselves. Each of us has the power to measure our height in meters and our mass in kilograms. We can all tell our weather widgets to give us the temperature in Celsius, and we can tell our navigation units to give us the distance to our destination in kilometers.

I believe strongly that society is the sum of its members. For every person who metricates, we push back the tide of Imperial units and bring their inevitable obsolescence closer. Within the US, this amounts to a unilateral decision to metricate. From a global perspective, however, it is simply a recognition that it is time to give up on the idea of American Exceptionalism, and join humanity in a species-wide pursuit of progress.

On Idealism

My politics are not comparable to those of a great majority of Americans’.  I identify as a Socialist because I don’t want to give false impression of moderation.  While I don’t believe in the most common conception of socialism, an economy administered by the government, I believe that my core principles put me far outside the norm for an American and are similarly divergent from most major ideologies in the Western world.  These are:

1) Society has no purpose other than to aid its members

2) Social structures should be changed wherever this is not the case

3) 1 and 2 taken together imply a society with drastically different order than exists anywhere on Earth

I believe that there is something deeply wrong with a world in which one billion go hungry while another billion suffer from an excess of food.  I think that it’s atrocious that at a time when technology allows some to live in unprecedented excess while they, through action and inaction make it hard to even begin to effect the changes that could create a better world.

It is unfortunate that people focus on money and possessions more than learning for its own sake, than on love and satisfaction and fraternity.  Where biological need has been eliminated, I don’t see a quest for enlightenment, but simply for more items.

I don’t know the solutions to these problems.  I know many don’t think that they’re problems.  I know many more don’t see their importance compared to more immediate concerns.  I know that, no matter what I say or do for the rest of my life, there will be people who disagree with me.  This puts us in a situation where, to use Machiavelli’s terminology, we have hard questions to answer about ends and means.

Every person who wants to change society has a fundamental question to answer with regards to strategy:  Do you act as a part of the society you seek to change or do you attack it as an outsider?  Depending on your choice, you are guilty either of appeasement or a willingness to fail.

The difficulty of formulating a coherent answer to this question only becomes harder when the realities of electoral politics are taken into consideration.  By placing your ideology on the ballot, you are putting yourself in a process which pushes back strongly against original thought and meaningful change.  Minority parties are often powerless, and achieving majority status usually means sacrificing your beliefs along the way.  My response to such realities is as follows:

Never compromise on your ideology or beliefs.  Never.

Always compromise on strategy.  Do everything you can to do something.  If your beliefs are important, partial successes matter, and a good compromise is success.

Eliminate the bourgeois capitalist system, one compromise at a time.

See wikified version at http://gammafactor.wikia.com/wiki/On_Idealism.