Ion Propulsion: A story of the Conservation of Momentum

     Behind every theory for future forms of spacecraft propulsion lies a terrible truth. They are all either too costly, to difficult, too dangerous, or too inefficient for practical use today for reaching towards the planets. But what if there were a form of propulsion for which the fuel would cost 33 times less than traditional fuel? What if there were a fuel that was non-toxic, non-volatile, and unlikely to cause an explosion or other catastrophic failure? What if there were an engine that had an average efficiency rating of 70%? And what if it were available and ready to launch today? Ladies and gentlemen, this is the truth behind the xenon ion propulsion drive.

     The ion thruster is a feasible solution for space travel in the not-so-distant future. The truth is, it has already been done. In 1998, NASA launched Deep Space 1, a probe designed to do a close flyby of an asteroid. What was supposed to be a fairly short, straightforward mission turned much more complicated when they realized that Deep Space 1 also had the opportunity to study the attributes of passing comet Borrelly first hand. Though the encounter with the asteroid was only considered a partial success, the information gathered from Deep Space One's flyby of comet Borrelly turned out to be a huge success. And it was all thanks to the highly efficient and decently cheap ion propulsion system mounted on board.

     The theory behind an ion drive is actually quite simple. Xenon atoms are initially stored in a containment unit. Since xenon has a full valence shell of electrons, it is non-volatile and non-toxic. This also means that xenon is very easy to ionize. Ionization is a process where a stream of electrons are shot at regular atoms in such a way that electrons get knocked loose from the atom, creating a net positive charge on the atom. Since positive charges are attracted to negative charges, these ions are then subjected to two powerful charged plates, the one behind the ions being positive, and the one in front of them being negative. As the positive plate forces the ions away from the craft, the negative plate pulls the ions out into space. The potential difference accelerates these positive ions out of the exhaust tube at speeds upwards of 40 kilometers per second! Since leaving a trail of positive ions behind the spacecraft can induce a charge upon the spacecraft, which can be dangerous, a stream of electrons is shot into outgoing ions to neutralize them so they have no net charge.

Theory Behind an Ion Drive

     However, at this point you may have noticed the problem. Though the exhaust velocity is 40 kilometers per second, we can only accelerate a few atoms of xenon at a time. This makes acceleration a very difficult and tedious process. It would be like instead of pressing the gas pedal to accelerate your car to work, you threw grains of sand at extreme velocities to slowly accelerate your car to its top speed. Granted, you would save gas money, and your car would actually achieve a higher speed (neglecting friction), but it would take you hours or days to even get your car to go a modest speed. An ion powered spacecraft would either need to be constructed in space, or blasted into orbit via chemical rockets. This is the only major downfall of ion propulsion.

     The ion propelled thruster works on a very simple level. The law of conservation of momentum states that the mass multiplied by the velocity of one object which exerts its energy upon a second object is equal to the other mass multiplied by the other velocity. In simpler terms:

m1v1 = m2v2

     Where m1 and v1 are the mass and velocity of the xenon atoms, and m2 and v2 are the mass and velocity of the spacecraft. We know that the mass of one xenon atom is 2.18*10^-25 kg, and that they are being expelled from the spacecraft at 40 km/s, or 40,000 m/s. Let's say that we wanted to launch a 500 kg probe to Mars. How much xenon would we need. Well:

m1 = 2.18E-25

v1 = 40,000

m2 = 500

v2 = ?

(2.18E-25)*(40,000) = (500)*(v2)

v2 = 1.74E-23 m/s

     Let's say we want our top velocity to reach 20,000 m/s, about 5 times the speed of a conventional rocket:

(20,000)/(1.74E-23) = 1.15E27 xenon atoms = 250.7 grams.

     This mass of xenon will get the craft to 20,000 m/s. The craft will then coast at that speed until it is time for it to begin to slow down to arrive at its destination. This does mean, however, that it will need another 250.7 grams of xenon to slow back down to starting velocity, bringing the total amount of fuel to about 502 grams. But how fast can we get up to this speed? Let's say that our engine has a diameter of half a meter. This will give it an output of about .044 Newtons of thrust. For our 500 kg craft, this is equal to an acceleration of  8.8*10^-5 m/s^2. This means that, starting from 0 m/s, it would take upwards of 6 hours to reach our desired speed of 20,000 m/s. So time is the only sacrifice of the ion engine.

     But what about the benefits? The cost of 501 grams of xenon fuel for a 500 kilogram spacecraft would cost just proud of $600. Compare that to the $20,000 of liquid fuel that it would take to get a space vessel from low Earth orbit to Mars's orbit. And even with all that expensive fuel, it would still only be able to achieve a peak net velocity of around 4000 m/s, though it may only take a few minutes to get there. This is only a fraction of the top speeds achievable by the ion drive. Plus, with no explosive or volatile chemicals on board, the journey would be much less prone to accidents. The far future may hold the keys of space exploration in nuclear and antimatter drives, but getting us to our celestial neighbors today may be much more feasible by means of the xenon ion drive.

A Trip to Mars: A New Form of Propulsion

     On November 6th, 2011, NASA launched the Atlas V space vehicle en route to the red planet, nearly a nine month trip ahead of it. On board the Atlas V, a small rover, about the size of a car, was curled up within the nose. That rover would later land on the surface of Mars to discover that the seemingly cold, dry dust of Mars once long ago had the ability to support life as we know it. The Curiosity project is regarded as one of the most informative and successful missions to Mars to date; but it didn't happen overnight. The unsatisfying truth of the Curiosity rover's trip to Mars is that it took 253 days from liftoff at Cape Canaveral, to touchdown at Gale Crater on Mars's dusty surface. Whats more is that this rocket requires 284,000 kg of fuel just to escape the strong pull of Earth's gravity and maintain a stable orbit. Since time is one of humanity's most valued possessions, and the cost of rocket fuel is not expected to drop in the near future, its time to start searching for new sources of fuel and propulsion to get us to the surfaces of our nearest celestial neighbors.

Atlas V Rocket

     At its closest approach, Mars is 55 million kilometers away. Since the Atlas V accelerates the Curiosity probe to about 20,000 kilometers per hour, one would think that you could just the distance and divide by the speed to get the length of time required for a trip to Mars. This, however, is not the case. Since everything that travels around the Sun goes in elliptical orbits, the rocket will not be an exception. It's path too will be curved by the gravitational pulls of the Earth, the sun, and Mars. At 20,000 kilometers per hour, a spacecraft launched from Earth when Earth and Mars are in sequence (in line with the Sun), the craft will have arrived at Mars after Mars has done one half orbit in its path around the Sun. So, the actual distance traveled by the craft itself is the average of the lengths of Earth's and Mars's orbits divided by 2. This gives us a total distance of about 593 million kilometers. 

     However, this is (clearly) not the most efficient way to launch a craft to Mars using conventional methods, seeing that it would take the craft more than three years to reach Mars. If the craft is launched at a higher velocity, the curved orbit from Earth to Mars will have flattened out. The only down side to launching spacecraft at these trajectories is that once at the red planet, the vessel will have to begin to slow down to enter her orbital velocity.

     In this image, the distance traveled between Earth and Mars is closer to 300 million kilometers, slicing the trip time down to only about 18 months. The truth is, there's no limit to how fast you can fly a ship from Earth to Mars (except the speed of light), thus the path between Earth and Mars will grow continually shorter with velocity. If you could accelerate a craft to the near the speed of light, the shortest possible distance would only be about 55 million kilometers. Take that divided by the speed of light and you get a trip time of about 3 minutes. So, if there is no limit, the only question is how fast can we go?

     There has been a lot of talk in recent years about the possibility of using nuclear fusion power to propel a rocket to Mars. The theory behind nuclear fusion is actually quite simple. Take, for example, the fusion of deuterium, an isotopes of hydrogen. Deuterium is simply a hydrogen atom with two neutrons. In the image below, blue spheres represent neutrons while red spheres represent protons. When heated to extreme temperatures, these hydrogen atoms will begin to gain velocity. Once a certain velocity is achieved, these atoms will slam into each other with enough force to combine into a single helium atom and release a tremendous amount of energy.

     Let's compare this to the energy released by liquid hydrogen, currently the most efficient and powerful rocket fuel in use today. The specific energy of a substance tells us exactly how many megajoules of energy are released per kilogram of whatever it is you are using as fuel. The specific energy of liquid hydrogen is about 141.86 megajoules per kilogram. That is a lot of energy. And that is in just 1 kilogram of liquid hydrogen. putting that into perspective, 1 kilogram of liquid hydrogen could power an average American household for 31 hours.

     So how about nuclear fusion? The net reaction of one deuterium-tritium fusion (tritium being another isotope of hydrogen, actually more efficient than a deuterium-deuterium reaction) creates about 12.5 megaelectron volts, or about 2*10^-12 joules. This may seem like a tiny amount of energy, but this is only in one reaction of just one deuterium and one tritium atom! Since one mole of hydrogen is equal to one gram, we just multiply that by Avagadro's number (the number of atoms in one mole of hydrogen) to obtain how much energy is released in one gram. This turns out to be 1.2*10^12 joules in one gram, or in a kilogram, 1.2*10^15 joules. Using nuclear chemistry, we find that one gram of reacted dueterium-tritium fuel generates 1.2 billion megajoules of raw energy! Modern experiments show that at best, a fusion engine could only utilize about 30% of the total energy created. This means that the energy density of the nuclear fusion of deuterium and tritium fuel at 30% efficiency is 361 million megajoules per kilogram. This is 2.5 million times more efficient than liquid hydrogen fuel. Recall from before that the Atlas V takes 284,000 kg of traditional fuel to launch off, leave Earth's orbit, slow to Mars's orbit, and land on the surface of the planet. Using a vessel of the same mass and a fusion powered rocket, we find that a voyage to Mars would only require about 113.6 grams of fuel! If it were a manned mission, we must assume that it would be nice for the crew of the craft to return home, bringing the grand total to about 227.2 grams of both deuterium and tritium fuel, or 454.4 grams of total fuel (about 1 pound for all you Imperial system goers).

Proposed NASA Nuclear Fusion Rocket

     Using the ideal gas law, we can determine the volume of two spherical fuel tanks to hold the deuterium and tritium as well. For a system at reasonable temperatures and pressures, PV = nRT, where:

P = Pressure
V = Volume
n = moles
R = .08206 (constant)
T = Temperature

     We want a pressure of 1 earth atmosphere, and since one gram equals one mole for hydrogen, n will equal 227.2. Room temperature in Kelvins is 294. Thus:

(1)*V = (227.2)*(.08206)*(294)

V = 5481.3 liters, or 5.48 cubic meters

For a sphere:

V = (4/3)*pi*r^3

Thus:

r = 1.09 meters

     So we would need two spherical fuel tanks just over 1 meter in radius to bring a manned crew to Mars and back in a vessel about the mass of the Atlas V. This is an incredibly small amount of fuel compared to the thousands of kilograms needed to launch the Atlas V towards Mars. Nuclear fusion is definitely a cheaper source of spaceflight, but what about faster? According to NASA, the exhaust velocity of a nuclear fusion powered rocket is likely to be upwards of 30 kilometers per second. Compared to that of the Atlas V rocket, whose exhaust velocity is approximately 4 kilometers per second, this would cut the trip time from here to Mars by more than 7 times, leaving the ETA at about 36 days. Not to mention that the only byproduct of nuclear fusion is helium and neutrons. So there you have it; cleaner, faster, and cheaper. So, why are we not using it?

     The sad fact of the matter is that fusion isn't easily achieved. For deuterium and tritium atoms to smash together with enough force to fuse and release energy, we must first heat them to 700 million degrees Kelvin, a temperature hotter than the core of the sun. Not only do we not have the means with current technology to heat anything to such a temperature, but no known material could contain such heat without deformation. Furthermore, though hydrogen is very abundant, deuterium and tritium only compose a fraction of a percent of the total amount of hydrogen in the universe.

Temperatures for Achieving Fusion (Blue line is Deuterium-Tritium)

     So fusion may not be practical as a source of rocket propulsion today. It is only a matter of time, however, until we are able to find a way to heat hydrogen to such temperatures, possibly by way of microwaves, and keep it contained maybe in an extremely rigid futuristic material or with a powerful electromagnetic field. In the near future, we may even be able to synthesize deuterium and tritium fuel by somehow combining excess neutrons with hydrogen atoms, using the strong nuclear force as a glue for the new isotopes. Though fusion may seem difficult today, we will, as a human race, find way to overcome these setbacks in order to further mankind. Mars is our nearest celestial neighbor besides our own moon, and we know so little about it under its mysterious, dusty red crust. With the nuclear fusion rocket, a trip to the red planet may be only a few pounds of fuel and 36 days away. So what are we waiting for?