Straight out of Star Trek: Antimatter Propulsion

An actual proposed antimatter rocket

     As a child, I grew up watching Star Trek. I grew up watching a lot of science fiction TV series, but Star Trek was different. Whereas Star Wars was a spectacular story aliens and spaceships of a different galaxy far, far away, Star Trek was closer to home. It depicted a future version of humanity, where we had all given up wars and savagery for a greater good; the exploration and discovery of the cosmos. With the recent death of renowned Star Trek actor Leonard Nimoy, many of us are wondering if the Star Trek legacy will continue to remain true to its standards. I cannot speak for Paramount on that issue, but I can say that the ideas and technology that Star Trek set the standards for are becoming closer to reality every single day. Communicators, computer pads, and even the universal translators first conceived by Star Trek are becoming analogous to the technology that we see around us every day. It is only a matter of time before our top scientists sit down and discuss the reality of a mission to another star. And the technology behind that mission was also predicted by, you guessed it, Star Trek.

Star Trek inspired many of today's technologies

     Antimatter propulsion has been on the scope as a potential form of space propulsion for decades. The idea is actually really simple. E = mc^2. Whereas other forms of space propulsion require muddy formulas that may give us the energy output of the engine estimating on a feasible efficiency, the formula for an antimatter rocket depends solely upon how much antimatter you are supplied. Okay, let's do a little experiment. Let's say that CERN has decided to use their atom smasher to collect antimatter for the use of energy yield. Slaving away for thousands of years (our current technology is not very suitable for collecting antimatter), they have amassed a grand total of one gram of antimatter. Plug that into our formula for energy yield:

E = mc^2
E = (.001)(3*10^8)^2 = 9,000,000,000,000 J = 90,000,000 MJ

     One measly gram of antimatter provides us enough energy to power a 100 watt light bulb for an astonishing 2854 years, providing that the light bulb doesn't burn out. Okay, but what about traversing the cosmos? If you can recall my post about nuclear fusion propulsion (you might want read that one first if you haven't yet), one gram of tritium-deuterium fuel gets you 3.6*10^11 joules of useful energy in one gram, after you multiply by efficiency. Simple math will tell you that antimatter is a whopping 250 times more efficient than nuclear fusion. Recall also that a mission to Mars required roughly 460 grams of total fuel. That's right, a trip to Mars using an antimatter propelled rocket would be able to make a round trip to the red planet and back with just 1.8 grams of total fuel, or just 0.92 grams of antimatter!

A matter-antimatter collision results in energy and gamma rays.

     Slap me twice and call me stupid, but that number is shockingly low. If the human race were to ever harness that kind of technology, traversing within our solar system would become a snap! Any spacecraft would only need a few grams of fuel to reach their destination, and return home with fuel to spare. But the real question remains; is it enough to get us to the stars? Mars is (on average) 225 million kilometers away. Our nearest neighbor, Proxima Centauri, is 4.22 light years. This equates to 4,000,000,000,000, or four-trillion kilometers. I could simply divide Mars's distance by Proxima Centauri's distance and multiply by 0.92 grams to find how much fuel it would take to perform a round trip, but let's be real. There are a lot of limitations and specifications that need to be addressed before we begin calculating our fuel consumption.

     First of all, traveling across the cosmos does not happen instantaneously. We are restricted by the speed of light. This means that any manned mission to another star would need enough volume for not only the crew, but also the food, the water, the engine, and all other electronics and systems on board. Secondly, a trip to another star is very different from a trip to a planet. We would most likely be dealing with relativistic speeds which require more complex math (which I will get to) to accommodate for the acceleration of the craft. Finally, due to the duration of such a mission, it is unlikely that our first manned mission to another star will involve a return trip. The crew of whatever craft leaves the Earth, is most likely destined to remain upon whatever planet they decide to go to.

What a habitable exoplanet might look like around Epsilon Eridani

     Now the year is 2150 and we are finally ready to send a manned mission to another star. Top scientists have identified a habitable planet in the Epsilon Eridani system, using gravitational microlensing, approximately 10 light years away. They have constructed a massive craft in high Earth orbit weighing in at over 840 million kilograms (roughly 420 times the mass of the space shuttle) that is suitable to support 1000 human passengers for a multi-year mission to this strange new world. The ship is designed so that it will accelerate at one Earth gravity until it reaches its maximum speed at 70% of the speed of light, where it will coast until it nears Epsilon Eridani. From there, it will spin around and reverse accelerate at 1*g, taking in hydrogen using a Bussard collector from interstellar space to use as matter for her matter-antimatter engines, until it comes into orbit around the young, orange star. The ship will maintain artificial gravity while coasting by having the crew live on a giant rotating hub, which spins just fast enough to simulate the gravitational forces of Earth. Now here's the kicker; how much fuel do they need? Imagine that one on your next physics exam!

     Let's start off small. How long will it take the crew to accelerate to 70% the speed of light (0.7*c) at 1 Earth gravity. The formula for that is decently simple:

v = a*t

     Where "v" is desired velocity, "a" is the acceleration, and "t" is time. 0.7*c is equivalent to 2.1*10^8 m/s accelerating at 9.81 m/s^2. Plugging in the numbers, we get that:

t = 21.4 million seconds = 0.68 years

     End of story? Not exactly. Because the ship will ultimately be traveling at relativistic speeds (which really just means speeds close to the speed of light), we will need to use a formula that compensates for the relativistic effects. This "relativistic acceleration" formula, is as follows:

v = (a*t)/(1+(a*t)/(c))^0.5

     Wait a minute, so why don't we see this formula when calculating the acceleration of a ball as it falls towards the Earth? Because the ball is not traveling near the speed of light. This formula is only useful for objects that accelerate to velocities close to "c". Once again solving for "t", we get:

t = 3.0*10^7 seconds = 0.95 years

     Assuming a constant acceleration of 9.81 m/s^2 (Earth gravity), this gives us that the total distance traveled is about 0.45 light years. Since it took the craft 0.45 light years to accelerate to 0.7*c, it will take that same distance to slow down. this means that the craft only needs to burn fuel for 0.9 light years of its 10 light year trip! 9.1 light years will be spent coasting at 0.7*c. Due to the fact that we are accelerating to relativistic speeds, our fuel percentages will shift as well. For any engine that is 100% efficient at converting mass to energy, the projected weight of the fuel for a 0.9 light year voyage is approximately 2.1 kg for every kg of spaceship. Recall that our spacecraft is 8.4*10^8 kg, meaning that our fuel mass is 1.8*10^9 kg! Yikes, this number seems pretty high. but let's break that down, quite literally. What I mean is, let's take that amount and split it into six spherical fuel tanks, each of which is equipped with powerful magnetic fields to ensure that the antimatter does not touch the walls of the tanks.

An antimatter rocket in the interstellar mod of Kerbal Space Program

     For the purpose of this calculation, I will assume that the vessel uses anti-hydrogen, which should have all of the same properties as regular hydrogen except its charge, which is reversed. The density, then, of compressed, liquid anti-hydrogen is the same as that of liquid hydrogen: 70.85 kg/m^3. Take our mass and divide by the density to find the volume of fuel we need:

(1.8*10^9)/(70.85) = 2.54*10^7 cubic meters

     Now we take that volume, and divide it into six identical spherical fuel tanks:

(2.54*10^7)/(6) = 4.23*10^6 cubic meters per fuel tank

     Finally, we set that volume equal to the formula for the volume of a sphere to find the radius of each tank:

4.23*10^6 = (4/3)*pi*r^3

r = 100 meters

     So that's the answer. Our proposed interstellar craft would need six spherical fuel tanks containing compressed anti-hydrogen each having a radius of about 100 meters. Is this realistic? Definitely not with today's technology. But who's to say that in a hundred years or so scientists won't come up with a way to produce antimatter in bulk? Criticisms aside, how long would this voyage really take? Assuming acceleration is constant at the beginning and end of the trip, and that speed is a constant 0.7*c for the 9.1 light year trip, calculation of duration becomes quite simple:

t = 2*((d_1/(0.5*a)))^.5 + (d_2)/(v)

d_1: distance traveled in ly during acceleration  phase(0.45)
d_2: distance traveled in ly during coasting phase (9.1)
a: acceleration (9.81)
v: velocity of coasting in "c" (0.7)

t = (((0.45)/((0.5)(9.81)))^.5)/(9.46*10^15) + (9.1)/(0.7) =  14.87 years

     That is, 14.87 Earth years. Due to the fact that the craft is traveling at relativistic speeds, strange things start to happen to the crew of this ship. Time actually passes slower for the crew of the ship than it does for a bystander on Earth. The factor of this time dilation, as it is called, can be derived as such:

y = 1/((1-(v^2)/(c^2))^.5)

y: time dilation factor
v: velocity
c: speed of light

     Plugging in 0.7*c for "v", we find that our time dilation factor, "y", is 1.40. This equates to time being 1.4 times slower for the crew of the ship at 70% of the speed of light. While central command at Earth would record the vessel as taking nearly 15 years to reach Epsilon Eridani, the crew of the spacecraft would have only aged less than 11 years. 15 (or 11) years is a very realistic time span for an interstellar crew. It is not so long that it would require a generational ship, but that is still 11 years of supplies that need to be carried along with the ship on its epic voyage to Epsilon Eridani. Alas, there are other problems with antimatter propulsion that need to be addressed.

Artist's rendition of the known Epsilon Eridani system

     Besides the fact that you need to amass a crew of 1000 who are willing to depart on a decade long mission to a planet light years away from which they will never return, there are many hurdles that need to be overcame before we make a trip to another star. As I have already mentioned, antimatter has proven difficult to produce. So far, we have only been able to produce a few atoms of anti-hydrogen, far short of the two billion kilograms needed for an interstellar mission. There is also the problem that matter-antimatter annihilation results in the formation of a lot of heat and dangerous gamma radiation that would be harmful to a crew that was near such an engine. Internal problems, however, are the least of the crew's worries when they realize that we need to come up with a material or energy field capable of protecting them from the dangerous blue-shifted mass of cosmic radiation that is capable of penetrating even the densest materials known to man.

     However, we have 100 or more years to devise ways to overcome these problems. If there is anything that the human race has taught me so far, it is that we don't stop when we believe that something is impossible, we try even harder. Though antimatter engines may not be able to take us to Vulcan, or fly us at Warp speeds, they are the first step in reaching humanity's ultimate goal of achieving a higher existence. It is only a shame that Leonard Nimoy will not be around to see the wonders that our brilliant race will create in the centuries to come. But I hope now that he can rest in peace knowing that his character has sparked the intuition multiple generations to reach for the stars, quite literally. Star Trek has already correctly predicted the invention of the cell phone, the iPad, and Google translate. Next stop: Epsilon Eridani.

In memory of Leonard Nimoy
-"Live long, and prosper..."