Back when the cold war was still hot, and everyone was searching for communists in the closet, nuclear was still fresh and awe inspiring. The fascination with everything nuclear spawned a tremendous variety of projects and ideas to realize the full potential of nuclear energy and find out its utility in many different applications. When looking back at those projects in this age, after being born into a "precautionary principle" ruled society, some of the ideas might seem like utter madness or amazing brilliance. Pretty much without exception all those projects involved solid engineering and the scientists back then dared to think big, really really big. It is fascinating to look back at those days and realise how many times the world was within centimeters of big revolutions in energy production or space travel. Many of the projects share the same depressing end, getting shut down by political, rather than technical, reasons. Thinking big might not be fashionable anymore in the west, but it will never cease to be educational and it gives hope for what we can accomplish in this century. That is why I will dedicate some time to write a series of “Nuclear History” blog posts that looks into the crazy, the fascinating and the plain ingenious projects of the first nuclear era. A maths warning is in its place, I will not be afraid to throw in equations into the blog posts if I feel it will explain something better than words. I am using MathJax to write the equations and it might not display properly if you read this through a RSS feed, in that case just jump to the blog. If you are put off by equations just skip them and read the text and graphs, they should be self explanatory anyway.

This first post will be about one of my favorites, the fission rocket!

Let us start back in the 50’s. In 1954 the first nuclear powered submarine, USS Nautilus, was launched into the seas and the development of a nuclear jet engine for bomber planes were under way. The grand space race had just started and it was only natural to ask what part nuclear energy could play. In 1955 the Atomic Energy Commission and the US Air Force got together and started the Rover program, the original goal of the program was to create a nuclear driven ICBM. Parallel to this a project called Pluto was started with the goal of creating a nuclear driven ramjet for a cruise missile that could potentially cruise for months on end carrying a large arsenal of nuclear weapons. Both programs where hedges against the possibility that conventional (chemical rocket driven) ICBM’s might not work as well as was hoped. After the success of traditional chemical ICBM’s in the end of the 50’s and beginning of the 60’s project Pluto became redundant and was cancelled (one must also mention that it was so dirty that one could have ignored weaponizing it and just letting it fly low over cities and the radiation from the darn thing would take care of business). Project Rover was left in a position where its military value was diminished but the possibility of a nuclear rocket was still intriguing. Therefor Rover was handed over from the Air Force to the Space Nuclear Propulsion Office (SNPO) which was a collaboration between NASA and ACE started in 1961. Rover continued as the development program for the rocket itself, regardless of what end use it would have, and a new program called NERVA (Nuclear Engine for Rocket Vehicle Application) was started to examine the utilization of the Rover rockets for civilian space exploration. I will a bit sloppily refer to both projects as NERVA.

But before we look into the developments that took place in those two programs, lets stop for a moment and ask what advantage does nuclear energy have in space exploration? After Gagarins first flight into space in 1961 it became blatantly obvious that it was possible to put people in space with chemical rockets, so why even bother with nuclear rocket? Was it simply because nuclear was the cool kid on the block? The traditional rocket engineers certainly did not want anything to do with nuclear, they understood chemicals perfectly well, thank you very much! The nuclear engineers on their side was equally oblivious to the demands of space flight.

The match between space and nuclear isn’t obvious until one starts to look into what is really important for good rocket performance. There are two key parameters that rule supreme, thrust and specific impulse. Thrust is just what it sounds like, the force the rocket is producing, good old fashion Newton’s second and and third laws (for those who have forgotten, the first law is force equals mass times acceleration and the second law is, every action has an equal and opposite reaction). You need a hell of a lot of thrust to overcome Earths gravity well! Specific impulse is a bit more complicated, it is a measure of the efficiency of a rocket engine. It tells you how much mass a rocket needs to expel in order to achieve a certain amount of velocity. In space the only mass you have to play with is the mass you bring and the only way to gain velocity is to throw some mass in the opposite direction of where you want to go. The less mass you need to bring to achieve a certain velocity the cheaper it is to send that bloody thing into orbit. Impulse is just another word for momentum (force) and specific impulse is the momentum gained per unit mass of propellant expelled. Total impulse given by the propellant to the rocket is just the mass of the propellant times the effective exhaust velocity of the propellant. If we assume constant thrust and constant exhaust velocity we can get the specific impulse by dividing the total impulse with the total propellant mass and all that is left is the effective exhaust velocity.

$I_{sp}= \frac{\int F dt} {m} = \frac{\int \frac{dM}{dt} V_e dt}{m} =\frac{MV_e}{M} = V_e$

Where:
$I_{sp}$ = specific impulse
F = Force
$V_e$ = effective exhaust velocity
M = total propellant mass

Now that is a lot of word simply to state that exhaust velocity is important. I go through all of this to explain the concept of specific impulse since it is a term one never gets away from when reading about rockets. Sometimes specific impulse isn't defined as above either, but it preserves its importance. In another definition, for some reason I don’t understand at all (after all I am only a physicist and not a rocket scientist), specific impulse is often defined per propellant unit of weight (on Earth) instead of unit of mass. The strict definition of weight is the force a mass experiences in a gravitational field. A scale doesn’t really measure your mass in kilos, it measures your weight in Newtons! Using that one then ends up with a definition if specific impulse that looks like this.

$I_{sp} = \frac{V_e}{g_o}$
Where:
$g_0$ = gravitational acceleration at earths surface (9.81 m/s^2)

In the first definition specific impulse has the unit of velocity, m/s, and in the second definition it has the unit seconds. So if you see people talking about specific impulse of this and that many seconds you know the reason. I explain this because I will consistently use the second definition of specific impulse from now on due to the fact that it is more common to find tables in units of seconds.

Now to realise why specific impulse is important lets have a look at the famous rocket equation formulated by Tsiolkovsky. This equation tells you how much velocity a rocket will gain from a given amount of propellant with a certain exhaust velocity.

$\Delta V=V_e*Ln(\frac{M0}{M0+Mr}) = I_{sp}*g_0*Ln(\frac{M0}{M0+Mr})$
Where:
$\Delta V$ = the speed given to the rocket
$V_e$ = rocket exhaust velocity
$M_0$ = Rocket mass without propellant
$M_r$ = propellant mass
$g_0$ = Gravitational acceleration at the earth surface

The higher the specific impulse the higher the $\Delta V$, that much is obvious. Looking at the masses involved is even more enlightening. So lets breaks out the $M_r$ term from the last equation and we get:

$M_r = M_0*[e^{\frac{\Delta V}{I_{sp}*g_0}}-1]$ = $M_0[e^{\frac{\Delta V}{V_e}}-1]$

Lets plot this function! Lets assume we want to go from low earth orbit to orbit around the moon. This will require a $\Delta V$ somewhere in the neighborhood of 4000 m/s (to get into low earth orbit in the first place one needs about 10 000 m/s, but lets assume we are already there). Lets also assume we want to deliver about 55 tons of material there. That is about the weight of the Apollo command module plus the lunar lander module plus the empty weight of the S-IVB last stage of the Saturn V rocket. This will give the resulting plot with Isp’s ranging from 100 to 1000 seconds (exhaust velocities of 981 m/s to 9810 m/s).

There are two blue X drawn on the plot. The first X is drawn at the $I_{sp}$ value 475, this happens to be the specific impulse that the third stage of the Saturn V rocket had, the part of the rocket that was supposed to give the final $\Delta V$ to go to the moon. It turns out that the reaction mass according to the plot above for $I_{sp}$ = 475 is 75 metric tons. In reality the S-IVB burned about 80 tons of fuel to reach the moon, so we are playing in the correct order of magnitude here! What about the second X drawn with a $I_{sp}$ of 925? To jump forward a bit in time, that happens to be the $I_{sp}$ of the final NERVA design, how much reaction mass does that correspond to? 30 tons! Less than half of the S-IVB, a dramatic reduction and a potential cost saver!

To show an even larger advantage for the nuclear rocket, lets look at missions requiring higher $\Delta V$. In the figure below I have plotted the reaction masses needed given an $I_{sp}$ of either 475 or 925 seconds. At the far right end of the graph one can see the propulsion mass needed to deliver 550 tons from Earth to landing on Mars.

For the nuclear rocket one would need a propellant mass of about 1200 tons while the chemical rocket needs 4900 tons. Given that the weight to launch something into low earth orbit right now is over 2000 US dollars per kg the cost saving on mass alone is close to 7.4 billions! To be fair to the chemical case, the cost to get things into orbit might be cut by a factor of 10 within the foreseeable future (if space x manages to make a reusable rocket), but even in such an optimistic case the potential cost saving might be close to one billion dollars.

The above plots shows why a nuclear rocket is desirable, but it doesn’t explain why a nuclear rocket performs so much better compared to chemical rockets. Why does a nuclear rocket have a much higher $I_{sp}$ ? Lets first consider how a chemical rocket works, in a chemical rocket the energy source and the reaction mass is one and the same. You mix two chemicals, they explode in a semi controlled fashion and the resultant products are sprayed out through the rocket nozzle and creates thrust. A common example of liquid rocket fuel is hydrogen and oxygen. There is also examples of solid fuels, the boosters for the space shuttle is one example that uses some kind of aluminum mixture. The chemical reaction heats the reaction products and throws them out of the rocket with a certain velocity. Temperature of a gas is proportional to the average energy of the gas molecules and energy is simply $E=\frac{mV^2}{2}$. Velocity of the particles are then $V=\sqrt{2E/m}$ and we instantly see that the smaller the mass, with a given temperature, the higher the particle velocity. Ideally, whatever we heat up, we want it to to be made of as light a particle as possible. In chemical rockets we don’t really have the luxury of choice, the reactions that gives the most energy doesn’t necessarily also give the reaction products with the smallest masses. The smallest possible mass is the hydrogen atom since it is the lightest element. The hydrogen + oxygen reaction is one of the most energetic chemical reactions, but the product of the reaction, water molecules, is 18 times heavier than the hydrogen atom. A heated hydrogen gas with the same temperature as a heated water gas will have a velocity more than 4 times higher.

A chemical rocket will never have the ideal propellant due to the fact that one has to introduce other compounds since the energy is generated by the compounds themselves. To have the ideal propellant the energy production has to be separate from the propellant. This is where a nuclear reactor finally enters the picture. If the heat source is nuclear fuel rods and the propellant is hydrogen heated by flowing over the rods. Then one can indeed get a pure flow of hydrogen out of the rocket. In that way one can maximise the $I_{sp}$ from the energy produced. Why can’t one do this with chemicals, may be by having some kind of contained chemical that produces heat that is transferred to a pure hydrogen gas? It is due to the fact that chemical reaction releases so little energy compared to nuclear reactions, this means the mass of the chemicals needed for the reaction would be as large or larger than the mass of the propellant. Fission however releases about a million times more energy from the same amount of mass compared to a chemical energy source. The energy required to put the space shuttle in orbit, of the order of $10^{13}$ joules, is contained in such a petty amount as roughly 100 grams of uranium. With fission it becomes feasible to separate the energy production from the propellant without having the energy production part being to massive. We can then have a rocket that runs with the same temperature as the best chemical rockets but have 4 time the $I_{sp}$ .

In reality everything isn’t quite so rosy, one can not expect to put 100 grams of uranium togheter with some hydrogen into a rocket and easily get a $I_{sp}$ that is 4 times higher than the space shuttle rocket. The $I_{sp}$ will rather be a bit more than double because hydrogen atoms form H2 molecules and thus the specific weight of the propellant is only one ninth of the weight of water. Hydrogen needs to be heated to over 5000 degrees Celsius before it forms free hydrogen atoms. Also a full reactor weights significantly more than 100 grams of uranium, even if only 100 grams needs to be fissioned to produce the total energy, one still needs a hefty amount of uranium for the reactor to go critical in the first place.

But even taking into accounts those pessimistic facts the nuclear rocket is still very promising. This first part of the series is already long enough. So lets save the fun stuff for the next part. Then we will look at what kind of things they actually built during the NERVA program and the basis for the reactor designs!

Johan