# Why do lower molar mass rocket exhaust products give higher specific impulse?

I tried to work this out. My thinking is as follows. If the same amount of energy is given to a low mass particle and a high mass particle, the low mass particle is faster. A particle with a mass of 1 will be twice as fast as a particle with a mass of 4 but the momentum of the particle with a mass of 1 will be twice that of the particle with mass 2, which seems to go against what is seen with regards to molar mass and specific impulse.

Suppose we have our choices of particles, we can convert all applied energy to exhaust velocity, and the energy is divided amongst the same number of particles whether they have an atom mass of 1 or 131. (Note well: Those are some big assumptions.) The velocity of an exhaust particle is $v=\sqrt{2E/m}$ where $E$ is the energy applied to an individual particle and $m$ is the mass of the particle. Multiplying by mass yields the momentum: $p=mv = \sqrt{2Em}$. So, yes you get higher velocity but less momentum with smaller particles.

Note that it's the exhaust velocity that counts when it comes to specific impulse, not the exhaust momentum. So why do ion thrusters inevitably use xenon, which has a very high atom mass, as exhaust?

One reason is the thrust to power ratio, which is ideally $2/v_e$. Attaining a high specific impulse engine needs more power than does a lower specific impulse engine. Ion thrusters are power hungry beasts; they would function best with Mr. Fusion. (Sadly, it's almost 2015 but Mr. Fusion is nowhere in sight.)

Another reason is that specific impulse isn't the be all and get all in rocket science. Even with chemical propulsion, where the energy source and the propellant are one and the same thing, a high specific impulse might be not be beneficial. Given a desired total $\Delta v$, there's an optimal exhaust velocity (specific impulse) that minimizes energy consumption. An exhaust velocity higher than that limit is just as detrimental as is an exhaust velocity under that limit. This becomes even more important with ion propulsion because the energy source and the propellant are very different things.

A third reason are those big assumptions made at the start. There are always inefficiencies in any system. In the case of ion thrusters, the energy needed to ionize the particles does not contribute to exhaust velocity. That energy ultimately goes to increasing the entropy of the universe. Big particles are a big win here because big atoms tend to hold their electrons less tightly than do smaller ones and because fewer atoms need to be ionized to attain a desired thrust. This makes ion thrusters considerably more efficient when they use larger particles, which in turn makes xenon the goto propellant for ion thrusters.

• Is the reason that ion engines have such a low thrust (although they have about 8 times the exhaust velocity compared to chemical rockets, if I've understood correctly some comparison with Rosetta), that high ion thrust requires high electrical effect, beyond the ability of solar panels today? But wouldn't a scaled up Russian TOPAZ fission reactor do the trick? Or are there more important problems with scaling up ion thrust (increasing the propulsive mass per second, if I've understood it)? – LocalFluff Jul 30 '14 at 15:29
• @LocalFluff I think you can decouple the physics of the solar panel from the ion engine. It's only a matter of sufficiency of the power production. The ion engine needs a certain electrical potential to accelerate the propellant, and this is greater when Xe. However, I think it's just not an issue. As long as you don't have electrical arcing, we can step up the voltage to what we need (many times what the PV produces). So by those parameters, the atomic mass doesn't even matter. The stability of the ion then becomes the determining factor. – AlanSE Aug 5 '14 at 17:54

For a chamber where the pressure is $p_c$ and contains a gas of molar mass $\mu$ and specific heat $\gamma$ and the ambient pressure condition be $p_e$. Then, the equation of the for the exhaust velocity is given by

$$V_j=\sqrt{\frac{2*\gamma*R_0*T_c}{(\gamma-1)*\mu}*(1-\frac{p_e}{p_c})^\frac{\gamma-1}{\gamma}}$$

where $V_j$= the velocity of the exhaust

and $\mu$ is the molar mass

the exhaust jet velocity is inversely proportional to the square root of the molar mass .

And there are many ways to define specific impulse in two ways (both of them are equivalent)

1) $I_{sp}=\frac{I}{m_p}$ where I=$m_p*V_j$ which leads you to $I_{sp}=V_j$

2)$I_{sp}=\frac{thrust}{mass..flow..rate}$

you can see that the specific impulse is directly depends upon the exhaust velocity which is inversely molar mass

hence lower molar mass rocket exhaust products give higher specific impulse :)

• That's assuming that the other variables in that equation remain unchanged. They don't. Burning fuel at anything but the stoichiometric ratio hits chamber temperature, and it's a net loss. The primary reason hydrogen-oxygen engines burn at a 4:1 to 5:1 ratio rather than stoichiometric 8:1 is because doing so reduces $\gamma$. This makes the net gain positive. You'd get the same change in $\gamma$ by burning lean, and now the inverse proportionality to mass does come into play. Burning lean doesn't make sense when you can kick out unburnt hydrogen by burning rich. – David Hammen Jul 30 '14 at 14:27

To give a hands-on explanation: Imagine you have a kilogram of a "light" gas like hydrogen and the same weight of a "heavier" gas like air in the same size containers of 1m^3. Now punch a hole into container and see how far you fly the other way in both scenarios.