Most energy efficient mass ratio of a rocket, or exhaust velocity

Question

In the process of thinking about some hard science fiction, I need to substitute some mass fractions that will minimize the amount of energy used in order to be maximally generous to the author. So this produces the following specifiers in terms of an academic problem:

1. Infinite propellant is available for free
2. The ejection velocity of the propellant can be selected by the pilot (and the engine is 100% efficient at all values, which is for the sake of argument)
3. A hard Delta V requirement exists due to the mission profile
4. What is the mass ratio that minimizes the energy used by the engine?

I'm thinking the mathematics might be sufficiently available here:

http://en.wikipedia.org/wiki/Rocket#Energy_efficiency

They have a notion of "overall efficiency", which I think directly maps to the concept of engine efficiency I have in mind. This metric has an absolute maximum, at a value of ve/v of about 1.6. But is it really 1.6? I can't find any reference to it. So I come here to ask:

1. What is an algebraic expression for maximum overall efficiency as defined here? No decimals. Just with math functions, like exp() or ln().
2. Does this necessarily correspond to an absolute most energy efficient mass ratio as well?

Answer 1

Yes, the Wikipedia answer applies to your problem and is correct. I get $v_e=0.6275\Delta V$. (Your "ve/v of about 1.6" is backwards -- it should be v/Ve, as shown correctly on the Wikipedia plot.) This gives a mass ratio of about $4.9$.

There is no closed form for that number, $x\approx 0.6275$. You need to iteratively solve ${1\over 2}=x\left(1-e^{-1/x}\right)$. (Unless you use the Lambert W-function, but you probably don't have that function on your calculator.)

The derivation is simple, combining the rocket equation and $E={m v^2\over 2}$, and is left as an exercise for the reader.