In popular culture one occasionally hears the claim “This isn’t rocket science!” The implication is that unlike rocket science, the subject at hand is simple. Or, put another way, that rocket science is forbiddingly difficult. And it truly is difficult. In spaceflight, nothing is easy or comes for free. Even the most mundane aspects of spacecraft design and management, such as regulation of temperature, are difficult.
Yet the fundamental equation that predicts the performance of a rocket is a marvel of simplicity, proof that complexity lies in the details, not the basic concepts. The equation is usually attributed to the Russian Konstantin Tsiolkovsky, the “father of astronautics”, who first published it in 1903.
Here is the Ideal Rocket Equation:
Delta V = Vex ln (Mf / Me)
Where:
Delta V is the overall change in velocity of the rocket
Vex is the exhaust velocity of the gases coming from the engine
Mf is the mass of the rocket fully fuelled (sometimes called the wet mass)
Me is the mass of the rocket with fuel expended (sometimes called the dry mass)
The term Ideal implies that certain dirty real-world considerations have been neglected. In this case, the equation is not accurate in the presence of air resistance or gravitation and applies only to a rocket in free-fall and in a vacuum. It must also be remembered that velocity is a vector and the multiplication in the Ideal Rocket Equation is vector multiplication.
But what does it really tell us? It says that only two things affect the final overall performance of a rocket: the exhaust velocity of the engine, and the percentage of the rocket given over to fuel tanks. Nothing else matters! Not the thrust of the engine, not the shape of the rocket, not the good intentions of its occupants or engineers. All that matters is exhaust velocity and mass ratio (the term (Mf / Me) is often referred to as the “mass ratio”).
We can make it even more interesting. Note the Vex in the equation, the exhaust velocity of the engine. It is not always easy to find the exhaust velocity for any given engine, but as it happens, the exhaust velocity is linked to the specific impulse of the propellants multiplied by the value of Earth’s sea-level gravitation. In other words, we can change the equation to look like this:
Delta V = (Isp * g) ln (Mf / Me)
So to calculate the net overall performance of an ideal rocket, all you need to know is the specific impulse of the propellants and the ratio of full to empty masses. Nothing else is required. Not even the thrust of the engine!
Thinking about this equation reveals certain truths about rocket science. Only two things affect the final velocity: the specific impulse and the mass ratio, Mf / Me. If we want the rocket to go faster, we can either increase the specific impulse by switching to more efficient propellants, or we can improve the mass ratio, usually by paring away excess structure, eliminating redundant subsystems, or using lighter materials in place of heavier ones. But once we’ve switched to the most efficient propellants (liquid oxygen and liquid hydrogen) we simply can’t increase the specific impulse. So in the end it becomes strictly a matter of the mass ratio.
But there is a limit on how sweet the mass ratio can become. The structure can become only so lean before the rocket becomes structurally unsound, and the mass of the payload can be assumed to be fixed – if you eliminate the payload, you eliminate the reason for using the rocket in the first place.
Thus there is a limit on the maximum velocity a rocket can achieve, determined by the specific impulse of the propellants and the skill of the engineers in reducing the empty mass to the smallest value possible. So the next step is to make the rocket physically bigger, launching the same payload with a rocket with much larger fuel tanks. But this increases the empty mass of the rocket (fuel tanks that have no mass have not yet been designed) so the rocket needs to carry even more fuel…
In the end, the Ideal Rocket Equation guarantees that no rocket powered by conventional chemical propellants can ever achieve anything like relativistic velocity, simply because the mass ratios become forbiddingly large – in the end one arrives at the situation of rockets the mass of small planets launching payloads the size of apples, and that clearly is no way to run a railroad.
William Hamman - I have been professionally involved in the aerospace field for 27 years, most of that time in the linked areas of guidance, navigation and ...
