# Gas Mileage and Horsepower On a Space Rocket

## Fuel efficiency and power of a rocket engine is measured in a very different way from cars and airplanes.

We measure the efficiency of a car engine by specifying how many kilometers it will get you per liter. Or alternatively if you are American, how many miles you get per gallon of gasoline.

The other metric we are usually interested in with a car engine is how much horsepower it provides. It does not matter how good mileage it has if it takes forever to get anywhere.

Now imagine you go into the used rocket dealership and ask for a great rocket to fly to Mars with. You want to figure out how good the rocket is, but you cannot ask about gas mileage, because that makes absolutely no sense for a rocket.

When you are in orbit, you keep moving in orbit, even when the engines are shut off. There is no friction in outer space. Thus you can move as many kilometers are you like without spending any fuel!

So instead we specify efficiency of a rocket engine using what is called specific impulse or Isp for short. Specific impulse tells us how much one unit of propellant or fuel will change the momentum of the rocket. Momentum, `p` is defined as:

`p = mv`

where `m` is mass and `v` is velocity. So a high specific impulse means the rocket engine requires less fuel to increase the velocity of the rocket. The reason why we talk about change in momentum rather than velocity, is because the mass of the rocket matters. The same rocket engine, cannot increase the velocity of a huge rocket as much as it can increase the velocity of a tiny one.

Specific impulse can be define in a number of ways:

In equation (1), we define the specific impulse, `vₑ` as a thrust relative to mass flow of propellant. Meaning for each unit of propellant spent, how much to we make the rocket move. This is also referred to as "effective exhaust velocity."

The alternative is to measure it as thrust relative to weight flow, as in equation (2).

Just a reminder, weight is a force caused by mass. We tend to abuse this terminology in daily speech. Saying: “I weigh 100 kg,” is actually wrong. You’re mass is 100 kg. Your weight would be `100 kg ⋅ 9.8 m/s² = 980 Newton`.

If you know how much you propellant you got, how much your rocket weighs and the specific impulse of your engine, then you can figure out where in the solar system you can travel to.

You do this using Tsiolkovsky rocket equation, which is used to calculate a quantity called delta-v. This is nothing more than a change in velocity. We say delta-v, because it can mean a combination of increases and reductions of velocity. However these all have to be added up.

In this equation the variables mean the following:

• `Δv` max change in velocity.
• `m₀` initial mass of rocket, including propellant.
• `mf` final total mass, dry mass. Mass of rocket without propellant. So combined mass of rocket engines, empty fuel tanks, fuselage etc.
• `vₑ` effective exhaust velocity, which is one way of measuring specific impulse.
• `Isp` specific impulse measured as force relative to weight flow.

You can think of this `Δv` as a budget available to your rocket, which you can spend parts of to get to different parts of solar system. Every so called orbital maneuver requires you to spend some `Δv`.

Here we got an example of an orbital maneuver called a Hohmann transfer. In this case the space ships starts out in the green orbit (1), and wants to get to the red orbit (3). The yellow orbit (2), is the Hohmann transfer orbit.

The black arrows are locations where the rocket engines are fired and we spend some of our total `Δv`, to make an orbital maneuver.

While specific impulse is really important, it is not the only thing that matters, how powerful the engine is also very important. Imagine driving a really fuel efficient car, but it has only half a horse power. Sure you may spend tiny amounts of fuel to go anywhere, but it will take an eternity to get anywhere.

Same goes for rocket engines. If the TWR is really low, then you will simply spend a lot of time to change your `Δv`. That may not matter that much to a robotic spacecraft exploring the solar system, but it matters to humans.

It is the force of the thrust `F` divided by the weight of the rocket, which is its mass `m` times the acceleration of gravity `g` on the planet you are trying to launch from.

So one way of looking at `TWR`, is to think about it as acceleration measured in Gs. If `TWR < 1` then your rocket cannot get off the ground, because the thrust is not powerful enough to counter gravity. At `TWR > 1`, your rocket will ascend.

On earth when they are launching rockets they aim for a TWR of 1.25 to 1.3. In the Kerbal Space program game, one aims for 1.5 to 3. Ideally TWR should be as high as possible because then you avoid wasting lots of `Δv` fighting gravity (gravitational drag). If the rocket goes vertical for a time `t`, then your actual `Δv` ends up being:

But this is a gross simplification, as real rockets spend very little time going completely vertical. Instead they perform a gravity turn, slowly tilting over until they are perpendicular to the earth. You cannot calculate total loss of `Δv` without taking the whole launch trajectory into account.

What prevents us from using as high TWR as possible is that air resistance (air drag) increased rapidly when you increase velocity. Thus if you go too fast, you start wasting more energy fighting air resistance. Hence one tries to keep thrust at 1.3 times the weight of the rocket. As propellant is spent, the weight of the rocket drops, which means you got to gradually throttle down as you ascent. Once in orbit TWR matters a lot less. On planets or moons without atmosphere you can launch with much higher TWR.

An interesting thing you’ll notice with the rocket equation, is that only the specific impulse and mass fraction matters. Meaning thrust per weight (TWR) is irrelevant with respect to how much `Δv` you got available. The exception being when you launch.

That means that if the rocket engines are of negligible mass, you can keep adding rocket engines to your rocket to increase thrust, but it will not alter the `Δv`. So more engines don't get you further or shorter in principle.

However engines do add mass, which screws up the mass fraction `m₀/mf`. Just to illustrate the problem, assume `mₑ` is the weight of all added engines. Assume we keep adding engines, what would happen with `Δv`?

From the equation, you can see that it would start approaching zero. Hence adding more engines will reduce `Δv`. That is why space ships in orbit will attempt to have as small engines as possible to improve `Δv` budget. However if you got them too small, your orbital maneuvers will take so long time to perform that it becomes impractical.

For launches however you want lots of massive engines to get out of earth’s gravity well as quickly as possible, before it gets to eat up lots of `Δv`.

Next time I may cover orbital mechanics and the orbital maneuvers a space ship will typically perform.

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