The Rocket Equation

Introduction

If you want to understand anything about rockets and spaceflight, the rocket equation is where to start. The rocket equation will give you insight in what limits the maximum speed of a rocket, and with that much more: why it is so hard to fly to Mars and back with our current rocket technology, why a satellite launch costs tens of millions, why it is impossible to make our current chemical rockets go faster just by bringing along more fuel, why a Nazi Germany could afford to launch thousands of V2 rockets to the UK for less than the cost a single moon rocket in the1960s, and so on. OK, there is much more to these questions than just the rocket equation (economics, slave labor in case of the Nazis,...) but as already said the rocket equation is the best place to start on the technical side of the matter.

Konstantin Tsiolkovski

It is always nice to tell a bit about the actual inventor first. Konstantin Tsiolkovski (1857-1935) was a Russian scientist, and is widely regarded to be one of the fathers of modern rocket science next to Hermann Oberth and the American Robert H. Goddard. He could not attend high school because of a hearing problem, but this did not hinder him to educate himself. He became a high school teacher (still with a hearing problem) and a passionate inventor.  Despite being stuck in a small town called Kaluga away from major learning centers, Tsiolkovsky managed to make scientific discoveries on his own.


Among his works are designs for rockets with steering thrusters, multistage boosters, space stations, airlocks for exiting a spaceship into the vacuum of space, and closed-cycle biological systems to provide food and oxygen for space colonies. Being truly ahead of his time only at the end of his life, after the Russian Empire became the Soviet Union (1918), Tsiolkovski got more recognition and later on a nice monument in Moscow.
Tsiolkovski derived the equation that bears his name in 1897.

 What is a Rocket?

Before we get to the rocket equation, it's good to recapitulate what a rocket actually is. Most vehicles (airplanes, cars, ships, ...) push against an external object (be it air, the road, water, ...) in order to gain speed. A rocket is different.  A rocket brings along disposable mass, called reaction mass, and pushes this mass backwards at a certain speed ve in order to accelerate forwards. Thus, a rocket does not need something external to push against in order to accelerate. Therefore, a rocket can accelerate even when there is nothing to push against, such as in space**. Two remarks:

  • The reaction mass can contain the source of the energy that is used to accelerate it. This is the case for the chemical rockets most people are familiar with: fuel is burned, and the hot gases resulting from the combustion are ejected backwards through a nozzle. However, in other types of rocket the energy source and the reaction mass are separate components! For example, an ion engine typically expels inert gas, accelerated by an electric field. In current ion engine driven spacecraft such as the Deep Space 1 probe, the energy to sustain this field is generated by solar cells.

    chem vs ion1 Chemical rockets vs. Ion engine rocket: in the chemical rocket, the energy source is also the reaction mass. In the latter the two are separate.
  • When the rocket is out of reaction mass, it cannot accelerate any more. Generally in space an empty rocket continues to float at the speed is has reached. There are no fuel stations in space! Even if an object with fuel would be floating somewhere between the rocket's starting point and its destination at zero speed with respect to starting point and destination, the rocket would need to slow down (requiring fuel), refuel and accelerate back. The whole operation would not help to increase the rocket's speed, unless the in-between-object would be moving in the direction of the rocket's target. This normally does not occur in space.
  • When there is nothing to push against, there is also nothing to brake against! An car can rely on friction of the road and the extra friction a break pad generates to slow down without having to expend energy for this. In empty space, a rocket that needs to slow down has to put its nozzle forwards and burn fuel. Thus, suppose a rocket would be capable of reaching a speed Δv when accelerating from 0 km/s to the point when it runs empty. If the rocket needs to slow down to 0 km/s again at the destination, it will only be capable of reaching a maximum speed of Δv/2. There are some exceptions to this: for example, the atmosphere of a nearby planet can provide the required friction for slowing down.

 

**OK, space is not completely empty as particles from the sun and other stars are constantly ejected into space and can be used to accelerate a sail. Space sailing is next to rocket riding the other known way to accelerate in space.It's also possible to beef up the particle stream, for example by aiming a large moon-based laser on the sail.  This is all stuff for another article, though. Real men use rockets, anyway. They don't rely on something they can't control.

The Equation

The rocket equation is not difficult, any science or engineering student can derive it. It is based on a simple but universal law of nature: conservation of momentum - which is directly related to Newton's laws (the momentum of an object is its speed multiplied by its mass). The rocket equation goes as follows:

 RocketEquation

As with all equations it is important to know what the different symbols mean. First, what is the Δv the rocket equation calculates? We've already touched this before: Δv (delta-v) is the maximum speed the rocket can obtain when launched from standstill in empty space. This is of course very interesting: we want our rocket to go fast. Not only to get it to its destination in a shorter time, also to get to some destinations at all: to escape the gravity well of a celestal object such as the Earth, you need a certain speed. To escape from Earth's pull, you need a speed of about 11.2 km/s.
When Δv is reached, the rocket can not accelerate any more because it has used all the reaction mass that it carried with it. As discussed above, take in mind that maneuvers such as course changes and deceleration also 'consume' Δv. If you have a rocket which can reach a Δv of 10 km per second and you want it to slow down to standstill again at the end of its trajectory, it will only reach a maximum speed of 5 km per second.

Note that Δv does not say anything about in what time your rocket will reach its maximum speed, about its acceleration in other words. For some tasks, like lifting off a planet, you'll need a rocket with high acceleration as well. A rocket without enough acceleration would be very unefficient to lift off a planet (if it can lift off at all), as it would spend most of its fuel fighting gravity. Once  in space, you can typically affort a much lower acceleration. Then, Δv becomes the main performance figure of your rocket.

Now, what is ve? ve is the speed at which the rocket ejects its reaction mass, as seen by an observer on the rocket. You can see from the equation that Δv is linear with ve: if you can double your rocket's ve, your rocket will go twice as fast. For chemical rockets, ve is determined by the chemical binding energy contained within the fuel that is burned. In the 1950s and 1960 all possible chemical fuels have been tried and the best practical combination turned out to be liquid hydrogen and liquid oxygen. These two combined yield a ve of about 4.4 km per second. Some more exotic chemicals were found to yield a somewhat higher ve, but this turned out to be not worth the extra risk and toxicity. - the rocket engine's designs are already close to the fundamental limits imposed by thermodynamics, so not much efficiency can be gained there anymore either. Therefore, we can confidently say chemical rockets are already at their limit with respect to ve. Luckily other types of rockets can be built, but we'll keep that for the end of this article.

Now, can't you just put more reaction mass in the rocket to get a higher maximum speed? That is what the rightmost part of the equation, ln(mfull / mempty) is about. mfull / mempty is called the mass ratio of the rocket: it is the mass of the rocket filled with fuel, devided by the mass of the entire rocket (cargo included) without fuel. Thus, this tells you how much fuel you take along.
But, there is a caveat here: when you want to take along more fuel, you will also need to make the fuel tank and the supports larger. Thus, the mass ratio cannot be increased to a very high number. In typical rockets the mass ratio is between 10 and 30. Imagine a rocket with a total mass of 1 ton and a mass ratio of 1000: when empty, it would only weight one kg, including its cargo, while when full it would need to contain 999 kg of fuel! This is of course almost impossible to make.

If this wasn't bad, Δv is not linearly proportional to the mass ratio. Rather, it is proportional to the natural logarithm of the mass ratio. Logarithms are very cruel functions, reducing large numbers to very small values. I've plotted the function below. You see that for a mass ratio of 20, Δv will be about 3 times ve. However, if you would try to make a rocket with an almost impossible mass ratio of say 60, Δv would still be only about 4 times ve. And even worse, if you would really let your magic work and produce a rocket with a mass ratio of 1000,  Δv would still be one about 6.9 times ve!

logaritmic rocketequation

This is all very logical: all the extra reaction mass that you bring along must be accellerated too until it's expelled from the rocket. Therefore, you get a law of diminishing returns. Therefore, in practice a single-stage (hold your breath) rocket can go only about 3.5 times ve.

 How fast do you need to go?

 Suppose you had a really powerful rocket (and by now you know that's one with a very high Δv and a reasonable acceleration) you could coast around the solar system almost as freely as in your favourite space cartoon. However, as it turns out  chemical rockets are not powerful enough to do this, a careful selection of the trajectory is needed in order get anywhere. Minimum-energy trajectories consisting of a short rocket burn at the start and another one to enter orbit at the destination are called Hohmann Transfer Orbits. Here's a list of approximate needed Δv's to get around. This list already takes into account the usual tricks, such as using a planet's atmosphere for braking.

 

Trajectory Δv in km/s
Earth -> Low Earth Orbit  9-10
Earth -> Geostationary orbit 13.5
Earth -> Moon surface 15

Earth -> Moon surface and back

18
Earth -> Mars surface 15
Earth -> Mars surface and back 22

 You'll immediately see that just getting from Earth to low earth orbit is being over halfway to the moon in terms of Δv!  Also, getting to Mars does not take a better rocket than getting to the surface of the Moon. Mars is much further away though, so how can this be? That's because most of the journey you'll be coasting in space without burning the rocket to change speed. Thus, you'll be away from home much longer but you won't need to spend more reaction mass to accelerate or slow down. However, getting to Mars and back is a lot harder. That's because Mars has a much higher gravity pull than the Moon, so basically you'll need to land a reasonably big rocket on Mars in order to get back up.

Not so long ago scientists found that even lower energy trajectories than the Hohmann transfer orbits exist in the solar system, forming a so-called interplanetary transport network that allows to move around in the solar system with much less Δv using seemingly chaotic orbits.  There is a downside: these trajectories are awfully slow and therefore only useful for automated probes. There's another part to this story: these chaotic orbits can be followed by debris shot off a planet by a meteor impact that can eventually, after thousands of years, end up on another planet. This way, even simple lifeforms such as bacteria could move from one planet to another! There is no proof that this has actually happened, but it's possible in theory.

For covering interstellar distances, the answer to the question 'how fast do you need to go' is basically as fast as you can. Even at a speed of 10% of the speed of light, the travel time to the nearest star is still over 40 years!

Staging

Looking at the Δv figures above you could think something like "Is there really so much difference in difficulty between going to low Earth orbit and going to Mars and back? You only need about 2 times the Δv??" But remember: it's very hard to make a rocket go more than 3 times the speed of its exhaust ve (and that's if you don't take much cargo with you), and for chemical rockets the best ve we have is about 4.4 km per second. After that, you're just out of fuel (reaction mass to be more precise) and there's not much you can do anymore. If you're not at your destination when you're out of reaction mass, you'll be lost in space forever.

But there is a way to boost up speed, and that way is called staging. Staging means placing a rocket on top of a rocket. If the first rocket burns out, it's cast away and the second rocket is ignited.
Let's give an example to clear things out: let's take a liquid hydrogen-liquid oxygen rocket with 20% cargo, 10% structural mass and 70% fuel.

Filling out the rocket equation leads to:

Δv=ve*ln(Mfull/Mempty) = 4400*ln(10/3) = 5297 m/s

Now, suppose that instead of the cargo, you put in a second rocket which also has with 20% of its mass consisting out of cargo, 10% of structural mass and 70% of fuel.
This rocket will start its engine when the first stage has bunt out and has been thrown off. Thus, it starts at a speed Δv and will accelerate to 2*Δv! In other words, the rocket will go twice as fast!

However, the cargo fraction the two-stage rocket will carry is just 20% of the cargo of the 1-stage version. The 1-stage version carried 20% useful cargo, the 2 stage version will carry 20% of 20% or just 4%! This decays exponentially with the number of stages: taking the same ratios, a three-stage rocket would only hold 0.8% useful cargo. Thus, you can multiply Δv by using staging, but only to a limit. Staging comes at a serious cost: for a certain cargo mass the size (and cost) of the rocket goes up exponentially!

If the rocket designer is stuck with a certain engine technology -and thus a certain Δv - and has exhausted all possbilities to get a higher mass ratio, staging is all that s/he can still do. That's why a moon rocket was over 100 meters tall at liftoff, and only about a 2 meter tall casule came back to Earth from the moon. Notice the difference with a cartoon rocket, which is not limited by the laws of physics - only by our imagination. Or perhaps cartoon rockets use a more advanced technology than chemcal fuels? More about that in the next part!

 

 cartoon rocket vs real rocket

Picture: comic book rocket vs. real rocket. Notice the difference in cargo space.

 


Conclusion: how to go really fast using rockets

The only way to go really fast using rockets is to use one with a high exhaust velocity ve and to take along more fuel than anything else. We already saw that taking along more and more fuel is a dead end, but a mass ratio of 10 to 30 is still advisable. Making it a 2 or 3 stage rocket also can help, but in the end only a high ve will really help.

So what kind of rockets can you use to get a high ve if burning the most energetic chemicals is not good enough? Nuclear rockets of course - nuclear reactions produce about 10 million times as much energy per kg of fuel as chemical reactions. As you can see in the table below, the reaction products from nuclear fission and fusion reactions go a significant fraction of the speed of light, about 10%c. If we can find a way to make these reactions happen in the rocket and -just as important- lead the reaction  products out though the nozzle without them losing too much energy, the rocket equation shows we'll have a decent interstellar-capable rocket.

There are different types of nuclear reactions useful for propulsion: nuclear fission, nuclear fusion and antimatter annihilation. All of them release energy, so no external energy source is needed. Nuclear fission reactions comprise basically splitting heavy atoms such as uranium. The advantage is definately that fission is technically feasisble right now. Nuclear fusion reactions comprise 'melting' light atoms such as hydrogen together, creating a bit larger atoms. Such reactions create about 10 times as much energy per unit of mass as fission reactions. Nuclear fusion research is now going on for more than 40 years, and is close to creating a reactor that produces much more energy than put in to ignite the reaction. Therefore, it is likely that humanity can build a fusion-based spaceship somewhere this century.

The ultimate rocket fuel would be antimatter, also known as the stuff that explodes and converts almost completely into energy when coming into contact with, well, anything made of normal matter It's probably the last fuel humanity will manage to control: even the fabrication of more than a few atoms is at this moment still a challenge, and to power a large interstellar rocket several tons of antimatter are needed.

Light particles (photons) itself go, well, at the speed of light and thus have the highest ve possible. Photon rockets have some drawbacks too, though: the impulse ('push') photons generate is very low: 300 megawatts of light power is needed for just a 100 grams of propulsion force! You'll need to bring along a large reactor to make it worthwhile, but the weight of course again will limit the maximum speed. Therefore, most concepts using photons propose to leave the light  and energy source at home, and to attach a sail to the spacecraft that is pushed by the laser beam.

highDvs

 

 All together, I hope now you have got a good idea about the possibilities and limits of rocket propulsion in space, with the help of the rocket equation. How to actually build a rocket that can eject  the fast particles from nuclear reactions out its nozzle to reach Δv's good enough for star travel is the topic of a next article, and of course one of the challenges for interstellar propulsion system researchers.