It’s Not Like It’s Rocket Science
The ideal rocket equation and its implications — explained
Over 400,000 Americans were involved with the Apollo program that sent NASA astronauts to the moon.
That’s roughly the current population of Minneapolis or New Orleans.
Of course, not all of these men and women worked on the rocket itself, but having this many people dedicated to a common goal is still impressive.
Elon Musk, CEO of the pioneering rocket company SpaceX, eloquently reiterates my point: (in a tweet nonetheless)
“Rockets are hard.”
Needless to say, rockets are very, very, complicated.
But hopefully, I can help you understand a little bit more about these amazing works of engineering and science.
So buckle up and prepare for launch ’cause this is about to get real.
Preface
Energy is the capacity to do work. Work is the force that causes the movement — displacement — of an object. On the most basic level, you need energy to move something.
Let’s take a look at how much energy we need in order to transport 1 kg to the ISS, which orbits at approximately 400 km above the surface of the Earth.
The energy of an object can be divided into 2 components: potential and kinetic energy. Potential energy is the energy an object possesses due to its position relative to other objects. Kinetic energy is the energy an object possesses due to its motion.
Potential Energy
From high school physics, we know that the potential energy of an object is the product of its mass (m), the acceleration due to gravity (g), and the height of the object (h). (We will be neglecting the change of the strength of gravity with height to keep the calculations simple.)
Our mass is a petite 1 kg. The acceleration due to gravity is 9.8 m/s². And our height is going to be the height of the orbit of the ISS or 400 km. But let’s write that as 400,000 meters to be consistent with our units of g.
Okay, great. But if you just put our little 1 kg object in the Earth’s atmosphere and let go, it’s just going to come falling back down to Earth.
Kinetic Energy
For an object to stay in orbit, it needs kinetic energy, or motion. Good old high school physics also taught us that an object’s kinetic energy is one half the product of its mass (m) and the square of its velocity (v).
Well, we know our 1 kg object’s mass (it’s in its name). But what about velocity?
To be in a stable orbit around the Earth, the object must be traveling at a certain speed. For this problem, we can estimate it to be about 8 km/sec. Again, let’s just write that in meters: 8 * 10³ m/s.
Total Energy
Finally, the total energy can be determined simply by taking the sum of the potential and kinetic energies.
Our answer can be rounded to 36 megajoules which is equivalent to 10 kilowatt-hours (kWh).
To put this into perspective, your average home air-conditioning system has an energy usage of about 3 kWh. So the energy needed to send 1 kg up into space is equivalent to the energy your air-conditioner uses to keep you cool for just about 3 hours.
If we look at the cost of this energy, it’s even more surprising. The national average electricity price is 13.19 cents per kWh so 10 kWh puts us at just over $1.30.
$1.30.
Merely a bit of spare change to give our little 1 kg object the ride of its life.
But wait.
The Curse
If we could just put that energy right into that 1 kg, it would be really inexpensive and completely revolutionize space travel.
But the thing is: we can’t do that. We need a rocket to get stuff into space. Bummer.
But our rocket isn’t just carrying our 1 kg object. It also needs to carry all of the propellant (made up of a fuel and an oxidizer) it will need on its journey, all the way up until the propellant is actually burned.
That doesn’t make sense? Basically, you need to burn propellant to put mass into orbit, but that propellant you just put into your rocket? That has mass, too. So you’ll need even more propellant to carry that propellant…
The whole thing just turns into an exponential runaway and is often referred to as the curse of the rocket equation.
When the space shuttle was in operation (1981–2011), it could bring our 1 kg friend to the ISS for $54,500. Now SpaceX charges a much lower $2,720 with their Falcon 9 rocket, but that’s still a whole lot more than the cost for just the energy required.
So why on God’s green (but mostly blue) earth does it cost so much to put stuff into orbit? Where does all the money go?
Enter — the rocket equation. 🚀
The Equation Itself
The infamous rocket equation was first published by Konstantin Tsiolkovsky, a Russian school teacher turned astronautics pioneer, in 1903.
It’s pretty simple to derive using a bit of calculus, but I am not going to derive it here because I like to keep my articles PG.
(You can find the full derivation on NASA’s website here if you are interested.)
Anyways, it looks like this:
Whoa. Okay. What do all the variables mean?
ΔV is how much the velocity of the rocket can increase or decrease. Vₑₓ is how fast exhaust is thrown out the back of the rocket. mᵢ is the initial mass of the rocket. m[sub]f is the final mass of the rocket.
With some simple algebra, you can rewrite it in these two forms:
Pause. We need to look at what the two masses mean.
The initial mass is simply the mass of the rocket as it sits on the launchpad.
It’s made up of the mass of the payload (whatever you are sending into space — a satellite, a rover, humans, or in our case, a generic 1 kg object), the rocket structure (the body, the engines, the fuel tanks, the fins, etc.), and the propellant.
It should make sense that the final mass of the rocket is just its initial mass minus the mass of the propellant (because it was all used up).
If you prefer a mathematical representation, their relationship can be modeled by this simple equation:
Ok. Now we understand the different masses. Back to the rocket equation:
Since the final mass is essentially the mass of non-propellant and the initial mass can be thought of as the total mass of the rocket, the second form of the equation above is particularly useful because we can use it to determine the percentage of a rocket’s mass that can exist as stuff that’s not propellant, such as payload and structure. We will be doing this shortly.
In simple terms, the rocket equation relates how much you can increase or decrease the velocity of your rocket with the fraction of your rocket’s total mass that is propellant.
The Implications
Okay… but what does all this mean?
Let’s take a look at an example:
Traditional chemical rockets (rockets that burn propellant) have a Vₑₓ in the neighborhood of 2000–4500 m/s.
The exhaust velocity (Vₑₓ) is conventionally described using a value called the “specific impulse” (Iₛₚ). It is defined as the value that, when multiplied with the acceleration due to gravity (g; we will estimate it as 10 m/s² here), yields Vₑₓ:
This also means that its units are in seconds because Vₑₓ is in m/s and g is in m/s². You can think of Iₛₚ as the “miles per gallon” of rockets.
First, let’s say a rocket has an Iₛₚ of 200 s:
Next, we’ve already established that we need to be traveling at 8,000 m/s to get into a stable low-earth orbit. But since ΔV is how much we need to change our velocity and not just the final velocity, this number is much closer to 9,000 m/s when you factor in drag and gravity so that’s the value we’ll be using.
With our ΔV and Vₑₓ, we can find the value of the fraction ΔV/Vₑₓ:
Remember the second form of the rocket equation we were talking about? Let’s just substitute this value in for ΔV/Vₑₓ:
What this tells us is that 1.1% of the rocket’s mass can be made up of the payload and structure. The other 98.9% has to be propellant or we won’t make it to low-earth orbit.
That’s a lot of fuel.
It should be pretty clear that having 99% of your rocket’s mass exist as propellant and the rest as everything else you need isn’t very feasible.
Hmm… Let’s try doubling how fast our engine spits out propellant so we have a Vₑₓ of 4,000 m/s. (This is the same as doubling the Iₛₚ to 400 s but I will omit the step using Iₛₚ to solve for Vₑₓ because it’s pretty much the same calculation as our first example.)
Now let’s do the same thing as our last example to find the percentage of our rocket that can be stuff that isn’t propellant.
10.5% of our rocket’s total mass can be made up of its payload and structure while “only” 89.5% needs to be propellant. We can kind of work with this!
We see that doubling our Vₑₓ (or Iₛₚ) increased our non-propellant mass fraction by an entire order of magnitude!
Since the critical ratio of ΔV/Vₑₓ is in the exponential (eˣ) in the rocket equation, its influence on the outcome, the mass fraction, is magnified.
And this reveals the true essence of the rocket equation:
Increasing the exit velocity of the exhaust makes a huge difference in the performance and efficiency of the rocket engine.
Looking Ahead
When it comes to venturing out farther into space, these traditional chemical rockets hit a performance ceiling. You see, generating thrust by combustion is not very efficient in the long term when it comes to traveling beyond just the closest celestial bodies to Earth.
One of the main reasons is that you need a lot of propellant. And that stuff can get pretty heavy. So lugging around tanks full of liquid oxygen and hydrogen or methane is not ideal.
Some alternatives that are generating interest from government agencies and private companies alike are ion propulsion, nuclear propulsion, antimatter propulsion, and solar sails.
Follow me and stay tuned for more articles that will explore these alternatives in more detail!
Takeaways
3…
The ideal rocket equation is actually quite simple and serves as a basis for understanding a lot about rockets such as the viability of rocket staging and refueling. But it is the ideal rocket equation so there are many factors that we did not take into consideration such as drag, the air pressure of the atmosphere, the change in the strength of gravity, and the rotation of the Earth.
2…
The rocket equation tells us that the speed at which burned propellant is thrown out the back of the rocket engine has a great impact on the performance of the rocket. That’s why rocket engines are constantly being improved and new forms of propulsion systems such as ion thrusters (Iₛₚ ~ 4,000 s) and antimatter propulsion (projected Iₛₚ of up to 100,000 s) are being researched, designed, and built.
1…
One thing’s for sure: the price of launching rockets will continue to be driven down as the technology improves. Elon Musk claims that launch costs with their super-heavy launch vehicle Starship could be as low as $10/kg! With the growing economic practicality of the space industry, space will become increasingly commercialized.
Liftoff!
Though space itself is quite dark, the future of space travel and exploration is bright with the increasingly cost-effective modes of transportation under development. We, humanity as a whole, can collectively look forward to a future where the formerly unreachable edges of our solar system inch closer and closer to our fingertips.
