Tyranny of the Rocket Equation
Austin Morris, Director of Engineering
In previous columns, I have covered several aspects of how objects in space behave. But I haven’t dived too deeply into how rockets actually get those objects to space. The answer is a combination of magic and math, with more emphasis on the latter. My intent with this column is to address some of the math and the concepts behind it, but leave the rest of the magic to the magicians. The primary equation that drives this problem was published in 1903 by Russian scientist Konstantin Tsiolkovsky, and has formed the basis of modern rocketry ever since.
My familiarity with the Tsiolkovsky rocket equation has varied over the years. It began with a passing awareness and a need to understand the concept of diminishing returns. It then eventually became relevant and even necessary to the company that I co-founded. For most people reading this, I don’t expect that the rocket equation itself is directly relevant to their daily lives, but I believe that some of the concepts contained within the equation actually are.
Here’s the scary part of the column: the actual equation. But don’t fret, we’ll break it down and explain the relevance in a moment.
\begin{equation}\large\Delta v = (v_e)(\ln\frac{m_0}{m_f}) = (I_{sp})(g_0)(\ln\frac{m_0}{m_f})\end{equation}Assuming the mathematicians in the audience forgive my overuse of parentheses to try and separate the terms, the above is our topic of discussion for today. For now, we are only going to worry about a few specific terms and we will take them one at a time.
$\large\Delta v$ is delta-v, or the total amount of how much you can change the velocity of your vehicle. If we imagined that your car has a delta-v of 60mph, for instance, that means that you can accelerate your car up to 60mph, or that you could accelerate up to 30mph and then slow down back to 0mph, or you could do any combination of acceleration and deceleration changes, so long as the total amount of changes add up to 60mph. When dealing with spacecraft, we typically discuss delta-v in terms of meters per second, or m/s.
$\large v_e$ is the effective exhaust velocity, or how much oomph you are spitting out to accelerate your vehicle. The oomph (definitely a technical term, trust me) is determined by the terms ($\large I_{sp}$) and ($\large g_0$). ($\large g_0$) is simply the constant standard gravity, while ($\large I_{sp}$) is the specific impulse, or the value that shows how the efficiency of engines is measured. But since I promised we wouldn’t get too deep into the math right now, let’s just leave it at oomph.
Lastly, ($\large\ln\frac{m_0}{m_f}$) is the natural logarithm (inverse of an exponential, the driver of diminishing returns) of the initial total mass including propellant (also known as wet mass) divided by the final total mass without propellant (also known as dry mass).
If you’ve made it this far, congratulations as this is the point at which I start making the point of how things relate to your daily life. The concept of diminishing returns is fairly simple: effort at the beginning of a task typically has more effect than effort later in the task. It is, as everything is, a bit more nuanced than that, but the simplified version can suffice. In other words, if you took a beginner, who has never played a guitar before, and gave them 40 hours of training, they will likely be able to play some notes and chords, and maybe even a song or a few, which is a whole lot more than the nothing that they could do before. If you gave Jimi Hendrix an additional 40 hours of guitar training, you likely wouldn’t notice as much of a difference (seeing as it’s difficult to improve upon the perfection that is All Along the Watchtower). This idea can be extrapolated to learning really any skill, be it language, woodworking, welding, etc.
It’s important to note that I’m not suggesting that someone at Hendrix’s level of skill shouldn’t keep practicing, just that they need to have the proper expectations for how much they will improve given a set amount of time. Additionally, this is where the efficiency term from the equation comes back into play. If you are practicing in a noisy, distracting environment or are being taught by someone who really doesn’t know what they’re teaching, you will not likely be as efficient in your practice as if you are given a studio with other talented musicians and without distractions.
These concepts really start to interact with the rocket equation when you think about the mass of the system. The concept of a rocket (generally) is to put some mass together, put some fuel behind it, and burn that fuel to put the mass somewhere else. If you want to put the mass further away, you need more fuel to do so. But the fuel itself has mass, so now you have more mass to push. So to push that mass, you add fuel. But now you’ve added more fuel mass, so you need to add more fuel mass to fuel the mass that you’ve already amassed atop your fuel. And herein lies the tyranny of the rocket equation: the more fuel you have, the more fuel you need.
But wait, it gets worse. When you add fuel to a rocket, you also have to increase the size of your fuel tank, and possibly add in some extra plumbing or make some structural changes to allow for this increase in size. All these changes add more weight to the rocket. For a variety of reasons, many missions launch aboard a multistage rocket, like the Saturn V that supported the Apollo program. If you add anything to the payload, you now need the third stage to have more fuel to carry that extra mass. And thus the second stage has to have more fuel to carry that mass, but not just the same amount of fuel, it needs even more fuel to support that change. And surprise surprise, that means the first stage now needs even MORE fuel to carry all of the above. So in adding a few pounds or kilograms to the payload, you’ve now had to add orders of magnitude more weight than that in adding to each stage as it cascades down the system.
The tyranny of the rocket equation certainly makes it seem like reducing the mass of a system should be the highest priority. But while it is important for KMI and other space operators to make their payloads as lean as possible, it shouldn’t come at the cost of reliability. After all, a heavy mission that succeeds is still far better than a lightweight mission that fails (and becomes debris). Operators and launch providers must excel on their respective portions so that when those portions combine into a collaborative mission, they combine into a sum greater than its parts with reliability, sustainability, and success in mind. Progress in different areas allows a full ecosystem of capabilities, from science missions to communications systems, with the actual rocket science left to those with the expertise. This parallel progress on the different needs for operation in space is what enables KMI to pursue our part: conducting active debris removal and #KeepingSpaceClearForAll.
Recommended column to read next: Orbits and the Cow that Jumped Over the Moon