Interplanetary Rocket Travel and the Rocket Equation

In rocket travel, one of the most essential elements is ∆v – the change in a ship’s velocity. The spaceship needs to accelerate to get out of the atmosphere, and then speed up to achieve orbit. If it’s going to another planet, it needs to achieve escape velocity from Earth, then speed up/slow down its orbit around the Sun to become intercepted by that other planet.

One of the most famous equations that governs this is (aptly) called the “rocket equation“: \Delta v = v_e ln\frac{m_0}{m_f} . This is complicated, so let’s break down each variable.

\Delta v in this equation is what we discussed above – the change in velocity.

v_e is the “effective exhaust velocity” – for our purposes, it just matters that this depends on the engine design. We’ll use SpaceX’s Raptor Engine, which has a “specific impulse” (or I_{sp} ) of 380s in a vacuum. You can get v_e by multiplying I_{sp} by the gravity at the Earth’s surface, which gives us v_e (Raptor Engine) = 380s \cdot 9.8m/s^2 = 3724m/s .

m_0 is the initial mass of the rocket, and m_f is the final mass of the rocket (after earlier stages have been discarded and fuel burned). The important thing to notice about these is they are within the logarithm ln\frac{m_0}{m_f} . That matters, because it means that as you want \Delta v to increase, the ratio \frac{m_0}{m_f} has to increase exponentially.

To give a bit of intuition for why that is: imagine you start with a probe. To get it to move, you only need a somewhat small engine – say we have a rocket with the same mass as the probe, and that gets us some acceleration. If we want it to go further, we need a rocket powerful enough to power both the probe and the engine we were using it before. So to get that same acceleration, we need to double the mass of the whole rocket again, leaving us with a new rocket four times as large. To get the same acceleration again, the rocket needs to be eight times as large. There’s the exponential climb.

Here’s a graph showing that exponential decline in more precision:

Source: Wikipedia

The interesting thing about this equation is that m_0 and m_f are the main things we control. Orbital mechanics gives us a required \Delta v if we want to orbit Earth, or go to Mars, or leave the solar system. The only thing we control is: how big is the probe we’re sending, and how big is the rocket we’re sending it on. So if we want to go to one of these places, we can find out exactly what percent of the mass of our first rocket we could theoretically keep. That’s what I calculated.

Below is the result of all of this. For each other world in our solar system, what percent of our mass can we keep? If we want to send 1 ton of stuff (cameras, sensors, food, people, etc.) to that world, how big of a rocket do we need?

WorldRequired ∆v% of Mass to Worldm_0 for m_f = 1t
Moon15.93 km/s1.392%72 tons
Mars21.3 km/s0.329%304 tons
Venus21.2 km/s0.338%296 tons
Mercury23.3 km/s0.193%519 tons
Juptier24 km/s0.160%627 tons
Saturn25 km/s0.122%819 tons
Neptune26 km/s0.093%1072 tons
Pluto26 km/s0.093%1072 tons
Escape Sun26.5 km/s0.082%1226 tons

Again, all of this assumes that we only use Raptor Engines to power our rocket, it ignores the different specific impulse in the atmosphere, and (most importantly) this is only enough to send your ship crashing into the other planet – not into orbit, and certainly not to land on it gently. All of those only take up more delta-v, which means you need a bigger rocket to start with.

Economics of Intra-Stellar Spacecraft

After seeing the discussion of spacecraft within our solar system in the textbook, and the impact the mission type has on the cost, I was curious to see how these discussions were reflected in actual data on these missions. So, I copied each mission from the book into Excel, and then researched each. Thanks mostly to Wikipedia and the Google search “How much did [MISSION NAME] cost?”, I came up with this table. Feel free to skip over it – the rest of the blog is more interesting.

NameLaunch YearCost (inf. adj.)Launch Mass (probe)TypePlanet Reached
MESSENGER2004 $504 Million 1110 kgOrbiterMercury
Magellan1989 $1155 Million 1035 kgOrbiterVenus
Venus Express2005 $128 Million 1270 kgOrbiterVenus
Spirit2003 $1094 Million 1063 kgRoverMars
Opportunity2004 $1600 Million 1063 kgRoverMars
Mars Express2003 $472 Million 1120 kgRoverMars
Mars Recon. Orbiter2005 $928 Million 2180 kgOrbiterMars
Phoenix2007 $469 Million 670 kgLanderMars
Curiosity2011 $2798 Million 3839 kgRoverMars
MAVEN2013 $725 Million 2454 kgOrbiterMars
Mars Insight2018 $832 Million 694 kgLanderMars
Voyager 11977 $1797 Million 825.5 kgFlybyNeptune
Voyager 21979 $1797 Million 825.5 kgFlybyNeptune
Galileo1989 $2766 Million 2562 kgOrbiterJupiter
Cassini2004 $4336 Million 5712 kgOrbiterSaturn
Juno2009 $1231 Million 3625 kgOrbiterJupiter

Then, with all the boring data out of the way, I could make (relatively) pretty graphs to see how each part of the mission related to cost:

Cost and Mass

The most intuitive finding of this data is that more massive probes are (quite literally) exponentially more expensive:

This makes sense. Here, “mass” refers to the mass of the final probe (including propellant for any maneuvers once it’s in orbit). So, a heavier probe would need a bigger rocket to bring it to space. But this is constrained by something called the rocket equation, which (in very rough terms) says that as you increase the final mass of what you’re bringing to space, the total mass of your rocket has to increase exponentially. So it makes sense that, the more massive your final probe already is, the greater the marginal cost of another kilogram.

Cost by Probe Type

The book suggested that flybys are less expensive than orbiters which are less expensive than landers (and, presumably, stationary landers are less expensive than rovers). In practice, this isn’t reflected in the actual costs of these missions.

Rover $1,490,865,000
Lander $650,132,000
Orbiter $1,471,548,000
Flyby $1,796,605,000

Of course, none of this shows the inherent cost of each mission type. But it may indicate how much priority different projects are given. A rover will be driving all over a planet’s surface, so it makes sense to have a lot of gadgets to comb through all that data. And if the best you can do is fly by a planet, you only have one chance to collect a lot of data, so you need to make it worth it (perhaps with very expensive equipment). Landers, meanwhile, can’t see as far as an orbiter, so they can only do a handful of experiments.

Cost by Launch Year

The book didn’t discuss this directly, but I was curious if the price (adjusted for inflation) of missions within the solar system had come down. It has.

Excel’s trendline suggests that (after adjusting for inflation), the average cost of a mission within the solar system comes down by about $22 million every year. But looking at the data, a lot of this seems to be caused by the missions from 2000-2010. It’s hard to say how much of this “trend” is caused by changes in technology vs. changes in available funding vs. changes in mission designs.

Cost by Planet

We’ve sent missions to every planet in the solar system. I was curious which were most expensive.

Overall, missions to Saturn have been the most expensive on average. It’s not hard to come up with an explanation for why this could be. We’ve sent far more missions to Mars, and it’s cheaper to get back there. That means it may not be worth sticking everything you can on the ship and increasing the price. But with Saturn, it takes a lot just to get the rocket there. So if you have something going there, then everything to make the mission better – every extra hour of design, every special instrument to get more knowledge, every unique material to bring down the mass – becomes worth the cost. This doesn’t explain the trend in the opposite direction – that Mercury’s missions are cheaper than Venus’s, even though Venus is closer to Earth. That could be because we aren’t as interested in Mercury as we are Mars or Venus, but I really couldn’t say.

Gravity-Assists in Reverse

Gravity assist maneuvers are incredibly useful for sending spacecraft far into space. The maneuver is like a skiing parent using a pole to give their kid a boost. The parent loses a bit of momentum, but leverage their larger size to quickly speed their kid up. The same is true for a gravity assist. There, instead of firing boosters to go directly to some far away place, the spacecraft flies towards a closer-by planet. Then, the spacecraft orbits around the planet, leveraging its gravity to launch it further into deep space. New Horizons was able to gain an additional 14,000 km/hr in velocity with a gravity assist around Jupiter.

But the laws of physics dictate that energy and momentum must be conserved. So, Jupiter slows down very slightly as the spacecraft speeds up. Effectively, the spacecraft is stealing some of Jupiter’s momentum and its energy. But this raises an interesting question: what would happen if you tried to do this in reverse? Gravity assists typically have the spacecraft and planet travel in the same direction (counter-clockwise). But if the spacecraft is orbiting in the opposite direction as the planet, can you send the spacecraft faster in one direction and the planet faster in the other? That would preserves momentum. But, if both increase their velocities, both would seem to have more energy, which would violate the laws of physics.

Unfortunately, there aren’t many good public-source orbital simulators that involve inter-planetary transfers. So I had to use the next best thing: Kerbal Space Program. KSP is a video game that uses a very basic model to simulate orbits. It assumes that every object is orbiting only one other object, and ignores the effect of other nearby planets. After all, in real life, the Moon’s distance from the Sun is very similar to the Earth’s distance from the Sun. So, you can get a good estimate of the Moon’s location by looking at its position relative to Earth as a function only of the Earth’s gravity, and then tracking the Sun’s effect on Earth separately.

So, using KSP, I did my best to plan a maneuver to do a reverse-gravity assist. Here are a series of pictures showing the gravity-assist I attempted:

This was the beginning orbit, on course to be intercepted by the Moon (or, in KSP, the “Mun”). Important to note are its velocity and altitude: 2799.2m/s at an altitude of 243,190m. Source: KSP
This is the spacecraft in its orbit, about to enter the Moon’s sphere of influence. Source: KSP
This was the orbital trajectory of the spacecraft as its was captured by the Moon. Source: KSP
This is the final orbit. Notably, the velocity increased (∆v = 6.9m/s) even at the higher altitude (∆d = 158m). Source: KSP

The maneuver did, technically, work. The spacecraft went faster despite its higher altitude. But it wasn’t close to the magnitude of effect from a traditional gravity-assist. And with a change in velocity of only 6.9m/s, KSP’s physics engine may have just shown something that a more accurate engine would show couldn’t happen. But it is worth noting that the spacecraft is further from the planet than the Moon when it enters the Moon’s sphere of influence. This yields a possibility: that the spacecraft accelerated towards the planet and the Moon accelerated away from the planet. This would decrease the Moon’s orbital velocity, conserving energy and momentum.

Fundamentally, though, I wanted to answer a simple question: could you do the sort of gravitational assist that’s typically possible if the spacecraft and planet orbit in opposite directions? And I got an answer: basically, no.

Nicholas Copernicus

Nicholas Copernicus – February 19, 1473 to May 24, 1543

Source: Wikimedia Commons

Historical Events During Life of Copernicus

In 1492, Christopher Columbus, trying to create a more direct trade route between India and Spain, becomes the first European explorer to discover the Americas.

In 1517, Martin Luther posts his theses, a list of grievances with the Catholic Church. This begins the protestant reformation.

Other Historical Figure During Life of Copernicus

Niccolò Machiavelli lived from 1469-1527. He was an incredibly influential philosophical thinker who is often considered to be the founder of political science, as he was one of the first to approach the subject in a systematic way that analyzed how to govern effectively as opposed to writing about normative judgments.

Reflection

What stuck out was how little happened during Copernicus’s lifetime. The periods before and after his lifetime were both far more eventful – more wars, more societal changes, more scientific advancement, etc. In part, this seems caused by him. He helped progress Europe from an era where ideas stemmed mostly from Aristotelian logic and introspection. He moved us towards looking more at external evidence and towards a more modern conception of what science is. And this matches with some of the other people who lived at the same time. Machiavelli and Martin Luther, like Copernicus, transformed how the West viewed (and views) the world. Above all else, Copernicus seemed to live during a time that planted the seeds for future changes and revolutions.

The Universe as the United States

If there’s one straightforward lesson from astronomy, it’s that we’re tiny. We’re small compared to the Earth’s vast size, which is small compared to the Sun, which is tiny compared to the space that contains our solar system, which is a tiny dot in one arm of the Milky Way galaxy, which is one of roughly two trillion galaxies in the observable universe.

It’s almost impossible to try to grasp the universe’s size directly. Because it’s far beyond the size of anything we interact with daily, it’s hard to comprehend. It’s easier if we scale it to the size of the United States, and see how big it makes things we’re more familiar with (e.g. the size of the Earth). We can go step by step through some of the steps between the size of the Earth and the size of the Milky Way.

To begin, we need the size of the United States. We can use the distance from Portland, Maine to Los Angeles (about 2,600 miles) as an estimate of its size. This lets us start to scale. The Earth’s diameter is about 12,700 km, but the distance to the Moon is 384,000 km. So, if we scaled down the Earth and the Moon to put the Earth in Los Angeles and the Moon in Maine, the Earth’s diameter would be about 86 miles. If one “corner” of the Earth were in Los Angeles, the opposite side would only reach about halfway to Nevada.

Source: Screenshot from Google Maps

The Sun is much further away. The Sun is about 150 million km from the Earth, almost 400x further than the Moon. If we scaled down the universe to fit the Sun in Maine while keeping the Earth in LA, the Earth would only be .22 miles wide. Viewed from above, a circle containing the Earth would be .6 square miles – about .1% of Los Angeles’s actual 487 square miles.

This shows how little space the Earth truly takes up, relative to its distance from the Sun. But what if we go a step further? What if we put one edge of the Milky Way in Los Angeles, and the other side in Maine? The short answer: things stop making sense. The Milky Way is 1,000,000,000,000,000,000 km across, or 621 million billion miles. To scale that down the to the size of the US (2,600 miles), the Earth would become 53 nanometers. A fingernail is somewhere around .4mm thick. So if we scaled down the Milky Way to the size of the US, you could fit 7,500 Earths side-to-side and it would only be as wide as your fingernail is thick.

We’re tiny.

A Bit About Me

Hi Reader!

My name is Andrew Talley, and I’m a sophomore Computer Science/Philosophy major studying at Vanderbilt University. After I graduate, I plan to go to law school. As a result, my activities on campus are both law-related. I’m a member of Vanderbilt’s Moot Court club, where we compete against other undergraduate schools in fake appellate cases. I’m also a member of Vanderbilt’s Mock Trial club, where we compete against other undergraduate schools in fake civil/criminal cases (you may notice a pattern).

In my spare time when I can’t find something more productive to do, I’m a big fan of TV. In particular, I can’t recommend The Good Place highly enough. It’s surprisingly witty and has a bewilderingly good plot for a network comedy.

Thank you for reading this, and expect more Good Place gifs in the future.

Andrew