Rocket Physics, the Hard Way: How to Go to Mars

“There is an art to flying, or rather a knack. The knack lies in learning how to throw yourself at the ground and miss.”

The Hitchhiker’s Guide to the Galaxy, Douglas Adams

In this series, Rocket Physics, the Hard Way, we will learn the basic science and engineering behind rocket propulsion and interplanetary travel. Knowledge of high school physics, algebra, and basic calculus is useful, but not required. Mathematics will be kept to an absolute minimum, and more in-depth resources will be provided at the end for those who want to dig deeper. This will not be a rigorous technical series; rather, it will focus on demystifying the science and engineering needed to put things and people on Mars.

In this installment, we will learn the basics of orbital mechanics. You will need to know the following concepts:

  • Newton’s Laws
  • delta-V
  • The rocket equation

If you need a refresher, check out The Tyranny of the Rocket Equation.

Perhaps the most unintuitive part of rocket science is orbital mechanics, alternatively known as astrodynamics – the science of how objects move in space. While orbits can get complicated, here are some simple key principles to remember:

  1. All objects move based on Newton’s laws of motion.
  2. There’s (almost) no air in space.
  3. All objects have gravity, and its strength diminishes with distance.

Newton’s Cannonball

Orbits were famously described by Isaac Newton in 1687 (get used to seeing his name a lot). His work Principia, often considered one of the most important works in the history of science, proposed a clever thought experiment. In it, he asks us to imagine a cannon on top of a very tall mountain – so tall that its peak is in space.

If one fires a cannonball from this cannon, it will travel horizontally until the Earth’s gravity pulls it back to the ground. But fire the cannonball fast enough, and it will travel so far that it will follow a complete circle and hit you in the back of the head (as shown in orbit C of this diagram):

Newton’s cannonball thought experiment, as described in Principia (image source: Wikimedia Commons.)

In the present day, this is essentially how spacecraft stay in orbit. In a circular orbit, centrifugal force1 balances the force of gravity. In other words, it’s travelling fast enough that the ground curves away under the spacecraft as it ‘falls’ around the Earth, but not so fast that it flies off into space forever.

In low Earth orbit, a spacecraft would need to orbit at about 7-8 km/s to achieve this. The speed of a spacecraft in its orbit depends on the orbit’s shape, the planet’s mass, and its current distance from the planet’s center. A spacecraft in a low circular orbit, for example, would need to travel faster than one in a high orbit, because a planet’s gravity is stronger the closer you get to it.

Hence, it is a misnomer to say that astronauts floating on the International Space Station are ‘weightless’ or in ‘zero gravity’ because they are still under the influence of the Earth’s gravity – at the ISS’s altitude, the force of gravity is about 90% as strong as it is at sea level. A more accurate (and perhaps pedantic) term is ‘freefall’. They seem weightless because the ISS is falling at exactly the same rate that their bodies are.

Why astronauts seem to be weightless in space (image by author, CC BY 4.0.)

What if we fired the cannonball slightly faster than required for a circular orbit? Then, it would enter into an elliptical orbit around the Earth, as shown in orbit D of the Newton’s Cannonball diagram. The orbits of all the planets in the Solar System are elliptical2. Periodic comets like Halley’s comet follow elliptical orbits too, but very eccentric (i.e. stretched) ones.

However, an object with a high enough speed in the right direction (i.e. not at the ground) will fly away and never return (see orbit E in the Newton’s Cannonball diagram.) These are known as hyperbolic3 orbits. Interstellar objects like ‘Oumuamua follow hyperbolic orbits around the Sun – they’re going too fast to ever fall back down.

Let’s Go to Mars Already!

Let’s return to our prior example of a Starship traveling to Mars. Previously, it was in low Earth orbit traveling at 7.8 km/s, then its engines fired briefly to accelerate it to 11.5 km/s, sending it on its way to the Red Planet. Now, relative to the Earth, it is leaving on a hyperbolic path. As it coasts away under its own momentum, the Earth’s gravity will slow it down somewhat, but not enough to cause it to fall back (not if we’ve done things correctly, anyway.)

As it leaves the Earth’s gravity, the shape of its path will come to be dominated by the Sun’s gravity. It will enter into an elliptical transfer orbit around the Sun, departing from Earth’s orbit en route to Mars’s orbit. We used the lowest- energy path possible, known as a Hohmann transfer, which will take 250 days to travel from Earth to Mars. Here is a diagram:

Hohmann transfer orbit diagram (image source: Wikimedia Commons.)

For our thought experiment, ‘O’ is the Sun, the green circle is Earth’s orbit, the yellow ellipse is the Hohmann transfer going from Earth to Mars, and the red arc is Mars’s orbit (not to scale.)

Here are the stages of an interplanetary transfer, along with the astrodynamics jargon used:

  1. Low Earth Orbit (LEO): At first, the Starship is travelling with the Earth around the Sun (green orbit.)
  2. Trans-Mars Injection (TMI): Then, it fires its engines (Δv) to eject itself from low Earth orbit.
  3. Interplanetary transfer: It ends up in an elliptical transfer orbit (yellow) around the Sun. In real life, a spacecraft sometimes has to make trajectory correction maneuvers (TCMs), which are small maneuvers to fine-tune its path.
  4. Mars encounter: After coasting for 250 days with the engines off, it intercepts Mars’s orbit (red.)
  5. Mars Orbital Insertion (MOI): It fires its engines again (Δv’) to match velocities with Mars, so that it can be captured by its gravity.

Each of these stages has to work correctly for the mission to be successful.

For the trans-Mars injection maneuver, the timing has to be precise so that the Starship and Mars will be in the same location on arrival. This is made more complex by the fact that both Earth and Mars are constantly moving, meaning that we need to hit a moving target. Otherwise, the spacecraft could miss and be stranded in a Solar orbit until it runs out of resources. This is why missions to Mars have to be launched during transfer windows or launch windows, which generally occur every 26 months.

On the other hand, if the Mars orbital insertion maneuver doesn’t brake the Starship sufficiently, it would remain on a high-energy hyperbolic trajectory, causing it to swing past and fly out into interplanetary space again. Mars’s gravity would fail to capture it, instead slingshotting it onto an orbit that doesn’t encounter any planet for centuries.

While the trans-Mars injection (leaving Earth) would be performed by firing the Starship’s engines, more options are available for the Mars orbital insertion (arriving at Mars) to save fuel. Instead of firing its engines to slow down, the Starship could cheat the Rocket Equation by flying through the upper layers of Mars’s atmosphere, using air resistance to brake. This technique is known as aerobraking and has been widely used by probes to both Venus and Mars to save fuel.

A Starship entering Mars’s atmosphere (image credit: SpaceX.)

Stopping to Refuel a Speeding Cannonball

Since there are no external forces4 on the Starship besides gravity during the interplanetary transfer, it can simply coast with the engines off, relying on its inertia and Newton’s First Law to keep it moving through space. This is unlike poorly-researched science fiction films that depict spacecraft firing their engines all the time (although there are cases where this is well-justified.)

This is why ‘stopping to refuel’ mid-way to Mars doesn’t make sense from an astrodynamics perspective. To stop at a refueling station, the spacecraft would have to fire its engines to match velocities, refuel, then fire its engines again to resume its original trajectory. It would make about as much sense as ‘stopping to refuel’ a speeding cannonball mid-flight. Refueling would make sense once the spacecraft has arrived at Mars, which could be done through techniques such as In-Situ Resource Utilization (ISRU.)

Getting There Faster

250 days is a long time to spend in interplanetary space. While this may be okay for cargo, a crewed mission may need to reach Mars more quickly to minimize radiation exposure and freefall-induced musculoskeletal degeneration – the Mars Direct plan called for a 180-day interplanetary transfer. This means a low-energy Hohmann transfer orbit is no longer good enough. Having a higher delta-V for the trans-Mars Injection will put the Starship into a more eccentric, higher-energy transfer orbit, meaning that it will reach Mars more quickly.

Here is one such higher-energy trajectory, computed using the NASA Ames Trajectory Browser (a great way to learn astrodynamics and procrastinate.) It departs Earth by executing a 4.34 km/s trans-Mars injection burn on November 2nd, 2024, and arrives at Mars after 176 days in space by shedding 2.44 km/s during the Mars orbital insertion maneuver:

A high-energy transfer from Earth to Mars. Earth’s orbit is blue, the transfer orbit is green, and Mars’s orbit is red (animation generated by author using NASA Ames Trajectory Browser.)

Note that the insertion delta-V is less than the injection delta-V, partially because the spacecraft slows down as it rises higher in the Sun’s gravity well.

Here is a Hohmann transfer for comparison. The spacecraft departs Earth by executing a 4.26 km/s trans-Mars injection maneuver on November 2nd, 2024, and arrives at Mars after 272 days in space by executing a 0.55 km/s Mars orbital insertion maneuver:

A minimum-energy Hohmann transfer from Earth to Mars. Earth’s orbit is blue, the transfer orbit is green, and Mars’s orbit is red (animation generated by author using NASA Ames Trajectory Browser.)

Overall, for this specific launch window, the Hohmann transfer requires 29% less delta-V than the high-energy transfer. This is significant because raising delta-V requirements by a small amount can raise fuel requirements by a large amount, due to the Rocket Equation. Note that the numbers will vary from launch window to launch window because Mars’s and Earth’s orbits are not perfect circles.

A shorter transfer time comes at the cost of higher fuel requirements and less payload mass. Not only is it more difficult to reach the required delta-V, it’s also more difficult to stop. The spacecraft will need to brake harder to match velocities with Mars upon arrival, meaning that either more fuel is required for Mars orbital insertion, or the aerobraking maneuver becomes more dangerous.

For real Starship missions, Elon Musk announced in 2016 that the real Starship would take between a mere 80 and 150 days to reach Mars, depending on the launch window. SpaceX chose to use such a high-energy transfer to minimize radiation and freefall exposure – especially because the Starship is (at the time of writing) not designed to have artificial gravity – at the cost of increased fuel requirements and a more difficult aerobraking maneuver. SpaceX can probably afford to have more extreme requirements due to in-orbit refueling of the Starship, more efficient engines and construction, and more advanced heat shield technology.

Ultimately, choosing a path to go to Mars is a question of balancing several competing design requirements. In this installment, we learned about orbits, interplanetary transfers, and aerobraking, which has hopefully provided a deeper understanding of the decisions made by mission planners.

Stay tuned for more rocket physics!

Footnotes and further reading

1 Yes, centrifugal force is a fictitious force. However, it’s a useful concept to gain an intuitive grasp of simple orbits. Incidentally, centrifugal force can be described as a ‘real force’ in a rotating frame of reference. One could also argue that due to general relativity, gravity is also a fictitious force.

2 Except that they don’t. Planetary orbits aren’t perfect ellipses due to relativity and perturbations caused by the gravities of all the other planets – nevertheless, an ellipse is a pretty good approximation for the orbits they do follow. An object will follow a perfectly circular, elliptical, parabolic, or hyperbolic orbit only when there are only two bodies in question. This approximation is okay for many situations, especially if great precision isn’t necessary. However, for problems involving three or more bodies and in situations where a celestial body’s gravity is uneven, we have to resort to brute-force computer simulations of the equations of motion. Real orbits exhibit all kinds of bizarre phenomena like Lagrange points, halo orbits, co-orbital moons, precessing sun-synchronous orbits, and orbital resonances, all of which are headaches to understand and model – accordingly, they will (probably) not be covered in this series.

3 Parabolic orbits are orbits with exactly the right amount of energy such that (to be pedantic) their velocity approaches zero as distance approaches infinity. Subtracting even the tiniest amount of energy would make it a closed elliptical orbit and adding even the tiniest amount of energy would make it a hyperbolic orbit. Hence, the concept of a parabolic orbit is only useful from a mathematical perspective and is ignored for simplicity.

4 In reality, you would have numerous small forces also playing a role such as radiation pressure. However, these forces are usually so tiny in comparison to the spacecraft’s momentum that we can ignore them – this is not the case for spacecraft like Solar sails, which rely on radiation pressure to operate.

In this installment, you may have noticed references to the amount of orbit an energy has. Due to the conservation of energy, a spacecraft has the same amount of energy at every point in its orbit. As it rises away from the object it’s orbiting, it slows down because it exchanges kinetic energy for potential energy. As it falls towards the object, the opposite happens. This is expressed mathematically through the vis-viva equation.

The movement of celestial bodies in orbit was first described in terms of mathematical laws by astronomer Johannes Kepler between 1609 and 1619 in Kepler’s Laws. These laws were derived purely from empirical data of the planets’ movements, but were later confirmed through Newton’s Laws in Principia. These laws are useful to know for astronomy and astrodynamics. For more details on calculating orbits and the parts of an orbit, see the Encyclopedia Britannica’s page on orbits.

The most effective (and fun!) way to learn orbital mechanics and rocket science is to play an accurate space mission simulator. The author personally recommends the video game Kerbal Space Program, having sunk well over 240 hours (of spectacular explosions and stranded Kerbals) and counting into it. However, you are strapped for cash and can’t wait for a sale, the author cut his teeth on orbital mechanics as a child with (probably several hundreds of hours on) Orbiter Space Flight Simulator. These are well-complemented by the educational videos on Scott Manley’s YouTube channel.

For curious readers who do not play Kerbal Space Program, the author learned most of what he knows about orbital mechanics from playing it and Orbiter. His current university sadly does not offer astrodynamics courses – nor any aerospace engineering courses at all.

For curious readers who do play Kerbal Space Program, the author’s proudest achievement is sending a twin-probe expedition to the outer Kerbolar system in Career mode. One probe was jettisoned at Jool and used a Tylo gravity assist for orbital insertion. The other probe used the Jool flyby as a gravity assist to reach Eeloo with plenty of propellant to spare. No phasing orbits were necessary – the planetary alignment was eyeballed.

The author just realized that he spent three paragraphs talking about Kerbal Space Program.

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3 thoughts on “Rocket Physics, the Hard Way: How to Go to Mars”

    1. Excellent question! It’s for the reason you highlighted – it would take a huge amount of energy without reducing transit time.

      All the planets in the Solar System orbit in one direction, or in ‘prograde’ anticlockwise orbits. Earth orbits the Sun prograde at ~30 km/s. To go into a ‘retrograde’ clockwise orbit, you have to completely reverse your direction of travel. i.e. we have to go from 30 km/s prograde to 30 km/s retrograde. So a spaceship would need to change its velocity by 60 km/s. To send it on a transfer orbit to Mars, we need to add another ~5 km/s to the orbit.

      Mars orbits prograde at ~24 km/s. On arrival, our spaceship would be travelling retrograde at ~23 km/s (it’s slowed down a little bit because it’s rising out of the Sun’s gravity well.) This means that to enter Mars orbit, the spaceship has to change its velocity by 47 km/s.

      So to go to Mars in the retrograde direction, the mission delta-V is 60 + 5 + 47 = 112 km/s, compared to ~5 km/s to go the prograde direction. Keep in mind that the transit time is exactly the same in both cases. 5 km/s is pretty achievable with modern chemical rockets, while 112 km/s is well beyond current technology.

      Hope that answers the question!

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