Rocket Physics, the Hard Way: How to go to Mars – Extra Credit

This is a follow-up to Rocket Physics, the Hard Way: How to go to Mars which covered the basics of orbital mechanics. Building on our foundation of intuitive understanding, we will now cover the more technical aspects. What is an apogee? What really is escape velocity? All these considerations and more will be covered in this ‘extra credit’ installment.

As a warning, this installment is targeted at more technically-minded readers who are interested in the nitty-gritty of orbital mechanics. A strong background in physics and mathematics is recommended.

We will be returning to less technical content in the next article.

Anatomy of a Keplerian Orbit

For most applications, we can model orbits as Keplerian orbits – a simple but powerful model created by German astronomer Johannes Kepler in the 17th century, originally used to accurately describe the motion of the planets. Elements of Kepler’s model are still used to this day, four hundred years later, by astronomers and space agencies.

In Keplerian orbits, we only consider two bodies – a primary (like a planet or star) and a satellite. We ignore relativity, air resistance, and the effects of all other bodies in the system. For most situations, this approximation works quite well.

Recall the shapes of Keplerian orbits from the last installment: circular (grey), elliptical (red), parabolic (green), and hyperbolic (blue):

Orbital Mechanics
Classes of Keplerian orbits (image source: Wikimedia Commons.)

Here, we have a satellite orbiting a massive object, like a planet. Circular and elliptical orbits are closed – a satellite in these orbits has been captured by the planet’s gravity. When an orbit is more stretched out (looks like a flattened circle), we say that it has a higher eccentricity, represented by the letter e.

An orbit with an eccentricity of zero is circular. An orbit with an eccentricity between zero and one is elliptical.

For example, Earth’s orbit has an eccentricity of 0.0167 – it’s more or less circular if you don’t look too closely. Mars’s orbit has an eccentricity of 0.0934 – the differences in distance to the Sun along its orbit begins to have significant effects on seasonal temperatures. On the other hand, Halley’s comet has an extremely eccentric orbit, with an eccentricity of 0.967. It comes between Mercury and Venus at its closest and reaches out to Pluto at its farthest from the Sun.

If we were to give Halley’s comet just a little bit more energy, we would be able to push its eccentricity up to 1. This would turn its orbit into a parabolic orbit – one with exactly enough energy to leave the Solar System and never return.

Any orbit with more energy than is required to leave the Solar System would follow a hyperbolic orbit and have an eccentricity exceeding 1.

In a stroke of mathematical beauty, all of these shapes can be represented as sections of a cone:

Conic Sections
Conic sections (image source: Wikimedia Commons.)

If you want to know why, check out this video from 3Blue1Brown:

Anatomy of an Orbit

A Keplerian orbit can be uniquely defined using a set of quantities known as orbital elements. We will focus on elliptical orbits for now. Here are the most important ones, because they govern the amount of energy an orbit has:

Rocket Physics
Basic elliptical orbital elements (image by author, CC BY 4.0.)

An ellipse is defined by having two foci (a circle is a special case where the two foci overlap.) At one focus is the object being orbited, or the primary. The dimensions of this ellipse are the:

  • Semi-major axis (a): One half of the ellipse’s major axis (the long dimension.) This is commonly used to quantify the orbit’s size.
  • Semi-minor axis (b): One half of the ellipse’s minor axis (the short dimension.) This is not commonly used – usually, the eccentricity (e) is given instead, and if needed, the semi-minor axis can be calculated from that.
  • Eccentricity (e): A measure of the ellipse’s ‘flatness’.
  • Periapsis (rp): The lowest point in the orbit.
  • Apoapsis (ra): The highest point in the orbit.

Usually, an orbit will be given in terms of semi-major axis and eccentricity or in terms of periapsis and apoapsis.

When the satellite falls towards its primary, it speeds up until it reaches its maximum velocity at periapsis. Once it passes through periapsis, it begins rising away under its own momentum and slows down until it reaches its minimum velocity at apoapsis. This is due to the conservation of energy – throughout its orbit, the satellite is continuously exchanging potential energy for kinetic energy or vice versa.

When considering the orbits of planets and moons about their primaries, periapsis and apoapsis usually refer to the distances between the bodies’ centres at closest and farthest approach, respectively. For example, Mars has a periapsis of 206.7 million km and an apoapsis of 249.2 million km.

However, beware! When referring to spacecraft orbiting a planet, periapsis and apoapsis are instead quoted as altitudes from the planet’s surface, not to the planet’s centre! For example, the International Space Station has a periapsis of 418 km above sea level and an apoapsis of 419 km above sea level.

All future writing will follow this convention.

You will hear alternative names for periapsis and apoapsis, depending on the body being orbited:

  • Perigee and apogee, for objects orbiting the Earth (from Gaia, from the Greek personification of the Earth.)
  • Perilune and apolune, for objects orbiting the Moon (from Luna, the Roman goddess of the Moon.)
  • Periareion and apoareion, for objects orbiting Mars (from Ares, the Greek equivalent of Mars.)
  • Perihelion and aphelion, for objects orbiting the Sun (from Helios, the Greek personification of the Sun.)

Maneuvering

Armed with our newfound knowledge of Keplerian orbits, let’s learn how to get from one orbit to another. We will use the 2020 flight of Bob Behnken and Doug Hurley in the SpaceX Dragon Endurance as a case study.

NASA Endurance Orbit
Stages of Endurance’s orbital transfer to the International Space Station (image source: NASA; see full-size original here.)

During the dramatic first leg of the flight, the Falcon 9 lifted off from Cape Canaveral. After jettisoning the first stage, the second stage inserted Endurance into a low circular orbit around the Earth. Then, as seen in step 2 of the diagram, the capsule fired its engines to raise its apogee, entering an elliptical orbit.

To raise a spacecraft’s orbit, it has to accelerate in its direction of travel, also known as the prograde direction. Alternatively, it could accelerate opposite to its direction of travel – or the retrograde direction – to lower its orbit:

Rocket Physics
Prograde and retrograde directions (image by author, CC BY 4.0.)

Note an important thing: There is no reason that a spacecraft has to orbit facing its direction of travel! In fact, when the Space Shuttle was still operating, it would usually orbit pointing retrograde to shield the crew cabin from potential orbital debris impacts.

In what is known as the ‘phasing burns’, Endurance conducted another prograde burn at apogee, raising its orbit further to intersect with that of the International Space Station. Once it reached its apogee of the new orbit, it made a final prograde burn to match orbits.

Later, at the end of its mission, Endurance undocked from the International Space Station, jettisoned its trunk, and made a retrograde burn to lower its perigee into the upper layers of the Earth’s atmosphere. Once inside the upper atmosphere, air resistance slowed the capsule further, causing it to fall deeper until it landed:

I Want to Break Free…

This section and the one that follows include some light algebra. The equations can be skipped safely.

Orbital velocity and escape velocity are commonly confused. Orbital velocity is the speed at which a spacecraft stays in orbit (usually a near-circular one), which depends on the orbit’s altitude. Because gravity is stronger at lower altitudes, lower orbits are faster than higher ones. Hence, why the International Space Station (altitude of ~400 km) travels at over twice the velocity of GPS satellites (altitude of ~36,000 km, or geostationary orbit.)

The velocity required to maintain a circular orbit is given by the following equation, which is derived by balancing centrifugal force and gravity with Newton’s second law:

However, to go fast enough and never come back, a spacecraft must travel at least as fast as escape velocity at the altitude of its orbit:

At the International Space Station’s altitude, escape velocity is 10.8 km/s. A spacecraft travelling directly away from the Earth at escape velocity would follow a straight line that extends to infinity. On the other hand, a spacecraft travelling tangentially to the Earth at escape velocity would follow a curved, parabolic orbit that extends to infinity. Travelling faster than escape velocity would put the spacecraft into a hyperbolic escape orbit.

An Aside on Black Holes

Black holes are undeniably cool and deserve a mention.

As an interesting aside, we can also calculate the size of a black hole’s event horizon – or its Schwarzschild radius – using the escape velocity equation. At a black hole’s event horizon, the escape velocity is the speed of light. Since it is impossible to travel faster than the speed of light, an orbit that escapes the black hole’s gravity at this distance from its singularity is impossible. With some algebra, we can rearrange the equation to make the orbital radius the subject:

At the Schwarzschild radius, the escape velocity is the speed of light:

Using this equation, we can calculate that were the Sun to be collapsed into a black hole, its event horizon would have a diameter of approximately 6 km.

Interplanetary Transfers

Up to now, we have only considered Keplerian orbits in situations consisting of a small satellite orbiting a massive planet. But what happens when we need to send a spacecraft to Mars? It will encounter the gravities of at least three different bodies on its way: Earth’s gravity, the Sun’s gravity, and Mars’s gravity. Keplerian orbits aren’t sophisticated enough to model the entire trajectory at once. However, piecemeal, they can provide a decent approximation.

Where a certain planet’s gravity dominates (also known as its sphere of influence), we can approximate the spacecraft’s path as a Keplerian orbit around it, neglecting the effects of the Sun and the other planets. We can do the same for the Sun, Earth, and Mars.

Orbital Mechanics
A diagram of Earth’s and Mars’s spheres of influence, not to scale. Whether the Sun has a sphere of influence is ambiguous, because our stellar neighbourhood has a complex gravitational field (image by author, CC BY 4.0.)

This is not a good enough approximation to plan a real interplanetary mission (you need numerical methods for that), but it can help us get a rough idea of how tough the mission is.

If astrodynamics and mission planning fascinates you beyond the level explored in this installment, perhaps you should consider pursuing a university-level education in physics (or purchasing a copy of the video game Kerbal Space Program)! We will return to regular, less mathematical content next installment.

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