Introduction

According to Wikipedia, Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to the practical problems concerning the motion of rockets and other spacecraft. In simpler words, it’s the field of study that allows us to calculate where and when something travelling through space will end up, and the changes in trajectory it’ll experience from a self-generated impulse (read: its rocket engines) or an external body (like a moon or planet for example), usually using Newton’s law of gravitation.

You see, getting from point A to point B in space isn’t as simple as just pointing at where you want to go and firing your engines. It isn’t even as simple as pointing at where your target will be in the time you will take to get there. You have to take into account both how you and your target will be curved by the parent body’s gravitational field, the fact that your velocity will be going down the entire time you’re moving away from the planet, and many, many other factors that make calculating orbital trajectories really hard.

In this course, though, we’re not going to be looking at how these trajectories are actually calculated. We’re going to be more focussed on how various parameters of an orbit affect its shape and other characteristics, and how a spacecraft trying to navigate through space can take advantage of various factors that allow it to save fuel and get to its destination more efficiently. I’m just going to say it off the bat: if you’re actually interested in orbital mechanics, the best way to get a good intuition for it is to play Kerbal Space Program, so consider doing just that.

Variation of speed and period with altitude

As discussed in the first module of Aerospace 101, we only looked at orbits from a very restricted one-dimensional view. While this was good to gain an intuition of what a basic orbit is, it doesn’t give you a good idea of how more complex orbits would look and behave. I am assuming here that you have a good basic understanding of what keeps a spacecraft in orbit, so if you’re still uncomfortable with the topic, I would recommend brushing up on it, perhaps by watching this excellent video I recently found by a rather small but great channel.

Let’s start off with the most basic parameter – altitude. As you (should) know, altitude refers to the height of the orbit above some point, commonly either the centre of mass of the celestial body or its surface. The most obvious effect of increasing altitude is reducing gravitational acceleration. According to Newton’s law of gravitation, the force of gravitation (and by extension the acceleration due to it) is inversely proportional to the square of the distance between the spacecraft and the celestial body. So, as altitude increases, the gravitational force rapidly decreases. This might not be apparent at lower altitudes since the radius of the Earth dwarfs changes of a few kilometres in altitude, but it is really obvious for larger changes.

Altitude directly affects the velocity required by a spacecraft to remain in a stable orbit. At lower altitudes, the acceleration due to gravity is relatively high, and consequently the speed at which a spacecraft must move laterally in order to keep “missing” the Earth is also high. However, as the altitude of the orbit increases, the acceleration due to gravity drops off as the distance squared. So, the spacecraft needs to travel sideways less and less quickly to maintain an orbit. This is why whereas a spacecraft in Low Earth Orbit or LEO needs to travel at nearly 8 km/s to remain in circular orbit, the Moon (or in fact any other object orbiting Earth at its altitude) only needs to move at about 1 km/s to maintain its circular orbit.

Three spacecraft on different circular orbits showing off the decrease in speed with increase in altitude

Another parameter that is affected by altitude (both directly and indirectly) is orbital period. As its name suggests, the orbital period of an object refers to the time it takes for it to complete one full orbit around its parent body. The easiest way to understand this is to just take the example of Earth – its period is 365.256 days, or about one year. In that time, the Earth will have returned back to the location it started at after completing one full orbit of the Sun. Its variation with altitude is also pretty obvious – as the height of the orbit increases, the orbital velocity of the spacecraft decreases and the distance it needs to traverse to complete an orbit increases. So, it’s pretty easy to guess what’s going to happen to the orbital period.

As a general rule, the farther the object is from the body it is orbiting, the longer is its period. Mercury has a period of 88 days or about 0.24 Earth years – Neptune on the other hand has a period of 165 Earth years. Similarly, the ISS at an altitude of roughly 400 km above the surface of Earth has a period of about an hour and a half – only slightly longer than the average trip home in Delhi. On the other hand, the Moon, at an average of 384,000 km from Earth, takes nearly a month to complete one orbit. And this isn’t anything special about the Moon – an object placed further out from Earth than the Moon would have an even longer orbital period. A spacecraft orbiting 1 million km from Earth would take 116 days to complete a single orbit.

Non-circular orbits

So far our discussion has been limited to circular orbits. Circular orbits are simple to deal with and easy to understand, since there is no variation in altitude or velocity over the length of the orbit, and so everything is uniform. You may have realized, however, that circular orbits aren’t the only stable orbits. What if you were just above or just below the velocity of a circular orbit at your altitude? What if you were to thrust facing directly away from the body you are orbiting? The force of gravity would still work and pull you down, but in what way?

Let’s consider a spacecraft in orbit of a celestial body (Kerbin in this case since it makes things easy for me) at a height of 100 km above the surface.

If this spacecraft were to accelerate along the direction of motion of its orbit (known as prograde), what do you think would happen? Well, initially, the spacecraft’s velocity would increase, but its altitude would remain the same. Soon, it would start moving sideways faster than it needs to in order to maintain its distance from Kerbin. Can you guess what this will do?

Let’s reason through this together. As the spacecraft accelerates, it is increasing its tangential velocity. This means that it is moving its trajectory closer towards a straight line by reducing the influence of Kerbin’s gravity on it relative to its own velocity. Since the spacecraft is following a straighter path, but the planet below is curving away at the same rate, its altitude will start rising.

Notice that in the second screenshot, the spacecraft is moving faster and its trajectory is straighter and separating from the planet

Now, as the spacecraft continues to move along its trajectory, it will experience a change in how gravity is acting on it. Whereas earlier, the entire force of gravity was acting purely perpendicular to the spacecraft’s velocity and hence not affecting it, now some component of that gravitational force is pulling the spacecraft back, slowing it down. This means that even as the altitude of the spacecraft rises, it is losing velocity and its rate of increase of altitude is accordingly dropping.

The result is that over time, the spacecraft exchanges its kinetic energy for potential energy as it gains altitude, all the way up to a point where the vertical component of its velocity once again becomes zero. Here, the spacecraft is at the highest point of its new orbit, known as its apoapsis. Now, since the spacecraft is travelling slower than the orbital velocity for this altitude, it will start falling back towards the planet, re-exchanging its potential energy for kinetic energy.

By the same mechanism it lost velocity going up, the spacecraft now gains velocity as it loses altitude. In fact, the trajectory it follows now is an exact mirror image of the trajectory it followed on the way up. So, it falls back exactly to the point where it earlier fired its engines to raise its apoapsis, (the lowest point of the orbit or the periapsis) at exactly the same velocity it attained after firing its engines. Now, the process repeats and the spacecraft continues to circle Kerbin.

Just by the way, (as you may have guessed from the image) the lowest part of the orbit is called the periapsis of the orbit. Also, sometimes the apsis part of both apoapsis and periapsis is replaced by a representative for the body the spacecraft is orbiting. This is why for example you might hear these points being referred to as “apogee” and “perigee” when talking about Earth-orbiting spacecraft, or “aphelion” and “perihelion” for planets and asteroids.

Defining the shape of an orbit

Now that you’re (hopefully) comfortable with the idea of elliptical orbits, you may be wondering how we can rigorously define the shape of these orbits. While altitude is perfectly fine for circular orbits, it kind of breaks down when talking about elliptical orbits. And although apoapsis and periapsis are fine for us here, defining these orbits in terms of shapes we know and understand is extremely useful when doing calculations regarding them. The contents of this section aren’t absolutely-need-to-know, especially if you have studied conic sections, but I still recommend you go through it so that you know what these terms are if you see them out in the wild somewhere.

First up is semi-major axis. This value represents half of the distance between the apoapsis and periapsis of the orbit of a spacecraft. For a circular orbit, this value is the same as the height of the orbit above the centre of the planet, but for an elliptical orbit it’s mostly unusable for anything on its own. This is because in an elliptical orbit, the parent body is actually located at one of the orbit’s foci, and so it isn’t at its centre, as you can see in the previous and next diagrams.

The speed of an object in an elliptical orbit varies greatly with its position – it travels faster near its periapsis than it does near its apoapsis. This means that it also takes varying amounts of time to travel through these sections of the orbit, which conventionally would make calculating its period really hard. However, semi-major axis makes it really easy to calculate the period through a formula that sets it proportional to the 3/2th power of orbital period, which is very convenient.

Let’s move on to eccentricity. This is essentially a number that gives you the elliptical-ness of the orbit. In other words (Wikipedia’s words), it determines the amount by which the orbit of an object around another body deviates from a perfect circle. Combined with semi-major axis, eccentricity allows you to define the shape of any orbit, as well as non-orbital parabolic and hyperbolic trajectories that don’t keep the object in a closed orbit and instead shoot if off to infinity.

An eccentricity of zero means that the orbit is perfectly circular. Anything between zero and 1 is an elliptical orbit, while an eccentricity of 1 is a parabolic escape trajectory; basically a trajectory that has the exact amount of energy an object needs to escape to infinity. Above eccentricity 1, the orbit is hyperbolic, which means it has enough energy to carry the object to infinity and still have velocity left over when it’s completely free of the parent body’s gravitational field.

Orbits in three dimensions

Using semi-major axis and eccentricity, we can define the shape of an orbit just fine, but we’re missing something. An orbit doesn’t necessarily have to be equatorial like the one we used in the example for the elliptical orbit above. It can be angled (or to use another word, inclined) to the equator at any angle, and can also be made to cross the equator at different longitudes. This means that there are a whole host of ways in which the orbit can be oriented around the parent body, and we need some other parameters which will help us define this orientation.

First, however, let’s try to visualise what the inclination of an orbit even refers to. Although we do have Kerbal Space Program to help up out in this aspect, it’s still good to reason through why an inclined orbit looks and behaves the way it does since that will help a lot in understanding many things related to inclination that may not be obvious otherwise, such as the fact that it is not possible to launch a rocket into an orbit with an inclination that is less than that of the launch site.

Let’s ignore the problem of changing inclinations for now and consider a spacecraft that is above the equator already in a 45° inclination orbit at its ascending node or travelling towards the North East. This “45°” means that the spacecraft’s velocity is at an angle of 45° with respect to the equator. So, as time passes, the spacecraft will initially “rise” away from the equator at an angle of 45°. However, as it continues to travel and the planet curves inwards below it, the angle it makes with the respective latitude lines it is passing over will start decreasing, all the way up to the point when it reaches 45° N, where it will be travelling parallel to the latitude line below it.

A roughly 45° inclined orbit around Kerbin – the flag you can see on the surface there sits on the equatorial line

After this, the spacecraft’s latitude will start “falling” until it once again crosses the equator after half of its orbital period has elapsed, this time crossing it at an angle of 45° going South East. Now, the spacecraft will go through the same cycle again, dropping in latitude all the way down to 45° S, becoming parallel, rising again and finally crossing the equator after one full orbital period, again on the same heading it was on earlier. This how the spacecraft would move over one full orbit. The higher will be its inclination, the higher will be the latitude it reaches before falling back towards the equator and the closer to the poles it will travel.

Notice that this means that it is impossible to calculate the inclination of an orbit by knowing just the cardinal direction of the orbit at any given time (except for a polar orbit), since the direction varies with latitude. In order to have a complete picture of the orbit’s inclination, both the latitude and direction must be known. What this also means is that the inclinations available to any spacecraft in orbit or on the launch pad is limited to the latitude of its location and higher. This is because going directly East at any given latitude will be equivalent to being at the highest deviation from the equator in the orbit, meaning that the spacecraft will “fall” towards the equator.

One thing that’ll become apparent if you start thinking about inclined orbits is that there’s actually a whole continuum of planes in which an orbit inclined at a particular value could be located. A perhaps simpler way to realize this is to just look at the points where the orbit crosses the equator (known as the ascending and descending node). Even while making the same angle with the equator, an orbit can be translated across such that the ascending and descending nodes get shifted from their original positions. This new orbit has the same inclination as the previous one, yet it is in a completely different plane relative to the planet.

Take a look at these four orbits in the foreground – they are all 30° inclination orbits since that’s the angle they make with the equator, yet they differ greatly in the points at which they cross the equator as well as their orientation relative to each other

The parameter used to define this orientation is the longitude of the ascending node. Formally, the ascending node is the point where the orbit of the object in question crosses a reference plane, like the equator in our case. The longitude of the ascending node or LAN as it is sometimes called is the angle at which the ascending node is located with respect to a reference longitude. This reference longitude is relatively arbitrary (like all longitudes really), so the actual angle is mostly useful for comparing two orbits. In the case of the Earth and the Sun, the First Point of Aries is used as the reference longitude since it’s relatively stable and easy to locate.

Here, the horizontal grey orbit is the reference plane, and if the yellow orbit is taken as the reference longitude, the blue orbit is has a longitude of ascending node of 30° i.e. the two orbits have a difference in longitude of ascending node of 30°

So now we can accurately define the size and shape of any orbit as well as the plane in which it is embedded. However, there’s still one thing we are missing. While the aforementioned parameters are enough for circular orbits, in the case of elliptical orbits there’s some additional variation that shows up. If you consider a spacecraft in a highly eccentric orbit, you will notice that there are different points where the periapsis of the orbit can be located, giving another whole continuum of orbital orientation. To best visualize this, take the example of two spacecrafts in eccentric polar orbits. Even if the semi-major axis and eccentricity of the orbits are identical, it is possible for them to be positioned completely differently by one of the orbits having its periapsis over the North pole while the other has its periapsis over the South pole. In this case, the orbits are (quite literally) polar opposites, despite having identical semi-major axis, eccentricity, inclination and longitude of ascending node. To define the orbit’s orientation in this last ring of freedom, we use a parameter known as the argument of periapsis.

In the above case, the yellow and blue orbits have identical orbital parameters except for argument of periapsis, which is 180° apart

The argument of periapsis is defined as the angle between the ascending node and the periapsis of the orbit, going along the direction of motion of the object in question. In the above image, the ascending node for both orbits is the orange line on the right side. That means that the argument of periapsis is 90° for the blue orbit and 270° for the yellow one (since the spacecraft started moving from their respective periapsides). Unlike longitude of ascending node, this is a value unique to the orbit irrespective of the reference frame it is measured from, and so it’s absolute value is actually pretty useful in defining and studying elliptical orbits. For circular orbits, however, argument of periapsis doesn’t really mean anything since there’s no real apoapsis or periapsis, and so in the case of circular orbits this value is usually just assumed to be zero.

These five parameters – semi-major axis, eccentricity, inclination, longitude of ascending node, and argument of periapsis – altogether allow us to uniquely define the size, shape, and orientation of any orbit in standard two body Newtonian mechanics. This is extremely useful as it gives us the ability to describe orbits in the language of mathematics, something that is essential for being able to understand and predict things regarding any phenomenon.

Navigating in space

Now that you have a good grasp of what exactly orbits are and how they can be described, let’s move on to how we can move spacecraft between different orbits. You already got a glimpse of how this works during the section about elliptical orbits, but here we’re going to go over the effects of thrusting in all directions, not just in the direction of motion of the spacecraft.

  1. Prograde and retrograde: Prograde refers to the vector along the orbit in the direction of motion of the spacecraft. Conversely, retrograde refers to the vector along the orbit in the direction against the motion of the spacecraft. These two are arguably the most important directions in spaceflight, since they give the spacecraft the oomph it needs to get wherever it wants to go. By thrusting prograde or retrograde, the spacecraft puts all of its thrust into changing its orbital velocity. This rapidly changes the apoapsis, periapsis, semi-major axis, eccentricity and period of the orbit. Thrusting along either of these vectors affects the exact opposite side of the orbit the most, with a prograde burn pulling the periapsis towards the location of the burn and vice versa for a retrograde burn.

    Quantities unaffected: Inclination, LAN
    Greatest effect on: Apoapsis, periapsis
    Most efficient at: Apoapsis (retrograde), periapsis (prograde)
  2. Normal and anti-normal: Normal and anti-normal refer to the vectors that are in-plane with prograde and retrograde but perpendicular to them. Both of them have the same effect on the orbit of the spacecraft – changing its inclination. On thrusting in either of these directions, the spacecraft is leaving its lateral orbital velocity at that instant untouched while introducing a new component of velocity that points in the direction the spacecraft is thrusting. This pulls the ascending and descending nodes towards a position where they will be exactly a quarter-orbit away from the spacecraft. Since the overall magnitude of orbital velocity is greater nearer to the parent body, changing the inclination is extremely inefficient at low altitudes. The opposite is true at high altitudes.

    Quantities unaffected: SMA, eccentricity, argument of periapsis
    Greatest effect on: Inclination
    Most efficient at: Low orbital velocity/high altitude
  1. Radial and anti-radial: These vectors are called by pretty varied names, but as Kerbal Space Program uses the terms radial and anti-radial, we’ll stick with those. These are the directions oriented perpendicular to both the prograde and retrograde plane as well as the normal and anti-normal plane. The radial vector points inwards towards the center of the planet (though not at it in elliptical orbits) while the anti-radial points in the other direction. Thrusting in either of these directions directly affects the spacecraft’s vertical velocity. The effect this has depends on the location of the spacecraft along its orbit, but in general this translates the apoapsis and periapsis around the orbit.

    Quantities unaffected: Inclination, LAN
    Greatest effect on: Eccentricity
    Most efficient at: N/A

Okay, so that’s a whole heap of information. How does it help you actually navigate between orbits? Well there’s a few rules of thumb you can keep in mind.

  • Thrusting prograde from a circular orbit will raise your apoapsis, making your orbit eccentric and increasing its period. If you want to circularize this orbit at the new altitude, you will need to perform another prograde burn at apoapsis to raise your periapsis to the same altitude as your apoapsis. This is known as a Hohmann transfer maneuver. The same procedure applies for reducing altitude, except with retrograde burns instead.
  • As an example, if you wanted to travel to the Mun in KSP, you would do a long retrograde burn to raise the apoapsis of your orbit to a height roughly matching the Mun. Here, you need to also make sure that you burn at a time such that when you arrive at the apoapsis of your orbit, the Mun is in roughly the same place and able to capture your spacecraft. As you pass by the Mun, you will have to do another burn at Mun periapsis to bring your apoapsis down to a stable Mun orbit. This is also a Hohmann transfer maneuver at heart. For leaving the Mun, you want to do the exact same thing but in reverse, doing a prograde burn to raise your apoapsis to infinity (relative to the Mun).
  • Normal and anti-normal burns are used for changing the plane of your orbit. As mentioned before, it is extremely inefficient to perform this maneuver at low altitudes, to the point where it takes less fuel to first do a prograde burn to raise the apoapsis to a considerable height, coast to the top and then perform your normal/anti-normal burn so that it is very short due to the low orbital velocity the spacecraft has at apoapsis.
  • A combination of prograde/retrograde, normal/anti-normal, and radial/anti-radial components usually make up a correction burn that happens en route to intercepting another object or body. Here, the spacecraft thrusts in a direction not aligned with any of these vectors to bring its trajectory closer to its target, whether that be by physically reducing the approach distance or changing the time of approach.

This is really a lot to take in, so I recommend that you take this moment to fire up Kerbal Space Program, press ALT + F12 to bring up the debug menu, enable unlimited fuel using the cheats tab, and then take a high TWR spacecraft into orbit and simply try thrusting in different directions and note the effect this has on your orbit. I am not joking when I say that this will give you a better intuition for orbital mechanics than any course at any level ever could. Being able to see live how an impulse delivered in a particular direction morphs the shape, size and orientation of an orbit, and then reasoning through exactly why it had the effect that it did is quite frankly enlightening. If this doesn’t make you fall in love with physics, I honestly don’t know what will. Good luck!

Further resources

  1. Wikipedia page on orbital elements — contains links to the individual parameters so you can read about them in more detail if you like, including calculating them.
  2. Scott Manley’s video on some of the unintuitive parts of orbital motion — discusses many things that may not be clear from this document; highly recommended.
  3. A simulator for the planetary motion of the solar system — especially recommend taking a look at the Mars missions section for their depiction of the Hohmann transfer.
  4. Quora thread on the Oberth effect — this is something that isn’t covered above but is really useful to know to understand some of the planning choices that are made.

3 Comments

Akshit Kumar · June 26, 2019 at 9:50 am

https://xkcd.com/1356/

Zayan Jamal · June 27, 2019 at 9:33 pm

I saw the 3rd week courses.I browsed through it but I could not find any questions related to it.Have the questions been released?If yes,then please send them on my given email address.Thank you

Aryan Gulati · July 3, 2019 at 3:58 pm

The form for the rendezvous assignment has still not been released so how are we supposed to submit the ‘Intro to Orbital Mechanics’ assignment (as today is the last day)?
Aryan Gulati
Class : 11G

Leave a Reply

Your email address will not be published. Required fields are marked *