This will be a pretty short post, at least by the standards I’ve set so far. I just want to make sure we’re all on the same page regarding how orbits function and what terms we use to describe them. Probably the easiest way to learn this is by playing a game like Kerbal Space Program, but that’s not everyone’s preference and they never bother defining a lot of the important terms. So:

Concept

Orbital Elements Back to Part II

Concept





The classical thought experiment to understand an orbit is Newton’s cannonball. Imagine that you have a cannon standing in a perfectly flat plane with the barrel perfectly parallel to the ground, and that there is no air resistance to worry about. To start out with, imagine that we let a cannonball tip out the end of the cannon. It has zero initial horizontal velocity and no horizontal forces acting on it, so it doesn’t move horizontally. It also has zero initial vertical velocity, but it has gravity acting on it to pull it down, so it gains vertical velocity and fall downward, hitting the ground after a specific period of time determined by its initial height and the acceleration of gravity.





Cannonball at the start and end of its flight. The red arrow shows the acceleration by gravity, the purple arrow the velocity at the end of the flight (and the start in later images) and the blue line shows the path of travel. Image by me; can you tell?





Now imagine that when we fire the cannonball, giving it some initial horizontal velocity but no vertical velocity. Once the cannonball leaves the barrel, it still has no horizontal forces acting on it and only gravity acting on it vertically, so it will take just as long to hit the ground as the first cannonball. But it continues to move horizontally as it falls, so it hits the ground at a different point.









Keep increasing the initial horizontal velocity, and it never increases the time it takes to hit the ground—but it does allow the cannonball to travel farther before it does.









Next imagine that the cannon is not on an infinite flat plane, but rather on a perfectly spherical planet. As the cannonball travels, the ground curves away, so in effect the cannonball has to travel a greater vertical distance to hit the ground the farther it travels from the cannon.









But of course the direction of “down” is changing as the ground curves, so the direction of gravitational acceleration changes as well. If we rotate our “vertical” and “horizontal” axes to compensate, then in effect, some of the cannonball’s horizontal velocity becomes upwards vertical velocity, and some of the downwards vertical velocity becomes more horizontal velocity. In this rotating reference frame, it’s as if there’s a constant upwards acceleration (centrifugal acceleration) opposing gravity.





Path of a cannonball fired at 6 kilometers/second from a high perch on Earth. Lookang, Wikimedia You can also play with making your own paths here.





Thus by increasing the horizontal velocity we can keep the cannonball aloft for longer, allowing it to travel further around the planet. As we keep increasing the initial velocity eventually we reach a point where the ground is curving away from the cannonball as fast as it falls—in the reference frame of the cannonball, centrifugal acceleration has matched gravitational acceleration—and the cannonball travels all the way around the planet and smashes into the back of the cannon (if the planet isn’t rotating). If we diligently moved the cannon out of the way before this happened, the cannonball would continue circling the planet forever (or as good as), losing neither altitude nor velocity. That is an orbit.





Cannonball fired at 7.3 km/s





Because the acceleration due to gravity falls with distance, the velocity necessary for such an orbit decreases with greater radius from the center of the planet. Near the planet’s surface the velocity can be quite high; orbital velocity just above Earth’s atmosphere is about 7.8 kilometers/second. This gives an object there an orbital period of 88 minutes. At greater orbital radii the orbital velocity drops and the length of the orbital path increases, leading to greater orbital periods. At some orbital radius—42,164 kilometers for Earth, which is an altitude of 35,786 km above the equator—the orbital period will match the rotational period of the surface, meaning the orbiting object will not appear to move west or east in the sky (though it may move north or south unless it orbits directly above the equator; all orbits pass over the equator). This is a geosynchronous orbit.





Returning to our thought experiment, consider what happens if we increase the initial velocity even more. Now the cannonball gains altitude as it travels. However, moving away from the planet requires the cannonball to lose kinetic energy—put another way, the rate at which the “down” direction is changing slows as the cannonball gets further from the planet, so centrifugal acceleration decreases until it falls below gravitational acceleration—and the upward velocity falls. At the exact opposite side of its orbit, across the planet from the cannon, the cannonball has lost so much velocity that it stops rising and instead starts falling back towards the planet. But in doing so it gains energy back, and after one more half-orbit it returns to the position of the cannon with the exact same velocity it started with.





Cannonball fired at 8 km/s





So this is still an orbit, with all energy ultimately conserved, but rather than a circle the path of the cannonball marks out an ellipse, with the center of the planet located at one of the foci. Kepler’s equations describe the movements of objects on these elliptical paths, but for now just remember that A, an orbiting satellite moves faster the closer it is to the parent body, and B, a satellite acted on only by the gravity of its parent will always stay on the same elliptical path.





Note that Kepler’s equations only work for a 2-body system with idealized point masses. Once you introduce a 3rd body with significant mass, they no longer apply (the equations also assume that the larger body remains stationary, which isn’t strictly true but is close to true if the bodies are of very different masses and can be compensated for where this is not the case). However, in most cases we find that the orbit of two nearby objects can be closely approximated as 2-body systems, which is fortunate because these systems have an analytical solution; that is, if you know the masses, positions, and velocities of both bodies at any one time, you can precisely calculate their positions and velocities at any other point of time. The same is not true for 3-body systems, and they are chaotic, meaning that small differences in initial position and velocity can lead to wildly different outcomes. We can never predict the outcome of these systems with perfect accuracy, though with powerful computers we can come fairly close.





There are a couple factors common to all elliptical Keplerian orbital paths, so long as no additional forces act on the orbiting object:

The object orbits within a plane that passes through the center of the parent body.

The object follows the same path with every orbit, and always takes the same amount of time.

The object moves fastest at closest approach to the planet and slowest when furthest away.

If any additional forces do act on the object, this will always change its orbital path, but the path will only continue to change so long as the force is acting.

Altogether, this means that an object in orbit is not in some fine equilibrium that it can easily be knocked out of; the velocity can change significantly and still leave the object in a perfectly stable orbit.





Before we move on, there is one more trajectory the cannonball can have besides circular and elliptical. As we keep increasing the initial velocity, the elliptical path becomes ever more stretched out, and the period required for the cannonball to complete one orbit ever longer. As the cannonball gets further from the planet, gravity becomes weaker, and it takes longer to accelerate the cannonball back towards the planet. Eventually the initial velocity is great enough that the cannonball leaves the planet so quickly that, even given infinite time, gravity can never completely stop the cannonball. In fact, the gravity can never remove more than a finite amount of the cannonball’s velocity. The cannonball never returns, and so follows a hyperbolic trajectory.





Cannonball fired at 11.2 km/s





The velocity required to achieve this is termed escape velocity, and though it depends on the total mass of the system and initial distance between bodies, it is independent of direction (so long as that direction doesn’t cause the bodies to collide). Even if a satellite starts with an escape velocity towards the parent, it will pick up enough speed as it falls to shoot back out and escape. In reality, if we have an object on a hyperbolic trajectory escaping from Earth, it will soon deviate from that trajectory and enter an elliptical orbit of the sun.





To save you all some time, here are the equations for the velocity necessary for a circular orbit and escape for given radii and system mass. If you set the altitude near the surface of a body, these should roughly tell you the delta-v needed for a spacecraft to reach orbit and escape that body (I’ll explain delta-v and other aspects of spacecraft navigation another time, but probably not for a while; for now I refer you to Atomic Rockets ). Note in both cases that if a satellite is much less massive than a parent body, the parent body’s mass can be used for the system’s mass and the satellite’s altitude plus the radius of the parent body can be used for the radius of the orbit.





v

= orbital velocity (meters/second)

G = gravitational constant; 6.67408*10-11 m3 kg-1 s-2

M = total mass of the system (kilograms)

r = radius from the barycenter (meters)

(unlike escape velocity this is not independent of direction, and must be perpendicular to the direction to the barycenter)



v e

= escape velocity (m/s)

otherwise as above

(so it’s about 1.41 times the circular orbit velocity)

Orbital Elements

The size and shape of an elliptical orbit can be described with two values:





Semi-Major Axis (a): Half the length of the ellipse on its longest axis. For a circular orbit, this is the distance between the center of the two bodies. When people say orbital distance, radius, or separation, they are generally referring to this value.





Eccentricity (e): A measure of how elongated the orbit is. e = 0 is a circular orbit, 0 < e < 1 is an elliptical orbit that is more elongated for higher values of e, e = 1 is technically a parabola but really just never occurs, and e > 1 is a hyperbolic escape trajectory. More precisely:





r a

= greatest separation between bodies

r p = least separation between bodies



From these values we can define two important points in the orbit:





Apoapsis (r a ): The point of greatest separation. Also refers to the distance between the bodies at that point. A.K.A. apocenter, apogee (for orbits around earth), aphelion (for orbits around the sun), apastron (for orbits around any star). Pretty much every major solar system body has its own term that is rarely used (perhaps we should use "apotea"). r a = a (1 + e)





Periapsis (r p ): The point of least separation, or distance at that point. Has all the same variations as above. r p = a (1 – e)





One more value helps us understand the orientation of an orbit relative to the parent body:









Inclination (i): The angle between the plane of the orbit and a reference plane. For orbits around planets and similar bodies this reference is the equatorial plane of the planet, but for orbits around the sun we tend to use the ecliptic, the plane of the Earth’s orbit (so-called because the moon passes through it during eclipses) even though this is inclined from the sun’s equator by 7.25°. For orbits around the Earth, at i = 0° an object remains permanently over the equator and moves east over the surface, such that when viewed from above the north pole it is moving counterclockwise around the planet; this is referred to as prograde motion. At i = 60° the object still moves west but also oscillates north and south until it is over points at 60° north or south latitude (that is, directly above them, so a high-orbiting satellite can be above the night side and still see over the Earth to the sun). at i = 90°, the object passes directly over the poles. If there is no simple fraction between the object’s orbital period and the planet’s rotational period, it will eventually pass over every point on the surface. At i > 90°, the object will move west over the surface, and is said to be in a retrograde orbit (technically objects with i < 90° in high orbits can also move west because their orbital motion lags behind the surface’s rotation, but they’re still classified as prograde). Otherwise they’re similar to orbits with inclinations equal to 180° - i, though once you take the other elements into account they’re orientated differently. At i = 180°, the maximum, an object orbits west over the equator.





We can now define two more important points in the orbit:





Ascending node : The point where the object passes through the reference plane heading “up”; i.e., where it passes over the equator heading from the southern hemisphere to the northern hemisphere.





Descending node : As above, but heading “down”, from north to south.





This tells you most of what you need about any given orbit. If you’re told these elements—semi-major axis, eccentricity, inclination—for a planet in a given star system, you can reasonably ballpark its habitability, surface climate, and likelihood of encountering other bodies. But if you want to know exactly how an orbit is orientated, such that you can compare it to other nearby orbits, you need two more values:









Longitude of the ascending node

(Ω): The angle between a reference direction (within the plane of the reference plane used to measure inclination) and a line connecting the barycenter to the ascending node, measured counterclockwise from the reference (as viewed from “above”). The standard reference direction in the solar system is towards the First Point of Aries, which is the same as the direction the sun appears from Earth during the vernal equinox (or it was when this system was defined). A.K.A right ascension of the ascending node, or RAAN, when used with this reference. For an orbit with i = 0°, the position of the ascending node is undefined, but by convention this is set to Ω = 0°.





Argument of periapsis (ω): The angle from the ascending node to the periapsis, moving in the direction of the objects motion, and within the plane of the orbit, not the reference plane. Has all the same name variations as periapsis.





This all tells us the shape and orientation of the orbital path, but if we want to know the position of an object within this path, we need one more value:





True anomaly ν, θ, or f): The angle from the periapsis to the object’s position, along its direction of motion. Often we define a reference time, called the epoch, and then define the “true anomaly at epoch”, allowing the object’s position to be calculated for any other time. This works even if the object wasn’t in its current orbit at epoch, because we can just extrapolate the current orbit backwards. By convention astronomers use a common epoch called J2000, corresponding to 12:00:00 midnight on January 1, 2000 on the Gregorian Calendar. Some people will use mean anomaly instead, which is similar but not the same (it’s the angle to where the object would be in a circular but otherwise identical orbit—thus it changes at a constant rate). They can be converted into each other, but it’s , or): The angle from the periapsis to the object’s position, along its direction of motion. Often we define a reference time, called the, and then define the “”, allowing the object’s position to be calculated for any other time. This works even if the object wasn’t in its current orbit at epoch, because we can just extrapolate the current orbit backwards. By convention astronomers use a common epoch called, corresponding to 12:00:00 midnight on January 1, 2000 on the Gregorian Calendar. Some people will useinstead, which is similar but not the same (it’s the angle to where the object would be in a circular but otherwise identical orbit—thus it changes at a constant rate). They can be converted into each other, but it’s annoying math I won’t reproduce here .





These 6 elements can define any 2-body orbit, and indeed no such orbit can be defined with less than 6 values. So if we want to construct a planetary system that can be loaded into simulations, we have to define these elements for every body within that system, along with their masses. Other characteristics of these bodies—radius, composition, etc.—can all be left for later if we desire.





When discussing a planet, however, there’s two more elements to its motion we should bear in mind:









Obliquity (ε): The angle between the object’s equatorial plane and the plane of the orbit, A.K.A axial tilt. As with inclination, there’s a convention regarding direction of spin, whereby an object with ε < 90° spins in the same direction as it orbits and is said to have prograde rotation (or maybe it’s the same direction as most of the system orbits? I’m not sure what the convention is for describing the spin of retrograde-orbiting objects) and an object with ε > 90° spins in the reverse direction and is called retrograde.

Argument of obliquity : The angle between a line from the object’s center to the center of its orbit (i.e. the star) at periapsis and the direction from the object’s center to the north pole within the orbital plane (so changing the obliquity to any nonzero value shouldn’t change this value) measured counterclockwise as viewed from above (in the direction of rotation for a prograde-rotating planet). If this value is 0, summer solstice in the northern hemisphere coincides with the periapsis; for greater values it will occur later in the year. This is only one of several conventions for describing an object’s rotational axis, but it’s the most convenient for use in this blog.



