The Keplerian Orbital Elements and Orbital Perturbations


In our last post, we constructed an abstract solar orbital hierarchy. This hierarchy contains no explicit orbital information: just the hierarchical structure of a solar system.

In order to ascribe actual orbital data to the various stars and barycenters of a solar system, we will need a way to describe orbits in PISES. In this post, we’ll get spun up on the 6 Keplerian Orbital Elements and what their limitations are.

The Keplerian Orbital Elements

An orbit can be fully described by the five Keplerian elements, named after Johannes Kepler who established three laws of planetary motion:

  1. The orbit of each planet is an ellipse, with the sun at a focus.
  2. The line joining the planet to the sun sweeps out equal areas in equal times.
  3. The square of a period of a planet is proportional to the cube of its mean distance from the sun.

In essence, all orbits are ellipses, and to fully describe an orbit, we must not only describe the shape of the ellipse, but also its orientation in space: two ellipses of the same shape may be both rotated or inclined at different angles.

To fully describe an orbital ellipse, we require the following five orbital elements:

  1. Eccentricity describes the shape of the ellipse. An ellipse with an eccentricity of 0 is a circle (red). An ellipse with a higher eccentricity becomes more elongated (purple).

2. The Semimajor Axis is the radial distance from the center of the ellipse to the furthest point along its curve. The semiminor axis is the radial distance from the center of the ellipse to the nearest point along its curve. The semiminor axis is not an orbital element because by using the semimajor axis and the eccentricity, we can always calculate the semiminor axis.

3. The Inclination Angle is the vertical tilt of the ellipse with respect to a referential plane: in the case of the earth, for example, the equatorial plane.

4. Now that the orbit is inclined, it has two nodes: the two places where it intersects the referential plane. At one of these nodes, the planet will be ascending (piercing the plane in an upward direction) and at the other node, the planet will be descending (piercing the plane in a downward direction). The Longitude of the Ascending Node is the angle that this ascending node creates with the chosen reference frame.

Basically, imagine that we are taking the inclined ellipse, and pinning it at its focus to the body that the orbit is around. Then, from above, looking down at the ellipse, we are rotating it like the hand of a clock. In this example, a few degrees counter clockwise.

5. The Argument of Periapsis is the angle drawn between the ascending node, the orbital focus, and the periapsis.

The Periapsis is the point in the satellite’s orbit in which it comes closest to the orbital focus. Alternatively, the Apoapsis is the point in a satellite’s orbit in which it is the furthest from the orbital focus.

Using our example above, we’ll highlight the argument of periapsis.

For the sake of example, let’s try raising the Argument of Periapsis to 135 degrees.

Basically, imagine we are taking the ellipse (which at this point we have both inclined (via inclination) and rotated (via the Longitude of Ascending Node)), Striking it through its focus with a pencil, and are now rotating the ellipse orthogonally around this pencil.

These five orbital elements (Semimajor Axis, Eccentricity, Inclination, Longitude of Ascending Node, and Argument of Perigee) are sufficient to completely describe any idealistic, two-body Keplerian orbit.

However, there is one thing that we have not described: the position of the satellite in this orbit. This is described by a sixth orbital element known as True Anomaly. The true anomaly is the angle between the periapsis point, the orbital focus, and the satellite.

Perturbations, Orbital Sinks, and Graveyard Orbits

Everything we have described so far describes the behavior of a perfect, idealized two-body system. If the Earth and Moon were absolutely perfect, uniform, unchanging spheres, and the Earth and Moon were the only two bodies in the universe, and the vacuum of space was fully devoid of all energy and radiation, then the Earth’s and Moon’s motions might be able to be described using only the five Keplerian Orbital Elements above.

Ellipsoidal Earth and its Orbital Sinks

However, such idealized orbits break down very quickly in real life. To begin, no astronomical body is a perfect sphere. The Earth is an ellipsoidal body, and thus its gravitational pull is not uniform. In fact, the Earth has “gravitational sinks” along its orbital plane toward which unmaintained and inactive satellites are drawn. Unmaintained satellites will oscillate (often dangerously!) around these orbital longitudes. For this reason, if a satellite expects to maintain its orbital position, it is required to “stationkeep;” that is, perform routine stabilizing maneuvers to keep it from falling into an orbital sink.

The light blue circle matches Earth’s seminimor radius. The heavy blue line is an ellipse matching the Earth’s true shape. The red line is the Karman Line (the barrier between Earth’s atmosphere and space). The orange line is the orbit of the ISS.

Graveyard Orbits

When satellites are preparing to run out of fuel, they will often boost upward, above the geo belt (a particular orbital region in which geostationary satellites reside) to something called a “graveyard orbit.” This is a region of space above the geo-belt filled with dead and inactive satellites which are no longer able to maintain their safe and stable orbits. By boosting away from active satellites, these retired spacecraft can hopefully avoid putting others at risk.

Solar Pressure

The ellipsoidal nature of the earth isn’t the only perturbation effecting the orbits of its satellites. Solar Pressure yields highly measurable impacts on the orbits of satellites and even on planetary bodies (over millions of years).

Radiation pressure is the mechanical pressure exerted on any surface due to the exchange of momentum between the object and its surrounding electromagnetic field. Suffice to say: the radiation of the sun can emit measurable forces on small bodies like satellites.

Using Solar Sails to harness radiation pressure has been a science fiction concept since the 60s (“The Lady Who Sailed The Soul,” Galaxy Magazine, 1960), but has been proposed numerous times in earnest as far back as the 80s.

Some very badass retro Solar Sails.

Atmospheric Drag

Believe it or not, Earth’s atmosphere extends well beyond the moon. A cloud of Hydrogen atoms known as the Geocorona extend 650,000 kilometers away from the Earth!

Closer to the Earth, atmospheric drag plays a very significant role in the motion of satellites. Like their need to stationkeep against Earth’s orbital sinks, LEO (Low Earth Orbit) satellites often need to compensate for atmospheric drag.

Earth’s Geocorona, as seen from the Moon in 1972.

N-Body Complications

Our solar system is comprised of hundreds of objects of varying mass. The Earth’s orbit around the Sun is substantially effected by the pull of the Moon: in fact, the Earth doesn’t orbit the sun at all, it orbits its shared barycenter with the moon. This barycenter, in turn, orbits the sun.

Additionally, Earth’s orbit is impacted by other massive bodies in the Solar System, like Jupiter and Saturn.

Maintaining Orbital Data

If it isn’t clear by now, orbits are a mess. Quite simply, we will never have the computational firepower nor sufficient knowledge of the multitude of perturbing forces at play in order to fully describe or predict an orbit indefinitely.

However, by using our knowledge of solar weather, of the elliptical Earth, and of atmospheric drag, combined with visual observation, we have gotten pretty good at predicting orbital paths in the short term. In industry, a precise satellite propagation can remain coarsely accurate for about two weeks; after that, it’s pretty much toast. The satellite needs to be re-observed and re-propagated.

How PISES Handles Orbital Perturbations

It absolutely does not.

Recommended Reading

This book has been massively helpful in building my understanding of orbits and learning how to interact with them in code at both work and in PISES. The book is not that advanced; if you managed to drag your carcass through a B.S. in some form of engineering, you can get through this book.

Bate, Mueller and White’s seminal “Fundamentals of Astrodynamics.”

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