• Orbital Resonances And Synodic Period
• Star Systems, Part 2: Binary Star Systems
• Planets and Worlds, Part 2: Planetary Pairs
The ellipses below have eccentricities of (l.-r.) e = 0.25, e = 0.50, and e = 0.75.
e = 0.7507, close to that of the right-most image above.
Among exoplanets, orbital eccentricities seem to be higher on the average than those in the Solar System ; the two highest eccentricities so far discovered belong to HD 20782 b (e = 0.97) and
HD 80606 b (e = 0.9336).
Mary Anne Limbach and Edwin L. Turner  have found that there seems to be a relationship between the number of planets in a system and the magnitude of their eccentricity: the larger the number of planets in a system, the lower the eccentricity of their orbits.
Kepler's Laws of Planetary Motion
ALL orbits—be they of stars around one another, planets around stars, planets around one another, or moons around planets—are ellipses, and the following terms and equations will be integral to determining the relationships between the motions of these objects.
It was Johannes Kepler who, between 1609 and 1618, crunching the numbers on Tycho Brahe’s copious observational data, first recognized and announced the elliptical nature of the orbits of the planets, with the Sun located at one focus. (There are non-Keplarian orbits; more on that elsewhere).
Equal Areas in Equal Time
Kepler also recognized that planets in orbit around the Sun move more slowly when they are at the farthest points of their orbit (aphelion), and more rapidly when they are closest to the Sun (perihelion), and they cover equal areas of their orbit in equal amounts of time.
In other words, if one draws a triangle between any two points on the planet's orbit and the third vertex at the center of the Sun, no matter how close or far the two points on the orbit are, the area of the triangle will always be the same.
Period and Distance
Kepler's final observation concerning planetary orbits was that the time it took the planet to orbit was proportional to the planet's distance from the Sun.
P = Planetary Period (in Earth years);
a = Orbital semi-major axis (in Astronomical Units)
Simply put, the square of the planet's orbit (expressed in Earth years), is proportional to the distance of the planet from the Sun (expressed in Astronomical Units, or multiples of the distance from the Earth to the Sun).
This is the "short form" of the more complete equation:
P = Planetary Period (in Earth years);
a = Orbital semi-major axis (in Astronomical Units);
MP = mass of the planet (in solar masses);
MS = mass of the star (in solar masses)
I discuss this in more detail in the blog Designing System Orbits, Part 6: Orbital Periods.
With stars and planets, it usually happens that the barycenter is at—or quite close to—the center of the star.
When two objects of similar mass, such as two stars or a planet with a massive moon, orbit one another, the barycenter around which they orbit usually lies at some point between the two of them, and they each orbit it separately.
The equations below allow calculation of the distance to the barycenter from either of two bodies orbiting one another.
Calculating the Barycenter
Calculating separations from the barycenter
When referring specifically to the orbits of Solar System planets, the terms aphelion and perihelion are generally employed; similarly, when referring to orbits around other stars, the terms are apastron and periastron. When referring to bodies orbiting the Earth, we frequently encounter apogee and perigee. Rarely (very), when referring to objects in Lunar orbit, the terms apocynthion and pericynthion may be employed. In all cases, the prefix apo- is from the Greek, meaning "away from", and peri- from the Greek meaning "near to".
The following equations allow calculation of the distances at which these points occur.
As can be seen from the above diagram, if the maximum separation of the primary (MaxP) is less than the minimum separation of the secondary (MinS), then the orbits are nested, with the primary orbiting entirely within the orbit of the secondary (Diagram A); if MaxP is equal to MinS, then the orbits touch at one point (Diagram C); and, if MaxP is greater than MinS, then the orbits cross (Diagram B).
This condition is consequent on the chosen eccentricity of the orbits, and we can determine exactly at what eccentricity value the orbits touch by setting the equation for MinS equal to the equation for MaxP, and solving for the value of the eccentricity (e).
The final eccentricity of the system overall is not determined by this equation; that still must be selected by the worldbuilder according to preference or need. If the worldbuilder wishes the orbits of the two bodies to cross, then she will select an eccentricity of this value or above; conversely, if she wishes the two orbits to be nested, then she will select an eccentricity less than, but not equal to, this value.
If we plot the graph of mass ratios against the graph of the calculated eccentricity at which the orbits touch, we see that as the mass ratio increases, the eccentricity at which the orbits touch decreases (remember that an eccentricity of e = 0 indicates a circle, but at e = 1, the ellipse “breaks open” and the path becomes a parabola).
Note that there is a point somewhere between 0.4 and 0.45 at which the two plots intersect. We can calculate what that exact value is.
So, the point at which the two plots cross in the graph has a value of e = √2 – 1 (or ~ 0.414214, which means that if the mass ratio of the two bodies is equal to √2 – 1, then the eccentricity of their orbits will also be √2 – 1, at the point where the orbits begin to cross one another.
Again, note that this is not saying that two bodies with a mass ratio of √2 – 1 must have an orbital eccentricity of √2 – 1; only that if the two bodies have that particular mass ratio and that particular orbital eccentricity, then their orbits will touch.
The two bodies may certainly have an eccentricity greater than √2 – 1, in which case their orbits will certainly cross; or the the two bodies may have an orbital eccentricity less than √2 – 1, in which case the more massive object will certainly orbit entirely within the orbit of the less massive object.
A Real-World Example
At what orbital eccentricity would their orbits touch? Using the equation:
A Fictional Example
MP = 0.86
MS = 0.21
(The units don't matter here, as long as both masses are expressed in the same units).
We can calculate the exact orbital eccentricity at which the orbits of these two bodies will touch one another, and beyond which the orbits will cross. (We’ll use the long version of the equation to show most clearly what is happening):
3. The below two versions of the same equation will also reveal the eccentricity value at which the orbits touch/cross: