The eccentricity (e) of a function (curve) is a measure of how much it varies from perfectly circular; an eccentricity of 0 indicates a perfect circle; for values of e between [>0, <1], the curve is an ellipse; for e = 1, the curve is a parabola; for e > 1, the curve is a hyperbola.

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 nonKeplarian 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.
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).
I'll discuss this in more detail when I cover planetary orbits in a later blog.
I'll discuss this in more detail when I cover planetary orbits in a later blog.
The Barycenter
When two objects are gravitationally bound, they both orbit a common point called the barycenterfrom the Greek barus, meaning “heavy” (the same place baryons get their name), and kéntron, meaning a “needle”, “spur”, and by extension, the pivot point in drawing a circle—is the center of mass (or center of gravity) around with which the orbits revolve.
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.
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
In all cases, there will be times when each body is at its farthest point from the barycenter (called the apoapsis or apocenter), and other times when it is at its closest approach (called the periapsis or pericenter).
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.
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.
In the above equations, rp and rs are the values for the average distances from the barycenter of each of the bodies, as calculated in the previous set of equations. The variable a is the average separation of the two bodies as they orbit one another, and e, of course, is the eccentricity of the body's orbit.
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).
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).
This equation [1], based on the ratio of the masses of the two bodies, reveals the orbital eccentricity at which the orbits of two bodies (stars, planets, etc.) touch one another.
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).
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.
Because we know that the value of e and 𝑓 (the ratio of the mass of the secondary divided by the mass of the primary) are the same at this point, we can substitute e for 𝑓 in our simplified equation:
Because we know that the value of e and 𝑓 (the ratio of the mass of the secondary divided by the mass of the primary) are the same at this point, we can substitute e for 𝑓 in our simplified equation:
A realWorld Example
A quick check at Wikipedia, tells us that the mass of Alpha Centauri A is ~1.1 times that of the Sun, and the mass of Alpha Centauri B is ~0.907 times that of the Sun.
At what orbital eccentricity would their orbits touch? Using the equation:
At what orbital eccentricity would their orbits touch? Using the equation:
... tells us that if the eccentricity of their orbits is anything greater than ~0.0962, then their orbits will cross. A second quickcheck at Wikipedia shows that their orbital eccentricity is measured at ~0.5179, which is certainly greater than ~0.0962, so their orbits must cross.
A Fictional Example
Let us assume we have two bodies such that their masses are:
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):
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):
So, for these two masses, if the eccentricity of their orbits is exactly 0.60747, then their orbits will touch; if it is greater than 0.60747, then their orbits will cross; conversely, if the eccentricity is anything less thanbut not equal to0.60747, then the orbits will be nested, and the primary will orbit entirely within the orbit of the secondary. Additionally, the smaller the eccentricity value, the more circular the orbits will be.
Thus, there are a wide variety of orbits possible for the two bodies, dependent entirely and only upon the chosen eccentricity of their orbits.
Thus, there are a wide variety of orbits possible for the two bodies, dependent entirely and only upon the chosen eccentricity of their orbits.
Note that the crossing of the orbits presents no danger to the system; the two objects will always be opposite one another on a line passing through the barycenter as they orbit (the dotted lines in the diagram above), so the two of them will never be at the crossing point(s) at the same time. They will never approach close enough to one another to risk a collision, unless the system becomes destabilized by some outside influence.
Note: I plan to discuss orbital periods in other posts, because the equations for determining them are slightly different for stars orbiting one another as compared to much smaller objects orbiting stars, or two small objects (like twin planets or planets and moons) orbiting each other.
Note: I plan to discuss orbital periods in other posts, because the equations for determining them are slightly different for stars orbiting one another as compared to much smaller objects orbiting stars, or two small objects (like twin planets or planets and moons) orbiting each other.
1. The below two versions of the same equation will also reveal the eccentricity value at which the orbits touch/cross: