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.Where:

*a*= semi-major axis;*b*= semi-minor axis;*c*= linear eccentricity (distance from center to either focus);*d*= distance from focus to apocenter;*e*= eccentricity;*f*= flattening (oblateness);*g*= distance from focus to pericenter;*h*= distance between the foci;*i*= major axis;

*j*= minor axisAt an eccentricity of

The ellipses below have eccentricities of (l. – r.)

*e**= 1*, the length of the semi-major axis would be*a**= 0*, because the curve has "popped open" into a parabola, which—by definition—has no semi-major axis.The ellipses below have eccentricities of (l. – r.)

*e*=*0.25*,*e*=*0.50*, and*e*=*0.75*.The elliptical orbits of the planets in the Solar system vary from 0.0068 (Venus), to 0.2056 (Mercury). Among Solar system moons, Triton's orbit has the lowest eccentricity at

Among exoplanets, orbital eccentricities seem to be higher on the average than those in the Solar system <1>; the two highest eccentricities so far discovered belong to HD 80606 b

(

*e*=*0.000016*—the closest to circular of any orbit in the Solar system—and Nereid's orbit has the highest eccentricity at*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 <1>; the two highest eccentricities so far discovered belong to HD 80606 b

(

*e*=*0.9336*) and HD 20782 b (*e*=*0.97*).Mary Anne Limbach and Edwin L. Turner <2> 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.

Between 1609 and 1618, Johannes Kepler, crunching the numbers on Tycho Brahe’s copious observational data, recognized and announced the elliptical nature of the orbits of the planets, with the Sun located at one focus. <3>

Between 1609 and 1618, Johannes Kepler, crunching the numbers on Tycho Brahe’s copious observational data, recognized and announced the elliptical nature of the orbits of the planets, with the Sun located at one focus. <3>

**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.

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.

**Semi-Major Axis and The Orbital Period**

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. <4>

Where:

Simply put, the square of the planet's orbital period (expressed in Earth years: I use the term “perannual” to describe this unit of measure), 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).

The above is the "short form" of the more complete equation:

*P*= planetary orbital period (perannual);*a*= orbital semi-major axis (in astronomical units)Simply put, the square of the planet's orbital period (expressed in Earth years: I use the term “perannual” to describe this unit of measure), 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).

The above is the "short form" of the more complete equation:

Where:

*P*= planetary orbital period (perannual);*a*= orbital semi-major axis (in astronomical units);*m*= the mass of the planet (in solar masses);

*M*= the mass of the star (in solar masses)**The Barycenter: The Two-Body Problem**

When two objects are gravitationally bound, they both orbit a common point called the barycenter—from the Greek

In single-star systems, it usually happens that the barycenter is at—or quite close to—the center of the star.

When two objects of similar mass orbit one another, such as two stars, a double-planet, or a planet with a massive moon, the barycenter lies at some point between the two individual centers of mass, and they each orbit it separately, on orbits which share the same eccentricity.

*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. The barycenter is the center of mass (or center of gravity) around with which the orbits revolve.In single-star systems, it usually happens that the barycenter is at—or quite close to—the center of the star.

When two objects of similar mass orbit one another, such as two stars, a double-planet, or a planet with a massive moon, the barycenter lies at some point between the two individual centers of mass, and they each orbit it separately, on orbits which share the same eccentricity.

The equations below allow calculation of the average distance to the barycenter from either of two bodies orbiting one another.

Where:

*α*= the average separation between the two bodies <5>;*d**p*= average distance from the primary to the barycenter (in the same units as*a*);*ds*= average distance from the secondary to the barycenter (in the same units as*a*);*M*= mass of the primary body;*m*= mass of the secondary body (in the same units as*M*)**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

When referring specifically to the orbits of Solar system planets, the terms

Note that the two orbits will always share a focus, and that the shared focus will be the location of the barycenter.

The following equations allow calculation of the distances at which these points occur.

*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”.Note that the two orbits will always share a focus, and that the shared focus will be the location of the barycenter.

The following equations allow calculation of the distances at which these points occur.

Where:

See the previous set of equations for how to calculate

*α*= the average separation between the two bodies;*d**p*= average distance from the primary to the barycenter (in the same units as*α*);*ds*= average distance from the secondary to the barycenter (in the same units as*α*);*e*= the eccentricity of the system;*G**p*= minimum distance from primary to barycenter;*D**p*= maximum distance from primary to barycenter;*G**s*= minimum distance from secondary to barycenter;*D**s*= maximum distance from secondary to barycenter;*G**t*= closest approach of the two bodies;*D**t*= widest separation between the two bodiesSee the previous set of equations for how to calculate

*d**p*and*d**s*.As can be seen from the diagram above, if the maximum separation of the primary (

The following table encapsulates the above information.

*D**p*) is less than the minimum separation of the secondary (*G**s*), then the orbits are fully nested with no crossover—the primary orbits entirely within the orbit of the secondary (Diagram A); if*D**p*is greater than*G**s*, then the orbits necessarily cross (Diagram B); finally, if*D**p*is equal to*G**s*, then the orbits are nested, but touch at one point (Diagram C).The following table encapsulates the above information.

**Crossing Orbits**

This condition of the orbits crossing is consequent on the chosen eccentricity of the orbits and the masses of the two bodies.

We can determine exactly at what eccentricity value the orbits touch by setting the equation for the apoapsis of the secondary (

Let:

Since we’re setting

We can determine exactly at what eccentricity value the orbits touch by setting the equation for the apoapsis of the secondary (

*G**s*) equal to the equation for the periapsis of the primary (*D**p*), and solving for the value of the eccentricity (*ε*). (I’m using*ε*for the eccentricity here, in order to highlight that it is**a known quantity in this case, but rather the quantity for which we are solving).***not*

Let:

*ε*= eccentricity at which the orbits begin to cross over (this is the variable we’re solving for);*M*= mass of the primary body;*m*= mass of the secondary body (in the same units as*M*)Since we’re setting

*D**p*and*G**s*equal,*d**p*and*d**s*will also be the equal, because the apastron of one orbit is the same as the periastron of the other, so we can leave them out of our starting equations, because they’ll immediately cancel out, anyway:So, our final equation for calculating the cross-over eccentricity is:

This value can also be calculated using the alternative equation:

Where:

*ε*= eccentricity of the system at which the orbits begin to cross over;*f*= the ratio of*m*/*M*The final eccentricity of the system overall isnot determinedby 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

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.

We showed above that:

*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.

We showed above that:

Because we know that the mass ratio and the eccentricity

*are the same value at this point on the graph*, we can substitute*e*for*m/M*, so:… and we can convert the equality into quadratic form:

… and solve for the positive root (since neither the eccentricity nor the mass ratio can be negative):

... which is √2 – 1, (or ≈

*1.414214··· – 1*≈*0.414214···*).Interestingly, this value also happens to be the conjugate of the Silver Ratio (2.414214···), which, is of course√2 + 1.

So, the point at which the two plots cross in the graph calculates to a value of

*e*= , or ≈ 0.414214···.Again, note that this isnotsaying that two bodies with a mass ratio of 0.414214musthave an orbital eccentricity of 0.414214; only that if the two bodies have that particular mass ratioandthat particular orbital eccentricity, then their orbits will touch.

The two bodies may certainly have an eccentricity greater than 0.414214, in which case their orbits will certainly cross; or the the two bodies may have an orbital eccentricity less than 0.414214, in which case the more massive object will certainly orbit entirely within the orbit of the less massive object.

**A Real-World Example**

A quick check at Wikipedia tells us that the mass of Alpha Centauri A is ≈ 1.1 solar and the mass of Alpha Centauri B is ≈ 0.907 solar.

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 quick-check 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:

We can calculate the

*M**= 0.86*

*m**= 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:So, for these two masses, if the eccentricity of their orbits is

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.

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.

*e**= 0.6075*, then their orbits*will touch*; if it is*e**> 0.6075*, then their orbits*will cross*; conversely, if the eccentricity is anything*e**<**0.6075*, then the orbits will be nested, and the primary will orbit entirely within the orbit of the secondary.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.

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.

- Orbital Eccentricity," Wikipedia, July 08, 2018, , accessed July 19, 2018, https://en.wikipedia.org/wiki/Orbital_eccentricity.
- http://www.pnas.org/content/112/1/20.full.pdf
- There
*are*non-Keplarian orbits; more on that elsewhere. - This can also be expressed as
*log(P)/log{a} = 1.50* - Note that
*α,*here, is different from*a*used in the equations for the ellipse;*a*refers to the semi-major axis of a single ellipse, whereas*α*, here, refers to the*average separation of the two bodies*in their mutual orbits around the system barycenter.