Blog Tie-ins:

• Orbital Resonances And Synodic Period

• Star Systems, Part 2: Binary Star Systems

• Planets and Worlds, Part 2: Planetary Pairs

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.

Below is a graph showing how the ratio of the semi-minor axis (

*b*) changes as the eccentricity increases. The graph assumes a constant value of*a = 1*.At an eccentricity of

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

*e = 1*, the length of the semi-major axis would be*b = 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), and 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 20782 b (

HD 80606 b (

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.

*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 20782 b (

*e = 0.97*) andHD 80606 b (

*e = 0.9336*).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. 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 TimeKepler 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.

Where:

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)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:

Where:

I discuss this in more detail in the blog Designing System Orbits, Part 6: Orbital Periods.

*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.

## The Barycenter

When two objects are gravitationally bound, they both orbit a common point called the

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.

*barycenter--*from 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.

## 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

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

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".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

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 (

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 (

*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*).This equation [3], 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.

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.

*The final eccentricity of the system overall is not*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.**determined**by this equation; that still must be selected by the worldbuilder according to preference or need.

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:... which is √2 – 1 (or ~ 1.414214 – 1).

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

Again, note that this is

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.

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

**saying that two bodies with a mass ratio of √2 – 1***not**must*have an orbital eccentricity of √2 – 1; only that if the two bodies have that particular mass ratio__that particular orbital eccentricity, then their orbits will touch.__**and**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

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 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:

MP = 0.86

MS = 0.21

(The units don't matter here,

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 than--*0.60747, 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.**but not equal to**--## Conclusion

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.

1. en.wikipedia.org/wiki/Orbital_eccentricity#Exoplanets

2. www.pnas.org/content/112/1/20.full.pdf

3. The below two versions of the same equation will also reveal the eccentricity value at which the orbits touch/cross:

2. www.pnas.org/content/112/1/20.full.pdf

3. The below two versions of the same equation will also reveal the eccentricity value at which the orbits touch/cross: