## Overview

There are a number of methods for designing planetary orbits for a star system. The process is identical for single-star systems and wide-binary systems, except that in the latter, the maximum orbit for a planet around either star is ⅕ the minimum separation of the two stars as they orbit one another.

There are special considerations for planetary orbits around close-binary systems (see the previous blog), especially with regard to Innermost Stable orbits and Habitable Zones. As I discuss each method, I'll make mention if and when special consideration must be taken for close-binary systems.

There are special considerations for planetary orbits around close-binary systems (see the previous blog), especially with regard to Innermost Stable orbits and Habitable Zones. As I discuss each method, I'll make mention if and when special consideration must be taken for close-binary systems.

**PLANET COUNT AND ORBITAL ECCENTRICITY**Mary Anne Limbach and Edwin L. Turner [1] have found that the larger the number of planets in a system, the lower the eccentricity of their orbits. For our Solar System, the eccentricities fall in the range [0.0068, 0.2056]. (Interestingly, these belong to Venus and Mercury, respectively.) The eccentricity of the Earth’s orbit is

*e = 0.0167*.

Be aware that the more eccentric an orbit, the more instability is introduced not only to the planet, but to the other star system bodies around it.

## Method 1: Follow the Giant

While some research [2] indicates that only about 40% of all star systems form gas giant planets, this method assumes at least one gas giant, and places the locations of the orbits of other planets in relation to it.

While it is also possible for any planet to occupy any orbit, a quick glance at the illustrations at Cosmic Diary and Centauri Dreams seems to indicate that in systems with varied planet sizes, the smaller-mass bodies tend to be located in the inner system and the larger-mass bodies in the outer system.

For this method, we first place the largest gas giant in the system relative to the Frost Line, using the following “equation”:

While it is also possible for any planet to occupy any orbit, a quick glance at the illustrations at Cosmic Diary and Centauri Dreams seems to indicate that in systems with varied planet sizes, the smaller-mass bodies tend to be located in the inner system and the larger-mass bodies in the outer system.

For this method, we first place the largest gas giant in the system relative to the Frost Line, using the following “equation”:

Where

What this means is that the first gas giant’s orbit (

The rest of the possible planetary orbits are then iteratively placed at interval ratios between [1.4, 2.0] AU inward and outward from the first gas giant’s orbit.

In the case of the Solar System, then, the first gas giant’s orbit would be in the range of (4.85 + 0.10) AU to (4.85 + 1.0) AU, or [4.95, 5.85] AU from the Sun (and Jupiter obligingly falls in place at 5.2 AU).

For planets inward from the gas giant, the orbits are found by the following iterative equation:

*Og*is the orbital distance of the first gas giant;*F*is the Frost Line [3] of the system, and the second addend should be read as “some value, randomly selected from the set of real numbers in the inclusive range of 0.100 to 1.000” [4].What this means is that the first gas giant’s orbit (

*Og*) will be located somewhere between 0.10 AU and 1.0 AU*beyond*the calculated distance of the Frost Line. The amount added to the Frost Line distance is purely a matter of choice or randomization.The rest of the possible planetary orbits are then iteratively placed at interval ratios between [1.4, 2.0] AU inward and outward from the first gas giant’s orbit.

In the case of the Solar System, then, the first gas giant’s orbit would be in the range of (4.85 + 0.10) AU to (4.85 + 1.0) AU, or [4.95, 5.85] AU from the Sun (and Jupiter obligingly falls in place at 5.2 AU).

For planets inward from the gas giant, the orbits are found by the following iterative equation:

… where

The denominator in this equation should be read as “some value randomly selected from the set of real numbers in the inclusive range of 1.4 to 2.0”.

For example; let the Frost Line be

*On+1*starts out as the calculated orbit of the first gas giant relative to the Frost Line, and then the calculated value of*On*is fed back into the equation until some lower bound is reached, such as the Innermost Stable Orbit.The denominator in this equation should be read as “some value randomly selected from the set of real numbers in the inclusive range of 1.4 to 2.0”.

For example; let the Frost Line be

*4.85 AU*(*F = 4.85*); let*rand[0.10, 1.0]*be*0.35*, such that:… thus yielding an orbital radius for the first gas giant of 5.20 AU. Our limiting value for the Sun is

*IS = 0.10*; we then begin calculating the orbits of the closer planets:… at which point the iteration stops because our function has returned a value

For planets outward from the gas giant, the process is similar, except that each successive orbit is iteratively

*On ≤ 0.10*. Thus, we have the following set of values for the seven inner system orbits, plus the first gas giant: {0.18, 0.27, 0.51, 0.81, 1.13, 2.04, 3.47, 5.20}.

*Note that the divisors in each iteration were randomly chosen numbers in the range of [1.4, 2.0], in the sequence: {1.5, 1.7, 1.8, 1.4, 1.6, 1.9, 1.5}*.For planets outward from the gas giant, the process is similar, except that each successive orbit is iteratively

*multiplied*by a random value in the range [1.4, 2.0], out to the maximum orbit desired. (See the discussion on maximum planetary orbits below).… where

The second addend should be read as “some value, randomly selected from the set of real numbers in the inclusive range of 1.4 to 2.0”.

So, we begin with the orbit of the first gas giant (

*On-1*starts out as the calculated orbit of the first gas giant relative to the Frost Line, and then the calculated value of*On*is fed back into the equation until some upper bound is reached (typically around 35.0 to 40.0 AU—I'll explain later).The second addend should be read as “some value, randomly selected from the set of real numbers in the inclusive range of 1.4 to 2.0”.

So, we begin with the orbit of the first gas giant (

*On-1 = 5.20*), and calculate successive orbits by iterating the equation:We’ll specify our limiting value as 35.0, and generate something like:

… at which point the algorithm stops, because the minimum possible next multiplier is 1.4 and 33.20 × 1.4 = 46.48, which is beyond our specified maximum orbit limit of 35.0 AU.

Thus we have generated a list of orbital distances for the outer system of: {7.80, 14.82, 20.75, 33.20} using the random sequence of multipliers {1.5, 1.9, 1.4, 1.6}.

This produces the following orbital parameters table (with the inner system orbits in ascending order now):

Thus we have generated a list of orbital distances for the outer system of: {7.80, 14.82, 20.75, 33.20} using the random sequence of multipliers {1.5, 1.9, 1.4, 1.6}.

This produces the following orbital parameters table (with the inner system orbits in ascending order now):

When using this method, bear in mind three things:

- No planet’s orbit can come closer to the host star than the innermost stable orbit distance (
*IS*)--*Oi*for close-binary systems—for the Sun, this is 0.10 AU. - No planet’s orbit can come closer than 1.4 AU to the already–established orbit of any other planet, so if you’ve already established a human–habitable planet in your system, you’ll have to tweak the orbits of its neighbors to remain outside this limit.
- Gaps (
*not intervals*—see below)be larger than 2.0 AU; an asteroid belt may exist in this region, or a planet previously occupying this orbit may have been ejected or migrated.*can*

## Maximum Planetary Orbital Distance

Technically, the maximum possible planetary orbit is limited only by a star’s Hill Sphere radius. But, for practical purposes, planets beyond about 30-35 AU are so distant as to be invisible to human-quality eyes from the region of the star's habitable zone without extreme magnification. Jupiter moved to Saturn's orbit (9.554 AU) would still be ~1.5 times brighter than Saturn, but moved to Uranus' orbit (19.22 AU), it would no longer be visible to the naked eye.

Still, there

Also, due to what is currently understood about protoplanetary disks and planet formation, it seems reasonable to say that the maximum distance at which a major planet

Planets found orbiting beyond this distance may have migrated outward after forming closer to the star, or—less likely, but more romantic—be former rogue planets which were captured by the star’s gravity and settled into orbit. Planet Nine in our own Solar System—if it actually exists—is estimated to have a semi–major axis of

In the example above, we stopped at an orbital value of 33.20 because the minimum possible next orbit was 46.48 AU, well beyond even a 40 AU maximum orbital distance.

If your culture is at least as technologically advanced as we are, (or more spacefaring), the likelihood is greater that they would be aware of planets farther out than 30-35 AU, and you should feel free to calculate orbits to your heart’s content.

Finally, if your system is a wide–binary, then remember that the outermost stable orbit is limited to one–fifth of the minimum distance between the two stars.

Still, there

*are*planets in our own Solar System beyond Saturn's orbit, even if we can't see them, so there's no reason to suppose that they wouldn't exist elsewhere; thus, selecting Neptune's orbit (30.11 AU) or slightly farther out isn't unreasonable for the most distant planetary orbits in a given system.Also, due to what is currently understood about protoplanetary disks and planet formation, it seems reasonable to say that the maximum distance at which a major planet

*might reasonably be expected to form*(around a Sun-like star) is perhaps slightly more distant from the Sun than the orbit of Neptune, which is at ~30.11 AU, and thus our maximum falls perhaps in the range of 35-40 AU.Planets found orbiting beyond this distance may have migrated outward after forming closer to the star, or—less likely, but more romantic—be former rogue planets which were captured by the star’s gravity and settled into orbit. Planet Nine in our own Solar System—if it actually exists—is estimated to have a semi–major axis of

*700 AU*out from the Sun, with a perihelion of 200 AU, an aphelion of 1200 AU, and an orbital period of 10,000-20,000 Earth years!)In the example above, we stopped at an orbital value of 33.20 because the minimum possible next orbit was 46.48 AU, well beyond even a 40 AU maximum orbital distance.

If your culture is at least as technologically advanced as we are, (or more spacefaring), the likelihood is greater that they would be aware of planets farther out than 30-35 AU, and you should feel free to calculate orbits to your heart’s content.

Finally, if your system is a wide–binary, then remember that the outermost stable orbit is limited to one–fifth of the minimum distance between the two stars.

## Intervals and Gaps

Again, note that the

This does result in a some suspiciously exact intervals (particularly 1.5); however, some (but not all) such intervals could be explained away by mean motion resonances, which are not unheard–of by any means.

*interval*denotes how many*times*bigger each successive orbit is than the previous one, whereas the*gap*simply indicates the spacing between the orbits.*In general, the orbital*We have an exception here; the gap between Planets 10 and 11 is smaller than the gap between Planets 11 and 12—but that just makes things appear more “natural”.**gap**will increase with each successive orbit.This does result in a some suspiciously exact intervals (particularly 1.5); however, some (but not all) such intervals could be explained away by mean motion resonances, which are not unheard–of by any means.

## Conclusion

This is a good, robust way of calculating planetary orbits around fictional stars.

This system has the potential problem of no planet falling at precisely 1.0 AU; thus, if there are any in the Habitable Zone, their orbital periods will necessitate creating a specialized calendrical system for any inhabitants (I'll do a blog on this later).

This system has the potential problem of no planet falling at precisely 1.0 AU; thus, if there are any in the Habitable Zone, their orbital periods will necessitate creating a specialized calendrical system for any inhabitants (I'll do a blog on this later).

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

2. iopscience.iop.org/article/10.1088/0004-637X/719/2/1454/meta

3. Recall that the Frost line is calculated by:

4. I’ve specified real numbers here rather than rational numbers, because we could arbitrarily decide to have the range be [ɸ, 𝘦], so that the lower bound is an irrational number (1.618033988…), and the upper bound is a transcendental number (2.71828…), neither of which is in the set of rational numbers.

2. iopscience.iop.org/article/10.1088/0004-637X/719/2/1454/meta

3. Recall that the Frost line is calculated by:

*F = 4.85 × √L*where*L*is the star’s luminosity in solar units.4. I’ve specified real numbers here rather than rational numbers, because we could arbitrarily decide to have the range be [ɸ, 𝘦], so that the lower bound is an irrational number (1.618033988…), and the upper bound is a transcendental number (2.71828…), neither of which is in the set of rational numbers.