I define ten fundamental orbital distances which serve to characterize the superstructure of a star system. These are related to the mass and/or luminosity of the star(s), and help to determine temperature zones.

**Forbidden Zone Limit**

This is a limiting distance in close-binary systems, discussed in detail below. In single-star systems, the innermost stable orbit serves the same purpose as the forbidden zone limit.

**Innermost Stable Orbit**

The innermost stable orbit is the closest distance at which a planetary body may exist in a stable orbit without being torn apart by gravitational tidal stresses. It is calculated in one of two ways: if the mass and/or luminosity of the host star is less than 1.0 solar, this orbit is calculated as one-tenth of the mass of the star; otherwise, it is calculated as one-tenth of the square root of the host star’s luminosity.

Why two calculation methods? Two reasons: 1) The luminosity based calculation produces orbits much too small for low-mass stars, on the order of thousandths of an astronomical unit; 2) Because the luminosity of stars below 0.430 solar masses is calculated differently, there is a downward jog in the luminosity based graph which is counter-intuitive.

Below is a graph showing the different results produced by the two calculation methods.

Below is a graph showing the different results produced by the two calculation methods.

Why is there a jog in the orange graph? Because, recall, for masses below 0.43 solar, a different equation is used to calculate the luminosity of the star.

Here is the graph of the two methods combined—mass-based below 1.0 solar mass and luminosity-based above 1.0 solar mass.

Here is the graph of the two methods combined—mass-based below 1.0 solar mass and luminosity-based above 1.0 solar mass.

While there is still a transition at 1.0 solar mass, the graph follows a generally smooth, increasing path over its span, with distances that are in keeping with the masses and luminosities of the host stars.

**The Habitable Zones And Their Limits**

The next five orbital distances are related.

Four of them are limiting distances, describing the

The fifth orbit I call the

Four of them are limiting distances, describing the

*innermost*, and*outermost*orbits at which a planet may orbit and yet receive enough light and heat from the star to have liquid surface water, and thus be potentially (in)habitable for Earth-type life. There are two commonly accepted habitable zones in current astronomy: the*optimistic habitable zone*and the*conservative habitable zone*. As their names imply, the first assumes a wider range of distances within which it is supposed that a planet would receive enough energy from its host star to maintain liquid water and an Earth-type climate. The second is narrower (*always*41.18% of the optimistic range), and takes a more cynical view of the habitable range of orbits.The fifth orbit I call the

*nucleal*<1>*orbit: this orbit is calculated to assume the most absolutely Earth-like conditions; that is, the orbit at which the star would have the same apparent brightness as the Sun does when seen from Earth, and at which the planet would receive the same amount of heat and light as the Earth receives from the Sun at 1.0 astronomical unit.*Note, thenucleal orbital distance(Earth's orbit in the Solar System) isthe average or median orbit between the inner and outer limits ofnoteitherhabitable zone. It is located at 11.91% of the conservative range of the Sun’s habitable zone, and at 24.51% of the optimistic range.

The following equations allow the determination of the inner, nucleal, and outer habitable zone distances based on the luminosity of the star. The nucleal habitable orbit is straightforward to calculate:

Where:

The validity of this equation can be shown by deriving from the equation for the apparent brightness:

*H**3*= the distance from the star of the nucleal orbit, in astronomical units;*L*= the luminosity of the single star in solar unitsThe validity of this equation can be shown by deriving from the equation for the apparent brightness:

Rearranging to solve for

*D*:Since we know we want the apparent brightness to be that of the Sun at this orbital distance,

*B**A*=*1*will always be the case, so this equation simplifies to:... which is exactly the same relation as in the equation above for the nucleal orbit distance.

The habitable zone limits are calculated by:

The habitable zone limits are calculated by:

We can simplify this to:

Where:

*n*= the ordinal of the habitable zone orbit;*m*= the multiplicand for the orbit with ordinal*n*, taken from the table belowNote that here, ordinal six corresponds to the frost line (

Below is an illustration of the theory of habitable zones, using figures for the Sun:

*F**L*) orbital distance (see below).

Below is an illustration of the theory of habitable zones, using figures for the Sun:

Here are the habitable zone figures for the Helion stars (see Part 3.8, below):

As the tables above show, while the

At the innermost limit, temperatures due to

*span*of the habitable zones increases with the luminosity of the host star, the*ratio*of the habitable zones remains constant, and the nucleal orbit is*always*located at 11.91% of the conservative range and at 24.51% of the optimistic range.At the innermost limit, temperatures due to

*insolation*<2> grow and remain too high for humans to tolerate (without artificial habitats or self-contained suits). At the outermost limit, temperatures due to insolation drop and remain too low for humans to tolerate (again, without artificial habitats or self-contained suits).Remember, the luminosity of the host star is the same quantity as the apparent brightness of the host star at theperannual distance only(see below). Apparent brightness forallother orbital distancesmustbe calculated as described above.

**The Frost Line (Ice Line) Limit**

The frost line (also sometimes called the ice line) is the distance at which the temperature is such that volatiles such as water solidify into ices.

The frost line distance is calculated by:

The frost line distance is calculated by:

Below is a graphical representation of these five fundamental orbital distances and limits:

**The Perannual Orbit**

The perannual orbit is that orbit which has a period of 1.0 solar year. It is less straightforward and more complicated to calculate. Also, the perannual orbit need not fall within a habitable zone; indeed, it may fall closer than the innermost stable orbit, or beyond the frost line limit. It should not be confused with the nucleal orbit!

As discussed in Ellipses And Orbits, the orbital period of a system is calculated by:

As discussed in Ellipses And Orbits, the orbital period of a system is calculated by:

Where:

Thus, to precisely calculate the perannual orbit, the masses of both the star

*P*= orbital period (in perannum);*a*= average separation of the two stars in astronomical units;*M*= the mass of the primary in solar masses;*m*= the mass of the secondary in solar masses

Thus, to precisely calculate the perannual orbit, the masses of both the star

*and the planet on that orbit*need to be known. However, since the mass of the planet is often negligible (for example, Earth’s mass is one three-millionth that of the Sun), the equation can be simplified to:… re-written to substitute the perannual orbit (

*O**p*) for*a.*Solving for*O**p*:Since we know the orbital period is 1.0 solar year, then the equation simplifies to:

… which we can use to

*approximate*the perannual orbit. (This approximation will be largely accurate as long as the mass of the planet is less than 1/3500 that of the host star(s)—see the discussion “Orbital Period Of Binary Systems” below for more information.)**The Outermost Orbit**

The maximum possible orbit in a single-star system is the outer extent of the star's Hill sphere <3>. For the Sun, this is ≈ 63,000 AU (≈ 1 light year), but no object other than a sizable star would ever be visible at that distance to any inhabitants of the either habitable zone.

- Yes, it’s a word; it means “of central importance; pivotal”—which this orbit is. (see https://en.oxforddictionaries.com/definition/nucleal).
- Solar radiation received at the Earth's surface.
- See The Hill Sphere And The Roche Limit, for a discussion of the Hill sphere and how to calculate it.