In a previous blog, I discuss solar analog stars; stars that are close to the Sun in composition, mass, radius, luminosity, etc. Generally speaking, these include stars from spectral classes K3 through F4.

Since the focus of this blog is on human-habitable and human-inhabitable environments, I thought I'd crunch the numbers and share some data on fundamental orbits and habitable zones of solar analog stars.

So, first, a table which lists—by spectral type—the fundamental orbits:

If you're not familiar with these orbits and how to calculate them, refer to my blog

Star Systems, Part 1: Fundamental Orbits.

Since the focus of this blog is on human-habitable and human-inhabitable environments, I thought I'd crunch the numbers and share some data on fundamental orbits and habitable zones of solar analog stars.

So, first, a table which lists—by spectral type—the fundamental orbits:

- Innermost Stable Orbit
- Inner Optimistic Habitable Zone Orbit
- Inner Conservative Habitable Zone Orbit
- Nucleal Habitable Zone Orbit
- Outer Conservative Habitable Zone Orbit
- Outer Optimistic Habitable Zone Orbit
- Frost Line Orbit

If you're not familiar with these orbits and how to calculate them, refer to my blog

Star Systems, Part 1: Fundamental Orbits.

Below is a graphical representations of the Habitable Zone data in the above table.

It is easy to see that the larger, more massive, and more luminous the star, the wider are both spans of the habitable zones. This increases the possibility of more than one planet orbiting in the habitable zone, especially for the optimistic range. It is also immediately clear that the conservative zone is always significantly narrower than the optimistic zone (in fact, the span of the conservative habitable zone is always 41.18% of the span of the optimistic habitable zone).

## Ideal Solar Analogs: Orbits With A Period of One Solar Year

Of the twenty spectral classes which we have defined as solar analogs, only 12 (60%)—G9 through F8—allow for planetary orbits with a period of exactly one Earth year (

In spectral classes F7 through F4, the optimistic innermost habitable zone orbit falls more than 1.0 AU from the star, so an orbit with a period of one Earth year would be closer than the innermost habitable zone limit. In spectral classes K3 through K0, the optimistic outermost habitable zone orbit falls less than 1.0 AU from the star, and thus an orbit with a period of one Earth year would be beyond the outermost habitable zone limit.

Remember that Apparent Brightness is calculated by the stellar luminosity over the distance squared:

*P =1*). Due to the relation*P²*∝*a³*, an orbital period of one Earth year requires a semi-major axis of 1.0 astronomical unit. We might think of these spectral classes as**for this reason.***ideal solar analogs*In spectral classes F7 through F4, the optimistic innermost habitable zone orbit falls more than 1.0 AU from the star, so an orbit with a period of one Earth year would be closer than the innermost habitable zone limit. In spectral classes K3 through K0, the optimistic outermost habitable zone orbit falls less than 1.0 AU from the star, and thus an orbit with a period of one Earth year would be beyond the outermost habitable zone limit.

Remember that Apparent Brightness is calculated by the stellar luminosity over the distance squared:

Where

Therefore, for the ideal solar analogs, the apparent brightness at 1.0 AU is equal to the luminosity in solar units, because

*AB*and*L*are in solar units and*D*is in astronomical units.Therefore, for the ideal solar analogs, the apparent brightness at 1.0 AU is equal to the luminosity in solar units, because

*D*is always 1.0 AU.*The ideal solar analog types are highlighted.*## Luminosity at the Nucleal Habitable Zone

However, the apparent brightness of the star

*at the nucleal habitable zone orbit*is—by definition, always 1.0 solar units, because of the equation:Which, remember from my blog Star Systems, Part 1: Fundamental Orbits, is equivalent to the equation:

... in which the value of

*AB*is always 1.0 solar units. Thus, solved for*L*:... in which—again--

*AB*is always 1.0 solar units, so the equation simplifies to:which is simply the reverse of the first equation given in this section. Thus, since we arrived at the nucleal habitable zone orbital distance by taking the square root of the luminosity, squaring the nucleal habitable zone orbital distance returns the luminosity figure we originally started with.

Thus, solving for

Thus, solving for

*AB*:... we see that, since

*HZn²*=*L*, the right side of the equation is equal to*L*-over-*L*, which is 1.0, so the apparent brightness always calculates to 1.0.## Orbital Period At The Nucleal Habitable Zone Orbit

Lastly, below are a table and a graph expressing the orbital period of a planet occupying the nucleal habitable zone orbit for all 20 of the solar analog spectral types, with the ideal solar analogs highlighted in green in the table.

Note that for F7-class stars, the nucleal habitable zone orbit has a period of just slightly shorter than a Martian year—which is 1.881 Earth years—and for K2-class stars, the nucleal habitable zone orbit is only 1.826 days longer than a Mercurian year—which is 0.241 Earth years.

As a reminder, below is a graph showing the apparent brightness range across the habitable zone orbital distances:

As a reminder, below is a graph showing the apparent brightness range across the habitable zone orbital distances:

*The black line indicates 1.0 AU and 1.0☉.*