The Habitable Zone of a solar system is self-explanatory - it is the spherical shell surrounding a star in which a planet must reside in to have liquid water. The habitable zone is primarily dependent on luminosity, defined as the power output of the star. These measurements are obviously going to be useful in detailing the Atos system.
However, before we discuss luminosity, let me suggest this: That we have a different habitable zone from Earth - in particular, one farther than Earth's. This would mean an increase in the luminoity of Atos. Why? Two reasons.
1) A larger habitable zone means that there is a possibility that more than one planet can fit in the zone. Whether in our case it is Indigo or Momiji is something to be decided later and not a concern of this thread.
2) It gives Micras some variety. Currently, Micras is a clone of Earth in pretty much all ways except geography. Having a further out habitable zone means that we will have a custom year with the same climate as Earth.
So, how do we go about this? Well, first, there is the matter of the star, which is the entire factor behind the habitable zone, after all. Let's start with the stellar classifcation.
O 30,000 - 60,000 K Ultraviolet-violet
B 10,000 - 30,000 K Blue
A 7,500 - 10,000 K White-green
F 6,000 - 7,500 K Yellow-white
G 5,000 - 6,000 K Yellow
K 3,500 - 5,000 K Yellow-orange
M 2,000 - 3,500 K Orange-red
For reference, the Sun has a surface temperature of 5780 K, and Alpha Centauri A (A close nearby star that may have the possibility of a habitable planet and used a lot in scifi) has a surface temperature of 5,800 K, although it is larger. As you can see, the temperature of the star is clearly related to the color of the star. The best candidate for a star with a habitable planet is obviously within the G classifcation, although late K and early F can work too. Here's a visualization of color temperature, related to the classification system:
Now, for the actual formulas. First of all, there is a way to calculate the peak wavelength of a black body at a particular temperature. As you know, wavelength determines color. This formula is:
Lambda is obviously the wavelength, T is the temperature of the blackbody, and that constant is, well, a constant.
This is the peak of the plank distribution of wavelength.So now we can do the color the star and its temperature. What now? Now comes the issue of the luminosity of the star. The luminosity of a star is simply the power output of a star. This is determined by the Stefan-Boltzmann law, which for the surface of a spherical black body is:
L is the luminosity of the star. R is the radius of the star. T is the temperature of the star. Sigma is the Stefan-Boltzmann constant. This is the absolute luminosity; e.g. the luminosity in SI, rather than comparing to the luminosity of the sun. However, for our cases, it probably is better to have a measure of luminosity relative to the sun. By dividing the absolute luminosity by the luminosity of the sun, you get the following:
L, R, and T are the same as before. The circle with a dot in the middle signifies that quantity of the sun (e.g. R_O means the radius of the sun.) This formula is a bit simpler, and changes unit systems into something that will be helpful for us. All we need is to pick a particular radius of the star and we can get the luminosity of Atos.
Now, how does this relate to other quantities? It so happens that for main sequence stars (the majority of stars, of which the sun is a part of), mass is related to the luminosity of a star through the following formula.
Thus, we can determine the mass of the star. Of course, all of this is well and good, but how does it relate to the habitable zone? Simple. Intensity, (power per area) is a measure of how much electromagnetic radiation reaches a body at a certain distance away. It so happens that intensity is an inverse square law, just like gravity. So, it's relatively simple here. Define the central habitable zone of the solar system to be the distance of the Earth from the sun, 1 AU. It is possible to come up with this formula for the central habitable zone of a star:
Now, this measure is all well and good, but there's a slight problem. We want a measurement for a shell, not a line. There must be a range. I found from someone else these formulas, although they relate to the mass of the star, and not the luminosity, but it can be calculated from the previous formulas anyway.
This is the maximum reach of the habitable zone.
And this is the minimum reach of the habitable zone.
It turns out that for Earth, the habitable zone is currently .95 AU - 1.35 AU, and will be .95 AU - 1.15 AU in the future. (Remember that the luminosity of a star will change over time - stars do evolve, after all) So we now have all the tools needed to find the habitable zone and distance of Micras from Atos. I strongly suggest something at least slightly different from Earth, even if it is something like 1.25 AU.
