Close-binary P-type systems and Habitable Zones
Generally, the higher the eccentricity of the orbits and/or the farther the average separation of the two stars, the less likely they will form a habitable system.
Why is this?
Well, remember that the minimum and maximum separations of the two stars are determined by their masses, the eccentricity of their orbits, and their average separation. However, the calculations for their habitable zones are determined only by their luminosities.
Let's revisit the equations:
In addition, the minimum and maximum separations of the two stars are a function of their average separations and the eccentricity of their orbits:
(True, the crossing eccentricity is calculated based on the stars' masses, but that only identifies the eccentricity at which their orbits necesarily will cross, it does not specify what the stars' eccentricity must be.)
The minimum and maximum separation of the two stars is a function of the sums of their minimum and maximum separations from the barycenter, determined by their masses (as seen in the first set of equations above) and the eccentricity of their orbits, thus:
In contrast, the locations of the Habitable Zone orbits are determined solely by the combined luminosities of the two stars, without regard to their masses:
The calculation of the Frost Line also relies on the luminosities:
Clarifying The Problem
OI ≤ HZi
... in order for a close-binary system to be "habitable".
In practice, as long as the nucleal habitable zone orbit is greater-than-or-equal-to the Innermost Stable Orbit:
OI ≤ HZn
... then the system is marginally habitable.
Let's look at an example to clarify.
The Megadar System
Let's specify two stars in a close-binary configuration with the following characteristics:
Megadar-A: M = 0.820 solar masses; L = 0.551 solar luminosities
Megadar-B: M = 0.783 solar masses; L = 0.480 solar luminosities
Average separation of the system: a = 3.075 AU (average of the range [0.15, 6.0])
Eccentricity of the orbits: e = 0.42 (arbitrarily chosen)
The Fly in the Ointment
This tells us that the Optimistic Innermost Habitable orbit of the Megadar system is 16.7 AU inside the Innermost Stable Orbit of the system.
Maybe the Optimistic Outermost Habitable Zone orbit will salvage the system's habitability?
Note 1: If e in this equation comes out negative, then the value of a is impossible for the HZi specified. Either the luminosity of the stars or their average separation will have to be changed.
Note 2: If e in this equation comes out equal-to-or-greater-than 1.0, then the orbit is parabolic and not elliptical. Some value will need to be tweaked to get e between zero and <1.0.
Note 3: These equations were derived by setting the equations for HZi and OI equal and solving for a and e, respectively. I've used OI instead of ZO (the Forbidden Zone limit), because I don't want the Optimistic Innermost Habitable Zone orbit to be precisely at the distance at which no stable planetary orbits can exist. Using OI, instead, gives us "breathing room".
Let's try these out on the Megadar system and see if we can salvage it.
We'll keep the masses and luminosities of the stars the same, and the orbital eccentricity at 0.42 and then re-work the value for a.
Let's calculate a new minimum separation first:
We can tweak either a or e, or—better—we can specify that the HZi of this system be some value greater than the Innermost Stable Orbit (OI) distance, and calculate a new average separation (a).
This is the better way to go, because it calculates the values based on a known limitation: the closest stable orbit any planet in this system could have.
So, the unbreakable rule says: HZi must be equal-to-or-greater-than Oi.
So, let's specify that our minimum separation (MINT) in this system is 0.125 AU; this is bigger than the minimum 0.10 to avoid stellar merger, but not overly large, either, which is what got us into trouble in the first place. Thus, we know that 4 times our MAXT (knowing that MAXT must be greater than MINT) will calculate out to some value greater than Oi.
We can now work out the new value of a, by deriving from the MINT equation:
Let's round it up to 1.30 AU, set HZi to this value, and calculate a new orbital eccentricity:
So, instead, we can derive the equation to tell us what the HZi must be for the specified average separation of 0.2215 and an eccentricity of 0.42:
Note that because the eccentricity value has not changed, the shape of the orbits remains the same, only their size as been reduced. The orbits still look the same as in Figure 1, above; they've just gotten about 19 times smaller.
What if we really want to keep the average separation at 3.075 AU? Is there some eccentricity we can assign that will allow this original average separation, along with the new HZi?
Also, having calculated the HZi (1.25812 AU), we should have known not to set the average separation to something as high as 3.074 AU; the average separation simply must be some value less-than-or-equal-to ¼ of the distance of the Optimistic Innermost Habitable Zone orbit, or the system is a failure from the start.
Characteristics of the Stars
Deriving the equation for the Optimistic Innermost Habitable Zone:
We can find the new luminosities by the cross-multiply-and-divide method.
The luminosity for Megadar-A was 0.551⊙; for Megadar-B it was 0.480⊙; the total luminosity was 1.031⊙, and the new total luminosity is 2.814⊙. Thus:
They are slightly more massive, larger, and more luminous than the Sun (hence their slightly shorter total lifetimes, but this system is imminently capable of supporting human-habitable planets!
Building habitable close-binary star systems requires some extensive regard for a number of considerations. In most circumstances, close-binary star systems will not have any planets orbiting closer than the Optimistic Innermost Habitable Zone orbit, so the first habitable planet in the system will also be the very first planet it the system.