## Close-binary P-type systems and Habitable Zones

The habitability of P-type star systems (when a system of planets orbits both stars as if they were a single, central mass) is a tricky business. The equations in the previous blog are generally applicable for close-binary pairs with low eccentricities and low average separations, but care must be taken with the initial values chosen.

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

Let's revisit the equations:

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,*their***and***average separation*. However, the calculations for their habitable zones are determined*by their***only***luminosities*.

Let's revisit the equations:

Note that while

In addition, the minimum and maximum separations of the two stars are a function of their

*a*, the average separation of the two stars, is used in these equations,*it is in no way determined**by the**masses**of the stars*. It is an*independent variable*chosen at will by the Worldbuilder.In addition, the minimum and maximum separations of the two stars are a function of their

*average separations*and the*eccentricity*of their orbits:Here, again, the value of

(True, the crossing eccentricity is calculated based on the stars' masses, but that only identifies

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:

*e*, the eccentricity of the stars' orbits, is*independent of the masses**of the stars*.*In no equation is the value of*.**e**__determined__in terms of the masses of the stars(True, the crossing eccentricity is calculated based on the stars' masses, but that only identifies

*the eccentricity at which their orbits*__, it does not specify what the stars' eccentricity__*necesarily will cross**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:

And it is

*MAXT*which*determines*the Innermost Stable Orbit and the Forbidden Zone:*Note that OI is different from the Innermost Stable Orbit (IS) for single-star systems, which is determined by the mass of the star for stars of ≤1.0 solar masses/solar luminosities and by the luminosity of the star for stars of >1.0 solar masses or solar luminosities. Here, the Innermost Stable Orbit (OI) is*always

* determined solely by the masses of the stars.*

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*:

These equations

The calculation of the Frost Line also relies on the luminosities:

*do not take into account the minimum, average, or maximum separations of the two stars*.The calculation of the Frost Line also relies on the luminosities:

... and ignores the masses of the stars.

## Clarifying The Problem

Ideally, the calculated distance of the Innermost Stable Orbit (

In practice, as long as the

... then the system is marginally habitable.

Let's look at an example to clarify.

Let's specify two stars in a close-binary configuration with the following characteristics:

Megadar-A:

Megadar-B:

Average separation of the system:

Eccentricity of the orbits:

*OI*) must be less-than-or-equal-to the Optimistic Innermost Habitable Zone limit (*HZi*):*OI*≤*in order for a close-binary system to be "habitable".*

...**HZ****i**...

In practice, as long as the

*nucleal*habitable zone orbit is greater-than-or-equal-to the Innermost Stable Orbit:*OI*≤**HZ****n**... 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 luminositiesMegadar-B:

*M*= 0.783 solar masses;*L*= 0.480 solar luminositiesAverage 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)We can find that the maximum separation of the stars is:

... and the Innermost Stable Orbit is four times this value:

This is a huge orbit, considering that Uranus is 19.0 AU from the Sun, but that just makes this system interesting, right?

## The Fly in the Ointment

What is the Optimistic Innermost Habitable Zone orbit for this system?

Uh, we begin to see the problem.

This tells us that the Optimistic Innermost Habitable orbit of the Megadar system is

Maybe the Optimistic Outermost Habitable Zone orbit will salvage the system's habitability?

This tells us that the Optimistic Innermost Habitable orbit of the Megadar system is

*16.7 AU**.***inside**the Innermost Stable Orbit of the systemMaybe the Optimistic Outermost Habitable Zone orbit will salvage the system's habitability?

Sadly, no. Even the Frost Line at 4.923 AU falls well closer than the Innermost Stable Orbit.

**CONCLUSION: THIS IS NOT A HABITABLE SYSTEM**

## The Solution

A couple of derived equations prove helpful:

Use the above equation to determine the

Alternatively:

*minimum**average separation*of the two stars, when you have an Optimistic Innermost Habitable Zone orbit*and*a specific orbital eccentricity already in mind, remembering that the two stars may never approach closer than 0.10 AU, or they will start to merge.Alternatively:

Use the above equation to determine the

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

Let's calculate a new minimum separation first:

*maximum eccentricity*of the stars' orbits, when you have an Optimistic Innermost Habitable Zone orbit and a specific orbital eccentricity in mind.*If***Note 1:***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.*If***Note 2:***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.*These equations were derived by setting the equations for***Note 3:***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:

... which is certainly smaller than our original value of 3.075; in fact it's actually below the specified 0.10 minimum separation for two stars, so these values for

We can tweak either

This is the better way to go, because it calculates the values based on a known limitation: the closest stable orbit

So

So, let's specify that our

We can now work out the

*a*and*e*cannot work together if we want a habitable system.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*MAX*T (knowing that*MAX*T*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:... and keeping the earlier value for

*e = 0.42*, just to see what happens.... which tells us that our

*average separation*must be at least 0.2215 AU. We can now use this to calculate*MAXT*:... and four times this value gives us our Innermost Stable Orbit distance:

... which tells us that we want our Optimistic Innermost Habitable Zone orbit to be some value greater than this.

Let's round it up to 1.30 AU, set

Let's round it up to 1.30 AU, set

*HZi*to this value, and calculate a*new*orbital eccentricity:... which gives us a whole new problem: an eccentricity of ≥1.0 indicates a parabola, so this value is not usable for our

So, instead, we can derive the equation to tell us what the

*HZi*.So, instead, we can derive the equation to tell us what the

*HZi***for the specified average separation of 0.2215 and an eccentricity of 0.42:***must be*This is not a lot less than the 1.3 we arbitrarily specified, but this value has been

What if we

*calculated*based on other calculated values, not just picked at random out of the interplanetary medium,*and*it is still (marginally) larger than the calculated*MAXT*for this system.*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*HZ**i*?Remember that it was specified earlier that if

Also, having calculated the

*e*ever comes out as negative in this equation, then the value for*a*is impossible, all else remaining the same. The fact that our answer comes out negative tells us that*there is*.**no**eccentricity value that can be paired with an average separation of 3.075 AU**and**this Optimistic Innermost Habitable Zone orbit for this systemAlso, 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

We can now work backward, to find out the details of our stars.

Deriving the equation for the Optimistic Innermost Habitable Zone:

Deriving the equation for the Optimistic Innermost Habitable Zone:

... yields the relation:

... which we can use to find the

*combined*luminosity of the two stars:Since this represents the

We can find the new luminosities by the cross-multiply-and-divide method.

The luminosity for Megadar-A

*sum of the luminosities of both stars*; we will need to determine their individual luminosities and thence the rest of their characteristics.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**2.814⊙. Thus:***is*These values fall into the second (Sun-like) luminosity regime, so we'd use this equation:

Where

*a = 4.0*and*k = 1.0*, which reduces the equation to:Adding the calculated radius, total lifetime, and temperature from these figures, we get the following description of the two stars:

These surface temperatures fall into the range of spectral type F: 6000-7500 K, so we can calculate the spectral class of the two stars:

We can also calculate their orbital period:

... or about 25.804 days.

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

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!

**Summary**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.