**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 sections 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, theless likely they will form a habitable system.

Why is this?

Recall 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 zone(s) are determined

Revisiting the equations for the barycenter:

Recall 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 zone(s) are determined

*only**by their luminosities*. This can lead to inconsistencies in the fundamental orbital distances.Revisiting the equations for the barycenter:

Where:

Note that while

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

*a*= the average separation between the two bodies;*d**p*= average distance from the primary to the barycenter (in the same units as*a*);*d**s*= average distance from the secondary to the barycenter (in the same units as*a*);*M*= mass of the primary body;*m*= mass of the secondary body (in the same units as*M*)Note that while

*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**from the barycenter*and the*eccentricity of their orbits*:Here, again, the value of

*e*, the eccentricity of the stars' orbits, is independent of the masses of the stars.*In*e**no equation**is the value of*determined**in terms of the masses of the stars.*True, thecrossing eccentricity(See Ellipses and Orbits) is calculated based on the stars' masses, but that only identifies the eccentricity at which their orbits necessarily will cross, it does not specify what the stars' eccentricitymustbe.

The minimum and maximum

*total*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

*D**t*which determines the forbidden zone and the innermost stable orbit:Recall thatOSis different from the innermost stable orbit(IS)for single-star systems; the latter 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(OS)isalways ultimately determined solely by the masses of the stars.

In contrast, the locations of the habitable zone limits are determined

This can cause a singular problem: one or more of the habitable zone limits can fall closer to the stars than the innermost stable orbit limit.

*solely*by the combined luminosities of the two stars,*without regard to their masses*. The equations*do not take into account the minimum, average, or maximum separations of the two stars.*This can cause a singular problem: one or more of the habitable zone limits can fall closer to the stars than the innermost stable orbit limit.

**Clarifying The Problem**

Ideally, the calculated distance of the innermost stable orbit (

*O**S*) must be less-than-or-equal-to the innermost optimistic habitable zone limit (*H**1*) in order for the system to be “habitable”:In practice, as long as the nucleal orbit is equal-to-or-greater-than the innermost stable orbit, then the system is at least marginally habitable:

Let’s look at an example.

**The Megadar System**

Megadar-A:

Megadar-B:

Average separation of the system:

Eccentricity of the orbits:

*M**= 0.820*solar*L**= 0.551*solarMegadar-B:

*M**= 0.783*solar*L**= 0.480*solarAverage separation of the system:

*a**= 3.075*AUEccentricity of the orbits:

*e**= 0.42*(arbitrarily chosen)The eccentricity value is relatively high, but within both sets of limits specified above.

Also, for story purposes, we want the stars to have an orbital period of about 4.25 years, so we calculate their average separation to be 3.075 AU.

The maximum separation of the stars is:

Also, for story purposes, we want the stars to have an orbital period of about 4.25 years, so we calculate their average separation to be 3.075 AU.

The maximum separation of the stars is:

… just over four-and-a-quarter astronomical units.

The innermost stable orbit is four times this value:

The innermost stable orbit is four times this value:

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

Well --

Well --

**The Fly In The Ointment**

What is the innermost optimistic habitable zone limit for this system?

Hmmmm, we begin to see the problem: This tells us that the innermost optimistic habitable limit of the Megadar system is 15.39 AU

This is because

Maybe the optimistic outermost habitable zone limit will salvage the system's habitability?

*inside the innermost stable orbit of the system*.This is because

*H**1*is calculated from the stars’ luminosities and average separation, but the*O**S*is calculated*only*from their average separation, which we selected based on a desired orbital period, and which has no relationship whatsoever to their luminosities.Maybe the optimistic outermost habitable zone limit will salvage the system's habitability?

Sadly, no. This orbit is still 14.658 AU inside the innermost stable orbit. Even the frost line orbit falls closer than the innermost stable orbit:

… by 12.448 AU.

Conclusion:

Conclusion:

**While this is a stable system,***it is not a**habitable**system*.**Solution 1: Altering Either The Eccentricity Or The Average Separation**

Our calculations have shown us that this system cannot be habitable with the set of parameters originally specified. Thus, solutions will necessarily involve modifying some of those parameters.

First, let’s look at allowing flexibility in the average separation or the orbital eccentricity.

We can derive a couple of equations that will help us not to choose initial values that will render our system uninhabitable from the get-go.

First, let’s look at allowing flexibility in the average separation or the orbital eccentricity.

We can derive a couple of equations that will help us not to choose initial values that will render our system uninhabitable from the get-go.

Where:

You can use the above equation to determine the minimum average separation of the two stars, when you have an innermost optimistic habitable zone limit 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.

Second:

*G**t*= the minimum separation of the two stars in astronomical units;*H**1*= the (pre-calculated) optimistic habitable zone inner limit in AU;*e*= the chosen eccentricity of the stars’ orbitsYou can use the above equation to determine the minimum average separation of the two stars, when you have an innermost optimistic habitable zone limit 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.

Second:

Where:

You can use the above equation to determine the

Third:

*a*= the average separation of the stars in astronomical units*e*= the eccentricity of the system;*H**1*= the (pre-calculated) optimistic habitable zone inner limit in AU;You can use the above equation to determine the

*average separation*between the stars in their orbits, when you have an innermost optimistic habitable zone orbit and a specific orbital eccentricity in mind.Third:

Where the variables are the same as in the second equation, above.

These three equations above were derived by setting the equations forH1andOSequal and solving foraande, respectively. I've usedOSinstead ofZFbecause I don't want the innermost optimistic habitable zone orbit to be precisely at the distance at which no stable planetary orbits can exist. UsingOSallows some "breathing room".

There are a couple of things to bear in mind relating to the third equation:

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:

- If
*e*in this equation comes out negative, then the value of*a*is impossible for the*H**1*specified. Either the luminosity of the stars or their average separation will have to be changed. - 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 0 <*e*< 1.0.

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*. (This will necessarily result in a different orbital period for the two stars.)Let's calculate a new minimum separation first:

... which is certainly smaller than our original value of 3.075; but it’s also below the specified 0.10 minimum separation for two stars, so these specific 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 any planet in this system could have.

*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*H**1*of this system be some value greater than the innermost stable orbit (*O**S*) distance, and calculate a new average separation (*a*) from that value.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.

Remember: Theunbreakable rulesays:Os ≤ H1; the innermost optimistic habitable zone orbitmust begreater-than-or-equal-tothe innermost stable orbit.

So, let's specify that our minimum separation (

We can now work out the new value of

*G**t*) 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*D**t*(knowing that*D**t*must be greater than*G**t*)*must*calculate out to some value greater than*O**S*.We can now work out the new value of

*a*, by rearranging the*G**t*equation:Keeping the earlier value for the eccentricity at

*e**= 0.42*, let’s see what happens.... which tells us that our average separation must be at least 0.2155 AU. We can now use this to calculate

*D**t*:... and four times this value gives us our innermost stable orbit distance:

... which tells us that we want our innermost optimistic habitable zone orbit to be some value greater than this.

Let's set

Let's set

*H**1*to 1.30 AU and calculate a new orbital eccentricity:… which is only slightly larger than the value we originally specified.

We can also derive an equation which tells us what

We can also derive an equation which tells us what

*H**1**must be*for the average separation value of 0.2155 and our original eccentricity of 0.42:… and in this case it calculates <1> to:

This is not a lot less than the 1.3 we arbitrarily specified in the previous equation, but this value has been

*calculated*based on other calculated values, not just picked at random out of the interplanetary medium,*and*it is still greater-than-or-equal-to the calculated*D**t*for this system.Note that because the eccentricity value has not changed, theshapeof the orbits remains the same, only theirsizeshave been reduced. The orbits still look the same as in the illustration at the start of this section—they’ve just gotten about 19 times smaller.

But — but, but …

What if we

We can use the eccentricity equation above, plugging in the new value of

What if we

*really*want to keep the average separation at 3.075 AU? Remember, we calculated this figure based on a story requirement to have the orbital period be about 4.25 perannum. Is there*some*eccentricity value we can assign that will allow this original average separation, along with the new*H**1*?We can use the eccentricity equation above, plugging in the new value of

*H**1**= 1.224*and the original value of*a*=*3.075*and see what eccentricity value we calculate:Remember that it was specified earlier that if

Also, having calculated

*e*ever comes out as negative in this equation, then the value for*a*is impossible, all else remaining the same. Thus,*there is no eccentricity value that can be paired with an average separation of 3.075 AU**and**an innermost optimistic habitable zone orbit of 1.224 for this system.*Also, having calculated

*H**1*(1.224 AU), we should have known not to set the average separation to something as high as 3.075 AU—the average separation simply*must be*some value less-than-or-equal-to ¼ of the distance of the innermost optimistic habitable zone orbit, or the system is a failure from the start.**The New Characteristics Of The Stars**

We can now work backward, to find out the details of our stars. From the equation for the innermost optimistic habitable zone orbit equation, we can rearrange for an equation that tells us what value the combined luminosities of the stars must be:

... which we can use to find the

*sum*of the luminosities of the two stars:Since this represents the

Our equation takes the form:

*sum of the luminosities of both stars*; we will need to determine their individual luminosities and thence the rest of their characteristics.Our equation takes the form:

Where:

The original total luminosity of the two stars was 1.031 solar. The new total luminosity we just calculated is 2.6634 solar.

Thus, for Megadar-A, which had an original luminosity of 0.551:

*L**n*= the new luminosity of the same star;*S**n*= the new sum of the two luminosities;*L**o*= the original luminosity of one of the stars;*S**o*= the original sum of the two luminositiesThe original total luminosity of the two stars was 1.031 solar. The new total luminosity we just calculated is 2.6634 solar.

Thus, for Megadar-A, which had an original luminosity of 0.551:

… the new luminosity is 4.738 solar.

For Megadar-B, which had an original luminosity of 0.480:

For Megadar-B, which had an original luminosity of 0.480:

… the new luminosity is 4.127 solar.

We don’t know their radii or temperatures, so we must use the alternative equation to calculate masses for the stars.

Both of these values fall into luminosity regime 2 (see above).

The basic equation is:

We don’t know their radii or temperatures, so we must use the alternative equation to calculate masses for the stars.

Both of these values fall into luminosity regime 2 (see above).

The basic equation is:

… and for luminosity regime 2,

*a = 4.0*and*k**= 1.0*, which reduces our equation to:Thus, for Megadar-A:

… and for Megadar-B:

Finally, what is their new orbital period:

… about 21.449 Earth days.

They are slightly more massive and much more luminous than the Sun (hence they will have shorter total lifetimes), but this system is imminently capable of supporting human-habitable planets!

They are slightly more massive and much more luminous than the Sun (hence they will have shorter total lifetimes), but this system is imminently capable of supporting human-habitable planets!

**Solution 2: Altering The Innermost Optimistic Habitable Zone Limit**

We start with the value we originally calculated for the innermost stable orbit: 17.466 astronomical units.

Next we set the innermost optimistic habitable zone limit to some value greater than 17.466. For convenience, I’ll choose 18.5 AU.

Then, as above, we calculate a new combined luminosity for the two stars using this value for

Next we set the innermost optimistic habitable zone limit to some value greater than 17.466. For convenience, I’ll choose 18.5 AU.

Then, as above, we calculate a new combined luminosity for the two stars using this value for

*H**1*:… which tells us immediately that the stars are going to be

Using the same system from above, and the original luminosities of the stars, we can calculate their new individual luminosities thus:

For Megadar-A:

*much*more luminous (and therefore larger and more massive) than what we originally specified.Using the same system from above, and the original luminosities of the stars, we can calculate their new individual luminosities thus:

For Megadar-A:

For Megadar-B:

We can calculate new temperature values from these new luminosities by first calculating the new masses of the stars. Referring to the earlier, we see that both stars fall into luminosity regime 3, so our fundamental equation for calculating their masses is:

For Megadar-A:

For Megadar-B:

Then, we can calculate their surface temperatures:

For Megadar-A:

For Megadar-B:

These temperatures are in solar units (hence the use of T), so we can multiply by the Sun’s surface temperature of 5840 to get the temperatures in Kelvin, and then discover the stars’ spectral classes:

For Megadar-A:

A surface temperature of 12,690.904 K is in the B spectral type range, so:

For Megadar-B:

This surface temperature of 12,440.952 is also in the B spectral type range, so:

Now that we know their luminosities and surface temperatures, we can calculate their radii:

For Megadar-A:

For Megadar-B:

This method has allowed us to keep the average separation of 3.075 AU and the orbital eccentricity of 0.42, by allowing the stars to become more massive and

The nucleal orbit for this system now becomes:

*much*more luminous.The nucleal orbit for this system now becomes:

… which is well beyond the 17.466 innermost stable orbit.

**Summary**

Building habitable close-binary star systems requires some extensive regard for a number of considerations. I am planning an entire series of blogs as a set of examples illustrating how to build single-star and close-binary systems from scratch, starting from a variety of known values.

## Notes

- Note that this is the same value we arrived at by calculating the new value for the innermost stable orbit from the newly determined
*D**t*.