The Australia Telescope National Facility web page on binary stars states, “Perhaps up to 85% of stars are in binary systems….” <1> As discussed above, binary star systems are defined either by their observational qualities (how they appear in the sky or in the field of view of a telescope), or by their physical relationship (how they actually interact with one another).

In the case of close-binary systems, there are two types which are

With semi-detached binaries, one of the stars has filled its Roche lobe (the actual physical material of the star has exceeded its own gravitational domain; see below), and begins to lose material to the companion.

In contact binaries, both stars have filled their Roche lobes and begin to share material between them, in some cases actually merging. When such a merger is stable, the resulting body can “…behave like a star [which is] at an earlier stage in its evolution…” because the two stars merge by “…mixing their nuclear fuel and re-stoking the fires of nuclear fusion.” <2> This is related to the way blue stragglers form in globular clusters. However, any planets which may have been orbiting the original two stars quite likely didn’t survive the “reboot”.

The remaining type, detached binaries, can be classed as either close- or wide-binaries, and these are what concern us here.

For close-binaries (also known as "Tatooine" systems), the stars orbit one another at distances between [0.15, 6.0] AU.

For wide-binaries, the separation is far greater, ranging from [120.0, 600.0] AU, for our purposes. (Much greater separations are possible; Proxima Centauri orbits Alpha Centauri AB at a distance of 10,000±700 AU, with a period of perhaps 500,000 years.)

In the case of close-binary systems, there are two types which are

*wholly unsuitable*for habitable systems: semi-detached binaries and contact binaries.With semi-detached binaries, one of the stars has filled its Roche lobe (the actual physical material of the star has exceeded its own gravitational domain; see below), and begins to lose material to the companion.

In contact binaries, both stars have filled their Roche lobes and begin to share material between them, in some cases actually merging. When such a merger is stable, the resulting body can “…behave like a star [which is] at an earlier stage in its evolution…” because the two stars merge by “…mixing their nuclear fuel and re-stoking the fires of nuclear fusion.” <2> This is related to the way blue stragglers form in globular clusters. However, any planets which may have been orbiting the original two stars quite likely didn’t survive the “reboot”.

The remaining type, detached binaries, can be classed as either close- or wide-binaries, and these are what concern us here.

For close-binaries (also known as "Tatooine" systems), the stars orbit one another at distances between [0.15, 6.0] AU.

For wide-binaries, the separation is far greater, ranging from [120.0, 600.0] AU, for our purposes. (Much greater separations are possible; Proxima Centauri orbits Alpha Centauri AB at a distance of 10,000±700 AU, with a period of perhaps 500,000 years.)

Both close- and wide-binary systems may be hosts to planetary systems. (Indeed, systems with three or more stars may have planetary systems by combining close- and wide-binary systems.) For both close- and wide-binary systems, the component stars must be described using the fundamental equations given early in the blog The Nature Of Stars.

Note: It will be necessary to choose either their average separation or their orbital period as a first step, and then calculate the other from the chosen value.

Under

Average separation and orbital period are

Both stars’ orbits will have the same eccentricity and the same orbital period, but the eccentricity must be arbitrarily chosen;

The farthest possible separation between the two stars is determined by the Hill sphere of the primary, and the farther away the secondary is, the more likely that its orbit will be catastrophically disturbed by other passing stars, the galactic bow shock, encounters with interstellar dust clouds, etc., so it is best for habitable systems to keep the maximum separation below about 600 AU, which would definitely be a wide-binary system.

For the Sun, the Hill sphere extends to about 1 light year (≈ 63,000 AU), which equates to a maximum orbital period for a secondary of nearly 16 million years. At such a distance, the orbital speed of the companion star would mean that it would take over 22000 years to move the width of a full Moon in the sky of a habitable planet in the primary’s system, so it would not move in the sky appreciably within the lifespan of any indigenous civilization. It would also have to be particularly large and/or bright to be distinguishable as anything other than just another star in the sky.

In the (numerous) sections that follow, we will look at both close- and wide-binary star systems.

*no circumstances*should close-binary stars be placed less than 0.10 AU from one another, or they risk becoming semi-detached or contact binary systems.Average separation and orbital period are

*independent*quantities; either can be calculated when the other is known, but neither is dictated by the fundamental properties of the stars. Thus, it will be necessary to choose either the average separation or the orbital period of the two stars as a first step, and then calculate the other from the chosen value.Both stars’ orbits will have the same eccentricity and the same orbital period, but the eccentricity must be arbitrarily chosen;

*nothing*in the fundamental data of the stars*dictates*the value of the orbital eccentricity. This gives the Worldbuilder a delightful range of configurations from which to choose.The farthest possible separation between the two stars is determined by the Hill sphere of the primary, and the farther away the secondary is, the more likely that its orbit will be catastrophically disturbed by other passing stars, the galactic bow shock, encounters with interstellar dust clouds, etc., so it is best for habitable systems to keep the maximum separation below about 600 AU, which would definitely be a wide-binary system.

For the Sun, the Hill sphere extends to about 1 light year (≈ 63,000 AU), which equates to a maximum orbital period for a secondary of nearly 16 million years. At such a distance, the orbital speed of the companion star would mean that it would take over 22000 years to move the width of a full Moon in the sky of a habitable planet in the primary’s system, so it would not move in the sky appreciably within the lifespan of any indigenous civilization. It would also have to be particularly large and/or bright to be distinguishable as anything other than just another star in the sky.

In the (numerous) sections that follow, we will look at both close- and wide-binary star systems.

**The Masses Of The Stars**

The primary star is always the more massive of the two, regardless of radius, luminosity, etc. The primary star can actually be the

The primary can be of any mass between 0.08 and ≈ 150.00 solar masses, but if you want a human-habitable system, then the mass range for the primary should fall between 0.744 and 1.162 solar masses. It is convenient to round these figures to a range of [0.7, 1.2].

The secondary star is always the less massive of the pair, regardless of radius, luminosity, etc., and can be any mass above 0.08 solar masses, but less-than-or-equal-to the mass of the primary.

*smaller*and*dimmer*of the two, as long as it is more massive than the secondary star.The primary can be of any mass between 0.08 and ≈ 150.00 solar masses, but if you want a human-habitable system, then the mass range for the primary should fall between 0.744 and 1.162 solar masses. It is convenient to round these figures to a range of [0.7, 1.2].

The secondary star is always the less massive of the pair, regardless of radius, luminosity, etc., and can be any mass above 0.08 solar masses, but less-than-or-equal-to the mass of the primary.

Where:

*M*= the mass of the primary star in solar units;*m*= the mass of the secondary star in solar unitsRemember that bodies of masses below ≈0.08 solar cannot sustain hydrogen fusion and are thus brown dwarfs. In addition, bodies of such a low mass would more likely orbit the Primary more like planets, rather than be considered in a stellar-binary relationship.

**Orbital Eccentricities**

The general concept of orbital eccentricity is covered in Fundamentals, but as a quick reminder, orbital eccentricities range from [0.0, < 1.0], with an eccentricity of zero being a perfect circle. At

Based on observation <3> there

*e**= 1.0*, the ellipse "breaks open" and becomes a parabola.Based on observation <3> there

*does*seem to be a relationship between the orbital eccentricity of binary systems and their orbital period:There is an overlap <4> in the two ranges; this is because of the different orbital configurations which result from how similar or dissimilar are the masses of the two stars. For similar masses, their average separation tends to be smaller, resulting in more nearly circular orbits and shorter orbital periods. However, even for components with significantly different masses, high eccentricities can produce low-eccentricity systems in which the primary orbits entirely within the orbit of the secondary. Consult Part 1: Fundamentals for a complete discussion of orbits and eccentricities.

**The Roche Lobe**

While the Roche limit has been discussed in The Hill Sphere And The Roche Limit, we are concerned here with the

**, which is the region around a star in a binary system within which orbiting material is gravitationally bound to that star. The Roche lobe forms a teardrop shape, with the apex pointing toward the other star, and located at the L1 point of the system. In most cases (for our purposes), it is convenient and sufficient to treat the Roche lobe as a sphere of constant radius and a volume equal to that of the teardrop.***Roche lobe***Calculating Roche Lobes**

The first set of following equations for calculating the Roche lobe are accurate to within about 2%. However, since knowing the radius of the Roche lobe is really only useful in determining if a pair of binary stars are semi-detached or contact binaries—and neither of those situations are suitable for habitable systems—it is largely a matter of academic interest to calculate the Roche lobes at all.

Nevertheless, I include a more complex and precise set of equations directly after the first set, for those who wish to have the information.

Nevertheless, I include a more complex and precise set of equations directly after the first set, for those who wish to have the information.

**Approximate Roche Lobe Calculation**

Where:

𝜆1 = radius of the primary’s Roche lobe in astronomical units;

𝜆2 = radius of the secondary’s Roche lobe in astronomical units;

*a*= average separation between the stars in astronomical units;𝜆1 = radius of the primary’s Roche lobe in astronomical units;

𝜆2 = radius of the secondary’s Roche lobe in astronomical units;

*M*= mass of the primary in solar units;*m*= mass of the secondary in solar units**More Precise Roche Lobe Calculations**

Where:

𝜆1 = radius of the primary’s Roche lobe in astronomical units;

𝜆2 = radius of the secondary’s Roche lobe in astronomical units;

*a*= average separation between the two stars in astronomical units;𝜆1 = radius of the primary’s Roche lobe in astronomical units;

𝜆2 = radius of the secondary’s Roche lobe in astronomical units;

*f*1 = M/m;*f*2 = m/M**Calculating The Barycenter Of Binary Systems**

While I have discussed the barycenter and minimum/maximum separations in detail in Ellipses and Orbits, I include the equations again here for the sake of convenience.

**Distances To The Barycenter**

Where:

α = the average separation between the two bodies <6>;

α = the average separation between the two bodies <6>;

*d**p*= average distance from the primary to the barycenter (in the same units as α);*d**s*= average distance from the secondary to the barycenter (in the same units as α);*M*= mass of the primary body;*m*= mass of the secondary body (in the same units as*M*)**Separations From The Barycenter**

Where:

α = the average separation between the two bodies;

See the previous set of equations for how to calculate

α = the average separation between the two bodies;

*d**p*= average distance from the primary to the barycenter (in the same units as α);*d**s*= average distance from the secondary to the barycenter (in the same units as α);*e*= the eccentricity of the system;*G**p*= minimum distance from primary to barycenter;*D**p*= maximum distance from primary to barycenter;*G**s*= minimum distance from secondary to barycenter;*D**s*= maximum distance from secondary to barycenter;*G**t*= closest approach of the two bodies;*D**t*= widest separation between the two bodiesSee the previous set of equations for how to calculate

*d**p*and*d**s*.**Orbital Period Of Binary Systems**

The orbital period, of course, is the time it takes the two stars to complete one orbit around their barycenter (or around each other, depending on your viewpoint). The equations below will reveal the orbital period when the semi-major axis (average separation) between the stars is known. If the orbital period is the known quantity, the second equation will return the necessary semi-major axis.

Remember, we have to predetermine either the orbital period

Below are the equations for calculating each value when the other is the known quantity.

As discussed in Ellipses And Orbits, the orbital period of a system is calculated by:

Remember, we have to predetermine either the orbital period

*or*the average separation between the two stars.Below are the equations for calculating each value when the other is the known quantity.

As discussed in Ellipses And Orbits, the orbital period of a system is calculated by:

Where:

From this, we can rearrange the equation to reveal the semi-major axis when the orbital period is the known quantity:

*P*= orbital period (perannum);*a*= average separation of the two stars in astronomical units;*M*= the mass of the primary in solar masses;*m*= the mass of the secondary in solar massesFrom this, we can rearrange the equation to reveal the semi-major axis when the orbital period is the known quantity:

**Wait, Wait, Wait! Why Not Use The Simple Relation P² ∝ a³?**

The simple version is adequate when the difference between the two masses is quite large, as it often is between a planet and its star. However, in the case of binary star systems, the two masses are more likely to be very similar, and this makes a significant difference in the outcome of the equation.

For instance, let’s look posit two stars, Marvor and Karus. Further, let’s specify that Marvor has a mass of 1.60 solar and Karus a mass of 0.62 solar. Their average separation is 0.382 astronomical units (remembering that a pair of stars cannot orbit closer than 0.10 AU).

Using the simple version of the orbital period equation, their orbital period calculates as

For instance, let’s look posit two stars, Marvor and Karus. Further, let’s specify that Marvor has a mass of 1.60 solar and Karus a mass of 0.62 solar. Their average separation is 0.382 astronomical units (remembering that a pair of stars cannot orbit closer than 0.10 AU).

Using the simple version of the orbital period equation, their orbital period calculates as

… just less than a quarter of a perannum.

However, using the long version of the equation, which takes into account their masses, their orbital period calculates as:

However, using the long version of the equation, which takes into account their masses, their orbital period calculates as:

… a 49% decrease in the orbital period!

In fact,anysecondary mass of ≥ 0.0002282 times the mass of the primary results in an orbital period shortfall of 0.01142% when using the short form of the equation instead of the long form of the equation. This amounts to a difference of one hour per year.

Jupiter’s mass is 0.0009551 solar, which is 3.38 times the limiting mass above, so to calculate Jupiter’s orbital period accurately, the long form of the equationmustbe used.

Saturn’s mass is 0.0002856 solar, which is 1.00835 times the limiting mass, so, again, using the long version is again the more accurate method.

This allows us to specify a general rule:

If the ratio of the primary mass to the secondary mass is less than 3500:1, (to wit: if the secondary is greater than1/3500the mass of the primary) then the long form of the equation is called for in calculating orbital periods.

**The Forbidden Zone**

In close-binary systems, there is a forbidden zone, which is is the region of gravitational instability in the immediate neighborhood of the binary pair within which no stable planetary orbits may exist. It is based not on the luminosity or masses of the stars, but upon the maximum distance which separates them as they orbit their common barycenter.

It is calculated by:

It is calculated by:

Where

Thus, the forbidden zone is 3.0 times the maximum distance between the stars as they orbit their barycenter. Nothing should approach closer than this limit or it risks destruction by gravitational stresses or being pulled directly into one of the stars.

*D**t*is the widest separation achieved by the two stars, calculated as shown above.

Thus, the forbidden zone is 3.0 times the maximum distance between the stars as they orbit their barycenter. Nothing should approach closer than this limit or it risks destruction by gravitational stresses or being pulled directly into one of the stars.

**Innermost Stable Orbit**

Above, I discussed the innermost stable orbit for single-star systems, specifying that it is calculated either by the mass or the luminosity of the star, depending on whether or not either exceeds 1.0 solar.

There is also an innermost stable orbit for close-binary star systems, but it is based—as the forbidden zone limit above—on the maximum distance which separates them as they orbit their barycenter. The equation is:

There is also an innermost stable orbit for close-binary star systems, but it is based—as the forbidden zone limit above—on the maximum distance which separates them as they orbit their barycenter. The equation is:

Thus, the innermost stable orbit— here designated as

*O**S*to distinguish it from the single-star version—is 4.0 times the maximum distance between the stars as they orbit their barycenter, or 1⅓ times the forbidden zone distance. Note also that the innermost stable orbit can also be calculated from a known forbidden zone distance by:**Habitable Zones And The Frost Line**

The habitable zones and the frost line are calculated for close-binary systems differently from single-star systems, because not only must the luminosities of the stars be taken into account, but also their average separation.

As can be seen in the above illustration, each star has its own habitable zones, and the habitable zones of the overall system are an average of the two sets of orbits.

The equation is:

The equation is:

Where:

*a*= the average separation between the stars in AU;*L*1 = the luminosity of the primary star, in solar units;*L*2 = the luminosity of the secondary star, in solar units;*n*= the ordinal number of the orbit;*m*= the multiplicand for the orbit with ordinal*n*, taken from the table below;Note that here, ordinal six corresponds to the frost line (FL) orbital distance.

Using a system with a primary of luminosity 1.1 solar and a secondary of luminosity 0.9 solar, with an average separation of 1.0 AU, we get an H3 distance from the primary of 1.049 AU, and an H3 distance from the secondary of 0.949 AU, which results in a system H3 distance (from the barycenter) of 1.499AU.

Using a system with a primary of luminosity 1.1 solar and a secondary of luminosity 0.9 solar, with an average separation of 1.0 AU, we get an H3 distance from the primary of 1.049 AU, and an H3 distance from the secondary of 0.949 AU, which results in a system H3 distance (from the barycenter) of 1.499AU.

Rotating the system around the barycenter to simulate the stars orbiting one another, we see that the system H3 distance covers all possible regions covered by the H3 regions of the individual stars.

Here are the equations of the individual orbits, for convenience:

**Maximum Orbital Distances**

**Close-Binary Systems**

For close-binary systems, the maximum orbit a planet (or distant companion star) can have is the combined Hill sphere of the two stars, calculated by:

Where:

*α*= distance between the barycenter and the orbiting object in astronomical units;*M*= mass of the primary star in solar units;*m*= mass of the secondary star in solar units;*n*= mass of the orbiting body in solar units;*H**S*= outer limit of the Hill sphere of the close-binary system, in astronomical unitsNote thatα, here, isnotthe average separation between the two stars, but theaverage separation between the orbiting body and the barycenter around which the two stars also orbit. Thus, I have chosen the Greek minuscule alpha (α) to use in this equation in place of the Latin minuscule (a).

**Wide-Binary Systems**

Whereas the maximum planetary orbit for single-star systems and close-binary systems is the Hill sphere of the star(s), the outermost safe orbit for each star in a wide-binary system must be closer to each star than ⅕ the minimum separation of the two stars:

Where

*G**t*is calculated as shown above.- CSIRO Australia Telescope National Facility and Epping NSW, "Australia Telescope National Facility," Introduction to Binary Stars, April 25, 2018, , accessed July 22, 2018, http://www.atnf.csiro.au/outreach/education/senior/astrophysics/binary_intro.html.
- Information@eso.org, "Vampires and Collisions Rejuvenate Stars," ESA/Hubble | ESA/Hubble, , accessed July 22, 2018, https://www.spacetelescope.org/news/heic0918/.
- Astrophysicsspectator.com, , accessed July 22, 2018, http://www.astrophysicsspectator.com/topics/stars/BinaryStars.html.
- Ibid.
- "Roche Lobe," Wikipedia, July 03, 2018, , accessed July 22, 2018, https://en.wikipedia.org/wiki/Roche_lobe.
- Note that α
*,*here, is different from*a*used in the equations for the ellipse;*a*refers to the semi-major axis of a single ellipse, whereas α, here, refers to the*average separation of the two bodies*in their mutual orbits around the system barycenter.