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Is there a habitability zone between the primary and secondary stars in a binary star system for a planet orbiting only the primary star at a distance less than that of the secondary star and, if so, what is the greatest energy output of the secondary star (in terms of energy received by the planet) that would permit the habitability zone?

  • Assume the primary star is equivalent to Sol.
  • Assume the planet is equivalent size and mass of the Earth.

Consider a 3D chart:

  • The X-axis is distance of the secondary from the primary.
  • The Y-axis is the energy output of the secondary
  • The Z-axis is habitability (0-100% liklihood of a habitability zone)

I can easily assume that as X approaches infinity habitability approaches 100%. Further, I can assume that as Y approaches zero, habitability approaches 100%. It's the space in between that I'm having trouble understanding.

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    $\begingroup$ You haven't provided enough information/said to ignore the stellar mechanics that would be necessary to determine the answer. $\endgroup$ – rek Feb 23 '18 at 4:54
  • $\begingroup$ @rek Can you reprise your comment. because the phrase(s) "You haven't provided enough information/said to ignore the stellar mechanics" doesn't seem to make sense. I think there is an excellent point buried deep within it. $\endgroup$ – a4android Feb 23 '18 at 5:51
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    $\begingroup$ I closed the question, so it could get re-opened with a clean slate. Perhaps you should edit your repose to comments into the main question? Either go with the last sentence of the 'response to comments' as the main question, or otherwise I was thinking go for a reality check where you say 'there are two sol-sized stars, is there any orbital configuration where an Earth-sized planet orbits in a habitable zone between them.' Edit it to re-open and I will vote to do so. $\endgroup$ – kingledion Feb 23 '18 at 11:40
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    $\begingroup$ Also, check out this answer. $\endgroup$ – kingledion Feb 23 '18 at 11:41
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    $\begingroup$ Related (I think; I'm having a little trouble understanding what you're asking for): worldbuilding.stackexchange.com/q/25166/28 $\endgroup$ – Monica Cellio Feb 23 '18 at 22:24
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Yes, there is a habitable zone.

Known examples of possibly-habitable binaries

I have to disagree with StephenG's answer; we have data that indicates that this is possible for similar, Sun-like stars. I talked about this in an answer I wrote a few months ago; searching this exoplanet catalog, I found several systems that might be of interest:

  • WASP-94, a pair of F-type main sequence stars, each with a close hot Jupiter orbiting it.
  • HD 20781/HD 20782, two G-type stars, each with 1-2 planets (one at 1.3 AU orbiting HD 20782, two at <1 AU orbiting HD 20781).
  • Kepler-132, a pair of G-type main sequence stars, although the structure of the system has been disputed.
  • XO-2, two cool K-type stars, with a confirmed hot Jupiter around one star and two possible planets around the other.

The HD 20781/HD 20782 system has me quite excited. Both are G-type stars, and each star has at least one planet within 2 AU of it. The planets are all more massive than Earth, but that's immaterial; the important thing is that the binary stars have a separation of 9080 AU! That's enormous, and it's absolutely enough for there to be relatively little effect on each planet from the other star in the system.

Some things to note:

  • HD 20782 b has a large eccentricity, and HD 20782 b and HD 20781 c also have larger eccentricities than normal, which could be due to the binarity of the system. The XO-2 system's planets seem to have smaller eccentricities, even though their separation is a mere 4600 AU.
  • In most of these systems, both stars are extremely similar in spectral type, which seems like a good thing. They're relatively Sun-like, not active red dwarfs or hot massive stars.
  • StephenG's requirements of a large separation is easily satisfied in several of these cases, by an order of magnitude or two. Given the inverse square law for flux, I would expect the contribution from the second star to be many orders of magnitude lower than the primary; it's essentially zero.

Calculating the habitable zone

I did some modeling (Python 3 code on Github) to give some numerical support to this answer, so I wrote a program that generates habitable zones around binary systems, with certain assumptions:

  • Both stars are on the main sequence
  • The stars' orbits have zero eccentricity
  • Any exoplanet orbiting the stars won't be massive enough to influence the stars' orbits, and the system is stable

I defined the habitable zone as the region where water is liquid on the surface of a planet. In other words, the planet's effective temperature - not taking into account greenhouse effects - must be between 273.15 K and 373.13 K. The formula for the effective temperature of a planet in a binary system is $$T=\left(\frac{1-a}{4\sigma}(F_1+F_2)\right)^{1/4}$$ where $a$ is the albedo of the planet and $F_1$ and $F_2$ are the fluxes from the stars.

Here are three basic plots, of a single Sun-like (G2V) star, two Sun-like stars separated by 2 AU, and two Sun-like stars separated by 5 AU. All assume a planetary albedo of 0. The habitable zone is shaded in black (the precise temperature is not shown):

A single G2V star

Two G2V stars, separation 2 AU

Two G2V stars, separation 5 AU

The effects from the second star are apparent with the 2 AU separation, but not with the 5 AU separation (although I can confirm that they're there). The form of the effective temperature formula means temperature only varies as $T\propto F^{1/4}$, where $F$ is flux, and thanks to the inverse-square law, even a separation of 5 AU produces minor results.

Here's a plot where the separation is 9080 AU, as in the HD 20781/HD 20782 system:

A G3V and a G9V star, separation 9080 AU

The other star is far off my screen. Zooming out makes it impossible to see. For the purposes of habitability calculations, each star is on its own.

Orbital stability

Now, a class of binaries I'm curious about are late-type dwarfs with relatively small separations (1-2 AU). Two M-type red dwarfs can orbit close together and still have their own individual habitable zones:

Two M0V stars, separation 1 AU

What I don't know is what the stable orbits are around these stars. I assume some stable orbits are possible for the above case, but I don't know the ranges, and would be interested to find out. Another interesting scenario is two K5V stars at 1 AU; their habitable zones are connected, but the zones are stable orbits may be much smaller:

Two K5V stars, separation 1 AU

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Yes, and right next door

Weigert and Holman, 1997 concludes that

The habitable zone for planets, as defined by Hart (1979), lies about 1.2–1.3 AU (1′′) from α Cen A. A similar zone may exist 0.73–0.74 AU (0.6 ′′) from α Cen B. From our investigations, it appears that planets in this habitable zone would be stable in the sense used here, at least for certain inclinations.

This is confirmed more recently and with more powerful software in Quarles and Lissauer, 2016:

Our simulations show that circumstellar planets (test particles), within the habitable zone of either α Cen A or α Cen B, remain in circumstellar orbit even with moderately high values of initial eccentricity or mutual inclination relative to the binary orbital plane

Reading through their paper, they simulated stability for > 1 billion years for particles in the habitable zones of both $\alpha$ Cen A and $\alpha$ Cen B (and also circumbinary orbits that would not be habitable).

So, as far as we know, there exists a habitable zone between the primary and secondary of the nearest (non-red dwarf) star to us. $\alpha$ Cen A is very similar to our sun (1.1 solar masses, 1.5 solar luminosities, spectral type G2V just like Sol). The luminosity of $\alpha$ Cen B is 0.5 times that of Sol, so the companion can be pretty bright in these circumstancs.

How close can they be?

To answer HDE's addendum/bounty, I fired up my trusty Rebound tool to find the closest orbits that suggest stability. I did a bunch of grid searches over various eccentricities, and the finding was that for the stellar masses below, eccentricity has relatively little effect (at least for low eccentricities < 0.1).

The computational demands of this problem proved to be much higher than in previous questions. I tried to test 10s of thousands of cases over 1 million years, integrating with a time step of 0.001 years (about 8 hours). I found some interesting cases and some generalizations about behavior, but take these answers with a grain of salt. 1 million years isn't enough to prove anything.

Case: Two stars, both of 1 solar mass

Here we have some very interesting behaviour. Some planets will break out their orbit and orbit the barycenter of the system. Starting with the companion 3.5 AU from the primary, planet 1 AU from primary, and all orbits with 0 eccentricity, the planet did a horseshoe orbit at ~0.87 AU from either star for a million years. It was actually very close to the setup in this question.

For the case of the companion star being n AU away from the primary, the effects on the planet are:

0 - 3 AU    Planet is quickly ejected
3 - 4 AU    Planet achieves an eccentric but stable orbit near the habitable zone
4 AU +      Planet achieves a stable orbit outside the habitable zone

The real finding here is that for suns of equal size, a planet is likely to end up near the barycenter of the two suns. The planet also very quickly achieves stability in the equal-mass sun setup, whereas in the following examples, the orbits are chaotic for more than a million years. I would suggest that in order to get the planet in an orbit in the habitable zone of the one of the suns, you would need to add other planets to the mix.

Case: Two stars, one of 1 solar mass, one a large red dwarf (M1V, m = 0.5 Sols)

In this case, there are a good variety of stable orbits once the companion is at least 5.5 AU away from the primary. I didn't find any orbits that were stable in the habitable zone, though. Stable orbits for the planet tended to start about 2.5 AU away from the main, in some sort of resonance with the companion. Unfortunately, my 1 year old powered off my computer before I could read the final results of the 8 hour grid search for stable outer orbits. That is what you get for writing to the console and not a file. Whose idea was it to make power buttons have LEDs anyways? Those things are toddler magnets.

For the case of the companion star being n AU away from the primary, the effects on the planet are:

0 - 3.5 AU    Planet is quickly ejected
3.5 - 5.5 AU  Planet enters eccentric orbit in vicinity of habitable zone. May be 
              eventually ejected, unlikely to be stable in habitable zone.
5.5 AU +      Planets in the habitable zone are pulled outwards into resonances 
              with the companion star

As with the last simulation, planets tend to be pulled towards the barycenter. This suggests that additional planets maybe necessary to straighten out eccentricities. However, it is also worth noticing that this simulation is close to the ones cited above related to Alpha Centauri, and it does not replicate the results. So, perhaps an extra big grain of salt needs to be taken with this entire endeavor.

Case: Two stars, one of 1 solar mass, one a small red dwarf (M6V, m = 0.1 Sols)

Beyond 2 AU from the primary star, the companion star is too small to immediately eject the planet from the system. However, almost all of the orbits I plotted remained unstable for 1 million years, implying that they will eventually lead to ejection (or collision with the primary star! which did happen in one case). The two important relationships appeared to be the distance of the barycenter of of the system from the main star, and orbital resonances between the companion star and the planet.

In general there were the following zones of interest based on distance between the primary and companion:

< 2 AU      The planet is quickly ejected 
2 - 4.5 AU  The planet finds a somewhat stable, but highly eccentric orbit
4.5 - 5 AU  The planet quickly enters a stable orbit at ~0.55 AU 
5 - 9 AU    The planet enters an eccentric orbit, and may stabilize in a resonant 
            orbit with the companion
9 - 12 AU   Same as above, but the barycenter is in the habitable zone, so a 
            stable orbit there is probably impossible
12 + AU     The planet enters an eccentric orbit, and may stabilize later (none of 
            these did within 1 million years)

I did not find a single orbit of the tens of thousands tried that ended up stable in the habitable zone within 1 million years. However, the 5-9 AU and 12 + AU cases both contained eccentric orbits with roughly the correct semi-major axis, so it would be possible for these to stabilize out given enough time.

In progress

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  • $\begingroup$ Thank you! I was really hoping you'd end up working on this. $\endgroup$ – HDE 226868 Aug 22 '18 at 1:28
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Not for two large, similarly sized stars, maybe for one large star and one dwarf star (like a brown dwarf).

The former case would not allow a stable orbit to form for long enough for the planet to develop a reasonably consistent climate (essentially driven by a nearly constant level of solar energy) over a period large enough to develop life.

The case with the brown dwarf (a very dim type of star) can be though of as a system with a very large Jupiter that's still much smaller than the Sun, and with a much, much lower energy output. Such a brown dwarf could, in principle, be far enough from the planet to not greatly affect it.

There is a quantity known as the effective temperature which let's us estimate the approximate temperature effect of a star on a planet. We can use this to relate the effect of the smaller star (the secondary) on the temperature of the planet (dominated by the primary, for stability).

We get a formula like this :

$$\frac {T_{sec}}{T_{prim}} = \left( \frac{L_{sec}D_{prim}^2}{L_{prim}D_{sec}^2} \right)^{\frac 1 4}$$

where the $L$ values are luminosity of the stars and the $D$ values are distance from the planet.

We want this to be a small number, like a percent or two at most for a reasonably stable climate.

So some very rough order of magnitude calculations :

Now a possible value for $L_{sec} \approx L_{prm} \times 10^{-3}$, and if we set $\frac {T_{sec}}{T_{prim}} \approx 2 \times 10^{-2}$ (about a 4% variation in temperature due to the secondary), we get :

$$D_{sec} \approx 79 D_{prim}$$

So the secondary has to be about 80 times further from the primary than the planet is. Both would be in roughly circular orbits at these ranges, and the brown dwarf could be of the order of about 50 Jupiter masses.

These are, of course, ballpark figures.

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    $\begingroup$ How would you define "large"? For two Sun-like stars - or maybe F-type stars, a bit more massive - there should be plenty of time for life to arise. It's only when you get to the massive B- and O-type stars that trouble begins. $\endgroup$ – HDE 226868 Feb 23 '18 at 5:44
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    $\begingroup$ The OP is talking about two sun-like stars in a binary system with a planet in between (which I'd call large, although strictly speaking the Sun is not an unusually large star). My view (and I'm happy to be corrected), is that no arrangement of that sort is going to make a stable orbit (for the planet) while keeping temperatures stable as well. The OP doesn't seem to mean circumbinary systems which makes it difficult I think. $\endgroup$ – StephenG Feb 23 '18 at 6:40
  • $\begingroup$ Question was edited. If anything, I think your answer fits it more, not less, but you may wish to review it. $\endgroup$ – Mołot Feb 23 '18 at 16:16
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    $\begingroup$ This doesn’t seem right, stars larger than brown dwarves should be possible simply by placing them farther way. Proxima Centauri is larger than a brown dwarf, yet it’s orbit around Alpha Centauri does not preclude planets with a stable orbit, AFAIK. $\endgroup$ – RBarryYoung Aug 20 '18 at 19:42
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    $\begingroup$ So apparently, even little Proxima Centauri has a planet in its habitable zone, even though it is in a ternary system (Alpha Centauri) with two other stars that are much larger than a single brown dwarf (they are approximately sun-sized), space.com/33834-discovery-of-planet-proxima-b.html $\endgroup$ – RBarryYoung Aug 20 '18 at 19:47
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First, to help clarify the orbit in the question posted, according to arXiv:0705.3444 that planets orbiting a single star within a binary system, is called an ‘S-type’ orbit (while a circumbinary orbit is of ‘P-type). This paper S-Type and P-Type Habitability in Stellar Binary Systems: A Comprehensive Approach explores various scenarios and their effect and limitation on possible habitable zones for said S-Type planets, though in heavily math-based, less conceptual-based terms.

However, as regarding to the orbital stability of an S-Type planet, Solstation.org on this page, referred other papers when indicating that planets with orbits that are less than 1/5 the closest approach of the secondary star are generally stable. It also mentioned that there were existing observed binaries in which dust rings and possibly planets appeared circling only one star, though far fewer exist in binaries with intermediate separations between 3-50 AU (below 3 AU there were some circumstellar P-Type rings) and it's above 50 AU that the S-type rings around a single star were observed).

So, the limitation on habitability is primarily restricted by the requirement of a steady insolation falling on it, not the stability of its orbit. Also, if you use StephenG’s equation as a ball park estimation, the closest the secondary star can be to the primary for various luminosities can be determined (with equal ball park accuracy) by maintaining the value in parenthesis. Therefore, as your change the luminosity of the secondary star L$_{sec}$ by a given percent, you have to change its distance (~80$D_{prim}$) by the square root of that percent (i.e. quadruple the luminosity of the secondary and it now has to be twice the original distance of about 80$D_{prim}$ to maintain the same relative insolation - up to some maximum limit below the primary’s luminosity, which I was not able to determine from the paper and not stated directly in it). Though it showed mathematically why equal stars could not support a habitable planet.

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