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Consider a G-class star (0.9457 Msol) as the primary and a M-class dwarf star (0.4335 Msol) as the secondary. If I want a stable planetary system around each (particularly the G-class), then how many astronomical units apart should I place them? I'm currently guessing around 80 AU or more.

Running the sphere of influence calculation for the M-star (with G-star as the primary 80 AU away) gives me ~58.56 AU, so the G-star's outermost planet could orbit at ~21.44 AU. Anything beyond this would be pulled into the M-star's influence. Where my knowledge lacks is in the 'stability of orbits' department. Is there a rough rule-of-thumb estimate that I may use?

Thanks!

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The two stars might have almost circular orbits around their center of gravity or highly elliptical orbits.

The two stars will always be on opposite sides of their center of gravity.

I think that each orbit's shape will be a mirror of the orbit's shape. So when one star is the closest in its orbit to the center of gravity, the other star should be the closet in its orbit to the center of gravity. When one star is the farthest in its orbit from the center of gravity, the other star should be the farthest in its orbit from the center of gravity.

The more massive star should have a smaller orbit, and the less massive star should have have larger obit.

And what is necessary for long term planetary orbital stability is that when the two stars are closest, and their gravitational effects on each other's planetary orbits are strongest, they are far enough apart not to pull the outermost planets of each star out of orbit.

If a star pulls the other star's outermost planet a tiny bit out of orbit each time the stars are their closest point, that will add up a little bit more in every orbit, until after thousands, or tens of thousands, or hundreds of thousands, or millions, of orbits that planet will be lost into interstellar space or collide with another planet, or drop into its star. And the planets in a star system will have to last for billions of years in stable orbits for any of the to develop an oxygen rich atmosphere and become habitable for multicellular land organisms.

The question asks about the stability of a system where two stars, each having planets around it in S-type orbits, have masses of 0.9457 the mass of the Sun and 0.4335 the mass of the Sun.

In the Alpha Centauri system Alpha Centauri A has a mass of 1.0788 Sun and Alpha Centauri B has a mass of 0.9092 Sun. So each component of the system has a larger mass than the corresponding component of the system asked about in the system. So if the separation between Alpha Centauri A and Alpha Centauri B is great enough for planets to stable S-type orbits around each of those stars it would be great enough for planets to have stable orbits if the stars had the lower masses mentioned in the question.

Their elliptical orbit is eccentric, so that the distance between A and B varies from 35.6 astronomical units (AU), or about the distance between Pluto and the Sun, to 11.2 AU, or about the distance between Saturn and the Sun.

https://en.wikipedia.org/wiki/Alpha_Centauri

And there is a third component in the system, Alpha Centauri C or Proxima Centauri.

Currently, the distance between Proxima Centauri and Alpha Centauri AB is about 13,000 AU (0.21 ly),[16] equivalent to about 430 times the radius of Neptune's orbit.

https://en.wikipedia.org/wiki/Alpha_Centauri

The vast distance between Alpha Centauri A & B, and Proxima Centauri means that the gravitational effects between the main pair and Proxima's planets and between Proxima and the main pair would be insignificant.

Could planets have stable orbits in the habitable zones of Alpha Centauri A and Alpha Centauri B? Where are their habitable zones? One way to calculate their habitable zones would to find in the inner and outer edges of the Sun's habitable zone and adjust them to compensate for the difference in the luminosity of their stars.

Stephen H. Dole, in Habitable Planes for Man, 1964 tried to calculate the properties of a planet and star system capable of having planets habitable for humans and other beings with similar requirements (and thus interesting for most types of science fiction stories).

https://www.rand.org/content/dam/rand/pubs/commercial_books/2007/RAND_CB179-1.pdf

On pages 63 to 66 Dole discussed the size of the "ecosphere" around a star where Earth like planets can have habitable temperatures - the modern term is the circumstellar habitable zone. Dole decided the habitable Zone extended from 0.723 Astronomical Units (AU) to 1.24 AU. And Dole calculated the sizes of the habitable zones of other stars by adjusting those distances to account from the relative luminosity of those stars.

In chapter 6 on pages 106 to 107 Dole calculated the probabilities of several nearby star systems having habitable planets. On pages 111-112 he made a rough estimate that planets orbiting Alpha Centauri A would have stable orbits within 2.68 AU and around B would have stable orbits within 2.24 AU, and that the habitable zones of both stars were within those limits.

Wikipedia discusses the possibility of planets in noncircumbinary or S-Type orbits having stable orbits.

In non-circumbinary planets, if a planet's distance to its primary exceeds about one fifth of the closest approach of the other star, orbital stability is not guaranteed.3 Whether planets might form in binaries at all had long been unclear, given that gravitational forces might interfere with planet formation. Theoretical work by Alan Boss at the Carnegie Institution has shown that gas giants can form around stars in binary systems much as they do around solitary stars.6

Studies of Alpha Centauri, the nearest star system to the Sun, suggested that binaries need not be discounted in the search for habitable planets. Centauri A and B have an 11 au distance at closest approach (23 au mean), and both have stable habitable zones.[2][7] A study of long-term orbital stability for simulated planets within the system shows that planets within approximately three au of either star may remain stable (i.e. the semi-major axis deviating by less than 5%). The habitable zone for Alpha Centauri A extends, conservatively estimated, from 1.37 to 1.76 au2 and that of Alpha Centauri B from 0.77 to 1.14 au2—well within the stable region in both cases.[8]

https://en.wikipedia.org/wiki/Habitability_of_binary_star_systems#Non-circumbinary_planet_(S-Type)

The list here:

https://en.wikipedia.org/wiki/Habitable_zone#Solar_System_estimates

Shows that there have been over a dozen estimates and calculations of the limits of the Sun's habitable zone since Dole wrote, some differing greatly from others.

I think that at the present time the most commonly used definition of the limits of the Sun's habitable zone is that by Kasting et al in 1993.

https://pubmed.ncbi.nlm.nih.gov/11536936/

At the present time Proxima Centauri or Alpha Centauri C has two confirmed planets.

Proxima Centauri d orbits too close and is too hot to be habitable.

Proxima Centauri b orbits its parent star at a distance of roughly 0.05 AU (7.5 million km; 4.6 million mi) with an orbital period of approximately 11.2 Earth days. Its other properties are only poorly understood, but it is believed to be a potentially Earth-like planet with a minimum mass of at least 1.07 M🜨 and only a slightly larger radius than that of Earth. The planet orbits within the habitable zone of its parent star; but it is not known whether it has an atmosphere.

There is a candidate planet of Alpha Centauri A>

Candidate 1 (also known as C1 or Alpha Centauri Ab) is an unconfirmed exoplanet candidate directly imaged around Alpha Centauri A in February 2021. If confirmed as an exoplanet, it would orbit at approximately 1.1 AU away from Alpha Centauri A with a period of about a year and would have a mass between that of Neptune and one-half that of Saturn and would therefore likely be a gas giant.1 Despite being a gas giant, due to its position in its orbit with Alpha Centauri A, it could have habitable moon(s) in its own orbit. The planet candidate is yet to be confirmed as an exoplanetary signal; additional observations are needed to confirm its true nature.

https://en.wikipedia.org/wiki/Candidate_1

There are no confirmed planets around Alpha Centauri b.

If Candidate 1 is conformed to be a planet, it would prove that a stable orbit is possible within 1.1 AU of Alpha Centauri A. Unfortunately the inner edge of Alpha Centauri's habitable zone is considered to be probably 1.37 AU from Alpha Centauri B.

Groombridge 34 is a nearby double star. Groombridge 34 A has two planets, and the farther, Groombridge 34 Ac, has an orbital period of 7,600 days. With a mass of 0.38 Sun for the star, and a mass of 35 Earths for the planet, an orbital period of 7,601 days would be at a distance of about 5.481 AU. the two stars are separated by about 93 AU and the habitable zone of Groombridge 34 A would be less than 1 AU from the star.

The two stars in EQ Pegasi are separated by about 36 AU. The known planet of EQ Pegasi A orbits at 0.643 AU which is beyond the habitable zone of the dim red dwarf star.

ADS 7251 or Gliese 338 is a double star system with the stars separated by about 107 AU. The known planet orbits at a distance of about 0.141 AU.

In short, there are examples of planets in S-type orbits of binary star systems where the orbits are stable.

The two stars in the system in question have masses of 0.9457 Sun and 0.4335 Sun.

The star with a mass of 0.9457 Sun would be about halfway between a G8V star with 0.94 the mass of the Sun and 0.68 the luminosity and a G7V star with 0.95 the mass of the Sun and 0.74 the luminosity.

If you want to have only one habitable planet around each star and for that habitable planet to the outermost planet of its star, you should probably put that planet at the distance where it gets exactly as much radiation from the star as Earth gets from the Sun. I call that the Earth Equivalent Distance or EED.

For a star with 0.68 the luminosity of the Sun the EED should be about 0.824 AU and for a star with 0.74 the luminosity of the Sun the EED should be about 0.876 AU.

The star with a mass of 0.4335 Sun would be much closer to a M2V star with 0.44 the mass of the Sun than to a M3V star with 0.37 the Mass of the Sun. So I will go with a M2V luminosity of 0.029 Sun, and thus an EED near 0.17 AU.

As a rough rule, the stars in a binary should be separated by at least 5 times the largest semi-major axis of any planet in an S-type orbit if the orbits are to be stable.

So if it is desired for planets at the EED to be the outermost of each star, the minimum separation between stars should be 5 times 0.876 AU, or 4.38 AU.

If it is desired that the farthest planet of either star orbits at 10 AU, the minimum separation of the two stars should be 50 AU. If it is desired that the farthest planet of either star orbits at 20 AU, the minimum separation of the two stars should be 100 AU. If it is desired that the farthest planet of either star orbits at 30 AU, the minimum separation of the two stars should be 150 AU. And so on.

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I don't have a value for you, but I am pretty sure 80 AU is not enough.

The calculation of the sphere of gravitational influence is not a good measure for estimating the stability of the system, since it only tells you where the gravitational pull of the body is the greatest among those present there.

Just to give you two counterexamples:

  • Earth is not in Jupiter Hill's sphere, yet its gravity affects its motion and orbit
  • think of a case where, the body's gravity being 1, the other's gravity is 0.9: its effects will be all but negligible.
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