Habitable zone of this red giant

Question

I have a Red Giant star of about ~4.15 solar masses (a number I generated from what I found to be the average size of red giants).

My question for right now would be "What would be the habitable zone of this star?"

I am aware that Red Giants don't live long enough to support the growth of life, so to resolve this particular issue I'm deciding that the planet that will sustain life was an exo-planet that was pulled into this star's gravitational reach, conveniently within the habitable zone, whatever that will be. How life came to be is not the issue, the problem is keeping it alive around this particular red giant.

It is important that the star remains a red giant, as it is narratively imperative that the characters face an imminent cosmic doom that they must try to escape.

I would really like this to work in a way that's as much as possible based in science, but, if need be, "it's just magic, bro" is always an option, if something just really, really, can't work.

Answer 1

This question should be answered in two steps:

1. What is the current (stable) luminosity of the star?
2. What is a habitable temperature range for the planet?

For question one I found this link:

https://pressbooks.bccampus.ca/.....evolution-from-the-main-sequence-to-red-giants

Note that not all stars evolve the same, so there are no exact numbers. In the diagram at the bottom we see that a star of 3 solar masses has a luminosity of about 100 suns before it goes boom and a star of 5 solar masses has a factor of almost 1000. So we could take a number of around 300 suns in luminosity. Note that also the lifespans are given, and as you can see a 5 solar mass sun lives only 70 million years.

For the habitable zone calculation also some rule of thumbs exist. Most link back to this page:

https://www.planetarybiology.com/calculating_habitable_zone.html

The light a planet receives drops of with the square of distance. So the relevant equations are these: $$r_i=\sqrt{\frac{L_{\text{star}}}{1.1}},\quad r_o=\sqrt{\frac{L_{\text{star}}}{0.53}}$$ Where $r_i$ is the inner and $r_o$ the outer boundary of the zone. Plugging in 300 gives: 16.5 AU and 24 AU respectively. These are not exact. The properties of the planet also play a huge role in the temperature it stabilizes at. But these are the ballpark figures.

Answer 2

Part One: Orbital Distance.

You need to find out a plausible luminosity for a red giant star with a mass of about 4.15 times the mass of the Sun.

If you find that such a star has a luminosity X times that of the Sun, you then find the square root of that ratio X. If the ratio is 400, for example, the square root will be 20.

So find the inner and outer edges of the Sun's habitable zone, and then multiply them by the square root of X to get the inner and outer edges of your star's habitable zone.

As can be seen here:

[https://en.wikipedia.org/wiki/Circumstellar_habitable_zone#Solar_System_estimates][1]

There have been over a dozen estimates and calculations of the inner, or outer, or both, edges of the Sun's habitable zone. And some of them greatly disagree with others.

So if a writer plans to have one and only one habitable planet in the star system in their story, they can put that planet at the distance from the star where it will receive exactly as much radiation from their star as Earth gets from the Sun. I call that distance the Earth Equivalent Distance or EED.

Since Earth orbits the Sun at one Astronomical Unit or 1 AU, to find the EED of a star with a specific luminosity, find the ratio of that star's luminosity divided by the luminosity of the Sun, then find the square root of that ratio, and then multiply 1 AU by that square root.

Part Two: The Age of the Planet.

Since the red giant stage of of a star's existence is usually shorter than the main sequence stage, and since a planet needs to have fairly stead illumination from its star for billions of years for life to produce a breathable oxygen rich atmosphere, it is uncertain whether a planet in the habitable zone of a red giant will have stead illumination from the star long enough to establish a breathable atmosphere. Maybe the first and most primitive forms of life will appear on the planet and then die out long before they can produce an oxygen rich atmosphere.

You suggested that your red giant star might capture a planet billions of years old from another star and that planet might chance to take up orbit right in the habitable zone of the red giant.

The odds against that happening to any random red giant would be astronomical. But of course the number of stars which were, or are, or will be, red giants is also astronomically vast. Thus Such a planet could exist somewhere in the universe.

And I would like to suggest an alternative. Maybe 50 million years ago and advanced society decided that your planet orbiting your red giant star would have about 60 milion years of temperatures suitable for liquid water using life. So they spent thousands of years terraforming that planet to be habitable, and then some of them lived there for 50 millions years. After fifty million years they ceased inhabiting that planet many thousands of years ago, and it now has about 10 million years of habitability left before changes in the red giant star will make it too hot or too cold, as the case may be, for life.

A writer should consider which of those those two possible explanations for a planet of a red giant having a habitable planet they consider most suited for their story.

Part Three: A Third Explanation.

Wikipedia's article on red giants discusses how long a planet in their habitable zone could have habitable temperatures.

Although traditionally it has been suggested the evolution of a star into a red giant will render its planetary system, if present, uninhabitable, some research suggests that, during the evolution of a 1 M☉ star along the red-giant branch, it could harbor a habitable zone for several billion years at 2 astronomical units (AU) out to around 100 million years at 9 AU out, giving perhaps enough time for life to develop on a suitable world. After the red-giant stage, there would for such a star be a habitable zone between 7 and 22 AU for an additional one billion years.[21] Later studies have refined this scenario, showing how for a 1 M☉ star the habitable zone lasts from 100 million years for a planet with an orbit similar to that of Mars to 210 million years for one that orbits at Saturn's distance to the Sun, the maximum time (370 million years) corresponding for planets orbiting at the distance of Jupiter. However, planets orbiting a 0.5 M☉ star in equivalent orbits to those of Jupiter and Saturn would be in the habitable zone for 5.8 billion years and 2.1 billion years, respectively; for stars more massive than the Sun, the times are considerably shorter.[22]

So possibly you might want to make you planet orbit a 0.5 solar mass red giant at the orbital distance of Jupiter to make it be habitable long enough.

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