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I am confused over the habitable zone, as I calculated the expected temperature of Earth (minus greenhouse effects), and it would be -17 C. Which is below the freezing point of water.

I also made a fictional A-type star with a luminosity of 20, and its habitable zone would be 4.26401432711 to 6.14295116834 AU given the formulae of sqrt(L/1.1) and sqrt(L/0.53) to define boundaries. However, given the albedo of 0.08, it would be 121.5919576 to 55.72744259900003 C. So, the end of the habitable zone is past the boiling point of water, and the furthest reaches would still be far, far above the freezing point of water.

Edit Additon: For context, the planet is a water planet (thus the Albedo of 0.08), with a relatively Earth-like atmosphere. If that can help find the habitable zone for it in regards to the star in question. Because I can't find one that works out well. Does 10.126 AU work out? That would make it recieve the same amount as Earth. It seems.

The temperature formula I am using is 4th sqrt(L(1-a)/16πd2ơ) with d being distance from the star in metres, L being the luminosity in watts, and ơ being the Stefan–Boltzmann constant.

This led me to questioning what exactly the habitable zone means, given it doesn't seem to actually follow the points where liquid water could exist. Can somebody try to elucidate me on what I am doing wrong here? Should I just try using the temperature formula for it and ignore the habitable zone calculation?

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  • $\begingroup$ Comments have been moved to chat; please do not continue the discussion here. Before posting a comment below this one, please review the purposes of comments. Comments that do not request clarification or suggest improvements usually belong as an answer, on Astronomy Meta, or in Astronomy Chat. Comments continuing discussion may be removed. $\endgroup$
    – Connor Garcia
    Sep 27, 2023 at 1:57

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The reason you get a too cold temperature for Earth is of course because you ignored the greenhouse effect. The atmospheric composition strongly affects the actual temperature of planets, which affects the atmospheric composition and albedo.

This is why the simple $k\sqrt{L}$-style formulas for the habitable zone are very rough simplifications and should not really be taken as an accurate range for liquid water to be able to exist.

Exactly what a stated zone means depends a fair bit on the author. Many use some standardized planet and calculate a range for liquid water using the energy balance equation with some assumed albedo, but it is not uncommon to see the simple approximation or more elaborate atmospheric models. Some authors bring in other aspects of livability. In short, it is a mess, and there is no standard "true" habitable zone across the literature.

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  • $\begingroup$ I am well aware, I mentioned that in my comment. But my confusion was just in the fact that the concept of a Habitable zone would just make no sense if the atmosphere was relevant to how it considers things. As, for instance, Venus is considered to be in the habitable zone, even though its atmosphere leaves it unhabitably hot (amongst other issues like toxcicity, but that isn't relevant to the discussion). This also doesn't answer the main thing I was discussing, which was about the calculation for the hypothetical planet I was doing the math for. $\endgroup$ Sep 24, 2023 at 20:34
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    $\begingroup$ I just added more details in my original post. (sorry for the double post, but Stack Exchange has a weird ban on editing messages after 5 minutes) $\endgroup$ Sep 24, 2023 at 20:41
  • $\begingroup$ @DanceroftheStars - Note that the albedo of the planet will strongly depend on how much ice there is on it. Further out there will be ice sheets, making it go up. Similarly there will be clouds reflecting light, and they likely increase as you go in towards the star. You can't just treat the albedo as constant in this case. And, yes, this means a non-waterworld may well have a somewhat different zone where water remains liquid. $\endgroup$ Sep 26, 2023 at 13:30
  • $\begingroup$ @DanceroftheStars Well, but you are completely correct, the concept of the HZ does make no scientific sense. It is only a testable hypothesis if you presume Earth-identical conditions, which no one believes in. All other atmospheric conditions are completely unknown, and hence the question "is it in the HZ for those unknown conditions" cant be tested, and is therefore unscientific. $\endgroup$ Sep 26, 2023 at 21:02
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You should take the inner and outer edges of the habitable zone of the Sun as your model. You say your star has luminosity of 20. If that is 20 times the luminosity of the Sun, there is a comparatively simple method to adjust the size of the Sun's habitable zone to fit your star.

The square root of 20 is 4.47213. So you multiply the inner and outer edges of the Sun's habitable zone by 4.47213 to get the inner and outer edges of your fictional star's habitable zone.

So what are the inner and outer edges of the Sun's habitable zone?

Here is a link to a list of about a dozen scientific estimates of the inner and outer edges, or both, of the Sun's habitable zone made in the last 60 years.

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

And some of the estimates are very different from some others.

So which estimates do you use in your story? The ones which are best for your story if you are a typical lazy science fiction writer, or the ones which make the most scientific sense to your if you are a conscientious science fiction writer.

And to find the estimates of the Sun's habitable zone which seem most scientifically plausible to you it is necessary to read the scientific articles proposing each estimate.

If you are certain you want one and only one habitable world in your fictional star system, you cans simply put that world at what I call the Earth Equivalent Distance or EED of the star. The EED of a star is the distance at which it receives the exact same amount of radiation from its star as Earth gets orbiting at a distance of one Astronomical Unit or AU from the Sun.

Divide the star's luminosity by that of the Sun, and find the square root of that ratio, and then multiply 1 AU by that square root of the ratio, and you will get the EED of that star. A star with 20 times the luminosity of the Sun would have an EED at 4.4721359 AU.

A star with 20 times the luminosity of the Sun would be in between an A3V star with 16.98 times the luminosity of the Sun and an A2V star with 22.99 times the luminosity of the Sun.

https://en.wikipedia.org/wiki/A-type_main-sequence_star#:~:text=They%20measure%20between%201.4%20and,expected%20to%20harbor%20magnetic%20dynamos.

Added 09-26-2023

You write:

For context, the planet is a water planet (thus the Albedo of 0.08), with a relatively Earth-like atmosphere.

I wonder what you mean by a "relatively Earth-like atmosphere."

Do you mean an atmosphere with a high oxygen content, so that humans can sail on the surface without wearing breathing gear, and so there can be multicellular plants and animals native to the planet?

If so, you should know that Earth has had a oxygen rich atmosphere for a "mere" 600 million years. It took four billion years before that for life to start and then for photosynthetic lifeforms to evolve and start producing free oxygen and for a large amount of oxygen to eventually build up in the atmosphere.

So a planet with an oxygen rich breathable atmosphere should be billions of years old, and its star should have been shining with a fairly steady luminosity for those billions of years. That means the star should have been in the main sequence phase of its existence for all those billions of years and has not yet become a red giant star, destroying all life on its planets.

As it happens, the amount of time that a star spends on the main sequence depends on its initial mass. The higher the mass the shorter the time spent on the main sequence.

Stephen H. Dole, in Habitable Planets For Man (1964), a very useful book for science fiction writers, discussed the types of stars suitable for having planets habitable for humans.

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

On pages 67 to 72 he discussed the s type of stars suitable for having habitable planets.

Dole decided that a planet might possibly develop a breathable oxygen rich atmosphere in only 3 billion years, 0.75 as long as it took Earth. And on page 68 he says that only main sequence stars of spectral class F2 and stars of lower mass can remain the main sequence for 3 billion or more years and so possibly have planets with oxygen rich atmospheres.

Spectral class A stars are more massive than spectral class F stars and so remain on the main sequence for short periods of time.

Vega (Alpha Lyrae) is a class A0V star with 40 times the luminosity of the Sun.

At present, the star is about 455 million years old. Vega will leave the main sequence in about 500 million years and bloat into a red giant before expelling its atmosphere and evolving into a white dwarf encircled by a planetary nebula.

https://skyandtelescope.org/astronomy-news/vega-the-star-at-the-center-of-everything/#:~:text=Vega%20is%20a%20bluish%2Dwhite,is%20about%2010%20times%20older.

Fomalhaut A is an A3V class star with 16.63 times the luminosity of the Sun.

Fomalhaut is a young star, for many years thought to be only 100 to 300 million years old, with a potential lifespan of a billion years.[37][38] A 2012 study gave a slightly higher age of 440±40 million years.[8]

https://en.wikipedia.org/wiki/Fomalhaut#Properties

A class A star with 20 times the luminosity of the Sun would be between Vega and Fomalhaut in mass and lifespan, and so would not remain on the main sequence for more than one billion years.

A writer who doesn't care about how low a score his story has in the scale of science fiction hardness

https://tvtropes.org/pmwiki/pmwiki.php/SlidingScale/MohsScaleOfScienceFictionHardness

will go ahead and put his planet in orbit around an spectral class A star without a thought.

A writer who wants a higher hardness score will have to imagine that a habitable planet with an oxygen rich atmosphere orbiting a class A star must have been terraformed by an advanced society in the past.

I note that as early as the 1950s Robert A. Heinlein mentioned that spectral class G stars were the best for having habitable planets in his juvenile novels Starman Jones (1953) and Time For the Stars (1956). While on the other hand Timothy Zahn's "Music Hath Charms", Analog, April, 1985 mentioned habitable planets orbiting Vega and Algol, both quite unsuitable.

https://scifi.stackexchange.com/questions/151973/demon-flute-story-in-analog

I also note that a habitable planet orbiting a spectral class A star needs to have a much greater ozone layer than Earth does, to protect the surface of the planet from the greater amount of ultra violent ultraviolet light it will get from its hotter star.

And the Worldbuilding Stack Exchange is a good place to ask questions about creating fictional worlds and societies.

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  • $\begingroup$ This is very detailed and helpful, thank you very much for putting so much effort into this answer. It is much appreciated. So I would just have to change the star it orbits, dissapointing, but reasonable (mostly annoying due to me having to change a lot of factors like year length and orbit and all of that). What spectral class is the highest luminosity that would be able to reasonably fit the required criteria. $\endgroup$ Sep 27, 2023 at 17:23
  • $\begingroup$ @Dancerofthestars Nobody knows for certain. On pages 51-53 Dole discussed the time it would take for a planet to become habitable for humans. And on page 53 he decided that it should take at least 2 or 3 billion years of relatively steady stellar radiation for a planet to develop an atmosphere breathable for humans. On page 68 Dole seems to consider 3 billion years the minimum possible time for a planet to develop a breathable atmosphere. not explaining why 2 billions years is no longer considered acceptable. Continued. $\endgroup$ Sep 29, 2023 at 20:32
  • $\begingroup$ @DanceroftheStars Continued. On page 68 Dole says that according to the understanding of stellar evolution (in that era) the most massive star capable of staying on the main sequence for 3 billion years would have mass 1.4 that of the Sun, and be an F2V class star, If you were to go for 2 billion years or more, the most massive suitable stars would be no more about A8V or A9V, I guess. Continued. $\endgroup$ Sep 29, 2023 at 20:54
  • $\begingroup$ @DanceroftheStars There is also some doubt whether F class stars can have planets with life. en.wikipedia.org/wiki/F-type_main-sequence_star#Habitability -- space.com/25716-alien-life-hotter-stars.html --- onlinelibrary.wiley.com/doi/10.1002/asna.201613279 -- cambridge.org/core/journals/… $\endgroup$ Sep 29, 2023 at 21:05
  • $\begingroup$ Thank you for your help, it is greatly appreciated. It is a bit dissapointing that I wouldn't be able to do it for more massive stars. One more question and I should be done, theoeretically, if we get the technology to terraform an atmosphere, would this bypass this limit and allow more massive stars? Or would it still have to be old enough to be able to sustain such a thing for any notable length of time? If so, how would this change the range? I perfectly understand if this question can't be answered, just thought I would ask. Continuned. $\endgroup$ Sep 30, 2023 at 14:08

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