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I did the orbital period for a planet orbiting a mass = 1 sol with a 24-hour year, it’s got a semi-major axis of 1.82 million miles - far and away beyond the Roche limit, still beyond the corona, but just a tad too hot. Problem!

So what sort of star would be a black body radiator generating earth-like irradiance at 1.82 million miles distance?

I’m also allowing for a strong magnetosphere to shield from solar winds, and an atmosphere sufficient to absorb excess gamma/X-rays.

Earth-like is a loose term with $\pm 10^{\circ}$F mean temperature, and a little tectonic activity never hurt anyone on Mustafar - except Annakin (that’s a joke). Seriously, gravity will shake things up but not break the planet, and that’s OK.

The star’s tidal force replaces the moon. The planet orbits clockwise every 24 hours and rotates anticlockwise from a synchronous rotation every 24 hours (the sun is observed to complete a $360^{\circ}$ altitude arc). You see the same sunrise/set cycle we do and a fairly similar tidal cycle as well - if not a bit more profound in variance - but the very concept of “year” ceases to exist. We just have days. No months, seasons, years, centuries, or millennia etc, etc.

If this is a no-go for an earth-like climate, I can work with a Venus-like planet if that makes anything more realistic.

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    $\begingroup$ Not really a direct part of your question, but something you probably need to keep in mind... is the planet tidally locked? Is the objective to have a "24 hour" orbital period, or for the planet's day/night period and orbital period to be the same? (I'm not sure this is even possible, but assuming it is, if you can make your "day" longer, that might help...) $\endgroup$ – Matthew Nov 8 at 17:39
  • $\begingroup$ At the end the anticlockwise rotation is 24 hours, so it’s not tidally locked. The goal is 1 cycle = 1 day = 1 year = 24 hours, with a solar irradiance reaching the surface of ~ 340 W/m$^2$ $\endgroup$ – Vogon Poet Nov 8 at 17:53
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    $\begingroup$ Rotating opposite the orbit at 24 hr sidereal will give a 12 hour "day". To have a 24 hour day in a 24 hour orbit, it would have to be sidereally locked (pretty unlikely). Maybe a 48 hour orbit with a 48 hour retrograde rotation? $\endgroup$ – Zeiss Ikon Nov 8 at 18:35
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    $\begingroup$ @VogonPoet Which, if you think about the way planets form, is far less likely than getting a habitable planet into a short-period orbit around a white dwarf or M8 red dwarf -- the whole thing can only be artificial. $\endgroup$ – Zeiss Ikon Nov 8 at 20:11
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    $\begingroup$ @VogonPoet I think it's well accepted that Uranus didn't form that that axial tilt, but was struck by something roughly the size of Pluto (if not a good bit bigger) after formation was complete or nearly complete. Just like Earth didn't form with that huge moon so far out, it was made by an immense impact and then crept out due to tidal drag. $\endgroup$ – Zeiss Ikon Nov 8 at 20:21
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You could use a brown dwarf

They are sort of 1/2 way between an actual star and gas giant. Unlike a normal star which can fuse any isotope of hydrogen, Brown dwarves only fuse the much more rare heavy isotope, deuterium (and in some cases lithium). Since most of their mass is un-fusible, their temperature typically ranges from about 250-2000 degrees Kelvin allowing for much lower orbits than you could have of other stars. They are also, as their name suggests, not very big. They all tend to be about the same size as Jupiter, but are more dense ranging from 13-90x Jupiter's mass (0.018-0.086 solar masses).

Using this http://www.1728.org/kepler3a.htm calculator, to determine orbital radius, and this one http://www.astro.indiana.edu/ala/PlanetTemp/index.html to determin temperatures, I could determine that an Earth like planet around a medium sized Brown dwarf could maintain Earth like temperatures with about a 24hr year.

  • Mass of Brown Dwarf: 0.037 solar masses
  • Orbital Distance: 0.0064 AU
  • Roche Limit: 0.0012AU (safe)
  • Average Surface Temp w/ an Earth Like Atmosphere: 30 Celsius
  • Apparent Size of star in sky: ~16 solar radii

How to beat Tidal Locking

That said, tidal locking at this orbital period is a very real problem. It is going to happen, and it is going to happen very quickly in geological time scales. The only way to maintain a day-night cycle on a planet with such a short orbital period is with spin-orbital resonance. This is a special kind of tidal locking were the orbit finds an optimal pattern of rotations to orbits.

Mercury does this in the form of a 3:2 resonance meaning that it rotates around its axis exactly 3 times for every 2 orbits. In this scenario, a day lasts for 2/3 of a year; so, you should instead have a 36hr year and a 24hr day. You would technically have seasons in this scenario, but they would be so short that they would be pretty unnoticeable in terms of day-to-day life.

This phenomenon is believed to be caused by a strong gravitational pull from a 3rd body which you could achieve by making this a binary star system, (brown dwarfs are not believed to support gas giants.)

  • Mass of Brown Dwarf: 0.047 solar masses
  • Orbital Distance: 0.0092 AU
  • Roche Limit: 0.0014AU (safe)
  • Average Surface Temp w/ an Earth Like Atmosphere: 30 Celsius
  • Apparent Size of star in sky: ~11 solar radii
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  • $\begingroup$ Due to higher density (from gravitational compression) brown dwarves are expected to be significantly smaller diameter than Jupiter, possibly smaller than Neptune, even with more than a dozen times Jupiter's mass -- and the heavier ones get smaller still. $\endgroup$ – Zeiss Ikon Nov 8 at 20:14
  • $\begingroup$ Opps. I said "size" where I meant "mass". This has been corrected. $\endgroup$ – Nosajimiki Nov 8 at 20:57
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    $\begingroup$ “Hey, how old are you?” “Oh, I jest celebrated my 16,491st birthday yesterday. Hey, you coming to my big 16,492 party tonight?” Oh, BIG sun. Cool! $\endgroup$ – Vogon Poet Nov 8 at 23:39
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    $\begingroup$ Or they'd just use metric dates. Instead of a 7 day week, they would schedule work in increments of a 10 day decidays, instead of paying bills once a month, they'd do it once every centiday, and instead of holidays being every year, they'd celebrate them once every kiloyear. So, instead it would be more like, " Hey, you coming to my big party tonight? I just turned 16 kiloyears." $\endgroup$ – Nosajimiki Nov 11 at 15:38
  • $\begingroup$ @VogonPoet Added a possible solution to your tidal locking problem. It means you technically no longer have exactly a 1-day year, days and years should be close enough to eliminate any noticeable seasons. $\endgroup$ – Nosajimiki Nov 11 at 17:57
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Interesting question, there should be a calculator around to check a scenario like that, only I could find none.

There is several other calculators and tables which can help answering this question:

Spectral type characteristics

Orbital period of a planet

Calculation of Habitable Zones

From a quick estimate, this (1-day orbit) is not possible with a main sequence star. If we take the smallish M8 star, it would have a mass of 0.17 suns, luminosity of 0.002 and 2700K surface temperature. A smaller star would be a brown dwarf and could not realistically support life as we know it on its planets.

The calculations would give us an orbit radius of 0.011 AU, while the habitable zone would span between 0.047 and 0.094 AU, which is at least 4 times farther from the star. In short, our 1-day planet there would be worse off than Venus.

However, if we don't restrict ourselves with main sequence stars, there are options. White dwarfs are known for very high mass to luminosity ratios as well as long term stability.

Let's pick Van Maanen 2 star, which is a white dwarf 14 light years away from Solar system and see how a planet can manage on an orbit around it. Van Maanen 2's mass is 0.68 suns, luminosity 0.00017 and surface temperature is a much more comfortable for us 6220K. The orbit for 1-day year results in 0.017 AU, while habitability zone ranges between 0.012 and 0.021 AU. Bingo!

Yes, our planet can orbit a white dwarf. However, it would likely be tidally locked, and at this point it's not entirely clear to the science if life can exist on a planet like that, even if it has the proper (on average) temperature.

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    $\begingroup$ A white dwarf can have a mass = 1 sol, that was my first thought. Also the M8 star would have a smaller orbital radius. $\endgroup$ – Vogon Poet Nov 8 at 18:33
  • $\begingroup$ That's a good point. I will rerun my calculations for a white dwarf. $\endgroup$ – Alexander Nov 8 at 18:34
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    $\begingroup$ it's not entirely clear to the science if life can exist on a planet like that : Are you sure you don't mean 'it's not entirely clear to the science if life can develop on a planet like that'? $\endgroup$ – Pelinore Nov 9 at 20:18
  • $\begingroup$ I was curious about what the light from such a white dwarf would be like, turns out 6220K will have a peak wavelength of 466 nanometers according to Wien's displacement law, which is in the blue light range, not the yellow light peak of our sun. Apparently photosynthesis does work fine under blue light so the different wavelength shouldn't be an evolutionary problem in that sense, but there might be other effects I didn't think of. $\endgroup$ – Hypnosifl Nov 10 at 0:13
  • $\begingroup$ @Hypnosifl 6220K dwarf was just an example. Sun-like dwarf with 5800K surface would yield a similar goldilock zone. $\endgroup$ – Alexander Nov 10 at 1:07
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Here are some findings

For a star with 1 solar mass, and at the distance you'd need (2,927,699.613 km) for a 24-hour year, you could have:

  • A star with a radius 0.019578 times that of the Sun, but with the same power density as the Sun.
  • A star with a power density about 0.0003834 times that of the Sun, but with the same radius as the Sun.

These are the two extremes, but they would give you the correct solar output.

The observed radii of white dwarfs have been about 0.8–2% of that of the Sun. This might possibly work for your solution, but you also have to take into consideration that white dwarfs have higher energy output per unit volume than the Sun does.

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You can not get a habitable planet with Earth like temperatures orbiting a star with a mass equal to the Sun with an orbital period or year one Earth day long. Thus you will have to settle for a star a lot different from the Sun for an Earth like planet to have a year one Earth day long.

The mass of a star (and its age to a lesser degree) will determine how much radiation the star will emit. The amount of a radiation a star emits will determine the distances of its habitable zone where a planet could orbit and have Earth like temperatures. The mass of the star and the orbital distance of the planet determine the planet's orbital speed and the length of its year.

There are known exoplanets with years less than one Earth day long. But they are not orbiting in the habitable zones of stars with masses equal to that of the Sun.

https://en.wikipedia.org/wiki/List_of_exoplanet_extremes#Orbital_characteristics1

There are exoplanets known to orbit in the habitable zones of their stars, making those exoplanets potentially habitable exoplanets.

https://en.wikipedia.org/wiki/List_of_potentially_habitable_exoplanets2

Their year lengths vary widely. The four planets in the habitable zone of TRAPPIST-1 included the three potentially habitable exoplanets with the shortest known days; 9.2 days, 6.1 days, and 4.05 Earth Days.

I haven't done calculations, but I think it is possible for a planet in the habitable zone of a star to have a year as short as one Earth day.

But all the potentially habitable worlds orbiting in the habitable zones that have very short years orbit around very dim red dwarfs of spectral class M. And those stars are much less massive than the Sun.

For a planet to orbit in the habitable zone of its star with a year one Earth day long, it would have to orbit around a class M star even dimmer than TRAPPIST-1 even closer than TRAPPIST-1d does. So there would seem to be absolutely no possible way for that star to have the same mass as the Sun.

It has been calculated that planets orbiting in the habitable zone of a red dwarf star would become tidally locked to their stars, with one side constantly facing the star and having eternal day and one side constantly facing away from the star and having eternal night. And it is controversial whether a planet could be Earth like and habitable if it was tidally locked to its star.

One way out of that problem would be to have the Earth-like planet actually be an Earth like giant moon of a giant planet. The moon would orbit the planet which would orbit the star. The Moon would become tidally locked to the planet and not to the star, and so it would have a daily cycle of light and dark equal to the period of its orbit around the planet.

If you try that, it would be more likely for the Earth like moon to have an orbital period of one Earth day around the planet, and for the planet to have an orbital period of at least several days around the star, instead of the planet having an orbital period of one Earth day around the star, and the moon having an orbital period of a fraction of an Earth day around the planet. So that would not be exactly what you want.

Of course it is always possible, depending on various theories about the subject, that a tidally locked planet orbiting close to a red dwarf star could remain habitable. Since Trappist-1 is spectral class M8V a habitable and Earth like planet with a year one Earth day long would probably be class M9V, or even dimmer.

If a red dwarf with spectral class M9V would still be too luminous to have a Earth-like planet with a year one Earth day long, you would have a problem selecting an even dimmer type of star.

You might need to go with a white dwarf star. White dwarf stars are much hotter than red dwarf stars, so each area of their surface emits a lot more radiation than the same area of a red dwarf star does. But white dwarfs can be very small, and so can have a lot less surface to emit radiation from.

Thus the dimmest white dwarf stars might be a lot dimmer than the dimmest red dwarf stars.

White dwarf stars, despite their smaller sizes, are also very dense and so are more massive than red dwarf stars. Thus a planet orbiting a white dwarf star might have to orbit faster to stay in orbit, and have a shorter year, than it would have if it orbited a red dwarf star at the same distance.

So white dwarf stars would be a good choice for an Earth like planet with a year on e earth day long. Except for the history of white dwarf stars.

White dwarf stars were once massive stars which had to burn their fuel very quickly, and so have run out of hydrogen to fuse, expanded into red giants, and then shrunk down to tiny white dwarfs shining with leftover heat.

That process would have destroyed any planet orbiting that close. So the Earth like planet would have had to have been moved from someplace else into its present orbit by rather unlikely natural forces or by a highly advanced civilization, and then remained in its new orbit for billions of years while it gradually developed Earth like conditions on its surface.

another possibility would be to have the planet orbit around a brown dwarf. A brown dwarf is an object more massive than a planet and less massive than a star, that would mostly emit invisible infrared radiation. A planet orbiting a brown dwarf of the right mass and luminosity might possibly have a year one Earth day long while orbiting within the habitable zone of the brown dwarf, though I haven't made any calculations.

So there are several possibilities for someone to calculate an orbit within the habitable zone with a length of one Earth day.

One) a red dwarf, probably spectral class M9V.

Two) a white dwarf, where the planet has been moved into its present orbit billions of years ago, after the star became a white dwarf.

Three) a brown dwarf.

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For a solar mass star with an Earth-like climate in a 1-day orbit, you'd need a white dwarf. This nevertheless presents a few problems.

Firstly, the white dwarf is the end state of stellar evolution. The progenitor star would have been much larger, especially during the giant stage, which would destroy the inner planetary system. You've also got the problem that a young white dwarf star is extremely hot and luminous, which would boil off all the water from a nearby planet long before the star cooled down enough (to a similar temperature to our Sun) for the 1-day orbit to be in the habitable zone.

The way to get around this is to form the planet at a later date. One way to do this is to have the orbital changes due to the mass loss from the progenitor star destabilise an existing planetary system, eventually kicking a planet into an orbit where it gets ripped apart by the white dwarf's gravity. This might produce a disc around the white dwarf that may be able to form a new generation of planets. If the instabilities take a sufficient amount of time to work themselves out, you might be able to avoid the worst of the early luminous stage of the white dwarf's evolution.

An alternative is that the white dwarf is part of a binary system, and when the second star goes through its red giant stage, the new white dwarf captures enough material to produce a planet-forming disc and create new planets. Of course, if too much material falls onto the white dwarf you'd run the risk of the system undergoing nova explosions or a type Ia supernova.

Once you've somehow got a planet into the "habitable zone" of a white dwarf, the next main problem is tidal heating. The tidal forces at that distance from a white dwarf will be severe: even tiny amounts of orbital eccentricity will result in a world that makes Io look stable, consequences of which could include a runaway greenhouse effect (a.k.a. a "tidal Venus"), magma oceans and other general unpleasantness. Tidal forces will act to circularise the orbit on short timescales, but if there's another planet in the system then the effect of its gravity on the orbit could easily trigger disaster.

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  • $\begingroup$ Given that planets have been found around pulsars, finding ones around white dwarves doesn't sound entirely far-fetched. The lack of water doesn't appear to be a problem, given the OP's constraints... they didn't request its presence at all. $\endgroup$ – Starfish Prime Nov 9 at 13:11

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