OTHER | SEEKING CREMATORIA-LIKE WORLDS | PART 2

PREVIOUSLY, ON HARD SCI-FI...

We want a planet that:

Is somewhat habitable, but only if lighting conditions are just right, that is - it has an overilluminated side and a dark side. In which case, life would need to move or cover itself when the surface passes through the overilluminated side, and do its thing when it is dark until day comes again. Like Crematoria from Chronicles of Riddick.

EXPLORING WAYS TO MAKE CREMATORIA REAL:

From the previous post...

• Trojan planets around single stars
• Trojan planets around binaries

 Now exploring...

• S-type planets in close binaries
• Dying binaries
• Luminous black holes

LIFE BETWEEN TWO SUNS

Since we've already discussed how binary systems work in the previous  post, let's jump straight into a sketch of my plan for this setting:

Now, although in my sketch the stars are set in a 1:1 mass ratio, that will definetly not be the case, both for being a lone binary, and because we will need a generous safe zone around our planet's orbit in order to have minimum gravitational disturbing from the other star.

Since our planet needs to be sufficiently illuminated by the other component, we can already infer it lives within a close binary, which means we will have other P-type planets in the system as well, and this world would be actually a captured planet by one of the pairs.

If we make the parent star sun-like and put the planet in the habitable zone, then we would have to put the other component at +3 AU away, which is observationally bad for us since the illumination of the secondary pair would drop down to <10% of the solar constant.

But if the parent star is small, such as a red dwarf or orange dwarf, we can put the other brighter star much closer, it also helps that we get a significantly larger planetary disk around stars with mass disparity like that. In that case we will explore two sub-scenarios, in which this mass ratio is 3:1 and 2:1, the most common mass ratios for binaries.

3:1 case, system must be smaller than 0.26x the orbital separation

A ~ 1.00 Msol, 1.00 Lsol, 1.00 Rsol, 5778 K - G2V yellow dwarf

    Mean HZ for A: 1.00 AU

B ~ 0.32 Msol, 0.03 Lsol, 0.40 Rsol, 3800 K - M0V red dwarf

    Mean HZ for B: 0.21 AU

Orbital separation set to be at least 4x that of component B's habitable zone, or about +0.80 AU.

With minimum requirements, the solid airless surface of the planet would be at 310 K or 37°C, but we can cool it down to about 20°C if the parent star orbits the brighter star at a distance of 0.93 AU. Slap it a moderately thick atmosphere at 0.5 bar for an Earth-sized planet, and we got ourselves a simmering 50°C of surface temperature, with some 70°C at the equator, for a 24 hour rotational period.

That average temperature will ocillate between 32°C and 79°C as the planet orbits and points towards or away from the brighter star, changing irradiance between 1.47x and 2.63x the solar constant. These changes occur very quickly over the course of 66 days, but the planet is tidally locked to its parent star, making it a seasonal eyeball planet.

If we want to make it so it is not tidally locked, then the we can place it a bit further away at some 0.30 AU, decreasing the average temperature from 50°C down to 33°C, with an average irradiation of 1.50x the solar constant. Ocillating instead between 4°C and 87°C.

Because our red dwarf parent is quite large, we don't have to worry about sudden violent X-ray and UV flares even though we have a thinner atmosphere than Earth does. Plus, given we know the Earth's atmosphere cools at roughly 0.5~1.0°C per hour (depends on place, at my region it is roughly 0.8°C/h), we can tune the planet's rotation period to cool down to an acceptable level before heating up during the day, so if we want the temperature to drop from some 60°C during the day, down to 20°C at night, we'd have to set night-time duration to some 50 hours, which means our planet should have a rotational period of at least 100 hours to have decent cooling.

The rotational period of 100 hours bring the equator temperature to some 50°C as well, which means that during the night we have it dropping to 10°C. But that's only on the side of the planet which faces the two stars at once, we might not even have enough exposure time to hit some 70°C for most of the year. When the planet finds itself between the two stars the brighter side heats may cool down to some 14°C on average, and the dimmer side to a freezing -60°C. Because of the planet does not stop in this position for long, these potential extremes might not be reached, instead staying between well 70°C and -60°C in the equator across the whole orbit.

But temperatures might actually be stable and comfortable towards the planets poles given the softer ever setting suns or just by being in a temperate zone, though in general the whole planet's temperature changes drastically over the year.

We should also give it vast oceans, so we have ices and decent temperature buffers, so we don't end up with a venusian planet. Now, depending of the latitude, one will still get hot deserts during the day and freezing cold nights, and one half of the planet will always be considerably colder than the half facing the brighter star. So far, it seems we've accomplished our goal with moderate success.

After days in the dark, the brighter sun finally rises, thawing glaciers and lakes to form rivers which will flow fresh for the next two days until it sets again. Clibanus' thin atmosphere allows for stars and the system's outer planets to be observed even during the day, despite the radical changes in irradiation, flooded plains allow consistent liquid water for life

Compared to the original Crematoria, unless the atmosphere can dissipate the heat twice as efficiently as the Earth does, the planet of the movie cannot have 52 hour days, not can have the extreme temperature differences we see in the movie from 370°C to -180°C.

By default, this arrangement is pretty common and actually pretty likely to occur in nature, HOWEVER one thing I've stated at the start is that given  this planet is very close to the secondary star, it is possible that it is a captured planet, thus it might also sport other quirks such as high orbital inclination, high eccentricity, retrograde motion as well. The high eccentricity in particular seems like an interesting way to vary the temperature extremes even more, as well as varying the amount of water present in the planet's climate system.

For planets in more circular orbits, ie, that formed around the red dwarf, we are limited in how massive we can make that planet the same way we are limited with gas giant moons, in this case topping between 2 to 10 Earth-masses, which we could actually distribute in a planetary system of Terran and Subterran worlds which suffer from the same condition, but varying climates according to their specific atmospheres, water content, and distance from the red dwarf.

2:1 case, system must be smaller than 0.22x the orbital separation

A ~ 1.00 Msol, 1.00 Lsol, 1.00 Rsol, 5778 K - G2V yellow dwarf

    Mean HZ for A: 1.00 AU

B ~ 0.50 Msol, 0.10 Lsol, 0.57 Rsol, 4334 K - K6V orange dwarf

    Mean HZ for B: 0.39 AU

Now because our habitable zone around the secondary component expanded to double the previous orbital radius, we have to change the distance between the stars just a bit to accommodate those changes. So the system's minimum size, given the planet has to orbit within 22% of the orbital distance, is about 1.82 AU, orbiting every 2 years. This way, the star's hill sphere is about 0.8 AU in radius, snuggly fitting our habitable zone at 0.4 AU, and given our parent body is larger than the previous one, our planets can be up to 6 Earth-masses in size, even giving space for superterran worlds in orbit.

At such distances from the stars, the planet gets 0.63x and 0.30x the solar constant from its parent and the brighter star, respectively, being close enough to barely rotate instead of being tidally locked. Its equilibrium temperature is around -24°C, and it warms up to 10°C under 1 Earth-atmosphere. By setting the planet's rotation to 500 hours the equator is able to heat up to some 23°C.

When the planet is between the stars, the orange side drops down to -17°C and -4°C on the equator, while the yellow side to some -50°C.

We can already stop right here, as it was concluded from the last example and from the last post that trying larger stars help, but not a lot because of the greater distances involved.

DYING BINARIES

It is not (observationally) uncommon for a bright star to have a dead or almost dead companion in a binary pair, see the Sirius system, an A-type white star and its white dwarf companion Sirius B. However, since the Nova which forms a white dwarf most likely destroy the planets around it (if any survived the red giant phase) is too dim to provide any significant illumination, we will be looking at red giant pairs.

If we want planets in those systems to be habitable for a long time, then such systems would have to be either very old and originating from sun-like stars, or very young containing a sun-like star and a heavier pair. Whatever the case we choose, it will be composed of a star still in its main sequence, and a bright red giant. For the distances we can be somewhat liberal, alerting only for the absence or presence of planetary disks. Let's say that by the time our primitive planet starts developing complex life, our heavier star finishes its MS phase and starts diving into the red giant phase. A sun-like star spends two billion years in the red giant phase, with variable luminosity as time passes, very variable in the last few hundred million years, which means such worlds are pretty ephemerous. Let's use that as a basis, with a star that enters the sub-giant phase at the age of 4 billion years, setting the system's age half-way through the process. The chosen mass ratio is 5:2, still close to your typical common binary system.

A ~ 1.28 Msol, 2.47 Lsol, 1.22 Rsol, 6566 K - F5V yellow-white star (nominal stats)

Mean HZ for A: 1.58 AU

SUB-GIANT STATS:

Current Age: 4.48 Gyr, 400 million years until the Helium Flash.

Current temperature: 4000 K, it will continue to drop to 3000 K until the Helium Flash. 

Current luminosity: 5 Lsol, it will continue to steadly rise up to 25 Lsol, before taking off to 2500 Lsol for the Flash in a short period of 100 million years.

Current radius: 2 Rsol, it will steadly rise to 10 Rsol, until inflating to 200 Rsol (1.9 AU) when the Flash happens. 

Mean HZ for L ~ 5.0 and L ~ 25.0: 2.2 AU and 5.0 AU

B ~ 0.50 Msol, 0.10 Lsol, 0.57 Rsol, 4334 K - K6V orange dwarf

    Mean HZ for B: 0.39 AU

Just so we keep the secondary pair, and thus the planet, under a significant illumination from the dying star, the orbital distance will be about 3 AU. So as the ages pass, the planet will go from receiving 0.5 to 2.8 solar constants from the dying star, while getting a consistent 0.6x solar constants from its parent, so the total illumination will rise from 1.1 to 3.4 over the course of 400 million years. Under 1 Earth-atmosphere, the mean temperature rises from 22°C to 105°C over this period, staying about 20 degrees hotter than those in the equator. The coldest tropical nights in such a world simmer at 65°C, while the planet is more temperate towards the tropics, however, with such high temperatures, we need to make this a waterworld to buffer it all, or a dry rock so the clouds don't start a runaway greenhouse effect. We could also balance the temperature by smaking the planet smaller, which decreases air pressure and volcanism so we don't turn this world into a venusian right away. A half as thin atmosphere puts the planet on pair with the previous world even on the hottest phase of its life.

By varying the amount of water in the atmosphere and a little bit of the secondary star's eccentricity around the main component, we can better control the surface conditions, still obeying the same principles as before, though we now know such worlds will be flash burned when the subgiant becomes a red giant, elevating the mean temperature to about 850°C.

Given what we know from how atmospheres react to the incoming light of different stellar spectra, it is safe to say that once the insolation coming from the red giant exceeds 0.7~0.8 solar constants, the planet will be no longer habitable, for the amount of infrared the atmosphere receives can no longer be irradiated away to space efficiently, thus, the surface temperature rises several dozen kelvin for even the tiniest increase in luminosity.

Reproduced from Habitable Zones around Main Sequence Stars, JF Kasting et al, 1993

We can say for sure that our planet around a red giant won't be habitable for much longer than half of its life between main sequence and the helium flash.

Hence why whenever I set up a planet around red dwarfs I try to keep their insolation below 1.0 and above 0.5, one can go even lower if they wish to add brighter stars to the system, if those stars are considerably hotter, even a few percent more incidence will the job at illuminating and heating the planet up without tipping it over the edge of a runaway greenhouse event.

Though I admit a future me or one of my readers will eventually find something I overlooked or winged for sake of argument for this post series, after all, this is supposed to be an expositional guide, an exercise for you to explore other particular variations more invested than I was with those quick examples.

DO NOT GO GENTLE INTO THAT GOOD NIGHT

Ah yes, luminous black holes and their blanets (yes with a B, that's a thing). For the ones not familiar with the concept, it's the kind of scenario as presented in the movie Interstellar (2014), but for the ones not familiar with the inner workings of such systems, I must warn such systems in nature would be so rare and ephemerous that one might as well regard them as legendary oasis, from what I could find and understand.

WHAT KIND OF BLACK HOLES ARE SUITABLE?

Black holes are a very interesting class of objects for their extreme variety of sizes and surrounding structures. The smallest black holes are created by stars above 23 solar masses, when their core collapses at the end of their lifes, compressing a good chunk of its matter into an infinetly dense point, at least some 2.9 to 3.0 solar masses, the infalling material and subsequent radiation burst ends up bouncing back up and eventually spewing all of the star's upper layers away into space, hence why the resulting black hole is rather small compared to the star's mass. Such small black holes are called Stellar-mass black holes, for they have the mass of typical high-mass stars, and typically, those black holes are the most dangerous for civilizations, for they are very small to the size of a few dozen kilometers, zipping through space at stellar speeds, gravitationally interfering with other systems as they fly by, and worst of all, they are often very dark - because the are rather small, it is very easy for matter to orbit and catapult around it without getting even a tiny bit close enough to be shredded or absorbed into it, thus they rarely emit or posess detectable signatures but the light-bending their gravity produces.

Stellar black holes are also much more aggressive than heavier ones for their small size, with a virtual density much higher than massive black holes, that is, the density you would expect if it was a solid object the size of its event horizon - a person standing 100 event horizon radii from a 5 Msol black hole would experience a gravitational gradient between their head and feet of ~750m/s, leading to instant material failure of the astronaut and resulting spaghettification. Whereas this same configuration with a 1000 solar mass black hole causes a difference of only 0.02m/s, far more tolerable.

Massive black holes are gentle giants.

Larger stars produce large black holes nearly twice the mass of these, but since stars rarely exceed masses above some 50~100 solar masses, stellar black holes cannot get any bigger than some 10 solar masses. From this point onward things get strange, black holes can only get bigger by absorbing lots of matter and other black holes, which means that the larger black holes are often much ancient than most stars, planets, or even the galaxy it currently inhabits.

When we look at massive black holes and supermassive blackholes, those which range from thousands to millions of times the mass of the Sun, we often find those which are surrounded by large disks of infalling matter, accretion disks. As matter accelerates towards the black hole, it rubs against other infalling molecules, heating up to thousands of degrees, generating all sorts of radiation, including light - those are the Luminous black holes. Non-luminous black holes include the ones such as Sagittarius A* at the center of the Milky Way, with 4 million solar masses, it has barely any accretion disk, existing in the dark, puppetteering nearby stars around a seemingly empty region of space. The feeding rate of a black hole, or accretion rate, is limited by its Eddington accretion limit, which is how much mass can fall into the black hole, before the resulting radiation pressure of the accretion disk counteracts the gravitational force of the infalling matter, the brightest luminous black holes such as quasars, blazars, and young radio galaxies find themselves near this limit or at super-Eddington limits, when the black hole also absorbs the extra radiation it would emit despite greater accretion rate. For obvious reasons, blanets and stars cannot reside near such monsters, because they would quickly be disintegrated into the accretion disk.

However, one detail we have to pay attention to while looking to settle black holes with habitable conditions, is that for most of their life, black holes will exist in their dark form, while luminous black holes are rather ephemerous. A single black hole might go through several luminous phases along its life, feeding on unlucky stars for a few million years, then waiting in the dark for the next prey, hence why radio galaxies are always young, as their central black holes did not have enough time to clear their surroundings, so not yet in their dark phase.

The amount of radiation released from a luminous black hole is directly proportional to the infall of matter, sometimes a black hole will traverse a region of space with little more gas than usual and shine very dimmly with a ghostly echo, or sometimes a whole rogue planet falls in, quickly spaghettified into a bright accretion disk which, like the rings of Saturn, will last a few million years.

Because we want blanets, moons, maybe even other stars around our luminous black hole, the gas around it which is the precursos to all of these bodies will be likely of solar-composition, with some sprikle of metals and not only hydrogen gas like the interstellar medium. The black hole would have to be near the end of its feeding / luminous stage, as we still want an accretion disk as energy source, but not so large of an accretion disk it just disintegrates any rocks with dense x-rays. So if the Eddington limit says the max accretion rate is a few billionths of a solar mass per year, then we will lean towards trillionths of a solar mass per year.

Because the surface area for an accretion disk around such black holes is immense, many times that of whole stars, the surface temperature of the disk should be star-like, between some 6000 to 2500 K, this works out to quite a headache of math when you're not familiar with the principles or equations behind it...

THE GENTLE GIANT

For an Interstellar-like scenario, we'll use a supermassive black hole about 100 million solar masses, spinning at 99.995% the speed of light, an ancient monster which hasn't fed upon anything for many millions of years, just now licking the breadcrumbs of its plate, that is, with a very thin ghostly accretion disk.

We're talking a 1.0 AU radii event horizon, with a disk that extends from 1.3 to 2.5 AU.

Our Eddington luminosity is around 4~6 trillion solar luminosities (depending on the gas makeup), with an accretion rate of 2 solar masses a year. So if we want the disk's Earth-like insolation zone to be at around 3 AU from the monster, we need an effective luminosity of 9 solar luminosities. So now we divide 9 by 4 trillion, we then get 2.25 trillionths of 2 solar masses, or 0.00012 Moon-masses a year. With a temperature between 288 thousand K near the ISCO down to 83 thousand K near the edge.

Even though the temperature is not enough for hard X-rays to be emmited through the ionization of metals within the disk, most of its emissions are still in the far UV spectrum. This can be avoided by increasing the opacity of the gas, making it partially ionized in a wider disk or toroidal cloud around the black hole. This makes the inner rim of the disk extremely hot while keeping the outer parts of the disk less hot, which means we need to lower the metal content of our gas cloud, or else the breaking radiation of relativistic electrons will increase the x-ray output of our accretion disk.

As for planetary formation around such objects, it would boil down to general rules of planetary formation, except the progenitor gas cloud would be the spewed guts of one or more stars devoured by the black hole, which for our purposes would have to be the black hole's last meal in a long time, or else the extreme x-rays would just photo-evaporate our blanets.

Those conditions will be very rare or even impossible to accomplish in real life, like, even a small rogue asteroid coming from interstellar space and falling in would increase the disk's luminosity by orders of magnitude - frying whatever life existed in the blanet surface. In the whole universe with its countless blackholes there might exist very few of those legendary oasis where conditions are just right, where life is possible hanging from a silk thread.

Because photon-matter interactions are rather too complex to bother going through, I'd admit handwaving most of them away would be the best course of action - for the sake of story telling, the habitable zone distances and time dilation regarding proximity to the black hole would have way more weight to it.

But realistically, given the many unknowns regarding radiation tolerances, a habitable blanet would look like the following:

Assuming the event horizon is some 10° wide in this image, the planet would find itself at 6 AU, receiving 1/4th of the Earth's insolation.

Would that be enough? I don't know, my IR correction equations are calibrated for stars, not massive accretion disks, the output for this case is right at the face of the event horizon inside the disk's inner radius, at 1.2 AU

An icy/oceanic superearth far far away from the black hole, some 10 Earth-masses and between 2.0 and 2.5 Earth-radii, the illumination is pretty dim compared to Earth's, but the incidence of x-rays and electron wind against the thick hydrogen/helium rich atmosphere reacts to produce scattered radiation, which warms it up to a tolerable temperature between 200 and 400 K. The atmosphere however would be rather anoxic, as the rays cannot penetrate very deep to react with the water or ammonia which pools on the surface as oceans, and any bacteria that develops here would be anaerobic, feeding on high energy or infrared light rays and minerals dissolved in the oceans, in a way, similar to Miller's planet - except much dimmer, much redder, and warmer.

For flavor we could add our habitable planet as a moon of a gas giant, or the more unlikely case - as the tidally locked planet of a red dwarf, working pretty much the same way as the first example from the start of the post, which seems to be the only viable way to obtain the desirable Crematoria-like effect

- M. O. Valent, 13/02/2023