WANNA BUY A PLANET?

SOMETHING THAT WEYLAND CORP. WOULD LIKE

In march 2009, after the Kepler Space Telescope was put into orbit, an astrophysicist posted an interesting equation.
It's initial purpose was to calculate how much the discovery of exoplanets worth, but in the end, you can basically price any planet with it.

Other than just label planets for sake of "fun", it is more of a way to measure the perceived potential of a world relative to our needs.

At the time, the planet Gliese 581c, thought to be the most Earth-like planet known outside the solar system, was worth merely 160 USD according to the calculation result.

So, how is this made? There are a lot variables here, but nothing we should be worried about, as I'll make sure to provide a spreadsheet for that at the end of his demonstration.

Let's use the Vol system and planet Paart as our inputs, and fill in as we advance.

V = (6*10^6) * (τ/0,5Gyr) * (Mplanet / Mstar)^(1/3)

The Value is the output in dollars, having the Kepler mission cost as baseline (about 600 million USD), divided by the 100 Earth-like planets they were initially expecting to find, ie, they initially evaluated each of those as ~6 million USD.
This is then multiplied by the star's age - divided by half a billion years, this way, older stars get the preference, as they systems and planets are more mature.
Then, multiplied by the ratio of the masses of the planet and the star (both in Msol), to the power of 1/3rd, this favor low mass stars, as the more massive ones live less.

exp - ((log10(Mplanet / MEarth)) / 0,2)^2

This section picks where in a Gaussian distribution this planets are located relative to Earth, the top of the bell-curve represents Earth's mass, this way, it favors planets that physically resemble Earth.

exp - ((Teff - 273) / 30)^2

Another Gaussian distribution, this time, evaluating how much lighting does the planet receive from it's star, this way, we discard terrestrials that are too far away like Mars, or too close to their stars, like Venus.

The 273 is the triple point of water in kelvins (~0ºC) at STP, the division by 30 means the equation will largely devaluate planets too hotter or colder than Earth.

exp - ((Tyr - 2009) / 4)

The next section evaluates how closer to Kepler's mission start was the planet discovered.
The value decreases every 4 years away from the mission start, as over time, is more likely the telescope finds more and more planets alike the one we're calculating.

(2,5^(12 - v))^(1/2)

This part kind of measures how close the star is to Earth - actually it's related to it's apparent magnitude of the star that harbors the planet.
Being V ≃ -26,7 the Sun at Noon as seen from Earth, and V≃ +30 the dimmest far away galaxies HST can see.
This section exists in case for future missions. A star may be particularly dim from Earth, but if you go there and establish an outpost, and later find another habitable planet in this system, the mere fact you're already there already raises the interest over the planet, so it's linked to how bright the star is in the sky.

The only problem with this equation so far is that it can over-evaluate inhospitable / hellish planets like Venus, or under-evaluate Earth-like planets, if we don't have enough data about them.

Filling the required values for Paart, we get about 416,6 million USD in 2020 values, or ~22,2 million if back in 2009.

I included a list of star luminosities and their respective magnitudes, in case you don't know where to convert your star's brightness.

>Planet Price Calculator<

- M.O. Valent, 31/05/2020

OTHER CONCEPTS | THE FIRST INHABITANTS OF THE UNIVERSE (REPOST)

FIRST INHABITANTS OF THE UNIVERSE (?)

This post is an updated version of a previous one (with corrected math and more research) from back in 11/12/2019.

In the blink of an eye, an infinitesimal point (nowhere, since time and space haven't been created yet) which contains everything... Suddenly becomes EVERYthing, just 1 second after the Big-Bang, the early Universe expanded to a trillion light-seconds across, or 33.889 light-years in diameter...

The ambient temperature of this baby universe is about 100 billion kelvin, but don't worry it will cool down over time, and grow bigger... 1 minute and 40 seconds later, the universe temperature is about a billion degrees kelvin... 56 thousand years after, the universe is now at about 9000 K, when it is ~380 thousand years old - the universe becomes transparent and the temperature go down to 3000 K... [source]

At about 15 million years after the Big-Bang, the room temperature of the universe is within the range to maintain liquid water [source]... Even so, there is no oxygen or heavy elements to bound together and make water or either rocky planets. Even if something of the kind did happen somewhere it is short lasted to even support the chemistry that leads to life for more than 2 million years.

THE OLDEST STARS

M37 star cluster

It's fairly dark for at least another 97 million years, until the first generation of stars is born, feeding from hydrogen, growing as heavy as 20 up to (theoretically) a thousand solar masses, stars this big may be up to 1,5AU across, given mass-diameter formula.

Those first stars were absent of metals, which were yet to be fused in their cores and spread through the cosmos as they died in the first supernovas.

As astrophysics tells us, bigger and hotter stars live shortly, and so it is almost certain that such called Population III stars died in a few thousand years after forming. We would need to wait for some more million years for metals to occur in sufficient amounts so planets and Population II stars appear in the record...

As we can take from the Life-Span formula, such a star with 20, 100 and 1000 solar masses would live for approximately, 5,5 million years, 100 thousand years and 316 years, respectively.

Even though second generation stars would be able to form in a hundred million years after the first supernovas, still we have plenty of heavy radiation emission like UV and Gamma Ray from nearby Population III stars and their dead remnants everywhere.

Then, we push the earliest life could have arisen to a billion years afte the beginning of time, the end of the Dark Ages...

The star in which may support such an early life-bearing planet is prolly a 11th generation star, stars like this are still alive today, see Cayrel and Sneden stars.

For Sneden's mass, we have to use a Magnitude Calculator in order to get it's luminosity, and the  derive the mass, which is ~99,12x that of the Sun, what gives us ~3,71 solar masses.

And for Cayrel we find it's luminosity around 94x that of the Sun. which gives us ~3,66 solar masses.

Our hypothetical star will be the average of those, and then we have 3,68 solar masses, 96x the luminosity of the Sun, and it burn at 10.720ºC (sky blue).

Such a star would emit only about 34,4% in visible light, 52% in UV, and 13,6% in IR and Radio, with a color index B-V of little less than ~0,10.

Still, this star would only live for about 340 million years, not even enough for the formation/cooling of Earth-sized planets.

A world around this star would not bear more than 53~55 elements (all the table down to Cesium and a few Uranium and Thorium), still it would have to orbit it's star as far as 9,8AU (as far as Saturn is from our Sun) to be in the Habitable Zone, and would have a 16 year orbit.

As we can see, neither Cayrel or Sneden stars share the same stats as our calculated star, which imply they work slightly differently. Maybe - being so poor in metals, neither all that mass is dense enough that it can get so compact it burns at insane 10.000ºC, ironically, they're more similar to our Sun, being around 4720ºC, (light orange/yellow).

The planet would still have to orbit in a Saturn-analogue orbit (9,2AU for 94 Lsol) to receive as nearly as Earthly possible of sunlight, any closer than 8,2AU and it falls inside the calculated Venus Zone for it's brightness.

I'm afraid to even wonder by what means such stars are still existing, if some star like Cayrel burns at the same rate as our sun, it's initial mass for a 12 billion year long run should have been about 10% more than it is today (about 4 Msol) - for a matter of curiosity.

If any civilization or life ever managed to live around these early stars, my best guess would be low mass stars in low density regions of space away from the larger star clusters, probably on the early galactic edge, and as the galaxies grew up they probably "moved" near the galactic center (as more stars are added on upper layers it "migrates" inward).

Low mass stars can live enough for being around for dozens of billions of years and still if they formed from low density clouds they're probably safe from the immense radiation from nearby novas and early population III stars energy output.

In order to be at least 12,7 billion years old and have at least 2 billion years of life remaining, such a star would need to be 0,8545 Msol or less... Not too far from our Sun's mass, yet, being so massive we can say it would follow a similar path to our Sun, being this old - an enormous and uninhabitable Red Giant or long gone White Dwarf...

Then our next best guess is to opt for a type of star that can live as long as the universe is currently old and still be "habitable"...

RED DWARFS

Red Dwarfs have amazing properties, they are small and can live for up to trillions of years, a hundred times more than the Universe's current age, the only two down sides of red dwarfs is that they are currently very active and thus they vary a lot in brightness, as well doubling it or halving it because of their giant spots generated by their active magnetic fields, which could potentially be harmful for life as we know it - the second down side is that they evolve very slowly, increasing in brightness as their hydrogen brns out - leaving helium behind, and thus when such a star is calm enough to not randomly burn their planets, most of the galaxy's stars will be long gone, and as far as we know it - no red dwarfs are near this phase.

So as we stick with a red dwarf that is as old as 11,5 billion years for example (I'm giving it some time to gather metals and stuff for it's planets, making it younger).

Make it 0,14 Msol and it will likely live for between ~3,5 trillion years, and only then, at the end of its life spend about 5 billion years as a blue dwarf and cool down to a white dwarf.

As we discussed before, red dwarf flares can be 10.000x as powerful as our Sun. Move away 10 AU, and still the atmosphere of an Earth-sized world would be blown away in 6,7 million years, double the world gravity and it will double the effort to rip the atmosphere away, still - 13,4 million years aren't enough, you may quadruple the amount of atmosphere and it will just get to 53,6 million years, even the activity attenuation in this time-scales is negligible.

Using the prediction model for a star like Barnard's Star, it will stay on the main sequence for 2.5 trillion years, followed by five billion years as a blue dwarf, during which the star would have one third of the Sun's Luminosity.

Let's assume that this model holds for the overall population with masses around 0,16 Msol, a little bit above and bellow that.

The model tells us that these stars will be on main sequence for between 559 billion years (0,20 Msol) and +10 trillion years (0,075 Msol). Which means that red dwarfs are on main sequence for a minimum of 99,991% of their lives (99.998% for Barnard's Star).

Going with 0,2 Msol, have about 30% of the Sun's diameter, it will be as hot as 2440 ºC and will likely be on main sequence for about 559 billion years, and this star will have about 9% the Sun's surface area, and shine with 0,357% the Sun's luminosity.

It's theoretical habitable zone would expand from 0,017AU to 0,025AU. While planet formation would not be much probable closer than 0,02 AU.

Any terrestrial planet would likely form between the Cold HZ (0,02~0,025AU) and 0,6AU.

Gliese 3470 - based on Kepler's Third Law and it's only known planet's orbital data, has a mass of ~0,35689 Msol, and then a luminosity of about 2,7% that of the Sun.

Our then calculated Red Dwarf (which I will refer to as AMB3R for now) is much more dimmer than Gliese 3470, 7,5x dimmer and thus could we say it is emitting much less radiation and particles to space than Gliese 3470, instead of being 10.000x as much active than the Sun, it could be about 1.326x as much active or even less -  a quick look at red dwarfs around 0,14~0,27 Msol show that they are flare stars.

Stellar classification based on temperature

RED DWARF FLARES

The closest in mass I was able to find was Ross 614, which may flare almost once per hour, though it had not much further information about it's flare period, Kruger 60 B whoever, doubles in brightness and returns to normal over 8 minutes.

So AMB3R flares once an hour, doubling it's overall brightness and returns to normal in 10 minutes after the pulse, it's overall brightness over long periods would be around 0,416% Lsol (1/6th of the time at double intensity and 5/6th at normal intensity), peaking at ~0,714% Lsol.

Set a 360ºx360º grid on the star's surface and we get a 129.600 square degrees of surface, and as our planet would only be exposed to 360 degrees along it's orbit, there is a 0,27% chance of it being hit if the star is equally likely to fire at any direction any time, 24 times a day - it is about 6,48% chance of being hit at any time.

Our imaginary grid around AMB3R

Let's put a planet, a Super-Earth 6 Earth masses by 2,5 Earth radii, orbiting on the cold HZ at 0,024AU, it's orbital period is then of ~3 Earth days.

Let's start our Red Dwarf habitability discussion here.

- So far, we see low mass red dwarfs have very narrow HZs, which decreases the chances of planets forming in the HZ already.

- Typical Earth-like orbital eccentricity is too radical for such a small orbit, at  0,016 eccentricity, the planet insolation would vary by 80%, alternating between nearly zero and Venus-like, the CO² in the atmosphere freezes one day and on the other, it puffs in the atmosphere just to fall as snow the next day, in order for planet climate being stable around such a low mass star, it's orbit have to be a nearly perfect circle.

- In this case, even considering 2x the greenhouse effect, surface temperatures are about -3ºC, which is bad for liquid water.

Now proceeding...

Now considering flares are pretty slow, about 1500km/s, for a planet that takes 3 days to orbit around it's star, ie, is at 3,6 million km away, it would take about 39~40 minutes for it to reach the planet, the moving of the planet of ~207 thousand km along it's orbit during this time is negligible for reasons I'll show you later.

Updating our chances... 6,48 * 120 = 777,6%

The planet runs through 120º of it's orbit every day, that's certainly 100% chance of being hit at least once a day, more likely around 7~8 times per day.

Now, a study in 2019 was published on the matter of Coronal Mass Ejections as potential Great Filters for advanced technological civilizations, it is possible that some CMEs may reach speeds up to 3000km/s, as we're talking about flare stars, I'll consider that a given, so first reason why I consider the planet movement at this scale is negligible is because this absurdly fast.

Second reason, Carrington-class events can reach levels up to X 1.1 (the one in 2012 did), which is one of the highest solar-flare classifications.

These kind of events, aren't just a blob like small blow through space, they can widen up to millions of kilometers and still be very damaging.

Computer model for the 2012 Carrington-class CME in case it had hit Earth

If we were considering a point-like directional emission for solar flares, we then now have flares that can wipe between 180~90º from the emission point.

If AMB3R is equally likely to fire in any give direction every hour, and considering how disperse and intense CMEs can be at this point, is certain, that any CME on the star's hemisphere that's facing the planet will hit it.

If the planet were stationary, the chances would be clear 50/50, but it does move about 120º in it's orbit every 24 hrs. Every day, the planet basically is exposed to 1,33 hemispheres of the star, if before the planet would be hit by 12 out of 24 flares per day, then now it can be hit by 16 out 24 flares per day.

Even though solar magnetic activity is more common and intense from the 30° latitude on both sides of the equator, at these small scales it is negligible where it does happen, as long is on the side facing the planet.

An X1 CME at 1AU generate an X-ray flux on the order of 0,0001 W/m², which is not so harmful  to humans, though it would be still equivalent to take 8,5 X-ray scans over the course of 10 minutes, a dose of about 0,85 mSv.

It can be lethal for creatures less than 25g in mass.

However, at 0,024AU, the flux is 1.736x more intense, on the order of 0,17 W/m², which is waaayyy more dangerous, a dose of 1,48 Sv if exposed to the full 10 minute flare, sufficient to cause an almost severe case of radiation poisoning, if such a human is around for a second dose after 1,5 hours later, he or she will be certainly dead.

Any living creature under 25kg will certainly die over the first flash.

It is even enough radiation to kill the Thermococcus gammatolerans - a type of Archea that get it's name from Gamma ray tolerance, about 30 thousand Grays, the problem is that for an organism which mass range on the picogram scale, it would be exposed to ≳6 trillion Grays in the first case at 1AU, ≳10,4 quadrillion Grays, on the second around AMB3R.

One may say that it is an overstatement to consider red dwarfs giving Carrington-class flares all the time, but, even with a B1 class flare, the absorbed radiation by present microorganisms is on the scale of ≳6 billion Grays.

There is no way, life could ever arise around such a star giving off such gargantuan levels of X-rays per hour.

And even if it arise in an ocean under a thick crust of ice, it could never go to the surface because there is probably kilometers of ice above, and neither survive in the surface due the intense radiation exposure.

Around higher weight-class red dwarfs, such as M1V and M0V class, is rather probable that life could indeed arise, but it's needed a proper analysis on the activities of these stars to know for sure.

Around M0V stars, a planet with Earth-analogue eccentricity in the mean HZ would have insolation variations of ~16%, however, with an Earth-analogue atmosphere, that means the planet loses water during summer, and nearly freezes during winter, but it is still more habitable than before, and may be even Earth-like if the orbital eccentricity is ≲60% that of Earth.

OLDER POPULATION I STARS

If Population III were the first stars, made from pure Hydrogen, and Pop.II are the metal-poor stars, then we should look at the oldest Pop.I stars around.

Our Sun, can be considered a mid-tier Pop.I star, while Mu Arae is on higher degree having a greater metallicity than our Sun.

High metallicity stars are more likely to have planets around then, tho we can't exactly say stars like Mu Arae or with even higher metallicity are IDEAL for planets, yes there can be planets in those, however, they're more likely to be gas giants or brown dwarfs.

That puts our search for habitable stars with an average metallicity, and slightly older than our sun, and more probably on a similar weight-class, in order to be on Main Sequence time enough for life to develop and flourish.

If we look at solar-analogues alone, we are looking for about 10% of all stars in the galaxy, in 50lyr from the Sun, there are about 10 out of 19 solar-analogue stars that are probably older than the Sun (52%).

There are 11 solar-type stars in 50ly from the Sun, 8 of which are probably older than the Sun (72%).

There are 16 solar-twins in 3000ly from the Sun, 9 of which are probably older than the Sun (56%).

With no repeating stars, we are looking for about 58~60% of the habitable stars around us, with ages ranging from a couple hundred million years older to up to double the Sun's age.

GameTheory did an amazing work about our next topic on this matter.

GREAT FILTERS

Considering 10 Great Filters (or Achievements if you wish):

Habitable Planet

RNA/DNA

Single celled life

Sexual Reproduction

Multicelullar life

Complex Multicelullar life

Tool-Using Animals/Species

Techonological Civilization

Kardashev I~II Civilization

Kardashev II~IIICivilization

Taking a coin-flip for each of those, renders any give planet a 0,1% chance of reaching galactic-scales.

Even so, comparing to the number of stars, such simplistic assumptions would still tell us that 1/1000 sun-like stars do have highly advanced civilizations, and that 20/1000 sun-like stars are inhabited by some sort of complex life-form.

Looking at the ~60% of sun-like stars that are older than ours, this should be about 12 billion stars, 1 billion which may have planets, from which 2,1 million may have life to some degree, using that coin-flip estimate to filter off, gives us 2100 places to look for high-tech-civs.

This may be a rough overestimate, but let's think of how old are these stars now.

Sun-like stars have this 1Gyr window for life to develop into higher degrees before they extinct their habitable planet's atmospheric carbon supply.

Anything older than 5~6 billions years, using the Sun as a model would have been dead, long gone for at least 500 million to 1 billion years.

Planets in the temperate HZ around a 6Gyr old star would have long entered a moist greenhouse effect, losing water to space and even entering Venus-like regimes as the eons go by, any traces of their existence would have been erased from the fossil record by severe weathering in their homeworld, the same applies to their outer colonies, if they have ever settled other habitable stars, they too would go through the same process.

The truth of this, is that if they ever existed, they belong to different time, a distant time almost like a parallel universe, and we're just talking about stars that are a couple hundred million years older than the Sun.

Stars that are billions of years older would have certainly obliterated any traces of civilization by now, they would be so old that isn't even possible for us the listen to their radio signature, as their active bubble would be hundreds of times wider than the Milky Way, and inconceivably dim.

WHAT ABOUT US?

For me, the sheer lifespan of a star can't be out-lived by a species, or clade if you wish.

We have been around for about 250 thousand years, and our genus for around 3~5 million years.

Tyrannosaurus was around for only 3 million years.

Trilobites were remarkably one of the most successful clade on Earth's history, surviving several mass extinctions, and persisting for about ~270 million years, until they finally perished at the end of the Paleozoic Era.

Ginkos been around for at least 270 million years.

Horseshoe crabs been around for about ~440 million years.

Jellyfish also been around for about 550 million years.

Sponges, or at least, sponge-like organisms are thought to be around for at least 760 million years, and counting.

Cyanobacteria been around for about 2,1 billion years already.

However, as you can see, it seems that the longer you can stand, the simpler you have to be, and even that doesn't grant you the geological-lifespan pass, it's easy to say Trilobites, Sponges, or Jellyfish in a general manner, but we are actually forgetting to mention the hundreds if not thousands of species in those clades that have come and gone over time, in order to keep up with the ever changing Earth.

For comparison, the Homo ergaster was one of the most successful hominids to ever walk upon Earth - considering of course it's other contemporaries and the general time-window of other species of the genus. Homo ergaster lived between 1,8 million ~ 750 thousand years ago, some older fossils date back to 2Mya, this is a species lifespan of a full blown million years - this was due a considerably greater ability to communicate, migrate and adapt to new environments than it's contemporaries, even with such low estimated lifespan per individual compared to us modern Homo sapiens.

If we are to ever be as successful as H. ergaster was, that means we probably burned ~1/4 of our natural lifespan as a species, however, the H. ergaster didn't have nuclear weapons, global warming or pollution to deal with, the price we pay for modern society is the ever increasing risk of us messing up everything, or at least, ending up with a self-destructive behavior.

This self-destructive behavior, or some other filter we don't currently know, understand, or don't have control upon, like Gamma-ray bursts, are more likely the things that stops many species from developing further - if we assume life is relatively common, but rare in higher levels - by either pushing them back or extinguishing them completely.

It may be that we humans start an interstellar civilization, on the best scenario - but we won't last enough to be a super ancient species, luckily, we will the be the first step on a long chain of interstellar civilizations, or at least, settle and diversify.

It is then - not far fetched, given our current understanding and view, that the universe is likely full of desert, inhospitable, but habitable worlds, awaiting to be settled, and maybe even, seeded with life.

It maybe that we are just, early arrivals in the great scheme of things, the new archosaurs before dinosaurs were a thing - or are we one of a few incredibly distant and incommunicable civilizations sparkling throughout the galaxy in our time, every one of us and them, empires and entire dynasties of kings, governors, dictatorships, benevolent leaders, thousand generations of archaic and advanced civilizations, all transient flashes in the long scale of our planet's own history, what to say about the universe itself.

If there is something we could leave, that will outlast Earth's habitable window, or even, the Sun's lifespan - are probes, some, remnants of our early days as a space-faring species, some, deliberate attempts to leave a mark away from the weathering and the flintiness of time - frozen in space and time, orbiting our dead Sun, our planets, our outer colonies, monuments to an ancient people that one day, wandered around these distant worlds, so we can say from the distant past...

...we fucked up at some points, please beware.

- M.O. Valent, 29/05/2020

PLANETARY MODEL | PART 4 | A MORE DETAILED APPROACH TO CLIMATE AND ATMOSPHERE MODELING

DETAILING APPROACHES TO CLIMATE AND ATMOSPHERE MODELING IN HARD SCI-FI

Over the course of this blog's history, I've made several attempts to tackle climate and atmospheric modeling, often involving some Greenhouse Gas considerations and other comparatives with Earth.

Basic Climate Model (comparing CO² levels)

Energy Budget (comparing energy availability)

Venus Zone (comparing atmosphere escape)

Atmospheric Modeling (trying to dimension an atmosphere)

So today, and over the course of next weeks, I'll try rectify and sum up everything tackled in those posts in a more formal, scientific and less speculative way. Which means this post may get long.

Alright, let's SCIENCE properly this time.

ENERGY BUDGET PROBLEM

Imagine an aquarium, the fish inside can only grow as large as the amount of food you give them, or a garden, where plants will only grow as large as the minerals available in the soil allow.

Our ENERGY BUDGET, is basically the measurement of how much is allowed to happen in a certain place given the available amount of energy.

No sunlight over Earth? No temperature gradient, no wind, no air, no circulation, and a lot of other processes which life needs to exist - ceases to exist, for the exception of the Earth's internal temperature, of course.

That means that in high energy places life would be more active, and in low energy places, less active. For an example of that, is the antarctic ice sheets or anaerobic environments, where only few species of microbes live of very low energy chemical reactions, compared to the African savanna or amazon rainforest, where the abundance of sunlight and carbon compounds allow for a very high biodiversity to be sustained.

This, of course would imply that the relatively closer to the star or the more output light a star produces, the better the planet, but, there are other variables to that, which we will explore later.

Let's just consider what Earth receives of sunlight a being 1E (Earth energy unit).

As sunlight spreads according the Inverse Square law, a planet twice the distance Earth-Sun would not receive 0,5E, but about 0,25E:

1solarLuminosity / 2AU^2 = 1/4 = 0,25

Planets at varying distances to their suns and varying suns would render very different available energies to their systems.

This 1E is equal to ~1.368 W/m² at the top of atmosphere, considering a rotating spherical Earth's surface - it is about 1/4th that, or about ~342 W/m².

Several parts of the Earth reflects and absorbs this energy in different ways, this is called Albedo, or how much light/energy does an object reflect back.

Earth's average albedo is about 0,30~0,31 - which means that Earth reflects 30~31% of sunlight back to space, and the closer this value approaches 1, or 100%, the more light it reflects back, and thus the colder it will be.

Even though the amount of energy available is similar to Earth's, the real availability of this energy mostly dependent on the existence of an atmosphere able to trap heat properly.

Considering Earth had no atmosphere, we could use Earth's distance to the Sun, and the Sun's output, to figure out Earth's temperature.

Simplifying this math results in a Temperature formula like this:

T = 279*(1-a)^(1/4) * 1/√d

Where a and d are Albedo and Distance to the Sun, T then  is given in Kelvin, this render us about 254~255K, which is about between -19ºC and -18ºC.

Simplifying further, and assuming the planet has this 0,3 bond albedo, we get

T = L^(1/4)

      D^(1/2)

Where L is the luminosity of your star, and D is the distance between your planet and star, and the final T is a multiple of 254,5K.

The atmosphere creates a greenhouse effect, being made of gases that are transparent to visible light, but pretty much opaque to infrared.

To understand that, we will recur to a simplified model of heat interactions, so, consider that:

- Hot objects lose heat faster than cold objects. And that happens to the 4th power of the temperature. Double the temperature, the rate at which heat is lost is 16x greater than before.

- Planets are found in their equilibrium temperature. They reached a point where the amount of heat energy lost is roughly equal to the energy they receive from their parent star.

Considering  no atmosphere, only oceans, grasslands and forests, and deserts albedo (~0,3), and a rotating sphere, we get ~240W/m².

The relationship (experimentally) between heat loss and temperature can be described by the equation:

T= (F/σ)^1/4

Where F is the rate of heat loss (heat flux), and σ is a fundamental physical constant (Stephan-Boltzmann constant) with a value of 5.67 x 10^-8 Watts/meter² Kelvin⁴s.

Using these values, we also get T=255K, or -18ºC

Now, let's consider a layer of atmosphere that's completely opaque to infrared light, in which case, when it absorbs sunlight, it re-emits it above and bellow, ie, back to the planet and back to space.

In this ideal case, when light enters the system, it warms the planet a little bit, it then also warms the atmosphere, the atmosphere, being this ideal opaque to IR substance re-emits it's heat to space above, and to the planet bellow. The total heat reaching the planet then is twice as before, half coming from sunlight, and half coming from the atmosphere.

T= (480/σ)^1/4 = 303K or 29,85ºC

Of course, this ideal model is an overestimation of the greenhouse effect on Earth, because the atmosphere elements themselves, like clouds contribute to the atmosphere being slightly reflective, that's why we see this blue haze around Earth for instance.

Bellow, a table of different surfaces's Albedo [source]

SURFACE	ALBEDO %
Ocean	2~10
Forest	6~18
City	14~18
Grassland	7~25
Soil	10~20
Desert / Sand	16~20
Ice	35~45
Cloud cover (Thin, Thick)	20~70
Snow (new)	30, 60~70
Snow (old)	75~95

Insolation can also be as low as 37,5% the Equatorial insolation, the further poleward you go.

For an extremely Earth-like atmosphere, the heat energy the atmosphere reflects back to Earth is roughly between +62,3% and +62,5% (assuming perfect black bodies).

The more Earthly your planet appear to be, the more closely it will distribute its energy like Earth does:

1/3 reflected back to space.

1/6 absorbed by the high atmosphere.

1/3 used by the water cycle.

1/6 directly absorbed and radiated to space by ground.

The balance between albedo and atmospheric composition will keep your planet Earth-like in a general way, still, it would be needed to pay further inspection to other aspects such as the...

GREENHOUSE PROBLEM

On Earth, the greenhouse effect is mostly caused by 80~60% H2O, 26% CO2, 5% CH4, 4% O3, 4% CFC/HFC, <1% NO2 and other trace pollutants.

Besides a number of other positive feedbacks, the water cycle is the worst.

Increase the temperature, more water in the atmosphere, the water will increase atmospheric pressure and trap more heat in the atmosphere, leading to more water evaporation, if this happened to Earth, and all of Earth's water evaporated into the atmosphere, the pressure would be over ~358atm (given average ocean depth), because all the Earth's oceans are above you in the atmosphere - and Earth would very probably turn into a planet like Venus this way.

However we should acknowledge that climate change is still debated, and we probably haven't seen it's effects to a full extent, so our current estimations are in risk of being rather underestimates of the real thing. Assuming our early/current estimates are somewhat correct, this means the world warms about 2~3ºC per doubling of CO2, however in times like the Carboniferous period, where CO2 levels were 2x what they are today, global averages were about 20ºC, in fact 1,3x hotter when it should theoretically be ~18ºC, of course, at the time Earth would have a lower albedo due large rainforests and large oceans, but it stops making that much direct sense when you consider that since the Cretaceous period, CO2 levels been decreasing drastically, and even so, temperatures are equivalent to that of the late Devonian when the CO2 were 5~6x current levels.

It would take about 15~16 doublings to make an entire Earth atmosphere out of just CO², with the 3ºC per 2xCO2 rule that's a 42~45ºC increase to about 65ºC avg temp - of course if such a disaster happened we would have water cycle feedback loop set long before it reaches 40ºC.

Talking about water cycle, there is this paper published in Nature that considered an oceanic planet like Earth in different scenarios. And despite the water vapor absorption bands overlap CO2 emission, thus negating some of the climatic change, they still managed to set a moist-greenhouse effect (1,10x Earth's current level) to heat up the planet to ~67ºC.

They've found that exists this threshold between 300K and 330K (26ºC and 56ºC) where the climate enters a rather unstable regime, ie, minor increases in the radiative intake can lead to drastic decreases and increases in temperature, and above 330K exists this warm regime, where for example at 340K it would be needed a full drop in the radiative index back to current Earth levels for temperatures to stabilize back at ~292K, but over a period of 120 years, 80 of which, things could have gone wrong again while in the unstable zone.

Things can be relatively safe up the mark of 1,05x current radiative intake, whereas it would take over 90~100yrs for it to reach an equilibrium at ~62ºC.

However, a minor 1,03x radiative intake is still safe inside the cold regime zone, bellow 300K, despite the temperature going up to 298K (~25 ºC).

This aspect of the thing leads us to the...

VENUS ZONE PROBLEM

The Venus Zone is related to a theoretical area around a star where Earth-like planets would eventually turn into Venus-relative planets through various processes, but mainly by triggering a runaway greenhouse effect.

This could also be related to the finding of a Mercury Zone, which is the area around a star where planet's are unable to retain an atmosphere due solar wind blowing it away from the planet.

Catching back what we talked previously on that matter, I kicked that extra 10% radiative intake could indeed push Earth into a runaway greenhouse effect - the Nature's paper just confirmed that point of view, however, it is fair to admit it didn't really turned Earth into a Venusian planet, in any case, a paper published by James F Kasting and his team in 1993, concluded that a conservative estimate for a continuously habitable Habitable Zone over the course of 4,6Gyr would be squeezed between 0,93AU and 1,37AU, surely between 0,95AU and 1,15 AU, but could as well be larger with different atmospheric conditions than those of Earth.

For them, the Venus Zone starts at 0,84AU, with a radiative intake of 1,41x.

Water loss threshold is on that 1,10x RI mark at 0,95AU.

Greenhouse effect can be very helpful up to the 1,67AU mark, where the RI is 0,36x that of Earth's.

However, if greenhouse effect is not addressed, the 1st condensation of atmospheric CO2 happens at the 1,37AU mark, with RI 0,53x.

Greater changes in albedo, atmospheric water and CO2 are not needed at all if the planet orbits a low-mass star, like a K or M class star, as their output light is redhshifted - H2O and CO2 proved to be better suited to absorb IR light, so the more red-shifted the light from the star, the more of it (proportionally) they will absorb.

They also calculated different planet sizes.

The minimum distance for a Mars-like planet is 0,88AU for runaway greenhouse, and 0,98 for water loss.

The minimum distance for a planet with 2,55 Earth's gravity, is 0,81AU for greenhouse, and 0,91 for water loss.

Funfact: They also believe that 5% of double S-type systems (planet orbits 1 of 2 binary stars) could be habitable, whereas 50% of P-type systems (planet orbits a binary star system) could be habitable, in which case, has great implications for SETI.

ATMOSPHERIC MODELING

Atmospheric modeling is something I must admit that I wasn't so confident about when doing my stuff, if I could say I really knew anything about it, it would mainly be something similar to what we would normally see in a 5~6th grade natural sciences book - and later on specific one about the atmosphere structure, in the latter, only the very basic.

It isn't something we are usually taught about for real, and it isn't really something I got to study in the college's library because of the Coronavirus Pandemic (2020) so far.

All I have looked for so far, is a peek in climate simulation / greenhouse correction, scale height, and habitability.

However after further research, I had worked on a general spreadsheet for the past 3 days non-stop, so you can derive basic data about your star, planet, planet's climate and atmosphere, all in one place - it can only do 1 star and 1 planet at once, but it was built over these research topics I have been discussing for long here on this blog, and updated.

Don't be shy to report bugs, corrections, or suggest more items to it in future versions.

Star & Planet Calculator v.1.0.0

- M.O. Valent, 24/05/2020