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What light spectrum would hit the surface of a planet orbiting a red dwarf star?
The planet is a humid greenhouse with a cloudy atmosphere and a strong ozone layer. Over 80% of the surface is water. The planet is tidally locked to its star.
I'm trying to determine plant life on a world, and the type of light it would absorb or reflect. Any additional considerations would be appreciated as well.
As most of the other answers say, the plants on this world would likely be purple-ish, using photosynthetic pigments that operate at the same wavelengths as bacteriochlorophylls. Chlorophyll a and chlorophyll b wouldn't receive as much light in the visible part of the spectrum as they do from the Sun, meaning that green plants would be inefficient and therefore comparatively rare. We would likely even see pigments that operate primarily in the infrared portion of the electromagnetic spectrum.
The overall spectrum of a star - ignoring emission and absorption lines - can be approximated by the Planck function, which describes the spectrum of a perfect black body and depends only on that body's temperature. From the Planck function, we can determine things like the wavelength of peak emission, or how much of the star's light is visible to humans.
The Sun's surface temperature is roughly 5800 Kelvin; a red dwarf might be closer to 3000 Kelvin. As such, its spectrum looks a little different - weaker, and shifted to longer wavelengths. This favors pigments with peak absorption at longer wavelengths. Here's a plot of two black body spectra (representing the Sun and a red dwarf), normalized to have the same maximum spectral radiance:
Feel free to skip this section if you want; it's not directly related to your question, but does elaborate on what a star's spectrum might look like.
Star's aren't perfect black bodies, but it turns out that the major deviations from a pure black body spectrum come from the composition of the star's atmosphere, which isn't uniform. There's hydrogen, yes, and a lot of it, but also heavier elements like helium and silicon and iron and magnesium. The elements lead to the formation of spectral lines, which appear as sharp dips and spikes in the otherwise smooth black body spectrum. These don't have a huge impact on the star's luminosity, but they do form the basis for distinguishing different types of stars from one another.
Spectral lines vary in strength and shape due to a number of factors:
In M-type stars like red dwarfs, titanium oxide bands are strong; as titanium oxide breaks down at higher temperatures, it's seldom seen as clearly in hotter stars like the Sun.
You say that your planet is largely cloud-covered, which could have an impact on what light comes through. The thing is, this is highly dependent on the composition of the clouds. Clouds of water transmit light differently than, say, clouds of carbon dioxide. It does seem that this transmission/scattering is (at least weakly) wavelength-dependent, and it could have an affect on pigment concentrations. This is something to keep in mind if you end up developing this world in more detail.
Here we get to the meat of your question. Once we have a star's spectrum, we can figure out what sort of photosynthetic pigments will thrive on a planet orbiting it. On Earth, the most successful of these pigments are chlorophyll a and chlorophyll b, which have absorption peaks near 400 and 600 nanometers, centered around the peak of the Sun's spectrum.
Now let's see how the stellar spectra compare to the absorption spectra of different photosynthetic pigments. I got data from omlc.org for chlorophyll A, chlorophyll B, and bacteriochlorophyll a, and plotted their extinction spectra superimposed over the spectra of the two stars:
Note the absorption peaks for the chlorophylls near the peak of the Sun's spectrum. At these wavelengths, there's little emission from the red dwarf, meaning that chlorophylls wouldn't thrive around such a cool star. Bacteriochlorophyll a, on the other hand, has peaks around 800 nm, closer to the wavelength of peak emission of the red dwarf. Bacteriochlorophylls, as well as related pigments that flourish in the purple part of the visible spectrum and at near-ultraviolet wavelengths, would likely be the dominant photosynthetic pigments on your world, making for flora that are much less green and much more purple.
Speaking of a hypothetical Earth-like planet (plus tidally locked) it would resemble something like this:
The planet itself would be divided into 4 zones, 3 of which would be technically uninhabitable: a perpetual hurricane on the side facing direct sunlight, a twilight zone (tormented by endless winds) and the icy biome, covering pretty much the other half of the planet. The only habitable zone would be the zone of indirect insolation, where life could thrive and evolve, in a range of temperature varying from over 104°F in the western regions to even less than 0°F in the Eastern part.
Now, Ozone is the major absorber of UV rays. If the layer was thicker than it is on Earth, then it could potentially absorb all UV rays. Less UV rays means less risks of skin cancers or insolations for us humans, but it could gradually lead our bodies to a Vitamin D deficiency (with consequent low calcium levels). Increased humidity levels could cause the hurricane located in the point of maximum insolation to grow bigger and thus more extreme temperatures in the transition "habitable" zone (with the tropical one becoming even more humid and the Eastern one colder).
Finally, as far as vegetation is concerned, we know that the chlorophyll in most plants here on Earth absorbs blue and red light, but (suprisingly!) less green light. Therefore, chlorophyll appears green to us. Green is still absorbed, but less than all the other colors. On a red dwarf world, since the light peak is in more in the infrared, leaves would try and catch every photon they could since it would require a larger amount of them to complete photosynthesis. I guess leaves would appear darker (even black) than here on Earth, but I'm not too sure about that. I don't even think there would be life in the oceans either, since light would be too weak to penetrate water below 10/20 meters.
The planet would have to be closer to the red dwarf star.
https://www.spaceanswers.com/solar-system/what-if-we-replaced-our-sun-with-a-red-dwarf/
Also, on flora, plants would be differently colored.
"As a result, astrobiologists have suggested that photosynthetic plants on worlds orbiting lone red dwarfs could take on hues of red, blue, yellow, purple, or even grayish-black to best absorb the starlight," (From link below).
Red dwarfs give off less light and because the planet has to be closer (than, for instance, Earth's distance from the sun) to the red dwarf.
There is also the issue of solar flares though. Red Dwarfs give off solar flares more powerful than the suns. With the planet being closer these flares could easily kill all plant life unless it could somehow shrink away and hide during these flares.
For the emission spectrum of the star you'll need two formulas. Wiens Displacement Law [1] and the Planks Law [2].
Wiens Displacement Law
$ λmax = (2.898 * 10^{-3}/T)*10^9$
$T$ = stellar temperature in Kelvin
Obtain temperature via this formula.
$T = M^{0.505}$
$T$ = stellar teperature relative to Sol (multiply by 5778 K)
$M$ = stellar mass relative to Sol
Planks Law
$Bλ(λ, T) = ((2*π*h*c^2)/λ^5)*(1/(e^{h*c/λ*kB*T}-1)$
Have fun figuring this one out. ;) Or use this calculator [3]
You mentioned Water Vapor, Ozone and CO2, as greenhouse gases. If you aim for a human breathable atmosphere consider these limits.
O2 = 0,16 - 0,5 atm, but only up to 35 % of atmosphere (runaway wildfires will keep it this low)
O3 = 0.0000001 atm
Co2 = 0.02 atm (gets uncomfortable at 0.005 atm)
Be aware that greenhouse or hothouse atmosphere are used to describe Venus like conditions. You don't seem to aim for that. Should you want to calculate the greenhouse effect on this world I'm gonna have to stop you right there. Nothing short of a PHD grade physics simulation will give you precise values. But I do have a list of linear approximations derived from calculating stuff backwards. These are by no means scientifically accurate, but they will do for worldbuilding.
$H2O = 677 K/atm$
$O3 = 19600000 K/atm$
$CO2 = 13784 K/atm$
Don't let the planets teperature get above an average temperature 47 Celsius, as this marks the beginning of a runaway greenhouse effect [4] leading to Venus like conditions. This again is just a ballpark number. You shouldn't go under an average temperature of - 56,6 C as CO2 will have frozen out at this point. These are the limits of habitability. Using these a way more elegant than calculating habitable zones and dropping the planet there.
You will get the planets temperature without greenhouse effect [5] via the following equation.
$T = (\frac{ L_{\odot}(1 - a)}{16 \pi d ^ 2 ơ}\;.)^{1/4}$
$L$ = Luminosity in Watts
Obtained via
$L = M^3$ ($M$ is relative to Sol, Sol L is 3.828×10$^{26} W$)
$a$ is the planets albedo, [6] and [7] should help you there.
$σ = 5.670373 × 10^{−8} \;\mathrm{W}\; \mathrm{m}^{−2}\; \mathrm{K}^{−4}.$ this the the Stefan-Boltzmann constant.
All that said a these formulars are tuned to an earth like sun and many things like the albedo and greenhouse effects of the gases will be altered by the new spectral class. But going for something more accurate is material for several scientific papers and not for a SE answer.
Assuming that the planet is tidally locked is sensible, but tidal locking does not always mean that the same side points towards the sun. Mercury is an example of a tidally locked world in a higher than 1:1 spin-orbit resonance [8]. As the eccentricity of the tidally locked planets orbit increases, the most likely spin-orbit resonances go up from 1:1 to 3:2 to 1:2 to 5:2. Just be aware that you will get strong distance based "seasons" as eccentricity increases. Just run the temperature formula for pericenter, apoapsis and apocenter. As the average eccentricity of discovered exoplanets is at 0.3, higher than 1:1 resonances seem very realistic and increase habitability.
It is very likely that you came across the concept of an eyeball planet [9] during your research for this. Sources and people who give you this information are not up to speed. The eyball planets are an artifact of early simulatiins without oceanic and atmospheric heat transfer. There will be strong, constant winds carrying warm air from the near point to the far one. Temperatures on the night and day sides will be close to equal. And if there is a frozen ocean the ice free hole won't be round, it will be lobster shaped. (If you want I'll go source hunting, tell me in the comments.)
Small stars like red dwarfs tend to be variable, flare or UV-Ceti stars [10]. They can increase their luminosity suddenly be orders of magnitude. Imagine the sun suddenly getting way hotter and brighter for a few hours and you see the problem. It isn't clear if all small stars are variable, as they become calmer as they age. Yet even ancient Barnards Star has been observed to flare. This needs to be considered while designing the biosphere.
Color
There is no definitive answer here, just a jungle of possibilities that might all be true. Youtuber Artifexion made a video on the subject suggesting that plants use either the peak radiation of their star for photosynthesis or reflect it to use the other, less intensive light [11]. On earth the secound approach is used, resulting in green plants. On red dwarf planets this would result in black plants as they would want to use all the light. Should the star be variable some kind of biological flare warning system, maybe an UV-detector and the ability to protect against or survive the flare will be crucial. Land plants will be more affected than sea plants. So rolling up like Shameplants, burrowing, using the stellar inferno for reproduction like Mammoth Threes or springing up rapidly after the flare like Eukalyptus does after fires seem like useful strategies.
The first approach is great but there is one huge caveat. Biochemistry isn't a wonderbox. Xenobiology might hold many wonders but a chlorophyll equivalent for every set of wavelenghts seems unlikely. Earths green plants don't really use all the blue and red light, chlorophyll a and b just have absorbtion spectra covering a part of both wavelenghts. The various chlorophylls c and chlorophyll d have other functional wavelenghts and there are various bacteriochlorophylls. Interesting for our purposes is the recently discovered chlorophyll f, capable of using infrared light with wavelengths between 707 and 800 nm [12]. This would lead to plants which ignore the entire visible spectrum or only use some red light via chlorophyll din addition to the infrared light. This kind of vegetation could be white or bright blue-green-metallic respectively. This kind of reflectiveness could allow for flare survivability.
[1] https://en.m.wikipedia.org/wiki/Wien's_displacement_law
[2] https://en.m.wikipedia.org/wiki/Planck's_law
[3] http://www.spectralcalc.com/blackbody_calculator/blackbody.php
[4] https://en.m.wikipedia.org/wiki/Runaway_greenhouse_effect
[5] https://en.m.wikipedia.org/wiki/Planetary_equilibrium_temperature
[6] https://en.m.wikipedia.org/wiki/Albedo
[7] https://youtu.be/y3Kb_ik5f-I
[8] http://www.skymarvels.com/infopages/vids/Mercury%20Spin-Orbit%20Resonance.htm
[9] https://en.m.wikipedia.org/wiki/Eyeball_planet
[10] https://www.aavso.org/vsots_uvcet
There is an episode of Stargate Universe which speculates that a red dwarf star would lead to purple plants.
If the planet is tidally locked to its star, then one side would always be facing the star and the other would get no light. This would create huge temperature differences between the side facing the star and the side not facing the sun. As hot air is less dense, this would also create enormous pressure differences. The winds resulting from these pressure differences would be immense, as well as a constant hurricane in the middle of the side facing the star created by a perpetual depression. The soil would be eroded so much by these winds that I doubt that any soil fit for plant life would remain in a goldilocks temperature area. Furthermore, any plants that grow would have to have thick, strong branches and trunks to stand against the winds. This may be difficult to evolve, as any smaller trunks or branches would immediately be broken and blown away.
