If the Earth circled a red dwarf/giant or a brown dwarf, would its sky still look blue? [duplicate]

Question

I have the impression that in brown dwarf and weak red dwarf systems, everything looks more reddish on a planet, including its atmosphere regardless of composition. Suppose there's a planet having an atmosphere at the same surface pressure and composition as that of Earth, circling either a brown dwarf or a below-average-luminous red dwarf at habitable distance, would this planet's atmosphere still look blue when seen from either the surface or outer space? Or would its atmosphere look more orange/red?

Answer 1

If we take 1 atmosphere of optical depth to mean looking though the Earth's atmosphere at zenith, then the optical depth to scattering is small - probably of order 0.3 for blue light and much smaller (according to $\lambda^{-4}$) for red light.

That means that when the Sun is at zenith, most of the light reaches the ground but some blue light is scattered out of the line of sight.

If we look to other sight lines then the optical depth will increase roughly as $\sec z$, where $z$ is the zenith angle (more complicated functions are available). When we look in those directions, we mainly see scattered (blue) light. If we look towards the Sun at large $z$ (don't look at the Sun) then the optical depth to scattering is considerably larger and blue light is scattered out of the line of sight, leaving red light to come towards us.

The amount of blue light we see in the sky then depends on the intrinsic spectrum of the Sun and the angle between the sightline and the Sun.

M-dwarfs and especially brown dwarfs, have very little blue light in their spectra. A typical $B-R$ colour for an M-dwarf would be about 3 magnitudes, whereas for the Sun is about 1. That means the flux ratio of red to blue light is a factor of 6.3 bigger in an M-dwarf. The ratio of the scattering cross-section of red to blue light is about $(400/700)^4 = 0.1$, which would just about compensate.

So I think the best you could get is sort of a yellow-ish scattered light at large angles from the direction of the M-dwarf, due to Rayleigh scattering.

However, this ignores the Mie scattering component. The cross-section for Mie scattering is almost flat, or slightly rising towards red wavelengths. This is caused by aerosols and larger particles in the atmosphere; and tends to wash out the colour and make it more similar to the illumination spectrum. From that point of view, and depending on the aerosol content, I think the yellow-ish sky due to Rayleigh scattering will end up a more salmon pink because of Mie scattering.

This is the view from the ground. I'm not sure what you mean by viewing it from space. The Earth's atmosphere is almost transparent except where there are clouds. I guess if you look right at the limb of the Earth you see Rayleigh back-scattered light; in which case my answer would be the "yellow-ish" answer I gave above, because Mie scattering is quite biased in the forward scattering direction.

Brown dwarfs really have almost no blue light in their spectrum. It is completely absorbed by molecules in their atmospheres and re-radiated at infrared wavelengths. However, what matters is I suppose the ratio of red to blue light and although this is very small, I doubt you can say it is zero. So I'll pass on that one - I'm not sure. But to the human eye it is going to be very dark (assuming that your planet is illuminated by 1.4 kW per square metre of infrared light).