Do enough photons from Proxima b make it to Earth that its surface could be imaged directly with a huge, but not astronomically so, telescope?

Question

I hope I didn't miss someone answering something similar. I was doing a small thought experiment. And I was wondering whether you kind people would check up on my math.

Say we want to build a telescope to see the surface of the extrasolar planet Proxima Centauri B. Well, it is $4 \times 10^{13}\ \mathrm{km}$ away and, if we assume it as big as Earth, approximately 12 742 km large in diameter. (*)

Let us not be greedy and just say we want from our telescope a 10 x 10 resolution, i.e. we need to be able to resolve 1274 km "points" on the planet's surface. The angular size of such a point is, therefore, $3.14 \times 10^{-12}$ radians.

Using $D = \frac{\lambda}{R}$ for the telescope's needed size, we get to rather comfortable 185 km, which is a lot for practical application, but theoretically reasonable.

Maybe we can see it in classical terms, but real light is quantum. Will we see it photon wise? That is, will enough photons get into our telescope to make a useful image in a time-frame we can afford to wait on? Let us forego reality for a second and say that PC b reflects as many photons per second per square meter as Earth or $\approx 1.26 \times 10^{21}$, considering our rather large "pixel" we should get somewhere around $1.23 \times 10^{29}$ photons per 1274 $\mathrm{km^2}$ area. Using area for a half-sphere at the distance of $4 \times 10^{13}\ \mathrm{km}$ we should still get about 12 photons per km^2/s.

Which again seem fairly reasonable.

Huh, I didn't expect it to be within the realm of realistic possibility. Is my conclusion correct, that we could still collect enough photons to get an image?

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(*) My (The_Sympathizer) note: Proxima b is actually somewhat larger.

Answer 1

I don't quite agree with your numbers. In order of magnitude terms, if the planet receives 1400 W m$^{-2}$ and reflects 30% (which is implicit in your question), then if the average photon is say 600 nm (questionable given spectral type of Proxima Cen), there are $\sim 4\times 10^{21}$ photons per second per square metre incident on the planet. The area this incident upon is $\pi R^2$, and with 30% albedo, means a luminosity of $1.6\times 10^{35}$ photons/s reflected into space.

The number heading towards us depends on the phase the planet is observed. A fully illuminated disc would be like a uniformly emitting sphere with twice the luminosity quoted above. The opposite phase would result in zero photons. I'll just assume half illumination and assume this is equivalent to the luminosity quoted above radiating into $4\pi$ solid angle.

In which case, dividing by $4\pi d^2$, we get 8 photons per square metre at the Earth.

Seeing planets around others stars is not a problem in terms of numbers of photons (and indeed directly imaged planets have already been seen), it is a problem of contrast and angular resolution. The nearby star would be separated from the planet by around 0.01 arseconds (to give the illumination level assumed) but would be a million times brighter.

Not to say it can't be done, but building a 180 km diameter optical telescope is more than 3 orders of magnitude bigger than what is possible now. The biggest telescope under construction has a diameter of about 40m. The largest optical interferometers (a more likely proposition) have baselines of about 100m.