Here are some thoughts adapted to an answer I placed on Phyiscs SE to a similar question some time ago. In order to observe the past we need to detect light from the Earth, reflected back to us from somewhere distant in space.
The average albedo of the Earth is about 0.3 (i.e. it reflects 30 percent of the light incident upon it). The amount of incident radiation from the Sun at any moment is the solar constant ($F sim 1.3 times 10^3$ Wm$^{-2}$) integrated over a hemisphere. Thus the total reflected light from the Earth is about $L=5times 10^{16}$ W.
If this light from the Earth has the same spectrum as sunlight and it gets reflected from something which is positioned optimally - i.e. it sees the full illuminated hemisphere. then, roughly speaking, the incident flux on a reflecting body will be $L/2pi d^2$ (because it is scattered roughly into a hemisphere of the sky).
Now we have to explore some divergent scenarios.
- There just happens to be a large object at a distance that is highly reflective. I'll use 1000 light years away as an example, which would allow us to see 2000 years into the Earth's past.
Let's be generous and say it is a perfect reflector, but we can't assume specular reflection. Instead let's assume the reflected light is also scattered isotropically into a $2pi$ solid angle. Thus the radiation we get back will be
$$ f = frac{L}{2pi d^2} frac{pi r^2}{2pi d^2} = frac{L r^2}{4pi d^4},$$
where $r$ is the radius of the thing doing the reflecting.
To turn a flux into an astronomical magnitude we note that the Sun has a visual magnitude of $-26.74$. The apparent magnitude of the reflected light will be given by
$$ m = 2.5log_{10} left(frac{F}{f}right) -26.74 = 2.5 log_{10} left(frac{4F pi d^4}{L r^2}right) -26.74 $$
So let's put in some numbers. Assume $r=R_{odot}$ (i.e. a reflector as big as the Sun) and let $d$ be 1000 light years. From this I calculate $m=85$.
To put this in context, the Hubble space telescope ultra deep field has a magnitude limit of around $m=30$ (http://arxiv.org/abs/1305.1931 ) and each 5 magnitudes on top of that corresponds to a factor of 100 decrease in brightness. So $m=85$ is about 22 orders of magnitude fainter than detectable by HST. What's worse, the reflector also scatters all the light from the rest of the universe, so picking out the signal from the earth will be utterly futile.
- A big, flat mirror 1000 light years away.
How did it get there? Let's leave that aside. In this case we would just be looking at an image of the Earth as if it were 2000 light years away (assuming everything gets reflected). The flux received back at Earth in this case:
$$ f = frac{L}{2pi [2d]^2} $$
with $d=1000$ light years, which will result in an apparent magnitude at the earth of
$m=37$.
OK, this is more promising, but still 7 magnitudes below detection with the HST and perhaps 5 magnitudes fainter than might be detected with the James Webb Space Telescope if and when it does an ultra-deep field. It is unclear whether the sky will be actually full of optical sources at this level of faintness and so even higher spatial resolution than HST/JWST might be required to pick it out even if we had the sensitivity.
- Just send a telescope to 1000 light years, observe the Earth, analyse the data and send the signal back to Earth.
Of course this doesn't help you see into the past because we would have to send the telescope there. But it could help those in the future see into their past.
Assuming this is technically feasible, the Earth will have a maximum brightness corresponding to $m sim 35$ so something a lot better than JWST would be required and that ignores the problem of the brightness contrast with the Sun, which would be separated by only 0.03 arcseconds from the Earth at that distance.
Note also that these calculations are merely to detect the light from the whole Earth. To extract anything meaningful would mean collecting a spectrum at the very least! And all this is for only 2000 years into the past.
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