From a purely geometric standpoint of the motion of the light involved, this could work. Reflected sunlight from our Earth could go out into space, be reflected off of a gigantic mirror, and back towards the Earth.
We actually do something like this currently (at a much smaller distance scale), because a few of the Apollo missions put mirror-like reflectors on the surface of the moon. They’re not specifically mirrors, but a special kind called a retroreflector, which directs light back the way it came from, which normal mirrors don’t do. By firing high powered lasers towards the surface of the Moon where the mirrors are situated, we can count the time delay for the round trip of the laser beam. This is primarily used to measure the distance between the Earth and the Moon to incredibly high precision, which is one of the reasons we know that the moon is receding from us by a little more than an inch every year.
For a laser, this round trip from the Earth to the Moon takes a relatively quick ~2.5 seconds to go the ~478,000 miles there and back. Even though we’re not dealing with large distances (on an astronomical scale) yet, we start to run into a problem already. We start to run out of photons.
Even for a laser beam, which starts out with all of its light focused into a very small beam, will spread out at larger and larger distances. Red lasers are more prone to doing this than green lasers, simply because the wavelength of light is shorter for green lasers, and this spreading is partially a function of wavelength. (Purple lasers would be even less prone to spreading out.) However, over several hundred thousand miles, even the highest wavelength lasers we can manufacture at the moment are going to spread out, and by the time this light gets to the Moon the laser is only able to faintly illuminate the surface of the moon, and only a tiny fraction of that light is going to be bounced off the reflector and back to Earth. The further away you put your mirror, the worse this problem gets, because light gets more and more spread out the further it’s traveled before it arrives at your mirror.
The Earth does reflect sunlight out into space, so we’re not in trouble there. However, this light begins to spread just as the laser light does, but it’s coming from a larger area to start with, so it’s never as compact as the laser light. By the time that we get to any distance away from the Earth, this reflected earthshine is very dim indeed. We can see the earthshine at night- the reason the dark part of the moon isn’t 100% black is that it’s getting some reflected light from the Earth. By the time this light has traveled several hundred light years, one can imagine that this light has become very, very diffuse. And then, of course, it would have to travel several hundred light years back, becoming even more diffuse on the return journey. (We’d also have to conveniently find a mirror out in space which has a clear light path between us and it – if we just put it out there now, we’d have to wait a few hundred years for anything to come back our way.)
But let’s say we managed to get a few photons back from our several hundred light year’s distant reflector, which we can arbitrarily make sufficiently enormous that this would happen – would we be able to identify them? Part of the reason we like using lasers for our moon experiment is because they’re all of a very particular color of light, so we can count up the returning photons at that color, relative to photons of any other color, which we know to be noise. The Earth is not a single color, and the atmosphere is incredibly complicated, so the set of photons that we would reflect would be a much more complicated set than the laser beam we’re firing at the Moon.
On top of this, when the Earth is showing the most reflected light, it’s because the angle between us and the sun is the smallest. So when the Earthshine is the brightest, we’re also most likely to be blinding our reflector with light from the Sun. The Sun, as you will recall, is really, really bright. Stars in general tend to be a big problem for taking direct pictures of planets around other stars, because they’re so bright that they swamp out any of the reflected light from a planet, and we have to get really clever with how we block out the light from the star without blocking out anything else.
That’s just the mirror method. If you want to throw gravitational lensing in here, it gets really complicated really fast. Most forms of gravitational lensing work more or less along a straight line, so if light from the Earth were gravitationally lensed, that wouldn’t help us see it, since the light would still be heading out away from us. There are objects which cause more extreme gravitational distortions, where light can do a U-turn before leaving; these are either going to be black holes or neutron stars. However, if the light is being distorted that much, so is all the other light coming from every other source in the universe. Gravitational lensing is difficult to untangle even in the mostly straight line case; trying to pick apart which photons have come from what source around a black hole or neutron star is difficult to impossible, unless it’s very isolated from other objects. Many neutron stars are in pretty complex areas, since they can be surrounded by the remains of the supernova that produced them. Trying to send some earthshine that way, and hoping that the photons do a U-turn and come back in the same direction is an even harder aiming job than getting light to a reflector and back, since there’s only a narrow range of angles that will allow the photon to do that. On top of this, we’d still have the same problems with the Earth’s light getting overwhelmed by the light from the sun, making it extra hard to untangle the Earth’s light from that of our star.
So in the end, while this is geometrically possible, given a pre-existing reflector, and a hypothetically massive telescope, from a practical standpoint we’re unlikely to observe ourselves this way.
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