Could the Earth be used as a gravitational lens to image very far distances with an effective aperture that would be far greater than any telescope? How far away from Earth would an observing spacecraft have to get to enter the focal point, and see very far at a resolution? Would that enable the direct imaging of small planets around other stars?
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1$\begingroup$ Using the Earth as a gravitational lens might be impractical, but a tangentially related idea of the Aragoscope might be more feasible. For example, a uniform wall built around the equator of the moon could be used to form the opaque disc of a Aragoscope. $\endgroup$– XavierJul 2, 2020 at 5:47
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Yes, it's possible in theory, but beyond current technology to achieve. The focal point of the Earth is 15300 AU away. By contrast, Neptune is about 30AU.
As a gravitational lens is not like a glass lens, you don't get an image formed and at any distance greater than 15300 AU an Einstein ring would be formed around the Earth. But at that distance, the Earth would be very close to the sun in the sky, and the sun has its own gravity. Moreover the Einstein ring would be have the same radius as the Earth at that distance. It would be very small. No Einstein ring is formed at less than 15300; the ring that would be formed by a body with Earth's mass is smaller than the diameter of the Earth at distances less than 15300
Larger bodies have a nearer focus, the Sun has a gravitational focus starting at 550 AU, which is perhaps just about within current solar sail technology. It might be possible to make direct observations using the sun as a gravitational lens. But you don't get a steerable telescope. It would still be a very challenging mission.
See a related question about the mission on space exploration stack and this forum discussion
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$\begingroup$ For scale, $15300$ AU is roughly a quarter of a light year (which is $63241$ AU). $15300$ AU is in the Oort cloud. $\endgroup$ Jul 1, 2020 at 20:42
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$\begingroup$ What do you mean by the focal point of a gravitational lens? AFAIK an Einstein ring can be formed by an object at any distance, as long as it's aligned with the Earth along the line of sight. $\endgroup$– JavierJul 2, 2020 at 12:51
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1$\begingroup$ @Javier The ring may be smaller than the object's visual size. $\endgroup$– fraxinusJul 2, 2020 at 16:48
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$\begingroup$ Even if not a proper gravitational lense, Earth can be converted into a lens by difraction of the atmosphere on a much more closer focus. Related: youtube.com/watch?v=jgOTZe07eHA $\endgroup$– SwikeJul 2, 2020 at 16:59