A solar gravitational lens will be humanity's most powerful telescope (2022)

How would this kind of thing work in practice? You move the telescope out to 500AU, and then... what? I assume the telescope would have to line up a potential star system with our sun in a direct line. So the telescope would have to move around until everything is lined up. Then I presume it would need to take a wide angle view of that star system to look for a likely planet. Would it need to move closer or farther away from our sun to get a wider field of view? Once it found a suitable planet, I presume it would need to collect light from that planet over an extended period of time, say 6 months. But, the planet is moving during those six months. How does the telescope keep the planet in view? Is the telescope constantly moving to keep pace with the planet? How much fuel would we have to send out there with the telescope?

The planet is also rotating, so how do you keep a specific 10 square kilometers part of it in focus?

If you want to use this method, you're basically going to require one spacecraft per target and a decade of lead time for a very short-duration observation. This is not necessarily a deal-breaker; Meaningful scientific inquiry would just require we mass produce them.

We already sabotage ourselves in astronomy by refraining from mass production approaches for political reasons.

There is no practical amount of fuel that can get you to 500AU on a simple trajectory. What seems to be the best option for setting massive vehicles on a solar escape is a sequence that looks something like:

* Launching to Jupiter propulsively

* Cancelling out most solar-orbital velocity there using a gravity assist in order to dive down into a sun-skimming orbit

* Burning through a large solid state rocket kick stage while at close approach to the sun from behind a heat shield. The Oberth Maneuver.

* (optional) unfurling an electric sail or solar sail once the rocket has finished as you're speeding away from Sol

Together that gets you the required ~100AU/year escape for a mass fraction that is tractable for our civilization.

There’s no wide angle in a gravitational lens. It’s only a couple of square km of view into the other system. So the telescope needs to lign up perfectly, and move around to follow the exoplanet. It’s a very difficult task.

How would they even find an exoplanet as small as earth if all they can see is a couple of square km? Point it at the system and hope the planet just happens to move across the field of view?

Maybe you can observe it over a long time, and you'll note that the data repeats every N hours. That must be the rotation time, and then you can find the corresponding "pixel" for each rotation and work with that data.

Sounds like several near impossible problems on top of each other though.

You need to already know it's there and specifically point at it. It's hard to point so exactly.