Would you get the same planet-discovering data for our own system at the same distance we're getting from Earth?

Question

So there's a boatload of planets being discovered around stars in our solar neighborhood.

Kepler 22, 638 ly away, was discovered to have a planet Kepler 22b, calculated to be twice the size of Earth.

The observations that make these measurements aren't high resolution images but periodic changes in luminosity (the transit method).

My question is: would a similar situation with similar telescope properties (Kepler) detect our own Earth around our Sun from a similar distance? If not, then how close would you have to be to see Earth?

Presumably there is some correspondence among: distance, luminosity/size of a star, size of a planet. Any other factors that make Kepler-22 different from us?

Note: this question asks about Alpha Centauri. I'm asking specifically about distances like Kepler-22.

Answer 1

There is a lot of detail in my answer to If Alpha Centauri A's solar system exactly mirrored our own, what would we be able to detect? I showed that current Doppler techniques and transit techniques would be unlikely to detect the Earth at the distance of Alpha Cen.

It does not get any easier if you are further away. Stars are point sources at 1pc or 200pc, so these spatially unresolved techniques just care what signal level is being received - further away is worse.

In the case of Doppler techniques we need an improvement in radial velocity precision and stability of a factor of a few.

With transits, Kepler was almost there. So another Kepler with a slightly bigger telescope that observed the Sun for say 5 years would do the trick (if the orbital inclination was about 90 degrees). The reason Kepler 22b was just detected is because it has twice the Earth's radius so eclipses 4 times the area and gives a deeper transit signal for a Sun-like star. It is this contrast that is the key to transit detectability.

Maybe the missing piece of information here is that there is a floor level to photometric precision that can't be bettered even if the star is close and bright. So long as you can reach this level in a time shorter than any transit, then the distance is irrelevant, all that matters is the fractional change in the brightness during transit and the number of transits you observe. The first of these grows as the square of the planet radius (for a fixed stellar radius), the second just depends on the length of your mission divided by the orbital period.

The bottom line here is that the Earth in an Earth-like orbit around a Sun-like star would not quite be detectable by Kepler in a Kepler-length mission, even if it were very close and nor if it were further away.