Suppose there’s a planet a distance of, say, 20 light years from here, and suppose there’s a civilization there with technology identical to ours. And suppose they aim their instruments at our sun. How much information about our solar system can they learn?
There are three relevant ways we can detect exoplanets:
Transit photometry – seeing a planet move across a star’s surface. This requires an edge-on view of our solar system for our alien, so I’ll assume it isn’t applicable.
Doppler spectrography – seeing the light from a star get Doppler shifted as planets tug the star back and forth as they orbit. This method favors large planets near the Sun. It won’t tell you the mass of a planet – just its minimum mass. This won’t work at all if the aliens are within 5 degrees or so of the poles of the Sun. However, assuming they aren’t:
They know about Jupiter (which makes the Sun wobble at a speed of 12.6 m/s) and probably have for close to a decade.
They strongly suspect Saturn’s (Sun wobble speed = 2.9 m/s) existence, but have only seen 40% or so of its orbit; they might want to see more before confirming its existence.
With current technology and enough time to observe a few of their orbits, they can detect Uranus (SWS = 310 mm/s; orbital period = 84 years) and Neptune (279 mm/s; 165 years).
Assuming alien technology improves at the same rate as ours, however, they’ll get spectrographs around 2017 that can detect Earth (SWS = 91 mm/s) and Venus (SWS = 85 mm/s) after a few orbits. Those spectrographs will help with the precision of the observation of the effects of Uranus and Neptune on the Sun – it would be apparent by 2060 (Uranus) and 2100 (Neptune), assuming technology hasn’t jumped even more by then.
Astrometry – directly measuring the location of the star and figuring out what’s causing it to wobble. This requires the observer to be relatively close to the star. Fortunately, 20 ly is close enough with current tech. This method favors large planets far away from the star – Neptune and Earth would be significantly easier to find than their twins.
The Gaia space telescope (launched in 2013) will be able to detect all the gas giants with high precision. However, as above, it will take a while to get enough data to confirm the orbits of the planets.
Finding Earth with Gaia would require the alien to be 2.something light years away. (I calculated it, but closed the tab).
(Disclaimers: I like astronomy. I am not an astronomer. My Intro to Astronomy class in college predates the discovery of exoplanets.)