How close to a star does a planet need to be, to generate a detectable transit
I've long had a question in my head, that I can't quite work out the answer to myself and I'm not sure I've seen it answered in any other post, so I'll ask the question here.
Posit two stars, one (star a) 1000ly away the other (star b) 500ly away, but b is incredibly dim, a brown dwarf say, star b has a couple of planets around it.
stars a & b almost perfectly line up, but never actually eclipse, as seen from Earth.
If the planets around b, transit star a (but perhaps not star b), would this generate a detectable transit in a Kepler LC?
I recognize this is probably very statistically unlikely,
I think it would be highly unlikely, though not impossible. Planets only reflect a small amount of light on their own, and to occlude a star 500ly past it, and be apparent from an equal distance would indicate an implausibly massive, or other cosmic structure such as an incredibly dense cloud of interstellar dust. Both would merit a closer look to begin ruling out possibilities in any effect.
This is an interesting question, but I haven't been able to find a definitive answer either. Here are my thoughts on this.
For planets in the same system as a star, the depth of the transit is independent of the distance between the planet and the star. It depends (essentially) only on the ratio of the radii of the planet and the star. So a planet orbiting further from a star makes the same sized transit as a planet of the same size which is closer.
I see no reason why this shouldn't in principle extend to planets in other systems which happen to lie in the same direction. Light travels in straight lines*, so a planet anywhere along the line of sight from Earth will also block part of the light from the star. Indeed the amount of light blocked should increase for planets closer to the Earth because the apparent size of the disk of the planet becomes larger.
This is maybe not all that unlikely either. There are plenty of stars that happen to appear in the same direction but at different distances. Indeed, when vetting transits, astronomers go to some effort to rule out the possibility of interference from other stars.
The question is: could such a transit from a planet around an invisible intermediary star be mistaken for a transit by an orbiting planet?
It seems to me that such chance configurations, if they do occur, probably only happen fleetingly. Both of the stars and the sun are moving relative to each to other and, assuming they are not gravitationally bound, their relative speeds are of a similar order of magnitude to the speed of an orbiting planet. This means that such a transit by an intermediate object will probably only happen once and not repeat regularly like an orbiting exoplanet.
In a quick search of the internet, I failed to find any mention of such transits having been observed, so maybe there is hole in this analysis, or the probability is very small, or such events are simply difficult to confirm because they don't repeat predictably.
*actually geodesics in space-time, but unless there are some are very massive objects in between, I doubt it will make much difference