Microlensing and exoplanets

Finally, let’s discuss how to represent amplification due to lensing. Previously we’ve seen that amplification due to microlensing is an important technique used to detect exoplanets (other techniques include radial velocity, transit, etc). But how can this be possible, since planets are so much smaller in mass than stars and galaxy clusters? Actually, even stars by themselves amplify light by focusing it, but the effect is very small. This GNU Octave plot illustrates the amplification factor around our Sun:

Things get interesting however when a planet orbits close to the star. While the planet itself exerts an even smaller gravitational influence than its star, when the two influences combine they can produce extreme, discontinuous jumps in brightness called caustics. Caustics can be seen as bright wavy lines in a pool of water, as illustrated in the excellent Optics Picture of the Day website: http://www.atoptics.co.uk/fz535.htm.

The following Octave light curve plot shows the amplification that is observed when a single exoplanet passes near the Einstein radius of its star. The Einstein radius is defined as \(2R_s\frac{D_{ds}}{D_s}\) where \(R_s\) is the Schwarzschild radius of the lens, a measure of how much the lens curves spacetime (and also the radius of the event horizon in a black hole). The Einstein radius can be thought of as the characteristic size of the lens, and is also the size by which an Einstein ring would appear around the lens.

Here is an actual light curve observed for an extrasolar system 4.1 kpc from Earth called OGLE-2012-BLG-0026.

Han, C. et al., 2013. The Second Multiple-planet System Discovered by Microlensing: OGLE-2012-BLG-0026Lb, c - A Pair of Jovian Planets beyond the Snow Line. The Astrophysical Journal, 762(2), p.L28.

I've implemented a real-time plotting feature in celestia.Sci using the Qwt framework. Here it is in action:

First to briefly describe how this was done, I had to create an add-on for the extrasolar system as stock celestia.Sci doesn't have it. Then I chose View > Plot, selected the star OGLE-2012-BLG-0026L, lined it up with a background star which I arbitrarily named OGLE-2012-BLG-0026L-SRC, and hit Refresh in the plot panel. The microlensing code then starts varying the impact parameter across the lens to produce the time-varying light curve.

The light curve of OGLE-2012-BLG-0026 plotted by celestia.Sci shows several differences such as higher peaks. This however could be explained by the fact that we are approximating lenses and source objects as points of zero area;  this results in singularities in the calculated amplification factors. Real microlenses have non-zero area, and this smooths out the singularities. However, qualitatively we can identify similarities between the simulated and real light curves, such as the presence of multiple peaks. Future improvements could try to increase the realism by taking into account the disc sizes of stars and planets.