by Jennifer Yee
Gaudi & Gould (1997) showed that the parameters of a planet (s and q) can be approximated analytically based on light curve observables.
First, determine the parameters of the underlying stellar event ( t_{0}, u_{0}, t_{E}) following Tutorial 1. Then, measure the time of the planetary perturbation t_{p}:
Combined with the point lens parameters, you can calculate τ and therefore, u at that time. This gives the source position relative to the lens. Since a planet must perturb one of the images to be detected, to first order, this means the planet must be at the location of one of the images:
The position of the images is given by: y_{±} = ± (½) (√(u^{2} + 4) ± u), so the planet location, s must be either y_{+} or y_{-}.
If the planet perturbs the minor image, it will tend to destroy that image, leading to a decrease in magnification. On the other hand, a planet will always further magnify a major image:
Therefore, the form of the perturbation will show which solution is correct.
The size of the Einstein ring is proportional to the square root of the mass:
Einstein Ring = | θ_{E} ∝ | M^{1/2} |
Therefore, the ratio of the duration of the planetary perturbation to the duration of the event should be proportional to the square root of the mass ratio. Equivalently,
Mass ratio = | q = | m_{p} | = | t_{E,p}^{2} |
M_{star} | t_{E,star}^{2} |
Try calculating the planetary parameters for one (or more) of the following events:
OGLE-2005-BLG-390L