Analytic Planet Parameters

Gaudi & Gould (1997) showed that the parameters of a planet (s and q) can be approximated analytically based on light curve observables.

Finding the Location of the Planet

First, determine the parameters of the underlying stellar event ( t₀, u₀, t_E) following Tutorial 1. Then, measure the time of the planetary perturbation t_p:

Estimating tp, the time of planetary perturbation

J. Yee

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:

Location of major and minor images during a lensing event

J. Yee

The position of the images is given by: y_± = ± (½) (√(u² + 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:

Lightcurve with planet close to the minor image

J. Yee

Lightcurve with planet close to the major image

J. Yee

Therefore, the form of the perturbation will show which solution is correct.

Finding the Mass Ratio of the Planet

The size of the Einstein ring is proportional to the square root of the mass:

Einstein Ring =

θE ∝

M1/2

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 =	mp	=	tE,p2
		Mstar		tE,star2

Examples:

Try calculating the planetary parameters for one (or more) of the following events:

OGLE-2005-BLG-390L
MOA-2009-BLG-266L
MOA-2010-BLG-328L
MOA-2010-BLG-353L
MOA-2011-BLG-028L
OGLE-2012-BLG-358L
OGLE-2012-BLG-406L
MOA-2013-BLG-605L
OGLE-2014-BLG-1760

References:

Gaudi, B.S. and Gould, A. (1997), ApJ, 486, 85