# Analytic Planet Parameters

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.

## Finding the Location of the Planet

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

#### 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:

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