# Point Lenses

by Jennifer Yee

### Parameterization of a Point Lens Lightcurve

#### Geometry

J. Yee

A point lens microlensing event is parameterized by three variables:

• u0 = the impact parameter between the source and the lens (i.e. minimum separation)
• t0 = the time of closest approach between the source and lens (i.e. t at u = u0)
• tE = the Einstein crossing time = the time for the source to travel 1 Einstein radius

The magnification of the source is given by the equation:

 Magnification = A(u) = u2 + 2 u (u2 + 4 )1/2

where

u = ( u02 + τ2)1/2

and

 τ = (t - t0) tE

## Exercise

In the limit that u0 << 1, what is A(u)?

# Examples

Below are discovery lightcurves of the first microlensing events. For each one, estimate its point lens parameters (t0, u0, tE). These basic parameters can be directly inferred from the lightcurve by measuring the height and time of the peak and tFWHM of the lightcurve (FWHM = Full Width Half Maximum). Then, tE can be calculated using the measured tFWHM and the above equations. In the limit where u0 << 1, tE ~ (1/2)tFWHM/u0.

J. Yee

J. Yee

## Example 1: The first MACHO event

From Alcock, Akerlof, Allsman, et al. 1993 Nature, 365, 621.

#### Lightcurve of MACHO-1, Alcock et al. Figure 2

Alcock et al. (1993)

t0:
u0:
tE:

## Example 2: The first EROS events

From Aubourg, Bareyre, Bréhin, et al. 1993 Nature, 365, 623.

### Event 1

#### Lightcurve of the first EROS event, Aubourg et al. Figure 1b

Aubourg et al. (1993)

t0:
u0:
tE:

### Event 2

#### Lightcurve of the second EROS event, Aubourg et al. Figure 2b

Aubourg et al. (1993)