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)
• t_E = 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)?
Answer

Examples

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

Estimating t0 and u0 for a point source, point lens event J. Yee

Estimating tE for a point source, point lens event 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)

	Your answers
t0:
u0:
tE:

Check

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)

Hint 1

Hint 2

	Your answers
t0:
u0:
tE:

Check

Event 2

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

Aubourg et al. (1993)

	Your answers
t0:
u0:
tE:

Check

Note

In each of the examples shown above, there are two panels shown: red and blue. Why? Gravitational lensing is achromatic meaning that all light is magnified equally, regardless of wavelength. The fact that the lensing signal is the same in both bands is further proof that the observed light curves are caused by lensing rather than some other astrophysical effect. (For example, a stellar flare would look different in red and blue light.) The bottom panel of the first figure, A_red/A_blue, demonstrates that the signal is achromatic by showing that the ratio of the magnifications is flat.

References:

Alcock, Akerlof, Allsman, et al. (1993), Nature, 365, 621
Aubourg, Bareyre, Bréhin, et al. (1993), Nature, 365, 623