Gravitational Microlensing - Mathematics

Mathematics

The mathematics of microlensing, along with modern notation, are described by Gould and we use his notation in this section, though other authors have used other notation. The Einstein radius, also called the Einstein angle, is the angular radius of the Einstein ring in the event of perfect alignment. It depends on the lens mass M, the distance of the lens d_L, and the distance of the source d_S:

(in radians)

For M equal to the mass of the Sun, d_L = 4000 parsecs, and d_S = 8000 parsecs (typical for a Bulge microlensing event), the Einstein radius is 0.001 arcseconds (1 milliarcsecond). By comparison, ideal Earth-based observations have angular resolution around 0.4 arcseconds, 400 times greater. Since is so small, it is not generally observed for a typical microlensing event, but it can be observed in some extreme events as described below.

Although there is no clear beginning or end of a microlensing event, by convention the event is said to last while the angular separation between the source and lens is less than . Thus the event duration is determined by the time it takes the apparent motion of the lens in the sky to cover an angular distance . The Einstein radius is also the same order of magnitude as the angular separation between the two lensed images, and the astrometric shift of the image positions throughout the course of the microlensing event.

During a microlensing event, the brightness of the source is amplified by an amplification factor A. This factor depends only on the closeness of the alignment between observer, lens, and source. The unitless number u is defined as the angular separation of the lens and the source, divided by . The amplification factor is given in terms of this value:

This function has several important properties. A(u) is always greater than 1, so microlensing can only increase the brightness of the source star, not decrease it. A(u) always decreases as u increases, so the closer the alignment, the brighter the source becomes. As u approaches infinity, A(u) approaches 1, so that at wide separations, microlensing has no effect. Finally, as u approaches 0, A(u) approaches infinity as the images approach an Einstein ring. For perfect alignment (u = 0), A(u) is theoretically infinite. In practice, finite source size effects will set a limit to how large an amplification can occur for very close alignment, but some microlensing events can cause a brightening by a factor of hundreds.

Unlike gravitational macrolensing where the lens is a galaxy or cluster of galaxies, in microlensing u changes significantly in a short period of time. The relevant time scale is called the Einstein time, and it's given by the time it takes the lens to traverse an angular distance relative to the source in the sky. For typical microlensing events, is on the order of a few days to a few months. The function u(t) is simply determined by the Pythagorean theorem:

The minimum value of u, called u_min, determines the peak brightness of the event.

In a typical microlensing event, the light curve is well fit by assuming that the source is a point, the lens is a single point mass, and the lens is moving in a straight line: the point source-point lens approximation. In these events, the only physically significant parameter that can be measured is the Einstein timescale . Since this observable is a degenerate function of the lens mass, distance, and velocity, we cannot determine these physical parameters from a single event.

However, in some extreme events, may be measurable while other extreme events can probe an additional parameter: the size of the Einstein ring in the plane of the observer, known as the Projected Einstein radius: . This parameter describes how the event will appear to be different from two observers at different locations, such as a satellite observer. The projected Einstein radius is related to the physical parameters of the lens and source by

.

It is mathematically convenient to use the inverses of some of these quantities. These are the Einstein proper motion

and the Einstein parallax

.

These vector quantities point in the direction of the relative motion of the lens with respect to the source. Some extreme microlensing events can only constrain one component of these vector quantities. Should these additional parameters be fully measured, the physical parameters of the lens can be solved yielding the lens mass, parallax, and proper motion as