by Jennifer Yee

How do caustics relate to light curve features?

The following animation shows what happens when a source crosses into a caustic (aka a caustic entrance):

Caustic entrance

Jennifer Yee

There is a sharp jump in magnification when the source crosses into the caustic (red). Once inside the caustic, the magnification declines but always stays above the level outside the caustic.

A caustic exit is the inverse of a caustic entrance: the magnification increases as the source approaches the caustic, then drops abruptly when the source crosses outside the caustic.

Caustic exit

Jennifer Yee

Because the caustic is a closed curve, there will always be two caustic crossings: an entrance and an exit. The figure below shows the magnification curve for a source crossing the caustic of a binary star lens. Can you identify the caustic entrance and the caustic exit in the light curve?

The caustic structure is often used as a kind of short-hand for the full magnification map because the features of the magnification map can be inferred from the caustic structure. Recall that a cusp is one of the points of the caustic. The animation below shows how the magnification is increased near the cusps.

Magnification map and caustic animation

Magnification map and caustic

Jennifer Yee

A source passing near a cusp can produce a magnified signal without actually crossing the caustic directly:

Caustic cusp

Jennifer Yee

Notice the single peak and rounded form of the cusp approach. A cusp crossing may also appear to be a single peak if the distance between the cusps is smaller than the diameter of the source. In that case, the caustic entrance and exit merge, producing a single peak but with sharp, rather than smooth shoulders.

Finite Source Effects:

At a caustic, the mathematical solution to the lens equation gives infinite magnification, but infinite magnification is never observed in practice. Why?

When evaluating the lens equation, we get infinite magnification if we assume the source is a point source, meaning that it is a mathematical point with zero size. In reality, the sources are stars, which have a physical size and therefore a finite extent. The observed magnification of the source is smeared out by the face of the source. If you wear glasses, this is similar to how things appear blurry if there is a smudge on the glasses. In practice we calculate the magnification of a source by integrating the magnification pattern across the face of the source.

The animation below compares a caustic entrance for a point source (dotted line) to a source with radius 0.01.

Effects of a finite source on caustics

The effects of a finite source on caustics

Jennifer Yee

If you are familiar with transiting planet lightcurves, this effect is directly analogous to the points of first and second contact for a transit. The images below show a caustic entrance and the first and second points of contact and a caustic exit and the third and fourth points of contact.

Points of contact with a caustic

Jennifer Yee

Points of contact with a caustic

Jennifer Yee

Exercise:

Match the following planetary light curves to their caustics and trajectories.

In each light curve identify any caustic entrances and exits and look for cusp approaches.
Also consider when the perturbation occurs: When is the peak of the underlying stellar event? Does the planetary perturbation occur before or after the peak?
Finally, consider the location of the caustic and source trajectory: Where is the closest approach between the source and lens (where is the source closest to 0,0)? Where is the caustic in relation to this?