X-ray telescopes usually also have gratings that can be inserted into the light path, so that photons of different energies get dispersed by different angles, and whose actual energies can then be inferred accurately by measuring how far away on the detector they ended up. The accuracy of the inference is usually limited by the width of the PSF. Thus, a major contributor to the LRF (Line Response Function) is the aforementioned scattering.

A correct accounting of the spread of photons of course requires a full-fledged response matrix (RMF), but as it turns out, the line profiles can be fairly well approximated with Beta profiles, which are simply Lorentzians modified by taking them to the power β –

where B(1/2,β-1/2) is the Beta function, and N is a normalization constant defined such that integrating the Beta profile over the real line gives the area under the curve as N. The parameter β controls the sharpness of the function — the higher the β, the peakier it gets, and the more of it that gets pushed into the wings. Chandra LRFs are usually well-modeled with β~2.5, and XMM/RGS appears to require Lorentzians, β~1.

The form of the Lorentzian may also be familiar to people as the Cauchy Distribution, which you get for example when the ratio is taken of two quantities distributed as zero-centered Gaussians. Note that the mean and variance are undefined for that distribution.

Almost all of astrophysics is about inferring some characteristic of the source from the light gathered by some instrument. And the instruments usually give us information on 3 quantities of interest: when the photon arrived, where did it land, and what was its color. In X-ray astronomy in particular, these are usually recorded as a lengthy table of “events”, where each row lists (t, p, E), where t are the arrival times of the photons (which can be measured with a precision anywhere from microseconds to seconds), p=(x,y) says where on which pixel of the detector the photon landed, and E represents how energetic the photon was. Generally, both p and E are blurred, the former because of the intrinsic limitations of how well the telescope can focus and how well the detector can localize, and the latter because of limitations in evaluating the total energy deposited in the detector by the incoming photon. So, in general, the expected signal at the detector can be written as follows:

where E is the energy of the incoming photon and E’ is what it is registered as, p is the true sky location and p’ is the registered location, S(E,p,t;theta) is the astrophysical signal, A(E,p’) is the effective area of the telescope/detector combination, P(p,p’;E,t) is the so-called Point Spread Function, and R(E,E’;p) is the energy response matrix.

The expected signal, M(E’,p’,t) is then compared with the collected data to infer the values and uncertainty in theta, the parameters that define the source model. The task of calibration scientists is to compute A, R, and P accurately enough that the residual systematic errors are smaller than the typical statistical error in the determination of the theta.

In general, the calibration products also vary with time, but usually these variations are negligible over the duration of an observation, so you can leave out the t from them. Not always though!