4.1.25. Lpro: spatial broadening model

This multiplicative model broadens an arbitrary additive component with an arbitrarily shaped spatial profile, in the case of dispersive spectrometers such as the RGS of XMM-Newton. In many instances, the effects of a spatially extended source can be approximated by making use of the fact that for small off-axis angles \theta the expression {\mathrm d}\lambda / {\mathrm d}\theta is almost independent of wavelength \lambda. This holds for example for the RGS of XMM-Newton (for which {\mathrm d}\lambda / {\mathrm d}\theta = 0.138 / m Å/arcmin, with m the spectral order).

We can utilize this for a grating spectrum as follows. Make an image I(\Delta\theta) of your source projected onto the dispersion axis, as a function of the off-axis angle \Delta\theta. From the properties of your instrument, this can be transformed into an intensity I(\Delta\lambda) as function of wavelength using \Delta\lambda = \Delta\theta {\mathrm d}\lambda / {\mathrm d}\theta. Assume that the spatial profile I(\theta) is only non-zero within a given angular range (i.e. the source has a finite extent). Then we can transform I(\Delta\lambda) into a probability distribution f(x) with f=0 for very small or large values of x (here and further we put x=\Delta\lambda). The auxilliary task rgsvprof (see Rgsvprof) is able to create an input file for the lpro component from a MOS1 image.

The resulting spatially convolved spectrum S_c(\lambda) is calculated from the original spectrum S(\lambda) as

S_c(\lambda) = \int f(\lambda-\lambda_0) S(\lambda_0) {\mathrm d}\lambda_0.

The function f(x) must correspond to a probability function, i.e. for all values of x we have

f(x)\ge 0

and furthermore

\int_{-\infty}^{\infty} f(x) {\mathrm d}x = 1.

In our implementation, we do not use f(x) but instead the cumulative probability density function F(x), which is related to f(x) by

F(x)\equiv \int_{-\infty}^{x} f(y){\mathrm d}y,

where obviously F(-\infty)=0 and F(\infty)=1. The reason for using the cumulative distribution is that this allows easier interpolation and conservation of photons in the numerical integrations.

If this component is used, you must have a file available which we call here vprof.dat (but any name is allowed). This is a simple ascii file, with n lines, and at each line two numbers: a value for x and the corresponding F(x). The lines must be sorted in ascending order in x, and for F(x) to be a proper probability distribution, it must be a non-decreasing function i.e. if F(x_{i})\le F(x_{i+1}) for all values of i between 1 and n-1. Furthermore, we demand that F(x_1)\equiv 0 and F(x_n)\equiv 1.

Note that F(x) is dimensionless. In addition, we allow for two other parameters: a scale factor s and an offset \lambda_o. Usually, s=1, but if s is varied the resulting broadening scales proportional to s. This is useful if for example one has an idea of the shape of the spatial profile, but wants to measure its width directly from the observed grating spectrum. In addition, the parameter \lambda_o can be varied if the absolute position of the source is unknown and a small linear shift in wavelength is necessary.

Warning

This model can be applied to grating spectra (like RGS), but if you include in your fit also other data (for example EPIC), the same broadening will also be applied to that other data SET. This can be avoided by using a separate sector for each detector type.

Warning

The above approximation of spatially extended sources assumes that there are no intrinsic spectral variations over the surface area of the X-ray source. Only total intensity variations over the surface area are taken into account. Whenever there are spatial variations in spectral shape (not in intensity) our method is strictly speaking not valid, but still gives more accurate results than a point-source approximation. In principle in those cases a more complicated analysis is needed.

The parameters of the model are:

s : Scale parameter s, dimensionless. Default value: 1.
dlam : Offset parameter \lambda_o, in Å. Default value: 0 Å.
file : Ascii character string, containing the actual name of the vprof.dat file

Recommended citation: Tamura et al. (2004)