4.1.34. Spln: spline continuum model

Sometimes the continuum of an X-ray source may be too complex to model with known physical components. A situation like that may be found in AGN continua, which are a complex superposition of hard power law, soft continuum excess, relativistically broadened and “normal” broad lines with a priori unknown line shape, etc., while in addition a superimposed warm absorber may have well defined narrow absorption lines. In that case it might be useful to fit the continuum with an arbitrary profile in order to get first an accurate description of the absorber, and then after having “removed” the absorber try to understand the underlying continuum spectrum.

For these situations the spln model introduced here is useful. It allows the user to model the continuum within two boundaries b_1 and b_2 with a cubic spline.

The algorithm works as follows. The user selects the limits b_1 and b_2 as well as the number of grid points n. SPEX then creates a grid x_1,\,x_2,\,\ldots,\,x_n with uniform spacing in b (see below for details). The spectrum at these grid points is contained in the corresponding array y_1,\,y_2,\,\ldots,\,y_n. These have the usual units of 10^{44} photons \mathrm{s}^{-1} \mathrm{keV}^{-1} used throughout SPEX, and is the spectrum emitted at the source. The parameters y_i can be adjusted during the spectral fitting, but b_1, b_2 and n and thereby x_i remain fixed. At intermediate points between the x_i, the photon spectrum is determined by cubic spline interpolation on the (x_i,y_i) data pairs. We take a natural spline, i.e. at x_1 and x_n the second derivative of the spline is zero.

Outside of the range b_1b_2 however, the photon spectrum is put to zero, i.e. no extrapolation is made!

Finally note that we have chosen to define the spline in logarithmic space of y, i.e. the \log of the photon spectrum is fit by a spline. This is done in order to guarantee that the spectrum remains non-negative everywhere. However, the y-values listed is the (linear) photon spectrum itself.

There are four different options for the energy grid x_i, indicated by the parameter type:

  • type=1: b_1 is the lower energy in keV, b_2 is the upper energy in keV, and the grid is linear in energy in between.

  • type=2: b_1 is the lower energy in keV, b_2 is the upper energy in keV, and the grid is logarithmic in energy in between.

  • type=3: b_1 is the lower wavelength in Å, b_2 is the upper wavelength in Å, and the grid is linear in wavelength in between.

  • type=4: b_1 is the lower wavelength in Å, b_2 is the upper wavelength in Å, and the grid is logarithmic in wavelength in between.

Note that the logarithmic grids can also be used if one wants to keep a fixed velocity resolution (for broadened line features for example). Further, each time that the model is being evaluated, the relevant values of the x_i grid points are evaluated.

Warning

Be aware that if you just set b_1, b_2 and n but do not issue the “calc” command or the “fit” command, the x_i values have not yet been calculated and any listed values that you get with the par show command will be wrong. After the first calculation, they are right.

Warning

If at any time you change one of the parameters type, b_1, b_2 or n, the y_i values will not be appropriate anymore as they correspond to the previous set of x_i values.

The maximum number n of grid points that is allowed is 999, for practical reasons. Should you wish to have a larger number, then you must define multiple spln components, each spanning its own (disjunct) b_1b_2 range.

It should be noted, however, that if you take n very large, spectral fitting may become slow, in particular if you take your initial guesses of the y_i parameters not too close to the true values. The reason for the slowness is two-fold; first, the computational time for each fitting step is proportional to the number of free parameters (if the number of free parameters is large). The second reason is unavoidable due to our spectral fitting algorithm: our splines are defined in \log photon spectrum space; if you start for example with the same value for each y_i, the fitting algorithm will start to vary each parameter in turn; if it changes for example parameter x_j by 1, this means a factor of 10; since the neighbouring points (like x_{j-1} and x_{j+1} however are not adjusted in third step, the photon spectrum has to be drawn as a cubic spline through this very sharp function, and it will show the well-known over-and undershooting at the intermediate x-values between x_{j-1} and x_j and between x_j and x_{j+1}; as the data do not show this strong oscillation, \chi^2 will be poor and the fitting algorithm will decide to adjust the parameter y_j only with a tiny amount; the big improvement in \chi^2 would only come if all values of y_i were adjusted simultaneously.

The parameters of the model are:

type : The parameter type defined above; allowed values are 1–4. Default value: 1.
n : The number of grid points n. Should be at least 2.
low : Lower x-value b_1.
upp : Upper x-value b_2. Take care to take b_2>b_1.
x001 : First x-value, by definition equal to b_1. x-values are not allowed to vary (i.e. you may not fit them).
x002 : Second x-value
x003 : Third x-value
... : Other x-values
x999 : last x-value, by definition equal to b_n. If n<999, replace the 999 by the relevant value (for example, if n=237, then the last x-value is x237).
y001 : First y-value. This is a fittable parameter.
y002 : Second y-value
y003 : Third y-value
... : Other y-values
y999 : last y-value. If n<999, replace the 999 by the relevant value (for example, if n=237, then the last y-value is 237).