Declaring Models

There are a couple of different philosophies about setting up the models, and the authors often disagree. However, since this section is being written while the other author skives off on his farm, the bias will be clear.

There are four different things that are to be fit to the data: the cosmic X-ray background (§6.2), the instrumental lines (§6.1), the solar wind charge exchange (§6.4), and the residual soft proton flares (§6.3). Of these, the first three all require the full photon redistribution/effective area functions while the last requires its own redistribution function and no effective area function. Since there are two different redistribution functions involved, there have to be at least two different model declarations. At the minimum:

model const*const*(apec+tbabs(apec+pow)
+gaus+gaus+gaus+gaus+gaus+gaus)
parameter 1, etc.

resp 2:1 mos1-diag.rsp
resp 2:2 mos2-diag.rsp
resp 2:3 pn-diag.rsp
resp 2:4 pn-diag.rsp
model 4:spf const(bknpow)
parameter spf:1, etc.

However, I find that having a large multi-component model, especially where some components do not apply to some of the spectra (i.e., the XMM instrumental lines certainly do not apply to the ROSAT spectrum), rapidly becomes confusing. Thus, I find it more useful to declare the models as follows:

model const*const*(apec+tbabs(apec+pow))
parameter 1, etc.

resp 2:1 mos1S001.rmf
arf 2:1 mos1S001.arf
resp 2:2 mos2S002.rmf
arf 2:2 mos2S002.arf
resp 2:3 pnS003_0.rmf
arf 2:3 pnS003_0.arf
resp 2:4 pnS003_4.rmf
arf 2:4 pnS003_4.arf
model 2:alsi const*const*(gaus+gaus)
parameter alsi:1, etc.

resp 3:1 mos1-diag.rsp
resp 3:2 mos2-diag.rsp
resp 3:3 pn-diag.rsp
resp 3:4 pn-diag.rsp
model 3:spf const*const*(bknpow)
parameter spf:1, etc.

resp 4:1 mos1S001.rmf
arf 4:1 mos1S001.arf
resp 4:2 mos2S002.rmf
arf 4:2 mos2S002.arf
resp 4:3 pnS003_0.rmf
arf 4:3 pnS003_0.arf
resp 4:4 pnS003_4.rmf
arf 4:4 pnS003_4.arf
model 4:swcx const*const*(gaus+gaus+gaus+gaus)
parameter swcx:1, etc.

This second way of doing things may take longer to set up, but it requires fewer statements for linking parameters and helps avoid errors when linking the parameters. In this formulation, the first model contains the cosmic X-ray background components, the second model contains the instrumental lines (the big Al K$\alpha$ and Si K$\alpha$ lines), and the third contains the residual SPF emission. Finally, there is the SWCX, which may be completely unneccessary for your work. Either way you set up the models there are at least 173 parameters. However, many of these parameters are linked, minimally used, or simply there in case they might be needed. We will now consider the models in order.

The Cosmic Background:

const*const*(apec+tbabs(apec+pow))

There are two constants. The first is the number of square arcminutes covered by the field of view. This value will be different for each instrument, and should be frozen once set. The solid angle of the region from which the spectrum was extracted can be determined from the BACKSCAL keyword of the SPECTRUM extension of the spectrum file. The BACKSCAL keyword is in units of $0.05\arcsec\times0.05\arcsec$ pixels.

The second constant is a fudge factor to account for the slight mismatches in the calibrations of the instruments. It should initially be set to unity and frozen. Once the fit is nearly correct, it can be thawed; it should change by less than 10%.

The apec+tbabs(apec+pow)) is the actual cosmic background; an unabsorbed Local Hot Bubble component, an absorbed Galactic halo component, and an absorbed contribution from the unresolved extragalactic background. Purists will note that the Galactic halo has multiple components; whether you include that complexity will depend upon the signal to noise ratio of the data, the location on the sky, and the objectives in fitting the data. If you are only interested in determining the residual SPF emission in order to do imaging, then this simple model may be appropriate. The absorbing column density can usually be frozen at the measured value. The index of the unresolved extragalactic background should be frozen. If you've been careful in correcting for the point source detection limit, the normalization of the extragalactic background can be frozen.

Note that all of these components are linked across the different instruments (and missions). Thus, there are only 18 active parameters from this part of the model.

Instrumental Lines:

const*const*(gaus+gaus)

Here the two constants are the same as for the previous model, and should be linked to the constants in the cosmic background model. The lines are described by their central energy, width, and normalization. Unfortunately the effective central energy is not constant across the different EPIC instruments due to small shifts in the gain. The normalizations vary over the different EPIC instruments. We find it a good idea, for the initial fit, to freeze the central energies at their expected values (1.486 keV for Al K$\alpha$ and 1.739 keV for Si K$\alpha$) and to freeze their widths to 0.0. Once a reasonable value has been found for the normalizations, the line centers and widths should be thawed. Note that there is no Si K$\alpha$ line for the pn.

Solar Wind Charge Exchange Lines:

const*const*(gaus+gaus+gaus+gaus)

Here the two constants are the same as for the previous model, and should be linked to the constants in the cosmic background model. Here all the parameters are linked between instruments. However, you will note that I have not included a SWCX component for ROSAT ; the bulk of the charge exchange in the ROSAT All Sky Survey was magnetospheric, and has been removed and the remainder is heliospheric and likely included in the LHB emission. Most users will probably not want to bother with the SWCX component when worrying about imaging analyses; it is usually small. However, there are examples where there are strong mismatches in the soft surface brightness between two overlapping observations. These are the cases where the possibility of SWCX should be checked in the spectra.

We've only fitted four charge exchange lines here. In principle, one could fit several more. In all cases, one should freeze the line centers to the expected values and the line widths to zero. Once a good fit has been obtained, then one can see if the energies need to be modified slightly, and in a self-consistent way.

Residual Soft Proton Flare Emission:

const*const*(bknpow)

The constants here are not actually needed since we are not using a real response matrix. However, I like using them in order to get a good feel for the relative amount of emission per square arcminute for the different instruments.

We do see differences between the two MOS detectors, both in normalization and spectral shape. Thus, the parameters for the two MOS observations should not be linked. However, we admit that we often do link them for the initial fit, and then separate the two instruments once we are in the vicinity of a good fit.

Similarly, the pn “sees” a very different spectral shape than the MOS detectors, so it can't be linked to the MOS. However, we are entering a bit of terra incognita here: we do not have extensive experience (yet) as to the relative behaviors of pn with PATTERN==0 and pn with PATTERN$<=$4. We expect that both the normalizations and the spectral shapes will be similar, but we have not fitted enough cases to know to what extent this isn't true.