XMM-Newton SciSim XMM-Newton Science Simulator
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Count rate calculation

The count rate is calculated by dividing the flux density, at wavelength $\lambda$, by the photon energy $\frac{hc}{\lambda}$, to give the photon flux. This is then integrated over the range of wavelengths $\lambda_{min}$ to $\lambda_{max}$ and multiplied by the telescope entrance area to give the total count rate:


$\displaystyle r$ $\textstyle =$ $\displaystyle A \int_{\lambda_{min}}^{\lambda_{max}}{F(\lambda) T(\lambda)
\eta(\lambda) \frac{\lambda}{hc}} d\lambda$ (7)

where $A$ is the entrance area of the telescope
  $\lambda$ is the wavelength
  $F$ is the stellar Flux per unit wavelength
  $T$ is the transmission of the optical system
  $\eta$ is the photocathode quantum efficiency
  $h$ is Planck's constant
and $c$ is the speed of light

The transmission $T(\lambda)$ is made up of the following components:


$\displaystyle T(\lambda)$ $\textstyle =$ $\displaystyle T_{p}(\lambda) T_{s}(\lambda) T_{d}(\lambda)
T_{f}(\lambda) T_{w}(\lambda) T_{a}(\lambda) T_{c}(\lambda)$ (8)

where $T_{p}$ is the primary mirror reflectance
  $T_{s}$ is the secondary mirror reflectance
  $T_{d}$ is the dichroic reflectance
  $T_{f}$ is the filter transmission
  $T_{w}$ is the detector window transmission
  $T_{a}$ is the open area ratio of the MCP
and $T_{c}$ is the ratio between measured and expected count rates

Each source in the input file specifies a spectral type $t$ and a magnitude $V$. OSIM has a set of tables, giving the flux $F_{0}$ of standard spectral types for magnitude V=0.

The source flux density is calculated as follows:


$\displaystyle F(\lambda)$ $\textstyle =$ $\displaystyle F_{0}(\lambda) 100^{\frac{-V}{5}}$ (9)

OSIM does not implement CCD readout frames or Poisson noise. Therefore, the total count is simply the count rate $r$ multiplied by the exposure time $\tau$:


$\displaystyle c$ $\textstyle =$ $\displaystyle r \tau$ (10)

Since the sources are regarded as point sources, this represents the number of counts at a single point in the focal plane. The application of the Point Spread Function subsequently distributes this total number of counts amongst a number of neighbouring pixels.

The symbols used above, correspond to the following keywords in the configuration file:

  $S$ area
  $F_{0}(\lambda)$ spectral files
  $T_{p}(\lambda)$ primary
  $T_{s}(\lambda)$ secondary
  $T_{d}(\lambda)$ dichroic
  $T_{w}(\lambda)$ window
  $T_{a}(\lambda)$ oa_ratio
  $T_{c}(\lambda)$ cr_ratio
  $\eta(\lambda)$ cathode
  $T_{f}(\lambda)$ filter files
  $\tau$ exposureTime
  $\lambda_{min}$ lambdaMin
  $\lambda_{max}$ lambdaMax

The source spectra are specified by tables giving the flux density, for various spectral types, as a function of wavelength. Similarly, the transmission and reflection coefficients of the optical components and the photocathode quantum efficiency are tabulated as a function of wavelength. The configuration file specifies the name of the files containing each of these tables.

Cubic-spline interpolation coefficients are calculated for each of the tables. The integration is then carried out numerically, by the trapezoidal method, using interpolated values for a sequence of closely-spaced wavelengths. The integration step size is controlled by the following additional parameter in the configuration file:

   lambdaStep

The tables should cover at least the range lambdaMin to lambdaMax, otherwise extrapolation will take place, leading to unreliable results (OSIM warns if this happens).

The wavelengths in the tables need not be equally spaced. More accurate interpolation will be achieved if the wavelengths are closely spaced in regions where the response is varying sharply.


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