9.4 Count Rates: Imaging

9.4.1 Point Source

For a point source, the count rate, C (e s−1), can be expressed as the following integral over the bandpass of the filter:

C=A\int F_\lambda\frac{\lambda}{hc}Q_\lambda T_\lambda \epsilon_f d\lambda=\int F_\lambda S\lambda \epsilon_f dh\ \ ,

where:

  • A is the area of an unobstructed 2.4-m telescope (i.e., 45,239 cm2)
  • Fλ is the flux from the astronomical source in erg cm2 s−1 Å1.
  • The factor λ/hc (where h is Planck’s constant and c is the speed of light) converts ergs to photons.
  • QλTλ is the system fractional throughput, i.e., the probability of detecting an electron per incident photon, including losses due to obstructions of the full 2.4-m OTA aperture. It is specified this way to separate out the instrument sensitivity Qλ and the filter transmission Tλ.
  • εf  is the fraction of the point-source energy encircled within Npix pixels.
  • Sλ is the total imaging point-source sensitivity in units of e s−1 Å1 
    per incident erg cm2 s−1 Å1.

The peak e− s−1 pixel−1 from the point source, Cpeak, is given by the following integral over the bandpass:

C_{peak}=\int F_\lambda S_\lambda \epsilon_f(1)d\lambda\ \ ,

where:

  • Fλ, and Sλ are as above.
  • εf(1) is the fraction of energy contained within the peak pixel.

If the flux from the source can be approximated by a flat continuum (Fλ = constant) and εf is roughly constant over the bandpass, then:

C=F_\lambda \epsilon_f \int S_\lambda d\lambda\ \ .

We can now define an equivalent bandpass of the filter, Bλ, such that:

\int S_\lambda d\lambda=S_{peak} B_\lambda\ \ ,

where:

  • Speak is the peak sensitivity.
  • Bλ is the effective bandpass of the filter.

The count rate from the source can now be written as:

C=F_\lambda \epsilon_f S_{peak} B_\lambda\ \ .

In Tables 9.1 and 9.2 above, we give the value of:

\int S_\lambda d\lambda\ \ ,

for each of the filters.

Alternatively, we can write the count-rate equation in terms of V magnitudes:

C=2.5\times10^{11} \epsilon_f \bigg( \int QTd\lambda/\lambda \bigg)\times10^{-0.4(V+AB_\nu)}\ \ ,

where V is the visual magnitude of the source, the quantity under the integral is the mean sensitivity of the detector+filter combination (also tabulated in Tables 9.1 and 9.2), and ΑΒν is the filter-dependent correction for the deviation of the source spectrum from a constant Fν spectrum. This latter quantity is tabulated for some representative astronomical spectra in Appendix A.

9.4.2 Diffuse Sources

For a diffuse source, the count rate, C (e s−1 pixel1), which is now per pixel, due to the astronomical source can be expressed as

C=\int I_\lambda S_\lambda m_x m_y d\lambda\ \ ,

where:

  • Iλ is the surface brightness of the astronomical source, in erg cm2 s−1 Å1 arcsec2.
  • Sλ is as above.
  • mx and my are the plate scales in arcsec pixel−1 along orthogonal axes.

9.4.3 Emission-Line Sources

For a source where the flux is dominated by a single emission line, the count rate can be calculated from the equation

C=2.28\times10^{12}\times(QT)_\lambda\times F(\lambda)\times\lambda\ \ ,

where C is the observed count rate in e s−1, (QT)λ is the system throughput at the wavelength of the emission line, F(λ) is the emission-line flux in units of erg cm−2 s−1, and λ is the wavelength of the emission line in angstroms. (QT)λ can be determined by inspection of the plots in Appendix A. See Section 9.9.4 for an example of count-rate estimation for an emission-line source.