9.4 Count Rates: Imaging

«Previous Next»

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 cm−2 s−1 Å−1.
  • 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 Å−1 
    per incident erg cm−2 s−1 Å−1.

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 \times 10^{11} \epsilon_{f} \bigg( \int{Q T d \lambda / \lambda } \bigg) \times 10^{-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 pixel−1), 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 cm−2 s−1 Å−1 arcsec−2.
  • 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

C = 2.28 \times 10^{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.

«Previous Next»