6.2 Determining Count Rates from Sensitivities

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In the simplest terms, the instrumental sensitivity (S) times the flux from your object of interest gives the counts/s (C) expected from your source times the gain (G) (i.e., it gives counts for the MAMA and electrons for the CCD):

\small{C\times G=S\times Flux~.}

Later in this chapter we provide specific formulae appropriate for imaging and spectroscopic modes, which can be used to calculate the expected count rates from your source and the signal-to-noise ratio. The formulae are given in terms of sensitivities, but we also provide transformation equations between the throughput (T) and sensitivity (S) for imaging and spectroscopic modes.

Sensitivities and throughputs are presented in graphical and tabular form as a function of wavelength for the spectroscopic modes in Chapter 13, and for the imaging modes in Chapter 14. Given the source characteristics and the sensitivity of the STIS configuration, calculating the expected count rate over a given number of pixels is straightforward. The additional information you will need for spectroscopic observations is the aperture transmission (TA), the encircled energy fraction (εf) in the direction perpendicular to the dispersion, the number of pixels per spectral resolution element (or line spread function FWHM) and the plate scale, which are provided in Chapter 13. For imaging observations you need only the encircled energies and plate scales. Below, we describe how to determine two quantities:

  1. The counts/s (C) from your source over some selected area of Npix pixels.
  2. The peak per-pixel count rate (Pcr) from your source—useful for avoiding saturated exposures and for assuring that MAMA observations do not exceed the local bright object limits.

We consider the cases of point sources and diffuse sources separately below.

6.2.1 Spectroscopy

Sensitivity Units and Conversions

The spectroscopic point source sensitivity, \small{S^p_\lambda,} has the following units, depending on the detector:

\small{\rm CCD\!: (e^{-}/s/pix_{\lambda})/(incident~erg/cm^2/s/Å)}
\small{\rm MAMA\!: (counts / s / pix_{\lambda}) / (incident~erg / cm^2 / s / Å)}

where:

  • pixλ = a pixel in the dispersion direction;
  • counts and electrons refer to the total received from the point source integrated over the PSF in the direction perpendicular to the dispersion (along the slit);
  • \small{S^p_\lambda} is corrected for time dependent and temperature dependent variations, aperture losses, and CTE losses by pipeline calibration.

The spectroscopic diffuse source sensitivity, \small{S^d_\lambda}, has the following units, depending on the detector:

\small{\rm CCD\!: (electrons /s /pix_λ /pix_s) / (incident~erg/ s /cm^2 / Å /arcsec^2)}
\small{\rm MAMA\!: (counts/s/ pix_λ/pix_s)/(incident~erg /s/cm^2/Å/arcsec^2)}

where:

  • pixλ = a pixel in the dispersion direction;
  • pixs = a pixel in the spatial direction.

\small{S^p_\lambda} and \small{S^d_\lambda} are related through the relation:

\small{S^d_\lambda \cong \big(S^p_\lambda \times m_s \times W\big)~,}

where:

  • ms is the plate scale in arcsec/pix in the spatial direction (i.e., in the direction perpendicular to the dispersion);
  • W  is the slit width in arcseconds;
  •  S^d_\lambda is corrected for time dependent and temperature dependent variations by pipeline calibration.

In general, we have assumed that the diffuse source has a uniform brightness over the area of interest and that the spectrum can be approximated as a continuum source (i.e., any emission or absorption lines are broader than the resolution after taking the effect of the slit into account).

Point Source

For a point source, the count rate, C, from the source integrated over an area of Npix = Nλpix × Nspix pixels can be expressed as:

\small{C=\frac{F_\lambda \times S^p_\lambda \times T_A \times \epsilon_f \times N_{\lambda pix}}{G}~,}

where:

  • G is the gain (always 1 for the MAMA, and 1 or 4 depending on the choice of CCDGAIN for the CCD);
  • Fλ = the continuum flux from the astronomical source, in erg/s/cm2/Å;
  • TA = the aperture transmission (a fractional number less than 1);
  • εf = the fraction of the point source energy contained within Nspix pixels in the spatial direction;
  • Nλpix = the number of wavelength pixels integrated over. For an unresolved emission line, Nλpix is just the number of pixels per spectral resolution element and Fλ is simply the total flux in the line in erg/s/cm2 divided by the product of the dispersion in Å/pix and Nλpix (i.e., divided by the FWHM of a resolution element in Å);
  • Nspix = the number of pixels integrated over in the spatial direction.

The peak counts/s/pix from the point source is given by:

\small{P_{cr}=\frac{\epsilon_f(1) \times F_\lambda \times S^p_\lambda \times T_A}{G}~,}

where:

  • εf (1) is the fraction of energy contained within the peak pixel;
  • Fλ, \small{S^p_\lambda}, and TA are as above.

Diffuse Source

For a diffuse continuum source over Npix = Nλpix × Nspix, the count rate, C, can be expressed as:

\small{C=\frac{I_\lambda \times S^d_\lambda \times N_{\lambda pix} \times N_{spix}}{G}~,}

where:

  • Iλ = the surface brightness of the astronomical source, in erg/s/cm2/Å/arcsec2;
  • Nλpix = the number of wavelength pixels integrated over in the dispersion direction. For an unresolved emission line, Nλpix is just the number of pixels per spectral resolution element, and Iλ is simply the total flux in the line in ergs/s/cm2/arcsec2 divided by the product of the dispersion in Å/pix and Nλpix, (i.e., divided by the FWHM of the resolution element in Å);
  • Nspix = the number of pixels integrated over in the spatial direction.

For a diffuse continuum source the peak counts/s/pix, Pcr, is given by:

\small{P_{cr}=\frac{I_\lambda \times S^d_\lambda}{G}~.}

For a diffuse, spectrally unresolved emission-line source the peak counts/s/pix, Pcr, is essentially independent of slit size and is given by:

\small{P_{cr}=\frac{I_{line} \times S^d_\lambda \times w}{G\times W\times FWHM}~,}

where:

  • Iline is the intensity in erg/s/cm2/arcsec2 in the line;
  • FWHM is the full width half max of the instrumental profile in Å, which for STIS is nearly always 2 pixels × d, where d is the dispersion in Å/pix;
  • w is the slit width in arcseconds which projects to n pixels in the detector plane, where n is the width of the resolution element in pixels. Note that w is numerically equal or close to twice the plate scale in the dispersion direction for all modes;
  • W is the actual slit width in arcseconds.

Thus, for STIS in particular, this expression reduces to:

\small{P_{cr}\sim\frac{I_{line} \times S^d_\lambda \times m_\lambda}{W\times d\times G}~,}

where:

  • d is the dispersion in Å/pix;
  • mλ is the plate scale in the dispersion direction;
  • All else is as above.

The counts from the emission line will be spread over Nλpix pixels where Nλpix is the slit width per plate scale in the dispersion direction (Nλpix = W/mλ).

6.2.2 Imaging

Sensitivity Units and Conversions

The imaging point source sensitivity, \small{S^p_\lambda}, has the following units, depending on the detector:

\small{\rm CCD\!: (e^{-}/s/Å) / (incident~erg/s/cm^2/Å)}
\small{\rm MAMA\!: (counts/s/Å) / (incident~erg /s/cm^2/Å).}

where:

  • counts and electrons refer to the total number received from the point source integrated over the PSF.

The imaging diffuse-source sensitivity, \small{S^d_\lambda}, has the following units, depending on detector:

\small{\rm CCD\!: (e^-/s/Å/pix) / (incident~erg/s/cm^2/Å/arcsec^2)}
\small{\rm MAMA\!: (counts/s/Å/pix) / (incident~erg/s/cm^2/Å/arcsec^2).}

Thus  \small{S^p_\lambda} and  \small{S^d_\lambda} are related through the relation:

\small{S^d_\lambda\cong \big(S^p_\lambda \times m^2_s\big)~,}

where:

  • ms is the plate scale in arcsec/pix.

Point Source

For a point source, the count rate, C, over an area of Npix pixels can be expressed as:

\small{C=\frac{\int F_\lambda\times S^p_\lambda\times\epsilon_f d\lambda}{G}~,}

where:

  • Fλ = the flux from the astronomical source, in ergs/s/cm2/Å;
  • εf = the fraction of the point source energy encircled within Npix pixels;
  • the integral is over the bandpass.

The peak counts/s/pix from the point source are given by:

\small{P_{cr}=\frac{\int F_\lambda\times S^p_\lambda\times\epsilon_f(1) d\lambda}{G}~,}

where:

  • εf(1) is the fraction of energy encircled within the peak pixel;
  • Fλ, and \small{S^p_\lambda} are as above;
  • the integral is over the bandpass.

If the flux from your source can be approximated by a flat continuum, then:

\small{C=\frac{F_\lambda \epsilon_f \int S^p_\lambda d\lambda}{G}~.}

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

\small{\int S^p_\lambda d\lambda=S^p_{peak} B_\lambda~,}

where:

  • \small{S^p_{peak}} is the peak sensitivity;
  • Bλ is the effective bandpass of the filter.

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

\small{C=\frac{F_\lambda \epsilon_f S^p_{peak} B_\lambda}{G} ~.}

In Chapter 14, we give the value of Bλ and \small{S^p_{peak}} for various filters.

Diffuse Source

For a diffuse source, the count rate, C, can be expressed as:

\small{C=\frac{\int I_\lambda \times S^d_\lambda \times N_{pix} d\lambda}{G}= \frac{\int I_\lambda \times S^p_\lambda \times m^2_s \times N_{pix} d\lambda}{G}~,}

where:

    • Iλ = the surface brightness of the astronomical source, in erg/s/cm2/Å/arcsec2;
    • Npix = the number of pixels integrated over;
    • the integral is over the bandpass.

For a diffuse source the peak counts/s/pix, Pcr, is given trivially by:

\small{P_{cr}=\frac{\int I_\lambda \times S^d_\lambda d\lambda}{G}~,}

where we have assumed the source to be uniformly bright.

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