Determining Count Rates from Sensitivities

In the simplest terms, the instrumental sensitivity (S) times the flux from your object of interest gives the counts sec^-1 (C) expected from your source, times the GAIN (i.e., it gives counts for the MAMA and electrons for the CCD):

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 (T_A), the encircled energy fraction (f) in the direction perpendicular to the dispersion, the number of pixels per spectral resolution element (or line-spread function FWHMs) and the plate scale, which are provided in Chapter 13. For imaging observations you need only the encircled energies and plate scales. The location of this information is summarized in Table 6.1 below.

Table 6.1: Location of Information Needed to Compute Expected Counts

Mode	Sensitivities	Slit Transmission	Line-Spread Function FWHM	Plate Scales	Encircled Energies
CCD first order	page 278	page 325	page 351	page 278	page 344
MAMA first order	page 298	page 325	page 352	page 298	page 344
MAMA echelle	page 310	page 325	page 353	page 310	page 344
PRISM	page 322	N/A	N/A	page 322	page 3441
CCD imaging	page 375	N/A	N/A	page 375	page 375
MAMA imaging	page 389	N/A	N/A	page 389	page 389

1 Numbers for G230M and G140M can be used for PRISM

Below, we describe how to determine two quantities:

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

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

Spectroscopy

Sensitivity Units and Conversions

The spectroscopic point-source sensitivity, , has the units

for CCD: electrons sec-1 pix-1 per incident erg cm-2 sec-1 Å-1

for MAMA: counts sec-1 pix-1 per incident erg cm-2 sec-1 Å-1

Where:

• pix = 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)

The spectroscopic diffuse source sensitivity, , has the units

for CCD: electrons sec-1 pix-1 pix_s-1 per incident erg sec-1 cm-2 Å-1 arcsec-2

for MAMA: counts sec-1 pix-1 pix_s-1 per incident erg sec-1 cm-2 Å-1 arcsec-2

Where:

• pix= a pixel in the dispersion direction
• pix_s= a pixel in the spatial direction

and are related through the relation:

Where:

• m_s is the plate scale in arcsec per pixel in the spatial direction (i.e. in the direction perpendicular to the dispersion)
• W is the slit width in arcseconds

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 N_pix = N_pix x N_spix pixels can be expressed as:

Where:

• G is the gain (always 1 for the MAMA, and 1 or 4 depending on the choice of CCDGAIN for the CCD)
• F = the continuum flux from the astronomical source, in erg sec^-1 cm^-2 Å^-1
• T_A = the aperture transmission (a fractional number less than 1)
• _f = the fraction of the point-source energy contained within N_spix pixels in the spatial direction
• N_pix = the number of wavelength pixels integrated over. For an unresolved emission line, N_pix is just the number of pixels per spectral resolution element and F is simply the total flux in the line in erg sec^-1 cm^-2 divided by the product of the dispersion in Å per pixel and N_pix (i.e., divided by the FWHM of a resolution element in Å)

The peak counts sec^-1 pixel^-1 from the point source is given by:

Where:

• _f(1) is the fraction of energy contained within the peak pixel.
• F, , and T_A are as above.

Diffuse Source

For a diffuse continuum source, the count rate C, for the astronomical source integrated over N_pix = N_pix x N_spix can be expressed as:

Where:

• I = the surface brightness of the astronomical source, in erg sec^-1 cm^-2 Å^-1 arcsec^-2.
• N_pix = the number of wavelength pixels integrated over in the dispersion direction. For an unresolved emission line, N_pix is just the number of pixels per spectral resolution element, and I is simply the total flux in the line in ergs sec^-1 cm^-2 arcsec^-2 divided by the product of the dispersion in Å per pixel and N_pix, (i.e., divided by the FWHM of the resolution element in Angstroms).
• N_spix = the number of pixels integrated over in the spatial direction.

For a diffuse continuum source the peak counts sec^-1 pixel^-1, P_cr, is given by:

For a diffuse, spectrally unresolved emission line source the peak counts sec^-1 pixel^-1, P_cr, is essentially independent of slit size and is given by:

Where:

• I_line is the intensity in erg sec^-1 cm^-2 arcsec^-2 in the line.
• FWHM is the full width half max of the instrumental profile in Angstroms, which for STIS is nearly always 2 x d, where d is the dispersion in Å per pixel.
• 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 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:

Where:

• d is the dispersion in Angstroms per pixel.
• m is the plate scale in the dispersion direction. All else is as above.

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

Imaging

Sensitivity Units and Conversions

The imaging point-source sensitivity, , has the units:

electrons sec-1 Å-1 per incident erg sec-1 cm-2 Å-1.

Where:

• electrons refer to the total number of electrons from the point source integrated over the PSF.

The imaging diffuse-source sensitivity, , has the units:

electrons sec-1 Å-1 pixel^-1 per incident erg sec-1 cm-2 Å-1 arcsec-2.

Thus and are related through the relation:

where m_s is the plate scale in arcsec per pixel.

Point Source

For a point source, the count rate, C, over an area of N_pix pixels due to the astronomical source can be expressed as:

Where:

• F = the flux from the astronomical source, in ergs sec^-1 cm^-2 Å^-1.
• _f = the fraction of the point source energy encircled within N_pix pixels.
• the integral is over the bandpass.

The peak counts sec^-1 pixel^-1 from the point source, is given by:

Where:

• _f(1) is the fraction of energy encircled within the peak pixel.
• F_, and are as above.
• the integral is over the bandpass.

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

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

Where:

• is the peak sensitivity.
• B is the effective bandpass of the filter.

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

In Chapter 14, we give the value of B and for various filters.

Diffuse Source

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

Where:

• I = the surface brightness of the astronomical source, in erg sec^-1 cm^-2 Å^-1 arcsec^-2.
• N_pix = the number of pixels integrated over.
• the integral is over the bandpass.

For a diffuse source the peak counts sec^-1 pixel^-1, P_cr, is given trivially by:

where we have assumed the source to be uniformly bright.