9.2 Determining Count Rates from Sensitivities

9.2.1 Imaging

9.2.2 Spectroscopy

In this Chapter, specific formulae appropriate for imaging and spectroscopic modes are provided to calculate the expected count rates and the signal-to-noise ratio from the flux distribution of a source. The formulae are given in terms of sensitivities, but we also provide transformation equations between the throughput (QT) and sensitivity (S) for imaging and spectroscopic modes.

Throughputs are presented in graphical form as a function of wavelength for the prisms and for the imaging modes in Chapter 10. Given your source characteristics and the sensitivity of the ACS configuration, calculating the expected count rate over a given number of pixels is straightforward, since the ACS PSF is well characterized. The additional required information is the encircled energy fraction (εf) in the peak pixel, the plate scale, and the dispersions of the grisms and prisms. This information is summarized in Table 9.1 and Table 9.2 for Side 1. For updates please see the ACS Web page.

Table 9.1: Useful Quantities for the ACS WFC, at -81C

Filter	Pivot λ (Å)	∫QλTλ dλ/λ	∫Sλ dλ	ABMAG zero point	Encircled Energy	Flux in Central Pixels	Background sky rate (per pixel)
F435W	4318.9	0.0739	3.14E+18	25.66	0.85	0.22	0.0302
F475W	4746.9	0.1070	5.49E+18	26.06	0.86	0.21	0.0566
F502N	5023.0	0.0033	1.87E+17	22.27	0.86	0.21	0.0019
F550M	5581.4	0.0366	2.60E+18	24.90	0.85	0.22	0.0275
F555W	5361.0	0.0782	5.12E+18	25.72	0.86	0.21	0.0538
F606W	5921.1	0.1593	1.27E+19	26.49	0.85	0.22	0.1270
F625W	6311.4	0.0930	8.44E+18	25.91	0.85	0.22	0.0827
F658N	6584.0	0.0051	5.05E+17	22.76	0.85	0.22	0.0047
F660N	6599.4	0.0019	1.89E+17	21.68	0.85	0.22	0.0018
F775W	7692.4	0.0744	1.00E+19	25.67	0.85	0.21	0.0779
F814W	8057.0	0.0962	1.42E+19	25.94	0.84	0.19	0.1026
F850LP	9033.1	0.0349	6.48E+18	24.84	0.74	0.15	0.0390
F892N	8914.8	0.0036	6.50E+17	22.37	0.77	0.15	0.0040
G800L	7467.8	0.1611	2.05E+19	26.50	--	--	--
CLEAR	6272.4	0.3857	3.46E+19	27.45	0.85	0.22	0.2934

 

Table 9.2: Useful quantities for the ACS SBC.

Filter	Pivot λ (Å)	∫QλTλ dλ/λ	∫Sλ dλ	ABMAG zero point	Encircled Energy8	Flux in Central Pixel	Background sky rate (per pixel)
F115LP	1406.6	0.0149	6.70E+16	23.92	0.82	0.11	0.0479
F122M	1273.7	0.0010	3.56E+15	20.95	0.82	0.09	0.0085
F125LP	1438.2	0.0123	5.81E+16	23.71	0.83	0.11	0.0053
F140LP	1528.0	0.0069	3.69E+16	23.09	0.83	0.13	0.0001
F150LP	1612.2	0.0038	2.28E+16	22.45	0.84	0.14	0.0000
F165LP	1762.5	0.0010	7.35E+15	21.03	0.85	0.16	0.0000
PR110L	1430.1	0.0121	5.62E+16	23.69	--	--	--
PR130L	1439.4	0.0120	5.67E+16	23.69	--	--	--

In each Table, the following quantities are listed:

•	The pivot wavelength, a source-independent measure of the characteristic wavelength of the bandpass, defined such that it is the same if the input spectrum is in units of Fλ or Fν:

•	The integral ∫QλTλ dλ/λ, used to determine the count rate when given the astronomical magnitude of the source.

•	The sensitivity integral, defined as the count rate that would be observed from a constant Fλ source with flux 1 erg/cm2/second/Å.

•	The ABmag zero point, defined as the AB magnitude of a source with a constant Fν that gives 1 count/second with the specified configuration.

•	The encircled energy, defined as the fraction of PSF flux enclosed in the default photometry aperture, 0.2 arcseconds for the WFC and 0.5 arcseconds for the SBC. These correspond approximately to 5 × 5 and 15 × 15 box-sizes respectively.

•	The fraction of PSF flux in the central pixel, useful for determining the peak count rate to check for overflow or bright object protection possibilities.

•	The sky background count rate, which is the count rate that would be measured with average zodiacal background, and average earthshine. It does not include the contribution from the detectors, tabulated separately in Table 3.1

Here, we describe how to determine two quantities:

1.	The counts/second (C) from your source over some selected area of Npix ­pixels, where a signal of an electron on a CCD is equivalent to one count.

2.	The peak counts/second/pixel (Pcr) from your source, which is useful for avoiding saturated CCD exposures, and for assuring that SBC observations do not exceed the bright-object limits.

We consider the cases of point sources and diffuse sources separately in each of the imaging and spectroscopy sections following.

9.2.1

Imaging

Point Source

For a point source, the count rate, C, can be expressed as the integral over the bandpass of the filter:

Where:

•	A is the area of the unobstructed 2.4 meter telescope (i.e., 45,239 cm2)

•	Fλ is the flux from the astronomical source in erg/second/cm2/Å

•	h is Planck’s constant

•	c is the speed of light

•	The factor λ/hc converts ergs to photons.

•	QλTλ is the system fractional throughput, i.e., the probability of detecting a count per incident photon, including losses due to obstructions of the full 2.4 meter 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 with units of counts/second/Å per incident erg/second/cm2/Å.

The peak counts/second/pixel from the point source, is given by:

Where:

•	Fλ and Sλ are as above.

•	εf(1) is the fraction of energy encircled within the peak pixel.

Again, the integral is over the bandpass.

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

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

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:

In Tables 9.1 to 9.2, we give the value of for each of the filters.

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

where V is the visual magnitude of the source, the quantity under the integral sign is the mean sensitivity of the detector+filter combination, and is tabulated in Tables 9.1 to 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 several different astronomical spectra in Tables 10.1 to 10.3 in Chapter 10.

Diffuse Source

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

Where:

•	Iλ = the surface brightness of the astronomical source, in erg/second/cm2/Å/ arcseconds2.

•	Sλ as above.

•	mx and my are the plate scales along orthogonal axes.

Emission Line Source

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

where C is the observed count rate in counts/second, (QT) is the system throughput at the wavelength of the emission line, F(λ) is the emission line flux in units of erg/cm2/second, and λ is the wavelength of the emission line in Angstroms. (QT)λ can be determined by inspection of the plots in Chapter 10. See Section 9.6.4 for an example of emission-line imaging using ACS.

9.2.2

Spectroscopy

Point Source

For a point source spectrum with a continuum flux distribution, the count rate, C, is per pixel in the dispersion direction, and is integrated over a fixed extraction height in the spatial direction perpendicular to the dispersion:

Where:

•	is the total point source sensitivity in units of counts/second per incident erg/second/cm2/Å; and

•	d is the dispersion in Å/pixel.

•	is the fraction of the point source energy within Nspix in the spatial direction.

•	the other quantities are defined above.

For an unresolved emission line at with a flux of in
erg/second/cm2 the total counts recorded over the Nspix extraction height is:

These counts will be distributed over pixels in the wavelength direction according to the instrumental line spread function.

In contrast to the case of imaging sensitivity , the spectroscopic point source sensitivity calibration () for a default extraction height of Nspix is measured directly from observations of stellar flux standards after insertion of ACS into HST. Therefore, the accuracy in laboratory determinations of for the ACS prisms and grisms is NOT crucial to the final accuracy of their sensitivity calibrations.

The peak counts/second/pixel from the point source, is given by:

Where:

•	is the fraction of energy contained within the peak pixel.

•	the other quantities are as above.