9.2 Determining Count Rates from Sensitivities

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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 (ef) in the peak pixel, the plate scale, and the dispersions of the grisms and prisms. This information is summarized in Table 9.1, Table 9.2, and Table 9.3 for Side 1. For updates please see the ACS Web page.

Table 9.1: Useful quantities for the ACS WFC.

Filter	Pivot l (Å)	ÚQlTl dl/l	AB mag zero point	ÚSl dl	encircled energy	Flux in central pixel	Background sky rate
F435W	4317.4	0.0744	25.67	3.18E18	0.85	0.22	0.0284
F475W	4744.3	0.1070	26.07	5.53E18	0.86	0.21	0.0609
F502N	5023.0	0.0032	22.27	1.88E17	0.86	0.21	0.0020
F550M	5581.2	0.0362	24.89	2.59E18	0.85	0.22	0.0280
F555W	5359.6	0.0775	25.72	5.11E18	0.86	0.21	0.0545
F606W	5917.7	0.1573	26.49	1.26E19	0.85	0.22	0.1358
F625W	6310.5	0.0915	25.90	8.37E18	0.85	0.22	0.0825
F658N	6583.9	0.0050	22.75	5.00E17	0.85	0.22	0.0045
F660N	6599.4	0.0019	21.67	1.87E17	0.85	0.22	0.0017
F775W	7693.0	0.0731	25.65	9.93E18	0.85	0.20	0.0780
F814W	8059.8	0.0948	25.94	1.41E19	0.83	0.19	0.1044
F850LP	9054.8	0.0352	24.86	6.63E18	0.73	0.14	0.0415
F892N	8914.9	0.0036	22.37	6.48E17	0.77	0.15	0.0039
G800L	7480.7	0.1591	26.50	2.04E19	0.86	0.17	0.1526
CLEAR	6269.8	0.3833	27.45	3.46E19	----	0.10	0.3580

Table 9.2: Useful quantities for the ACS HRC.

Filter	Pivot l (Å)	ÚQlTl dl/l	AB mag zero point	ÚSl dl	encircled energy	Flux in central pixel	Background sky rate
F220W	2255.3	0.0106	23.55	1.23E17	0.79	0.17	0.0001
F250W	2715.8	0.0123	23.72	2.09E17	0.82	0.16	0.0003
F330W	3362.7	0.0172	24.08	4.47E17	0.84	0.16	0.0006
F344N	3433.7	0.0017	21.59	4.67E16	0.84	0.16	0.0001
F435W	4311.0	0.0436	25.09	1.86E18	0.86	0.17	0.0043
F475W	4775.7	0.0650	25.53	3.40E18	0.85	0.17	0.0120
F502N	5022.9	0.0021	21.82	1.24E17	0.85	0.17	0.0003
F550M	5579.8	0.0234	24.42	1.67E18	0.85	0.15	0.0048
F555W	5356.0	0.0503	25.25	3.31E18	0.85	0.16	0.0099
F606W	5888.0	0.0983	25.98	7.82E18	0.85	0.14	0.0238
F625W	6295.5	0.0557	25.36	5.07E18	0.84	0.13	0.0148
F658N	6583.7	0.0030	22.18	2.98E17	0.83	0.13	0.0008
F660N	6599.1	0.0011	21.11	1.11E17	0.83	0.13	0.0003
F775W	7665.1	0.0380	24.94	5.13E18	0.78	0.11	0.0111
F814W	8115.3	0.0521	25.29	7.88E18	0.76	0.10	0.0161
F850LP	9145.2	0.0228	24.39	4.37E18	0.68	0.08	0.0075
F892N	8916.1	0.0022	21.87	4.09E17	0.73	0.09	0.0007
G800L	7506.4	0.0924	25.91	1.20E19	0.81	0.07	0.0250
PR200L	5650.2	0.2756	27.10	2.02E19	0.80	0.16	0.0624
CLEAR	5472.0	0.3205	27.26	2.20E19	----	0.12	0.0683

Table 9.3: Useful quantities for the ACS SBC.

Filter	Pivot l (Å)	ÚQlTl dl/l	AB mag zero point	ÚSl dl	encircled energy	Flux in central pixel	Background sky rate
F115LP	1406.1	0.0147	23.92	6.70E16	0.83	0.11	0.0238
F122M	1273.8	0.0010	20.95	3.56E15	0.82	0.09	0.0042
F125LP	1437.5	0.0122	23.71	5.80E16	0.83	0.12	0.0027
F140LP	1527.0	0.0069	23.09	3.68E16	0.84	0.13	0.0000
F150LP	1610.7	0.0038	22.45	2.27E16	0.84	0.14	0.0000
F165LP	1757.9	0.0010	21.02	7.29E15	0.85	0.16	0.0000
PR110L	1429.4	0.0120	23.69	5.62E16	0.86	0.26	0.0129
PR130L	1438.8	0.0120	23.68	5.66E16	0.87	0.28	0.0027

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 Fl or Fn:


The integral ÚQ_lT_l dl/l, used to determine the count rate when given the astronomical magnitude of the source.
The ABmag zero point, defined as the AB magnitude of a source with a constant F_n that gives 1 count/second with the specified configuration.
The sensitivity integral, defined as the count rate that would be observed from a constant F_l source with flux 1 erg/cm2/second/Å.
The encircled energy, defined as the fraction of PSF flux enclosed in the default photometry aperture 0.2 arcseconds for the WFC and HRC, and 0.5 arcseconds for the SBC. These correspond approximately to 5 ¥ 5, 9 ¥ 9, 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:

The counts/second (C) from your source over some selected area of N_pix pixels, where a signal of an electron on a CCD is equivalent to one count.
The peak counts/second/pixel (P_cr) 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)
F_l 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 l/hc converts ergs to photons.
Q_lT_l 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 Q_l and the filter transmission T_l.
e_f = the fraction of the point source energy encircled within N_pix
pixels.
S_l 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:

F_l, and S_l are as above.
e_f(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 (F_l = constant) and e_f is roughly constant over the bandpass, then:

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

Where:

S_peak is the peak sensitivity.
B_l is the effective bandpass of the filter.

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

In Tables 9.1 to 9.3, 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 9.3, and AB_n is the filter-dependent correction for the deviation of the source spectrum from a constant F_n 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:

Il = the surface brightness of the astronomical source, in
erg/second/cm^2/Å^/ arcseconds².
S_l as above.
m_x and m_y 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(l) is the emission line flux in units of erg/cm²/second, and l is the wavelength of the emission line in Angstroms. (QT)_l 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.