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 system throughput, QλTλ, and sensitivity, Sλ, for imaging and spectroscopic modes. Qλ is the instrument sensitivity and Tλ is the filter transmission.

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 central 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 observations after the temperature decrease in July 2006. For updates please see the ACS webpage.

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. Qλ is the instrument sensitivity and Tλ is the filter transmission.

(1) \lambda_p = \sqrt{\frac{\int Q_{\lambda} T_{\lambda} \lambda d\lambda}{\int Q_{\lambda} T_{\lambda} \lambda^{-1} d\lambda)}}

  • The bandpass "efficiency", the integral \int Q_{\lambda} T_{\lambda} \lambda^{-1} d\lambda, is used to determine the count rate when given the astronomical magnitude of the source.

  • The sensitivity integral, \int S_{\lambda} d\lambda, defined as the count rate that would be observed from a constant Fλ source with flux 1 erg/cm2/s/Å. The sensitivity integral is equivalent to PHOTFLAM-1, where PHOTFLAM is the header keyword defined as the inverse sensitivity, representing the flux density (erg/cm2/s/Å) of a source that produces a count rate of 1 count/s.

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

  • The encircled energy, defined as the fraction of PSF flux enclosed in the specified aperture, which is 0.2 arcsec radius for the WFC and 0.4 arcsec radius for the SBC.

  • 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.

Table 9.1: Useful quantities for the ACS WFC (at –81°C since July 2006) extrapolated to MJD 60401.

Filter

Pivot λ (Å)

Bandpass Efficiency ∫QλTλ λ-1 dλ

Sensitivity ∫Sλ dλ
(counts/s / erg/cm2/s/Å)

ABMAG Zero point

Encircled Energy (0.2 arcsec circular aperture)

Fraction of Flux in Central Pixel

Average Background Sky Rate (counts/s/pixel)

F435W

4329.9

0.0731

3.12E+18

25.65

0.84

0.22

0.0308

F475W

4747.0

0.1051

5.39E+18

26.04

0.84

0.21

0.0568

F502N

5023.0

0.0033

1.88E+17

22.27

0.84

0.21

0.0020

F550M

5581.5

0.0349

2.48E+18

24.84

0.84

0.22

0.0268

F555W

5361.0

0.0768

5.02E+18

25.70

0.84

0.21

0.0540

F606W

5921.9

0.1581

1.26E+19

26.48

0.84

0.22

0.1289

F625W

6311.8

0.0915

8.30E+18

25.89

0.84

0.22

0.0832

F658N

6584.0

0.0050

4.98E+17

22.74

0.84

0.22

0.0047

F660N

6599.4

0.0019

1.91E+17

21.69

0.84

0.22

0.0018

F775W

7693.5

0.0736

9.91E+18

25.65

0.84

0.21

0.0787

F814W

8045.5

0.0952

1.40E+19

25.93

0.83

0.19

0.1037

F850LP

9031.5

0.0350

6.50E+18

24.84

0.78

0.15

0.0400

F892N

8914.8

0.0036

6.59E+17

22.39

0.78

0.15

0.0042

G800L

7471.4

0.1586

2.02E+19

26.49

–

0.39

–

CLEAR

6278.5

0.3786

3.40E+19

27.43

0.84

0.22

0.2949


Table 9.2: Useful quantities for the ACS SBC.

Filter

Pivot λ(Å)

Bandpass Efficiency ∫QλTλ λ-1 dλ

Sensitivity ∫Sλ dλ
(counts/s / erg/cm2/s/Å)

ABMAG Zero point

Encircled
Energy (0.4 arcsec circular aperture)

Fraction of Flux in
Central
Pixel

Background sky rate (counts/s/pixel)

F115LP

1392.9

0.0202

8.95E+16

24.25

0.76

0.11

0.0707

F122M

1267.1

0.0012

4.44E+15

21.20

0.76

0.09

0.0109

F125LP

1426.5

0.0167

7.73E+16

24.04

0.76

0.11

0.0065

F140LP

1519.3

0.0088

4.60E+16

23.34

0.76

0.13

4.95E-05

F150LP

1605.7

0.0048

2.84E+16

22.70

0.77

0.14

3.58E-07

F165LP

1758.1

0.0012

8.30E+15

21.16

0.77

0.16

3.38E-07

PR110L

1416.8

0.0181

8.29E+16

24.13

–

0.31

–

PR130L

1427.6

0.0179

8.33E+16

24.12

–

0.31

–

Here, we describe how to determine two quantities:

  1. The counts/s, C, from your source over some selected area of Npix pixels, where a signal of one electron on a CCD is equivalent to one count.
  2. The peak counts/s/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:

(2) C = A \int{F_{\lambda} \frac{\lambda}{hc} Q_{\lambda} T_{\lambda} \varepsilon_{f} d\lambda} = \int{F_{\lambda} S_{\lambda} \varepsilon_{f} d\lambda}

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/s/cm2/Å.
  • h is Planck's constant.
  • c is the speed of light.
  • The factor λ/hc converts units of 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/s/Å per incident erg/s/cm2/Å.

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

(3) C_{\mathrm{peak}} = \int F_{\lambda} S_{\lambda} \varepsilon_f(1) d\lambda

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:

(4) C = F_{\lambda} \varepsilon_f \int{S_{\lambda} d\lambda}

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

(5) \int{S_{\lambda} d\lambda} = S_{\mathrm{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:

(6) C = F_{\lambda} \varepsilon_f S_{\mathrm{peak}} B_{\lambda}

In Tables 9.1 and 9.2, we give the value of:

(7) \int{S_{\lambda} d\lambda}

for each of the filters.

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

(8) C = 2.5 \times 10^{11} \varepsilon_f \left( \int{Q T d\lambda/\lambda} \right) \times 10^{-0.4(V + \Delta_{f-V})}

where V is the visual magnitude of the source in Vega magnitudes, the quantity under the integral sign is the mean sensitivity of the detector+filter combination, and is tabulated in Tables 9.1 and 9.2, and \Delta_{f-V} is the correction for the deviation of the source spectrum from a constant Fν spectrum for filter f. This latter quantity is tabulated for several different astronomical spectra in Table 10.1 to Table 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:

(9) C = \int I_{\lambda} S_{\lambda} m_x m_y d\lambda

where:

  • Iλ is the surface brightness of the astronomical source, in erg/s/cm2/Å/arcsec2.
  • 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:

(10) C = 2.23 \times 10^{12} Q_{\lambda}T_{\lambda} F_{\lambda} \lambda

where C is the observed count rate in counts/s, QλTλ is the system throughput at the wavelength of the emission line, Fλ is the emission line flux in units of erg/cm2/s, and λ is the wavelength of the emission line in Angstroms. QλTλ can be determined by inspection of the plots in Chapter 10.3. 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 Nspix in the spatial direction perpendicular to the dispersion:

(11) C = F_{\lambda} S^{\prime}_{\lambda} \varepsilon^{\prime}_{N_{\mathrm{spix}}} = F_{\lambda} A \frac{\lambda}{hc} T_{\lambda} \varepsilon^{\prime}_{N_{\mathrm{spix}}} d

where:

  • S^{\prime}_{\lambda} is the total point source sensitivity in units of counts/s per incident erg/s/cm2/Å; and S^{\prime}_{\lambda} = S_{\lambda} d.
  • d is the dispersion in Å/pixel.
  • \varepsilon^{\prime}_{N_{\mathrm{spix}}} 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 λ = L with a flux of FL in erg/s/cm2 the total counts recorded over the Nspix extraction height is:

(12) C = F_{\lambda} S^{\prime}_{\lambda}/d

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 Sλ, the spectroscopic point source sensitivity calibration (S^{\prime}_{\lambda} \times \varepsilon^{\prime}_{N_{\mathrm{spix}}}) 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 Tλ for the ACS prisms and grisms is NOT crucial to the final accuracy of their sensitivity calibrations.

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

(13) P_{\mathrm{cr}} = \varepsilon_F^{\prime}(1) F_{\lambda} S_{\lambda}^{\prime}

where:

  • \varepsilon^{\prime}_f(1) is the fraction of energy contained within the peak pixel.
  • the other quantities are as above.



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