To estimate the sensitivity of IRAC in flight, where possible we use the measured properties of Spitzer and IRAC from the IOC/SV and IWIC; otherwise we use the required performance based on the design specifications. The sensitivity (σ) to point sources (in flux density units) is based on the following formula:
(2.6)
where the scale factor is
,
(2.7)
the background current is
,
(2.8)
and the effective exposure time is
(2.9)
In these equations, the spectral resolving power is from Table 2.2 ; the detector quantum efficiency Q (electrons per photon) is 0.87, 0.86, 0.45, and 0.70 in channels 1 – 4, respectively; the instrumental throughput ηI is from Table 2.2; the telescope throughput ηT = [0.889, 0.902, 0.908, 0.914] for channels 1 to 4, respectively (with Be primary, Al-coated secondary, and 50 nm ice contamination); the telescope area (including obstruction) A = 4636 cm2; the equivalent number of noise pixels Npix is from Table 2.1 (and defined in Section 2.2.2); h is the Planck constant; Ibg is the background surface brightness in MJy/sr; fS= 1.2 is the stray light contribution to the background; the dark current D is < 0.1, 0.28, 1, and 3.8 e-/second for channels 1, 2, 3, and 4, respectively; the read noise R is from Table 2.3; Ωpix is the pixel solid angle (see Table 2.1); is the in-flight estimated throughput correction for point sources (Table 2.4); is the in-flight estimated throughput correction for the background (Table 2.4); TF is the frame time from Table 2.6 and NF is the Fowler number from Table 2.6. Table 2.4 lists some useful combinations of IRAC instrument parameters. The quantity νF is the flat-field pixel-to-pixel variance, which depends on the observing strategy. In what follows, we will set νF= 0, which would apply strictly in the case of stable detectors with perfect flat-field measurements, and should apply practically for highly-dithered observations.
The “throughput corrections” and were determined by comparing the observed to expected brightness of stars and zodiacal light. Stars were measured in a 10-pixel radius aperture, and the zodiacal light was measured in channels 3 and 4 for comparison to the COBE/DIRBE zodiacal light model. (This measurement was not possible in channels 1 and 2 because we could not use the shutter for absolute reference.)
Early in the mission, we found that the throughput in channels 3 and 4 was lower than expected, both for the extended emission and point sources (but more so for point sources). Measurements of diffuse Galactic emission confirmed the deficit, and measurements of the PRF using bright stars showed that a considerable amount of stellar flux was being spread all across the arrays. The deficit in throughput from the diffuse emission was due to the fact that the quantum efficiency (QE) of the arrays had been overestimated. No reliable measurements of the QE existed for channels 3 and 4, so the QE was based on a theoretical model that incorrectly assumed that all the flux was reflected at the front of the detector diode chip (the detector arrays are backlit). Some of the light that passed through the detector was scattered widely across the array. Measurements on a sister array confirmed this internal scattering, and showed that it was strongly wavelength dependent. There was considerable evidence that the “loss” of the QE and the scattered light were not due to contamination, or damaged optical coatings, etc.
Table 2.5 gives the background brightnesses, in useful units, for three nominal observing directions. The low-background model applies near the ecliptic pole; the high-background case is in the ecliptic plane; and the medium-background case is intermediate. The background model includes contributions from emission and scattering from the zodiacal dust and emission from the Galactic dust. The near-infrared cosmic infrared background radiation is not included because it was partially resolved by Spitzer.
Table 2.4: Useful quantities for IRAC sensitivity calculations. The second set of numbers in channels 1 and 2 are for the warm mission.
Wavelength
3.6 μm
4.5 μm
5.8 μm
8.0 μm
Conversion factor (electrons/seconds)/(MJy/sr)
25/29.5
29/25.3
14
29
S (electrons/seconds)/( μJy)
0.77/0.84
0.89/0.73
0.42
0.91
Gain (electrons/DN)
3.3/3.7
3.7/3.7
3.8
3.8
(throughput correction for point sources)
1.06/1
0.84/1
0.45
0.61
(throughput correction for background)
1
1
0.72
0.88
Table 2.5: Background brightness in IRAC channels. The second set of numbers in channels 1 and 2 are for the warm mission. Iν fS is the expected background brightness on the detectors, FνBG is the flux density on each pixel due to the background brightness, and B is the background brightness on the detectors in electrons per second units, using the FLUXCONV values given in Section 4.3 of this Handbook, and the Gain values in Table 2.4.
3.6 μm
4.5 μm
5.8 μm
8.0 μm
low background model
Iν fS (MJy/sr)
0.093
0.32
1.7
6.6
FνBG (μJy)
3.2
11
57
220
B (electrons/second)
2.5/2.7
9.9/8.9
18
184
medium background model
Iν fS (MJy/sr)
0.15
0.44
2.3
9.3
FνBG (μJy)
5.1
15
79
320
B (electrons/second)
4.1/4.4
14/11
25
260
high background model
Iν fS (MJy/sr)
0.52
1.0
5.6
22
FνBG (μJy)
18
35
190
750
B (electrons/second)
14/15
32/26
60
620
An observation with no dithering is limited by correlated noise. The accuracy of a flat-field derived from a single observing campaign was measured as 2.4%, 1.2%, 1.0%, and 0.3% in channels 1 – 4, respectively, by comparing flats in several campaigns. Using combined flats (“superskyflats”) from the entire cryogenic (warm) mission, the estimated νF was 0.14% (0.17%), 0.09% (0.09%), 0.07%, and 0.01% in channels 1 – 4, respectively. Using these values for νF in equation 2.6, single frames were dominated by background and read noise. When combining multiple frames to generate a mosaic, the background and read noises averaged down (as square root of the number of frames), while the flat-field noise only averaged down for dithered observations. For N undithered observations on the medium background, flat-field noise dominated when the total exposure time, , exceeded approximately 420 seconds (using individual campaign flats) or 2.5 hours (using the superskyflat). For dithered observations, the flat-field noise also averaged down, and was only important for the very deep observations of high background fields.
For the frame times used in IRAC operations in flight, Table 2.6 gives the readout mode and Fowler number. For full array readout mode, only the 2, 12, 30, and 100 second (and initially 200 second) frame times could be chosen in the IRAC AOT in the cryogenic mission. In the warm mission, 0.4 and 6 second frame times were added. The 0.6 and 1.2 second frame times came as part of the High Dynamic Range (HDR; see Section 3.1) sequences. The 0.4 second full frame time was only available for channels 1 and 2 in the Stellar Mode (see Section 3.1) in the cryogenic mission, but it was available in the regular full array mode in the warm mission. The 2 second subarray frame time was only available in the warm mission. The frame sets that were taken for each pointing in the HDR mode are shown in Table 2.7. Long frame times at 8.0 µm were background-limited. Therefore, there was a maximum frame time of 50 seconds at 8.0 µm, and the 100/200 second frames were automatically converted into two/four repeats of 50 second frames in channel 4. The last column, Th, gives the extra time spent taking the HDR frames (used in the observing time estimate equation below).
Table 2.6: Fowler numbers for IRAC frames. The numbers in parentheses are for channels 1 and 2 in the warm mission if different from the cryogenic values. Wait ticks are defined in Section 2.4.3.
Frame Time (seconds)
Exposure
Time (seconds)
Readout Mode
Fowler Number
Wait Ticks
200
193.6 (187.2/193.6)
Full (not 8.0 μm)
32
936
100
96.8 (93.6/96.8)
Full (not 8.0 μm)
16 (32/16)
468 (436/468)
50
46.8
Full (8.0 μm only)
16
218
30
26.8 (23.6/26.8)
Full
16 (32/16)
118 (86/118)
12
10.4
Full
8
44
(6)
(4.4/4.4)
Full
(8/8)
(14/14)
2
1.2
Full
4
2
2
(1.92/1.92)
Subarray
(8/8)
(184/184)
1.2
1.0 (0.8/0.8)
HDR
1 (2/2)
4 (2/2)
0.6
0.4
HDR
1
1
0.4a
0.2
Stellar/Full
1
0
0.4
0.32 (0.36/0.36)
Subarray
8 (4/4)
24 (32/32)
0.1
0.08
Subarray
2
6
0.02
0.01
Subarray
1
0
aavailable only in the stellar photometry mode in the cryogenic mission, full array in the warm mission.
Table 2.7: IRAC High Dynamic Range (HDR) framesets.
Long Frame Time
List of frames taken
Th (seconds)
200
0.6, 12, 200
15
100
0.6, 12, 100
15
30
1.2, 30
3
12
0.6, 12
2
6*
0.6, 6
2
*Warm mission only.
The sensitivities for the four IRAC channels (for each of the three background models) are shown in Table 2.8, Table 2.9, and Table 2.10 and Figure 2.8, Figure 2.9, and Figure 2.10. The sensitivities in the tables are for point sources extracted from single images (but perfectly flat-fielded). In the figures, the sensitivities are for point sources extracted from coadded images (perfectly registered). We do not include “confusion noise” (due to overlapping images of distant galaxies or other sources of background structure) in the sensitivity estimates. The detectors were assumed to perform according to the ground-based IRAC detector measurements of read noise, dark current, and quantum efficiency. The first eight rows in each table show the sensitivity for full array readouts, and the last four rows show the sensitivity for subarray readouts.
Table 2.8: IRAC point-source sensitivity, based on Equation (2.6), low background (1σ, μJy). The second set of numbers in parentheses for channels 1 and 2 are for the warm mission.
Frame Time (seconds)
3.6 μm
4.5 μm
5.8 μm
8.0 μm
200
0.40
0.84
5.5
6.9
100
0.60 (0.86)
1.2 (1.25)
8.0
9.8
30
1.4 (2.2)
2.4 (2.5)
16
18
12
3.3 (4.9)
4.8 (5.0)
27
29
6
(12.2)
(9.8)
-
-
2
32 (44)
38 (39)
150
92
0.6a
180 (185)
210 (222)
630
250
0.4b
360 (369)
430 (443)
1260
450
2c
(19)
(21)
-
-
0.4c
81 (102)
89 (102)
609
225
0.1c
485 (552)
550 (566)
2010
690
0.02c
7300 (8374)
8600 (8447)
25000
8100
aavailable only in the high dynamic range (HDR) mode.
bavailable only in the stellar photometry mode in the cryogenic mission, and full array in the warm mission.
csubarray mode (set of 64 32×32 images). Sensitivity is per frame, not the sensitivity of a 64-frame coadd.
Table 2.9: Same as Table 2.8 but for medium background (1σ, μJy).
Frame Time (seconds)
3.6 μm
4.5 μm
5.8 μm
8.0 μm
200
0.49
0.97
6.4
8.2
100
0.73 (1.0)
1.4 (1.44)
9.3
12
30
1.6 (2.5)
2.8 (2.9)
18
21
12
3.6 (5.2)
5.3 (5.5)
31
34
6
(12.4)
(10.4)
-
-
2
32
38
150
110
0.6a
180 (185)
210 (222)
640
260
0.4b
360 (370)
430 (443)
1260
460
2c
(19)
(21)
0.4c
82 (102)
89 (103)
610
250
0.1c
490 (553)
550 (567)
2020
720
0.02c
7300 (8374)
8600 (8447)
25000
8100
aavailable only in the high dynamic range (HDR) mode.
bavailable only in the stellar photometry mode in the cryogenic mission, and full array in the warm mission.
csubarray mode (set of 64 32×32 images). Sensitivity is per frame, not the sensitivity of a 64-frame coadd.
Table 2.10: Same as Table 2.8 but for high background (1σ, μJy).
Frame Time (seconds)
3.6 μm
4.5 μm
5.8 μm
8.0 μm
200
0.89
1.5
9.8
12
100
1.3 (1.48)
2.1 (2.16)
14
18
30
2.5 (3.3)
4.1 (4.2)
27
32
12
4.8 (6.1)
7.1 (7.4)
44
52
6
(13.4)
(12.8)
-
-
2
34 (45)
41 (42)
180
156
0.6a
180 (186)
220 (224)
660
330
0.4b
360 (370)
430 (445)
1280
540
2c
21
24
-
-
0.4c
84 (104)
93 (106)
650
340
0.1c
490 (555)
560 (569)
2100
860
0.02c
7300 (8375)
8600 (8848)
25000
8200
aavailable only in the high dynamic range mode.
bavailable only in the stellar photometry mode in the cryogenic mission, and full array in the warm mission.
csubarray mode (set of 64 32×32 images). Sensitivity is per frame, not the sensitivity of a 64-frame coadd.
Figure 2.8: IRAC point source sensitivity as a function of frame time, for low background. The cryogenic values are on the left, and the warm mission values are on the right. To convert to MJy/sr, see equation (2.13). Subarrays are plotted in blue.
Figure 2.8, Figure 2.9, and Figure 2.10 show the point source sensitivity as a function of the integration time for each background model. The bottom axes in the plots represent the frame time for the images, which does not include time for moving the telescope. The IRAC full array frame times were 0.4, 0.6, 2, 6, 12, 30, and 100 seconds (0.4 and 6 second frames were available only in the warm mission; 200 seconds was also available in the early mission). Other times plotted are assumed to use multiple exposures of those fixed times.
For bright sources, shot noise due to counting statistics in electrons from the source itself became the dominant source of noise. One could estimate the total noise by adding the shot noise in quadrature, so that
(2.10)
Figure 2.9: Similar to Figure 2.8, but for medium background.
where σ is the noise from Equation 2.6 and
.
(2.11)
In the bright source limit, F >> Fb, the signal-to-noise ratio became
.
(2.12)
If the exposure time is in seconds and the source flux is in μJy, then for IRAC channels 1, 2, 3, and 4, respectively, the SNR was 0.88 (0.86), 0.95 (0.85), 0.65, and 0.95 times . Warm mission values are in parentheses.
Figure 2.10: Similar to Figure 2.8, but for high background.
In the sensitivity figures, the dashed line at 0.6 μJy is the confusion limit predicted by Franceschini et al. (1991). This does not represent a hard sensitivity limit, but rather indicates where source confusion affected the reliability of source extractions for low background regions. Data from the IOC/SV showed noise decreasing as to 0.25 μJy (channels 1 and 2) or 0.6 µJy (channels 3 and 4). Moderately deep source counts indicated that a source density equivalent to 36 beams/source was reached at 20.5 mag, or 1.8 and 1.1 μJy at 3.6 and 4.5 μm, respectively (Fazio et al. 2004). The confusion estimates by Franceschini et al. and Fazio et al. are for low background, extragalactic observations only. For observations of higher background or more dense regions (such as the Galactic Plane), the confusion noise was much more significant.
For diffuse emission, the surface brightness sensitivity per pixel (in MJy/sr) was
The saturation limit for IRAC was calculated as follows. Using the same notation as earlier in this section,
,
(2.14)
where W is the well depth (discussed in Section 2.3.2), fw =0.9 is the fraction of the well depth to which we can linearize the intensities, and fcen is the fraction of the source flux falling onto the central pixel (Table 2.1). Table 2.11 shows the point source saturation limits of IRAC at each frame time. In an extremely bright area of the sky, such as an H II region, the saturation limit was lower. Note that the saturation value was conservatively computed from the worst case in which the PSF is directly centered on a pixel. To apply Table 2.11 for extended sources,
for compact (diameter < 30”) sources
(2.15)
for more extended sources,
(2.16)
where is the total surface brightness (in MJy/sr) at which a pixel saturates; fp, fcen, and fex are as defined above, and Fsat is the saturating point source flux density (in mJy) from Table 2.11, appropriate for the channel and integration time. For 8.0 μm observations at low ecliptic latitude, an estimate of the zodiacal light should be included in the surface brightness.
Table 2.11: Maximum unsaturated point source (in mJy), as a function of IRAC frame time. The second set of numbers in parentheses are for channels 1 and 2 are in the warm mission.
Frame Time (seconds)
3.6 μm
4.5 μm
5.8 μm
8.0 μm
200
1.9
1.9
14
28
100
3.8 (2.8)
3.9 (3.5)
27
28
30
13 (10)
13 (12)
92
48
12
32 (25)
33 (30)
230
120
6
(50)
(60)
2
190 (160)
200 (200)
1400
740
0.6
630 (≈ 600)
650 (≈ 750)
4600
2500
0.4*
950 (≈ 1050)
980 (≈ 1350)
6950
3700
2**
(150)
(180)
0.4**
1000 (700)
820 (900)
3100
2300
0.1**
4000 (3000)
3300 (3700)
13000
9000
0.02**
20000 (≈ 23000)
17000 (≈ 31500)
63000
45000
*stellar mode (cryogenic mission), full array (warm mission); **subarray mode
The zodiacal background only made a difference for long frames in channel 4 when observing near the ecliptic plane. If the bright extended source extended well beyond the 5.2 arcminutes × 5.2 arcminutes FOV, then the saturation brightness was lower by the factor fs( = 1.2; defined near the beginning of Section 2.5.1).
,
,
is from 
is the in-flight estimated throughput correction for point sources (
is the in-flight estimated throughput correction for the background (
, exceeded approximately 420 seconds (using individual campaign flats) or 2.5 hours (using the superskyflat). For dithered observations, the flat-field noise also averaged down, and was only important for the very deep observations of high background fields.
.
.
. Warm mission values are in parentheses.
to 0.25 μJy (channels 1 and 2) or 0.6 µJy (channels 3 and 4). Moderately deep source counts indicated that a source density equivalent to 36 beams/source was reached at 20.5 mag, or 1.8 and 1.1 μJy at 3.6 and 4.5 μm, respectively (Fazio et al. 2004). The confusion estimates by Franceschini et al. and Fazio et al. are for low background, extragalactic observations only. For observations of higher background or more dense regions (such as the Galactic Plane), the confusion noise was much more significant.
× the point source sensitivity [in μJy].
,
for compact (diameter < 30”) sources
for more extended sources,
is the total surface brightness (in MJy/sr) at which a pixel saturates; fp, fcen, and fex are as defined above, and Fsat is the saturating point source flux density (in mJy) from