The best empirical estimate of the system performance is the observation that
for imaging with 0.5'' pixels after 6 hr of on-source integration, objects
spread over 
with 
19.5 mag are detectable with a
signal-to-noise ratio of 
5, depending on how they are measured. This
is confirmed by similar observations with 4 min on-source integration times
reaching 17.0 mag with similar signal-to-noise ratio.
The following describes measurements of basic system parameters, and then calculations of the theoretical performance based on these parameters. These calculations can be used to estimate system performance in other configurations, but should be normalized by comparison with the above observed sensitivities.
System zero point offsets are based on the total ADUs in a sky-subtracted
stellar image after correction for airmass effects and are defined by the
equation 
. Typical values of the zero
point offsets for each filter (mainly with the 0.5'' pixel scale) are listed
in Table 5. These can be used to calculate the total
signal expected on an object of a given brightness or the signal/pixel on an
object of a given surface brightness.
Typical background brightnesses measured with CASPIR are listed in Table 6. The expected background photon fluxes can be calculated from the tabulated background brightnesses and the system zero point offsets.
Table 6: Background Brightnesses (mag/arcsec 
)
The noise in an image is a combination of photon shot noise from the sky and telescope, photon shot noise from the object, shot noise from the dark current, read noise, and other systematic noise sources that are difficult to quantify. For small signals, the noise per pixel can be estimated from the equation
where RN is the read noise in e 
, T is the integration time in
sec, 
is the background signal in e /sec, and 
is the
dark current in e /sec. The read noise for the double sample
readout methods (methods 2-4, and method 5 with FNDR=1) is 60
e . The dark current is typically < 30 e /s/pixel for most of
the array, but there are a significant number of detectors with dark
currents of > 50 e /s/pixel (see Fig. 3).
With these data, theoretical performance figures for CASPIR can be calculated from the measured background brightness and the camera throughput as quantified by the system zero point.
For example, for a Kn background brightness of 12.4 mag/arcsec
and a Kn zero point offset of 20.5 mag, the Kn background flux for
0.5'' pixels is
In a 5 sec integration, the background count is 2170 ADU/pixel,
or 19550 e /pixel (1 ADU = 9 e ). The shot noise of this
background signal is 
which dominates
the typical readout noise of 60 e . Consequently, Kn images with
0.5'' pixels and an integration time of 5 sec should be background
limited. The total noise per cycle is 
e /pixel or 152/9 = 16.9 ADU/pixel which is reduced to 4.9
ADU/pixel when 12 cycles are averaged. The measured value is 5
ADU/pixel. We assume here that sufficient sky frames are averaged so
that sky subtraction is essentially noiseless.
For a 5 
detection of an object spread over 
pixels, we require an average signal in each of these 
pixels of
five times the noise per pixel. The total object signal is then 
ADU. This can be converted to a Kn magnitude
after dividing by the integration time of 5 sec and using the Kn
zero point offset of 20.5 mag. In this way, we can estimate limiting
magnitudes at Kn for a range of seeing or object sizes. The results
of these calculations are shown in detail in Table 7.
Table 7: Performance Predictions at Kn (0.5''/pixel)
Predicted performance figures for 5 detections in 1 min of
on-source integration in different seeing conditions for various
filters are listed in Table 8 for the
0.5''/pixel scale and in Table 9 for the
0.25''/pixel scale. The 1 min integration time does not include
time for sky measurements and the dead time between frames of
20 sec. It is recommended to limit individual frames to 1 min
exposures, effectively making the elapsed time 80 sec, so that
an adequate number of sky frames can be obtained in the timescale of
15 min on which the sky level is observed to change
significantly. If off-source sky measurements are necessary, it is
recommended that equal time be spent on the object and sky positions.
Table 8: Predicted System Performance (0.5''/pixel)
Table 9: Predicted System Performance (0.25''/pixel)
Relative performance figures for each filter are of interest in deciding which
passband is most sensitive for a particular observation. These calculations
for the 0.5''/pixel scale in 1.5'' seeing and an integration time of 5 sec
with 12 cycles are listed in Table 10 for a 15.0 mag star with
zero color (S/N 
), a typical unreddened late-type star with K =
15.0 mag, J-K = 1.0, and H-K = 0.2 (S/N 
), and a typical AGN
power law spectrum with 
and K = 15.0 mag
(S/N 
).
Table 10: Relative Performances for Different Objects With K = 15 mag
Performance figures for the grisms can be estimated assuming a slit
transmission 
and grism transmissions,
, as shown in Tables 11 &
12. For example, consider an observation
using the HK grism and a 1'' (2 pixel) wide slit of an object spread
over 2'' ( 
pixels) along the slit and recorded in two
frames ( 
) each with an integration time T = 180 sec
and with the object placed at different positions along the slit in
both images. The spectral resolution at 2.2 
m is 2.2/2200 =
0.001 m/pixel, or a factor of 
lower
than for Kn. In the spectral direction, each pixel sees signal from
an area of sky 
= 0.5 arcsec ,
dispersed by a factor of 
more than for Kn, but reduced in
flux by . The background current per pixel (averaged
over features in the sky emission spectrum) is therefore
Here we have used the observed Kn background brightness, 
,
of 12.4 mag/arcsec , and zero point offset, Z.P., of 20.5 mag,
and conversion between electrons and ADU, Gain, of 9 e /ADU. The
noise per pixel is then
where we use a read noise of 60 e and an average long-integration
dark current of 10 e /s/pixel (see Fig.
3). We see from this that shot noise from the
dark current makes a dominant contribution to the total noise, and
that integration times of order 180 sec are required to minimise the
read noise contribution. Integration times significantly longer than
this are not recommended because sky intensity variations make
accurate sky subtraction increasingly difficult, and significant
numbers of hot pixels saturate as the integration time is increased
further. Normally, spectra are recorded as a nodded pair to allow sky
subtraction, and preferably as an ABBA sequence. In the case where a
single sky frame is subtracted from an object frame to perform the sky
subtraction, the total noise per pixel is increased by 
because the sky frame has the same noise per pixel as the object
frame. For our example, the final noise becomes 123 e /pixel.
We define a 5 detection per pixel by requiring a
signal-to-noise ratio of 5 per pixel after averaging 
pixels
along the slit and averaging the 
object frames. The
average object signal per pixel in the dispersion direction is then
The equivalent Kn imaging signal rate would be
which corresponds to a Kn brightness of
From this we predict that a signal-to-noise ratio of 50:1 can be
achieved with the HK grism on a Kn = 8.1 mag object in the continuum
in 6 min of on-source integration. Similar calculations for each of
the grisms are presented in Tables 11 &
12. Note that integrations with the K
grism are limited to 120 sec duration by the background flux at 2.4
m.
The 5 line detection sensitivity in the same time can be
estimated by assuming the line occupies two pixels in the dispersion
direction. The line flux is then 
W cm 
.
Table 11: Predicted Long-Slit Grism Performance
Table 12: Predicted Cross-Dispersed Grism Performance
The signal-to-noise ratio achived with the HK grism has been determined empirically to be given by
This reproduces the predicted HK grism sensitivity in Table 12.
