figure table

# Warm Spitzer IRAC Photometry: dependencies on observing mode and exposure
time

Jessica E. Krick  IPAC, MC 330-6, Caltech, 1200E. California Blvd. Pasadena,
CA 91125  Patrick J. Lowrance  IPAC, MC 330-6, Caltech, 1200E. California
Blvd. Pasadena, CA 91125  Sean Carey  IPAC, MC 330-6, Caltech, 1200E.
California Blvd. Pasadena, CA 91125  Jason Surace  IPAC, MC 330-6, Caltech,
1200E. California Blvd. Pasadena, CA 91125  Carl J. Grillmair  IPAC, MC 330-6,
Caltech, 1200E. California Blvd. Pasadena, CA 91125  Seppo Laine  IPAC, MC
330-6, Caltech, 1200E. California Blvd. Pasadena, CA 91125  Schuyler D. Van
Dyk  IPAC, MC 330-6, Caltech, 1200E. California Blvd. Pasadena, CA 91125
James G. Ingalls  IPAC, MC 330-6, Caltech, 1200E. California Blvd. Pasadena,
CA 91125  Matthew L. N. Ashby  Center for Astrophysics $|$ Harvard &
Smithsonian, 60 Garden St. MS-66, Cambridge MA 02138  S.P. Willner  Center for
Astrophysics $|$ Harvard & Smithsonian, 60 Garden St. MS-66, Cambridge MA
02138

###### Abstract

We investigate differences in Spitzer/IRAC 3.6 and 4.5 $\mu$m photometry that
depend on observing strategy. Using archival calibration data we perform an
in-depth examination of the measured flux densities (“fluxes”) of ten
calibration stars, observed with all the possible observing strategies. We
then quantify differences in the measured fluxes as a function of 1) array
mode (full or subarray), 2) exposure time, and 3) dithering versus staring
observations. We find that the median fluxes measured for sources observed
using the full array are 1.6% and 1% lower than those observed with the
subarray at [3.6] and [4.5], respectively. Additionally, we found a dependence
on the exposure time such that for [3.6] observations the long frame times are
measured to be lower than the short frame times by a median value of 3.4% in
full array and 2.9% in subarray. For [4.5] observations the longer frame times
are 0.6% and 1.5% in full and subarray respectively. These very small
variations will likely only affect science users who require high-precision
photometry from multiple different observing modes. We find no statistically
significant difference for fluxes obtained with dithered and staring-modes.
When considering all stars in the sample, the fractional well depth of the
pixel is correlated with the different observed fluxes. We speculate the cause
to be a small non-linearity in the pixels at the lowest well depths where
deviations from linearity were previously assumed to be negligible.

###### keywords:

Infrared, Space Telescopes, Instrumentation, Detectors, Photometry

*Jessica Krick<EMAIL_ADDRESS>
##  1 Introduction

The Infrared Array Camera (IRAC) [1] was operational on the Spitzer Space
Telescope [2] from 2003 - 2020 with four broad mid-infrared bands with
response covering 3.15 - 9.25 $\mu$m (at 3.6, 4.5, 5.8, and 8.0 $\mu$m; also
denoted channels 1 – 4, respectively). These filter names are labels and not
the actual effective wavelengths (for more detailed information on filter
transmission see Ref. 3). After depletion of the cryogens, from mid 2009 until
January 2020, only [3.6] and [4.5] were available for observations.

In order to efficiently calibrate the telescope, a discrete set of observing
templates was enabled for IRAC observers. Available options to define the
templates included array mode, frame time, and dithering strategy. The array
mode could be specified either as full array (meaning, the entire IRAC
256$\times$256 array of 1.2 arcsecond pixels was read out after each exposure)
or subarray, in which case 64 consecutive 32$\times$32 pixel images were taken
at a higher readout rate without moving the telescope (0.01 s single readout
for subarray vs. 0.2 s for full array). A number of preset frame times were
available; those and their corresponding exposure times are listed in Table 1.
Frame time is a measure of time elapsed between successive array resets. The
exposure time (effective integration times) is the time elapsed between the
first pedestal and the first signal read, not including resets or multiple
reads. High dynamic-range exposures, in which paired short and long exposures
were acquired sequentially within a single full-array observing template, were
also available to IRAC observers, but did not add diversity to the range of
possible observations. Lastly, observers were able to choose from a discrete
set of dither patterns, make their own mapping strategy, or employ a staring-
mode. Dithering (small position changes to reduce noise) and mapping (position
changes of order the size of the array to cover an area greater than the size
of the array) were both possible in the full array mode. Dithering was
available in the subarray mode between 64-frame sets, but not within the sets.
Observations which do not move positions between frames are known as “staring-
mode.” Staring-mode was most often used for high-precision time-series
observations of brown dwarfs and exoplanets.

Table 1: Available IRAC Frame Times Frame Time | Exposure Time | Observing Mode
---|---|---
(seconds) | (seconds) |
0.02 | 0.01 | subarray
0.1 | 0.08 | subarray
0.4 | 0.36 | subarray
2 | 1.92 | subarray
0.4 | 0.2 | full array
2 | 1.2 | full array
6 | 4.4 | full array
12 | 10.4 | full array
30 | (23.6, 26.8) 111 Entries with two exposure times are for [3.6] and [4.5] respectively. | full array
100 | (93.6, 96.8) | full array

The rich Spitzer/IRAC archive contains many observations of the same target
taken with different array modes, frame times, or dithering strategies. This
is often the case for serendipitous observations but also for observations
designed intentionally in different modes. Possible science cases for this
include, but are not limited to, 1) initial dithered observations to find
proper motions of brown dwarfs followed by staring-mode data to refine the
characterization of their atmospheres, 2) dithered debris disk searches
looking for IR excess followed by a staring-mode study of variations in
specific debris disks, 3) archival dithered observations followed by targeted
subarray staring observations for confirmation, or 4) observations that had
either a too low signal-to- noise ratio (SNR) or were saturated at one
exposure time were followed up later in the mission using a different exposure
time. These science cases therefore require combining photometry from
different modes to arrive at scientific conclusions.

In this work we examine how IRAC photometry depends on observing strategy. We
emphasize that these are very small measured variations (a few percent at
most) and so will likely only affect science users who require high-precision
photometry from multiple different observing modes. A full set of calibration
observations to test for variations in measured fluxes among observing modes
were only taken during the warm IRAC mission. We consequently do not discuss
data taken during the cryogenic mission (which ended in 2009), or in the [5.8]
or [8.0] channels.

In Section 2 we describe the archival data used for this project. Section 2.2
covers our methods for reducing the data, carrying out photometry, and
applying photometric corrections. Section 3 discusses the different potential
correlations and photometry effects. We make concluding remarks in Section 4.

##  2 The Data

###  2.1 Observations

For this work we used the calibration observations taken for Spitzer Program
ID (PID) 1336 and PID 1367. Specifically, we observed a set of ten stars with
varying exposure times, in both subarray and in full array, and in staring and
dithering modes. Almost always, staring-mode observations were taken on the
same pixel (”the sweet spot”), as that pixel was the best characterized pixel
on the array. Table 2 lists the stars. We searched the archive for additional
data usable for this analysis but did not find anything suitable, combining
non-variable stars with observations in all available modes and having a
sufficient number of images to achieve statistical significance.

Figure 1 shows a visualization of this dataset. We plot the exposure time(s)
vs. aperture flux for just the [4.5] channel. Similar observations were made
for [3.6]. All stars were observed with multiple exposure times. Not all of
the ten stars could be observed at all exposure times due to SNR and
saturation concerns. This plot illustrates how the range of stellar
brightnesses and frame times in the sample filled the available phase space.

We considered how close to saturation(well depth) a star is as a means of
interpreting our results. Looking at our sample as a whole, the range of
possible well depths is not well sampled; having a median fractional well
depth of 0.04 and 0.02 at [3.6] and [4.5], where a fractional well depth of
1.0 indicates saturation. The right panel of Figure 1 shows well depths of the
sample. The median well depths of our sample correspond to a SNR of 39 and 24
at [3.6] and [4.5], respectively. This dataset was not designed with well
depth in mind, and was instead designed to find stars which would sample the
available exposure time parameter space.

Although available, we reject the 0.02 s photometry because of its large
scatter. We have also rejected any data where the well depth is greater than
the listed saturation limit in the IRAC Instrument
Handbook.222https://irsa.ipac.caltech.edu/data/SPITZER/docs/irac/iracinstrumenthandbook/
The saturation limit results in the rejection of around 250 photometry points
at [4.5]. All observations used in this work have SNRs of six or greater.

For a subset of three stars, we made a larger number of observations in all
possible modes with 2 s frame times in both channels. This allowed us to build
up statistically meaningful samples where we hold the frame time constant
while varying the other observing parameters. The three stars are
NPM1+66.0584, NPM1+66.0578, and KF03T2.

The 2 s frame time is the closest set of IRAC frame times available for
holding exposure time constant between the full and subarray. Unfortunately,
the 2 s subarray and 2 s full-array frame times do not actually correspond to
the same exposure times; 1.92 s and 1.2 s, respectively. Similarly, frame
times of 0.4 s in full and subarray also have different exposure times, so
that frame time was not a good alternative. This difference in the exposure
times for the same frame time comes from a different number of Fowler numbers
(N), wait ticks, and readout times for the different modes as listed in the
IRAC Instrument Handbook. Briefly, Fowler sampling is a way of observing by
taking N non-destructive reads at the beginning of the observation and another
N non-destructive reads at the end of the observation, the difference of which
is the flux measured per pixel.

All of the stars chosen for this work were vetted as potential calibration
stars for IRAC. Seven of the ten stars are published as primary or secondary
calibrators for IRAC known to not exhibit flux variability in the IRAC bands
[4]. The three stars not included in that reference are NPM1+74.0514,
NPM1+57.0835, and NPM1+66.0584. Specifically NPM1+66.0584 is one of the three
stars in the subset taken with more observations in 2 s frame times. We cannot
use this dataset to determine both if the stars are time variable, and if they
vary as a function of the other parameters studied herein. While vetted,
because these have not specifically been published as calibration stars, we
experimented with removing these three stars from our sample. All plots look
similar (albeit with larger scatter due to fewer data points), and conclusions
remain the same if we remove those stars from the sample. We therefore choose
to keep these in the sample for the remainder of this work.

Each of the [3.6] and [4.5] datasets include about 80,000 total individual
observations. Specifically, for each channel we have roughly 65,000 subarray
and 15,000 full array observations.

---
Figure 1:  Description of the dataset used in this work for the [4.5] channel, showing the aperture flux as a function of the exposure time on the left panel and aperture flux as a function of well depth on the right panel. The [3.6] channel has a similar observation set but for brevity is not shown. Points are color coded by star names, and symbol shapes denote full (filled circles) or subarray (crosses). Each data point represents one observation, and data are not binned, but there are overlapping data points.  Table 2: The Calibration Star Sample Name | R.A. | Dec. | Spec. | [3.6] | [4.5] | Alternate
---|---|---|---|---|---|---
| (h m s) | (deg’ ”) | Type | (mJy) | (mJy) | Name
NPM1+61.0569 | 17 23 25.9 | +61 12 40.7 | K0.5 III | 363 | 208 |
NPM1+74.0514 | 19 02 53.5 | +74 14 43.6 | K0.5 III | 323 | 194 | BD+74 804
NPM1+61.0582 | 17 36 55.6 | +61 40 58.1 | K1 III | 274 | 162 |
NPM1+66.0507 | 17 31 22.1 | +66 46 35.3 | K2 III | 250 | 145 |
NPM1+66.0578 | 19 25 32.2 | +66 47 38 | K1 III | 127 | 75 |
NPM1+57.0835 | 18 04 12.6 | +57 42 18.6 |  | 95 | 54 |
TYC4213-241-1 | 18 03 45.53 | +66 56 03.7 | K1 III | 74 | 42 | KF01T3
KF03T2 | 17 57 51.4 | +66 31 03.0 | K1.5 III | 70 | 40 |
NPM1+66.0584 | 19 36 07.39 | +66 21 54.2 | F0 | 56 | 35 | BD+66 1222
NPM1+60.0581 | 17 24 52.3 | +60 25 50.8 | A1 V | 37 | 23 | BD+60 1753

###  2.2 Photometry

We briefly describe our pipeline here, emphasizing where it is differs from
previous work. For an overview of the Spitzer/IRAC absolute photometric
calibration, see Refs. 4, 5. Specifics of the photometry pipeline used here
are included in Ref. 6. We used the basic calibrated data (BCD) FITS files
(suffix bcd.fits) from the Spitzer Heritage Archive (SHA). These BCD files
have had a dark correction, flat-field correction, linearity correction, and
conversion to flux units already applied.

One somewhat novel aspect of this work is that we apply a different dark
correction to staring-mode data than we do to dithered observations. Staring-
mode data are processed in the standard pipeline in the same manner as
dithered data, including using a dark image which was made by dithering. The
delay time between frames affects the bias level in the frame (including in
the dark); this is known as the “first-frame effect.” Therefore, dithered
observations will have different bias levels and patterns than staring-mode
observations since staring-mode observations have shorter delay times between
consecutive frames, although that pattern is constant if the delay time is
constant. This effect adds both noise and systematics to the photometry. For
this reason, dithered darks are inappropriate for staring-mode science frames
when precision photometry is required.

We used PID 1345 to make our own staring-mode dark suite for all subarray
frame times. We began by removing the dithered dark correction from the PID
1345 data. Because the dark correction is not the last correction made to the
BCD files, care was taken to first back out the other corrections, apply the
staring-mode dark, then re-apply the other pipeline corrections. We then
created a median image for each exposure time (0.02, 0.1, 0.4, 2.0 s) and used
this median frame as the staring-mode dark. Applying this new staring-mode
dark to the staring-mode data has a measurable impact on derived fluxes. We
recommend that anyone doing precision photometry with staring-mode data use a
staring-mode dark instead of the pipeline-provided, dithered dark. While the
IRAC pipeline will not include these starting-mode darks, code is available
for users to to change which darks are used in a BCD frame on the contributed-
code section of the Spitzer IRAC website.
333https://irsa.ipac.caltech.edu/data/SPITZER/docs/dataanalysistools/tools/contributed/irac/change_dark_calibrate/

To measure flux, we use our appropriately dark-corrected BCD exposures and
make the following corrections in order. We first convert images into units of
electrons to enable a statistical calculation of uncertainties. Second, we use
a center-of-light method to find stellar centroids.
444https://irsa.ipac.caltech.edu/data/SPITZER/docs/irac/calibrationfiles/pixelphase/box_centroider.pro
Third, we do aperture photometry with a three-pixel radius aperture and (3-7)
pixel background annulus. The small aperture size is chosen to reduce noise
and the number of cosmic rays in the aperture. Fourth, we make a correction
for pixel-phase using pixel_phase_correct_gauss.pro.
555https://irsa.ipac.caltech.edu/data/SPITZER/docs/irac/warmfeatures/pixel_phase_correct_gauss.pro
The pixel-phase correction accounts for gain changes as a function of the
position within a pixel, coupled with the undersampling of a point source by
IRAC. Fifth, we make a correction for array location. The array location-
dependent correction takes into account the variation in system response of
the instrument across the field of view, which is primarily due to the change
in the angle of incidence of light through the bandpass filter as a function
of position on the array. Lastly, we discard the first frame of every subarray
FITS file and of every full array AOR. These frames are affected by the first-
frame effect discussed above and are likely to have measured fluxes that
differ from those of subsequent images. We do not apply an aperture correction
since the same aperture is used for all photometry regardless of observing
mode.

To compare photometry for all stars on the same plots, we normalize the stars
to the same absolute level by dividing all photometry by the median stellar
fluxes. The distributions of fluxes per star are somewhat skewed, so a mean
does not capture the peak of the distribution. Having skewed distributions
causes the mean levels in the subsequent plots to differ from unity.

##  3 Results and Discussion

This section describes how the measured IRAC photometry differs as a function
of observing mode and exposure time. Sec. 3.3 examines the effects of
observing mode and exposure time together. Finally, we consider the impact of
staring and dithering on IRAC photometry.

We choose to present our results mainly with box plots to show the
distributions per star as a function of both array mode (full or subarray) and
exposure time for both channels, discussed individually below in Sections 3.1
and 3.2. A box plot shows the median of the distribution as the solid line in
the middle of the box. The box top and bottom indicate the top and bottom
quartiles and the caps at the end of the lines show the maximum and minimum
values in the distributions. These box plots include the entire dataset (all
stars, all exposure times, full and subarray, staring and dithering). On
average, each box contains a few thousand data points. Generally the boxes
with smaller quartile ranges and max/min values have fewer data points. The
plots are color-coded by star for each of the ten stars. Stars are listed in
order from the brightest (NPM1+61.0569) to the faintest (NPM1+60.0581). Color-
coding is consistent for all box plots and labeled in Figure 2.

###  3.1 Array Mode - Full vs. Subarray

Figures 2 shows the [3.6] and [4.5] photometry (top and bottom respectively)
of the stars in Table 2 as a function of array mode, revealing that the
measured fluxes appear on average to be lower for the full-array observations
than the subarray observations of the same stars. We attempt to confirm this
statistically by using an Anderson two-sample test, per star, to see if the
distributions for full array and subarray are drawn from the same population.
This statistical method considers the vertical distance between the two
cumulative distributions. For all ten stars we can say that the full array and
subarray data are not drawn from the same distribution at the 25% significance
level (maximum possible significance). Thus, the difference between the fluxes
measured in full array and subarray modes are statistically significant. Using
all ten stars, the median difference between the full and subarray flux
measurements is 1.6% at [3.6] and 1.0% at [4.5].

Figure 2:  Box plot of the normalized fluxes of all the stars divided into
full and subarray observations for [3.6] on the top and [4.5] on the bottom.
Stars are normalized by dividing by the median flux. The solid line in the
middle of the boxes shows a median value, while the top and bottom of the box
are quartiles, and the endcaps indicate the full range of the data. Stars are
color coded according to the legend and listed in order of brightness from
brightest to faintest. On average, the measured fluxes in full-array
observations are smaller than the measured fluxes in the subarray
observations.

###  3.2 Frame Time

Figure 3 shows the distributions of normalized calibrator star fluxes as a
function of Frame time. [4.5] includes more data on the six-second full-array
frame time, so the bottom plot includes that frame time whereas the [3.6] plot
does not. The median normalized flux of all the stars changes as a function of
frame time such that the measured fluxes are larger at lower frame times. This
is most strongly evident in [3.6]. Figure 4 shows a measure of the strength of
this effect as the median difference between the 2 s and 0.4 s frame time
fluxes for all the stars, per channel, per observing mode.

As in Section 3.1, we used the Anderson two-sample test to compare the full
array 0.4 s observations with the full array 2 s observations. For both
channels, we find the difference between the normalized flux distributions at
the two frame times to be statistically significant. The same is true for
fluxes from the subarray in 0.4 s and 2 s frame time observations.

We tested whether the trend seen in flux as a function of exposure time
depends on stellar brightness such that the brighter or the fainter stars
would be more or less likely to show this effect. We do this by calculating
the difference between the 2 s and the 0.4 s normalized fluxes. A difference
between the fluxes implies that different fluxes are measured using different
exposure times. Because we know there is an effect with array mode, we divide
the sample into full and subarray data. Finally, looking for a trend in the
stellar brightness, we divided the sample into bright and faint stars, and re-
calculated the median values. Figure 4 shows the results, where the circular
points are the median values of the difference in fluxes between the 2 s and
0.4 s frame-times for both the full array only data (blue) and the subarray
only data (red). The bright and faint subsets are shown with the appropriately
colored triangles. The differences between the bright and faint samples are
within one sigma of each other. Here sigma is calculated as the standard
deviation between stars, which means that each bin contains only a few stars
(and not thousands of data points). Our sample size is not large enough to
make conclusions based on star brightness.

Figure 3:  Box plot of the distributions of normalized fluxes of all the stars
as a function of exposure time at [3.6] on the top and [4.5] on the bottom.
Frame time labels also indicate full or subarray. Color coding is the same as
for Figure 2. On average, fluxes measured at longer frame times are lower than
those measured at shorter frame times.
---
Figure 4:  Exposure-time effect for both channels divided into bright and
faint stars to look for an effect with star brightness. [4.5] is shown in the
top half of the diagram, while [3.6] plotted in the lower half. Blue circles
show the median difference for all the stars observed in the full array
between the 2 s and 0.4 s frame time normalized fluxes, including one sigma
error bars. Blue right and left triangles show the median values for the
bright and faint populations of stars. The same quantities are plotted for
subarray data in red.

###  3.3 Disentangling Exposure Time and Array Mode Effects

We examine the possibility of disentangling the effects of exposure time and
array mode. Instead of dividing the dataset by star, here we consider all
stellar photometry as a single dataset, and divide the dataset into four
categories depending on the combination of array mode (full array vs.
subarray) and exposure time (short vs. long). Each distribution has between
3000 and 50000 stars (subarray exposure times have lots more frames than full
array). Exact exposure times cannot be compared in this analysis because they
differ significantly for full and subarray modes (see Table 1). For that
reason, the division between the short and long exposure times is set at 0.3
s, to construct statistically significant samples.

Figure 5:  Distribution of normalized star fluxes for [3.6] on the top and
[4.5] on the bottom. The legend lists the different observing modes (full
array vs subarray and long exposure times vs. short exposure times) as well as
the mean well depth per distribution. Solid dark lines show the kernel density
estimation (KDE) overplotted on the histograms. We see a difference in the
absolute photometry of stars taken in available observing modes and exposure
times shown here in the colored distributions.

Figure 5 reveals differences between the overall distributions of photometry
taken in full and subarray mode while accounting for exposure time, with
enough data points to accurately reflect the shapes of the distributions.
Longer exposure times are in red and light blue; shorter exposure times are in
orange and dark blue. Especially at [3.6], an effect is apparent with both
exposure time and observing mode. For the stars which have full-array
observations in both short and long exposure times, the median difference in
the flux is 3.4% at [3.6] (for six stars) and 0.6% at [4.5] (for ten stars).
For subarray we find the median difference in flux between short and long
exposure times to be 2.9% at [3.6] and 1.5% at [4.5].

###  3.4 Well Depth

The legend to Figure 5 lists median well depths for each of the distributions.
Well depth is one physical feature which correlates with the difference
between short and long and full and subarray exposures. We do expect that
longer exposure times on the same set of stars will have larger fractional
well depths, so it makes sense that the full long and sub long distributions
have the larger median well depths in both channels. Also, subarray mode has
the possibility of shorter frame times than full array, so we would expect
that subarray would have lower well depths than full array. Both channels show
this behavior.

Figure 6 shows the distributions of normalized fluxes for the entire sample of
stars in [3.6] and [4.5] divided into three well-depth bins. These bins do not
have the same number of photometry points in them as we have many more low
well-depth observations than intermediate well-depth observations. Because of
this change in the number of data points per bin, we do not quote values for
the peak of each bin, but rather address the trends revealed by the data in
hand [3.6] shows a clear trend such that observations at intermediate well
depths have lower normalized fluxes. [4.5] shows little, if any trend,
consistent with trends seen in the observing mode and exposure time plots.
While there are sources at higher well depth than 50% full well, their
distribution overlaps those of the 20-50% bin. We conclude that only
fractional well-depths less than a few tens of percent are affected by this
well-depth effect. Consideration of well-depth does not help to disentangle
the effects we have seen with observing mode and exposure time, but it is a
clue that potentially the source of some of what we are seeing is non-
linearity in the low well-depth regime.

Figure 6:  Distributions of fluxes in three fractional well depth bins for
[3.6] on the left and [4.5] on the right. Colors for the distributions get
darker as the fractional well depth increases.

###  3.5 Dithering vs. Staring-mode

We do not detect a difference in photometry between the dithered and staring-
mode observations. For three stars we have observations in all the modes for
the 2 s exposure times (sub staring, sub dithering, full staring, full
dithering). Recall that subarray observations taken with 2 s frame-time have
1.92 s exposure times and 2 s frame time full-array observations have 1.2 s
effective exposure time, so the 2 s exposure time subarray and full array
observations are not equivalent, but they are the closest to equivalent that
exist in the archive.

Figure 7 shows a Cleveland dot plot of the median and standard deviation of
the normalized fluxes for each mode, for each of the three stars in this
sample. The legend also lists the number of data points per mode. The full and
subarray measurements differ as before( see Section 3.1). The staring-mode
photometry is consistent with the dithered mode photometry within one sigma in
both channels.

---
Figure 7:  Cleveland dot plot comparing normalized fluxes from different
observing modes and dithering vs. staring for three stars listed on the
y-axis. [3.6] is on the left, [4.5] is on the right. The legend is for both
plots, and lists the number of data points for each mode.

###  3.6 Multiple Regression

Finding a relation among the observing parameters (array mode, exposure time,
staring/dithering) may help both to a) understand the effects found in this
paper and b) correct for them. We therefore use the statistical technique of
multiple regression to search for any relations. Specifically, we used
Multiple Linear Regression, i.e., multiple independent variables, to predict
the value of the dependent variable. To correct IRAC photometry for these
effects, we have tried ordinary least squares (OLS) as a multiple linear-
regression technique. We used array mode, exposure time, and stare/dither as
independent variables, and flux as the dependent variable. We used two
different modules in Python for this work, statsmodels[7] and sklearn[8]. We
are unable to find successful models. The $R^{2}$ goodness of fit is 0.037,
when ”good” models should have values close to 1.0. The failure of OLS in this
situation could imply that the relationship between the independent and
dependent variables is nonlinear.

##  4 Conclusions

We document exposure time, array mode (full vs. subarray), and fractional well
depth flux dependence in IRAC photometry. The full array has median fluxes
higher than the subarray by 1.6% at [3.6] and 1.0% at [4.5]. Dividing the
sample further, the long exposures have a lower median flux than shorter
exposures. The difference at full array is 3.4% at [3.6] and 0.6% at [4.5].
For subarray we find the median difference in flux is 2.9% at [3.6] and 1.5%
at [4.5]. These noted effects are only relevant for a small fraction of IRAC
high precision users with data in multiple modes who should include these
values in their uncertainty calculations. We posit two potential causes of the
noted low-level differences in photometry.

Overall, the normalized fluxes decrease as well depth increases up to a few
tens of percent full well. Well depth is correlated with the differences in
the flux distributions between full and subarray and long and short exposures.
While these correlations with array mode and exposure time are expected, it
potentially indicates the presence of non-linearities at the low well depths
sampled in this work.

Linearity corrections were made for warm IRAC to correct for a known effect
where an increase in incoming photons does not correspond to an increase in
counts (or data numbers DN). This occurs because filling the well decreases
the potential, which in turn results in a less responsive system. The
linearity correction for the IRAC InSb arrays inherently assumes no linearity
correction at low well depths. Therefore, time-intensive observations were not
made during the mission to include extremely low well-depth observations.
Instead, calibration observations focused on the moderate ($>$ 20%) to high
well depths where the linearity deviated most significantly from a straight
line correlation between photons and DN (see IRAC data handbook for a
description of the derivation of the warm linearity correction.
666https://irsa.ipac.caltech.edu/data/SPITZER/docs/irac/iracinstrumenthandbook/)
One explanation for why this low count linearity effect could exist at [3.6],
but not [4.5] is that the applied biases are different between [3.6] and [4.5]
implying that the electric potentials are different, which could explain the
stronger effect at [3.6] than [4.5].

Understanding the root cause of non-linearities is beyond the scope of this
paper. Many complicating issues are hiding under that designation including
the 3D structure of a pixel, how the linearity interacts with Fowler sampling
(specifically how to translate corrections derived at one set of Fowler
sampling parameters to those in another, which would get worse as the
integration times become comparable to the time spent reading the detector),
the speed at which the electric fields are changing in these very short
exposures compared to the timescales of the exposures or the readouts and
persistent image trap filling.

Linearity can in some detectors depend on the flux of the source in the sense
that it matters not only how many photons come in to the detector, but also
the rate at which they fill the well. We do not have enough observations to
know if the effect we are seeing is dependent on the brightness of the stars.

A second possible explanation for the difference between full and subarray
photometry is that the array is resetting faster between consecutive frames
for the subarray. A faster reset applies a stronger reverse bias on the array
more frequently, which could affect the distribution of photoelectron traps,
and therefore could have a low-level effect on photometry. We know that other
similar effects ( e.g., ”first-frame effect”) are more significant at [3.6]
than
[4.5].777https://irsa.ipac.caltech.edu/data/SPITZER/docs/files/spitzer/som12.2.pdf

No flux difference is apparent between the staring and dithering mode
observations after correcting the staring-mode data by a staring-mode dark.
Anyone doing absolute photometry with staring-mode data should be using a
staring-mode dark. The full and subarray staring-mode datapoints considered
here are all taken on the same array pixel (the sweet spot of the subarray),
which is not true for the dithered positions. Thus, residual pixel-phase
uncertainties cannot be the cause of the measured flux differences between the
full and subarray staring-modes, otherwise we would see this effect in this
work when comparing staring and dithering modes.

We have no evidence for low level persistent images being the cause of the
differences in photometry measured here. We know that the persistent images
are stronger at [3.6]. However, for [3.6] the same fraction of observations
were taken at the sweet-spot pixel in both full and subarray images. The sweet
spot is the pixel at the center of the array where most observations are
conducted in the subarray because it has been the best characterized for the
pixel phase effect. This means that the sweet spot pixel is more likely to
have frequent low-level persistent images than other pixels on the array. If a
larger fraction of observations in the subarray had been taken at the sweet
spot than the full array, we might have expected persistent images to be the
culprit. On the contrary, we see no evidence for this.

Our analysis implies that these differences in fluxes are systematic; they do
not average out with more observations. All full-array photometry will be
different than all subarray photometry, no matter how many observations are
taken in any given observing mode.

### Acknowledgments

We thank the anonymous referees for their time and care in providing very
useful comments on this manuscript. This work is based in part on observations
made with the Spitzer Space Telescope, which is operated by the Jet Propulsion
Laboratory, California Institute of Technology under a contract with NASA.
This research has made use of NASA’s Astrophysics Data System, the NASA/IPAC
Infrared Science Archive, which is operated by the Jet Propulsion Laboratory,
California Institute of Technology, under contract with the National
Aeronautics and Space Administration, and the SIMBAD database, operated at
CDS, Strasbourg, France. The acknowledgements were compiled using the
Astronomy Acknowledgement Generator.

## References

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  * [5] S. Carey, J. Ingalls, J. Hora, et al., “Absolute photometric calibration of IRAC: lessons learned using nine years of flight data,” in Space Telescopes and Instrumentation 2012: Optical, Infrared, and Millimeter Wave, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series 8442, 84421Z (2012).
  * [6] J. E. Krick, P. Lowrance, S. Carey, et al., “Spitzer IRAC Photometry of JWST Calibration Stars,” AJ 161, 177 (2021).
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###### List of Figures

  1. 1  Description of the dataset used in this work for the [4.5] channel, showing the aperture flux as a function of the exposure time on the left panel and aperture flux as a function of well depth on the right panel. The [3.6] channel has a similar observation set but for brevity is not shown. Points are color coded by star names, and symbol shapes denote full (filled circles) or subarray (crosses). Each data point represents one observation, and data are not binned, but there are overlapping data points.
  2. 2  Box plot of the normalized fluxes of all the stars divided into full and subarray observations for [3.6] on the top and [4.5] on the bottom. Stars are normalized by dividing by the median flux. The solid line in the middle of the boxes shows a median value, while the top and bottom of the box are quartiles, and the endcaps indicate the full range of the data. Stars are color coded according to the legend and listed in order of brightness from brightest to faintest. On average, the measured fluxes in full-array observations are smaller than the measured fluxes in the subarray observations.
  3. 3  Box plot of the distributions of normalized fluxes of all the stars as a function of exposure time at [3.6] on the top and [4.5] on the bottom. Frame time labels also indicate full or subarray. Color coding is the same as for Figure 2. On average, fluxes measured at longer frame times are lower than those measured at shorter frame times.
  4. 4  Exposure-time effect for both channels divided into bright and faint stars to look for an effect with star brightness. [4.5] is shown in the top half of the diagram, while [3.6] plotted in the lower half. Blue circles show the median difference for all the stars observed in the full array between the 2 s and 0.4 s frame time normalized fluxes, including one sigma error bars. Blue right and left triangles show the median values for the bright and faint populations of stars. The same quantities are plotted for subarray data in red.
  5. 5  Distribution of normalized star fluxes for [3.6] on the top and [4.5] on the bottom. The legend lists the different observing modes (full array vs subarray and long exposure times vs. short exposure times) as well as the mean well depth per distribution. Solid dark lines show the kernel density estimation (KDE) overplotted on the histograms. We see a difference in the absolute photometry of stars taken in available observing modes and exposure times shown here in the colored distributions.
  6. 6  Distributions of fluxes in three fractional well depth bins for [3.6] on the left and [4.5] on the right. Colors for the distributions get darker as the fractional well depth increases.
  7. 7  Cleveland dot plot comparing normalized fluxes from different observing modes and dithering vs. staring for three stars listed on the y-axis. [3.6] is on the left, [4.5] is on the right. The legend is for both plots, and lists the number of data points for each mode.

###### List of Tables

  1. 1 Available IRAC Frame Times
  2. 2 The Calibration Star Sample