# The James Webb Space Telescope Absolute Flux Calibration. I. Program Design
and Calibrator Stars

Karl D. Gordon Space Telescope Science Institute, 3700 San Martin Drive,
Baltimore, MD, 21218, USA  Sterrenkundig Observatorium, Universiteit Gent,
Gent, Belgium  Ralph Bohlin Space Telescope Science Institute, 3700 San Martin
Drive, Baltimore, MD, 21218, USA  G. C. Sloan Space Telescope Science
Institute, 3700 San Martin Drive, Baltimore, MD, 21218, USA  Department of
Physics and Astronomy, University of North Carolina, Chapel Hill, NC
27599-3255, USA  George Rieke Steward Observatory, University of Arizona,
Tucson, AZ 85721, USA  Kevin Volk Space Telescope Science Institute, 3700 San
Martin Drive, Baltimore, MD, 21218, USA  Martha Boyer Space Telescope Science
Institute, 3700 San Martin Drive, Baltimore, MD, 21218, USA  James Muzerolle
Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD,
21218, USA  Everett Schlawin Steward Observatory, University of Arizona,
Tucson, AZ 85721, USA  Susana E. Deustua formerly at Space Telescope Science
Institute  Sensor Science Division, National Institute of Standards and
Technology, 100 Bureau Dr., Gaithersburg, MD 20899-8444, USA  Dean C. Hines
Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD,
21218, USA  Kathleen E. Kraemer Institute for Scientific Research, Boston
College, 140 Commonwealth Avenue, Chestnut Hill, MA 02467, USA  Susan E.
Mullally Space Telescope Science Institute, 3700 San Martin Drive, Baltimore,
MD, 21218, USA  Kate Y. L. Su Steward Observatory, University of Arizona,
Tucson, AZ 85721, USA

###### Abstract

It is critical for James Webb Space Telescope (JWST) science that instrumental
units are converted to physical units. We detail the design of the JWST
absolute flux calibration program that has the core goal of ensuring a robust
flux calibration internal to and between all the science instruments for both
point and extended source science. This program will observe a sample of
calibration stars that have been extensively vetted based mainly on Hubble
Space Telescope, Spitzer Space Telescope, and Transiting Exoplanet Survey
Satellite observations. The program uses multiple stars of three different,
well understood types (hot stars, A dwarfs, and solar analogs) to allow for
the statistical (within a type) and systematic (between types) uncertainties
to be quantified. The program explicitly includes observations to calibrate
every instrument mode, further vet the set of calibration stars, measure the
instrumental repeatability, measure the relative calibration between subarrays
and full frame, and check the relative calibration between faint and bright
stars. For photometry, we have set up our calibration to directly support both
the convention based on the band average flux density and the convention based
on the flux density at a fixed wavelength.

calibration

##  1 Introduction

The goal of absolute flux calibration is to convert astronomical measurements
in instrumental units to physical units. Answering most science questions
requires measurements in flux densities or surface brightnesses. While many
science programs require 5–10% flux calibration accuracy, higher accuracy
significantly enhances supernovae, dark energy, stellar populations, and
stellar structure investigations (Kent et al., 2009). A common method for such
calibration is to measure the flux density of a star in instrumental units and
compare it to its predicted flux density, which is often based on a model of a
dust extinguished stellar atmosphere, with their parameters determined by
fitting previous calibrated measurements (e.g., Cohen et al., 1999; Rieke et
al., 2008; Engelke et al., 2010; Bohlin et al., 2014). While stars are the
sources most often used for absolute flux calibration, other sources have been
used including laboratory calibrated blackbodies (Fixsen et al., 1994; Price
et al., 2004) and asteroids (Stansberry et al., 2007; Müller et al., 2014).

The ideal flux calibration source is one that has been directly calibrated
against laboratory standards at all wavelengths of interest. A few sources
have been calibrated this way, but only at specific wavelengths. For example,
the star Vega has been calibrated at the monochromatic wavelength of 5556 Å
through observations using ground-based telescopes and laboratory calibrated
sources (e.g., Megessier, 1995). In the mid-infrared (MIR), the space-based
Mid-Course Space Experiment (MSX) observed a number of bright stars and small
calibrated spheres (Price et al., 2004). In particular, Sirius was observed
many times by MSX, which provides direct laboratory calibrated measurements of
this star in four photometric bands from 8–22 µm. Thus a few bright stars have
measurements directly tied to laboratory standards, but these stars are too
bright for almost all of the James Webb Space Telescope (JWST, Gardner et al.,
2006) observing modes. Hence, these measurements need to be transferred to
fainter targets that are observable by JWST, and models need to be used to
provide predictions at all the JWST wavelengths.

The transfer of measurements of bright stars to fainter stars requires
instruments that can observe stars with a large range of flux densities with
high relative accuracy. Fortunately, this is possible both in the optical and
MIR. In the optical, the Space Telescope Imaging Spectrometer (STIS) on the
Hubble Space Telescope can obtain spectra of stars with a very large range of
flux densities; specifically, it can observe stars as bright as Sirius
($V=-1.46$ mag) to stars as faint as $V\sim 15$ mag (Bohlin, 2014; Bohlin et
al., 2014). In the NIR and mid-infrared (MIR), this is possible using ground-
based and Spitzer Space Telescope Infrared Array Camera (IRAC) observations.
For the IRAC observations, bright-star photometry can be derived from the
extended wings of a star’s point spread function (Su et al., 2021). This
allows the relative measurement of stars with NIR and MIR flux densities from
that of Sirius to stars as faint as $K_{s}\sim 12$ mag (Rieke et al., 2022, G.
Rieke et al., in preparation).

The design of the JWST absolute flux calibration program builds directly on
the absolute calibration programs of Hubble and Spitzer which both have direct
ties to the laboratory standard measurements and wavelength ranges that
overlap with JWST. The flux calibration of Hubble instruments is based on
three white dwarf stars whose flux densities over the UV, optical, and NIR are
set by non-LTE models fitting the Balmer lines (Bohlin et al., 2014). The
absolute level of the three white dwarf spectra is set by comparison with the
previously measured flux densities of Vega at 5556 Å and Sirius in the MIR
(Bohlin et al., 2020). For the Near Infrared Camera and Multi-Object
Spectrometer (NICMOS) instrument (Thompson, 1992), the solar analog GSPC
P330-E was also used (Bohlin et al., 2001; Dickinson et al., 2003).

The flux calibration of Spitzer (Werner et al., 2004) was independently
determined for each of the three instruments. For the Infrared Spectrograph
(IRS, Houck et al., 2004), the calibration was based on a sample of $\sim$20 A
dwarfs and $\sim$30 K giants, from which a subset of well-behaved stars were
chosen as primaries (Decin et al., 2004; Sloan et al., 2015). The IRS
observations confirmed difficulties first seen in spectra from the Infrared
Space Observatory with K giants used as primary standards in the MIR, where
molecular absorption from CO, SiO, and OH were not well predicted by models
(Heras et al., 2002; Price et al., 2004; Sloan et al., 2015). For IRAC (Fazio
et al., 2004), four A dwarfs and six K giants served as the primary standards.
Their flux densities were set by LTE models fit to ground-based photometry and
spectroscopy (Reach et al., 2005; Carey et al., 2012). For the Multiband
Imaging Photometer for Spitzer (MIPS, Rieke et al., 2004), the flux
calibration targets varied with wavelength. At 24 µm, the flux calibration was
based on 22 A dwarfs with predicted flux densities based on MSX measurements
of A stars, confirmed with solar analog stars, and then extrapolated to 24 µm
using a stellar atmospheric model of an A dwarf (Engelbracht et al., 2007;
Rieke et al., 2008). At 70 µm, 66 stars with spectral types from B to M were
used with predicted flux densities determined in the same way as at 24 µm
(Gordon et al., 2007). For the 53–99 µm Spectral Energy Distribution mode, the
calibration was based on 22 stars with K to M spectral classes from the 70 µm
imaging calibration sample (Lu et al., 2008). At 160 µm the calibration was
based on asteroids, with their flux density predictions based on the
contemporaneous measurements at 24 and 70 µm and a thermal emission model
(Stansberry et al., 2007).

The relative calibration between Hubble and Spitzer instruments and
observatories are in good agreement either by the design of their calibration
programs or through empirical measurements. The calibrations of Hubble and
IRAC were studied using white dwarfs, A dwarfs, and solar analogs and found to
be consistent within 2% (Bohlin et al., 2011). All the instruments on Hubble
have accurate relative calibration as they are all based on the three primary
white dwarfs. The relative calibration of MIPS and IRS on Spitzer are formally
tied together at 24 µm (Sloan et al., 2015). Rieke et al. (2008) found that
the calibration between MIPS and IRAC is consistent within 1.5% using a sample
of A dwarf and solar analog stars. A comparison of IRAC observations of the
sample of A dwarfs and K giants revealed that the IRAC and IRS calibrations
agree to within $\sim$1% (Kraemer et al. 2022, submitted).

The goal of the JWST absolute flux calibration program is to provide flux
calibration for all JWST observing modes that supports the broadest range of
science. This paper focuses on the overall program design and the calibration
stars that support this goal. The calibration stars are picked to be traceable
to laboratory standards via Vega and Sirius at select optical and infrared
wavelengths. In addition, these stars can be well modeled allowing the flux
densities at these select wavelengths to be transferred to all the wavelengths
observed by JWST. The details of and uncertainties in predicting the flux
densities of the JWST calibration stars are beyond the scope of this paper,
they can be found in (Bohlin et al., 2014, 2017, 2020; Rieke et al., 2022, ;
G. Rieke et al., in preparation).

JWST operates from the visible through the MIR (0.6–28.3 µm), with instruments
that have many capabilities similar to those on large ground-based telescopes.
This requires an approach to absolute calibration that unifies this full
spectral range, and doing so with improved accuracy compared to previous work.
This approach should have significant benefits for all of optical and infrared
astronomy, benefits that we will support through cross-calibrating with
databases including the CALSPEC
catalog111https://www.stsci.edu/hst/instrumentation/reference-data-for-
calibration-and-tools/astronomical-catalogs/calspec of stellar calibrators
(Bohlin et al., 2014) and the 2MASS infrared sky survey (Skrutskie et al.,
2006).

Gordon et al. (2009) and Gordon & Bohlin (2012) have described early work on
designing the JWST program. This work has been refined by the JWST Absolute
Flux Team, which includes experts in absolute flux calibration and the JWST
instruments. Sec. 2 details the program design, Sec. 3 discusses the sample of
calibration stars, and Sec. 4 describes the calculation of the calibration
factors, including a discussion of conventions for photometric systems.
Finally, Sec. 5 summarizes the plan. Future papers will give the detailed flux
density predictions for each star and the observations and calibration factors
for each instrument.

##  2 Program Design

The core design principle for this program is to provide a robust, cross-
instrument absolute flux calibration for all JWST observations where the
statistical and systematic uncertainties of the calibration can be empirically
quantified. As a result, this program will observe a network of calibration
stars with all JWST instruments, both in photometric and spectroscopic modes.
The four JWST instruments are the Near-InfraRed Camera (NIRCam, Rieke et al.,
2005), the Near-InfraRed Spectrometer (NIRSpec, Jakobsen et al., 2022), the
Fine Guidance Sensor/Near-Infrared Imager and Slitless Spectragraph
(FGS/NIRISS, Doyon et al., 2012), and the Mid-InfraRed Instrument (MIRI, Rieke
et al., 2015). The FGS/NIRISS instrument includes FGS that is used for guiding
and NIRISS that is focused on science observations. The calibration stars have
different spectral properties and a range of flux densities. Observing a
sample of calibration stars will allow for an empirical measurement of the
statistical accuracy. Observing stars with different spectral types will
enable testing for systematic biases in our understanding of stars as encoded
in the stellar atmosphere models used. Stars with a range of flux densities
provide empirical tests of the calibration at different flux density levels,
and a robust cross-calibration between instruments having different
sensitivity ranges.

Extended design principles for this program include directly connecting to the
calibration of Hubble and Spitzer. We have satisfied these principles by
including in our calibration sample (see Sec. 3) stars that were used in those
flux calibration programs (Reach et al., 2005; Engelbracht et al., 2007; Carey
et al., 2012; Bohlin et al., 2014; Sloan et al., 2015). In addition, over the
last two decades we have worked to obtain new Hubble and Spitzer observations
for the JWST calibration stars that had not already been observed (Bohlin &
Cohen, 2008; Bohlin, 2010; Bohlin et al., 2011; Bohlin & Deustua, 2019; Krick
et al., 2021).

The required accuracies for the flux calibration of JWST are 5% for photometry
and 10–15% for spectroscopy (Gordon et al., 2019). The budgets for these
requirements include multiple terms not directly related to predicting the
flux densities of calibration stars (e.g., stability of point spread functions
and flat field uncertainties). The photometry requirement is the most
stringent, and there the term is a 2.8% uncertainty in the average flux
density prediction for a sample of stars (Gordon et al., 2019). The stretch
goal for the absolute flux calibration is to have as accurate a calibration as
possible using a reasonable amount of JWST observing time and effort.
Achieving the required level of accuracy, much less higher accuracy, requires
(1) averaging the measurements of multiple calibration stars, (2) having
stellar atmospheric models of high quality, (3) accurately transferring
measurements tied to laboratory sources to fainter sources observable with
JWST, and (4) obtaining JWST observations where measurement noise is small.
Averaging multiple stars of each type is needed as stellar atmosphere models
are quite good at modeling a class of stars, but likely have a 2% or somewhat
worse accuracy modeling a specific star. This accuracy limit is based on the
results for modeling Hubble CALSPEC stars from the UV through the MIR (Bohlin
et al., 2011, 2014). In addition, models of specific types of stars are
subject to systematic uncertainties (Bohlin et al., 2014). We estimate that we
will need on the order of five calibration stars of each type to account for
issues with individual stars and to quantify the systematics by type. The
number five is based on the goal of 1% accuracy per type, allowing for one of
the five stars to be found unsuitable for calibration after analysis of the
JWST observations, based on experience with previous calibration projects. In
addition, we have been working with modelers to provide new stellar
atmospheric models that explicitly include predictions in the NIR and MIR
(Bohlin et al., 2017, 2020). The models will be constrained by archival and
dedicated Hubble and Spitzer observations of the JWST calibration star sample
(Engelbracht et al., 2007; Bohlin & Cohen, 2008; Bohlin et al., 2011, 2014;
Krick et al., 2021), providing both the transfer from laboratory measurements
and allowing robust cross calibration between all three observatories.
Finally, to ensure that measurement noise is not the limiting factor, the
signal-to-noise goal of the JWST observations has been set to 200 (i.e.,
0.5%). We expect that this program will produce the level of accuracy needed
to support the JWST requirements and has a strong chance of achieving the
stretch calibration goal of higher accuracy. This stretch goal would enhance
the science results for many programs including supernovae, dark energy,
stellar populations, and stellar structure investigations (Kent et al., 2009).

###  2.1 Detailed Plan Components

Table 1: JWST Observing Modes Summary Instrument | Mode | Details
---|---|---
NIRCam | Imaging | 29 filters, 0.6–5 µm
| Coronagraphy | 5 masks, 1.8–5 µm
| Slitless spectroscopy | 2.4–5 µm
NIRSpec | MOS spectroscopy | 9 dispersers, 0.6–5.3 µm
| IFU spectroscopy | 9 dispersers, 0.6–5.3 µm
| Fixed slit spectroscopy | 9 dispersers, 0.6–5.3 µm
NIRISS | Wide field spectroscopy | 2 gratings, 0.8–2.2 µm
| Single object spectroscopy | 1 grating, 0.6–2.8 µm
| Aperture masking interferometry | 4 filters, 2.8–4.8 µm
| Imaging | 12 filters, 0.8–5.0 µm
FGS | Imaging for Guiding | open, 0.5–5.5 µm
MIRI | Imaging | 9 filters, 5–28 µm
| Coronagraphy | 4 masks, 5–12, 23 µm
| Low resolution spectroscopy | 5–12 µm
| IFU spectroscopy | 12 gratings, 4.9–28.8 µm
Figure 1: Example spectra for each of the three types of stars included in our
sample are plotted in Rayleigh-Jeans units. The spectra are shown at spectral
resolving powers of 3,000 and 150 as these match the approximate maximum and
minimum spectral resolving powers of JWST spectroscopy. For reference, the
highest resolution model spectra available are shown.

JWST has four science instruments and the FGS to calibrate, and each
instrument has a number of observing modes (Table 1). In order to efficiently
calibrate all JWST observing modes and achieve the goals described above, we
will observe three types of stars and have separated the observing program
into five parts. The JWST calibration stars (Sec. 3) include hot, A dwarf, and
solar analog stars, and Fig. 1 shows example spectra of each type. The five
parts of the observing program are: (1) observe at least one star of each type
in all observing modes, (2) establish the average calibration and refine the
calibration star sample by observing at least five stars of each type in a
select set of modes, (3) measure the baseline instrumental uncertainty and
track secular trends by observing one specific star repeatedly throughout the
mission, (4) provide the transfer between subarrays and full-frame
observations to give the largest dynamic range possible, and (5) investigate
any dependence on brightness in the flux calibration by observing calibrator
stars spanning the sensitivity range of each instrument.

Part 1 focuses on the most basic goal of this program: calibration of all
observing modes. Explicitly, this means one calibration star has to be
observed with every filter, optical element, and detector. To ensure that the
calibration of any one mode is robust to issues discovered after the
observations, at least one star of each type is observed in each mode. Thus,
at least one of the observations should be acceptable, with the expectation
that all are likely to be acceptable given the vetting carried out for the
calibration star sample.

Part 2 establishes the average calibration by observing a larger sample of
stars with a subset of the observing modes. The larger sample of stars
contains at least five of each type and often a few more given that the
sensitivities of all the observing modes do not fully overlap. Five stars per
type was picked to allow for one of the five stars to be removed from the
average and still obtain a factor of two improvement in the accuracy. This
step also allows us to potentially refine the sample of stars usable for JWST
absolute calibration. Based on past experience (e.g., Gordon et al., 2007;
Bohlin et al., 2014), a small number of the stars in the initial calibrator
sample may exhibit undesirable properties (e.g., winds, weak disks, or star
spots) that will only be revealed with the JWST observations themselves as
these will be the most sensitive measurements of these stars ever taken at
JWST wavelengths. The subset of observing modes to be used include 10 NIRCam
filters, all MIRI filters, the NIRSpec fixed-slit prism (R $\sim$ 100 from
0.6–5.3 µm), the NIRISS SOSS grating (R $\sim$ 700 from 0.6–2.8 µm), and the
MIRI LRS prism (R $\sim$ 100 from 5–12 µm). This subset was chosen to provide
observations over the widest range of wavelengths possible, include both
photometry and spectroscopy, within a reasonable amount of observing time.

Part 3 measures the minimum uncertainty of observations using repeated
measurements of the same star taken over the JWST mission. This measurement is
made after correcting for all known instrumental effects. It is an empirical
repeatability measurement for a point source and is simply the scatter in the
measurements in instrumental units. This repeatability uncertainty is thought
to originate in the detectors, so that the observations only need to be taken
in one filter or grating per detector. For example, such measurements made for
the Spitzer IRAC 3.6 and 4.5 µm and MIPS 24 µm bands resulted in repeatability
scatter of 0.3 % to 0.6 % (Engelbracht et al., 2007; Bohlin et al., 2022).

Part 4 empirically measures any responsivity variations between full frame and
subarray exposures by measuring the same star with both types of exposures.
Given the large range in sensitivity between full frame and the smallest
possible subarrays, multiple stars may be used with observations that overlap
for certain subarrays. While such responsivity variations are expected to be
small or non-existent, it is critical to quantify them because the large range
of sensitivities across the JWST instruments requires the use of subarrays,
especially for Part 2 of the calibration program.

Part 5 checks that the absolute flux calibration applies to a wide range of
flux densities. Correction for known instrumental non-linearities as a
function of measured signal will be done as part of the standard data
reduction for all the instruments. The goal of this part is to confirm the
standard non-linearity correction and check for any possible remaining non-
linearities that depend on count rate. Observations of calibration stars with
a range of flux densities are used to measure and, if needed, correct for any
brightness dependence in the flux calibration. We will also check that the
flux densities we measure for the same stars agree between different
instruments with overlapping wavelengths, as disagreements can also indicate
residual non-linearities. Like Part 4, any flux density dependence to the
calibration is expected to depend on detector, and hence observations are only
needed for one filter or grating per detector. This work will use many of the
observations taken for Parts 1 and 2 with additional observations needed to
fill in the areas of flux density space not already covered.

Figure 2: The spectra of all the calibration stars compared to the
instrumental sensitivities. The spectra are plotted in Rayleigh-Jeans units.
The calibration min/max ranges for the NIRCam instrument modes are plotted as
vertical lines. The wavelengths of the sensitivities of some of the modes have
been shifted to minimize overlap.  Figure 3: The spectra of all the
calibration stars compared to the instrumental sensitivities. The spectra are
plotted in Rayleigh-Jeans units. The calibration min/max ranges for the
NIRISS/FGS, NIRSpec, and MIRI instrument modes are plotted as vertical lines.
The wavelengths of the sensitivities of some of the modes have been shifted to
minimize overlap.

Section 3 presents the sample of JWST calibration stars. This sample was
chosen in order to provide five stars of each type for each instrument mode.
For each instrument mode, the maximum flux density observable that does not
saturate with the minimum possible integration time with the smallest
supported subarray was tabulated. Similarly, the faintest flux density
observable in an exposure time of one hour with a signal-to-noise of 200 was
tabulated. Figs. 2 and 3 illustrate the minimum and maximum flux densities for
each mode along with the the predicted spectra of most of the calibration
stars. These figures provide a graphical summary of how the range in flux
densities of our calibration stars cover the minimum and maximum flux
densities targeted in this program. Fig. 2 shows that our calibration stars do
a good job covering the needed flux density range for NIRCam wide field
slitless spectroscopy and coronagraphy, but do not cover the fainter range for
many of the NIRCam imaging filters. Fig. 3 shows good coverage by all NIRSpec
modes, all NIRISS modes except the faint range for imaging, the one FGS mode,
and all MIRI modes except for the bright range at wavelengths longer than 15
µm. We reach the goal of five stars of each type for most instrument modes.
However, there were some modes where this was not possible with the current
sample of calibration stars. In addition, finding both fainter and brighter
calibration stars is a long-term and challenging goal. Thus, the number of
calibration stars will likely increase as more suitable stars are identified.

###  2.2 Cycle 1 Plan

The comprehensive plan detailed above will take multiple years to fully
execute, both due to the need to monitor the flux calibration over the full
JWST mission lifetime, and because the total amount of observation time needed
is beyond what is reasonable for absolute calibration in any one year (=
cycle). One of the main driving factors in the time needed to execute this
program is directly related to the extensive capabilities of the JWST
instruments, as this results in many filters, gratings, etc. that require
calibration. Here we give the portions of the full plan that are planned for
execution in Cycle 1, with the focus on enabling the broadest range of
science.

Part 1: For the NIR instruments, one star of each of the three calibration
types will be observed in every filter, grating setting, detector, etc. For
the MIR, observations of one star each from the samples of A dwarfs and solar
analogs will be obtained. Observations of the hot stars in the MIR will be
obtained in future cycles once the use of such stars for MIR flux calibration
is validated using observations from Part 2.

Part 2: Observations of at least three stars of each type are planned for
Cycle 1. Three stars should be enough to identify if one of the three shows
non-ideal behavior. Observations of the three types of stars will provide
indications if one of the types has non-ideal behavior. These observations
will be obtained in the NIR and MIR for all three types. In addition to the
overall goals for Part 2, this will allow us to quantitatively investigate the
suitability of hot stars for MIR flux calibration.

Part 3: For the majority of the observations, BD+60 1753 will be observed 10
times over the year. For MIRI MRS observations, HD 163466 will be observed as
it is significantly brighter, and will also be repeated 10 times over the
year.

Part 4: One star or appropriate set of stars of different flux density levels
will be observed with each detector where subarrays are supported.

Part 5: The stars picked for Parts 1 and 2 will be used to perform a
preliminary analysis of the flux density linearity of the calibration. In
future cycles, stars with a wider range of flux densities will be observed to
cover the largest range possible.

The calibration stars used for Parts 1 and 2 were chosen to minimize the total
amount of time required in Cycle 1. Using the same star for observations for
multiple instruments reduces the overheads, especially those due to slewing to
the star. A simple algorithm was used to pick stars, with the first star being
the one that was observable by most modes, then the second the next most, with
this repeated until all the modes were covered. This resulted in 3 hot stars,
8 A dwarfs, and 7 solar analogs with Table 3 indicating the specific stars.
The program IDs for the JWST programs that contain the full details of the
planned observations are 1536, 1537, 1538 (Parts 1, 2 & 5), and 1539 (Part 3).
For Part 4, the observations are spread over multiple programs including, but
not limited to, 1094, 1096, and 1550. As with all JWST data, all the
observations taken for this work will be processed with the JWST pipeline and
publicly available through the Mikulski Archive for Space Telescopes (MAST)
archive.

##  3 Calibration Star Sample

Our sample of calibration stars includes hot stars, A dwarfs, and solar
analogs. The use of hot stars is motivated by the fact that these stars have
simple atmospheres dominated by radiative transport. Three of the white dwarfs
that are the basis of the Hubble flux calibration (Bohlin et al., 2014) form
the core of this sample. No similar, simple white dwarfs that are bright
enough for some of JWST observation modes are available (e.g., MIRI
spectroscopy), so we added bright OB dwarf stars that were used in the
calibration program of the International Ultraviolet Explorer (IUE). The A
dwarfs are included because they also have simple atmospheres dominated by
radiative transport, and they provided the basis of the Spitzer calibration
(Reach et al., 2005; Engelbracht et al., 2007; Sloan et al., 2015). The final
type included in the sample is solar analogs, because they are similar to our
Sun, which is the star that we have the most extensive observations and
knowledge of, and they are primary calibrators for ground-based NIR
observations (Johnson, 1966) and the Hubble NICMOS instrument (Dickinson et
al., 2003).

The sample includes multiple stars of each type. Using an average of multiple
stars of the same type has a long history in absolute flux calibration, going
back to at least the work of Johnson who used a sample of A dwarf stars
(Johnson & Morgan, 1953; Johnson et al., 1966). The star Vega has been used as
the single calibrator for observations based on a stellar atmospheric model of
an A dwarf. Vega provides a cautionary tale, however, as it has been found to
be a pole-on rapid rotator with a significant surface temperature gradient
(Aufdenberg et al., 2006), and it has a circumstellar debris disk with
significant MIR emission above the stellar photosphere (Aumann et al., 1984;
Su et al., 2005; Bohlin, 2014).

The criteria that make a star a good absolute flux calibrator are a mixture of
properties of the star itself, the line-of-sight to the star, and the use of
the star in previous absolute calibration efforts. The stellar properties
start with the star being a single star or in wide binary to avoid biases in
measuring the star’s flux density; close binaries require more complex models
to account for complex mixed spectra and possible heating by the companion. In
addition, the star should have a spectral type that is straightforward to
model. Hot stars and A dwarfs are relatively straightforward to model as their
atmospheric structures are driven by radiative transport as opposed to those
dominated by convective transport. Solar analogs can be modeled accurately as
they are similar to our Sun for which we have the most detailed observations
and understanding. The star should not vary photometrically at or below the
0.25% level ($1\sigma$), not have a circumstellar disk of gas or dust, rapid
stellar rotation, or a significant wind. Finally, the line-of-sight dust
extinction should be low and well modeled. Stars used in previous calibration
efforts are always a good place to start given they have already undergone
significant vetting. A star does not have to pass every criterion, but as many
as possible should be satisfied.

Table 3 gives the calibration stars in our sample. This list may be updated as
additional data are obtained including from JWST itself, but we fully expect
that the majority of these sources will be suitable for JWST absolute flux
calibration given the extensive vetting they have undergone. The coordinates
are from Gaia DR2 (Gaia Collaboration, 2018). The table gives references for
the spectral types, which were determined mostly using standard spectral
typing techniques where optical spectra are visually compared with spectra of
spectral standards. Those stars with the ”A” reference were determined from
the spectral fitting of the Hubble/STIS spectroscopy to stellar atmosphere
models (Bohlin et al., 2017). The $K_{s}$ band magnitudes are from 2MASS
(Skrutskie et al., 2006) unless noted in the table. The $E(B-V)$ values are
from spectral fitting of the Hubble/STIS spectroscopy (Bohlin et al., 2017)
except for NGC2506-G31 that is from Knudstrup et al. (2020) and HR 7018 that
is estimated from the spectral type and measured $(B-V)$ color (Oja, 1991).
Stars that have observations planned for the first year (Cycle 1) of JWST
operations are designated. The variability is given as $1\sigma$ values for
both the variability detections and upper limits. The $1\sigma$ values were
determined from the Transiting Exoplanet Survey Satellite (TESS) variability
measurements by Mullally et al. (2022) or from more recent equivalent TESS
measurements made specifically for this paper. The $1\sigma$ values are
computed as $V_{95}/4$ or $V_{99.7}/6$, where $V_{95}$ measures the envelope
of all data within $\pm 2\sigma$ of the median and $V_{99.7}$ is a similar
measurement within approximately $\pm 3\sigma$. The TESS measurements are
sensitive to variations with timescales from 2 weeks to 4 min except for a few
stars where the shortest timescale is 20 or 60 min (Mullally et al., 2022).
For three stars (HD 2811, 18 Sco, and HD 142331) we give Hipparcos HPSCAT
variability measurements for reference until new TESS variability measurements
are obtained. For 18 Sco, , the variability is from Hipparcos (Adelman, 2001).
For the stars without existing TESS variability measurements, three have lower
time resolution measurements that we will analyze in future work (HD 2811,
SNAP-2, and C26202). For all the stars, we have submitted a TESS proposal to
observe them at the highest available cadence (2 min) to provide continued
monitoring or their first such measurements. For the stars not detected as
variable, all had even lower upper limits that given in Table 3 for periodic
variability with periods less than one day. The notes give additional
information on a star including alternative names. All of the stars have
Hubble UV/optical STIS spectroscopic observations, except for HR 7018 and
NGC2506-G31. For these two stars, a Hubble calibration proposal to obtain the
STIS observations has been submitted and accepted. Most of the sources have
Spitzer IRAC and/or MIPS 24 µm photometry (Engelbracht et al., 2007; Bohlin et
al., 2011; Krick et al., 2021). The NGC2506 cluster has 19 solar analogs that
are candidates for absolute flux calibration, and we have picked G31 as the
best candidate as it is the most isolated (Schlawin et al., in prep.).

Table 2: Calibration Program Stars

Name | RA | DEC | SpType | Ref | $K_{s}$ | $E(B-V)$ | Var | Cycle 1 | Notes
---|---|---|---|---|---|---|---|---|---
Hot Stars |
$\lambda$ Lep | 05 19 34.5240 |  $-$13 10 36.441 | B0.5IV | 15 | 5.09 | 0.01 | 0.09% |  | HD 34816
10 Lac | 22 39 15.6786 | +39 03 00.971 | O9V | 15 | 5.50 | 0.07 |  $<$0.13% |  | HD 214680
$\mu$ Col | 05 45 59.895 |  $-$32 18 23.165 | O9.5V | 15 | 5.99 | 0.01 | 0.03% |  | HD 38666
G 191-B2BaaHubble standard (Dickinson et al., 2003; Bohlin et al., 2014) | 05 05 30.6183 | +52 49 51.921 | DA0.8 | 1 | 12.76 | 0.0 |  $<$0.13% | ✓ |
GD 71aaHubble standard (Dickinson et al., 2003; Bohlin et al., 2014) | 05 52 27.6197 | +15 53 13.229 | DA1.5 | 1 | 14.12 | 0.0 |  $<$0.32% | ✓ |
GD 153aaHubble standard (Dickinson et al., 2003; Bohlin et al., 2014) | 12 57 02.3325 | +22 01 52.634 | DA1.2 | 1 | 14.31 | 0.0 |  $<$0.41% | ✓ |
LDS 749B | 21 32 16.2328 | +00 15 14.398 | DB4 | 2 | 15.22 | 0.01 |  |  | TESS planned Aug 2022
WD 1057+719 | 11 00 34.2433 | +71 38 02.920 | DA1.2 | 1 | 15.47 | 0.0 |  $<$1.0% |  |
WD 1657+343 | 16 58 51.1126 | +34 18 53.321 | DA0.9 | 1 | 17.4ccCalculated from a model fit to UV/optical STIS spectroscopy | 0.0 |  $<$6.3% |  |
A Dwarf Stars |
$\delta$ UMibbSpitzer standard (Reach et al., 2005; Engelbracht et al., 2007; Sloan et al., 2015) | 17 32 12.9967 | +86 35 11.258 | A1Van | 3 | 4.26 | 0.01 | 0.01% | ✓ | HD 166205
HR 701bbSpitzer standard (Reach et al., 2005; Engelbracht et al., 2007; Sloan et al., 2015) | 02 22 54.6753 |  $-$51 05 31.659 | A5V | 4 | 5.44 | 0.01 | 0.01% |  | HD 14943
$\eta^{1}$ DorbbSpitzer standard (Reach et al., 2005; Engelbracht et al., 2007; Sloan et al., 2015) | 06 06 09.3815 |  $-$66 02 22.630 | A0V | 5 | 5.75 | 0.0 | 0.05% |  | HD 42525
HR 7018bbSpitzer standard (Reach et al., 2005; Engelbracht et al., 2007; Sloan et al., 2015) | 18 37 33.5178 | +62 31 35.660 | A0V | 6 | 5.75 | 0.05 | 0.03% |  | HD 172728
HR 5467bbSpitzer standard (Reach et al., 2005; Engelbracht et al., 2007; Sloan et al., 2015) | 14 38 15.2219 | +54 01 24.025 | A1V | 6 | 5.76 | 0.0 | 0.01% |  | HD 128998
HR 6514 | 17 26 04.8370 | +58 39 06.831 | A4V | 6 | 6.15 | 0.05 | 0.10% |  | HD 158485
HD 163466bbSpitzer standard (Reach et al., 2005; Engelbracht et al., 2007; Sloan et al., 2015) | 17 52 25.3757 | +60 23 46.940 | A7Vm | B | 6.34 | 0.02 | 0.05% | ✓ |
HD 101452bbSpitzer standard (Reach et al., 2005; Engelbracht et al., 2007; Sloan et al., 2015) | 11 40 13.6509 |  $-$39 08 47.674 | F0Vm | B | 6.82 | 0.02 |  $<$0.02% |  |
HD 2811bbSpitzer standard (Reach et al., 2005; Engelbracht et al., 2007; Sloan et al., 2015) | 00 31 18.4899 |  $-$43 36 23.000 | A3V | 4 | 7.04 | 0.02 | 1.1%ddHipparcos HPSCAT variability measurement (ESA, 1997) | ✓ |
HD 37725 | 05 41 54.3697 | +29 17 50.957 | A3V | B | 7.90 | 0.05 | 0.13% |  |
HD 116405 | 13 22 45.1238 | +44 42 53.911 | A0V | 7 | 8.48 | 0.0 |  $<$0.02% |  |
HD 180609 | 19 12 47.1996 | +64 10 37.175 | A1V | B | 9.12 | 0.04 |  $<$0.02% | ✓ |
HD 55677 | 07 14 31.2897 | +13 51 36.786 | A2V | 8 | 9.16 | 0.05 | 0.06% |  |
BD+60 1753 | 17 24 52.2772 | +60 25 50.781 | A1V | B | 9.64 | 0.01 |  $<$0.03% | ✓ |
J1757132bbSpitzer standard (Reach et al., 2005; Engelbracht et al., 2007; Sloan et al., 2015) | 17 57 13.2333 | +67 03 40.774 | A8Vm | B | 11.16 | 0.0 |  $<$0.06% | ✓ | 2MASS J17571324+6703409
J1802271bbSpitzer standard (Reach et al., 2005; Engelbracht et al., 2007; Sloan et al., 2015) | 18 02 27.1631 | +60 43 35.542 | A0V | B | 11.83 | 0.02 |  $<$0.14% | ✓ | 2MASS J18022716+6043356
J1805292bbSpitzer standard (Reach et al., 2005; Engelbracht et al., 2007; Sloan et al., 2015) | 18 05 29.2755 | +64 27 52.12 | A1V | B | 12.01 | 0.03 |  $<$0.08% |  | 2MASS J18052927+6427520
J1743045bbSpitzer standard (Reach et al., 2005; Engelbracht et al., 2007; Sloan et al., 2015) | 17 43 04.4857 | +66 55 01.663 | A5IIIm | B | 12.77 | 0.03 |  $<$0.22% | ✓ | 2MASS J17430448+6655015
Solar Analog Stars |
18 Sco | 16 15 37.2704 |  $-$08 22 09.982 | G2Va | 9 | 3.99 | 0.0 | 0.4%ddHipparcos HPSCAT variability measurement (ESA, 1997) |  | HD 146233
16 Cyg B | 19 41 51.9732 | +50 31 03.086 | G3V | 9 | 4.66 | 0.0 |  $<$0.04% | ✓ | HD 186427
HR 6538 | 17 32 00.9923 | +34 16 16.131 | G1V | 10 | 5.05 | 0.0 |  $<$0.02% |  | HD 159222
HD 205905 | 21 39 10.1510 |  $-$27 18 23.666 | G1.5 IV-V | 9 | 5.32 | 0.0 | 0.07% |  |
HD 106252 | 12 13 29.5100 | +10 02 29.889 | G0V | B | 5.93 | 0.0 |  $<$0.01% | ✓ |
HD 37962 | 05 40 51.9659 |  $-$31 21 03.985 | G2V | 11 | 6.27 | 0.01 |  $<$0.02% | ✓ |
HD 142331 | 15 54 19.7884 |  $-$08 34 49.369 | G5V | 12 | 7.13 | 0.01 | 1.8%ddHipparcos HPSCAT variability measurement (ESA, 1997) |  |
HD 167060 | 18 17 44.1430 |  $-$61 42 31.623 | G3V | 5 | 7.43 | 0.02 |  $<$0.02% | ✓ |
HD 115169 | 13 15 47.3883 |  $-$29 30 21.184 | G3V | 13 | 7.71 | 0.01 |  $<$0.02% |  |
GSPC P330-EaaHubble standard (Dickinson et al., 2003; Bohlin et al., 2014) | 16 31 33.8125 | +30 08 46.398 | G0V | B | 11.42 | 0.03 |  $<$0.13% | ✓ | 2MASS J16313382+3008465
GSPC P177-D | 15 59 13.5786 | +47 36 41.905 | G0-1V | B | 11.86 | 0.04 |  $<$0.17% | ✓ | 2MASS J15591357+4736419
SNAP-2 | 16 19 46.1029 | +55 34 17.863 | G3V | B | 14.49 | 0.02 |  |  | 2MASS J16194609+5534178
C26202 | 03 32 32.843 |  $-$27 51 48.58 | F7V | A | 14.82 | 0.06 |  |  | 2MASS J03323287$-$2751483
SF1615+001A | 16 18 14.2397 | +00 00 08.609 | G1V | A | 15.31 | 0.10 |  |  | 2MASS J16181422+0000086
NGC2506-G31 | 08 00 14.2126 |  $-$10 47 29.467 | G1V | 16 | 16.25 | 0.08 |  | ✓ | Gaia EDR3 3038045185547143936

References. — (1) Gianninas et al. (2011), (2) Oswalt et al. (1988), (3) Gray
& Garrison (1987), (4) Houk (1978), (5) Houk & Cowley (1975), (6) Cowley et
al. (1969), (7) Woolley et al. (1969), (8) Fehrenbach (1966), (9) Keenan &
McNeil (1989), (10) Gray et al. (2003), (11) Gray et al. (2006), (12) Houk &
Swift (1999), (13) Houk (1982), (14) Johnson & Morgan (1953), (15) Morgan et
al. (1955), (16) Schlawin et al., in preparation, (A) based on modeling by
Bohlin et al. (2017), (B) MK type determined from the STIS spectra

Some stars that were initially included in our sample have been removed
because they were later found to not fulfill the calibration star criteria
given above. Stars removed due to excessive variability are 2MASS
J17325264+7104431 (TESS $\sigma=0.35$%), 2MASS J18083474+6927286 (TESS
$\sigma=0.41$%), 2MASS J18120957+6329423 (TESS $\sigma=0.40$%), HD 38949 (TESS
$\sigma=0.30$%), and HD 209458 as it varies by $\sim$2% due to a transiting
exoplanet (Charbonneau et al., 2000; Deming et al., 2005). Other removals
include GSPC P041-C as it has a nearby (0.57$\arcsec$) faint companion
(Gilliland & Rajan, 2011), HD 27836 as it is a double star separated by
0.45$\arcsec$ in Hubble STIS observations,${}^{\mathrm{\ref{calspec}}}$ HD
60753 as it has a STIS spectrum that is not well modeled as a single
star,${}^{\mathrm{\ref{calspec}}}$ and $\xi^{2}$ Cet as it is a possible
spectroscopic binary and has a very late B spectral class making it too
similar to A dwarf stars (Johnson & Morgan, 1953; Buscombe, 1963).

##  4 Calibration Calculation

The observations of the calibration stars combined with their predicted flux
densities will be used to calculate the calibration factors that convert a
measurement in instrumental units to physical units. In general, the
calibration factor $C_{FD}$ can be calculated for a source with an expected
flux density $F$ and total integrated data numbers222Data Numbers (DN) and
Analog to Digital Units (ADU) are equivalent and both are used in the JWST
community. per second (DN s-1) of $N$ with

$C_{\mathrm{FD}}=\frac{F}{N}.$ (1)

Using a star as the calibration source, $N$ would have to be measured with an
infinite aperture to capture all of the signal. Hence, measurements in
appropriate finite apertures are corrected to infinite apertures using
aperture corrections calibrated from observations of isolated bright stars and
optical models of the telescope and instrument.

The calibration factor for extended sources in surface brightness units is
then

$C=\frac{C_{\mathrm{FD}}}{\Omega_{\mathrm{pix}}}$ (2)

where $\Omega_{pix}$ is the average solid angle of a pixel. The region used to
determine $\Omega_{pix}$ should be the same region of the array that was used
to normalize the flat field, as the flat field corrects for the different
pixel areas as well as different pixel responsivities. These equations and
those in the rest of this section are based on the work of Bohlin et al.
(2014).

For JWST, the basic instrumental measurement is DN s-1
$\mathrm{(average\>pixel)}^{-1}$ and, thus, the natural physical unit for
calibration is surface brightness. The basic measurement is per average pixel
as dividing by the normalized flat field corrects for differences in
responsivity and area between pixels. Calibrating in surface brightness units
explicitly supports science for both point and extended sources. Calibrating
in surface brightness units does not compromise the accuracy of point source
measurements. Measuring a point source using the same aperture, aperture
correction, and $\Omega_{pix}$ value as used for calculating the calibration
factor results in an uncertainty in point source flux density that does not
include the aperture correction or $\Omega_{pix}$ uncertainties.

JWST will generally calibrate observed images in mega-Janskys per steradian
(MJy sr-1), and these are the surface brightness units common in the IR
community. For measurements integrated over spatial regions (e.g., extracted
source photometry and spectrosocopy), the units will be given in Janskys (Jy).
For photometry, magnitudes in the AB and “Vega” systems will be provided,
where the “Vega” system zero magnitude flux densities will be based on using
Sirius as the color reference (Rieke et al., 2022).

###  4.1 Spectroscopy

For spectroscopy, the calibration factors are given as a function of
wavelength, where

$C_{\mathrm{FD}}(\lambda)=\frac{F(\lambda)}{N(\lambda)}$ (3)

and

$C(\lambda)=\frac{C_{\mathrm{FD}}(\lambda)}{\Omega_{\mathrm{pix}}(\lambda)}.$
(4)

The prediction of a source’s $F(\lambda)$ includes the source spectrum and the
appropriate convolution by the instrumental line-spread function. The
measurement of $N(\lambda)$ is corrected to an infinite extraction height
perpendicular to the dispersion and for slit observations to an infinite slit
width. Finally, $\Omega_{\mathrm{pix}}(\lambda)$ is the average area of the
pixels perpendicular to the dispersion direction at each wavelength. In
general, bandpass effects for a single wavelength pixel are small and the
measured signal is attributed to the average wavelength of that pixel. The
assumption of small bandpass effects will be checked where possible. Bandpass
effects may be important, for example, in the case of observations at low
spectral resolution, especially for narrow unresolved spectral lines or
anywhere the sensitivity is changing rapidly across a pixel.

###  4.2 Photometry

For photometry, the bandpass effects on flux calibration can be quite
significant. The measurement in a photometric band can be calibrated as an
integrated flux or as the average flux density, with the latter used more
commonly. The two common average flux density conventions are (A) to give the
average flux density at the effective wavelength, and (B) to give the flux
density at a fixed, reference wavelength for a reference spectral shape. In
both conventions, knowledge of the spectral shape of the source is needed to
either compute the effective wavelength or correct the flux density for the
difference between the actual source and reference spectral shapes. Over the
wavelength range of JWST, both conventions have been used for calibration for
ground- and space-based observations (e.g., Johnson, 1965, 1966; Beichman et
al., 1988; Bessell et al., 1998; Reach et al., 2005; Engelbracht et al., 2007;
Bohlin et al., 2014). The following subsections give the equations for both
conventions and show how it is possible to have one calibration that applies
for both conventions.

####  4.2.1 Convention A: Average Flux Density at $\lambda_{\mathrm{eff}}$

A photometric measurement in a filter can be given as the photon-weighted
average flux density, and this can be computed using

$\left<F(\lambda)\right>=\frac{\int F(\lambda)R(\lambda)\lambda d\lambda}{\int
R(\lambda)\lambda d\lambda}$ (5)

where $F(\lambda)$ is the flux density, $R(\lambda)$ is the bandpass function,
and the integration is done over $\lambda d\lambda$ in photon units to match
the detection method (i.e., photons $=F(\lambda)/h\nu=F(\lambda)\lambda/hc$,
and $hc$ cancels out since it is in both integrals). The bandpass function
$R(\lambda)$ is the fractional transmission of the telescope and instrument
including the detectors. The effective wavelength $\lambda_{\mathrm{eff}}$ of
this average flux density is then

$\lambda_{\mathrm{eff}}=\frac{\int\lambda F(\lambda)R(\lambda)\lambda
d\lambda}{\int F(\lambda)R(\lambda)\lambda d\lambda}.$ (6)

The number of DN detected per second is

$N_{\mathrm{A}}=g^{-1}\int F(\lambda)R(\lambda)\lambda d\lambda$ (7)

where $g$ is the electronic gain in e-/DN. Thus, the calibration factor is

$\displaystyle C_{\mathrm{FD,A}}$ $\displaystyle=$
$\displaystyle\frac{\left<F(\lambda)\right>}{N_{\mathrm{A}}}$ (8)
$\displaystyle=$ $\displaystyle\frac{g}{\int R(\lambda)\lambda d\lambda}.$ (9)

Note that these quantities can also be derived using bandpass functions in
photon units where the conversion to photon units is shifted from the
integration to the bandpass function.

####  4.2.2 Convention B: Flux Density at $\lambda_{\mathrm{ref}}$

The photometry in a filter can also be given as the flux density at a
reference wavelength $\lambda_{\mathrm{ref}}$ for a source with a reference
spectral shape. For sources that have spectral shapes different than the
reference shape, then the flux density will need to be corrected for the
difference in spectral shapes. The number of DN per second for a source with
$F_{\mathrm{ref}}(\lambda)$ is then

$N_{\mathrm{B}}=\frac{F_{\mathrm{ref}}(\lambda_{\mathrm{ref}})}{g}\int\frac{F_{\mathrm{ref}}(\lambda)}{F_{\mathrm{ref}}(\lambda_{\mathrm{ref}})}R(\lambda)\lambda
d\lambda$ (10)

and the calibration factor is

$\displaystyle C_{\mathrm{FD,B}}$ $\displaystyle=$
$\displaystyle\frac{F_{\mathrm{ref}}(\lambda_{\mathrm{ref}})}{N_{\mathrm{B}}}$
(11) $\displaystyle=$
$\displaystyle\frac{g}{\int\frac{F_{\mathrm{ref}}(\lambda)}{F_{\mathrm{ref}}(\lambda_{\mathrm{ref}})}R(\lambda)\lambda
d\lambda}.$ (12)

The flux density for an arbitrary source at the reference wavelength is

$\displaystyle F(\lambda_{\mathrm{ref}})$ $\displaystyle=$
$\displaystyle\frac{C_{\mathrm{FD,B}}N_{\mathrm{B}}}{K}$ (13) $\displaystyle=$
$\displaystyle\frac{1}{K}\frac{\int F(\lambda)R(\lambda)\lambda
d\lambda}{\int\frac{F_{\mathrm{ref}}(\lambda)}{F_{\mathrm{ref}}(\lambda_{\mathrm{ref}})}R(\lambda)\lambda
d\lambda}$ (14)

where the color correction333The term “color correction” is also used to
describe the corrections needed to convert measurements between different
photometric systems (e.g., Bessell et al., 1998). $K$ (Beichman et al., 1988;
Reach et al., 2005; Stansberry et al., 2007) is

$K=\frac{\int\frac{F(\lambda)}{F(\lambda_{\mathrm{ref}})}R(\lambda)\lambda
d\lambda}{\int\frac{F_{\mathrm{ref}}(\lambda)}{F_{\mathrm{ref}}(\lambda_{\mathrm{ref}})}R(\lambda)\lambda
d\lambda}.$ (15)

Thus, the given value of a source in this convention should be divided by $K$
for the appropriate filter and spectral shape to produce the flux density at
the reference wavelength of the filter. One way to think of the color
correction is that it adjusts the calibration factor to align the reference
spectral shape with the current source, which results in the correct flux
density at the reference wavelength. For JWST we will provide color
corrections for a reasonable range of spectral shapes different from the
adopted $F_{\mathrm{ref}}(\lambda)$. Custom color corrections for specific
spectral shapes can be computed with the JWST bandpass functions.

The choice of $\lambda_{\mathrm{ref}}$ is arbitrary, but generally picked to
be near the middle of the bandpass. For JWST we adopt the pivot wavelength for
$\lambda_{\mathrm{ref}}$ where

$\lambda_{\mathrm{ref}}=\lambda_{p}=\left(\frac{\int R(\lambda)\lambda
d\lambda}{\int R(\lambda)\lambda^{-1}d\lambda}\right)^{1/2}.$ (16)

The pivot wavelength has the useful properties of being source independent,
and the associated pivot frequency is directly related (Bohlin et al., 2014).

####  4.2.3 Conversion between Conventions

For JWST, we could pick one of the two conventions and provide conversion
factors between them for every filter. Fortunately, it is possible to pick
$F_{\mathrm{ref}}(\lambda)$ for convention B that allows a single calibration
factor per filter to work for both conventions. This can be shown by examining
the equation to convert between the two conventions. The multiplicative
conversion from convention A to convention B is the ratio of equations 12 and
9 and is

$R_{\mathrm{AB}}=\frac{C_{\mathrm{FD,B}}}{C_{\mathrm{FD,A}}}=\frac{C_{\mathrm{B}}}{C_{\mathrm{A}}}=\frac{\int
R(\lambda)\lambda
d\lambda}{\int\frac{F_{\mathrm{ref}}(\lambda)}{F_{\mathrm{ref}}(\lambda_{\mathrm{ref}})}R(\lambda)\lambda
d\lambda}.$ (17)

This equation is equivalent to the color correction for convention B (Eq. 15)
where $F(\lambda)=\mathrm{const}$. This clue shows that convention A is
mathematically equivalent to convention B with
$F_{\mathrm{ref}}(\lambda)=\mathrm{const}$. For JWST we adopt
$F_{\mathrm{ref}}(\lambda)=\mathrm{const}$, resulting in the same calibration
factors for both conventions A and B (i.e., $R_{\mathrm{AB}}=1$).

##  5 Summary

We have presented the motivation and details of the absolute flux calibration
program for JWST. This unified, efficient program for all JWST instruments has
the goal of providing accurate absolute and relative calibration. We designed
this program to quantify and minimize the statistical and systematic
uncertainties through observations of three different types of stars and
multiple stars of each type having a range of flux densities. The three types
are hot stars, A dwarfs, and solar analogs, and the program stars have been
extensively vetted using Hubble, Spitzer, and TESS observations along with
other ancillary data. The program includes observations to calibrate every
instrument mode, improve the vetting of the program stars using the JWST
observations themselves, measure the instrument repeatability for each
detector, measure the calibration between subarray and full frame
observations, and confirm that observations of bright and faint sources have
the same calibration. The program calibration stars have a wide range of flux
densities, providing good coverage of the minimum and maximum observable range
except for the faint flux densities for NIRCam and NIRISS imaging and bright
flux densities for MIRI observations beyond 15 µm. Carrying out the full
objectives of this program will require observations taken throughout the JWST
mission, and we give the specifics of the planned first year of observations.
How the calibration factors will be calculated is given for photometry and
spectroscopy. For JWST we will use a photometric calibration convention that
directly supports the two commonly used conventions. Future papers on the JWST
absolute flux program and related efforts will refine the detailed flux
predictions for all stars and provide calibration factors for each for the
many modes for each JWST instrument.

## References

  * Adelman (2001) Adelman, S. J. 2001, A&A, 367, 297, doi: 10.1051/0004-6361:20000567
  * Aufdenberg et al. (2006) Aufdenberg, J. P., Mérand, A., Coudé du Foresto, V., et al. 2006, ApJ, 645, 664, doi: 10.1086/504149
  * Aumann et al. (1984) Aumann, H. H., Gillett, F. C., Beichman, C. A., et al. 1984, ApJ, 278, L23, doi: 10.1086/184214
  * Beichman et al. (1988) Beichman, C. A., Neugebauer, G., Habing, H. J., Clegg, P. E., & Chester, T. J., eds. 1988, Infrared Astronomical Satellite (IRAS) Catalogs and Atlases.Volume 1: Explanatory Supplement., Vol. 1
  * Bessell et al. (1998) Bessell, M. S., Castelli, F., & Plez, B. 1998, A&A, 333, 231
  * Bohlin (2010) Bohlin, R. C. 2010, AJ, 139, 1515, doi: 10.1088/0004-6256/139/4/1515
  * Bohlin (2014) —. 2014, AJ, 147, 127, doi: 10.1088/0004-6256/147/6/127
  * Bohlin & Cohen (2008) Bohlin, R. C., & Cohen, M. 2008, AJ, 136, 1171, doi: 10.1088/0004-6256/136/3/1171
  * Bohlin & Deustua (2019) Bohlin, R. C., & Deustua, S. E. 2019, AJ, 157, 229, doi: 10.3847/1538-3881/ab1b50
  * Bohlin et al. (2001) Bohlin, R. C., Dickinson, M. E., & Calzetti, D. 2001, AJ, 122, 2118, doi: 10.1086/323137
  * Bohlin et al. (2014) Bohlin, R. C., Gordon, K. D., & Tremblay, P. E. 2014, PASP, 126, 711, doi: 10.1086/677655
  * Bohlin et al. (2020) Bohlin, R. C., Hubeny, I., & Rauch, T. 2020, AJ, 160, 21, doi: 10.3847/1538-3881/ab94b4
  * Bohlin et al. (2022) Bohlin, R. C., Kirck, J. E., Gordon, K. D., & Hubeny, I. 2022, ApJ, submitted
  * Bohlin et al. (2017) Bohlin, R. C., Mészáros, S., Fleming, S. W., et al. 2017, AJ, 153, 234, doi: 10.3847/1538-3881/aa6ba9
  * Bohlin et al. (2011) Bohlin, R. C., Gordon, K. D., Rieke, G. H., et al. 2011, AJ, 141, 173, doi: 10.1088/0004-6256/141/5/173
  * Buscombe (1963) Buscombe, W. 1963, Mount Stromlo Observatory Mimeographs, 6, 0
  * Carey et al. (2012) Carey, S., Ingalls, J., Hora, J., et al. 2012, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 8442, Space Telescopes and Instrumentation 2012: Optical, Infrared, and Millimeter Wave, ed. M. C. Clampin, G. G. Fazio, H. A. MacEwen, & J. Oschmann, Jacobus M., 84421Z, doi: 10.1117/12.927183
  * Charbonneau et al. (2000) Charbonneau, D., Brown, T. M., Latham, D. W., & Mayor, M. 2000, ApJ, 529, L45, doi: 10.1086/312457
  * Cohen et al. (1999) Cohen, M., Walker, R. G., Carter, B., et al. 1999, AJ, 117, 1864, doi: 10.1086/300813
  * Cowley et al. (1969) Cowley, A., Cowley, C., Jaschek, M., & Jaschek, C. 1969, AJ, 74, 375, doi: 10.1086/110819
  * Decin et al. (2004) Decin, L., Morris, P. W., Appleton, P. N., et al. 2004, ApJS, 154, 408, doi: 10.1086/422884
  * Deming et al. (2005) Deming, D., Seager, S., Richardson, L. J., & Harrington, J. 2005, Nature, 434, 740, doi: 10.1038/nature03507
  * Dickinson et al. (2003) Dickinson, M., Sosey, M., Rieke, M., Bohlin, R. C., & Calzetti, D. 2003, in HST Calibration Workshop : Hubble after the Installation of the ACS and the NICMOS Cooling System, 231
  * Doyon et al. (2012) Doyon, R., Hutchings, J. B., Beaulieu, M., et al. 2012, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 8442, Space Telescopes and Instrumentation 2012: Optical, Infrared, and Millimeter Wave, ed. M. C. Clampin, G. G. Fazio, H. A. MacEwen, & J. Oschmann, Jacobus M., 84422R, doi: 10.1117/12.926578
  * Engelbracht et al. (2007) Engelbracht, C. W., Blaylock, M., Su, K. Y. L., et al. 2007, PASP, 119, 994, doi: 10.1086/521881
  * Engelke et al. (2010) Engelke, C. W., Price, S. D., & Kraemer, K. E. 2010, AJ, 140, 1919, doi: 10.1088/0004-6256/140/6/1919
  * ESA (1997) ESA, ed. 1997, ESA Special Publication, Vol. 1200, The HIPPARCOS and TYCHO catalogues. Astrometric and photometric star catalogues derived from the ESA HIPPARCOS Space Astrometry Mission
  * Fazio et al. (2004) Fazio, G. G., Hora, J. L., Allen, L. E., et al. 2004, ApJS, 154, 10, doi: 10.1086/422843
  * Fehrenbach (1966) Fehrenbach, C. 1966, Publications of the Observatoire Haute-Provence, 8, 25
  * Fixsen et al. (1994) Fixsen, D. J., Cheng, E. S., Cottingham, D. A., et al. 1994, ApJ, 420, 457, doi: 10.1086/173577
  * Gaia Collaboration (2018) Gaia Collaboration. 2018, VizieR Online Data Catalog, I/345
  * Gardner et al. (2006) Gardner, J. P., Mather, J. C., Clampin, M., et al. 2006, Space Sci. Rev., 123, 485, doi: 10.1007/s11214-006-8315-7
  * Gianninas et al. (2011) Gianninas, A., Bergeron, P., & Ruiz, M. T. 2011, ApJ, 743, 138, doi: 10.1088/0004-637X/743/2/138
  * Gilliland & Rajan (2011) Gilliland, R. L., & Rajan, A. 2011, WFC3 UVIS High-resolution Imaging Performance, Space Telescope WFC Instrument Science Report
  * Gordon & Bohlin (2012) Gordon, K., & Bohlin, R. 2012, JWST Absolute FluxCalibration II: Expanded Sample of PrimaryCalibrators, Tech. Rep. JWST-STScI-002540, STScI
  * Gordon et al. (2009) Gordon, K., Bohlin, R., Fullerton, A., Beck, T., & Robberto, M. 2009, JWST Absolute Flux Calibration I. Proposed Primary Calibrators, Tech. Rep. JWST-STScI-001855, STScI
  * Gordon et al. (2019) Gordon, K., Boyer, M., Muzerolle, J., Sloan, G., & Volk, K. 2019, An End-to-End Calibration Error Budget for the JWST Science Instruments, Tech. Rep. JWST-STScI-001007-RevC, STScI
  * Gordon et al. (2007) Gordon, K. D., Engelbracht, C. W., Fadda, D., et al. 2007, PASP, 119, 1019, doi: 10.1086/522675
  * Gray et al. (2006) Gray, R. O., Corbally, C. J., Garrison, R. F., et al. 2006, AJ, 132, 161, doi: 10.1086/504637
  * Gray et al. (2003) Gray, R. O., Corbally, C. J., Garrison, R. F., McFadden, M. T., & Robinson, P. E. 2003, AJ, 126, 2048, doi: 10.1086/378365
  * Gray & Garrison (1987) Gray, R. O., & Garrison, R. F. 1987, ApJS, 65, 581, doi: 10.1086/191237
  * Heras et al. (2002) Heras, A. M., Shipman, R. F., Price, S. D., et al. 2002, A&A, 394, 539, doi: 10.1051/0004-6361:20021124
  * Houck et al. (2004) Houck, J. R., Roellig, T. L., van Cleve, J., et al. 2004, ApJS, 154, 18, doi: 10.1086/423134
  * Houk (1978) Houk, N. 1978, Michigan catalogue of two-dimensional spectral types for the HD stars
  * Houk (1982) —. 1982, Michigan Catalogue of Two-dimensional Spectral Types for the HD stars. Volume_3. Declinations -40_f0 to -26_f0.
  * Houk & Cowley (1975) Houk, N., & Cowley, A. P. 1975, University of Michigan Catalogue of two-dimensional spectral types for the HD stars. Volume I. Declinations -90_ to -53_f0.
  * Houk & Swift (1999) Houk, N., & Swift, C. 1999, Michigan catalogue of two-dimensional spectral types for the HD Stars ; vol. 5, Vol. 5
  * Jakobsen et al. (2022) Jakobsen, P., Ferruit, P., Alves de Oliveira, C., et al. 2022, arXiv e-prints, arXiv:2202.03305.  https://arxiv.org/abs/2202.03305
  * Johnson (1965) Johnson, H. L. 1965, Communications of the Lunar and Planetary Laboratory, 3, 73
  * Johnson (1966) —. 1966, ARA&A, 4, 193, doi: 10.1146/annurev.aa.04.090166.001205
  * Johnson et al. (1966) Johnson, H. L., Mitchell, R. I., Iriarte, B., & Wisniewski, W. Z. 1966, Communications of the Lunar and Planetary Laboratory, 4, 99
  * Johnson & Morgan (1953) Johnson, H. L., & Morgan, W. W. 1953, ApJ, 117, 313, doi: 10.1086/145697
  * Keenan & McNeil (1989) Keenan, P. C., & McNeil, R. C. 1989, ApJS, 71, 245, doi: 10.1086/191373
  * Kent et al. (2009) Kent, S., Kaiser, M. B., Deustua, S. E., et al. 2009, in astro2010: The Astronomy and Astrophysics Decadal Survey, Vol. 2010, 155.  https://arxiv.org/abs/0903.2799
  * Knudstrup et al. (2020) Knudstrup, E., Grundahl, F., Brogaard, K., et al. 2020, MNRAS, 499, 1312, doi: 10.1093/mnras/staa2855
  * Krick et al. (2021) Krick, J. E., Lowrance, P., Carey, S., et al. 2021, AJ, 161, 177, doi: 10.3847/1538-3881/abe390
  * Lu et al. (2008) Lu, N., Smith, P. S., Engelbracht, C. W., et al. 2008, PASP, 120, 328, doi: 10.1086/533459
  * Megessier (1995) Megessier, C. 1995, A&A, 296, 771
  * Morgan et al. (1955) Morgan, W. W., Code, A. D., & Whitford, A. E. 1955, ApJS, 2, 41, doi: 10.1086/190016
  * Mullally et al. (2022) Mullally, S. E., Sloan, G. C., Hermes, J. J., et al. 2022, ApJ, in press
  * Müller et al. (2014) Müller, T., Balog, Z., Nielbock, M., et al. 2014, Experimental Astronomy, 37, 253, doi: 10.1007/s10686-013-9357-y
  * Oja (1991) Oja, T. 1991, A&AS, 89, 415
  * Oswalt et al. (1988) Oswalt, T. D., Hintzen, P. M., & Luyten, W. J. 1988, ApJS, 66, 391, doi: 10.1086/191263
  * Price et al. (2004) Price, S. D., Paxson, C., Engelke, C., & Murdock, T. L. 2004, AJ, 128, 889, doi: 10.1086/422024
  * Reach et al. (2005) Reach, W. T., Megeath, S. T., Cohen, M., et al. 2005, PASP, 117, 978, doi: 10.1086/432670
  * Rieke et al. (2022) Rieke, G. H., Su, K., Sloan, G. C., & Schlawin, E. 2022, AJ, 163, 45, doi: 10.3847/1538-3881/ac3b5d
  * Rieke et al. (2004) Rieke, G. H., Young, E. T., Engelbracht, C. W., et al. 2004, ApJS, 154, 25, doi: 10.1086/422717
  * Rieke et al. (2008) Rieke, G. H., Blaylock, M., Decin, L., et al. 2008, AJ, 135, 2245, doi: 10.1088/0004-6256/135/6/2245
  * Rieke et al. (2015) Rieke, G. H., Wright, G. S., Böker, T., et al. 2015, PASP, 127, 584, doi: 10.1086/682252
  * Rieke et al. (2005) Rieke, M. J., Kelly, D., & Horner, S. 2005, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 5904, Cryogenic Optical Systems and Instruments XI, ed. J. B. Heaney & L. G. Burriesci, 1–8, doi: 10.1117/12.615554
  * Skrutskie et al. (2006) Skrutskie, M. F., Cutri, R. M., Stiening, R., et al. 2006, AJ, 131, 1163, doi: 10.1086/498708
  * Sloan et al. (2015) Sloan, G. C., Herter, T. L., Charmandaris, V., et al. 2015, AJ, 149, 11, doi: 10.1088/0004-6256/149/1/11
  * Stansberry et al. (2007) Stansberry, J. A., Gordon, K. D., Bhattacharya, B., et al. 2007, PASP, 119, 1038, doi: 10.1086/521880
  * Su et al. (2021) Su, K. Y. L., Rieke, G. H., Marengo, M., & Schlawin, E. 2021, arXiv e-prints, arXiv:2111.10054.  https://arxiv.org/abs/2111.10054
  * Su et al. (2005) Su, K. Y. L., Rieke, G. H., Misselt, K. A., et al. 2005, ApJ, 628, 487, doi: 10.1086/430819
  * Thompson (1992) Thompson, R. 1992, Space Sci. Rev., 61, 69, doi: 10.1007/BF00212476
  * Werner et al. (2004) Werner, M. W., Roellig, T. L., Low, F. J., et al. 2004, ApJS, 154, 1, doi: 10.1086/422992
  * Woolley et al. (1969) Woolley, R., Asaad, A. S., Candy, P., & Penston, M. J. 1969, Royal Greenwich Observatory Bulletins, 156, 35