# Using the Sun to Measure the Primary Beam Response of the Canadian Hydrogen
Intensity Mapping Experiment

Mandana Amiri Department of Physics and Astronomy, University of British
Columbia, Vancouver, BC, Canada  Kevin Bandura Department of Computer Science
and Electrical Engineering, West Virginia University, Morgantown WV, USA  Anja
Boskovic  Department of Physics and Astronomy, University of British Columbia,
Vancouver, BC, Canada  Jean-François Cliche Department of Physics, McGill
University, Montreal, QC, Canada  Meiling Deng Dominion Radio Astrophysical
Observatory, Herzberg Astronomy & Astrophysics Research Centre, National
Research Council Canada, Penticton, BC, Canada  Perimeter Institute for
Theoretical Physics, Waterloo, ON, Canada  Department of Physics and
Astronomy, University of British Columbia, Vancouver, BC, Canada  Matt Dobbs
Department of Physics, McGill University, Montreal, QC, Canada  Mateus Fandino
Department of Physics and Astronomy, University of British Columbia,
Vancouver, BC, Canada  Department of Physical Sciences, Thompson Rivers
University, Kamloops, BC, Canada  Simon Foreman Perimeter Institute for
Theoretical Physics, Waterloo, ON, Canada  Dominion Radio Astrophysical
Observatory, Herzberg Astronomy & Astrophysics Research Centre, National
Research Council Canada, Penticton, BC, Canada  Mark Halpern Department of
Physics and Astronomy, University of British Columbia, Vancouver, BC, Canada
Alex S. Hill Department of Computer Science, Math, Physics, and Statistics,
University of British Columbia-Okanagan, Kelowna, BC, Canada  Dominion Radio
Astrophysical Observatory, Herzberg Astronomy & Astrophysics Research Centre,
National Research Council Canada, Penticton, BC, Canada  Gary Hinshaw
Department of Physics and Astronomy, University of British Columbia,
Vancouver, BC, Canada  Carolin Höfer Department of Physics and Astronomy,
University of British Columbia, Vancouver, BC, Canada  Joseph Kania Department
of Physics and Astronomy, West Virginia University, Morgantown, WV, USA  T.L.
Landecker Dominion Radio Astrophysical Observatory, Herzberg Astronomy &
Astrophysics Research Centre, National Research Council Canada, Penticton, BC,
Canada  Joshua MacEachern Department of Physics and Astronomy, University of
British Columbia, Vancouver, BC, Canada  Kiyoshi Masui MIT Kavli Institute for
Astrophysics and Space Research, Massachusetts Institute of Technology,
Cambridge, MA, USA  Department of Physics, Massachusetts Institute of
Technology, Cambridge, MA, USA  Juan Mena-Parra MIT Kavli Institute for
Astrophysics and Space Research, Massachusetts Institute of Technology,
Cambridge, MA, USA  Laura Newburgh Department of Physics, Yale University, New
Haven, CT, USA  Anna Ordog Department of Computer Science, Math, Physics, and
Statistics, University of British Columbia-Okanagan, Kelowna, BC, Canada
Dominion Radio Astrophysical Observatory, Herzberg Astronomy & Astrophysics
Research Centre, National Research Council Canada, Penticton, BC, Canada
Tristan Pinsonneault-Marotte Department of Physics and Astronomy, University
of British Columbia, Vancouver, BC, Canada  Ava Polzin Department of
Astronomy, Yale University, New Haven, CT, USA  Alex Reda Department of
Physics, Yale University, New Haven, CT, USA  J. Richard Shaw Department of
Physics and Astronomy, University of British Columbia, Vancouver, BC, Canada
Seth R. Siegel Department of Physics, McGill University, Montreal, QC, Canada
Saurabh Singh Department of Physics, McGill University, Montreal, QC, Canada
Raman Research Institute, Sadashivanagar, Bengaluru, India  Keith Vanderlinde
David A. Dunlap Department of Astronomy & Astrophysics, University of Toronto,
Toronto, ON, Canada  Dunlap Institute for Astronomy and Astrophysics,
University of Toronto, Toronto, ON, Canada  Haochen Wang MIT Kavli Institute
for Astrophysics and Space Research, Massachusetts Institute of Technology,
Cambridge, MA, USA  Department of Physics, Massachusetts Institute of
Technology, Cambridge, MA, USA  James S. Willis  Dunlap Institute for
Astronomy and Astrophysics, University of Toronto, Toronto, ON, Canada  and
Dallas Wulf Department of Physics, McGill University, Montreal, QC, Canada

###### Abstract

We present a beam pattern measurement of the Canadian Hydrogen Intensity
Mapping Experiment (CHIME) made using the Sun as a calibration source. As
CHIME is a pure drift scan instrument, we rely on the seasonal North-South
motion of the Sun to probe the beam at different elevations. This semiannual
range in elevation, combined with the radio brightness of the Sun, enables a
beam measurement which spans $\sim$7,200 square degrees on the sky without the
need to move the telescope. We take advantage of observations made near solar
minimum to minimize the impact of solar variability, which is observed to be
$<$10% in intensity over the observation period. The resulting data set is
highly complementary to other CHIME beam measurements—both in terms of angular
coverage and systematics—and plays an important role in the ongoing program to
characterize the CHIME primary beam.

Radio telescopes — Interferometers — Calibration — Quiet sun

††facilities: CHIME (CHIME Collaboration et al., 2022a), Learmonth Solar
Observatory (Australian Bureau of Meteorology, 2017), John A. Galt Telescope
(Locke et al., 1965)††software:  NumPy (Harris et al., 2020), SciPy (Virtanen
et al., 2020), HDF5 (The HDF Group, 1997), h5py (Collette et al., 2021),
Matplotlib (Hunter, 2007), Skyfield (Rhodes, 2019), caput (Shaw et al.,
2020a), ch_pipeline (Shaw et al., 2020b), draco (Shaw et al., 2020c), kotekan
(Renard et al., 2021)

##  1 Introduction

The Canadian Hydrogen Intensity Mapping Experiment (CHIME) is a drift scan
radio interferometer array operating between 400 and 800 MHz. It consists of
four parabolic cylindrical reflectors. Each reflector is 20 m in diameter, 100
m in length, oriented North-South, and is instrumented with 256 dual
polarization feeds that are sensitive to linear polarizations in the North-
South and East-West directions (1024 total feeds). This configuration provides
a large instantaneous field of view, with the main lobe spanning $\sim$300
square degrees along the local meridian. CHIME is thus optimized for high
mapping speed, making it a powerful instrument for 21 cm cosmology, transient
detection, and source monitoring. For a more detailed description of CHIME, we
refer the reader to CHIME Collaboration et al. (2022a) and references therein.

One of the challenges of working with CHIME is characterizing the large
primary beam without the ability to steer the telescope. Due to the fixed
pointing of the telescope, astrophysical radio sources that are typically used
for beam measurements only transit the beam at a single declination. Thus,
each source probes only a small fraction of the total beam solid angle.
Moreover, few of these sources are bright enough to rise above the confusion
level and be seen in the far side lobes. These challenges are shared by all
drift scan instruments with limited pointing capability, such as the Molonglo
Observatory Synthesis Telescope (MOST, Bailes et al., 2017), the Deep Synoptic
Array (DSA-110, Kocz et al., 2019), the Canadian Hydrogen Observatory and
Radio transient Detector (CHORD, Vanderlinde et al., 2019), the Tianlai
Project (Phan & Collaboration, 2020), and the Hydrogen Intensity and Real-time
Analysis eXperiment (HIRAX, Crichton et al., 2022). While some of these
instruments are exploring the use of drones to overcome these challenges (E.g.
Zhang et al., 2021), they are not feasible for CHIME, which has a far-field
distance greater than one kilometer.

These considerations motivated us to use the Sun as a calibration source for
measuring the primary beam response of CHIME. The Sun has been used for beam-
mapping purposes before (E.g. Pauliny-Toth & Shakeshaft, 1962; Murphy, 1993;
Higgs & Tapping, 2000; Chang et al., 2015). However, these measurements were
made with fully steerable dishes, capable of measuring the full beam in a
single observation by employing a raster scan strategy. In contrast, the
measurement reported here relies on the seasonal motion of the Sun and
combines observations spanning one year. Between the winter and summer
solstices, the declination of the Sun moves between $-23.5^{\circ}$ and
$+23.5^{\circ}$, enabling a quasi-continuous measurement of the beam over this
range. Moreover, the radio brightness of the Sun exceeds 200 kJy in our band,
enabling a high signal-to-noise measurement of the beam into the far side
lobes. These combined properties enable the Sun to probe the beam over
$\sim$7,200 square degrees on the sky without the need to move the telescope.

In what follows, we describe a measurement of the CHIME primary beam made
using the Sun. In Sec. 1.1, we review the emission characteristics of the Sun
that are relevant to its use as a calibration source at decimeter wavelengths.
In Section 2 we describe the observations and analysis techniques used to
produce a beam measurement. Results are presented in Sec. 3, followed by a
discussion of this measurement in the context of the broader CHIME beam
measurement program in Sec. 4.

###  1.1 Solar Emission at Decimeter Wavelengths

Solar emission at radio wavelengths is conventionally divided into three
components: a minimum/quiet component, a slowly varying component, and a burst
component. The relative contributions of the latter two fluctuate according to
the well-known 11-year solar cycle. The observations reported here were made
in late 2019 and early 2020, near the minimum point between Solar Cycles 24
and 25111National Research Council Canada, https://www.spaceweather.gc.ca/.
The quiet component is produced by thermal free-free (Bremsstrahlung)
emission, which, at decimeter wavelengths, originates in the lower corona. Due
to steep temperature and density gradients in this part of the solar
atmosphere, the brightness temperature of the Sun is a strong function of
frequency, with higher frequencies probing lower altitudes and thus lower
temperatures.

There is little direct imaging of the Sun in the CHIME band (Shibasaki et al.,
2011). However, 432 MHz images from Mercier & Chambe (2012) made during the
previous solar minimum (2008–2011) show that emission is nearly uniform over
the face of the Sun and is confined to $\lesssim 1.1R_{s}$, where $R_{s}=16$
arcmin is the optical radius of the Sun as viewed from Earth. At higher
frequencies, the apparent radius is even closer to the optical radius. We
therefore conclude that the emission profile of the Sun throughout the CHIME
band can be approximated as a disk with radius $\sim R_{s}$. Given this radius
and the temperature range of the lower corona, the flux density of the quiet
Sun is a few $\times 10^{5}$ Jy in the CHIME band, as shown in Figure 1.

Even at solar minimum, there is still a contribution to the solar flux from
slowly varying and burst components. To quantify the amplitude of these
components, we refer to total intensity data from the Learmonth Solar
Observatory (Australian Bureau of Meteorology, 2017), available at 1 second
cadence at 410 MHz and 610 MHz. Between 31 May 2019 and 11 July 2020, the
standard deviation of the daily mean flux at these frequencies was found to be
7% of the mean value, with comparable variation on intra-day timescales.
Instrumental instability may account for a non-negligible fraction of the
variability observed in the Learmonth data, though this contribution is likely
small. Nevertheless, we consider this to be an upper limit on the intrinsic
solar variability. During this interval, CHIME made calibrated measurements of
the solar spectrum four times on days when the Sun was at the declination of
either Tau A or Vir A (31 May 2019, 20 August 2019, 21 April 2020, and 11 July
2020). The beam response at these declinations is independently measured using
source fluxes from Perley & Butler (2017), enabling an absolute flux
measurement of the Sun on these days. Averaged across the band, these four
measurements have a range of 5% relative to the mean. Both the Learmonth and
CHIME measurements are plotted in Figure 1. More details about the CHIME flux
measurements are described in Section 2.4.

Figure 1: Total solar intensity calibrated against Tau A on 31 May 2019,
before (dashed blue) and after (solid blue) correcting for quantization bias
(Section 2.5). The shaded blue region covers the range of three other CHIME
measurements made at the declinations of Tau A and Vir A through 11 July 2020.
Independent measurements of the mean and standard deviation of the daily mean
solar flux are shown in orange (Australian Bureau of Meteorology, 2017). The
gray bars mask frequencies of persistent RFI. The green contours indicate the
flux density of a 32 arcmin diameter disk with a given (frequency-independent)
brightness temperature, and are shown to emphasize the frequency dependence of
the effective temperature of the Sun in this band. Both CHIME and Learmonth
observations indicate solar variability $<$10% over the observation period.

##  2 Methods

###  2.1 Data Set

Observations of the Sun are extracted from the CHIME stack data set described
in detail in CHIME Collaboration et al. (2022a). CHIME is an interferometer
which gathers data in the form of correlation products between pairs of feeds.
We will therefore discuss the CHIME data in the language of interferometry,
where a baseline is the separation between two feeds and a visibility is the
correlation product of two voltage streams. By design, the CHIME baseline
configuration is highly redundant, enabling data compression by averaging over
redundant baselines. Prior to averaging over baselines, data are averaged to
10 s cadence, calibrated for gain variations, and baseline pairs containing
bad feeds are masked. Time and frequency bins impacted by radio frequency
interference (RFI) are also masked. As these data are recorded as part of
CHIME’s primary science objectives, no dedicated observing time or specialized
data products are needed for this measurement.

###  2.2 Baseline Selection

As described in Section 1.1, the emission profile of the quiet Sun in the
CHIME band is well described by a disk approximately 0.5∘ in diameter. Due to
its finite size, only baseline separations between 30 cm and 10 m are used to
avoid resolving out the Sun. Zero-spaced baselines (autocorrelations) are
masked to remove the receiver temperature bias. Since the minimum East-West
separation between feeds is 22 m (set by the spacing of the cylindrical
reflectors), this baseline selection only includes purely North-South
separations. The corresponding synthesized beam has a North-South width of
$\sim 2^{\circ}$ FWHM at 800 MHz and attenuates the solar flux by $<2\%$ at
all frequencies. Finally, only co-polar visibilities are used, with North-
South polarized data analyzed separately from East-West polarized data. After
all baseline selection criteria have been applied, 32 unique baselines are
used for each polarization.

The selection of baselines also affects the confusion noise limit. Confusion
noise is estimated by taking the median absolute deviation (MAD) of the night-
time sky brightness over the solar elevation range, using a map made using the
same baseline selection as the solar measurements (Figure 2). Full sidereal
coverage of the map is achieved by stitching together right ascension ranges
observed over the course of the year. Comparing Figures 1 and 2, the flux of
the Sun is seen to be about four orders of magnitude greater than the
confusion noise across most of the band (ignoring beam effects, which are of
order unity). Due to the geometry of the reflector, the East-West polarized
beam is slightly wider than the North-South polarized beam, resulting in
slightly higher confusion noise. The ratio of the solar flux to the confusion
noise sets the dynamic range of the beam measurement, which is $\sim$40 dB.

Figure 2: Estimate of confusion noise versus frequency for East-West polarized
data (blue) and North-South polarized data (orange). The gray bars mask
frequencies of persistent RFI. The difference between the two polarizations is
primarily due to the different widths of the primary beams. Confusion noise is
estimated as the median absolute deviation (MAD) of the night-time sky
brightness over the solar declination range. On average, the confusion noise
is about 4 orders of magnitude lower than the brightness of the Sun (Fig. 1).
There is a ripple with period 30 MHz due to standing waves between the
reflector and the focal line (CHIME Collaboration et al., 2022a).

###  2.3 Beamforming

The visibilities from selected baselines (after averaging over redundant
baselines) are beamformed to the direction of the Sun. Given the short
baselines and large synthesized beam, we treat the Sun as a point source,
beamforming only to its center position. The Sun is observed for the full
duration that it is above the horizon on each available day between 31 May
2019 and 11 July 2020. Beamformed visibilities are then averaged over the
selected baselines, weighted by the inverse noise variance of each visibility.
Visibilities in the stack data set have been calibrated in units of Jy/beam,
such that this average also has units of Jy/beam. This interferometric view of
the Sun results in a measurement of the average primary beam across the
instrument. Variations in the primary beam over individual feeds are not
measured, as this would require the full set of redundant visibilities, which
are not saved at all frequencies due to data volume considerations.

###  2.4 Flux Calibration

The flux of the Sun was calibrated once, when the Sun transited at the
declination of Tau A ($+22.7^{\circ}$) on 31 May 2019. At peak transit (0∘
hour angle) at this declination, the beam response for each polarization is
independently measured using CHIME observations of Tau A and the flux of Tau A
reported in Perley & Butler (2017). These beam measurements allow us to
calibrate the solar spectrum at this point in time. As discussed in Section
1.1, the average solar flux in the CHIME band varied by $\lesssim$10% between
31 May 2019 and 11 July 2020. Without simultaneous solar monitor data
available across the band, we are unable to track the true flux of the Sun to
better than this precision and so we treat the flux as constant in this
analysis. Among the potential calibrators within the solar declination range,
Tau A is chosen for its high signal to noise. Tau A is the brightest radio
source in this range and transits at an elevation where the CHIME primary beam
response is also high. As a cross-check, the flux of the Sun was calibrated
again against Tau A on 11 July 2020 and against Virgo A on 20 August 2019 and
21 April 2020. As shown in Figure 1, these measurements agree to within 5%
across the band.

###  2.5 Correcting for Quantization Bias

The dynamic range of the CHIME correlator is limited by the bit depth of the
channelized voltages, which are rounded to four real bits and four imaginary
bits prior to correlation (Bandura et al., 2016). While this bit depth is
sufficient for CHIME’s primary science objectives, it introduces a bias due to
digital compression when the Sun is in the main lobe of the primary beam. If
left uncorrected, this effect can bias the solar spectrum by as much as
$\sim$30% in amplitude, as shown in Figure 1. We account for this effect using
an approximation of the analytical corrections derived in Benkevitch et al.
(2016) and Mena-Parra et al. (2018) for complex digital correlators. For a
complete description, the reader should refer to these publications, though we
will briefly describe the process here.

Equation 21 of Benkevitch et al. (2016) provides an analytical expression for
the quantized correlation $\hat{\kappa}_{ij}$ given the true, unquantized
correlation, $\kappa_{ij}$, as well as the standard deviations of inputs $i$
and $j$. We will refer to this function as $g$.

$\hat{\kappa}_{ij}=g(\kappa_{ij},\sigma_{i},\sigma_{j})$ (1)

Similarly, Equation 8 of Mena-Parra et al. (2018) provides an expression for
the quantized standard deviation as a function, $f$, of the true standard
deviation.

$\hat{\sigma}=f(\sigma)$ (2)

The objective is to recover the true correlation, $\kappa$, when only the
quantized quanities, $\hat{\sigma}$ and $\hat{\kappa}$, are known. This is
accomplished by inverting Equations 1 and 2 as follows.

$\displaystyle\kappa_{ij}$ $\displaystyle=$ $\displaystyle
g^{-1}(\hat{\kappa}_{ij},\sigma_{i},\sigma_{j})$ (3) $\displaystyle=$
$\displaystyle
g^{-1}(\hat{\kappa}_{ij},f^{-1}(\hat{\sigma}_{i}),f^{-1}(\hat{\sigma}_{j}))$

Our analysis makes two further simplifying assumptions. The first assumption
is to ignore the phase error of $\hat{\kappa}$ and only to correct for
amplitude. At peak signal level, the phase error in a single baseline due to
compression is $\sim$10 arcmin and becomes negligible ($\lesssim$10 arcsec)
when averaged over redundant baselines. Second, because the CHIME stack data
set has been averaged over redundant baselines, we need to assume that
$\hat{\sigma}^{2}_{i}=\hat{\sigma}^{2}_{j}=\langle\hat{\sigma}^{2}\rangle$ and
$\hat{\kappa}_{ij}=\langle\hat{\kappa}\rangle$ for all inputs $i,j$ in each
unique baseline. However, since the CHIME off-line data set does include the
full correlation matrix (which has not been averaged over redundant baselines)
at four frequencies, we can compare the exact analytical correction to our
simplified version at these frequencies. We find that the error incurred by
averaging over redundant baselines is $\lesssim$1%—less than the intrinsic
flux variation of Sun.

##  3 Results

The power beam ratio is formed by dividing the measured response to the Sun in
each direction by the calibrated flux of the Sun. We normalize the beam to
unity at the transit of Cyg A ($0^{\circ}$ hour angle, $+40.7^{\circ}$
declination) to be consistent with how the data are normalized by the complex
gain calibration procedure (for details, see CHIME Collaboration et al.
2022a). As a consequence, the beam will have values greater than one in some
directions.

Figure 3: North-South (top) and East-West (bottom) polarized beam measurements
from a single solar transit on 20 August 2019, when the Sun was at the
declination of Vir A ($+12.4^{\circ}$). On large frequency scales, the beam
width is proportional to wavelength, as expected due to diffraction. There is
also a modulation in the beam amplitude and width on 30 MHz scales, due to
standing waves between the reflector and the focal line. By convention in
CHIME, the beam is normalized to be 1 at the transit of Cyg A ($0^{\circ}$
hour angle, $+40.7^{\circ}$ declination)

Each solar transit traces a line of approximately constant declination. The
transit on 20 August 2019, when the Sun was at the declination of Vir A
($+12.4^{\circ}$), is shown in Figure 3. Throughout the year, the daily change
in solar declination is less than the $\sim$0.5∘ diameter of the Sun, enabling
consecutive transits to be stitched together to achieve continuous declination
coverage. The full solar declination range was observed twice: first in late
2019 and again in early 2020. Due to planned software deployments and
telescope maintenance resulting in instrument downtime, certain declination
ranges were missed in 2019. In 2020, deployments were planned to avoid missing
the same declination ranges twice, enabling complete declination coverage in
the combined data set.

The combined data set at 600 MHz is shown in Figure 4 using an orthographic
projection with its origin at zenith. This projection has the advantage of not
distorting the apparent beam width at different elevations and the projected
coordinates $x$ and $y$ are always parallel to East and North, respectively.
The coordinates $x$ and $y$ are the projections of the unit vector pointing to
hour angle $h$ and declination $\delta$, given by

$x=-\cos{\delta}\sin{h}$ (4)

and

$y=\cos{\delta_{0}}\sin{\delta}-\sin{\delta_{0}}\cos{\delta}\cos{h}$ (5)

where $\delta_{0}$ is the latitude of the observer ($+49.3^{\circ}$ for
CHIME). The basic shape of the beam in Figure 4 can be described as a 1∘– 4∘
wide stripe along the local meridian, with moderate ($-20$ dB) side lobes
extending up to $20^{\circ}$ on either side. As shown in Figure 5, the
amplitude and width of the main lobe varies considerably ($\sim$25%) over
modest changes in frequency and elevation angle. There is also significant
structure in the side lobes of both polarizations. Though difficult to
quantitatively predict, these variations are qualitatively expected due to
multiple reflection paths within the telescope, cross talk between feeds, and
blockages in the aperture. For more in-depth discussion of these beam
features, we refer the reader to CHIME Collaboration et al. (2022a).

Figure 4: Orthographic projection of the North-South (top) and East-West
(bottom) polarized beam measurements at 600 MHz, made by combining one year of
solar transits. The modulation in beam width on small angular scales is real
and arises from multiple reflection paths within the telescope. The white
dashed line marks the path of the Sun made during the transit shown in Figure
3. The black curved line near the bottom of the plot marks the Southern
horizon. The zenith (off scale) is located at the coordinate (0,0). Figure 5:
Slices of the North-South polarized beam as a function of orthographic
coordinate $x$, emphasizing the evolution of the beam profile over frequency
and elevation angle. Beam slices at 600 MHz (618 MHz) are plotted in blue
(orange). Beam slices at $y=-0.57$ ($-0.60$) are plotted with solid (dashed)
lines. The variations in beam amplitude and width arise due to interference
between multiple reflection paths within the telescope. Rays with multiple
reflections have broader illumination compared to the primary ray, resulting
in narrower beam widths at frequencies and angles corresponding to
constructive interference (solid blue and dashed orange lines).

Where repeat observations are available, the two measurements are averaged and
their difference is used to assess measurement repeatability (Figure 6). In
regions where the response to the Sun is much greater than the confusion
noise, we expect the difference between measurements to be dominated by
variations in the flux of the Sun. Comparing Figures 4 and 6, where the beam
response is greater than $10^{-3}$ ($\gtrsim$10$\times$ the confusion limit),
the fractional difference is found to be 9% at 600 MHz, which is comparable
with our estimate of solar variability over the observation period. Where the
beam response is less than $10^{-3}$, the measurement becomes increasingly
confusion-noise limited, causing the fractional error to increase.

Figure 6: The fractional difference between measurements made in late 2019 and
early 2020, shown for the North-South (top) and East-West (bottom) polarized
beam at 600 MHz. White space corresponds to where only a single measurement
exists. Where the beam response in Figure 4 is greater than $10^{-3}$ (cyan
contour), the measurement uncertainty is dominated by intrinsic solar
variability. Below this level, confusion noise is comparable to or larger than
the effect of solar variability. For both polarizations, the fractional
differences within the contour approximately follow a normal distribution with
$\sigma=9\%$.

###  3.1 Comparison With Other Beam Measurements

The solar beam measurements are only part of the broader CHIME beam
measurement program. Although the Sun probes about 7,200 square degrees,
additional measurements are needed to span the full sky. Where spatial overlap
exists between measurements, we can make comparisons. The other measurements
all rely on astrophysical point source observations and thus have
complementary systematics to the solar measurement. In general, the fluxes of
these sources are more stable than the Sun (though they may have large
uncertainties) and are small enough to not require correction for quantization
bias. Additionally, point source observations can be made with a complementary
set of baselines, since these sources are unresolved by CHIME’s longest
baselines.

The angular coverage of the different beam measurements is summarized in
Figure 7. There are two categories of measurements which rely on astrophysical
point sources: source tracks and deconvolutions. The former are made by
beamforming visibilities to the location of bright sources as they transit
through the primary beam, analogously to how the solar measurement is made.
Beamforming may be done with CHIME-only baselines, or with baselines between
CHIME and the John A. Galt 26 m tracking telescope (Locke et al., 1965) using
the holographic technique described in Berger et al. (2016). Holographic
measurements have the added benefit of measuring the beam phase in addition to
amplitude. The second type of point source beam measurement is made by
deconvolving a model of the point source sky from a CHIME map as described in
CHIME Collaboration et al. (2022b). Unlike the source tracks, which probe
discrete declinations over a moderately wide range in hour angle, the
deconvolved beam has continuous declination coverage, but is limited to the
width of the main lobe in the East-West direction.

Figure 7: Orthographic projection of multiple North-South polarized beam
measurements at 600 MHz. Despite spanning more than 7,200 square degrees, the
solar measurement (cyan outline) only probes the main lobe between $y=-0.44$
and $y=-0.95$. In contrast, the beam measurement derived by deconvolving a
point source model of the sky (magenta outline) spans the full elevation range
of the main lobe, but does not probe the side lobes. Holography tracks of the
brightest point sources (yellow outline) probe the beam at discrete elevations
over a wide range in hour angle. The black circle marks the horizon.

In Figure 8, we show the difference between the solar measurement at the
declination of Vir A ($+12.4^{\circ}$) and source tracks of Vir A. Note that,
unlike Tau A, Vir A was not explicitly used to calibrate the flux of the Sun
and thus provides a truly independent measurement of the beam. While the
CHIME-only observation of Vir A provides slightly better agreement with the
solar measurement than does the holographic measurement, both comparisons
reveal systematic differences which are small—typically a few percent of the
peak beam response. Residuals are dominated by systematic differences in the
East-West width of main lobe. For the North-South polarized beams shown in
Figure 8, the solar beam is on average 2% wider than the CHIME-only
measurement of Vir-A, and 6% wider than the holographic measurement.

In Figure 9 we plot the North-South polarized beam at $x=0$ as a function of
$y$ and frequency, as well as the difference between this beam and the
deconvolved point source beam. Currently, the deconvolved point source beam
has only been evaluated at frequencies greater than 585 MHz. Again, the level
of agreement is typically a few percent of the peak response, though there are
some large excursions around 600 MHz. We also find that the solar beam
measurement is systematically wider than the deconvolved point source beam by
$\sim$5% in both polarizations. These differences in width are larger than can
be explained by the angular extent of the Sun and efforts to understand the
discrepancies between measurements are ongoing.

Figure 8: The difference between the solar measurement of the North-South
polarized beam at the declination of Vir A ($+12.4^{\circ}$) and a CHIME-only
measurement of Vir A (top), and a holographic measurement of Vir A (bottom).
Systematic differences between beam measurements are typically less than a few
percent of the peak beam response, and are primarily due to percent-level
differences in measured beam width. Systematic differences between the East-
West polarized beam measurements (not shown) are approximately a factor of two
larger.

Figure 9: The solar measurement of the North-South polarized beam as a
function of orthographic coordinate $y$ and frequency (top), and the
difference compared the corresponding measurement made by deconvolving a point
source model of the sky from a long-baseline CHIME map (bottom). The white
dashed line in the top panel marks the position of Sun during the transit
shown in Figure 3. For both the North-South and East-West polarized beams
(latter not shown), the differences are approximately normally distributed
with $\sigma=0.04$ in Cyg A normalized units.

##  4 Discussion

In this section we will discuss the utility of this beam measurement in the
context of CHIME science. As its name suggests, one of CHIME’s primary science
objectives is to map the distribution of neutral Hydrogen in the low-redshift
($z\sim 1$) Universe. It is difficult to generalize the beam knowledge
requirements to meet this objective, since any statement on beam accuracy
depends on both the details of the analysis and the form of the beam errors.
To give one example, the analysis presented in Shaw et al. (2015) using
$m$-mode formalism and a Karhunen–Loève (KL) transform for foreground removal
concluded that the beam width had to be known to an accuracy of $10^{-3}$ in
order to avoid foreground contamination of the 21 cm power spectrum. For
comparison, we found in Section 3.1 that the East-West width of the solar beam
measurements were discrepant with other measurements at the level of few
percent, or about one and a half orders of magnitude worse than the desired
accuracy for this particular analysis.

However, there are reasons for optimism. First, the modeled beam width errors
in Shaw et al. (2015) were drawn from a random distribution, resulting in
power leakage from low-$k$, foreground-dominated modes to higher $k$ signal
modes. In contrast, the errors in this measurement are highly correlated in
frequency, affecting the high-$k$ modes to a lesser degree. Second, the use of
alternative foreground removal techniques, instead of or in conjunction with
the KL transform, may relax requirements on instrumental knowledge. The degree
to which both of these factors impact the ultimate analysis is a subject of
ongoing investigation.

Nevertheless, these solar beam measurements bring us significantly closer to
achieving CHIME’s cosmology goals and, in combination with other beam
measurements, have enabled the first 21 cm detection with CHIME (CHIME
Collaboration et al., 2022b). These measurements can also be used to aid in
interpolating between the comparatively sparse point source beam measurements
that are outside the solar declination range, bringing us closer to a full
$2\pi$ description of the beam.

As shown in Figure 7, the solar measurement is the only data set to provide
significant information about the 2D spatial structure of the primary beam.
While the structure appears rather complicated, in fact it is highly separable
in $x$ and $y$ and its variance is dominated by a few separable spatial modes
at each frequency. This relative simplicity is being used to guide us in how
to interpolate between point source measurements over the rest of the sky. One
approach we are currently exploring is to derive a set of basis functions in
$x$ from the solar data, which are then fit to the point source data near
meridian, at each $y$ and frequency. These efforts show promising early
results.

In addition to such data-driven analyses, the solar data can also be used to
improve our physical understanding of the telescope. As mentioned in Section
3, the complicated spatial structure of the beam is qualitatively expected due
to multi-path and receiver cross-talk effects within the telescope. We have
been developing a “coupling model” similar to that described by Kern et al.
(2019) that may be fit to our full set of beam measurements. The solar data
play a key role in constraining the off-meridian properties of this model,
which account for 10-20% of the CHIME’s total primary beam solid angle.

These solar beam measurements are also applicable to CHIME observations of
fast radio bursts and pulsars, which have considerably less stringent
requirements on beam uncertainty compared to the cosmological measurement. The
beam models used in CHIME/FRB Collaboration et al. (2021) and Good et al.
(2021) were each informed in part by these data. Also, in combination with a
holographic measurement of Tau A, these data were used to calibrate the
spectral flux density of the flare observed from Galactic magnetar SGR
1935+2154 reported in CHIME/FRB Collaboration et al. (2020).

##  5 Conclusions

We have presented a measurement of the CHIME primary beam pattern using the
Sun as a calibration source. This data set is an integral part of the broader
CHIME beam measurement program, complementing more conventional beam
measurements using astrophysical point sources. This data set provides a
direct measurement of the CHIME beam at low declination and is indeed the only
measurement over much of the sky. Where overlap exists with other beam
measurements, comparisons can be made to constrain the systematics of
different measurement techniques. The rich spatial structure revealed by this
measurement has enormous qualitative and quantitative value for understanding
the source of beam features and may enable extrapolating beyond the current
measurement. Efforts to combine beam measurements will be the topic of future
publications.

The semiannual motion of the Sun enables measuring a large solid angle without
the need to move the telescope, which is particularly useful for driftscan
instruments. Despite the intrinsic variability of the Sun, the uncertainty on
the resulting beam measurement is estimated to be $<10\%$ where the response
is above the confusion limit, which is $\sim 40$ dB below the peak beam
response. While CHIME was able to take advantage of the solar minimum period
for these observations, future instruments wishing to use the Sun for beam
measurements may find themselves poorly situated with respect to the 11-year
solar cycle. However, solar beam measurements may still be viable in times of
increased solar activity. Whereas this analysis made no attempt to correct for
variations in the solar flux, even simple adjustments based on independently
available solar monitor data may be sufficient to account for the slowly
varying component of solar emission.

We thank the Dominion Radio Astrophysical Observatory, operated by the
National Research Council Canada, for leasing the CHIME site and for their
gracious hospitality and support. DRAO is situated on the traditional,
ancestral, and unceded territory of the Syilx Okanagan people. We are
fortunate to live and work on these lands. CHIME is funded by grants from the
Canada Foundation for Innovation (CFI) 2012 Leading Edge Fund (Project 31170),
the CFI 2015 Innovation Fund (Project 33213), and by contributions from the
provinces of British Columbia, Québec, and Ontario. Long-term data storage and
computational support for analysis is provided by
WestGrid222https://www.westgrid.ca and Compute
Canada333https://www.computecanada.ca. Additional support was provided by the
University of British Columbia, McGill University, and the University of
Toronto. CHIME also benefits from NSERC Discovery Grants to several
researchers, funding from the Canadian Institute for Advanced Research
(CIFAR), and from the Dunlap Institute for Astronomy and Astrophysics at the
University of Toronto, which is funded through an endowment established by the
David Dunlap family. This material is partly based on work supported by the
NSF through grants (2008031) (2006911) and (2006548) and by the Perimeter
Institute for Theoretical Physics, which in turn is supported by the
Government of Canada through Industry Canada and by the Province of Ontario
through the Ministry of Research and Innovation.

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