# A New Flaring Black Widow Candidate and Demographics of Black Widow
Millisecond Pulsars in the Galactic Field

Samuel J. Swihart  National Research Council Research Associate, National
Academy of Sciences, Washington, DC 20001, USA,
resident at U.S. Naval Research Laboratory, Washington, DC 20375, USA  Jay
Strader  Center for Data Intensive and Time Domain Astronomy, Department of
Physics and Astronomy,
Michigan State University, East Lansing, MI 48824, USA  Laura Chomiuk  Center
for Data Intensive and Time Domain Astronomy, Department of Physics and
Astronomy,
Michigan State University, East Lansing, MI 48824, USA  Elias Aydi  Center for
Data Intensive and Time Domain Astronomy, Department of Physics and Astronomy,
Michigan State University, East Lansing, MI 48824, USA  Kirill V. Sokolovsky
Center for Data Intensive and Time Domain Astronomy, Department of Physics and
Astronomy,
Michigan State University, East Lansing, MI 48824, USA  Sternberg Astronomical
Institute, Moscow State University, Universitetskii pr. 13, 119992 Moscow,
Russia  Paul S. Ray  Space Science Division, U.S. Naval Research Laboratory,
Washington, DC 20375, USA  Matthew Kerr  Space Science Division, U.S. Naval
Research Laboratory, Washington, DC 20375, USA

###### Abstract

We present the discovery of a new optical/X-ray source likely associated with
the _Fermi_ $\gamma$-ray source 4FGL J1408.6–2917. Its high-amplitude periodic
optical variability, large spectroscopic radial velocity semi-amplitude,
evidence for optical emission lines and flaring, and X-ray properties together
imply the source is probably a new black widow millisecond pulsar binary. We
compile the properties of the 41 confirmed and suspected field black widows,
finding a median secondary mass of $0.027\pm 0.003\,M_{\odot}$. Considered
jointly with the more massive redback millisecond pulsar binaries, we find
that the “spider” companion mass distribution remains strongly bimodal, with
essentially zero systems having companion masses between $\sim
0.07-0.1\,M_{\odot}$. X-ray emission from black widows is typically softer and
less luminous than in redbacks, consistent with less efficient particle
acceleration in the intrabinary shock in black widows, excepting a few systems
that appear to have more efficient “redback-like” shocks. Together black
widows and redbacks dominate the census of the fastest-spinning field
millisecond pulsars in binaries with known companion types, making up
$\gtrsim$80% of systems with $P_{\rm{spin}}<2\,\rm{ms}$. Similar to redbacks,
the neutron star masses in black widows appear on average significantly larger
than the canonical $1.4\,M_{\odot}$, and many of the highest-mass neutron
stars claimed to date are black widows with $M_{\rm{NS}}\gtrsim
2.1\,M_{\odot}$. Both of these observations are consistent with an
evolutionary picture where spider millisecond pulsars emerge from short
orbital period progenitors that had a lengthy period of mass transfer
initiated while the companion was on the main sequence, leading to fast spins
and high masses.

##  1 Introduction

In the standard “recycling” scenario for forming millisecond pulsars (MSPs), a
rotating neutron star is spun up to rapid spin periods through accretion from
a binary companion (Alpar et al., 1982). Once mass transfer ends, the
resulting binary system consists of a MSP with a low-mass white dwarf
companion in a relatively wide ($P\gtrsim 2$ d) orbit (Tauris & van den
Heuvel, 2006). Prior to the launch of the _Fermi_ -LAT satellite in 2008,
nearly all of the known MSP binaries in the Galactic field (i.e., outside of
globular clusters) were in these end-stage systems.

Since 2008, dozens of field MSPs with short-period ($P\lesssim 1$ d), low
mass, non-degenerate companions have been discovered, many by conducting
multiwavelength follow-up of _Fermi_ $\gamma$-ray sources. These compact
binaries are typically referred to as “spiders” due to the evaporative effects
the high-energy pulsar wind has on the companion, and are further subdivided
into either “black widows” or “redbacks” depending on whether the companion is
less or more massive than $\sim 0.1\,M_{\odot}$ (Roberts, 2013).

In these spider binaries, the radio pulsar is often obscured or eclipsed due
to ionized material being blown from the companion by the relativistic pulsar
wind, making it difficult to find pulsations using typical radio search
techniques, especially at binary phases when the companion lies between Earth
and the MSP (e.g., Camilo et al., 2016; Cromartie et al., 2016; Deneva et al.,
2016; Corongiu et al., 2021, Swihart et al. in prep.). However, nearly all
MSPs appear to be $\gamma$-ray emitters (Abdo et al., 2013), and since these
high-energy photons flow through the diffuse companion material unimpeded,
multiwavelength follow-up of unidentified _Fermi_ sources continues to be a
successful way to discover and characterize these binaries (e.g., Nieder et
al., 2020; Li et al., 2021; Swihart et al., 2021; Ray et al., 2022; Swihart et
al., 2022).

Optical and near-IR light curves of these spider systems show characteristic
variability modulated on the orbital period of the binary. For most redbacks,
this modulation has two peaks per orbital cycle (i.e., ellipsoidal variations)
corresponding to when we are the viewing the maximum surface area of the
tidally distorted companion as it orbits the MSP (e.g., Salvetti et al., 2015;
Bellm et al., 2016; Li et al., 2016). In the black widows (and some redbacks),
irradiation from the energetic pulsar wind dominates over tidal distortion
effects, causing one face of the tidally locked companion to be heated to a
much higher temperature than the side facing away from the MSP (e.g., Breton
et al., 2013; Romani et al., 2016; Linares et al., 2018a; Draghis et al.,
2019; Swihart et al., 2019). In these cases only one broad peak in the light
curve is observed per orbit when viewing this heated face at companion
superior conjunction, along with one minimum when we see the unheated
“nightside” of the companion as it crosses between Earth and the MSP.

The tidal synchronization timescale for typical spider MSPs is short
($\lesssim$ few Myr), so the orbital period equals the rotation period of the
secondary. This rapid rotation along with irradiative heating of the
photosphere from the pulsar wind can drive strong winds from the secondary
(e.g., Morin, 2012; Romani et al., 2015). At the intersection between the
stellar wind and the pulsar wind, an intrabinary shock can form that emits
nonthermal X-rays, often modulated on the orbital period of the binary (e.g.,
Bogdanov et al., 2014; Romani & Sanchez, 2016; Wadiasingh et al., 2017; Al
Noori et al., 2018).

Balmer and Helium optical emission lines can also be attributed to the stellar
winds directly, or in combination with the intrabinary shock that serves as a
natural location for a region with high enough temperature to produce
H$\alpha$ and He I photons (Romani et al., 2015; Halpern et al., 2017; Swihart
et al., 2018; Strader et al., 2019). The combination of all these effects can
cause significant flaring and variability on rapid timescales, both in the
overall brightness of the system and in the strength and location of emission
lines (e.g., Cho et al., 2018; Halpern, 2022).

In addition to the significant radio eclipses seen in the spider binaries,
these systems also show other differences from the traditional population of
MSPs, including higher neutron star masses and faster spins (Strader et al.,
2019). Three of the most massive neutron stars known to date that have
relatively well-constrained neutron star mass measurements are in spider
binaries, each with neutron star mass estimates that exceed $2\,M_{\odot}$
(Kandel & Romani, 2020a; Romani et al., 2021, 2022a). We note however that the
component mass estimates in many spider MSPs often rely on binary inclination
measurements derived from fitting irradiation-dominated light curves, which
are difficult to model as described in the following sections.

In this paper, we present the discovery of a new compact binary associated
with the _Fermi_ source 4FGL J1408.6–2917 as part of our ongoing program
following up X-ray/optical matches within unassociated _Fermi_ regions. The
X-ray properties and variable optical light and radial velocity curves
strongly suggest this is a new black widow MSP binary. We introduce this
source as part of a compilation of the confirmed and candidate field black
widows. This census of black widow MSPs, totalling 41 systems as of mid-2022,
is compared to the closely related redbacks, showing strong bimodality in some
of their observed and intrinsic multiwavelength properties, despite (possibly)
sharing a similar evolutionary path.

Figure 1: Left: Optical Digital Sky Survey image of the field showing the 95%
error ellipses from the 3FGL (blue) and 4FGL (red) catalogs corresponding to
the $\gamma$-ray source 4FGL J1408.6–2917. The relative position of the two
X-ray sources in the 4FGL region (see Sec. 2.2) are marked with magenta
circles. The likely black widow counterpart to the _Fermi_ source is labeled
as J1408. Right: Gemini-South/GMOS $i^{\prime}$ image zoomed in on the
position of J1408. The _Swift_ and _XMM_ X-ray positions and their 90%
confidence regions are shown with blue and magenta circles, respectively. The
variable optical source discussed in Sec. 3.3 is spatially coincident with the
X-rays and is the likely companion in a black widow MSP binary.

##  2 Observations/Data

###  2.1 $\gamma$-rays

The gamma-ray source associated with a new MSP binary candidate presented here
for the first time is listed in the third incremental data release of the
fourth full catalog of _Fermi_ -LAT sources (4FGL-DR3; Fermi-LAT collaboration
et al., 2022a) as 4FGL J1408.6–2917. Based on twelve years of survey data, the
source has an overall detection significance of 11.3$\sigma$ in the 0.1–100
GeV energy range, and has been detected in the previous 1FGL, 2FGL, and 3FGL
catalogs. The 4FGL 95% confidence error region lies entirely within the 3FGL
region and is $\sim$80% smaller in area (Figure 1). The source shows no
significant variability over the lifetime of _Fermi_ , consistent with most
other MSPs. The $\gamma$-ray spectrum shows marginal evidence for curvature;
while some confirmed MSPs show strongly curved GeV spectra, others show a
similar lack of strong spectral curvature (e.g., Strader et al., 2015).

###  2.2 X-rays

####  2.2.1 Swift

4FGL J1408.6–2917 was targeted with _Swift_ /XRT on seven epochs between 2019
Jun to Sep. In an automated analysis of these data (Stroh & Falcone, 2013),
one X-ray source is present within the 4FGL ellipse and this source also
matches to a faint ($G=20.2$) _Gaia_ source, which we later determined was a
high-amplitude optical variable (Sec. 2.3), motivating additional follow-up.
For the remainder of the paper, we refer to this optical/X-ray source as
J1408.

We also obtained a $\sim$1.4 ksec _Swift_ ToO observation of the region on
2021 Feb 25. Including the data described above, the total on-source XRT
exposure time used in the following analysis is 8.0 ksec.

We used the online _Swift_ -XRT analysis
tools111https://www.swift.ac.uk/user_objects/ (Evans et al., 2020) to analyze
these data, finding J1408 with a ICRS position (R.A., Dec.) of (14:08:26.77,
–29:22:22.7) and a 90% positional uncertainty of 3.9″ (Figure 1). With a net
count rate of only $\sim 3\times 10^{\rm{-3}}$ ct s-1, there were not enough
source counts to assess variability with these data.

This analysis also revealed a second _Swift_ source, closer to the center of
the 4FGL region, but fainter than J1408, with only $\sim$ 8 net counts, at a
(R.A., Dec.) position of (14:08:35.40, –29:18:57.4) and a 90% positional
uncertainty of 6.4″. We classify this source as a distant active galactic
nucleus (AGN) unrelated to the 4FGL $\gamma$-ray source, and defer additional
discussion of this source to Appendix A.

####  2.2.2 XMM

We obtained a ToO observation centered on the 4FGL J1408.6–2917 region with
_XMM-Newton_ /EPIC on 2021 Jul 23. We reprocessed these data using standard
tasks in the Science Analysis System (SAS, v20.0.0).

We found no evidence of strong, extended background flares, resulting in a
total on-source exposure time of $\sim$24 ksec. Standard flagging and
selection criteria were used for the pn and MOS cameras, respectively, as
described in the online analysis threads222https://www.cosmos.esa.int/web/xmm-
newton/sas-threads.

For the spectral analysis, we extracted individual spectra from each EPIC
camera using a circular source extraction radius of 25″ and local background
regions three times larger. Individual pn, MOS1, and MOS2 spectra were then
combined using epicspeccombine333https://www.cosmos.esa.int/web/xmm-
newton/sas-thread-epic-merging. We grouped the final combined spectrum into a
minimum of 30 counts per bin and fit the resulting background-subtracted
spectrum using XSPEC v12.12.1 (Arnaud, 1996). The best position of the _XMM_
source associated with J1408 is overlaid onto an optical image in the right
panel of Figure 1.

As part of compiling the multiwavelength properties of the field black widows,
we found three confirmed systems with archival _XMM_ data from observations
taken in 2016 (PI: M. Roberts). These data were reduced in the same manner as
described above and represent the first X-ray analyses of these systems. The
results are presented in Section 4.

###  2.3 Optical Counterpart

There is one optical source matching the _Swift_ and _XMM_ X-ray positions of
J1408, and it is listed in _Gaia_ DR3 (Gaia Collaboration et al., 2022) with
brightness $G=20.21\pm 0.06$ mag and a ICRS position of (R.A., Dec.) =
(14:08:26.789, –29:22:21.21). No parallax or proper motion information is
available in the current _Gaia_ data release. This source is also listed in
Pan-STARRS DR2 with a brightness of $i^{\prime}=20.87\pm 0.04$. No other
optical sources are present within 4.4″ of J1408 down to a limiting magnitude
of $i^{\prime}\sim 22.5$. This is confirmed with our deep Gemini exposures
(Sec. 2.3.2).

The _Gaia_ photometric uncertainty is large for an isolated star at this
brightness, implying a variable source (Andrew et al., 2021; Mowlavi et al.,
2021). The association between this variable optical source and the X-ray
source are confirmed by our spectroscopy and photometry presented below.

####  2.3.1 SOAR Imaging and Spectroscopy

We obtained spectroscopy of J1408 with the red camera of the Goodman
Spectrograph (Clemens et al., 2004) on the SOAR telescope over several epochs
from 2021 Feb 18 to 2021 Aug 6, with 11 usable spectra obtained over 7
different nights. In all cases we used a 400 l mm-1 grating with wavelength
coverage from $\sim 3900$ to $7850~{}\AA$. Depending on the seeing, we used
either a 0.95″ or 1.2″ longslit, giving full-width at half-maximum (FWHM)
spectral resolutions of $\sim 5.8$ Å or 7.3 Å, respectively. Exposure times
per spectrum ranged from 1200 s to 1800 s. Due to the low resolution of the
spectra, smearing due to the motion of the star during the exposure has only a
marginal effect on the final data. All spectra were reduced and optimally
extracted in the standard manner using IRAF (Tody, 1986).

We also performed imaging observations with SOAR/Goodman during three nights
on 2021 Feb 19, Mar 1, and Mar 3 centered on the position of J1408. On each
night we took a series of exposures using the SDSS $i^{\prime}$ filter. On Feb
19, exposures were 300 sec in length, while on the March nights the exposures
were 400 sec. Typical seeing was 1.1”, 1.2”, and 1.4” on Feb 19, Mar 1, and
Mar 3, respectively.

Raw images were reduced with combinations of bias and flat fields in the usual
manner using IRAF. For each nightly dataset, we performed differential
aperture photometry and calibrated the instrumental magnitudes of the target
using the Pan-STAARS DR2 $i^{\prime}$ magnitudes of 22 nearby, non-variable
comparison stars (see also, Swihart et al., 2020).

The most obvious feature of these light curves is the rapid brightening of the
source by $>$2 mag over the course of $\sim$1 hour. The source was too faint
to be detected in a subset of these images, presumably when we are viewing the
“nightside” of the companion. We removed images where the source was
undetected at $\gtrsim$1.8 sigma above the background, giving uninterrupted
intervals of 0.87 hr, 1.62 hr, and 1.83 hr on Feb 19, Mar 1, and Mar 3,
respectively.

The final sample of SOAR photometry consists of 11, 15, and 17 measurements on
these nights. The brightness peaks at $i^{\prime}\sim 20.35$, and is
undetectable in these images around $i^{\prime}\sim 22.5$.

####  2.3.2 Gemini/GMOS Imaging and Spectroscopy

Since J1408 was too faint to measure at high signal-to-noise throughout its
full orbit with SOAR, we obtained a 4.5-hr series of photometric observations
using Gemini-South/GMOS as part of the Gemini Fast Turnaround Program (ID:
GS-2021A-FT-112). These observations were performed on 2021 Jun 06 using the
$i^{\prime}$ filter with a frame time of 220 sec. The seeing varied throughout
the night but was typically good, around $\sim 0.7\arcsec$. We performed
standard GMOS-imaging data reduction routines using DRAGONS v2.1.1 (Labrie et
al., 2019). The light curve was extracted using IRAF in a similar manner to
the SOAR data, and calibrated using the same 22 nearby, non-variable
comparison stars.

J1408 was significantly detected in all images. The source shows a clear
periodic modulation of $\sim$3.42 hrs, rising to a maximum brightness of
$i^{\prime}\sim 20.5$ and dimming to a minimum of $i^{\prime}\sim 22.9$. The
final Gemini photometric sample consists of 66 measurements.

We show the Gemini and SOAR data folded on the best-fit period ($P\sim 3.42$
hr; see Sec. 3) in Figure 2, where we have set the phase of maximum brightness
to $\phi=0.75$, consistent with the radial velocity peaking at $\phi=0.5$
(i.e., pulsar phase convention). Overall, the amplitude and period of this
variability are fully consistent with a black widow MSP that is being heated
substantially on one side facing the pulsar.

We also obtained 7 additional optical spectra of J1408 with Gemini-South/GMOS
(ID: GS-2022A-FT-202) on 2022 Apr 11 over a time range of about 3.15 hr, which
is nearly a full orbital cycle (3.42 hr). These data used a 1″ slit with the
R400 grating and the GG455 order-blocking filter centered at 6800 Å, with per
spectrum exposure times of 25 min. These spectra have a mean FWHM resolution
of about 7.2 Å and cover a nominal wavelength range of $\sim 4500$–9150 Å,
though the signal-to-noise toward the blue is very poor, and the data in the
wavelength range $\sim 7160$–7660 Å is unusable. As for the SOAR/Goodman data,
we reduced and optimally extracted these spectra using IRAF.

###  2.4 Radio Pulsar Search Data

We obtained a short series of pulsar search observations with the Robert C.
Byrd Green Bank Telescope on 2021 Jun 14 in an effort to detect the suspected
radio pulsar (Project ID: GBT21A-428). The first pointing lasted $\sim$30 min
using the Prime Focus 820 MHz receiver, immediately followed by a $\sim$35 min
pointing in S-band centered at 2.2 GHz.

##  3 Results for J1408: A Flaring Black Widow

###  3.1 X-ray Flux and Spectrum

####  3.1.1 Swift

To fit the _Swift_ X-ray spectrum of J1408, we used XSPEC (v12.12.1) to fit a
simple absorbed power-law model and assumed Wilms et al. (2000) abundances.
The neutral hydrogen column density was held fixed to the Galactic value of
$3.4\times 10^{20}$ cm-2 (HI4PI Collaboration et al., 2016). The best-fit
photon index is $\Gamma=2.4^{+1.3}_{-0.9}$ (here and throughout the paper,
uncertainties on the X-ray properties are quoted at the 90% confidence level),
and the unabsorbed 0.5–10 keV X-ray flux is $F_{X}=(8^{+56}_{-5})\times
10^{-14}$ erg s-1 cm-2 .

####  3.1.2 XMM

The deeper _XMM_ dataset allows for a much more precise determination of the
X-ray spectrum, with a EPIC count rate of $0.010\pm 0.002$ ct s-1 and a total
of $\sim$244 net counts. We fit a similar power-law model as with the _Swift_
data (TBabs*powerlaw). Leaving $N_{H}$ free in the fit resulted in very low
values, so we held $N_{H}$ fixed to the Galactic value. The best-fitting
photon index from these data is $\Gamma=1.7^{+0.6}_{-0.5}$ with an unabsorbed
0.5–10 keV flux of $(2.4^{+1.6}_{-1.1})\times 10^{-14}$ erg s-1 cm-2. Although
this median flux value is $\sim$70% less than the flux from the _Swift_ data,
the two values are consistent within their uncertainties. The _XMM_ data
provides $\gtrsim 10\times$ more counts than the _Swift_ dataset so we assume
the results from the _XMM_ analysis for the remainder of the paper.

In an attempt to constrain the orbital variability, we divided the 0.2–10 keV
background-subtracted light curve into four time bins, each spanning about 6
ksec. Within the large uncertainties, there is no evidence for significant
variability, though we note that even factor of $\sim 2$ variability could not
be detected at high significance given the low count rate.

Some black widow MSPs show dramatic orbital variability associated with the
intrabinary shock (e.g., Huang et al., 2012), with the largest effects
occurring in systems that are more edge-on (e.g., Romani & Sanchez, 2016;
Wadiasingh et al., 2017). A more face-on orbit would weaken the orbital
effects of the intrabinary shock emission, so the fact that we see little
evidence for X-ray variability may be indicative of a relatively low orbital
inclination angle. A somewhat face-on orbit is supported by both the light
curve models and our spectroscopy, so deeper X-ray observations to detect
variability at the low levels expected here are well-motivated.

###  3.2 Pulsar Search

For the Green Bank Telescope pulsar search data obtained on 2021 Jun 14, we
excised RFI and searched the data for periodic signals using typical
procedures in PRESTO v3.0.1 (Ransom, 2011). The short orbital period of the
system relative to our observation duration may cause “smearing” of the pulsar
signal in the Fourier domain. We therefore implemented the “jerk” search
feature in PRESTO (i.e., -wmax) to improve the search sensitivity (Andersen &
Ransom, 2018). We searched both the full datasets as well as $\sim$15 min
subsets of each pointing with accelsearch up to -zmax 200 and with and without
-wmax 600.

These searches did not produce any clear detections of a radio pulsar. Using
the orbital period and ephemerides presented below, the 820 MHz observation
occurred during the orbital phase range $0.69<\phi<0.81$, and $0.88<\phi<1.03$
for the S-band pointing. These orbital phases (especially during the 820 MHz
observation) should be favorable for minimizing absorption of the radio pulsar
by ionized material associated with the likely black widow companion, though
we note that these represent a single search epoch.

Pulsar non-detections have also been found for some other likely spider MSPs:
systems which have strong optical and/or X-ray evidence for the presence of a
pulsar, but in which no radio pulsar has been detected despite numerous
efforts (e.g., Swihart et al., 2020; Corbet et al., 2022; Halpern, 2022). One
possibility is that even at higher frequencies, in some spider MSPs the
material lost from the companion may eclipse the pulsar to an even greater
degree than the 5–15% of a typical spider (e.g., Ryba & Taylor 1991; Stappers
et al. 1996a; Polzin et al. 2020). Another possibility is that in a subset of
systems, the radio pulsar beams do not sweep over our line of sight. Given the
strong evidence that J1408 is indeed a black widow, additional pulsar search
observations would be valuable.

###  3.3 Optical Light Curve

The uninterrupted Gemini photometry shows a strong periodic signal at $P\sim
3.42$ hr. When combined with the SOAR data, the period that best agrees with
the full photometric dataset is $P=0.14261385\pm 0.0000015$ d, which we take
as the best period. When folded on this period (Figure 2), the photometry
shows a single, broad, large-amplitude peak, consistent with the light curves
of other black widow binaries. In this context, the broad peak corresponds to
when we are viewing the hot irradiated face of the companion that is being
heated substantially by the pulsar on its tidally locked “dayside,” whereas
the narrower minimum corresponds to when we are viewing the much cooler
“nightside” of the companion when it lies between Earth and the suspected
neutron star primary on its orbit (i.e., companion inferior conjunction).

Figure 2: SOAR and Gemini $i^{\prime}$-band photometry of the likely black
widow companion to 4FGL J1408.6–2917. The data are folded on the best fit
period, $P\sim 3.42$ hr. The ephemeris is set such that $\phi=0.75$ at maximum
brightness, the presumed superior conjunction of the heated companion. While
the Gemini data are relatively smooth and well-behaved, the SOAR data shows
evidence of flares that vary from night to night, especially after $\phi\sim
0.65$ (Sec. 3.3). Two representative light curve models are shown (green and
blue lines/text) that only differ significantly in the orbital phase range
$\phi\sim 0.2-0.35$ (see Sec. 3.3.1).

Overall, the Gemini light curve is rather smooth, showing no evidence for
flaring. This is in contrast to the SOAR data, which shows what appears to be
minor flares as the source rises to peak brightness. This is especially
apparent in the 2021 Mar 3 data where the final two points are significantly
brighter ($\gtrsim$ 0.1 mag) than the preceding measurements at a time when
the light curve is expected to have turned over towards fainter values. The
2021 Mar 3 photometry also seems to be slightly brighter overall than the
other datasets throughout the orbit. This is also apparent during the end of
the 2021 Mar 1 epoch, where the photometry agrees well with the Gemini data
until $\phi\sim 0.65$ where the source rises quickly to be $\sim$0.1–0.15 mag
brighter than the Gemini data at corresponding phases. A similar effect is
observed at the end of the 2021 Feb 19 SOAR epoch starting just after
$\phi\sim 0.5$.

The observing conditions were similar during each SOAR night, and we used the
exact same set of comparison stars in our analysis, so we conclude the
variations we see in the photometry are real. These flaring optical light
curves are not uncommon in spider binaries, with similar phenomenology seen in
the black widow candidate 4FGL J0935.3+0901 (Halpern, 2022) and in a few
confirmed and candidate redbacks (Cho et al., 2018; Halpern et al., 2022).

The exact mechanism causing these flares in spider systems is still unclear,
but is likely caused by a combination of variable intrabinary shock emission
and anisotropic heating due to ducting of the pulsar wind or shock emission
along the companion magnetic field lines (e.g., Romani et al., 2015; Sanchez &
Romani, 2017). In this context, rapid structural variations in the heating
mechanism or a variable companion magnetic field can provide natural
explanations for fast flaring activity on timescales of days to weeks (Cho et
al., 2018). If the heating occurs high enough in the companion’s atmosphere
that it creates a thin layer of ionised material in the chromosphere, these
effects may also contribute to the rapid appearance and disappearance of H and
He emission lines in the spectra.

These flaring effects can operate simultaneously with other processes that
give rise to slower variations in the continuum emission from the companion,
which can cause large heating asymmetries in the light curves, such as
hot/cold spots on the stellar surface, or diffusion and convection within the
photosphere due to the large temperature asymmetry on the companion surface
(e.g., Swihart et al., 2019; Kandel & Romani, 2020a; Voisin et al., 2020a).

We also observe flaring in the optical spectra, which we describe in detail in
Sec. 3.4.

####  3.3.1 Light Curve Modeling

We used the Eclipsing Light Curve code (ELC; Orosz & Hauschildt, 2000) to
provide an initial model of the optical light curve of the system. While this
model includes the effect of direct heating of the companion, it does not
include other potentially relevant physical effects such as zonal heat flow
that have been shown to improve light-curve fits for spider companions (e.g.,
Kandel & Romani 2020a). At present we only have a rather limited, single-band
light curve for J1408, so more sophisticated modeling is not yet justified.

The SOAR photometry has incomplete phase coverage, large uncertainties when
$i^{\prime}\gtrsim 21.5$, and suffers from irregular flaring. We therefore
only model the Gemini light curve, which covers the full orbit, is more
precise, and shows no evidence for significant flaring.

Absent a pulsar timing solution or constraints on the binary mass ratio, we
assumed a primary mass consistent with a neutron star ($\sim
1.4-2.0\,M_{\odot}$) and fit for the binary inclination $i$, Roche lobe
filling factor of the companion $f_{2}$, base (nightside) temperature of the
companion $T_{2}$, the isotropic irradiating luminosity from the pulsar (we
characterize this quantity as the maximum dayside temperature of the heated
secondary $T_{\rm{day}}$), and the binary mass ratio $q=M_{2}/M_{\rm{NS}}$. We
also assumed the orbital period from the photometry.

In general, a wide range of models fit the data equally well, all with
component masses fully consistent with a black widow MSP binary. The
statistical uncertainties on the Gemini photometry are very small, so after
finding a range of good-fitting models, we inflated the photometric errors by
a factor of $2.6\times$ so that the total reduced $\chi^{2}$ of the final
model was closer to 1.0, a method commonly used to model the light curves of
spider MSPs (e.g., Linares et al., 2018a; Swihart et al., 2020). We note that
the best-fitting model values are not sensitive to the exact multiplicative
value used to inflate the photometric errors (we did not fine-tune this value
so our final models have reduced $\chi^{2}$ slightly greater than 1.0). Within
the uncertainties, the overall results are identical for models with and
without inflating the photometric errors.

Overall, the best-fitting model has $i=58.0^{\circ}$, $f_{2}=0.95$,
$T_{2}=3837$ K, $T_{\rm{day}}=8797$ K, with a primary mass of
$M_{1}=1.40\,M_{\odot}$ and secondary mass of $M_{2}=0.058\,M_{\odot}$. This
is a good fit statistically with a reduced $\chi^{2}$ ($\chi^{2}_{\rm{red}}$ =
$\chi^{2}$/dof) of 69.4/58 = 1.2. However the $K_{2}$ value associated with
this fit is 376 km s-1, larger than the value derived from our SOAR and Gemini
spectroscopy (Sec. 3.4).

There are a range of other models that have similarly good statistical fits
but that agree better with our spectroscopy-derived $K_{2}$ value. For
example, a model with $i=44.0^{\circ}$, $f_{2}=0.98$, $T_{2}=2895$ K,
$T_{\rm{day}}=6246$ K, and $M_{1}=1.5\,M_{\odot}$, $M_{2}=0.07\,M_{\odot}$
results in a fit with $\chi^{2}_{\rm{red}}$ = 1.6, but with $K_{2}=317$ km
s-1.

Setting $T_{2}=2400$ K, the approximate mean nightside temperature of the nine
black widows fit by Draghis et al. (2019), returns a best-fitting model with
$\chi^{2}_{\rm{red}}$ = 1.2. The model parameters for this fit are
$i=55.0^{\circ}$, $f_{2}=0.98$, $T_{\rm{day}}=3750$ K, $M_{1}=1.4\,M_{\odot}$,
$M_{2}=0.020\,M_{\odot}$, and $K_{2}=369$ km s-1.

All of these models produce nearly identical fits to the data between binary
phases $\phi=0.4-1.2$. The largest differences between models occur near
minimum brightness ($\phi\sim 0.2-0.35$) where the fit residuals are highest,
especially near $\phi\sim 0.3$. Figure 2 shows the small differences in the
model light curves near these phases.

Given the wide range of generally well-fitting models, the single photometric
filter of the dataset, and the incomplete inclusion of physical effects in the
models, we do not quote formal uncertainties on the fitted and derived model
values. Broadly, we find that the light curve is consistent with that of a
near-Roche-lobe-filling companion to a neutron star with an intermediate
inclination $i\sim 44-58^{\circ}$ and potentially a companion mass that is on
the high side for black widows ($\sim 0.05-0.07\,M_{\odot}$). We defer a
closer comparison with the spectroscopic results to Section 3.4.

Table 1: Summary of SOAR & Gemini Spectroscopic Observations Date | Binary Phase | Emission Lines?
---|---|---
SOAR |  |
2021 Feb 18 | 0.90 | ✓
2021 Feb 20 | 0.75 | –
2021 Jun 09 | 0.07 | –
2021 Jul 16 | 0.92 | –
2021 Jul 16 | 0.07 | –
2021 Jul 28 | 0.71 | ✓
2021 Jul 28 | 0.84 | ✓
2021 Jul 29 | 0.99 | ✓
2021 Jul 30 | 0.80 | ✓
2021 Jul 31 | 0.98 | ✓
2021 Aug 06 | 0.76 | –
Gemini |  |
2022 Apr 11 | 0.76 | –
2022 Apr 11 | 0.89 | –
2022 Apr 11 | 0.02 | ✓
2022 Apr 11 | 0.15 | ✓
2022 Apr 11 | 0.30 | ✓
2022 Apr 11 | 0.43 | ✓
2022 Apr 11 | 0.56 | ✓

###  3.4 Optical Spectroscopy

####  3.4.1 SOAR Spectroscopy

Due to the extreme faintness of J1408 at optical minimum ($i\sim 23$), most of
the SOAR spectra were obtained at phases closer to optical maximum
($\phi=0.75$), from $\phi=0.71$ to 0.99, with two additional spectra at
$\phi\sim 0.07$. This means that the orbital phase coverage of the SOAR
spectra is generally poor.

Figure 3: Two SOAR spectra of J1408 taken at similar orbital phase on
different days, showing the variations observed at different epochs. In the
bottom spectrum, taken on 2021 Feb 20 ($\phi=0.75$), no emission lines are
seen, but instead primarily hydrogen Balmer lines in absorption. In the top
spectrum, from 2021 Jul 30 ($\phi=0.80$), the dominant features are instead
broad/double-peaked H and He emission lines.

Of the 11 SOAR spectra, 6 clearly show broad luminous emission lines of H and
in some cases He, while in the other spectra no emission is seen. Given the
narrow range of phases in the SOAR data ($\phi=0.71-1.07$), it is immediately
clear that the presence of the emission lines is not primarily related to the
orbital phase. The Gemini spectra (Sec. 3.4.2) provide additional support for
this conclusion (Table 1).

In the first SOAR spectrum of J1408, obtained on 2021 Feb 18, both H
(H$\alpha$ and H$\beta$) and He I (5875 and 7065 Å) emission is observed. The
FWHM of H$\alpha$ is quite broad at $\sim 1770$ km s-1. No emission lines were
visible in four subsequent spectra taken from 2021 Feb 20 to Jul 16, nor in
the final SOAR spectrum obtained on 2021 Aug 6.

Emission lines are clearly visible in all three SOAR spectra obtained on 2021
Jul 28/29, which were the next data taken after 2021 Jul 16. In the first two
spectra ($\phi=0.71$ and 0.84) the emission lines are clearly double-peaked,
with H$\alpha$ peak separations of $\sim 1425$ and 1150 km s-1, respectively.
Fitting a Gaussian to the full double-peaked line shapes gave FWHMs of $\sim
2820$ and 2290 km s-1, respectively. In the subsequent spectrum ($\phi=0.99$)
the emission lines are broad but no longer double-peaked, with an H$\alpha$
FWHM of $\sim 1510$ km s-1. A similar evolution occurs in the pair of SOAR
spectra obtained on the night of 2021 Jul 30/31, where emission lines
including H$\alpha$ are double-peaked in the first spectrum ($\phi=0.80$) but
not in the second ($\phi=0.98$).

The signal-to-noise in the continuum of the SOAR spectra ranges from poor to
modest. Metal-line absorption at Mg$b$ is visible in a few spectra, even
including several on 2021 Jul 28/29 and 30/31 where emission lines are also
present. No spectra showing broad M star absorption lines are seen, but this
may be primarily due to the poor orbital phase coverage. In one spectrum (2021
Feb 20) no metal lines are seen, but H$\beta$, H$\gamma$, and H$\delta$ are
instead observed in absorption (H$\alpha$ is marginal, and likely partially
filled in by emission), suggesting a warmer surface temperature at this epoch.

Figure 3 compares the emission line spectrum of 2021 Jul 30 to the absorption
line spectrum of 2021 Feb 20, which were each taken during similar binary
phases.

####  3.4.2 Gemini Spectroscopy

The seven Gemini/GMOS spectra were taken in series, covering nearly a full
orbit of J1408. Continuum emission was present for the first two spectra in
the series ($\phi=0.76$, 0.89). Absorption lines associated with the donor
were seen clearly in the first spectrum and marginally in the second, and
there were no emission lines visible in either spectrum. The continuum was
very faint in the third ($\phi=0.02$) and fifth ($\phi=0.30$) spectra and was
undetectable in the fourth ($\phi=0.15$). This general trend in the continuum
flux is expected, as the mean photometric maximum is expected around
$\phi=0.75$ and the minimum at $\phi=0.25$.

Over the timespan of the $\phi=0.02$ to 0.30 spectra, strong, broad emission
lines grew in prominence. In the $\phi=0.02$ spectrum only H$\alpha$ is
visible, but it is joined by marginally detectable He I 5785 Å in the
$\phi=0.15$ and $\phi=0.30$ spectra. In the final two spectra ($\phi=0.43$,
0.56), the continuum re-brightened toward the expected maximum at $\phi=0.75$.
However, the emission lines did not fade, but continued to increase in flux.
In the $\phi=0.56$ spectrum, in addition to H$\alpha$ and He I 5785 Å,
H$\beta$ and part of the Paschen series in the red, as well as He I 6678 and
7065 Å, were all observed clearly. No photospheric absorption lines were seen
in these final two spectra despite the re-emergence of the continuum.

The H$\alpha$ FWHM increased from $\sim 640$ km s-1 at $\phi=0.02$ to $\sim
1660$ km s-1 for $\phi=0.30$. The FWHM in the final spectrum ($\phi=0.56$) is
similar to this highest value, but is substantially lower in the penultimate
($\phi=0.43$) spectrum at $\sim 1160$ km s-1, suggesting a non-monotonic
trend. The emission lines were well resolved, but not double-peaked, in all
the Gemini/GMOS spectra in which they were apparent.

####  3.4.3 Emission Lines and Their Origin

Between the SOAR and Gemini spectroscopic datasets, broad emission lines were
observed at all orbital phases in at least one epoch. However, at most phases,
J1408 shows emission in some epochs, but not in others, suggesting there is no
simple relationship between orbital phase and the presence of emission (the
exception is $\phi=0.2$–0.6, which only are covered by a single Gemini epoch).
The FWHM of the emission lines also show no clear relationship to orbital
phase.

Instead, it is likely, that as observed in some other black widows and
redbacks with sufficiently extensive optical spectroscopy, that the emission
lines are primarily associated with an intrabinary shock between the stellar
wind of the secondary and the pulsar wind of the primary. This shock is not
constant in time, but instead shows flaring activity.

Figure 4: Circular Keplerian fit to the SOAR (filled circles) and Gemini (open
circles) radial velocities.

####  3.4.4 Absorption Lines, Radial Velocities, and Orbital Solution

For the SOAR spectra that showed evidence of a continuum and metal absorption
lines, we attempted to derive radial velocities of the companion star through
cross-correlation around the Mg$b$ region with bright template stars of early
to mid-K. For the 2021 Feb 20 spectrum, the strongest features were Balmer
lines in absorption, and a warmer template was used for cross-correlation in
the regions of H$\beta$, H$\gamma$, and H$\delta$. Six of the eleven SOAR
spectra yielded absorption-line radial velocities.

Only two of the seven Gemini spectra showed evidence for absorption lines.
Since the continuum signal in these spectra was higher in the red than in the
blue, radial velocities for these two spectra were derived through cross-
correlation in the Ca triplet region.

We next fit a circular Keplerian model to these eight radial velocities. Owing
to the small number of velocities and their substantial uncertainties, we
fixed the orbital period and epoch of the ascending node of the pulsar to the
best-fit photometric values. The latter assumption may not be correct due to
asymmetric heating, but in practice does not seem obviously wrong.

The best-fit model has $K_{2,obs}=317\pm 31$ km s-1 and $v_{sys}=-148\pm 16$
km s-1, with the uncertainties inferred via bootstrap. Owing to the poor phase
coverage of the velocities, these quantities are correlated, with a lower
$K_{2,obs}$ implying a more negative $v_{sys}$. This best fit is shown in
Figure 4, and has a $\chi^{2}$/d.o.f = 6.8/6.

Taken at face value, the mass function implied by the measured $K_{2,obs}$ and
orbital period is only $0.47\pm 0.14\,M_{\odot}$, suggesting a likely
inclination of J1408 of $i\lesssim 45^{\circ}$. This more face-on inclination
could also help explain why metal absorption lines are observed even at
superior conjunction of the secondary, when in a typical black widow system
this is when the warm irradiated face should dominate the spectrum.

However, there is a mild tension between this $K_{2,obs}$ and the predicted
$K_{2}$ from the best-fitting light curve model (376 km s-1, so different at
the $1.7\sigma$ level). The uncertainty in $K_{2,obs}$ is larger than typical
due to the poor phase coverage of the spectra for which radial velocities
could be derived. Another possibility is that the correction from a center-of-
light to center-of-mass $K_{2}$ might partially address this tension (though
the corrections for black widows typically go the other direction, e.g.,
Kandel & Romani 2020a).

We conclude that the kinematics of the secondary are generally consistent with
expectations for a black widow companion, but that the joint spectroscopic and
light curve evidence is equivocal between a typical intermediate inclination
and a somewhat more face-on inclination. Future multi-band light curves and
“lucky” spectroscopy when the flaring happens to be less prominent, and/or a
pulsar timing solution, can help address these uncertainties.

##  4 Black Widow Census

###  4.1 The Black Widow Sample

We have compiled many of the multiwavelength properties of the confirmed and
candidate black widows in the Galactic field in Tables 2–5. Confirmed systems
are defined here as those where a radio millisecond pulsar
($P_{\rm{spin}}<8\,\rm{msec}$) has been detected, and which have a companion
mass $\lesssim 0.1\,M_{\odot}$ (sans PSR J1908+2105, see below). As shown in
Sec. 4.3.1, a small change to this mass cutoff has no effect on the sample as
there are essentially no spider MSPs with companion masses between
$0.07-0.1\,M_{\odot}$.

There are 37 systems that meet these criteria. Of these, 7 have companions
that are extremely low mass, with minimum companion masses $\lesssim
0.01\,M_{\odot}$ (implying very large mass ratios, $M_{\rm{NS}}/M_{c}\gtrsim
150$, assuming typical neutron star masses). It may be the case that the
evolutionary paths leading to these extreme-mass-ratio systems differ from the
bulk of the black widow distribution (e.g., Romani et al., 2016), but given
their broad similarities to the black widows we include them here.

At the bottom of the tables, we also include four candidate systems for which
radio pulsations have not yet been detected, but that have strong evidence for
a black widow classification based on the multiwavelength data. These are 4FGL
J1408.6–2917 (this work), 4FGL J0335.0+7502 (Li et al., 2021), 4FGL
J0935.3+0901 (Wang et al., 2020; Halpern, 2022), and ZTF J1406+1222 (Burdge et
al., 2022), a recently discovered candidate black widow in a hierarchical
triple.

We do not include PSR J1908+2105 in this list of black widows. Its likely
companion mass ($>0.055\,M_{\odot}$) is in a sparsely populated mass range,
consistent with either the most massive black widow companions or the low-mass
end of redbacks, and the extensive radio eclipses observed in this system are
more characteristic of redbacks than black widows (Cromartie et al., 2016;
Strader et al., 2019).

###  4.2 Measured Properties

A vast majority of the black widows have a precise pulsar timing solution that
enables tight constraints on the typical pulsar parameters, namely the spin
period and its derivative ($P_{\rm{spin}}$, $\dot{P_{\rm{obs}}}$), projected
semimajor axis ($a\,\rm{sin}\,i$), dispersion measure (DM), and pulsar spin-
down power ($\dot{E}$). For the candidates with no pulsar detection the _Gaia_
position is used, while for the others the best position is taken from the
ATNF database444https://www.atnf.csiro.au/research/pulsar/psrcat/ (Table 2).

For compiling distances to each source, we take a hierarchical approach. The
most accurate and precise distances come from radio timing parallax
measurements. For the four systems with a reliable timing parallax we adopt
those distances and associated uncertainties. One source, PSR J1653–0158, has
a moderately precise _Gaia_ parallax measurement
($\widetilde{\omega}/\sigma_{\widetilde{\omega}}\sim 2.3$), and we adopt the
associated geometric distance for this source (Bailer-Jones et al., 2021).

For systems with modeled optical light curves, an estimate of the distance to
the binary is possible by comparing the observed fluxes to the flux predicted
by the light curve model assuming some radius and temperature for the
companion (e.g., Breton et al., 2013; Swihart et al., 2017; Draghis et al.,
2019). For the spider MSPs with a significant _Gaia_ parallax measurement, and
therefore a precise geometric distance estimate, these optical light curve
derived distances have been shown to be more accurate than the DM-based
distance estimates (Jennings et al., 2018). For sources without parallax
measurements, we adopt these optically-derived distances unless the authors in
the cited reference suspect the distance is unreliable (for example, Draghis
et al. (2019) suspect the distance they estimate to PSR J0251+2606 may be
dubious due to the incomplete light curve coverage). Lastly, for sources with
no parallax or light-curve derived distance estimates, we adopt the Yao et al.
(2017) DM model distances since they have been shown to be more accurate than
the Cordes & Lazio (2002) model for pulsars, especially those outside the
Galactic plane (Jennings et al., 2018). In these cases we assume 30%
uncertainty on the DM-based distance model value. We list these distance
estimates in Table 3.

For the 20 systems that have been observed in X-rays, we list the unabsorbed
0.5–10 keV X-ray fluxes ($F_{X}$) and best-fit power-law photon index
($\Gamma$) in Table 4. We corrected all the X-ray fluxes to this standard
energy range using WebPIMMS555https://heasarc.gsfc.nasa.gov/cgi-
bin/Tools/w3pimms/w3pimms.pl assuming the best-fit flux and photon index in
the reference cited (see also, Lee et al., 2018). For most sources, the listed
$\Gamma$ assumes a simple power-law model, unless a combined thermal plus
power-law model was a significantly better fit. If a combined thermal plus
power-law model was comparable statistically to the simple power-law, we
assumed the simpler model. For some of the brighter sources, a phase-resolved
X-ray analysis was performed to determine whether the X-ray properties differ
based on binary phase (e.g., Kandel et al., 2021). In these cases we assumed
the properties of the phase-averaged (i.e., full orbit) spectrum for
consistency.

For three systems, PSR J1946–5403, PSR J2052+1219, and PSR J2115+5448, we
found unpublished archival _XMM_ observations and analyzed these with the same
procedures as described in Sec. 2.2.2. To our knowledge, this work presents
the first analysis of these systems in X-rays.

Using the final adopted distance and associated uncertainty, we list the
0.5–10 keV X-ray luminosity ($L_{X}$) and 0.1–100 GeV $\gamma$-ray luminosity
($L_{\gamma}$) if the source is detected with _Fermi_ (Table 4).

Binary orbital periods from pulsar timing are adopted when they are available,
otherwise we assume the best period from the optical photometry and/or
spectroscopy (Table 5).

####  4.2.1 MSP Spin Distributions

In Figure 5 we show the spin period versus the orbital period for the field
black widows and redbacks along with field MSP–He white dwarf binaries. This
figure highlights what a large fraction of the fastest spinning MSPs are in
spider binaries. Among the 34 systems with $P_{\rm{spin}}\leq 2.5\,\rm{ms}$,
24 (71%) are confirmed spiders. At even shorter spin periods
($P_{\rm{spin}}\leq 2.0\,\rm{ms}$), 11 of 13 (85%) are black widows/redbacks.

The short orbital period pre-MSP binaries that begin transferring mass on the
main sequence will naturally mass transfer for a longer period of time
relative to systems that don’t start mass transfer until the companion has
evolved off the main sequence (e.g., Tauris & Savonije, 1999). The spin
distribution we observe in Figure 5 is therefore direct confirmation of the
behavior expected if the progenitors of spiders indeed had very short initial
orbital periods compared to the progenitors of typical MSP–He white dwarf
binaries.

Figure 5: Spin period versus orbital period for the redbacks (red) and black
widows (black) along with the field MSPs with He white dwarf companions (blue)
highlighting the large fraction of spiders among the fastest spinning systems.

####  4.2.2 Masses

Reliable constraints on the neutron star masses in black widow binaries are
difficult to obtain because these estimates often rely on the binary
inclination derived from modelling the optical light curves, which can be
plagued by systematic uncertainties when the light from the companion is
dominated by irradiation. Similar heating effects make it hard to estimate the
binary mass ratio, since this often relies on an accurate measurement of the
semi-amplitude of the companion radial velocity curve, which must be carefully
corrected for the difference between the system’s center-of-mass and its
center-of-light. Given the large (and often poorly characterized) systematic
uncertainties associated with these effects, we refrain from including neutron
star mass estimates in this bulk catalog. However, we do explicitly list the
systems for which photometry/spectroscopy exists, which allow for more
accurate estimates of the component masses when coupled with a precise pulsar
timing solution (Table 5). While we recognize the uncertainties involved, we
proceed with discussion of the companion mass estimates in the following
section. Although we do not list the neutron star mass estimates explicitly in
the catalog, we summarize some of the recent literature about neutron star
masses in black widows in Sec. 4.4.

Figure 6: Median (or best-fit, if available) companion mass vs. orbital period
for the field redbacks (red) and black widows (black). MSPs with He white
dwarf companions are also shown (blue) along with a binary evolution model
which assumes an initial secondary mass of $1.0\,M_{\odot}$ and denotes the
endpoints of an ensemble of systems with varying initial period (Tauris &
Savonije, 1999). The recently discovered MSP–proto white dwarf binary
associated with 4FGL J1120.0–2204 (Swihart et al., 2022) (green circle) and
the two long-period “huntsmen” systems with (sub)giant companions (orange) are
progenitors of “normal” MSP–He white dwarf binaries. Despite the growing
number of discoveries made in the last several years, there is still a notable
absence of sources with companion masses in the range $0.07-0.1\,M_{\odot}$.

###  4.3 Black widows vs. Redbacks

####  4.3.1 Companion Masses

In order to compare the companion mass distribution of black widows versus
redbacks, we first collated the black widow companion masses. The most
accurate way to directly measure the companion mass in a MSP is through the
relativistic Shapiro delay (Shapiro, 1964), which is typically only possible
in nearly edge-on systems with precise and long-term pulsar timing solutions
(e.g., Cromartie et al., 2020a). The significant radio eclipses and orbital
variability observed in most black widows typically make them poor targets for
long-term timing. Therefore, typically the best way to infer companion masses
in these systems is from the pulsar orbital parameters ($P_{\rm{orb}}$ and
$a\,\rm{sin}\,i$) in conjunction with light curve modeling of the companion to
constrain the inclination, despite the substantial uncertainties associated
with the latter measurements.

For most black widows, a lower limit on the mass of the companion is available
via pulsar timing, assuming an edge-on inclination ($i=90^{\circ}$) and a
neutron star mass of $1.4\,M_{\odot}$. We list these lower limits in Table 5.
If no inclination constraints are available from optical light curve modeling,
we adopt the companion mass assuming $i=60^{\circ}$ (hereafter referred to as
the “median” mass), since for randomly aligned orbits the systems should be
uniformly distributed on the sky in cos($i$). For the systems with modeled
optical light curves, we assume the best-fit companion mass derived from these
models.

These median or best-fit (if available) masses are listed in Table 5. Owing to
the large systematic uncertainties associated with deriving inclinations from
black widow light curves, we do not include formal uncertainties on these mass
estimates.

We assumed the black widow companion star masses were drawn from a normal
distribution, and we modeled this sample using a Bayesian MCMC model. We find
a median mass $M_{c}=0.027\pm 0.003\,M_{\odot}$ with $\sigma=0.015\pm
0.002\,M_{\odot}$. This can be compared to the distribution of redback
companion masses, which have a median mass $M_{c}=0.39\pm 0.05\,M_{\odot}$
with $\sigma=0.12\pm 0.05$ (Strader et al., 2019). We note that since we do
not include formal uncertainties on the individual black widow companion mass
estimates in this analysis, the observed dispersion quoted for the black
widows is likely an overestimate of the intrinsic dispersion.

We fixed the inclination for a large number of the systems, so as a check on
how sensitive these conclusions are to our inclination assumptions, we
repeated this analysis after randomly assigning inclination values drawn from
a prior that is flat in cos($i$) (i.e., assuming random orientations), finding
a median mass $M_{c}=0.029\pm 0.004\,M_{\odot}$ with $\sigma=0.021\pm
0.003\,M_{\odot}$, very similar to our original analysis but with a larger
spread. Finally, we analyzed the distribution for only the 17 black widow
systems with mass estimates from modeling the photometry, finding
$M_{c}=0.033\pm 0.004\,M_{\odot}$ with $\sigma=0.016\pm 0.003\,M_{\odot}$.

We plot the orbital period and median (or best-fit, if available) companion
masses for the field black widows and redbacks in Figure 6. We also show the
white dwarf companions to field MSPs (blue circles) along with a binary
evolution model that shows the well-known relation between the final period
and white dwarf mass for systems with a range of initial orbital periods
(black line, Tauris & Savonije, 1999).

Despite the continued discovery of several new black widow and redback systems
in the past few years, the mass distributions appear bimodal and there is a
noticeable absence of sources in the companion mass range $\sim
0.07-0.1\,M_{\odot}$.

####  4.3.2 Orbital Periods

Similar to our comparison of the masses in the previous section, we also
analyzed the orbital period distributions of the black widows and redbacks.
For both subclasses, we included all confirmed and candidate systems, with the
exception of the two “huntsman” systems that have giant companions in wide
($\gtrsim$5 d) orbits, which were left out of the redback distribution. For
the black widows, we find a median orbital period of
$P_{\rm{median}}^{\rm{BW}}=0.229\pm 0.032$ d with $\sigma=0.178\pm 0.024$ d.
For the redbacks, we find the orbital periods are longer in the mean, with
$P_{\rm{median}}^{\rm{RB}}=0.389\pm 0.056$ d and $\sigma=0.252\pm 0.043$ d.
Despite the significant difference in the mean orbital period, the
distributions overlap substantially, so it is challenging to classify a spider
on the basis of orbital period alone except at the shortest periods.

####  4.3.3 Emission Lines

As a class, redback companions are significantly brighter than the black
widows, so optical spectroscopic follow-up is easier and is much more complete
for the redback population. The initial interpretation of emission lines in
the optical spectra of some redbacks (e.g., Strader et al. 2015) was colored
by the existence of transitional millisecond pulsars, which sometimes show
emission lines from accretion disks (e.g., de Martino et al. 2014) and which
are all redbacks. However, the appearance of similar highly-broadened or
double-peaked lines in black widows, as well as more detailed study of
emission lines in redbacks, suggests that in most cases the emission lines in
optical spectra of spiders are more likely associated with the companion
(either directly from the chromosphere or a stellar wind), or via emission
from the intrabinary shock itself, rather than an accretion disk.

For example, in the candidate redback 1FGL J0523.5–2529, Halpern et al. (2022)
infer the optical flaring and spectral emission features are nonthermal,
coming from above the photosphere of the secondary and presumably associated
with the companion wind outflow and/or the intrabinary shock. A similar
conclusion was reached for the origin of the emission lines in the “huntsmen”
MSP 1FGL J1417.7–4402 (Swihart et al., 2018). As another explanation, Romani
et al. (2015) attributed the emission lines in the black widow PSR J1311–3430
to thermal emission below the stellar photosphere, likely due to pulsar wind-
triggered magnetic reconnection that provides a localized heating source.

Setting aside disk-dominated spectra of transitional MSPs or candidate members
of this class, there are 15 redbacks with published optical spectroscopy. Of
these, 5 (33%) show prominent emission lines: 1FGL J0523.5–2529 (Halpern et
al., 2022), 3FGL J0838.8–2829 (Halpern et al., 2017), PSR J1048+2339 (Strader
et al., 2019), PSR J1306–40 (Swihart et al., 2019), and PSR J1628–3205
(Strader et al., 2019). Both of the huntsman sources 1FGL J1417.7–4407
(Strader et al., 2015; Swihart et al., 2018) and 2FGL J0846.0+2820 (Swihart et
al., 2017) also show emission, as does the subgiant–MSP binary PSR J1740-5340A
in the globular cluster NGC 6397 (Sabbi et al., 2003).

For the black widows, only 10 systems have published spectra, but of these 6
(60%) show H and/or He in emission (Table 5). Although the statistics are low,
these results suggest that as a class, emission lines are at least as common
among black widows as among redbacks, and indeed may be more common.

If the emission lines are associated with an irradiation-driven wind from the
companion or an intrabinary shock, it is not immediately clear whether this
should be stronger in redbacks or black widows. Owing to their larger relative
Roche Lobe radii, redbacks intercept a larger fraction of emission than do
black widows from a source centered on the pulsar. But if the high-energy
emission responsible for the optical emission lines is due primarily to X-rays
mediated by the intrabinary shock, rather than $\gamma$-rays, the situation
becomes more complex. If the shock wraps around the companion in black widows,
but around the pulsar in redbacks (e.g., Romani & Sanchez 2016; Wadiasingh et
al. 2017), then the weaker shock in black widows might be compensated for by a
closer location to the secondary.

To add to the complexity, the appearance of emission lines in these systems is
not always stable and predictable. For example, in the redback 3FGL
J0838.8–2829 (Halpern et al., 2017) and the black widow 4FGL J1408.6–2917
(this work), the emission lines come and go irregularly on short
($\sim$minutes to hours) timescales, with varying strength and phenomenology
(i.e., sometimes double-peaked, sometimes single-peaked), and have no clear
trend with orbital phase. Other systems, like the huntsman 1FGL J1417.7–4407
(Swihart et al., 2018), show persistent, orbital phase-dependent emission that
is consistent in data taken over timespans of years. Furthermore, a range of
temperatures and/or compositions in the emission line-producing regions is
implied by the fact that some systems show only Balmer emission in their
spectra (e.g., PSR J1306–40; Swihart et al. 2019) while others also display
prominent He I lines that are occasionally double-peaked (e.g., PSR
J1311–3430; Romani et al. 2015).

Given the large range of phenomenologies, it is possible that the exact origin
of the emission lines differ from one system to the next. Future spectroscopic
monitoring of new and existing spider MSPs to determine the connections
between emission features and the properties of the companion and the
intrabinary shock is needed.

####  4.3.4 X-ray Emission

We compare the 0.5–10 keV X-ray luminosities ($L_{X}$) and best-fit photon
indices ($\Gamma$) for the redbacks and black widows in Figure 7. The X-ray
properties of the redbacks were taken from the cited references in Lee et al.
(2018) or Strader et al. (2019) and scaled to 0.5–10 keV as described in Sec.
4.2. Redback X-ray luminosities were derived assuming the distances from
Strader et al. (2019).

The average properties of the redback sample are $L_{X}=2.3\times
10^{32}\,\rm{erg\;s^{-1}}$ and $\Gamma=1.44$, while for the black widows
$L_{X}=1.4\times 10^{31}\,\rm{erg\;s^{-1}}$ and $\Gamma=2.51$. Lee et al.
(2018) found that X-ray emission in redbacks is brighter and harder than in
black widows, although their sample only consisted of X-ray properties from 12
black widows and 8 redbacks. With our larger sample (18 black widows and 17
redbacks/candidates), we again observe evidence for redbacks being brighter
and harder than black widows, but also confirm a clear trend in the spider
distribution as a whole, with the softer sources typically being intrinsically
fainter in X-rays with a nearly continuous distribution spanning over three
orders of magnitude in X-ray luminosity.

In Figure 8 we show the relation between the X-ray photon index and the X-ray
luminosity as a fraction of the pulsar spin-down power $\dot{E}$. Although the
statistics are limited, this figure suggests that, in general, the X-ray
luminosity in redbacks represents a much larger percentage of the pulsar spin-
down power than in black widows. This figure also shows an apparent trend,
whereby systems with a softer X-ray spectrum also convert a smaller fraction
of the pulsar spin-down power into X-rays.

The simplest explanation for these figures is that the intrabinary shock
acceleration is vastly more efficient in redbacks. From the perspective of the
pulsar, the solid angle subtended by the companion is significantly larger in
redbacks than in the smaller black widows. The interplay between the pulsar
and companion’s wind or magnetosphere then controls the overall geometry of
the shock. In general, if the companion’s pressure dominates over the pulsar
wind, then the shock standoff radius will move farther from the companion and
the shock will wrap around the pulsar, and vice-versa if the pulsar wind
dominates (e.g., Romani & Sanchez, 2016; Wadiasingh et al., 2017). In redbacks
with well-sampled phase-resolved X-ray light curves, the shock is almost
always wrapped around the pulsar, implying stronger companion winds or
magnetopheres in these systems compared to the black widows (Wadiasingh et
al., 2018; van der Merwe et al., 2020). As such, in redbacks, a larger
fraction of the pulsar’s $\dot{E}$ is captured by the shock, naturally
resulting in higher intrinsic non-thermal X-ray luminosities.

Figure 7: Photon index versus the 0.5–10 keV X-ray luminosity for redbacks
(red) and black widows (black). Luminosities were derived as described in Sec.
4.2. X-ray emission from redbacks tends to be brighter and harder than in
black widows, likely due to more efficient acceleration/compression of the
intrabinary shock (see text).

More interestingly, the results in Figures 7 and 8 for the spectral index
suggest that the particle acceleration is also systematically more efficient
in redbacks. The pulsars themselves influence the injection spectrum of
electron/positron pairs into the shock. Since the pulsars are not
intrinsically very different in redbacks from black widows this implies the
systematic differences in the spectral index and particle acceleration
efficiency results from some conditions at the shock that must differ between
the two classes. The spectrum of accelerated particles at the shock is
expected to be modified by geometric differences of compression of the pulsar
wind’s stripes, the local shock magnetic obliquity, and the resultant shock-
driven reconnection (e.g., Summerlin & Baring, 2012).

Close inspection of Figure 8 shows there are at least two black widows with
redback-like $\Gamma$ and $L_{X}/\dot{E}$. These are PSR J1311–3430 and PSR
J1653–0158, which share many observed and intrinsic properties. Among the
confirmed black widows in our sample, these two systems have the shortest
orbital periods. Both systems also have very low-mass H-depleted companions,
show prominent emission lines in their optical spectra, and are two of the
brightest high Galactic latitude _Fermi_ sources. In addition, both their
light curves show evidence of nonthermal flux that dominates the optical light
near minimum brightness, implying a strong evaporating wind.

Figure 8: Photon index versus the X-ray luminosity as a fraction of the pulsar
spin-down power $\dot{E}$. In general, redbacks convert a larger fraction of
their pulsar spin-down power to X-rays. The few black widows with redback-like
X-ray properties suggests that in lieu of stellar winds some of these pulsar
companions have very strong magnetospheres (see Sec. 4.3.4). Despite the low
statistics, a rough trend appears that suggests systems with softer X-ray
spectra also convert less of their pulsar spin-down power to X-rays.

One explanation may be that these two black widows have redback-like shocks
that wrap around the pulsar instead of around the companion. Since the
extremely low-mass companions in black widows likely do not support strong
winds, this argues for very strong companion magnetospheres in some spider
systems in order to achieve a pressure balance with the relativistic pulsar
wind (Wadiasingh et al., 2018); some recent observations support the existence
of these strong companion magnetospheres (e.g., Li et al., 2022).

We note there are selection biases involved when analyzing these systems as a
population. Typical follow-up observations of _Fermi_ sources with _Swift_ are
a few ksec in duration, reaching X-ray flux limits of a few $\times\,10^{-14}$
erg s-1 cm-2, so very low-luminosity or more distant systems cannot be
detected in X-rays.

Among the 18 confirmed black widows detected in X-rays and that also have a
distance estimate listed in Table 3, the median distance is 2.20 kpc. This can
be compared with the median distance of the 19 non-detected sources, which is
2.24 kpc, suggesting that larger distances may not be the primary factor
leading to the non-detections.

Even for many of the nearby sources, follow-up observations have just simply
not gone deep enough to detect the expected faint X-ray emission. For example,
two black widows that are very nearby and that have reliable radio timing
parallax distances are PSR J2322–2650 and PSR J2234+0944, which have distances
of $0.23^{+0.09}_{-0.05}$ kpc and $0.8^{+0.3}_{-0.2}$ kpc, respectively. PSR
J2322–2650 was observed with _Swift_ for $\sim$4.7 ksec and has a 0.5–10 keV
flux upper limit of $<1.0\times 10^{-13}$ erg s-1 cm-2, assuming a power law
with $\Gamma=2.5$. Using its parallax distance, this flux corresponds to an
X-ray luminosity upper limit $L_{X}<6.4^{+6.0}_{-2.5}\times 10^{29}$ erg s-1.
An upper limit for PSR J2234+0944 using similar Swift data corresponds to a
luminosity limit $L_{X}<4.3^{+3.8}_{-1.9}\times 10^{30}$ erg s-1. These limits
are near the bottom envelope of, but not far outside, the range of
luminosities currently observed for black widows (Figure 7).

With future deeper and more sensitive X-ray observations, it will be
interesting to see if the tentative correlations seen in Figure 7 and Figure 8
hold at the lowest X-ray luminosities. Since new systems have recently been
discovered with distances $\lesssim$3 kpc, the census of these binaries is
clearly far from complete, especially at larger distances, and future
multiwavelength programs to find new spider MSPs both inside and outside of
$\gamma$-ray source regions could reveal a wider range of phenomenologies.

###  4.4 Neutron Star Masses in Black Widows

Despite the complex systematic effects involved in modeling the heated
companions of black widows, measuring the radial velocity of the secondary
with optical spectroscopy provides valuable dynamical constraints that allow
for estimates on the mass of the neutron star when coupled with light curve
fitting and pulsar timing models.

To date, the most massive neutron star with a precise and direct measurement
comes from the relativistic Shapiro delay pulsar timing measurement from the
neutron star–white dwarf binary PSR J0740+6620 (Fonseca et al., 2021;
Cromartie et al., 2020b), which has a mass of $M_{\rm{NS}}=2.08\pm
0.07\,M_{\odot}$. There are claims of neutron star masses from some black
widows (and at least one redback) that are higher than this value. Perhaps the
most notable of these is PSR J0952–0607, the fastest spinning black widow in
our sample, which was recently reported to have the most massive well-measured
neutron star mass to date at $M_{\rm{NS}}=2.35\pm 0.17\,M_{\odot}$ (Romani et
al., 2022a). Other black widows that have been claimed to host neutron stars
$\gtrsim 2.1\,M_{\odot}$ include the original black widow PSR B1957+20
($2.40\pm 0.12\,M_{\odot}$; van Kerkwijk et al. 2011), PSR J1311–3430 ($2.2\pm
0.4\,M_{\odot}$; Romani et al. 2015), PSR J1653–0158 ($2.17\pm
0.2\,M_{\odot}$; Nieder et al. 2020), and PSR J1810+1744 ($2.13\pm
0.04\,M_{\odot}$; Romani et al. 2021). To these can be added the highly-
irradiated redback PSR J2215+5135, which has a claimed mass as high as
$2.27\pm 0.16\,M_{\odot}$ (Linares et al. 2018b; Kandel & Romani 2020b, though
see also Voisin et al. 2020b).

Despite the increasing sophistication of the light curve modeling in the most
recent papers cited above, which include an improved treatment of gravity
darkening and non-standard heat transport across the surface of the companion,
it has not been established that these models accurately incorporate all
relevant physical effects. Given, for example, that an orbital inclination
change of only $1.3^{\circ}$ is the difference between a derived mass of
$2.1\,M_{\odot}$ and $2.2\,M_{\odot}$ at values around the median inclination
of $60^{\circ}$, it is clear that the conclusions drawn about the most massive
neutron stars in spiders are affected by even small systemic uncertainties in
light curve modeling. In this context, it is relevant that the highest-mass
neutron stars have all been found in highly irradiated spiders (black widows
or the strongly irradiated redback PSR J2215+5135), rather than in comparably
recycled neutron stars in redbacks that have light curves less affected by
irradiation, which all have inferred masses $\lesssim 2.1\,M_{\odot}$ (Strader
et al., 2019).

While it is unclear the extent to which specific individual measurements of
black widow neutron star masses are fully reliable, a more secure claim, made
already in many of the papers cited above, is that the black widow neutron
star mass distribution has a median value significantly larger than the
canonical $1.4\,M_{\odot}$; a similar result was found for the redbacks
(Strader et al., 2019). This is consistent either with these neutron stars
having been born massive or having accreted a substantial amount of mass.
Given the fast spins and long predicted accretion lifetimes of these binaries
as discussed above, accretion seems likely to have played a substantial, if
not dominant role.

##  5 Discussion and Conclusions

###  5.1 The Properties of Black Widows

In the context of our discovery of a likely new black widow associated with
4FGL J1408.6–2917, we compiled the properties of known black widows. J1408 is
now one of 41 confirmed and candidate black widows in the Galactic field. We
showed that the spider companion mass distribution is still strongly bimodal,
split between the lower-mass black widows and the more massive redbacks. We
also find the orbital periods of the black widows are slightly shorter than
for redbacks. Optical emission lines are common in both systems, and although
the statistics are limited, they are seen more frequently in the spectra of
black widows.

We compared the X-ray properties of the spiders showing that the harder and
more luminous X-rays in redbacks implies that acceleration/compression of the
intrabinary shock is more efficient in these systems. We also observe the
broad relation that spider binaries with harder X-ray spectra tend to convert
a larger fraction of the pulsar spin-down power to X-rays. Some black widows
show spectral indices and $L_{X}/\dot{E}$ values that are comparable to
redbacks. If these black widows are unable to power significant winds due to
their low masses, an implication could be that the companion magnetospheres in
at least some spider binaries are the dominant source of pressure balance
supporting the intrabinary shock from the companion side, rather than a wind
from the companion.

###  5.2 The Origin of Black Widows

As mentioned above, one of the most puzzling observational findings is that of
a bimodal companion mass distribution among spider MSPs, with additional
emerging evidence for systematic differences in their X-ray properties.

This bimodality does not emerge naturally from binary evolution models. Most
spiders are expected to have evovled from close binaries where the secondary
filled its Roche Lobe on the main sequence or early in its post-main sequence
evolution, recycling the neutron star. Compared to the well-studied
cataclysmic variables with white dwarf primaries, the subsequent evolution is
strongly affected by irradiation by the pulsar wind, which continues even when
accretion has ceased. This can occur “naturally” if magnetic braking shuts off
when the donor becomes fully convective or perhaps even earlier in the
evolution, if accretion-induced irradation leads to a Roche Lobe underfilling
donor (e.g., Benvenuto et al. 2012; Chen et al. 2013; De Vito et al. 2020). In
either case, the proximate cause for the companion to become a redback is a
high level of irradiation, leading to faster evaporation, compared to the
black widow case.

However, in the more recent models from Ginzburg & Quataert (2020),
evaporation alone cannot cause significant enough mass loss to explain the
observed spider populations. Instead, the irradation is proposed to change the
internal structure of the companion, allowing it to maintain a strong magnetic
field even down to very low masses. The irradition-driven evaporative wind
couples to this magnetic field and the companion can maintain stable Roche-
lobe overflow for a much longer time and at longer orbital periods, giving
more efficient magnetic braking and evolutionary timescales that agree better
with the observed spider population (Ginzburg & Quataert, 2021).

In this model, the two parameters that control the evolution are the MeV
$\gamma$-ray pulsar luminosity ($L_{\rm{MeV}}$) that evaporates the companion,
and the broader spectrum irradiating $\gamma$-ray luminosity
($L_{\rm{irr}}>L_{\rm{MeV}}$) that is deposited in the companion atmosphere.
$L_{\rm{irr}}$ lengthens the thermal timescale of the companion, allowing it
to maintain Roche-lobe overflow at longer orbital periods as described above.
In this model, different values of $L_{\rm{irr}}$ give a range of observed
periods, while the mass gap between the two populations is proposed to
originate from a bimodal distribution of companion magnetic fields (weaker for
redbacks, stronger for black widows).

A key prediction of this model is that $L_{\rm{irr}}$ is correlated with the
orbital period of the binary ($L_{\rm{irr}}\propto P_{\rm{orb}}^{2.5}$).
Ginzburg & Quataert (2021) found that the measured pulsar spin-down luminosity
does not agree with this relation, but that the high-energy $\gamma$-ray
luminosities from _Fermi_ do (see their Figure 12), providing support for this
theoretical model.

Figure 9: The $\gamma$-ray luminosity of black widows does not clearly
correlate with orbital period. The figure shows the 0.1–100 GeV _Fermi_
$\gamma$-ray luminosity versus the binary orbital period, adapted from Figure
12 in Ginzburg & Quataert (2021). Black points are the systems plotted in the
original figure with updated values for $L_{\gamma}$ from Table 4,
supplemented by 7 additional systems (blue points) with reliable distance
estimates (see text). The two sets of error bars on each point represent the
uncertainty associated with the flux and distance, respectively. Evolutionary
models from Ginzburg & Quataert (2021) predict the pulsar’s irradiating
luminosity, potentially tracked by the _Fermi_ GeV luminosity, is correlated
with the orbital period (dashed line). With our updated luminosities and
additional systems, we find the data do not fit the predicted relation.

We have revisited this proposed relation, adding an additional 7 systems that
have well-constrained distance estimates from either a parallax measurement or
from light curve models and updating the data for the others where relevant
(Figure 9). With the updated data, the observational support for this
predicted relation is weaker. For most systems the $\gamma$-ray luminosity
provides sufficient energy to power the theoretically inferred $L_{\rm{irr}}$,
with all but two systems lying above the dashed line in Figure 9. But, unlike
in Ginzburg & Quataert (2021), we see no clear relation between $L_{\gamma}$
and orbital period. In the context of this model, this finding is consistent
with the idea $L_{\gamma}$ is a poor proxy for the irradiating luminosity
$L_{\rm{irr}}$. Alternatively, the fact that almost all the observations lie
above the predicted line might result from inefficient heat transport between
the two hemispheres of the tidally locked black widow companion. According to
Ginzburg & Quataert (2021), it is the fraction of $L_{\rm{irr}}$ that is
transported to the companion’s non-irradiated nightside which sets its thermal
timescale and therefore correlates with the orbital period. The lack of
relation between $L_{\gamma}$ and $P_{\rm{orb}}$ could also potentially be
explained by a beaming or another efficiency factor, leading to a model that
is more akin to the assumptions made in some previous models (e.g., Benvenuto
et al. 2012; Chen et al. 2013).

Beaming, or more generally a variation in the irradiation efficiency, may also
be relevant for understanding the relationship between black widows and the
related class of accreting millisecond X-ray pulsars (AMXPs). Some AMXPs, such
as SAX J1808.4–3658, have very similar MSP spin periods, orbital periods, and
companion masses to black widows (e.g., Patruno & Watts 2021). One possibility
is that AMXPs come from initially similar systems to black widows but have a
lesser or minimal degree of MSP irradiation, leading to lower mass loss rates
and shorter orbital periods as observed (e.g., Ginzburg & Quataert 2021;
D’Antona & Tailo 2022). The difference in orbital periods indicates that AMXPs
and black widows cannot be exactly the same population of systems at different
points in a cyclical accretion cycle. However, in principle a difference in
irradiation efficiency among individual systems could have been present either
early in their evolution, or could have developed over time. At least for some
systems, an evolutionary connection is also possible: many binary evolution
models of the SAX J1808.4–3658 suggest that this system will eventually stop
accreting and become a detached black widow (e.g., Chakrabarty & Morgan, 1998;
Ergma & Antipova, 1999; Chen, 2017; Tailo et al., 2018; Goodwin & Woods, 2020)

In any case, the present work emphasizes the continuing evidence for a bimodal
spider companion mass distribution: not only the existence of redbacks, but
also the “gap” in companion masses in the range $0.07$–$0.1\,M_{\odot}$.
Notably, longer orbital period AMXPs with hydrogen-rich donor stars also
appear to avoid this mass gap (Patruno & Watts, 2021; D’Antona & Tailo, 2022),
consistent with a common physical cause that must be addressed by a successful
model.

###  5.3 Concluding Remarks

The explosion in the number of spider MSPs found by _Fermi_ and associated
follow-up was seen as notable even from earliest discoveries (e.g., Ray et al.
2012; Roberts 2013). Subsequent work has shown that these spiders are not
simply a curious “add-on” to the existing subpopulations of MSPs: spiders are
now relatively common, making up $\gtrsim 15-20$% of the fully recycled MSPs
with known companion classifications. In addition, they represent a dominant
fraction (70–80%) of the fastest spinning field MSPs, some of which may also
be among the most massive neutron stars known. Additional observational and
theoretical work to understand the formation and evolution of black widows and
redbacks is an extremely promising route to understand the extremes of neutron
star behavior.

## Acknowledgements

Special thanks to Z. Wadiasingh for thoughtful discussion on the X-ray
emission and to S. Ginzburg for helpful insights on the evolutionary models.

This research was performed while SJS held a NRC Research Associateship award
at the Naval Research Laboratory. Work at the Naval Research Laboratory is
supported by NASA DPR S-15633-Y.

We also acknowledge support from NSF grants AST-1714825 and AST-2205550 and
the Packard Foundation.

Based on observations obtained at the Southern Astrophysical Research (SOAR)
telescope, which is a joint project of the Ministério da Ciência, Tecnologia,
Inovações e Comunicações (MCTIC) do Brasil, the U.S. National Optical
Astronomy Observatory (NOAO), the University of North Carolina at Chapel Hill
(UNC), and Michigan State University (MSU).

We acknowledge the use of public data from the Swift data archive.

The Green Bank Observatory is a facility of the National Science Foundation
operated under cooperative agreement by Associated Universities, Inc

This work used observations obtained with XMM–Newton, an ESA science mission
with instruments and contributions directly funded by ESA Member States and
NASA.

This research has made use of data and software provided by the High Energy
Astrophysics Science Archive Research Center (HEASARC), which is a service of
the Astrophysics Science Division at NASA/GSFC and the High Energy
Astrophysics Division of the Smithsonian Astrophysical Observatory.

This work has made use of data from the European Space Agency (ESA) mission
Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing
and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement.

Based on observations obtained at the international Gemini Observatory, a
program of NSF’s NOIRLab, which is managed by the Association of Universities
for Research in Astronomy (AURA) under a cooperative agreement with the
National Science Foundation on behalf of the Gemini Observatory partnership:
the National Science Foundation (United States), National Research Council
(Canada), Agencia Nacional de Investigación y Desarrollo (Chile), Ministerio
de Ciencia, Tecnología e Innovación (Argentina), Ministério da Ciência,
Tecnologia, Inovações e Comunicações (Brazil), and Korea Astronomy and Space
Science Institute (Republic of Korea).

Table 2: Black widow Catalog ID | Other ID | R.A. (ICRS)aaCoordinates are taken from the ATNF pulsar database if pulsations have been detected, otherwise the best position of the optical counterpart is used. | Decl. (ICRS) | $P_{\rm{spin}}$ | $\dot{P_{\rm{obs}}}$ | $a\,\rm{sin}\,i$ | $\dot{E}$
---|---|---|---|---|---|---|---
|  | (h:m:s) | (${}^{\circ}:\arcmin:\arcsec$) | (ms) | ($10^{-20}$ s) | (lt-s) | ($10^{34}$ erg s-1)
PSR J0023+0923 | 4FGL J0023.4+0920 | 00:23:16.877498(8) | +09:23:23.8604(3) | 3.050203104754390(4) | 1.142345(14) | 0.03484136(5) | 1.6
PSR J0251+2606 | 4FGL J0251.0+2605 | 02:51:02.5537(5) | +26:06:09.97(2) | 2.5415543469461(3) | 0.7572(8) | 0.065681(8) | 1.8
PSR J0312–0921 | 4FGL J0312.1–0921 | 03:12:06.2 | –09:21:56 | 3.704 | 1.972 | $\cdots$ | 1.5
PSR J0610–2100 | 4FGL J0610.2–2100 | 06:10:13.595462(17) | –21:00:27.9313(4) | 3.8613247042986(6) | 1.235(6) | 0.0734891(4) | 0.85
PSR J0636+5129 | PSR J0636+5128 | 06:36:04.847464(5) | +51:28:59.96547(11) | 2.868952846616078(8) | 0.34483(7) | 0.00898636(6) | 0.58
PSR J0952–0607 | 4FGL J0952.1–0607 | 09:52:08.32141(5) | –06:07:23.490(2) | 1.41379835502312(12) | 0.4773(8) | 0.0626670(9) | 6.7
PSR J1124–3653 | 4FGL J1124.0–3653 | 11:24:01.116(3) | –36:53:19.10(4) | 2.41 | $\cdots$ | $\cdots$ | $\cdots$
PSR J1301+0833 | 4FGL J1301.6+0834 | 13:01:38.26 | +08:33:57.5 | 1.84 | $\cdots$ | $\cdots$ | $\cdots$
PSR J1311–3430 | 4FGL J1311.7–3430 | 13:11:45.7242(2) | –34:30:30.350(4) | 2.5603710316720(3) | 2.0964(14) | 0.010581(4) | 4.9
PSR J1446–4701 | 4FGL J1446.6–4701 | 14:46:35.712054(8) | –47:01:26.78210(14) | 2.194695780881595(15) | 0.98075(3) | 0.06401226(15) | 3.7
PSR J1513–2550 | 4FGL J1513.4–2549 | 15:13:23.32059(6) | –25:50:31.285(3) | 2.1190675651177(1) | 2.161(2) | 0.0408132(7) | 9.0
PSR J1544+4937 | 4FGL J1544.0+4939 | 15:44:04.48722(2) | +49:37:55.2545(2) | 2.15928839043292(5) | 0.2933(5) | 0.0328680(4) | 1.2
PSR J1555–2908 | 4FGL J1555.7–2908 | 15:55:40.6586(10 | –29:08:28.426(13) | 1.78750176696926(20) | 4.45502(4) | 0.1514468(1) | 31.0
PSR J1641+8049 | 4FGL J1641.2+8049 | 16:41:20.8381(4) | +80:49:52.9142(8) | 2.02117938468221(9) | 0.895(9) | 0.0640793(3) | 4.3
PSR J1653–0158 | 4FGL J1653.6–0158 | 16:53:38.05381(5) | –01:58:36.8930(5) | 1.9676820247057(2) | 0.2402(3) | 0.01071(1) | 1.2
PSR J1719–1438 |  | 17:19:10.07293(5) | –14:38:00.942(4) | 5.7901517700238(5) | 0.8043(7) | 0.0018212(7) | 0.16
PSR J1720–0533 |  | 17:19:55 | –05:30:05 | 3.27 | $\cdots$ | $\cdots$ | $\cdots$
PSR J1731–1847 | 4FGL J1731.7–1850 | 17:31:17.609823(17) | –18:47:32.666(3) | 2.34455954688568(11) | 2.5407(5) | 0.1201611(6) | 7.8
PSR J1745+1017 | 4FGL J1745.5+1017 | 17:45:33.8371(7) | +10:17:52.523(2) | 2.6521296710897(4) | 0.2729(15) | 0.088172(1) | 0.58
PSR J1745–2324 | PSR J1745–23 | 17:45:30(24) | -23:25(7) | 5.41669986(14) | $\cdots$ | 0.06247(6) | $\cdots$
PSR J1805+0615 | 4FGL J1805.6+0615 | 18:05:42.39969(3) | +06:15:18.606(13) | 2.1289064590218(5) | 2.2758(9) | 0.087728(15) | 9.3
PSR J1810+1744 | 4FGL J1810.5+1744 | 18:10:37.28(1) | +17:44:37.38(7) | 1.66 | $\cdots$ | 0.095 | $\cdots$
PSR J1833–3840 | 4FGL J1833.0–3840 | 18:33:04.6 | –38:40:46 | 1.87 | 1.773 | $\cdots$ | 11.0
PSR J1928+1245 |  | 19:28:45.39360(6) | +12:45:53.374(3) | 3.0216063479651(6) | 1.680(10) | 0.018951(1) | 2.4
PSR J1946–5403 | 4FGL J1946.5–5402 | 19:46:34.497(3) | –54:03:42.51(4) | 2.710 | $\cdots$ | 0.0435 | $\cdots$
PSR J1959+2048 | 4FGL J1959.5+2048 | 19:59:36.76988(5) | +20:48:15.1222(6) | 1.60740168480632(3) | 1.68515(9) | 0.0892253(6) | 16.0
| PSR B1957+20 |  |  |  |  |  |
PSR J2017–1614 | 4FGL J2017.7–1612 | 20:17:46.1478(8) | –16:14:15.51(5) | 2.3142872649224(4) | 0.245(5) | 0.043655(5) | 0.78
PSR J2047+1053 | 4FGL J2047.3+1051 | 20:47:10.246(3) | +10:53:07.80(4) | 4.29 | $\cdots$ | $\cdots$ | $\cdots$
PSR J2051–0827 | 4FGL J2051.0–0826 | 20:51:07.519768(18) | –08:27:37.7497(8) | 4.50864182000643(11) | 1.2733(7) | 0.0450720(3) | 0.55
PSR J2052+1219 | 4FGL J2052.7+1218 | 20:52:47.77803(15) | +12:19:59.022(5) | 1.98525628181868(8) | 0.67037(20) | 0.061377(4) | 3.4
PSR J2055+3829 |  | 20:55:10.306550(4) | +38:29:30.90571(6) | 2.08929030219107(3) | 0.09996(5) | 0.0452618(2) | 0.43
PSR J2115+5448 | 4FGL J2115.1+5449 | 21:15:11.7678(1) | +54:48:45.154(2) | 2.602876738872(2) | 7.49(1) | 0.044846(1) | 17.0
PSR J2214+3000 | 4FGL J2214.6+3000 | 22:14:38.853711(10) | +30:00:38.19160(14) | 3.119226581323024(12) | 1.47285(4) | 0.0590813(3) | 1.9
PSR J2234+0944 | 4FGL J2234.7+0943 | 22:34:46.854176(7) | +09:44:30.2224(3) | 3.627027895734199(12) | 2.00998(6) | 0.06842966(13) | 1.7
PSR J2241–5236 | 4FGL J2241.7–5236 | 22:41:42.0269841(10) | –52:36:36.239590(11) | 2.1866997725548446(10) | 0.689656(3) | 0.025795324(11) | 2.6
PSR J2256–1024 | 4FGL J2256.8–1024 | 22:56:56.39294(7) | –10:24:34.385(3) | 2.29453181696499(3) | 1.13535(10) | 0.08296575(5) | 3.7
PSR J2322–2650 |  | 23:22:34.64004(3) | –26:50:58.3171(6) | 3.46309917908790(11) | 0.05834(15) | 0.0027849(6) | 0.055
4FGL J0336.0+7502 |  | 03:36:10.1811 | +75:03:17.268 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
4FGL J0935.3+0901 |  | 09:35:20.719 | +09:00:35.90 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
ZTF J1406+1222 |  | 14:06:56.173(4) | +12:22:43.398(3) | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
4FGL J1408.6–2917 |  | 14:08:26.789 | –29:22:21.212 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
Table 3: Black widow Catalog ID | DM | CL02aaDistance using the pulsar dispersion measure and the Cordes & Lazio (2002) electron density model. | Y17bbDistance using the pulsar dispersion measure and the Yao et al. (2017) electron density model. | Other dist.ccParallax or light curve derived distance estimate. | References | Final Dist.ddFinal adopted distance.
---|---|---|---|---|---|---
| pc cm-3 | (kpc) | (kpc) | (kpc) |  | (kpc)
J0023+0923 | 14.3 | 0.69 | 1.25 | 1.1(2); 2.23(8) | (1); (2) | 1.1(2)
J0251+2606 | 20.2 | 0.82 | 1.17 | 3.26(10) | (2) | 1.2(4)
J0312–0921 | 20.5 | 0.87 | 0.82 |  $\cdots$ |  $\cdots$ | 0.8(2)
J0610–2100 | 60.7 | 3.54 | 3.26 | $2.24^{+0.70}_{-0.57}$ | (3) | $2.24^{+0.70}_{-0.57}$
J0636+5129 | 11.1 | 0.49 | 0.21 |  $1.1^{+0.6}_{-0.3}$; 1.05(1) | (1); (2) | $1.1^{+0.6}_{-0.3}$
J0952–0607 | 22.4 | 0.97 | 1.74 | $6.26^{+0.36}_{-0.40}$ | (57) | $6.26^{+0.36}_{-0.40}$
J1124–3653 | 44.9 | 1.72 | 0.99 | $2.72^{+0.10}_{-0.08}$ | (2) | 1.0(3)
J1301+0833 | 13.2 | 0.67 | 1.23 | $2.23^{+0.08}_{-0.13}$ | (2) | $2.23^{+0.08}_{-0.13}$
J1311–3430 | 37.8 | 1.41 | 2.43 | 2.6 | (4) | 2.4(7)
J1446–4701 | 55.8 | 1.46 | 1.57 |  $\cdots$ |  $\cdots$ | 1.6(5)
J1513–2550 | 46.9 | 1.95 | 3.96 |  $\cdots$ |  $\cdots$ | 4.0 $\pm$ 1.2
J1544+4937 | 23.2 | 1.23 | 2.99 | 2.0–5.0 | (5) | 3.0(9)
J1555–2908 | 75.9 | 2.65 | 7.56 | 5.1(2) | (6) | 5.1(2)
J1641+8049 | 31.1 | 1.65 | 3.04 |  $\cdots$ |  $\cdots$ | 3.0(9)
J1653–0158 |  $\cdots$ |  $\cdots$ |  $\cdots$ | 0.84(40); $1.00^{+1.31}_{-0.46}$ | (7); (8) | $1.00^{+1.31}_{-0.46}$
J1719–1438 | 36.9 | 1.21 | 0.34 |  $\cdots$ |  $\cdots$ | 0.3(1)
J1720–0533 | 36.8 | 1.35 | 0.19 |  $\cdots$ |  $\cdots$ | 0.2(1)
J1731–1847 | 106.5 | 2.55 | 4.78 |  $\cdots$ |  $\cdots$ | 4.8 $\pm$ 1.4
J1745+1017 | 244.9 | 1.26 | 1.21 |  $\cdots$ |  $\cdots$ | 1.2(4)
J1745–2324 | 24.0 | 4.48 | 7.94 |  $\cdots$ |  $\cdots$ | 7.9$\pm$2.4
J1805+0615 | 64.9 | 2.48 | 3.89 |  $\cdots$ |  $\cdots$ | 3.9$\pm$1.2
J1810+1744 | 39.7 | 2.00 | 2.36 | 3.03(1) | (9) | 3.03(1)
J1833–3840 | 78.6 | 2.05 | 4.65 |  $\cdots$ |  $\cdots$ | 4.7$\pm$1.4
J1928+1245 | 179.2 | 6.08 | 6.08 |  $\cdots$ |  $\cdots$ | 6.1$\pm$1.8
J1946–5403 | 23.7 | 0.87 | 1.15 |  $\cdots$ |  $\cdots$ | 1.2(4)
J1959+2048 | 29.1 | 2.49 | 1.73 |  $2.22^{+0.03}_{-0.02}$;$2.57^{+1.84}_{-0.77}$ | (10); (11) | $2.57^{+1.84}_{-0.77}$
J2017–1614 | 25.4 | 1.10 | 1.44 |  $\cdots$ |  $\cdots$ | 1.4(4)
J2047+1053 | 34.6 | 2.05 | 2.79 |  $\cdots$ |  $\cdots$ | 2.8(8)
J2051–0827 | 20.7 | 1.04 | 1.47 | 2.5(2) | (58) | 2.5(2)
J2052+1219 | 42.0 | 2.44 | 3.92 | 3.94(7) | (2) | 3.94(7)
J2055+3829 | 91.8 | 4.36 | 4.59 |  $\cdots$ |  $\cdots$ | 4.6$\pm$1.4
J2115+5448 | 77.4 | 3.39 | 3.11 |  $\cdots$ |  $\cdots$ | 3.1(9)
J2214+3000 | 22.5 | 1.54 | 1.67 | $0.4^{+0.2}_{-0.1}$ | (1) | $0.4^{+0.2}_{-0.1}$
J2234+0944 | 17.8 | 1.00 | 1.59 | $0.8^{+0.3}_{-0.2}$ | (1) | $0.8^{+0.3}_{-0.2}$
J2241–5236 | 11.4 | 0.51 | 0.96 | $1.24^{+0.04}_{-0.05}$ | (2) | $1.24^{+0.04}_{-0.05}$
J2256–1024 | 13.8 | 0.65 | 1.33 | 2.0(6) | (12) | 2.0(6)
J2322–2650 | 6.1 | 0.32 | 0.76 | $0.23^{+0.09}_{-0.05}$ | (13) | $0.23^{+0.09}_{-0.05}$
J0336.0+7502 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |
J0935.3+0901 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |
J1406+1222 |  $\cdots$ |  $\cdots$ |  $\cdots$ | 1.14(20) | (28) | 1.14(20)
J1408.6–2917 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |
Table 4: Black widow Catalog ID |  $F_{\gamma}$aa$\gamma$-ray flux from _Fermi_ -LAT over the energy range 0.1–100 GeV (Fermi-LAT collaboration et al., 2022b). | $L_{\gamma}$ |  $F_{X}$bbUnabsorbed 0.5–10 keV X-ray flux. | $\Gamma$ | ReferencesccReference for X-ray properties. | $L_{X}$
---|---|---|---|---|---|---
| ($10^{-12}$erg s-1 cm-2) | ($10^{33}$ erg s-1) | ($10^{-14}$erg s-1 cm-2) |  |  | ($10^{30}$ erg s-1)
J0023+0923 | 7.8(6) | 1.1(1) | $4.6^{+1.6}_{-1.1}$ | 3.3 $\pm$ 0.5 | (18) | $1.72^{+2.92}_{-1.28}$
J0251+2606 | 4.9(4) | 0.8(1) |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
J0312–0921 | 5.5(4) | 0.4(1) |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
J0610–2100 | 7.2(5) | 4.3(3) |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
J0636+5129 |  $\cdots$ |  $\cdots$ | $1.61^{+0.24}_{-0.24}$ | 2.4 $\pm$ 0.2 | (19) | $0.08^{+0.04}_{-0.03}$
J0952–0607 | 2.4(3) | 11.3 $\pm$ 1.4 | 0.652 | $2.51^{+0.53}_{-0.39}$ | (20) | $30.6^{+3.6}_{-3.8}$
J1124–3653 | 12.5(6) | 1.5(1) | $5.5^{+1.3}_{-1.0}$ | 2.1 $\pm$ 0.3 | (21) | $7.3^{+10.7}_{-5.2}$
J1301+0833 | 7.7(5) | 4.6(3) |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
J1311–3430 | 60.6 $\pm$ 1.2 | 41.8(8) | $6.04^{+0.28}_{-0.27}$ | $1.67^{+0.09}_{-0.09}$ | (22) | $55.7^{+74.3}_{-39.7}$
J1446–4701 | 7.7(7) | 2.4(2) | $1.9^{+2.2}_{-0.8}$ | $2.93^{+0.50}_{-0.42}$ | (18) | $8.5^{+36.1}_{-7.0}$
J1513–2550 | 7.6(6) | 14.6 $\pm$ 1.1 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
J1544+4937 | 2.4(3) | 2.5(3) |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
J1555–2908 | 4.7(6) | 14.5 $\pm$ 1.9 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
J1641+8049 | 2.0(3) | 2.1(3) |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
J1653–0158 | 34.3 $\pm$ 1.0 | 4.1(1) | $21.5^{+10.2}_{-6.8}$ | $1.65^{+0.39}_{-0.34}$ | (23); (24) | $18.1^{+11.2}_{-6.8}$
J1719–1438 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
J1720–0533 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
J1731–1847 | 5.2 $\pm$ 1.1 | 14.2 $\pm$ 3.0 | $0.63^{+0.62}_{-0.34}$ | $1.9^{+1.5}_{-1.3}$ | (18) | $17.2^{+51.3}_{-14.5}$
J1745+1017 | 7.6(6) | 1.3(1) |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
J1745–2324 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
J1805+0615 | 5.3(5) | 9.6 $\pm$ 1.0 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
J1810+1744 | 23.2(9) | 25.5(9) | $1.72^{+0.44}_{-0.35}$ | 2.2 $\pm$ 0.4 | (18) | $11.4^{+17.4}_{-8.3}$
J1833–3840 | 2.8(5) | 7.5 $\pm$ 1.2 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
J1928+1245 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
J1946–5403 | 9.8(5) | 1.7(1) | $2.87^{+1.39}_{-0.96}$ | $1.82^{+0.41}_{-0.40}$ | (27) | $4.95^{+2.39}_{-1.66}$
J1959+2048 | 15.7(9) | 12.4(7) | $5.60^{+0.24}_{-0.26}$ | 1.96 $\pm$ 0.12 | (25) | $44.3^{+91.7}_{-23.6}$
J2017–1614 | 6.5(6) | 1.5(1) |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
J2047+1053 | 4.3(6) | 4.0(5) | $1.2^{+0.8}_{-0.5}$ | 0.87 $\pm$ 0.68 | (18) | $11.2^{+25.7}_{-9.0}$
J2051–0827 | 2.5(3) | 1.9(4) | $0.30^{+0.24}_{-0.13}$ | 4.1 $\pm$ 0.7 | (18) | $2.24^{+1.80}_{-0.97}$
J2052+1219 | 4.6(6) | 8.5 $\pm$ 1.0 | $0.34^{+0.43}_{-0.27}$ | $2.9^{+1.6}_{-1.2}$ | (27) | $6.24^{+7.96}_{-5.00}$
J2055+3829 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
J2115+5448 | 7.0(7) | 8.1(8) | $2.5^{+1.2}_{-0.8}$ | $3.4^{+1.1}_{-0.9}$ | (27) | $28.4^{+13.8}_{-9.1}$
J2214+3000 | 32.6(7) | 0.6(1) | $1.81^{+0.93}_{-0.56}$ | 3.8 $\pm$ 0.4 | (18) | $0.78^{+1.59}_{-0.59}$
J2234+0944 | 10.0(6) | 0.8(1) |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
J2241–5236 | 25.0 $\pm$ 1.1 | 4.6(2) | $2.60^{+0.83}_{-0.58}$ | 2.8 $\pm$ 0.4 | (26) | $2.87^{+4.72}_{-2.11}$
J2256–1024 | 8.2(5) | 3.9(2) | $2.39^{+0.42}_{-0.36}$ | 2.9 $\pm$ 0.3 | (18) | $5.07^{+6.83}_{-3.59}$
J2322–2650 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
J0336.0+7502 | 8.08(52) |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
J0935.3+0901 | 4.56(54) |  $\cdots$ | 12.75 $\pm$ 1.32 | $1.88^{+0.25}_{-0.22}$ | (31) |  $\cdots$
J1406+1222 |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
J1408.6–2917 | 5.21(69) |  $\cdots$ | $2.4^{+1.6}_{-1.1}$ | $1.66^{+0.55}_{-0.48}$ | (27) |  $\cdots$
Table 5: Black widow Catalog

ID | Orbital Period | Modeled | Spectroscopy? |  $M_{c}^{\rm{min}}$aaMinimum companion mass assuming $i=90^{\circ}$ and a neutron star mass of $1.4\,M_{\odot}$. |  $M_{c}$bbBest-fit companion mass from modeling the optical photometry. If no light curve exists we adopt the median companion mass assuming $i=60^{\circ}$. | References | Discovery | References
---|---|---|---|---|---|---|---|---
| (days) | Photometry? | Emission Lines (Y/N) | ($M_{\odot}$) | ($M_{\odot}$) |  |  |
J0023+0923 | 0.13879914382(4) | ✓ | – | 0.016 | 0.018 | (2) | Radio follow-up of _Fermi_ | (32)
J0251+2606 | 0.2024406403(9) | ✓ | – | 0.024 | 0.032 | (2) | Radio follow-up of _Fermi_ | (33)
J0312–0921 | 0.0975 | – | – | 0.009 | 0.010 |  $\cdots$ | Radio follow-up of _Fermi_ | (55)
J0610–2100 | 0.2860160068(6) | ✓ | – | 0.021 | 0.022 | (3) | Parkes survey | (34)
J0636+5129 | 0.066551340763(16) | ✓ | – | 0.007 | 0.018 | (2) | GBT survey | (35)
J0952–0607 | 0.2674610347(5) | ✓ | ✓(N) | 0.019 | 0.026; 0.032 | (14); (57) | Radio follow-up of _Fermi_ | (36)
J1124–3653 | 0.2291666 | ✓ | – |  $\cdots$ | 0.041 | (2) | Radio follow-up of _Fermi_ | (32)
J1301+0833 | 0.27 | ✓ | ✓(N) |  $\cdots$ | 0.045 | (2) | Radio follow-up of _Fermi_ | (37)
J1311–3430 | 0.0651157335(7) | ✓ | ✓(Y) | 0.008 | 0.0104 | (4) | Opt/X-ray search of _Fermi_ | (38)
J1446–4701 | 0.27766607699(15) | – | – | 0.019 | 0.022 |  $\cdots$ | Parkes survey | (39)
J1513–2550 | 0.1786354505(8) | – | – | 0.016 | 0.019 |  $\cdots$ | Radio follow-up of _Fermi_ | (40)
J1544+4937 | 0.1207729895(1) | ✓ | – | 0.017 | 0.025 | (5) | Radio follow-up of _Fermi_ | (41)
J1555–2908 | 0.23350026854(11) | ✓ | ✓(N) | 0.051 | $0.060^{+0.005}_{-0.003}$ | (6) | Radio follow-up of _Fermi_ | (42)
J1641+8049 | 0.0908739634(1) | – | – | 0.040 | 0.047 |  $\cdots$ | GBT survey | (35)
J1653–0158 | 0.0519447575(4) | ✓ | ✓(Y) | 0.010 | 0.013 | (7) | Opt/X-ray search of _Fermi_ | (24); (43)
J1719–1438 | 0.0907062900(12) | – | – | 0.0011 | 0.0013 |  $\cdots$ | Parkes survey | (44)
J1720–0533 | 0.131666 | – | – |  $\cdots$ |  $\cdots$ |  $\cdots$ | FAST survey | (56)
J1731–1847 | 0.3111341185(10) | – | – | 0.033 | 0.039 |  $\cdots$ | Parkes survey | (45)
J1745+1017 | 0.730241444(1) | – | – | 0.014 | 0.016 |  $\cdots$ | Radio follow-up of _Fermi_ | (46)
J1745–2324 | 0.165562(10) | – | – | 0.027 | 0.030 |  $\cdots$ | Parkes survey | (47)
J1805+0615 | 0.3368720310(48) | – | – | 0.023 | 0.027 |  $\cdots$ | Radio follow-up of _Fermi_ | (33)
J1810+1744 | 0.14817083 | ✓ | ✓(N) | 0.043 | 0.065 | (17); (9) | Radio follow-up of _Fermi_ | (32)
J1833–3840 | 0.900 | – | – |  $\cdots$ |  $\cdots$ |  $\cdots$ | Parkes survey |  $\cdots$
J1928+1245 | 0.1366347269(8) | – | – | 0.009 | 0.010 |  $\cdots$ | Arecibo survey | (48)
J1946–5403 | 0.130 | – | – | 0.021 | 0.025 |  $\cdots$ | Radio follow-up of _Fermi_ | (49)
J1959+2048 | 0.3819666069(8) | ✓ | ✓(Y) | 0.021 | 0.036 | (2) | Arecibo survey | (50)
J2017–1614 | 0.0978252578(4) | – | – | 0.026 | 0.030 |  $\cdots$ | Radio follow-up of _Fermi_ | (40)
J2047+1053 | 0.12 | – | – | 0.036 | 0.042 |  $\cdots$ | Radio follow-up of _Fermi_ | (37)
J2051–0827 | 0.09911025490(4) | ✓ | – | 0.027 | $0.039^{+0.010}_{-0.011}$ | (58) | Parkes survey | (51)
J2052+1219 | 0.1146136251(2) | ✓ | – | 0.033 | 0.042 | (2) | Radio follow-up of _Fermi_ | (33)
J2055+3829 | 0.12959037294(1) | – | – | 0.022 | 0.027 | (16) | NRT survey | (16)
J2115+5448 | 0.135322188(3) | – | – | 0.022 | 0.025 |  $\cdots$ | Radio follow-up of _Fermi_ | (40)
J2214+3000 | 0.41663294591(20) | – | – | 0.013 | 0.015 |  $\cdots$ | Radio follow-up of _Fermi_ | (52)
J2234+0944 | 0.41966003706(17) | – | – | 0.015 | 0.018 |  $\cdots$ | Radio follow-up of _Fermi_ | (37)
J2241–5236 | 0.14567224025(2) | ✓ | – | 0.012 | 0.016 | (2) | Radio follow-up of _Fermi_ | (53)
J2256–1024 | 0.21288263050(7) | ✓ | – | 0.030 | 0.032 | (17) | Radio follow-up of _Fermi_ | (32)
J2322–2650 | 0.322963997(6) | – | – | 0.00074 | 0.00086 |  $\cdots$ | Parkes survey | (13)
J0336.0+7502 | 0.15492408(38) | ✓ | – |  $\cdots$ |  $\cdots$ | (29) | Opt/X-ray search of _Fermi_ | (29)
J0935.3+0901 | 0.10153276(36) | ✓ | ✓(Y) |  $\cdots$ |  $\cdots$ | (30) | Opt/X-ray search of _Fermi_ | (54)
J1406+1222 | 0.043056621(2) | ✓ | ✓(Y) |  $\cdots$ |  $\cdots$ | (28) | Optical survey | (28)
J1408.6–2917 | 0.14261385(150) | ✓ | ✓(Y) |  $\cdots$ |  $\cdots$ | (27) | Opt/X-ray search of _Fermi_ | (27)

††footnotetext: References. (1) Arzoumanian et al. (2018); (2) Draghis et al.
(2019); (3) van der Wateren et al. (2022); (4) Romani et al. (2015); (5) Tang
et al. (2014); (6) Kennedy et al. (2022); (7) Nieder et al. (2020); (8) Gaia
Collaboration et al. (2022); (9) Romani et al. (2021); (10) Kandel & Romani
(2020a); (11) Romani et al. (2022b); (12) Crowter et al. (2020); (13) Spiewak
et al. (2018); (14) Nieder et al. (2019); (15) Stappers et al. (1999); (16)
Guillemot et al. (2019); (17) Breton et al. (2013); (18) Arumugasamy et al.
(2015); (19) Spiewak et al. (2016); (20) Ho et al. (2019); (21) Gentile et al.
(2014); (22) An et al. (2017); (23) Cheung et al. (2012); (24) Romani et al.
(2014); (25) Huang et al. (2012); (26) An et al. (2018); (27) this work; (28)
Burdge et al. (2022); (29) Li et al. (2021); (30) Halpern (2022); (31) Zheng
et al. (2022); (32) Hessels et al. (2011); (33) Cromartie et al. (2016); (34)
Burgay et al. (2006); (35) Stovall et al. (2014); (36) Bassa et al. (2017);
(37) Ray et al. (2012); (38) Romani (2012); (39) Keith et al. (2012); (40)
Sanpa-Arsa (2016); (41) Bhattacharyya et al. (2013); (42) Ray et al. (2022);
(43) Kong et al. (2014); (44) Bailes et al. (2011); (45) Bates et al. (2011);
(46) Barr et al. (2013); (47) Cameron et al. (2020); (48) Parent et al.
(2019); (49) Camilo et al. (2015); (50) Fruchter et al. (1988); (51) Stappers
et al. (1996b); (52) Ransom et al. (2011); (53) Keith et al. (2011); (54) Wang
et al. (2020); (55) Tabassum et al. (2021); (56) Wang et al. (2021); (57)
Romani et al. (2022a); (58) Dhillon et al. (2022)

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##  Appendix A An Unrelated AGN

Here we discuss the properties of the other X-ray source found within the
error ellipse of 4FGL J1408.6–2917. Based on the results presented below, we
classify this source as an AGN that is unrelated to the $\gamma$-ray source.

In the _Swift_ observations described in Sec. 2.2.1, this source was detected
with $\sim$8 counts, corresponding to a 0.3–10 keV flux of
$1.0^{+2.0}_{-0.8}\times 10^{-13}$ erg cm-2 s-1. In our _XMM_ observation of
this region, this source is also present, with an unabsorbed 0.3–10 keV flux
of $1.5^{+0.3}_{-0.3}\times 10^{-13}$ erg cm-2 s-1. This is marginally
brighter than in the _Swift_ data, but within the large uncertainties of that
shallower data. Fitting the _XMM_ /EPIC spectrum as described in Sec. 2.2.2,
we find the source has a very hard spectrum with $\Gamma=1.18\pm 0.18$.

The X-ray source matches spatially to a single candidate optical counterpart,
a faint Pan-STARRS (Flewelling et al., 2020) source with $r^{\prime}=21.6$,
listed as Pan-STARRS ID 72822121474471499. The optical color of this source is
nominally blue ($g^{\prime}-r^{\prime}=-0.09\pm 0.26$) though with a large
uncertainty. The source is detected in _WISE_ $W1$ and $W2$ filters and has an
extremely red Pan-STARRS/WISE color of $r^{\prime}-W1=5.0\pm 0.3$ and a WISE
color of $W1-W2=0.85$. These photometric properties and the hard X-ray photon
index are consistent with the classification of this source as a background
AGN (e.g., Stern et al. 2012).

Since this source (Pan-STARRS 72822121474471499) is somewhat closer to the
center of the 4FGL error ellipse than the black widow J1408, we considered the
possibility that this AGN was instead the counterpart of the _Fermi_ -LAT
$\gamma$-ray source. This is very unlikely solely based on the existence of
the black widow candidate J1408: such binaries are rare and there is no other
variable class that can mimic the short period, large amplitude, and emission
lines we observe. To serendipitously find one within a _Fermi_ -LAT error
ellipse, but have it not be associated with the GeV source, would be
extraordinarily unlikely.

Focusing on the properties of the AGN, it has a lower X-ray flux and softer
$\gamma$-ray spectrum than most _Fermi_ -detected AGN, and the lack of
$\gamma$-ray variability and slight evidence for $\gamma$-ray spectrum
curvature also provide additional evidence that the 4FGL source is not
associated with an AGN (Ajello et al., 2020). Since the high-latitude source
density of X-ray sources down to the flux of Pan-STARRS 72822121474471499 is
$\sim 10$–20 per deg2 (e.g., Carrera et al. 2007), there is a $\sim 15-30$%
probability of a chance coincidence between a _Fermi_ error ellipse the size
of that for 4FGL J1408.6–2917 and an unrelated X-ray source of this flux
level. Since the bulk of these X-ray sources are background AGN, it is
reasonable to expect that many high-latitude _Fermi_ -LAT sources will have
error ellipses that encompass unrelated background AGN, as appears to be the
case for 4FGL J1408.6–2917.