# A light redback companion of PSR J1622$-$0315 and irradiation power in
spider systems

Y. X. Jane Yap  Institute of Astronomy, National Tsing Hua University, Hsinchu
30013, Taiwan R.O.C Albert K. H. Kong  Institute of Astronomy, National Tsing
Hua University, Hsinchu 30013, Taiwan R.O.C Kwan-Lok Li  Department of
Physics, National Cheng Kung University, Tainan 701, Taiwan R.O.C

###### Abstract

We report optical observations of the millisecond pulsar binary system PSR
J1622$-$0315 with the Lulin 1m telescope in Taiwan and the Lijiang 2.4m
telescope in China between 2019 and 2021. The companion of the pulsar, which
is of V$\sim$19 mag, showed ellipsoidal-distorted orbital variations in its
light curves. The best-fit model to the light curves, with the binary code
PHOEBE, gives a companion mass of 0.122$\pm 0.006\;\\!{\rm M}_{\odot}$. This
places PSR J1622$-$0315 in the spider-system subclass. We compared the
properties of PSR J1622$-$0315 with other spider pulsar binaries for the
scalings between the spin-down luminosity derived for the pulsar, irradiation
luminosity of the companion, and X-ray luminosity of the binary. We find that
pulsar irradiation in PSR J1622$-$0315 is insignificant and the irradiation
luminosity of the transitional millisecond pulsars PSR J1023$+$0038 and PSR
J1227$-$4853 are the highest among the redback systems.

Compact binary stars(283); Low-mass x-ray binary stars(939); Millisecond
pulsars(1062)

##  1 Introduction

Spider pulsar systems are compact binaries containing a millisecond pulsar
(MSP), with a low-mass companion star, orbiting around each other in a period
of $P_{\rm b}\lesssim 24~{}{\rm hr}$. They are usually classified as black
widows (BWs) or redbacks (RBs). The companion stars generally have masses
$\lesssim 0.1\;\\!{\rm M}_{\odot}$ and $\sim 0.1-0.4\;\\!{\rm M}_{\odot}$ for
BW and RB, respectively (see e.g. Chen et al., 2013; Roberts, 2013). A rapid-
spinning MSP is believed to be a phenomenon caused by the accretion of
material from the companion star111known as the recycling scenario (Alpar et
al., 1982). This scenario is supported by observations which showed state
transition(s) between accretion-powered and rotation-powered (pulsar) state in
three transitional MSPs (tMSPs): PSR J1023$+$0038 (Archibald et al., 2009),
PSR J1227$-$4853 (Bassa et al., 2014) and PSR J1824$-$2452I (Papitto et al.,
2013).

The light curve of the companion star in spider systems contains information
about the irradiation of the system and the companion’s stellar properties.
The effect of strong irradiation is observed in a few RBs and BWs (e.g.,
Breton et al., 2013; Draghis et al., 2019). Pulsar irradiation also causes
evaporation of the companion star and results in the mass loss of the star
(van den Heuvel & van Paradijs, 1988). The evolution history of a companion
star in spider systems evolving from a low-mass X-ray binary (LMXB) system
under ablation was discussed in Chen et al. (2013). The interaction between
the magnetodipole radiation and a disk was proposed to explain the neutron
star rotation properties in these systems, and the model was applied in
transitional systems during the rotation-powered state. (Burderi et al., 2001;
Papitto & Torres, 2015).

In this work, we report new optical observations of PSR J1622$-$0315\. We use
Markov Chain Monte Carlo (MCMC) sampling to explore the parameter space in the
models we use to fit the light curves, and to estimate the uncertainties of
the masses of the components and the orbital properties of the system. We
constructed a table of archival values to compare the spider systems in terms
of their irradiation luminosities ($L_{\rm irr}$) inferred from the light
curves, X-ray luminosities ($L_{\rm x}$) of the systems, and spin-down
luminosities ($\dot{E}$) of the pulsars.

We introduce our target and summarize our observations in Section 2. In
Section 3, we describe our light curve analysis using the eclipsing binary
modeling code PHOEBE (Prša et al., 2016). In Section 4, we discuss the
empirical relationship between $L_{\rm irr}$, $L_{\rm x}$ and $\dot{E}$ in
BWs, RBs and tMSPs. We summarize and discuss the implications of this work in
Section 5.

Table 1: Observation details of MSP J1622$-$0315

Telescope |  Date |  filter (SDSS) |  exposure time (s) |  duration (mins)
---|---|---|---|---
Lulin-1m |  2019 February 2 |  $r^{\prime},~{}g^{\prime}$ |  180, 300 |  120
|  2019 February 3 |  $r^{\prime},~{}g^{\prime}$ |  180, 300 |  120
|  2019 April 6 |  $r^{\prime},~{}g^{\prime}$ |  180, 300 |  180
|  2021 July 4† |  $r^{\prime},~{}g^{\prime}$ |  180, 300 |  36
Lijiang-2.4m |  2019 March 10 |  $r^{\prime},~{}g^{\prime}$ |  180, 300 |  80

† Opportunity allowed for the collection of additional quality data on July 4,
2021 with the LOT for slightly more than half an hour.

##  2 Target and observations

###  2.1 PSR J1622$-$0315

PSR J1622$-$0315 is a binary MSP discovered by Sanpa-Arsa (2016) using the
Green Bank Telescope (GBT) and the Nançay telescope. It has a spin period of
3.845 ms, and it is located at coordinates RA=16h22m59s.6 and
DEC=-03°15′37.3″(J2000). The system has an orbital period of 3.9 hr and the
dispersion measure is 21.4pc cm-3 ($\sim$20% uncertainties; Sanpa-Arsa 2016).
Radio flux variations, possibly caused by scattering with the ISM or ejected
particles from the companion were also reported (Sanpa-Arsa, 2016). An optical
counterpart was identified with the MDM Observatory located in Arizona, United
States, and its light curve exhibits an ellipsoidal variation. From the pulsar
timing, the mass function was obtained and a companion mass of
$>$0.1$\;\\!{\rm M}_{\odot}$ was derived, assuming a neutron star (NS) mass of
1.35$\;\\!{\rm M}_{\odot}$ and an edge-on orbit. No strong emission lines were
observed in the optical spectrum of the companion of PSR J1622$-$0315 (Strader
et al., 2019). Measurements of the companion radial velocity further allowed
the neutron star mass to be constrained to $>$1.45$\;\\!{\rm M}_{\odot}$ for
an edge-on orbit or, alternatively, constrained the inclination to $>$64° if
the pulsar is less massive than 2.0$\;\\!{\rm M}_{\odot}$. Gentile (2018)
studied the X-ray light curve and spectrum of PSR J1622$-$0315 with XMM-
Newton. Although the X-ray count rate was the lowest at the superior
conjunction of the pulsar, the orbital modulation was not obvious. The X-ray
spectrum was fitted with a power law of photon index $\Gamma$=2.0 and there
was no strong indication of a thermal component. Additionally, folded Fermi-
LAT observation indicated plausible $\gamma$-ray pulsation (Sanpa-Arsa, 2016).

###  2.2 Observations and timing analysis

We performed a series of targeted observations using the Lulin 1-m telescope
(LOT) in Taiwan and the Lijiang 2.4m telescope in mainland China during the
first quarter of 2019 (complemented with a shorter epoch obtained in July
2021). We used the SDSS r’ and g’ filters, and alternated them between each
exposure. The details of the observations are listed in Table 1. The
combination of our observations allowed us to cover all orbital phases, with
data spanning over a few months. We only analyzed the observations which
occurred when the weather conditions allowed a seeing better than
FWHM$\lesssim$ 3″. The reported light curves include 68$\times$180 s exposures
of ${r^{\prime}}$ images and 66$\times$300 s exposures of ${g^{\prime}}$
images. Raw images were calibrated with bias and flat frames using the IRAF
V2.15 standard pipelines. We performed photometry for all image frames using
the IRAF package PHOT. We selected eight spherical, unsaturated point sources
that are relatively isolated, as comparison stars, for all image frames.
Source counts were extracted from circular apertures with 4″ radii for the
target and comparison stars in the field. We extracted background counts from
annular regions centered at the positions of the stars, with inner radii of 8″
and width of 2″. The photometry was performed using constant weighting, and
the background rate was calculated using the centroid sky fitting algorithm.
We then computed the differential photometry of PSR J1622-0315 by subtracting
the average magnitudes of the comparison stars from the instrumental
magnitudes of the target. The propagated errors of the measurements are shown
in the error bars.

To convert the differential magnitude into apparent magnitude, we selected
another stable bright star222GAIA EDR3 ID 4358429221667653632 as a reference
star and calculated its differential magnitudes with the same differential
photometry process described above. The differential magnitudes of the
reference star varied less than 0.1 mag among the frames, which is sufficient
for magnitude calibration. The apparent magnitude of the reference star is
obtained from the Sloan Digital Sky Survey (SDSS) Data Release 14 catalog. The
difference between the differential magnitude and the apparent magnitude of
the reference star is used to calibrate the light curve of PSR J1622-0315.

The date is converted to Barycentric Julian Day (BJD) time before performing
timing analysis. We performed a Lomb-scargle periodogram analysis on the
selected light curves using astropy.timeseries. We found a period of 3.88(5)
hr in the light curves (c.f. Figure 1). This period is consistent with the
more precise pulsar timing solution previously reported (0.1617006798 d;
Sanpa-Arsa 2016), therefore we adopted the latter in the following modeling
analysis. We use $T_{0}$ (ascending node of the companion) published in
Strader et al. (2019) to fold the data. The light curve obtained is shown in
Figure 2.

Figure 1:  Lomb-Scargle periodogram of the combined light curves taken in 2019
for PSR J1622$-$0315\. The peak with the strongest amplitude is located at a
frequency that corresponds to a period of 0.162(2) days, indicating the
orbital period of the system. Our result is consistent with the previously
reported orbital period obtained using radio timing, which is indicated by a
dotted line.  Figure 2: The folded light curve of PSR J1622-0315 companion in
${r^{\prime}}$(top) and ${g^{\prime}}$(bottom) band. Phase 0 corresponds to
the inferior conjunction of the companion. The black solid line is the best-
fit model from PHOEBE. The most recent dataset (2021 July 4) is not included
in the fit. The gray solid line demonstrates the effect of irradiation on the
companion star surface due to heating from the pulsar; the effect is most
obvious at the companion superior conjunction where most of the extra
irradiation is seen. Figure 3: Projected view of the companion star across
different orbital phases, including its effective temperature distribution.
This model corresponds to the parameter values reported in Table 3.

##  3 Light curve modeling

We conducted an inverse modeling analysis following the method described in
Conroy et al. (2020) using PHOEBE 2.3. To begin, the apparent magnitudes are
converted to the integrated fluxes using the asinh softening parameters and
zero points provided in the SDSS
documentation.333http://classic.sdss.org/dr6/algorithms/fluxcal.html The
${r^{\prime}}$ and ${g^{\prime}}$ light curves are loaded into PHOEBE as
separate light curve datasets with the passband set to their respective
colors. We used data taken between February and April 2019, which covered the
entire orbit for our analysis. The most recent dataset (2021 July 4) is not
included in the modeling because it was taken substantially later compared to
other datasets, to avoid possible changes in the system during this period.
The position of the target with respect to the location of the telescope
prevented us from achieving full orbital coverage in a single observing
session of one night. While there were fluctuations between sessions, the
overall ellipsoidal modulation is clearly seen in the folded light curves. We
fixed the orbital period (Pb) and the superior conjunction of the pulsar
($T_{0,\rm{sup\\_conj}}$) using the adopted values mentioned in Sections 2.2,
with an additional phase shift of 0.25 to account for the different definition
of phase 0 used in PHOEBE. We also adopted the projected semi-major axis of
0.219258 lt-s from Sanpa-Arsa (2016).

We modeled PSR J1622$-$0315 to resemble a semi-detached binary that consists
of one primary (subscript ‘1’; pulsar) and one secondary (subscript ‘2’;
companion) component, where the companion star has a radius approaching the
maximum allowed value before overflowing the Roche-lobe. We assumed no eclipse
in the system (“eclipse method”=“only horizon”), since the light curve
variation mainly comes from ellipsoidal distortion. The passband luminosity
mode is set to “absolute” to avoid any flux rescaling. For the passband limb-
darkening correction, we use a logarithmic law for the secondary star, and the
coefficient is derived from the atmosphere model (Castelli & Kurucz, 2004)
“ck2004” implemented in PHOEBE. Since the optical emission from a pulsar is
negligible, we can turn off the primary star contribution by setting
“distortion method” to “none”. We also assumed a convective envelope for the
companion star where the bolometric gravity darkening coefficient $\beta$ is
0.08. This value is also used in the studies of similar systems (e.g. Bellm et
al., 2016). The modeled system and the temperature distribution of the
companion is shown in Figure 3. The inner tip of the companion star appears
coldest compared to the rest of the surface; this is because the effective
gravity near the tip is among the lowest under the influence of the tidal
effect. The inner tip would appear brighter if significant pulsar irradiation
were present.

Figure 2 also illustrates a light curve (in gray) with irradiation on the
surface of the companion star caused by the pulsar. This is done by setting
the irradiation method to “wilson” (Wilson, 1990). For the primary star
inputs, we changed the distortion method back to “roche”, and applied the
effective temperature and the radius of a neutron star to allow the
irradiation effect to take place. The flux of the primary star heats the
surface of the secondary star, and results in an increase of temperature,
especially near the inner tip of the secondary star. As shown in the light
curve, the irradiation in this system is not significant (the difference in
the computed loglikehoood is smaller than 0.01%); if added to the model, the
effect is mostly observed at the superior conjunction of the companion star. A
more severe irradiation will not reproduce the data well. Due to the
uncertainty in the irradiation physics for spider MSPs, this effect is not
included in the following MCMC analysis.

###  3.1 MCMC sampling

We performed a MCMC study for six system parameters: mass ratio (qbinary;
$M$comp/$M$NS), companion’s effective temperature (Teff,2), system inclination
(ibinary), companion radius (Requiv,2), extinction parameter (AV) and distance
(d). We used the EMCEE (Foreman-Mackey et al., 2019) solver implemented in
PHOEBE. We note that although the filling factor is more commonly used in the
spider community, it is not a default parameter for detached binary models in
PHOEBE. However, a volume-averaged filling factor can be derived using the
critical Roche radius provided by the model.

We assigned uniform priors for qbinary, Teff,2, ibinary, AV and d, and a
Gaussian prior for Requiv,2. The ranges for each of the uniform priors are
shown in Table 2. We chose a uniform prior for qbinary with limits ranging
from 0.04 to 0.12 for pulsar mass ranges from 1.4 ${\rm M}_{\odot}$ to 2.0
${\rm M}_{\odot}$. Based on the binary mass function value of 1.27 ${\rm
M}_{\odot}$ (Strader et al., 2019), we set the lower limit for the system
inclination at 50$\arcdeg$ to avoid fitting a large neutron star mass ($>$3
${\rm M}_{\odot}$). For Teff,2, the limits are chosen based on the wide range
of temperatures observed in RB companions. The $E$(B$-$V) color excess is 0.25
mag at the given coordinate and distance, by examining the Bayestar3D dust map
through the python package dustmaps (Green et al., 2019). The color excess is
then converted to AV using $R_{\rm V}=3.1$. We set a uniform prior between 0.7
and 0.9 mag. Due to the degeneracy between AV, Teff,2 and d, it is necessary
to incorporate more than one color information in order to achieve a better
fit. The distance obtained from Bailer-Jones et al. (2021) is 5843 pc with
uncertainties range from 1664 pc to 7766 pc. For Requiv,2, we use a Gaussian
prior with a mean value of 0.27$\;\\!{\rm R}_{\odot}$, which corresponds to
roughly 90$\%$ of the critical Roche radius.

An initial sample of fitting parameters is drawn from Gaussian distributions
centered around the mean of the priors. The initial sample distributions and
prior distributions are chosen such that the desired parameter space can be
fully explored within a reasonable computational time. We exposed the failed
walkers (due to physical or backend limitations) to make sure that the
parameter space is successfully covered. As demonstrated in Conroy et al.
(2020), these limitations are used to constrain the parameter search space if
no priors are provided explicitly.

Lower limit | Parameter | Upper limit
---|---|---
0.04 | qbinary | 0.12
3500 |  Teff,2 (K) | 7000
50 |  ibinary (∘) | 90
0.7 |  AV (mag) | 0.9
1.664 |  d (kpc) | 7.766
Table 2: Uniform priors are used for the above fitting parameters. The prior
limits are discussed in the text.  Figure 4: Corner plot of the posterior
distributions for the fitted parameters. The median values are displayed with
the respective 1$\sigma$ uncertainties, indicated by dotted lines. Degeneracy
is observed in the following parameters: i) $q_{\rm binary}$ and $i_{\rm
binary}$ ii) $T_{\rm eff,secondary}$ and $A_{\rm V}$, iii) $R_{\rm secondary}$
and $d$. Given the data quality, we were unable to constrain the extinction
parameter, $A_{\rm V}$. Figure 5: Propagated distribution of parameters
$M_{\rm primary}$ (NS) and $M_{\rm secondary}$ (companion). The median values
are displayed with the respective 1$\sigma$ uncertainties.

###  3.2 MCMC results

We ran 30 walkers with 10600 iterations, where around 3000 steps were dropped
as burn-in. Figure 4 shows the corner plot of the posterior distributions. We
obtained a mass ratio of $\sim$0.066, which is similar to that reported in
Strader et al. (2019) using a radial velocity measurement. The existing
degeneracy between the mass ratio and the inclination parameter is observed.
The sampled density for the companion’s effective temperature peaks around
6400 K. We note that the fitted effective temperature is higher than most
known RB companions ($\sim$3300$-$5500 K; see references in Table 4). The
extinction parameter, $A_{\rm V}$, is not well constrained due to the large
uncertainties in our data. We obtained a system distance of $\sim$2.1 kpc and
a companion star radius of $\sim$0.28 ${\rm R}_{\odot}$. The determination of
the distance of the system is affected by the companion star radius parameter.
If the radius increases, a larger distance is required to fit the light curve.
The maximum Roche radius, as a result of the orbital parameters and the mass
ratio, is 0.283 ${\rm R}_{\odot}$. Hence, the volume-averaged filling factor
is close to 1. A semi-detached system with a close-to-unity filling factor is
common among spider systems (De Vito et al., 2020). The binary masses are
derived from the fitted mass ratio and inclination (c.f. Figure 5). The mass
ratio implies a 1.84-$\textup{M}_{\odot}$ pulsar and a
0.122-$\textup{M}_{\odot}$ companion with a high inclination of
$\sim$80$\arcdeg$. All best-fit parameters are shown in Table 3. This result
suggests that the PSR J1622$-$0315 companion is the lightest RB companion
known so far.

Fit parameter(s) |  Result(s)
---|---
qbinary [$M$comp/$M$NS] |  $0.066_{-0.003}^{+0.005}$
Teff,2 (K) |  $6383_{-98}^{+89}$
ibinary (∘) |  $78.1_{-6.9}^{+7.9}$
Requiv,2 (${\rm R}_{\odot}$) |  $0.279_{-0.004}^{+0.004}$
AV (mag) |  $0.81_{-0.07}^{+0.06}$
d (kpc) |  $2.06_{-0.04}^{+0.04}$
Derived parameter(s) |
Mprimary (${\rm M}_{\odot}$) |  $1.84_{-0.19}^{+0.19}$
Msecondary (${\rm M}_{\odot}$) |  $0.122_{-0.006}^{+0.007}$
Table 3: Best-fit results with 1$\sigma$ uncertainties.

Item |  Designation |  BW/RB (a) |  P |  $\dot{P}$ |  Companion |  $L_{\rm irr}~{}$ |  $\log_{10}\;\\!(L_{\rm x}/$ |  $\log_{10}\;\\!(\dot{E}/$ |  References
---|---|---|---|---|---|---|---|---|---
|  |  |  (ms) |  ($\times 10^{-21}$ s s-1) |  mass (${\rm M}_{\odot}$) |  ($\times 10^{34}$ erg s-1) |  erg s-1) |  erg s-1) |
1 |  PSR J0023$+$0923 |  BW (1) |  3.05 |  9.61 |  $0.018^{+0.002}_{-0.001}$ |  $0.296^{+0.016}_{-0.013}$ |  $29.35^{+0.47}_{-0.53}$ |  34.13 |  1,2
2 |  PSR J0251$+$2606 |  BW (1) |  2.54 |  7.57 |  $0.032^{+0.006}_{-0.003}$ |  $0.252^{+0.012}_{-0.011}$ |  - |  34.26 |  1,4
3 |  PSR J0636$+$5128 |  BW (1) |  2.87 |  3.35 |  $0.018^{+0.001}_{-0.001}$ |  $0.097^{+0.001}_{-0.001}$ |  $27.93^{+1.18}_{-0.82}$ |  33.75 |  1,5
4 |  PSR J0952$-$0607 |  BW (1) |  1.41 |  4.56 |  $0.029^{+0.001}_{-0.001}$ |  $2.043^{+0.077}_{-0.076}$ |  30.47 |  34.81 |  1,7,8,9
5 |  PSR J1124$-$3653 |  BW (1) |  2.41 |  1.41 |  $0.041^{+0.001}_{-0.003}$ |  $0.286^{+0.008}_{-0.010}$ |  $30.56^{+0.45}_{-0.68}$ |  33.60 |  1
6 |  PSR J1301$+$0833 |  BW (1) |  1.84 |  - |  $0.045^{+0.001}_{-0.002}$ |  $0.697^{+0.046}_{-0.058}$ |  - |  34.82 |  1,10
7 |  PSR J1311$-$3430 |  BW (1) |  2.56 |  20.92 |  $0.016^{+0.001}_{-0.001}$(b) |  $\sim$20 |  $31.63^{+0.37}_{-0.54}$ |  34.69 |  11,12
8 |  PSR J1544$+$4937 |  BW (1) |  2.16 |  2.93 |  $0.025^{+0.006}_{-0.006}$ |  $0.161^{+0.035}_{-0.035}$ |  - |  34.06 |  15,16
9 |  PSR J1555$-$2908 |  BW (1) |  1.79 |  44.5 |  $0.060^{+0.005}_{-0.003}$(d) |  $9.92^{+0.002}_{-0.002}$(d) |  - |  35.49 |  3,43
10 |  PSR J1653.6$-$0158 |  BW (1) |  1.97 |  2.40 |  $0.013^{+0.001}_{-0.001}$(e) |  $0.333^{+0.039}_{-0.034}$(e) |  $31.07^{+0.04}_{-0.04}$ |  33.64 |  17,18,51
11 |  PSR J1810$+$1744 |  BW (1) |  1.66 |  4.60 |  $0.071^{+0.027}_{-0.027}$(k) |  2.389 |  $30.68^{+0.44}_{-0.52}$ |  34.60 |  19,2
12 |  PSR J1959$+$2048 |  BW (1) |  1.61 |  10.63 |  $0.036^{+0.001}_{-0.001}$ |  $3.01^{+0.036}_{-0.036}$ |  $31.04^{+0.32}_{-0.48}$ |  35.00 |  1,20,21
13 |  PSR J2017$-$1614 |  BW (1) |  2.31 |  2.45 |  -(c) |  - |  - |  33.85 |  22
14 |  PSR J2051$-$0827 |  BW (1) |  4.51 |  12.13 |  $\sim$0.04 |  - |  $28.62^{+0.55}_{-0.70}$ |  33.72 |  23,24
15 |  PSR J2052$+$1219 |  BW (1) |  1.99 |  6.70 |  $0.042^{+0.001}_{-0.001}$ |  $0.944^{+0.024}_{-0.024}$ |  - |  34.53 |  1,25
16 |  PSR J2241$-$5236 |  BW (1) |  2.19 |  6.65 |  $0.016^{+0.001}_{-0.001}$ |  $0.282^{+0.005}_{-0.006}$ |  $29.88^{+0.42}_{-0.57}$ |  34.40 |  1,26
17 |  PSR J2256$-$1024 |  BW (1) |  2.29 |  12.10 |  $>$0.030 |  0.277 |  $30.08^{+0.37}_{-0.53}$ |  34.60 |  2
18 |  PSR J1023$+$0038 |  RB (1) |  1.96 |  11.81 |  $0.205^{+0.008}_{-0.008}$(j,f) |  0.884 |  $31.90^{+0.31}_{-0.75}$(i) |  34.40 |  27,29,2
19 |  PSR J1048$+$2339 |  RB (1) |  4.67 |  30.37 |  0.24-0.35 |  $0.759^{+0.210}_{-0.175}$ |  $31.52^{+0.18}_{-0.18}$(h) |  34.08 |  30,31,32
20 |  PSR J1227$-$4853 |  RB (1) |  1.69 |  11.11 |  $0.17^{+0.01}_{-0.01}$(j,f) |  $\sim$1.21(f) |  $32.12^{+0.31}_{-0.47}$(i) |  34.96 |  27,33,34
21 |  PSR J1306$-$40 |  RB (1) |  2.20 |  - |  $0.59^{+0.01}_{-0.01}$(g) |  $\sim$5.90(g) |  31.94 |  - |  35,36,47
22 |  PSR J1431$-$4715 |  RB (2) |  2.01 |  14.11 |  0.13 - 0.19 |  - |  - |  34.83 |  37,38
23 |  PSR J1622$-$0315 |  RB (2) |  3.85 |  11.6 |  $0.122^{+0.007}_{-0.006}$ |  - |  $30.61^{+0.07}_{-0.07}$(h) |  33.89 |  39,40,41
24 |  PSR J1628$-$3205 |  RB (2) |  3.21 |  15.06 |  0.17 - 0.24 |  - |  $30.96^{+0.33}_{-0.50}$ |  34.26 |  10
25 |  PSR J1723$-$2837 |  RB (2) |  1.86 |  7.56 |  $0.36^{+0.08}_{-0.06}$ |  - |  $31.92^{+0.31}_{-0.48}$ |  34.67 |  42,44,38
26 |  PSR J2039$-$5617 |  RB (2) |  2.65 |  14.16 |  0.162 - 0.18 |  $0.150^{+0.027}_{-0.018}$ |  $31.18^{+0.03}_{-0.03}$(h) |  34.40 |  38,28,13,49
27 |  PSR J2129$-$0429 |  RB (2) |  7.62 |  335.6 |  $0.44^{+0.04}_{-0.04}$ |  $0.091^{+0.018}_{-0.018}$ |  $31.79^{+0.34}_{-0.51}$ |  34.48 |  45,46
28 |  PSR J2215$+$5135 |  RB (1) |  2.61 |  33.37 |  $0.345^{+0.008}_{-0.007}$(k) |  0.794 |  $31.92^{+0.41}_{-0.61}$ |  34.87 |  19,2
29 |  PSR J2339$-$0533 |  RB (1) |  2.88 |  6.68 |  $0.30^{+0.02}_{-0.02}$(l) |  $0.316^{+0.073}_{-0.073}$ |  $31.44^{+0.33}_{-0.49}$ |  34.04 |  48,50

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et al. (2018), (15) Tang et al. (2014), (16) Bhattacharyya et al. (2013), (17)
Kong et al. (2014), (18) Nieder et al. (2020), (19) Schroeder & Halpern
(2014), (20) Guillemot et al. (2012), (21) Huang et al. (2012), (22) Sanpa-
Arsa (2016), (23) Stappers et al. (2001), (24) Espinoza et al. (2013), (25)
Zharikov et al. (2019), (26) Keith et al. (2011), (27) Stringer et al. (2021),
(28) (0.5-8 keV) Bogdanov et al. (2021), (29) Bogdanov et al. (2011), (30)
Deneva et al. (2016), (31) (0.3-10 keV) Miraval Zanon et al. (2021), (32) Yap
et al. (2019), (33) Roy et al. (2015), (34) Bogdanov et al. (2014), (35)
Swihart et al. (2019), (36) (0.5-10 keV) Linares (2018), (37) Bates et al.
(2015), (38) Strader et al. (2019), (39) this work, (40) Sanpa-Arsa (2016),
(41) (0.2-10 keV) Gentile (2018), (42) Crawford et al. (2013), (43) Ray et al.
(2022), (44) van Staden & Antoniadis (2016), (45) Bellm et al. (2016), (46)
Hui et al. (2015), (47) Keane et al. (2018), (48) Kandel et al. (2020), (49)
Romani (2015), (50) Romani & Shaw (2011), (51) Long et al. (2022).

Notes.
-$\dot{P},\tau,L_{\rm x},\dot{E}$ are taken from reference (14) and the references therein, unless otherwise stated.

-Irradiation power is calculated using $L_{\rm irr}=4\pi a^{2}\sigma({T_{2}}^{4}-{T_{1}}^{4})$ or $L_{\rm irr}=\epsilon\dot{E}$, depending on the available information.

Items: (a) Number of optical peak(s). (b) Parameters from the ‘Eq Hot Spots’
model are used. (c) No light curve derived parameters. (d) 1$\sigma$ value is
used. (e) Parameters from the ‘Veiled model’ are used. (f) Parameters from the
‘HR2’ model are used; $a$ = 0.97$\;\\!{\rm R}_{\odot}$ obtained from reference
(33). (g) Temperatures from the ‘No spot’ model are used; $a$ is calculated
using $T=2\pi\sqrt{a^{3}/G/(M_{1}+M_{2})}$. (h) $\Delta(\log L_{\rm
x})=(\Delta L/L)/\ln{10}$. (i) Rotation-powered state. (j) Companion mass is
calculated using the reported pulsar mass and binary mass ratio. (k)
Parameters from the ‘NextGen’ model are used. (l) Parameters from the ‘HS’
model are used.

Table 4: A compilation of spider MSPs with published optical counterparts.

##  4 Pulsar irradiation and other system properties

The interplay between the energy loss of a neutron star that gives rise to
radiation and the mass transfer from the companion (which, in turn, increases
the rotational energy of the pulsar) determines the evolution state of a NS
low-mass X-ray binary. Many dedicated observations have been conducted in the
last decade to study the multi-wavelength properties of BWs and RBs. However,
the transient nature of these systems increases the uncertainty in
constraining binary evolution models. An empirical method is often used to
understand the behavior of different astronomical systems. We study the
relationship between the irradiation luminosity ($L_{\rm irr}$), the X-ray
luminosity ($L_{\rm x}$), and the spin-down luminosity ($\dot{E}$) in BWs, RBs
and tMSPs using an empirical method. The relationship between the X-ray
properties and spin-down luminosities of BWs and RBs has been previously
investigated (see, e.g. Arumugasamy et al., 2015; Lee et al., 2018). In
particular, using a simple power-law model, Lee et al. (2018) showed that the
X-ray emission of RBs is brighter and harder compared to that of BWs. Among
the spiders, tMSPs are brighter X-ray sources ($\sim\\!10^{32}$ erg s-1
during rotation-powered state; or see Li et al. 2020).

In general, for optical modeling of BWs and RBs, the pulsar irradiation is
computed as $L_{\rm irr}\propto({T_{2}}^{4}-{T_{1}}^{4})$, where $T_{1}$ and
$T_{2}$ are often the base or night-side temperature and the day-side
temperature of the companion. The companion is usually a low-mass star, and,
depending on its spectral type, the increased temperature due to the pulsar’s
irradiation on its inner face ($\Delta T=T_{2}-T_{1}$) can be more than a few
thousand K for some BW systems.

We compiled a list of BWs and RBs in the Galactic field (c.f. Table 4), for
which optical light curves have been previously reported. We conducted a
literature review and compiled the reported irradiation luminosities. If
either the spin-down energy conversion efficiency or the characteristic
irradiation temperature is reported, we computed the irradiation luminosities
as described in the footnotes. We dedicated a column to record the number of
peaks observed in the optical light curve of each source. A tidally distorted
companion without strong irradiation from the pulsar shows a two-peaked
ellipsoidal light curve variation. This allows us to quickly identify the
systems that are dominated by pulsar irradiation, which would otherwise show a
1-peak modulated light curve. All BWs in our sample have 1-peak modulation. We
note that the detection of BW companions is more challenging since they are
fainter, and we may not be able to detect them if there are less irradiated.
The RBs in our sample have 1 or 2 peaks, or a combination of these two
effects. The X-ray luminosity and spin-down luminosity are also included in
the table; these values, where available, are based on the study of Lee et al.
(2018). We also added the companion mass, spin and spin-down period to the
table for completeness. The tMSP parameters in the table correspond to the
radio pulsar state.

As mentioned in the Introduction, spiders systems are believed to have evolved
from LMXB systems. Therefore, it is useful to compare the X-ray properties of
these systems to the irradiation luminosity experienced by the companions.
Such a comparison allows us to evaluate the total energy available, and the
evolution of the luminosity in these systems that suffer constant ablation
(Levinson & Eichler, 1991). The left panel of Figure 6 shows the relationship
between irradiation luminosity and X-ray luminosity. The irradiation
luminosity is of about 2 orders of magnitude higher than the X-ray luminosity,
and therefore the X-ray emission, regardless of its origin, is unlikely to be
the main source of irradiation in these systems. A RB system with higher X-ray
luminosity could be experiencing weak accretion activity at some stages; on
the other hand, BW systems are dominated by the pulsar irradiation, hence they
are not accreting systems. We performed a linear regression analysis for the
two variables using the scipy package linregress. The Pearson correlation
coefficients are 0.56 and 0.75, for RBs and BWs respectively. The two
variables are related by $\log_{10}L_{\rm irr}=0.74~{}\log_{10}L_{\rm
x}+10.11$ for RBs, and $\log_{10}L_{\rm irr}=0.50~{}\log_{10}L_{\rm x}+18.60$
for BWs. In our samples, all four highly irradiated BWs (ID: 4, 7, 11, 12)
have high X-ray and spin-down luminosity compared to other BWs. The
irradiation luminosity gap observed in the $L_{\rm irr}$-$L_{\rm x}$ plot for
BWs (i.e. $0.3\times 10^{34}$ erg s-1$\lesssim L_{\rm irr,BW}\lesssim 2\times
10^{34}$ erg s-1) corresponds to characteristic irradiation temperatures of
5400 K $\lesssim T_{\rm irr}\lesssim$ 8700 K, assuming an orbital separation
of one solar radius. By applying the empirical relation we obtained for BW
systems and substituting for the rough upper and lower bounds of the observed
gap, we found that the resulting X-ray luminosity is $L_{\rm x}\in(6\times
10^{29},2\times 10^{31})$ erg s-1. In the plot, we show that the X-ray
counterpart of some of the BWs falls within this range, suggesting that the
irradiation luminosity gap is likely not an observational bias. We note that
the derived irradiation temperature is dependent on the model used, and is
subject to the variable nature of spiders (e.g. the pulsar heating in PSR
J1048$+$2339 became significant in less than two weeks; Yap et al. 2019),
however this should not affect the general trend reported. The tMSPs are the
brightest among the RBs in both irradiation and X-ray luminosity.

The right panel of Figure 6 shows the relationship between irradiation
luminosity and spin-down luminosity. The order of magnitudes between the two
luminosities are similar ($\sim\\!10^{33}-10^{35}$ erg s-1). For PSR
J1622-0315, while the system is dominated by ellipsoidal distortion, we cannot
rule out the presence of pulsar irradiation. Therefore, we present the
irradiation luminosity of PSR J1622-0315 as an upper limit in the same figure,
where the effective temperature is used as the characteristic irradiation
temperature. The error bar corresponds to 1$\sigma$ uncertainty from the
fitting results. Given that a system requires more energy to raise the
temperature of a hotter star compared to a cooler star, and the effective
temperature in PSR J1622-0315 is relatively higher among other spider
companions, an irradiation temperature of $T_{\rm irr}\approx$ $T_{\rm
eff}$=$6400~{}{\rm K}$ corresponds to a day-side temperature ${T_{2}}$=$7300$
K (if $T_{1}$=$6000$ K). A considerable amount of energy is required to
achieve this temperature difference, hence the effective temperature is used
as an approximation to estimate the upper limit of pulsar irradiation in PSR
J1622-0315.

In analogy to the conversion efficiency from spin-down power to X-ray
luminosity (Fig. 10 in Arumugasamy et al., 2015) and $\gamma$-ray luminosity
(Fig. 11 in Strader et al., 2019), we added analytic lines that correspond to
$L_{\rm irr}/\dot{E}$ for comparison. A conversion efficiency would suggest
that the heating mechanism involves some form of reprocessing of the spin-down
energy. A theoretical study of this would be useful, but beyond the scope of
this paper. The $L_{\rm irr}$-$\dot{E}$ relationship for BWs and RBs are
similar in that they scatter across the parameter space despite the companion
stars having masses well separated into two groups. However, one of the BWs,
PSR J1311-3430 has an irradiation luminosity that exceeds its spin-down
luminosity ($L_{\rm irr}>\dot{E}$; conversion efficiency greater than
100$\%$), in contrast to other systems that have $L_{\rm irr}<\dot{E}$. It is
likely that a less irradiated BW system would have a low conversion efficiency
of only a few percent, and would be faint and challenging to observe. To
complement the $L_{\rm irr}$-$\dot{E}$ plot, in Figure 7 we show the
$P$-$\dot{P}$ diagram of the pulsars in known spider systems. The spin periods
($P$) of the pulsar in RBs and BWs are similar, but the pulsars in a RB system
occupy the space with higher $\dot{P}$ values compared to the pulsars in a BW
system. We note that the tMSPs (marked red stars) have very similar $P$ and
$\dot{P}$ values. We added the MSP spin-up line, reproduced from a recent
study reported by Liu et al. (2022). This information is useful for comparing
the spin profile of irradiated transitional systems, which is outside the
scope of this paper.

Figure 6: (Left) A comparison between the irradiation luminosity and the X-ray
luminosity. (Right) A comparison between the irradiation luminosity and the
spin-down luminosity. The ID of BWs are labelled in blue while RBs are
labelled in red. tMSPs (ID: 18, 20) are marked as red stars. The upper limit
of irradiation luminosity of PSR J1622-0315 (marked by a red square box) is
calculated using the effective temperature. The RBs and BWs occupy two
distinct regions in the $L_{\rm irr}$-$L_{\rm x}$ plot. The dotted lines in
$L_{\rm irr}$-$\dot{E}$ plot correspond to different $L_{\rm irr}/\dot{E}$
ratios. PSR J1311$-$3430 (ID: 7) is the only source in our study that has
$L_{\rm irr}>\dot{E}$. Figure 7: A complimentary $P$-$\dot{P}$ diagram of the
pulsars in spider systems shown in logarithmic scale where
$\dot{P}_{-21}=\dot{P}/10^{-21}$. The characteristic magnetic field strengths
are indicated by dashed gray lines. The ID of BWs are labelled in blue while
RBs are labelled in red. tMSPs are marked as red stars. The magenta line shows
the MSP spin-up line according to Liu et al. (2022).

##  5 Summary

PSR J1622$-$0315 was discovered in the pulsar search of Fermi unassociated
sources using the GBT and the optical counterpart was identified (Sanpa-Arsa,
2016). The light curve showed a clear orbital variation and fluctuations at
the same orbital phase. Strader et al. (2019) published the radial velocity
and spectra of PSR J1622$-$0315, and the companion mass was constrained to
0.10$-$0.14 $\textup{M}_{\odot}$.

In this work, we modeled the light curve of PSR J1622$-$0315 using the PHOEBE
code. We performed MCMC sampling on six system parameters and found that the
pulsar irradiation on the companion is not significant in this system (c.f.
Fig. 2). The result obtained constrained the pulsar mass to
$1.84^{+0.19}_{-0.19}~{}\textup{M}_{\odot}$ and the companion mass to
$0.122^{+0.007}_{-0.006}~{}\textup{M}_{\odot}$. From the MCMC result, we find
that PSR J1622$-$0315 is the lightest known redback system with a relatively
hot companion. We note that PSR J1622$-$0315 has an orbital period of 0.16 d,
which is slightly smaller than other RBs ($\gtrsim$ 0.2 d).

Additionally, we studied the energetic relationship between irradiation
luminosity ($L_{\rm irr}$), X-ray luminosity ($L_{\rm x}$) and the pulsar’s
spin-down luminosity ($\dot{E}$) of spider systems. Both BWs and RBs show a
correlation in their $L_{\rm irr}$ and $L_{\rm x}$ values, with BWs having a
higher Pearson correlation of 0.75. We find that PSR J1622$-$0315 lies well
within the RB group with insignificant pulsar irradiation. PSR J1622$-$0315
also has the smallest X-ray and spin-down luminosity among the RBs in the
sample. Observations of borderline-mass (companion) spiders like PSR
J1622$-$0315 and PSR J1810$+$1744 (Schroeder & Halpern, 2014) are important to
constrain the binary evolution model. Recent findings suggest that both NS
low-mass X-ray binaries and spider systems tend to have an ionized and clumpy
environment (Knight et al., 2023), implying that pulsar irradiation and mass
loss in the form of winds are common in different evolution stages. Assuming
the presence of pulsar irradiation, it is useful to compare the companion star
behavior to the neutron star properties as they co-evolve with each other. For
instance, in this study, both tMSPs are the RB systems with the highest pulsar
irradiation luminosity and X-ray luminosity compared to other spider systems,
and this suggests that the transitional systems are in general more energetic.
Provided the system is bright enough to observed, both light curve modeling of
the irradiation properties and spectral observation can be used together to
study the evolution of the companion under pulsar irradiation. Finally, the
estimation of irradiation luminosity is model-dependent and is subject to the
variable nature of these systems, which may not be fully reflected in this
study.

We thank the referee for his/her many useful scientific suggestions and
comments. This work has made use of data collected at Lulin Observatory,
partly supported by the National Science and Technology Council of the
Republic of China (Taiwan) through grant 105-2112-M-008-024-MY3. Y.X.J.Y and
A.K.H.K. are supported by the National Science and Technology Council of the
Republic of China (Taiwan) through grants 110-2628-M-007-005 and
111-2112-M-007-020. KLL is supported by the Ministry of Science and Technology
of the Republic of China (Taiwan) through grant 110-2636-M-006-013, and he is
a Yushan (Young) Scholar of the Ministry of Education of the Republic of China
(Taiwan). This work used high-performance computing facilities operated by the
Center for Informatics and Computation in Astronomy (CICA) at National Tsing
Hua University. This equipment was funded by the Ministry of Education of
Taiwan, the Ministry of Science and Technology of Taiwan, and National Tsing
Hua University. We thank Kinwah Wu and Kaye Li for their advice to improve the
readability of the manuscript.

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