# Reheated Sub-40000 Kelvin Neutron Stars at the JWST, ELT, and TMT

Nirmal Raj<EMAIL_ADDRESS>Prajwal Shivanna<EMAIL_ADDRESS>Rachh
Gaurav Niraj<EMAIL_ADDRESS>Centre for High Energy Physics, Indian
Institute of Science, C. V. Raman Avenue, Bengaluru 560012, India

###### Abstract

Neutron stars cooling passively since their birth may be reheated in their
late-stage evolution by a number of possible phenomena: rotochemical, vortex
creep, crust cracking, magnetic field decay, or more exotic processes such as
removal of neutrons from their Fermi seas (the nucleon Auger effect), baryon
number-violating nucleon decay, and accretion of particle dark matter. Using
Exposure Time Calculator tools, we show that reheating mechanisms imparting
effective temperatures of 2000–40000 Kelvin may be uncovered with excellent
sensitivities at the James Webb Space Telescope (JWST), the Extremely Large
Telescope (ELT), and the Thirty Meter Telescope (TMT), with imaging
instruments operating from visible-edge to near-infrared. With a day of
exposure, they could constrain the reheating luminosity of a neutron star up
to a distance of 500 pc, within which about $10^{5}$ (undiscovered) neutron
stars lie. Detection in multiple filters could overconstrain a neutron star’s
surface temperature, distance from Earth, mass, and radius. Using publicly
available catalogues of newly discovered pulsars at the FAST and CHIME radio
telescopes and the Galactic electron distribution models YMW16 and NE2001, we
estimate the pulsars’ dispersion measure distance from Earth, and find that
potentially 30$-$40 of these may be inspected for late-stage reheating within
viable exposure times, in addition to a few hundred candidates already present
in the ATNF catalogue. Whereas the coldest neutron star observed (PSR
J2144$-$3933) has an upper limit on its effective temperature of about 33000
Kelvin with the Hubble Space Telescope, we show that the effective temperature
may be constrained down to 20000 Kelvin with JWST-NIRCam, 15000 Kelvin at ELT-
MICADO, and 9000 Kelvin with TMT-IRIS. Campaigns to measure thermal
luminosities of old neutron stars would be transformative for astrophysics and
fundamental physics.

##  I Introduction

Neutron stars, celestial objects extraordinaire, are versatile natural
laboratories Nättilä and Kajava (2022); Bramante and Raj (2024). The cooling
of neutron stars (NSs) provides an observational handle on their mass, radius,
composition of core, crust and envelope, the equation of state (EoS) of NS
matter, and their magnetic fields; data on NS cooling curves has helped to
show that nucleons in the NS core exhibit weak superfluidity Potekhin  _et
al._ (2020). While there is much agreement between passive cooling models and
observations of NS thermal emission and ages down to $10^{5}$ Kelvin and
$10^{6}$ years, certain NSs anomalously hot for their spin-down age have also
been identified Potekhin  _et al._ (2020). Meanwhile, the coldest NS on
record, PSR J2144$-$3933, has its effective temperature bounded at $<$ 33000
Kelvin by Hubble Space Telescope (HST) observations Guillot  _et al._ (2019);
its spin-down age is $3\times 10^{8}$ years, implying that if passive cooling
were in place its temperature would be a few hundred Kelvin in “minimal
cooling” models Page  _et al._ (2004); Ofengeim and Yakovlev (2017); Bramante
_et al._ (2022), those that incorporate only modified Urca processes of
neutrino cooling.

In light of these, observational studies of late-time reheating mechanisms of
NSs become important. These mechanisms include astrophysical effects Gonzalez
and Reisenegger (2010) such as rotochemical heating, vortex creep heating,
crust-cracking, and magnetic field decay, and those involving a hidden sector
of particles Bramante and Raj (2024) such as dark matter capture and heating
through scattering and annihilations, Bondi accretion of dissipative-fluid
dark matter as well as minimal accretion of collisionless dark matter residing
in micro-halos, accretion through a long-range force between dark matter and
neutron stars, the nucleon Auger effect via neutron oscillations, and baryon
number-violating neutron decays; see Appendix A for more discussion on these
mechanisms.111Despite the name “reheating”, these mechanisms do not heat the
NS more than once. They continuously deposit energy in the NS material over a
long time, their effects only becoming apparent when the NS cools down to a
luminosity comparable to the energy deposition rate. It is timely that the
currently operational James Webb Space Telescope (JWST) Gardner  _et al._
(2006) that is delivering a raft of discoveries JWS (a), the soon-to-appear
Extremely Large Telescope (ELT) Neichel  _et al._ (2018), and the less
imminent Thirty Meter Telescope (TMT) Skidmore  _et al._ (2015) can span
wavelengths corresponding to blackbody peak temperatures of 1300–4300 Kelvin
with their imaging instruments, respectively the Near Infrared Camera
(NIRCam), the Multi-AO Imaging Camera for Deep Observations (MICADO), and the
InfraRed Imaging Spectrograph (IRIS). That is, all three telescopes can cover
from the far-optical down to the infrared.

In this study we estimate the sensitivities of these telescopes to NS
temperatures in this range. We find that $>10^{5}$ pulsars may be potentially
measured at a reheated temperature of $\lesssim$ 40000 Kelvin. (As the
distribution of pulsar ages $t_{\rm NS}$ is roughly weighted as $t_{\rm
NS}/t_{\rm MW}$ Sartore  _et al._ (2010), where $t_{\rm MW}=5$ Gyr is the age
of the Milky Way, we expect $>1000$ K pulsars merely cooling passively, i.e.
pulsars with $t_{\rm NS}<10^{7}$ yr, to be rare. Hence our emphasis on
potentially reheated neutron stars.) Statistically significant measurements of
the spectral flux density of thermal emission of an NS in multiple filters
could help to simultaneously pin down the NS surface temperature, distance,
mass, and radius. To launch a campaign of observing NS reheating, some
fraction of this NS population must be discovered as pulsars in order to train
the infrared and optical telescopes at the right sky position. However, the
ATNF catalogue Manchester  _et al._ (2005) already contains a few hundred
pulsar candidates close enough to be amenable to luminosity measurements.
Further, in this work we avail of publicly available catalogues of radio
pulsars discovered in the currently operational Five-hundred-meter Aperture
Spherical Telescope (FAST) and Canadian Hydrogen Intensity Mapping Experiment
(CHIME) to extract their (dispersion measure-based) distances from Earth,
showing that a few tens of freshly spotted pulsars are close enough to be good
targets for observations of thermal emission at JWST, ELT, and TMT.

Moreover, we argue that there is already a clear observational target for
these telescopes to improve our understanding of the NS cooling and reheating,
namely, the coldest NS observed, PSR J2144$-$3933, that was mentioned earlier.
While an upper limit on its thermal luminosity exists thanks to HST, this
limit can be improved by a factor of 7 by JWST-NIRCam, a factor of 20 by ELT-
MICADO, and a factor of 180 by TMT-IRIS. Thus there is immediate opportunity
to push the “neutron star temperature frontier” and revise constraints on NS
reheating mechanisms offered by astrophysics and particle physics.

Yet another campaign may be mounted: the observation of (already discovered)
X-ray sources that are almost certainly neutron stars. These are “central
compact objects” (CCOs), located near the centers of supernova remnants that
lack pulsar wind nebulae De Luca (2017). As described in detail in Sec. III,
CCOs could be surrounded by a disk of debris left behind by the neutron star’s
progenitor after supernova explosion. This disk could source an infrared flux
that is obtained in addition to the X-ray thermal peak. A search for such disk
emission has already been undertaken recently using JWST-NIRCam in the Cas A
system, yielding a null result Milisavljevic  _et al._ (2024). At the time of
writing, there are nine other confirmed CCOs Cat  for which similar searches
can be performed.

This document is organized as follows. In Section II we set up the basic
framework of imaging sensitivities, describe the tools that estimate exposure
times for our telescopes and our choices of input parameters, and present our
results. We also discuss the future reach of these telescopes on the
temperature of PSR J2144$-$3933\. In Section III we provide some discussion
and conclude. In the appendices we review various late-stage reheating
mechanisms in the literature, and tabulate our derived dispersion measure
distances of pulsars in recently available catalogues made public by FAST and
CHIME.

Figure 1: Blackbody spectral flux densities of neutron stars with various
effective temperatures at various distances compared with filter sensitivities
at JWST-NIRCam and ELT-MICADO at SNR=2 for the indicated exposure times.
Luminosity measurements of late-time-reheated neutron stars at these imaging
instruments are eminently feasible. See Sec. II for further details.

Figure 2: The maximum distance to which neutron stars with effective
temperatures of 40000 K, 20000 K and 10000 K can be constrained at SNR 2 in
the filters of JWST-NIRCam, ELT-MICADO and TMT-IRIS, with net exposure times
of $10^{4}$, $10^{5}$, $10^{6}$ seconds. These temperatures are within an
order of magnitude of the upper limit on the effective temperature of the
coldest neutron star on record (PSR J2144$-$3933), about 33000 K, and are also
above the blackbody peak temperatures corresponding to the central wavelength
of the filters here. Not shown for visual clarity are astrophysical modelling
uncertainties at the 10% level, comparable to uncertainties in distance
measurements via radio parallax and much smaller than uncertainties in
distance estimates via pulsar dispersion measure. The plot data are tabulated
and uploaded to arXiv as an ancillary file. See Sec. II for further details.

Figure 3: Same as Fig. 2 but for effective temperatures of 4000 K and 2000 K.
These temperatures are below the blackbody peak temperatures corresponding to
the central wavelength of the filters here. See Sec. II for further details.

##  II Signals and Results

###  II.1 General set-up

The blackbody spectral flux density of an NS of mass $M_{\rm NS}$, radius
$R_{\rm NS}$, surface temperature $T$ (so that the radius and effective
temperature as seen by a distant observer are $R_{\rm\infty}=(1+z)R_{\rm NS}$
and $T_{\rm\infty}=T/(1+z)$, where $1+z=1/\sqrt{1-2GM_{\rm NS}/R_{\rm NS}}$ )
at a distance from Earth $d$ is given by

$\displaystyle f_{\nu}$ $\displaystyle=$ $\displaystyle\pi
B(\nu,T_{\rm\infty})\bigg{(}\frac{R_{\rm\infty}}{d}\bigg{)}^{2}~{}\times
10^{-0.4A_{\nu}},$ (1) $\displaystyle{\rm with}\ B_{\nu}$ $\displaystyle=$
$\displaystyle\frac{2h\nu^{3}}{c^{2}}\frac{1}{\exp(h\nu/kT_{\rm\infty})-1}~{},$

where $A_{\nu}$ is the extinction factor determined by the column density of
Galactic dust along the line of sight to the NS. Extinction will modify our
results at worst by about 10%, which is within the uncertainties of
measurements of NS distances. Therefore we show our main results by setting
$A_{\nu}=0$, and discuss the effect of extinction in Sec. II.5.

The signal-to-noise ratio (SNR) as a function of exposure time $t_{\rm exp}$
roughly goes as $F_{\rm sig}A_{\rm SNR}t_{\rm exp}/\sqrt{(F_{\rm bg}A_{\rm
SNR}+\Gamma_{\rm noise})t_{\rm exp}}$, where $A_{\rm SNR}$ is the SNR
reference area in the detector, $F_{\rm sig}$ and $F_{\rm bg}$ are signal and
background fluxes, and $\Gamma_{\rm noise}$ is the rate of noise from non-sky
sources. Thus $t_{\rm exp}\propto$ (SNR)2. As we will see soon, further
subtleties enter the calculation of exposure time, but the above expression
helps us understand some salient features. It tells us that for a given
$T_{\rm\infty}$, there is a maximum distance $d_{\rm max}$ up to which NSs can
be observed in a given instrument filter that comes with a flux sensitivity
corresponding to some SNR and exposure time. For a solar neighborhood NS
number density $n_{\rm NS,\odot}$, there is a maximum of $N_{\rm max}=4\pi
n_{\rm NS}d^{3}_{\rm max}/3$ potential NS targets. From Ref. Sartore  _et al._
(2010), $n_{\rm NS,\odot}\simeq(1-5)\times 10^{-4}(N_{\rm NS}/10^{9})~{}{\rm
pc}^{-3}$, the factor of 5 variation arising from modelling of the local NS
population. The Milky Way NS population $N_{\rm NS}$ could be $10^{8}$ for a
constant supernova rate, however it could be as high as $10^{9}$ via a higher
past supernova rate as suggested by Galactic nucleosynthesis constraints
Arnett  _et al._ (1989). Given these uncertainties, we take $n_{\rm
NS,\odot}=10^{-4}~{}{\rm pc}^{-3}$ as a representative value to show our
results.

In Fig. 1 we plot $f_{\nu}$ for $T_{\rm\infty}=4000$ K, $d=30$ pc (left panel)
and $40000$ K, $d=100$ pc (right panel), and show for comparison the SNR = 2
sensitivities of filters at JWST-NIRCam (for $t_{\rm exp}=10^{4}$ s) and ELT-
MICADO (for $t_{\rm exp}=10^{5}$ s). This figure illustrates that observations
of thermal emission of NSs of temperatures and distances considered here can
indeed be made with the right choice of filters. In Figs. 2 and Fig. 3, we
visually tabulate $d_{\rm max}$ – and the corresponding $N_{\rm max}$ on the
right-hand y-axis – against various filters, indicating their central
wavelengths in the lower x-axis and the corresponding peak blackbody
temperatures in the upper x-axis. We do this for an SNR of 2 (corresponding to
a 95% C.L. limit in Gaussian statistics) and total exposure times of
$10^{4},10^{5},10^{6}$ seconds. The first of these choices
($\sim\mathcal{O}$(hours)) is a reasonable telescope time to request, as seen
in several JWST proposals JWS (b). The last choice ($\sim\mathcal{O}$(months))
is achievable in deep exposures as undertaken by HST. The intermediary
exposure timescale (about a day) has been quoted as a viable exposure time at
JWST in the literature Chatterjee  _et al._ (2023). The panels of Figs. 2 and
3 correspond to different $T_{\rm\infty}$ as indicated. The case of
$T_{\rm\infty}=40000$ Kelvin is near the observational upper limit on NS
temperatures Guillot  _et al._ (2019), and is hence inherently interesting.
The cases of lower temperatures are those corresponding to reheating
mechanisms in the literature discussed in the Introduction and Appendix A. To
show our results we had assumed a benchmark NS mass and radius of

$M_{\rm NS}=1.4~{}M_{\odot}~{},~{}~{}R_{\rm NS}=11~{}{\rm km}~{}.$ (2)

These values are typical of NSs and were assumed by Ref. Guillot  _et al._
(2019) to illustrate the temperature measurement of PSR J2144$-$3933\. (Note
that, with this choice, the limit on the redshifted temperature is
$T_{\rm\infty}<$ 33000 K.) These values also happen to fall in the mass-radius
curve predicted by the EoS of Wiringa, Fiks, Fabrocini (WFF) Wiringa  _et al._
(1988); Datta  _et al._ (1995). We will discuss in Sec. II.5 the effect of
varying the NS mass and radius along the WFF EoS prediction.

JWST/NIRCam, DEEP8 readout
---
net exposure time ($s$) | dithers | integrations per exposure | groups per integration
$10^{4}$ | 17 | 2 | 2
105 | 21 | 21 | 2
106 | 21 | 22 | 11
Table 1: Our choice of parameters to achieve desired net exposure times in the
JWST Exposure Time Calculator tool. See Sec. II.2 for further particulars.

###  II.2 JWST-NIRCam

Our NIRCam results are obtained using the online tool JWST Exposure Time
Calculator (ETC) version 3.0. Detailed documentation of the working of the ETC
is available JWS (c), and here we summarize relevant points. In the ETC one
must make an optimal choice from among 9 available readout patterns, which fix
the number of usable detector readout samples (“frames”) in a “group” of them.
For the full detector, the sampling time $t_{\rm frame}$ is 10.737 s. A
collection of groups is an “integration”, and a collection of integrations is
an “exposure”. To achieve a net exposure time, the ETC user can tune the
groups per integration $N_{\rm group}$ and the integrations per exposure
$N_{\rm int}$, but not the frames per group, which is fixed by the readout
pattern. Within a readout pattern, a certain number of frames are averaged for
signal-fitting in a group ($N_{\rm frame}$), while others are skipped ($N_{\rm
skip}$). Skipped samples at the end of an integration are not processed, so
the integration time $t_{\rm int}$ is given by

$t_{\rm int}=t_{\rm frame}[(N_{\rm group}-1)(N_{\rm frame}+N_{\rm
skip})+N_{\rm frame}]~{}.$ (3)

For faint sources such as those we are interested in, increasing $N_{\rm int}$
while decreasing $N_{\rm group}$ is the preferred strategy, to reduce the data
volume that is downlinked to the ground. Finally, the user must suitably
choose the number of “dithers”, small shifts in telescope pointing made
between exposures to improve sky coverage and image quality by combining
resultant images and thereby removing detector artifacts and cosmic ray hits,
filling inter-detector gaps, and calibrating the point spread function.

In Table 1 we show our choices of the number of dithers, $N_{\rm int}$ and
$N_{\rm group}$ to achieve our net exposure times. As appropriate for long
integration times, we used DEEP8, the readout pattern with the maximum number
of frames averaged in a group (which is 8, with 12 other frames skipped) and
with the maximum number of groups allowed. Such a readout pattern is
susceptible to pixel loss due to cosmic ray impacts, hence we select a high
number of dithers. As in Ref. Chatterjee  _et al._ (2023), we picked a
reference background model at RA=$03^{\rm h}\ 32^{\rm m}\ 42.397^{\rm s}$ and
Dec= $-27^{\circ}\ 42^{\prime}\ 7.93^{\prime\prime}$, corresponding to a blank
field. Due to how the data volume is handled and the use of dithering, the SNR
grows with exposure time at a rate slightly smaller than $\sqrt{t_{\rm exp}}$.

NIRISS sensitivities.

In addition to NIRCam, JWST can perform imaging with the Near-Infrared Imager
and Slitless Spectrograph (NIRISS) instrument. The sensitivities of NIRISS’
filters are generally slightly weaker than NIRCam’s and for this reason we do
not display NIRISS results. By comparing NIRCam and NIRISS filter
sensitivities in the JWST Pocket Guide JWS (d) and by using Eq. (1), we find
that the ratios of $d_{\rm max}$ obtained in NIRISS versus NIRCam in various
filters are F090W: 0.87, F115W: 0.89, F140M: 0.89, F150W: 0.84, F158M/F162M:
0.88, F200W: 0.85, F277W: 1.07, F356W: 0.99, F430W: 0.90, F444W: 0.93, F480M:
0.90. In the JWST Pocket Guide the NIRCam sensitivities are based on the
MEDIUM8 readout pattern with 10 exposures and $N_{\rm group}=10$, while the
readout pattern of NIRISS sensitivities is not specified but is presumably NIS
NIR  as appropriate for faint sources. If the DEEP8 readout pattern were used
at NIRCam instead (as done in our study), the NIRCam sensitivities would be
even better than those of NIRISS.

###  II.3 ELT-MICADO

For the terrestrial telescopes ELT and TMT imaging through the atmosphere,
other considerations come into play. As per the ELT ETC documentation ELT ,
the signal-to-noise ratio is given by

${\rm SNR}=\frac{N_{\rm exp}N_{\rm obj}}{\sqrt{N_{\rm exp}[N_{\rm obj}+N_{\rm
sky}+N_{\rm pix}(N_{\rm ro}^{2}+\Gamma_{\rm DC}t_{1})]}}~{},$ (4)

where $N_{\rm exp}$ is the number of exposures, $N_{\rm pix}$ is the number of
pixels in the SNR reference area, $N_{\rm ro}$ is the detector read-out noise
(in $e^{-}$/pixel), $\Gamma_{\rm DC}$ is the detector dark current (in
$e^{-}$/s/pixel), and $t_{1}$ is the detector exposure time for one exposure.
The number of detected electrons per exposure from the source in the SNR
reference area is $N_{\rm obj}=\hat{\Phi}_{\rm
obj}\varepsilon\Omega\alpha_{c}t_{1}$, where $\hat{\Phi}_{\rm
obj}=10^{-0.4\chi k}\Phi_{\rm obj}$ is the source flux at the telescope
entrance with $\Phi_{\rm obj}$ the source flux at the top of the atmosphere
and the pre-factor the atmospheric extinction characterized by the airmass
$\chi$ and the extinction coefficient $k$, $\varepsilon$ is the ensquared
energy in the SNR reference area as tabulated in the ETC documentation,
$\alpha_{c}$ is the conversion factor from energy flux at the telescope
entrance to detected number of photo-electrons, and $\Omega=10^{-6}N_{\rm
pix}p_{\rm sc}^{2}$ is the size of the SNR reference area (in arcsec2);
$p_{\rm sc}$ is the pixel scale (in mas/pixel). The number of detected
electrons per exposure from the background in the SNR reference area is
$N_{\rm sky}=\hat{\Phi}_{\rm sky}\Omega\alpha_{c}t_{1}$, with $\hat{\Phi}_{\rm
sky}$ the sky surface brightness at the telescope entrance.

In the ELT ETC (version 6.4.0) we chose the observatory site as “Paranal”,
which has an altitude of 2635 m that is close to that of the actual proposed
site, Cerro Armazones at 3046 m. We also set the telescope diameter to 39 m as
currently designed, the adaptive optics (AO) mode to multi-conjugate AO as
available on ELT’s near-infrared imaging instrument MICADO, and the airmass
$\chi$ to 1.5. MICADO has fields of view $50.5^{\prime\prime}\times
50.5^{\prime\prime}$ with pixel scale 4 mas/pixel and $19^{\prime\prime}\times
19^{\prime\prime}$ with pixel scale 1.5 mas/pixel, both implying 160 pixels
each Padovani and Cirasuolo (2023). Thus we set the SNR reference area to
10$\times$10 pixels and pixel scale to 5 mas/pixel, the closest available
options.

###  II.4 TMT-IRIS

In the Exposure Time Calculator of TMT’s versatile instrument IRIS TMT , we
take the default zenith angle of observation of 30 degrees, the default
“average” atmospheric conditions, and the PSF location at
($8.8^{\prime\prime},8.8^{\prime\prime}$), corresponding to the centre of the
detector field of view. The imager’s field of view is $34^{\prime\prime}\times
34^{\prime\prime}$ with pixel scale 4 mas/pixel.

###  II.5 Discussion of results

As mentioned in Sec. II.1 for deep exposures the SNR $\propto\sqrt{t_{\rm
exp}}$ approximately, and thus from Eq. (1), $d_{\rm max}\propto t_{\rm
exp}^{1/4}$. This is just what we see in Figs. 2 and 3 as we scan vertically
up for each filter. In Fig. 2, where $T_{\infty}>4000$ K, the $d_{\rm max}$
sensitivity decreases with increasing filter central wavelength more or less
monotonically, as expected for spectral flux densities falling with wavelength
(and as seen in Fig. 1). Deviations from monotonic decrease are due to
differences in filter sensitivities. For $T_{\infty}=4000$ K in Fig. 3, the
peak $d_{\rm max}$ reach for each instrument is obtained for one of the
intermediary filters with central wavelength corresponding to peak blackbody
temperature somewhat below 4000 K. This is also not surprising as the best
result obtainable is an outcome of interplay between the location of the
spectral peak and filter sensitivities. For $T_{\infty}=2000$ K in Fig. 3 the
$d_{\rm max}$ reach for each instrument is seen to roughly increase with the
central wavelength of the filter choice, as the low-wavelength filters now
catch the low wavelength tail of the NS’s blackbody spectrum.

Across instruments, it may seen that for comparable filter ranges ELT-MICADO
is somewhat more sensitive than JWST-NIRCam, and TMT-IRIS generally yet more
sensitive. In fact for large $T_{\infty}$ NSs the $Z$ filter of TMT-IRIS gives
conspicuously better reach than all other filters considered.

PSR J2144$-$3933 effective temperature sensitivities in Kelvin
---
distance (pc) | exposure (s) | JWST-NIRCam: F707W | ELT-MICADO: I band | TMT-IRIS: Z band
157 | $10^{4}$ |  43450 95130 |  29240 61350 |  15370 29070
$10^{5}$ |  17880 32940 |  13820 24630 |  8510 13350
172 | $10^{4}$ |  50150 112350 |  33580 71970 |  17240 33500
$10^{5}$ |  19910 37550 |  15330 28100 |  9250 14860
192 | $10^{4}$ |  60670 137240 |  39940 87600 |  19940 39990
$10^{5}$ |  23350 45360 |  17510 33160 |  10290 17050
Table 2: The SNR = 2 (underlined) and SNR = 5 (italicized) sensitivities, for
two practically feasible exposure times 104 and 105 s, of the effective
temperature in Kelvin of the PSR J2144$-$3933, the coldest neutron star known
with a current upper bound on its effective temperature of 33000 Kelvin set by
non-detection at HST Guillot  _et al._ (2019). Here a 1.4 $M_{\odot}$ mass and
11 km radius are assumed. Three pulsar parallax distances as estimated in Ref.
Deller  _et al._ (2009) are considered: the central one, and the ends of the
$\pm 1\sigma$ range. We see that JWST, ELT and TMT are capable of pushing the
neutron star temperature frontier lower, potentially improving limits on late-
time reheating mechanisms. See Sec. II.6 for further particulars.

One may wonder if there is a minimum NS distance to which the imaging filters
here are sensitive. An NS that is too close may be so bright that it saturates
the detector pixels within an integration time. We checked for this effect in
the JWST ETC, and found that saturation occurs if we artificially increase the
spectral flux density by 4$-$5 orders of magnitude. From Eq. (1), this implies
that NS distances smaller by an $\mathcal{O}(100)$ factor than the $d_{\rm
max}$ values shown in Figs. 2 and 3 would result in detector saturation. As
even the closest NS is theorized to be about 10 pc from Earth, and the closest
observed is about 100 pc away, we may conclude that our results apply to all
known NSs whose thermal luminosities are unmeasured.

While we have used the benchmark NS in Eq. (2), other mass-radius
configurations will introduce only $\mathcal{O}(10\%)$ variation in the
spectral flux density, hence in $d_{\rm max}$. For example, if we consider a
low-mass configuration on the mass-radius curve predicted by the WFF EoS, with
$M_{\rm NS}=0.95~{}M_{\odot}$ and $R_{\rm NS}=11$ km, the spectral flux
density in Eq. (1) reduces by 0.92 and hence $d_{\rm max}$ increases by 1.04.
For a high-mass configuration with $M_{\rm NS}=2~{}M_{\odot}$ and $R_{\rm
NS}=10.6$ km, $f_{\nu}$ increases by 1.18 and hence $d_{\rm max}$ reduces by
0.92. We had also set the interstellar extinction factor $A_{\nu}$ to zero,
but since $d_{\rm max}\propto 10^{-0.2A_{\nu}}$ as obtained from Eq. (1), and
since typically $A_{\nu}\ll 1$, our estimate of $d_{\rm max}$ will again only
reduce by a small factor if extinction is present along the line of sight. The
average Milky Way extinction factor in the $V$ band (centred at 0.55 $\mu$m)
is $A_{V}=0.0186$ as derived from the average visual extinction-to-reddening
ratio of 3.1 Schultz and Wiemer (1975), with smaller $A_{\nu}$ predicted for
larger wavelengths by extinction curves Gordon  _et al._ (2003) since
scattering by interstellar grains becomes weaker. Setting our $A_{\nu}$ even
to this maximal extinction, $d_{\rm max}$ decreases but by a factor of 0.92.
These uncertainties are comparable to those of pulsar distance measurements
from radio parallax, which achieve a precision of 10-20% Verbiest  _et al._
(2010, 2012). Where parallax data is unavailable, the radio dispersion measure
is used to estimate neutron star distances, which suffer from much larger
uncertainties as we will discuss in Sec. III.

Now suppose an NS is observed to shine with detectably high spectral flux
density in some filter. Due to uncertainties in its distance from Earth, its
mass and radius, the NS’ luminosity and temperature cannot be reliably
inferred. However a follow-up observation in a second filter (perhaps nearby
in wavelength range) can immediately take the distance (and the corresponding
uncertainty) out of the equation if the ratio of $f_{\nu}$ in two filters is
taken, as seen from Eq. (1). This would constrain the temperature to a narrow
range, which in fact is how temperatures are inferred from the “color”, i.e.
the difference in magnitudes obtained in two filters, as in a color-magnitude
diagram. In fact observations in four different filters would help
simultaneously fit $T_{\rm\infty}$, $d$, $M_{\rm NS}$ and $R_{\rm NS}$ in Eq.
(1), and even more filters could confirm and improve the fit, or in other
words, overconstrain the parameters. The precision achievable in this exercise
would depend on how much exposure time is allotted to each filter, the NS
luminosity, and so on, a careful study of which we leave to future work.

###  II.6 Implications for PSR J2144$-$3933 and similar pulsars

Let us recollect that PSR J2144$-$3933, with spin-down age $t_{\rm
NS}=3.3\times 10^{8}$ yr, is the coldest neutron star observed with a
temperature upper limit of $T_{\rm\infty}<$ 33000 K as obtained in Ref.
Guillot  _et al._ (2019). From Fig. 2, the $d_{\rm max}$ sensitivities of
JWST-NIRCam, ELT-MICADO and TMT-IRIS in the F070W, $I$ and $Z$ filters to a
$T_{\rm\infty}=20000$ Kelvin NS for an exposure time of $10^{5}$ seconds are,
respectively, 170 pc, 210 pc, and 340 pc. These are comparable to or greater
than the parallax distance of PSR J2144$-$3933, 157–192 pc, which is the
1$\sigma$ range obtained after correcting for the Lutz-Kelker bias in radio
parallax measurements Deller  _et al._ (2009); Verbiest  _et al._ (2010). This
implies that a stronger upper limit on the temperature of this pulsar (or a
potential measurement) is within the sensitivity of these filters.

In Table 2 we show the SNR = 5 “discovery” reach and SNR = 2 “upper limit”
reach in the effective temperature $T_{\infty}$ of PSR J2144$-$3933 for these
filters with exposure times of $10^{4}$ s and $10^{5}$ s, and for distances of
157 pc, 172 pc, 192 pc (the central value and the 1$\sigma$ edges of the
parallax distance measurement). We see that in the best case, the future TMT-
IRIS in the $Z$ band can reach down to 13350 K at SNR = 5 and 8510 K at SNR =
2, a significant improvement over the current upper bound. Pressingly, today’s
JWST-NIRCam already has the sensitivity to set an SNR = 2 upper bound of
17880–23350 Kelvin, an improvement over the 27000–36000 Kelvin range of upper
bounds set in Ref. Guillot  _et al._ (2019) by varying the pulsar distance and
radius. This is one of the main results of our study.

We note in passing that PSR J2144$-$3933 is not only the coldest NS observed,
but is also the pulsar with the smallest known spin-down power Deller  _et
al._ (2009), and is notable for lying far beyond the so-called pulsar death
line on the $P$-$\dot{P}$ plane, calling into question current models of
pulsar radio emission Kou  _et al._ (2019). Any opportunity for closer
scrutiny of this maverick pulsar should therefore be seized. Observations of
anomalous reheating could potentially lead to new astrophysical insights that
explain the unusual properties of such slowly rotating radio pulsars.

While we have shown the imminent reaches of the temperature of PSR
J2144$-$3933, other middle-aged pulsars with a measured upper bound on their
temperatures are also ripe for observation at JWST, ELT and TMT, albeit with
lower sensitivities. One such is PSR J1932+1059 (a.k.a. PSR B1929+10), with an
upper limit on its effective temperature $T_{\infty}\lesssim 5\times 10^{5}$ K
Potekhin  _et al._ (2020); coo . For its spin-down age of $3.1\times 10^{6}$
yr, one expects from minimal cooling models the effective temperature to be
about $2\times 10^{4}$ K Ofengeim and Yakovlev (2017); Bramante  _et al._
(2022). As the distance to this pulsar is $310\begin{subarray}{c}+90\\\
-60\end{subarray}$ pc Verbiest  _et al._ (2012), we see from Fig. 2 that its
passive-cooling temperature – or a slightly reheated temperature – may be
cornered at JWST and ELT with exposure times of $10^{6}$ s and at TMT with
$\lesssim 10^{5}$ s. Of course, a reheated temperature in excess of 40000
Kelvin (via a mechanism that is non-universal to pulsars so as to satisfy the
upper limit on PSR J2144$-$3933’s temperature) could be unveiled at these
telescopes with much shorter exposure times.

###  II.7 Potential neutron star candidates and pulsar catalogues

Although about $10^{5}$ NSs lie in the solar neighborhood of the Galaxy, fewer
than 4000 have been found (see Sec. III), and among these fewer than 100 have
had their luminosities measured Potekhin  _et al._ (2013). On the other hand,
the ATNF catalogue lists about 280 pulsars within a distance of kpc as derived
from dispersion measure data and the YMW16 model of electron column densities:
see Sec. III for a discussion. Thus hundreds, if not thousands, of discovered
NSs wait for their temperature to be taken.

In addition, the recently instated radio telescope FAST in Guizhou, China has
listed, at the time of writing, 825 newly found pulsars [188 in the Commensal
Radio Astronomy FAST Survey (CRAFTS) pul (a) and 637 in the Galactic Plane
Pulsar Snapshot (GPPS) survey v3.2.0 pul (b); *pulsarcatFASTGPPS1Han:2021ekd;
*pulsarcatFASTGPPS2Zhou:2023nns; *pulsarcatFASTGPPS3Su:2023vcw], and CHIME in
British Columbia, Canada has listed 25 pulsars pul (c) (of which 14 are
rotating radio transients Dong  _et al._ (2023)). From these catalogues, we
have taken the information on these pulsars’ dispersion measures and
equatorial co-ordinates, and used the online tool in Ref. dis  (that is based
on PyGEDM Price  _et al._ (2021)) to extract their distances from Earth as per
the YMW16 Yao  _et al._ (2017) and NE2001 Cordes and Lazio (2002) models of
Galactic electron densities. We have tabulated these distances in Appendix B
(Tables 3, 4, 5), in ascending order of the minimum of the two distances
obtained from either model. Our exercise shows that there are potentially 24
pulsars in the FAST-CRAFTS catalogue, 7 in the FAST-GPPS catalogue, and 5 in
the CHIME catalogue that are within a kpc of Earth, for which luminosity
measurements can be performed at JWST, EMT, and TLT with not-unreasonable
exposure times. With continued observation, the rate of change of spin period
$\dot{P}$ of these pulsars may be measured, from which their spin-down age
$P/2\dot{P}$ inferred. Combining the luminosity and age measurements will
inform us about late-time reheating in these NSs. It must be noted that the
distances we have inferred from the dispersion measure are far from certain, a
point to which we turn in the next section.

##  III Discussion

In this study we have estimated the maximum distance to neutron stars that
JWST, ELT and TMT can observe with viable exposure times for stellar
temperatures ranging between 2000–40000 Kelvin, which would make them the
coldest NSs detected. One challenge to note here is that the measurement of NS
distances is fraught with large uncertainties. These come particularly in the
form of disagreement between methods of radio parallax and dispersion measure,
and among inferences of dispersion measure distances from various Galactic
electron distribution models. To illustrate with an example, the parallax
estimate of PSR J2144$-$3933 (discussed in Sec. II.6), 157$-$192 pc, is
consistent with the dispersion measure method in the TC93 model giving 180 pc
but not in the NE2001 model giving 264 pc Deller  _et al._ (2009) and YMW16
model giving 289 pc Manchester  _et al._ (2005). An even more egregious
example is PSR J1057-5226, listed in the ATNF catalogue as the closest pulsar
at 93 pc via the YMW16 model, but is put at 1530 pc and $730\pm 150$ pc by the
TC93 and NE2001 models respectively Posselt  _et al._ (2015), and at
350$\pm$150 pc by a simultaneous analysis of the optical and X-ray spectrum
Mignani  _et al._ (2010); a parallax estimate of its distance is lacking to
our knowledge. Due to the highly uncertain and mutually inconsistent models of
electron column densities Price  _et al._ (2021), dispersion measures yield
less accurate distance measurements than radio parallax data, when available.
Such are the dispersion measure distance systematics that the YMW16 model can
only guarantee that the uncertainty on 95% of its estimates is less than 90%,
in itself a 50% improvement over the NE2001 model (if the pulsar sample size
of NE2001 were artificially increased to that of YMW16, that is) Yao  _et al._
(2017). As a reminder, typical uncertainties in parallax distance measurements
are 10%$-$20%. Refinement of dispersion measure methods for nearby pulsars
warrants further study Bramante  _et al._ . In any case, as discussed in Sec.
II.5 the distance uncertainty may be eliminated from the temperature
measurement if an NS’ thermally emitted photons are observed in two or more
filters with sufficient statistics.

As indicated by the right-hand y-axis of Fig. 2, there lurk a few million NSs
within a kpc distance, yet only about 3500 pulsars have been discovered in all
so far Manchester  _et al._ (2005). While this is partly due to the limitation
in sensitivity of radio telescopes, it is also likely due to suppression in
the pulsing mechanism itself. It is generally believed that as pulsars get
older than $10^{7}$ yr their rotational energy gets too weak to generate the
pulsar beam. The pulsar is then said to cross the “pulsar death line”, a.k.a.
“death valley”, as drawn on a $P$-$\dot{P}$ diagram. However, numerous pulsars
(including PSR J2144-3933, as mentioned in Sec. II.6) have been recently
observed to lie beyond different models of the death line Kou  _et al._
(2019); Caleb  _et al._ (2022); Young  _et al._ (1999); Kou and Tong (2019).
Further, around $10^{5}$ NSs in the Milky Way are estimated to similarly lie
beyond the death line Young  _et al._ (1999). These anomalies underline the
fact that the precise mechanism generating pulsar beams is as yet poorly
established. They also suggest that with better radio sensitivity, the
fraction of NSs discoverable as pulsars would be larger than previously
supposed.

If late-stage reheating mechanisms such as those outlined in Appendix A were
in place, one other way to discover large populations of neutron stars would
be sky surveys. In the near future, Rubin/LSST Peterson (2016); Ivezić  _et
al._ (2019) that is designed to achieve a sensitivity of g-band AB magnitude
27.5 (corresponding to 36 nJy) will be able to detect NSs with effective
temperatures a few times 104 Kelvin that are about 100 pc away Bramante  _et
al._ (2022). This means that there are potentially a few hundred
$\mathcal{O}(10^{4})$ Kelvin-hot NSs that may be discovered by sky surveys
with Rubin/LSST, which would be a spectacular discovery. In the future,
infrared surveys such as Roman/WFIRST WFIRST (2012) may also uncover fainter
neutron stars in the solar vicinity. Getting such observational handles on
reheating mechanisms gains significance when we consider that NSs have already
been discovered that are hotter than expected for their age, notably the
millisecond pulsars PSR J0437$-$4715 Kargaltsev  _et al._ (2004); Durant  _et
al._ (2012) and PSR J2124$-$3358 Rangelov  _et al._ (2017), and the regular
pulsars PSR J0108$-$1431 Mignani  _et al._ (2008); *NStemp2:J0108-1431:HST and
PSR B0950$+$08 Pavlov  _et al._ (2017). These and other similarly over-warm
pulsars are consistent with rotochemical and vortex creep heating mechanisms;
see Appendix A.

As mentioned in the Introduction, infrared imaging instruments may shed light
on (after taking it from) poorly understood central compact objects, sources
of steady X-rays with luminosities larger than their spin-down power. CCOs are
also marked by their relatively small magnetic field strengths
($\mathcal{O}(10^{10-11})$ Gauss), for which one explanation could be that an
expected $10^{12-14}$ Gauss magnetic field of an NS at birth was buried in its
crust by supernova debris falling back. This fallback could form around the NS
a disk that could emit in the infrared, detection of which would favour models
predicting the re-surfacing of the (small) magnetic field, and explain the
non-detection of later-stage versions of young CCOs Gotthelf  _et al._ (2013).
The infrared luminosity of the disk, if heated by the CCO’s thermal X-rays,
could be proportional to the X-ray luminosity, and the infrared spectrum could
be power-law or blackbody. Non-thermal emission from the NS could also result
in a power law spectrum. A recent search for these possibilities using NIRCam
at JWST resulted in non-detection Milisavljevic  _et al._ (2024), however
there are 9 confirmed and 3 suspected CCOs Cat  for which analogous
measurements may be taken at JWST, ELT, and TMT.

Since their discovery, neutron stars have served as excellent testing grounds
for fundamental physics. Today’s and tomorrow’s telescopes now have the
opportunity to be pioneering thermometers, with the potential to make
transformative discoveries.

## Acknowledgments

We thank Joseph Bramante, Raghuveer Garani, Pranav Kalsi, Avinash Kumar
Paladi, and Akshank Tyagi for discussion, Jasnoor Singh for a script to
extract distances to pulsars in the FAST catalogues, and Koichi Hamaguchi and
Natsumi Nagata for comments on the manuscript.

pulsar |  dispersion measure (pc/cm3) |  distance YMW16 (pc) |  distance NE2001 (pc)
---|---|---|---
J0745-0411 | 10 | 149.2 | 670.3
J0623+0051 | 20 | 153.3 | 1073.7
J1720-0533 | 37.8 | 191.4 | 1380.1
J1721-0854 | 49.1 | 201.0 | 1616.0
J1708-0424 | 49.3 | 230.3 | 1858.7
J1712-1124 | 69.3 | 232.5 | 2253.3
J1020+4056 | 9.9 | 1050.5 | 418.0
J1115-0958 | 9.9 | 506.3 | 467.1
J0844-0329 | 15 | 516.1 | 815.8
J1751-0542 | 64 | 522.3 | 1950.9
J2355+0049 | 11 | 943.8 | 545.7
J0846-1049 | 13.7 | 562.4 | 866.3
J1737-0514 | 78 | 568.1 | 2525.5
J0631+4147 | 24.5 | 574.0 | 823.7
J2230-0336 | 10.3 | 851.7 | 587.5
J0756-0517 | 21.7 | 673.6 | 1272.2
J0941+4542 | 17.6 | 1794.6 | 680.5
J0203-0150 | 19.2 | 1841.9 | 772.2
J0803-0937 | 21.5 | 840.0 | 1336.5
J0521+5647 | 21.8 | 982.9 | 863.3
J1742-0559 | 96.2 | 910.1 | 3035.3
J0924+6102 | 21.8 | 1989.0 | 919.2
J0447+2446 | 32.8 | 920.5 | 1066.6
J1611-0113 | 19.5 | 1183.7 | 990.2
Table 3: A subset of the 188 pulsars in the publicly available FAST-CRAFTS catalogue pul (a), along with dispersion measure distances derived in this work using the pulsars’ equatorial co-ordinates (indicated in their names) and the YMW16 and NE2001 electron column density models. The pulsars are listed in ascending order of the minimum of the distance obtained in either model. Only pulsars with this minimum smaller than 1 kpc are listed here, while the rest of the catalogue is uploaded to arXiv as an ancillary file. Distances smaller than 600 pc (1 kpc) are highlighted in blue (bold). The uncertainties on most of the YMW16 distances are estimated to be $<$90%, but for some it could be larger in both models. See Secs. II.7 and III for further details. pulsar |  dispersion measure (pc/cm3) |  distance YMW16 (pc) |  distance NE2001 (pc)
---|---|---|---
J0653+0443g | 27.3 | 181.3 | 1251.1
J0623+0220g | 32.1 | 437.5 | 1399.5
J1908+1035g | 10.9 | 670.9 | 593.4
J1854+0704g | 10.8 | 647.2 | 973.3
J2011+3006g | 14 | 943.9 | 1225.0
J1844+0315g | 23 | 974.2 | 1334.9
J1857+0642g | 21.6 | 993.5 | 1586.7
Table 4: Same as Table 3 but a subset of the 637 pulsars in the publicly available FAST-GPPS catalogue pul (b); *pulsarcatFASTGPPS1Han:2021ekd; *pulsarcatFASTGPPS2Zhou:2023nns; *pulsarcatFASTGPPS3Su:2023vcw. The rest of the catalogue is uploaded to arXiv as an ancillary file. FAST adds the suffix “g” to these pulsar names to indicate a position uncertainty of about 1.5′, which may propagate into the dispersion measure distance estimate. pulsar |  dispersion measure (pc/cm3) |  distance YMW16 (pc) |  distance NE2001 (pc)
---|---|---|---
J0746+55 | 10.5 | 410.4 | 468.5
J1105+02 | 16.5 | 1579.6 | 690.6
J0854+54 | 17.8 | 1330.7 | 710.3
J0658+29 | 40.05 | 891.8 | 1350.5
J1252+53 | 20.7 | 3190.6 | 995.7
J1130+09 | 21.9 | 5184.6 | 1023.5
J1541+47 | 19.4 | 2033.8 | 1037.8
J2215+4524 | 18.5 | 1131.2 | 1579.2
J0741+17 | 44.3 | 1206.8 | 1711.1
J2355+1523 | 26 | 3422.3 | 1362.1
J1838+5051 | 21.8 | 1702.6 | 1539.6
J0227+33 | 27.56 | 2050.7 | 1741.7
J0209+58 | 56 | 1751.2 | 2142.3
J0653-06 | 83.7 | 2140.3 | 2984.7
J2138+69 | 46.6 | 2486 | 2204.4
J2237+2828 | 38.1 | 3989.9 | 2279.5
J0121+5329 | 87.35 | 2529.2 | 3213.6
J2116+37 | 44 | 2914.7 | 2803.5
J2208+46 | 63 | 3148.4 | 3095.9
J2108+45 | 84 | 3592.3 | 3778.3
J1943+58 | 71.2 | 7874.5 | 4722.8
J2057+46 | 218.962 | 4939.3 | 7105
J0012+5431 | 131.3 | 5425 | 5727.3
J2008+3758 | 143 | 5923.7 | 5825.4
J2005+38 | 192.8 | 7647 | 7633.1
Table 5: The 25 pulsars (including 14 rotating radio transients) in the CHIME
catalogue pul (c) along with dispersion measure distances derived in this work
using the pulsars’ equatorial co-ordinates (indicated in their names) and the
YMW16 and NE2001 electron column density models. The pulsars are listed in
ascending order of the minimum of the distance obtained in either model.
Distances smaller than 600 pc (1 kpc) are highlighted in blue (bold). The
uncertainties on most of the YMW16 distances are estimated to be $<$90%, but
for some it could be larger in both models. See Secs. II.7 and III for further
details.

##  Appendix A Reheating mechanisms

In this appendix we briefly review neutron star late-stage reheating
mechanisms that arise in plausible astrophysical settings as well as via more
exotic agents such as a hidden sector of states and dark matter. It must be
remembered that none of these mechanisms have been unambiguously observed, nor
are they soundly established in theory. Non-observation of NS luminosities
would place upper limits on all their strengths.

###  A.1 Astrophysical mechanisms

Rotochemical heating Fernandez and Reisenegger (2005);
*Rotochem:Reisenegger:2006ky; *Rotochem:PetrovichReis:2009yh;
*Rotochem:GonzalezJimenezReis:2014iia; *Rotochem:Gusakov2021:Refined. As an NS
spins down, the decreasing centrifugal forces drive NS material out of beta
equilibrium. Processes restoring the balance in chemical potentials would then
deposit heat. It is for certain nucleon pairing models of superfluidity in NSs
with $\lesssim\mathcal{O}(10)$ ms natal periods that this mechanism will take
effect Yanagi  _et al._ (2020); Hamaguchi  _et al._ (2019), which must be
contrasted against findings that initial spin periods are likely
$\mathcal{O}(10-100)$ ms Faucher-Giguere and Kaspi (2006); *NatalPeriod:2010;
*NatalPeriod:2012; *NatalPeriod:2013; *NatalPeriod:Igoshev:2013yua;
*NatalPeriod:Gullon:2014dva; *NatalPeriod:Gullon:2015zca;
*NatalPerod:Muller:2018utr, with the caveat that NS initial periods depend on
the details of the supernova event. Rotochemical heating has been invoked to
explain the temperatures of pulsars that are otherwise too high for their
spin-down age, and is also shown to satisfy the upper bound on the effective
temperature of PSR J2144$-$3933 Yanagi  _et al._ (2020). One variant of this
mechanism is reheating due to upsetting of nuclear statistical equilibrium in
the NS crust as opposed to chemical equilibrium in the core Gusakov  _et al._
(2015).

Vortex creep Anderson and Itoh (1975); *VortexCreep:1984; *VortexCreep:1989;
*VortexCreep:1993; *VortexCreep:VanRiper:1994vp; *VortexCreep:Larson:1998it.
Nucleon-superfluid vortex lines in the inner crust may be pinned to the
background lattice of nuclei. In addition to this pinning force, vortex lines
experience a Magnus force due to differences in rotational speed between the
normal and superfluid matter, which causes them to move outward. This creep
motion induces friction, which then dissipates energy. The heating luminosity
of this mechanism is proportional to the angular deceleration $\dot{\Omega}$,
and the proportionality constant is expected to be the same for all NSs.
Intriguingly, the proportionality constant reconstructed from observation
assuming thermal equilibrium, $J_{\rm obs}\equiv 4\pi
R_{\infty}^{2}\sigma_{\rm SB}T^{4}_{\infty}/|\dot{\Omega}|$, is indeed found
to be roughly uniform across an ensemble of millisecond and ordinary pulsars
that are too hot for their spin-down age Fujiwara  _et al._ (2023). The fitted
range $J\simeq 10^{35.9-36.8}$ Joule sec is also consistent with theoretical
predictions in a mesoscopic approach of estimating the vortex pinning force.

Magnetic field decay Goldreich and Reisenegger (1992). Through such processes
as ohmic decay, ambipolar diffusion and Hall drift, the magnetic field of the
NS could dissipate energy into the stellar material. Rough estimates of the
heating luminosity Gonzalez and Reisenegger (2010); Köpp  _et al._ (2023)
indicate that old isolated NSs have magnetic field strengths in the right
range to produce $T_{\infty}\lesssim 10^{4}$ K.

Crust cracking Baym and Pines (1971). As the NS spins down, the ellipsoidal
crust gradually becomes spherical, acquires stress, and breaks, liberating the
energy of the accumulated strain. Crust cracking can result in reheating
temperatures of $T_{\infty}\simeq 10^{4}$ K Gonzalez and Reisenegger (2010).

We remark that accretion of interstellar material (ISM) is unlikely to reheat
old NSs to detectable luminosities. Around old, isolated NSs with periods
$<10^{3}$ sec, the NS magnetic field drives a “pulsar wind” of ISM outflow
that is much denser than inflowing material. Thus kinetic pressure prevents
accretion, and the NS is said to be in an ejector phase Treves  _et al._
(2000). Were the ISM to overcome a weak pulsar wind, its accretion would still
be thwarted by the centrifugal acceleration provided by the co-rotating
magnetosphere, which would disperse the ISM. The NS is then said to be in the
propellor phase. For unusually slow-rotating NSs with periods $>1000$ seconds,
infalling ISM would travel along the NS’ magnetic field lines so that only a
small polar region accretes, which may be discriminated from thermal emission.

###  A.2 Non-standard mechanisms

Dark matter capture Bramante and Raj (2024); Baryakhtar  _et al._ (2022);
*snowmass:Carney:2022gse. Perhaps the most minimal theoretical setup to reheat
old, isolated NSs, this mechanism suggests that the incident energy flux of
dark matter particles ambient to a neutron star in the solar vicinity has a
heating luminosity corresponding to blackbody temperatures of about 2000 K.
The steep gravitational potential of NSs (with surface escape speeds $\gtrsim
0.5c$) serves to both enhance gravitational focusing of the dark matter flux
and accelerate the dark matter to a kinetic energy comparable to its mass
energy. Via kinetic energy transfer Baryakhtar  _et al._ (2017), scattering of
the dark matter on the NS’ nucleons, electrons, muons, possible hyperons, and
even crustal components like nuclear pasta and superfluid neutrons can impart
heat Raj  _et al._ (2018); Bell  _et al._ (2018, 2019); Joglekar  _et al._
(2019, 2020); Bell  _et al._ (2021); Garani and Heeck (2019); Chen and Lin
(2018); Hamaguchi  _et al._ (2019); Garani  _et al._ (2019); Camargo  _et al._
(2019); Bell  _et al._ (2020a, b); Anzuini  _et al._ (2021); Keung  _et al._
(2020); Dasgupta  _et al._ (2020); Zeng  _et al._ (2022); Maity and Queiroz
(2021); Fujiwara  _et al._ (2022); Hamaguchi  _et al._ (2022); Coffey  _et
al._ (2022); Alvarez  _et al._ (2023); Fujiwara  _et al._ (2024). In addition,
in some models captured dark matter can thermalize Bertoni  _et al._ (2013);
Garani  _et al._ (2019); Bell  _et al._ (2023) with the NS and self-annihilate
or co-annihilate with nucleons to deposit more heat, helping observation
prospects Kouvaris (2008); de Lavallaz and Fairbairn (2010); Baryakhtar  _et
al._ (2017); Raj  _et al._ (2018); Acevedo  _et al._ (2020); Chatterjee  _et
al._ (2023); Davoudiasl  _et al._ (2010, 2011); Jin and Gao (2018); Keung  _et
al._ (2020); Herrero  _et al._ (2019); Bai  _et al._ (2020). JWST may also
reveal the overheating of exoplanets by self-annihilations of captured dark
matter Leane and Smirnov (2021).

Higher temperatures up to 40000 K can be obtained via Bondi accretion of
strongly-interacting dark matter particles with dissipative fluid behavior in
clumps/microhalos that periodically encounter NSs Bramante  _et al._ (2022).
In the absence of Bondi accretion, these clumps could temporarily impart
temperatures $>10^{3-4}$ K to NSs, which can be observed in a sky survey
looking for a fraction of overheated NSs in an ensemble. These temperatures
may also be achieved by a combination of tidal heating and kinetic energy
transfer by dark matter states that experience a new long-range force with
baryons, with the heating energy sourced mainly by the long-range potential
that effectively enhances the gravitational potential of the neutron star
Gresham  _et al._ (2023).

Nucleon Auger effect McKeen  _et al._ (2021a); *NSheat:Mirror:McKeen:2021jbh.
Nucleons may disappear from NSs through scattering and decay processes,
notably to final states involving dark sector states carrying baryon number,
“dark baryons”. The resulting hole in the Fermi sea gets filled rapidly by
surrounding nucleons with higher energy, accompanied by spillage of
electromagnetic and kinetic energy, reminiscent of the Auger effect in semi-
conductors McKeen  _et al._ (2021a); *NSheat:Mirror:McKeen:2021jbh; Goldman
_et al._ (2022a); *NSheat:Mirror:Goldman:2022rth;
*NSheat:Mirror:Berezhiani:2020zck. The heat liberated in this process can
raise NS temperatures to 40000 K.

Baryon number-violating decay Davoudiasl (2023). The decay of the neutron to a
pion and positron via an ultra-light scalar mediator is better constrained by
neutron star overheating due to neutron mass energy deposition than by
terrestrial experiments like Super-Kamiokande. This process too can impart NS
temperatures of 40000 K.

##  Appendix B Pulsar catalogues

In this Appendix we tabulate pulsars from the catalogues of FAST pul (a, b);
*pulsarcatFASTGPPS1Han:2021ekd; *pulsarcatFASTGPPS2Zhou:2023nns;
*pulsarcatFASTGPPS3Su:2023vcw, and CHIME pul (c) and their dispersion measure
distances we obtained using the online tool in Ref. dis , as discussed in Sec.
II.7.

## References

  * Nättilä and Kajava (2022) J. Nättilä and J. J. E. Kajava,  (2022), 10.1007/978-981-16-4544-0105-1, arXiv:2211.15721 [astro-ph.HE] .
  * Bramante and Raj (2024) J. Bramante and N. Raj, Phys. Rept. 1052, 1 (2024), arXiv:2307.14435 [hep-ph] .
  * Potekhin  _et al._ (2020) A. Y. Potekhin, D. A. Zyuzin, D. G. Yakovlev, M. V. Beznogov, and Y. A. Shibanov, Mon. Not. Roy. Astron. Soc. 496, 5052 (2020), arXiv:2006.15004 [astro-ph.HE] .
  * Guillot  _et al._ (2019) S. Guillot, G. G. Pavlov, C. Reyes, A. Reisenegger, L. Rodriguez, B. Rangelov, and O. Kargaltsev, Astrophys. J. 874, 175 (2019), arXiv:1901.07998 [astro-ph.HE] .
  * Page  _et al._ (2004) D. Page, J. M. Lattimer, M. Prakash, and A. W. Steiner, Astrophys. J. Suppl. 155, 623 (2004), arXiv:astro-ph/0403657 [astro-ph] .
  * Ofengeim and Yakovlev (2017) D. D. Ofengeim and D. G. Yakovlev, Mon. Not. Roy. Astron. Soc. 467, 3598 (2017).
  * Bramante  _et al._ (2022) J. Bramante, B. J. Kavanagh, and N. Raj, Phys. Rev. Lett. 128, 231801 (2022), arXiv:2109.04582 [hep-ph] .
  * Gonzalez and Reisenegger (2010) D. Gonzalez and A. Reisenegger, AAP 522, A16 (2010), arXiv:1005.5699 [astro-ph.HE] .
  * Gardner  _et al._ (2006) J. P. Gardner _et al._ , Space Sci. Rev. 123, 485 (2006), arXiv:astro-ph/0606175 .
  * JWS (a) “JWST Discovery Timeline,” https://www.jameswebbdiscovery.com/discoveries/all-james-webb-telescope-discoveries (a).
  * Neichel  _et al._ (2018) B. Neichel, D. Mouillet, E. Gendron, C. Correia, J. F. Sauvage, and T. Fusco, “Overview of the european extremely large telescope and its instrument suite,” (2018), arXiv:1812.06639 [astro-ph.IM] .
  * Skidmore  _et al._ (2015) W. Skidmore _et al._ (TMT International Science Development Teams & TMT Science Advisory Committee), Res. Astron. Astrophys. 15, 1945 (2015), arXiv:1505.01195 [astro-ph.IM] .
  * Sartore  _et al._ (2010) N. Sartore, E. Ripamonti, A. Treves, and R. Turolla, Astronomy and Astrophysics 510, A23 (2010).
  * Manchester  _et al._ (2005) R. N. Manchester, G. B. Hobbs, A. Teoh, and M. Hobbs, Astron. J. 129, 1993 (2005), arXiv:astro-ph/0412641 .
  * De Luca (2017) A. De Luca, J. Phys. Conf. Ser. 932, 012006 (2017), arXiv:1711.07210 [astro-ph.HE] .
  * Milisavljevic  _et al._ (2024) D. Milisavljevic _et al._ , (2024), arXiv:2401.02477 [astro-ph.HE] .
  * (17) “Central Compact Object Catalogue,” https://www.iasf-milano.inaf.it/~deluca/cco/main.htm.
  * Arnett  _et al._ (1989) W. D. Arnett, D. N. Schramm, and J. W. Truran, Astrophys. J. Lett. 339, L25 (1989).
  * JWS (b) “JWST approved proposals,” https://www.stsci.edu/jwst/science-execution/approved-programs (b).
  * Chatterjee  _et al._ (2023) S. Chatterjee, R. Garani, R. K. Jain, B. Kanodia, M. S. N. Kumar, and S. K. Vempati, Phys. Rev. D 108, L021301 (2023), arXiv:2205.05048 [astro-ph.HE] .
  * Wiringa  _et al._ (1988) R. B. Wiringa, V. Fiks, and A. Fabrocini, Phys. Rev. C 38, 1010 (1988).
  * Datta  _et al._ (1995) B. Datta, A. V. Thampan, and D. Bhattacharya, Journal of Astrophysics and Astronomy 16, 375 (1995).
  * JWS (c) “JWST exposure times,” https://jwst-docs.stsci.edu/jwst-exposure-time-calculator-overview (c).
  * JWS (d) “JWST Pocket Guide,” https://jwst-docs.stsci.edu/ (d).
  * (25) “NIRISS readout patterns,” https://jwst-docs.stsci.edu/jwst-near-infrared-imager-and-slitless-spectrograph/niriss-instrumentation/niriss-detector-overview/niriss-detector-readout-patterns.
  * (26) “ELT Exposure Time Calculator,” https://www.eso.org/observing/etc/doc/elt/helpelt.html.
  * Padovani and Cirasuolo (2023) P. Padovani and M. Cirasuolo, Contemp. Phys. 64, 47 (2023), arXiv:2312.04299 [astro-ph.IM] .
  * (28) “TMT-IRIS Exposure Time Calculator,” https://www.tmt.org/etc/iris.
  * Deller  _et al._ (2009) A. T. Deller, S. J. Tingay, M. Bailes, and J. E. Reynolds, The Astrophysical Journal 701, 1243–1257 (2009).
  * Schultz and Wiemer (1975) G. V. Schultz and W. Wiemer, aap 43, 133 (1975).
  * Gordon  _et al._ (2003) K. D. Gordon, G. C. Clayton, K. A. Misselt, A. U. Landolt, and M. J. Wolff, Astrophys. J. 594, 279 (2003), arXiv:astro-ph/0305257 .
  * Verbiest  _et al._ (2010) J. P. W. Verbiest, D. R. Lorimer, and M. A. McLaughlin, Monthly Notices of the Royal Astronomical Society (2010), 10.1111/j.1365-2966.2010.16488.x.
  * Verbiest  _et al._ (2012) J. P. W. Verbiest, J. M. Weisberg, A. A. Chael, K. J. Lee, and D. R. Lorimer, apj 755, 39 (2012), arXiv:1206.0428 [astro-ph.GA] .
  * Kou  _et al._ (2019) F. F. Kou, H. Tong, R. X. Xu, and X. Zhou, APJ 876, 131 (2019), arXiv:1901.00300 [astro-ph.HE] .
  * (35) “A. Potekhin’s Neutron Star Thermal Emission Catalogue,” https://www.ioffe.ru/astro/NSG/thermal/index.html.
  * Potekhin  _et al._ (2013) A. Y. Potekhin, A. F. Fantina, N. Chamel, J. M. Pearson, and S. Goriely, Astron. Astrophys. 560, A48 (2013), arXiv:1310.0049 [astro-ph.SR] .
  * pul (a) “FAST-CRAFTS pulsar catalogue,” http://groups.bao.ac.cn/ism/CRAFTS/202203/t20220310_683697.html (a).
  * pul (b) “FAST-GPPS pulsar catalogue,” http://zmtt.bao.ac.cn/GPPS/GPPSnewPSR.html (b).
  * Han  _et al._ (2021) J. L. Han _et al._ , Res. Astron. Astrophys. 21, 107 (2021), arXiv:2105.08460 [astro-ph.HE] .
  * Zhou  _et al._ (2023) D. J. Zhou _et al._ , Res. Astron. Astrophys. 23, 104001 (2023), arXiv:2303.17279 [astro-ph.HE] .
  * Su  _et al._ (2023) W. Q. Su _et al._ , Mon. Not. Roy. Astron. Soc. 526, 2645 (2023), arXiv:2305.16754 [astro-ph.HE] .
  * pul (c) “CHIME pulsar catalogue,” https://www.chime-frb.ca/galactic (c).
  * Dong  _et al._ (2023) F. A. Dong _et al._ , Mon. Not. Roy. Astron. Soc. 524, 5132 (2023), arXiv:2210.09172 [astro-ph.HE] .
  * (44) “Galactic dispersion measure tool,” https://pulsar.cgca-hub.org/compute.
  * Price  _et al._ (2021) D. C. Price, A. Deller, and C. Flynn,  (2021), 10.1017/pasa.2021.33, arXiv:2106.15816 [astro-ph.GA] .
  * Yao  _et al._ (2017) J. M. Yao, R. N. Manchester, and N. Wang, APJ 835, 29 (2017), arXiv:1610.09448 [astro-ph.GA] .
  * Cordes and Lazio (2002) J. M. Cordes and T. J. W. Lazio, (2002), arXiv:astro-ph/0207156 .
  * Posselt  _et al._ (2015) B. Posselt, G. Spence, and G. G. Pavlov, Astrophys. J. 811, 96 (2015), arXiv:1507.07924 [astro-ph.HE] .
  * Mignani  _et al._ (2010) R. P. Mignani, G. G. Pavlov, and O. Kargaltsev, Astrophys. J. 720, 1635 (2010), arXiv:1007.2940 [astro-ph.HE] .
  * (50) J. Bramante, K. Mack, N. Raj, L. Shao, and N. Tyagi, “To appear,” .
  * Caleb  _et al._ (2022) M. Caleb _et al._ , Nature Astron. 6, 828 (2022), arXiv:2206.01346 [astro-ph.HE] .
  * Young  _et al._ (1999) M. D. Young, R. N. Manchester, and S. Johnston, Nature 400, 848 (1999).
  * Kou and Tong (2019) F. F. Kou and H. Tong, Research Notes of the AAS 3, 175 (2019).
  * Peterson (2016) J. R. Peterson,  (2016), 10.2172/1272167.
  * Ivezić  _et al._ (2019) v. Ivezić _et al._ (LSST), Astrophys. J. 873, 111 (2019), arXiv:0805.2366 [astro-ph] .
  * WFIRST (2012) WFIRST, (2012), arXiv:1208.4012 [astro-ph.IM] .
  * Kargaltsev  _et al._ (2004) O. Kargaltsev, G. G. Pavlov, and R. W. Romani, Astrophys. J. 602, 327 (2004), arXiv:astro-ph/0310854 .
  * Durant  _et al._ (2012) M. Durant, O. Kargaltsev, G. G. Pavlov, P. M. Kowalski, B. Posselt, M. H. van Kerkwijk, and D. L. Kaplan, Astrophys. J. 746, 6 (2012), arXiv:1111.2346 [astro-ph.HE] .
  * Rangelov  _et al._ (2017) B. Rangelov, G. G. Pavlov, O. Kargaltsev, A. Reisenegger, S. Guillot, C. Reyes, and M. H. van Kerkwijk, Astrophys. J. 835, 264 (2017), arXiv:1701.00002 [astro-ph.HE] .
  * Mignani  _et al._ (2008) R. P. Mignani, G. G. Pavlov, and O. Kargaltsev, Astron. Astrophys. 488, 1027 (2008), arXiv:0805.2586 [astro-ph] .
  * Abramkin  _et al._ (2021) V. Abramkin, Y. Shibanov, R. P. Mignani, and G. G. Pavlov, Astrophys. J. 911, 1 (2021), arXiv:2103.00332 [astro-ph.HE] .
  * Pavlov  _et al._ (2017) G. G. Pavlov, B. Rangelov, O. Kargaltsev, A. Reisenegger, S. Guillot, and C. Reyes, Astrophys. J. 850, 79 (2017), arXiv:1710.06448 [astro-ph.HE] .
  * Gotthelf  _et al._ (2013) E. V. Gotthelf, J. P. Halpern, and J. Alford, APJ 765, 58 (2013), arXiv:1301.2717 [astro-ph.HE] .
  * Fernandez and Reisenegger (2005) R. Fernandez and A. Reisenegger, Astrophys. J. 625, 291 (2005), arXiv:astro-ph/0502116 .
  * Reisenegger  _et al._ (2006) A. Reisenegger, P. Jofre, R. Fernandez, and E. Kantor, Astrophys. J. 653, 568 (2006), arXiv:astro-ph/0606322 .
  * Petrovich and Reisenegger (2010) C. Petrovich and A. Reisenegger, Astron. Astrophys. 521, A77 (2010), arXiv:0912.2564 [astro-ph.HE] .
  * González-Jiménez  _et al._ (2015) N. González-Jiménez, C. Petrovich, and A. Reisenegger, Mon. Not. Roy. Astron. Soc. 447, 2073 (2015), arXiv:1411.6500 [astro-ph.SR] .
  * Kantor and Gusakov (2021) E. M. Kantor and M. E. Gusakov, MNRAS 508, 6118 (2021), arXiv:2110.02881 [astro-ph.HE] .
  * Yanagi  _et al._ (2020) K. Yanagi, N. Nagata, and K. Hamaguchi, Mon. Not. Roy. Astron. Soc. 492, 5508 (2020), arXiv:1904.04667 [astro-ph.HE] .
  * Hamaguchi  _et al._ (2019) K. Hamaguchi, N. Nagata, and K. Yanagi, Phys. Lett. B795, 484 (2019), arXiv:1905.02991 [hep-ph] .
  * Faucher-Giguere and Kaspi (2006) C.-A. Faucher-Giguere and V. M. Kaspi, Astrophys. J. 643, 332 (2006), arXiv:astro-ph/0512585 .
  * Popov  _et al._ (2010) S. B. Popov, J. A. Pons, J. A. Miralles, P. A. Boldin, and B. Posselt, mnras 401, 2675 (2010), arXiv:0910.2190 [astro-ph.HE] .
  * Popov and Turolla (2012) S. B. Popov and R. Turolla, apss 341, 457 (2012), arXiv:1204.0632 [astro-ph.HE] .
  * Noutsos  _et al._ (2013) A. Noutsos, D. H. F. M. Schnitzeler, E. F. Keane, M. Kramer, and S. Johnston, mnras 430, 2281 (2013), arXiv:1301.1265 [astro-ph.GA] .
  * Igoshev and Popov (2013) A. P. Igoshev and S. B. Popov, Mon. Not. Roy. Astron. Soc. 432, 967 (2013), arXiv:1303.5258 [astro-ph.HE] .
  * Gullón  _et al._ (2014) M. Gullón, J. A. Miralles, D. Viganò, and J. A. Pons, Mon. Not. Roy. Astron. Soc. 443, 1891 (2014), arXiv:1406.6794 [astro-ph.HE] .
  * Gullón  _et al._ (2015) M. Gullón, J. A. Pons, J. A. Miralles, D. Viganò, N. Rea, and R. Perna, Mon. Not. Roy. Astron. Soc. 454, 615 (2015), arXiv:1507.05452 [astro-ph.HE] .
  * Müller  _et al._ (2019) B. Müller, T. M. Tauris, A. Heger, P. Banerjee, Y. Z. Qian, J. Powell, C. Chan, D. W. Gay, and N. Langer, Mon. Not. Roy. Astron. Soc. 484, 3307 (2019), arXiv:1811.05483 [astro-ph.SR] .
  * Gusakov  _et al._ (2015) M. E. Gusakov, E. M. Kantor, and A. Reisenegger, MNRAS 453, L36 (2015), arXiv:1507.04586 [astro-ph.HE] .
  * Anderson and Itoh (1975) P. W. Anderson and N. Itoh, Nature 256, 25 (1975).
  * Alpar  _et al._ (1984) M. A. Alpar, D. Pines, P. W. Anderson, and J. Shaham, APJ 276, 325 (1984).
  * Shibazaki and Lamb (1989) N. Shibazaki and F. K. Lamb, APJ 346, 808 (1989).
  * Umeda  _et al._ (1993) H. Umeda, N. Shibazaki, K. Nomoto, and S. Tsuruta, APJ 408, 186 (1993).
  * Van Riper  _et al._ (1995) K. Van Riper, B. Link, and R. Epstein, Astrophys. J. 448, 294 (1995), arXiv:astro-ph/9404060 .
  * Larson and Link (1999) M. B. Larson and B. Link, Astrophys. J. 521, 271 (1999), arXiv:astro-ph/9810441 .
  * Fujiwara  _et al._ (2023) M. Fujiwara, K. Hamaguchi, N. Nagata, and M. E. Ramirez-Quezada, (2023), arXiv:2308.16066 [astro-ph.HE] .
  * Goldreich and Reisenegger (1992) P. Goldreich and A. Reisenegger, apj 395, 250 (1992).
  * Köpp  _et al._ (2023) F. Köpp, J. E. Horvath, D. Hadjimichef, C. A. Z. Vasconcellos, and P. O. Hess, Int. J. Mod. Phys. D 32, 2350046 (2023), arXiv:2208.07770 [astro-ph.HE] .
  * Baym and Pines (1971) G. Baym and D. Pines, Annals of Physics 66, 816 (1971).
  * Treves  _et al._ (2000) A. Treves, R. Turolla, S. Zane, and M. Colpi, Publ. Astron. Soc. Pac. 112, 297 (2000), arXiv:astro-ph/9911430 .
  * Baryakhtar  _et al._ (2022) M. Baryakhtar _et al._ , in  _2022 Snowmass Summer Study_ (2022) arXiv:2203.07984 [hep-ph] .
  * Carney  _et al._ (2023) D. Carney _et al._ , SciPost Phys. Core 6, 075 (2023), arXiv:2203.06508 [hep-ph] .
  * Baryakhtar  _et al._ (2017) M. Baryakhtar, J. Bramante, S. W. Li, T. Linden, and N. Raj, Phys. Rev. Lett. 119, 131801 (2017), arXiv:1704.01577 [hep-ph] .
  * Raj  _et al._ (2018) N. Raj, P. Tanedo, and H.-B. Yu, Phys. Rev. D 97, 043006 (2018), arXiv:1707.09442 [hep-ph] .
  * Bell  _et al._ (2018) N. F. Bell, G. Busoni, and S. Robles, JCAP 1809, 018 (2018), arXiv:1807.02840 [hep-ph] .
  * Bell  _et al._ (2019) N. F. Bell, G. Busoni, and S. Robles, JCAP 1906, 054 (2019), arXiv:1904.09803 [hep-ph] .
  * Joglekar  _et al._ (2019) A. Joglekar, N. Raj, P. Tanedo, and H.-B. Yu, (2019), arXiv:1911.13293 [hep-ph] .
  * Joglekar  _et al._ (2020) A. Joglekar, N. Raj, P. Tanedo, and H.-B. Yu, (2020), arXiv:2004.09539 [hep-ph] .
  * Bell  _et al._ (2021) N. F. Bell, G. Busoni, S. Robles, and M. Virgato, JCAP 03, 086 (2021), arXiv:2010.13257 [hep-ph] .
  * Garani and Heeck (2019) R. Garani and J. Heeck, Phys. Rev. D100, 035039 (2019), arXiv:1906.10145 [hep-ph] .
  * Chen and Lin (2018) C.-S. Chen and Y.-H. Lin, JHEP 08, 069 (2018), arXiv:1804.03409 [hep-ph] .
  * Garani  _et al._ (2019) R. Garani, Y. Genolini, and T. Hambye, JCAP 1905, 035 (2019), arXiv:1812.08773 [hep-ph] .
  * Camargo  _et al._ (2019) D. A. Camargo, F. S. Queiroz, and R. Sturani, JCAP 1909, 051 (2019), arXiv:1901.05474 [hep-ph] .
  * Bell  _et al._ (2020a) N. F. Bell, G. Busoni, S. Robles, and M. Virgato, (2020a), arXiv:2004.14888 [hep-ph] .
  * Bell  _et al._ (2020b) N. F. Bell, G. Busoni, T. F. Motta, S. Robles, A. W. Thomas, and M. Virgato, (2020b), arXiv:2012.08918 [hep-ph] .
  * Anzuini  _et al._ (2021) F. Anzuini, N. F. Bell, G. Busoni, T. F. Motta, S. Robles, A. W. Thomas, and M. Virgato, (2021), arXiv:2108.02525 [hep-ph] .
  * Keung  _et al._ (2020) W.-Y. Keung, D. Marfatia, and P.-Y. Tseng, JHEP 07, 181 (2020), arXiv:2001.09140 [hep-ph] .
  * Dasgupta  _et al._ (2020) B. Dasgupta, A. Gupta, and A. Ray, JCAP 10, 023 (2020), arXiv:2006.10773 [hep-ph] .
  * Zeng  _et al._ (2022) Y.-P. Zeng, X. Xiao, and W. Wang, Phys. Lett. B 824, 136822 (2022), arXiv:2108.11381 [hep-ph] .
  * Maity and Queiroz (2021) T. N. Maity and F. S. Queiroz, (2021), arXiv:2104.02700 [hep-ph] .
  * Fujiwara  _et al._ (2022) M. Fujiwara, K. Hamaguchi, N. Nagata, and J. Zheng, (2022), arXiv:2204.02238 [hep-ph] .
  * Hamaguchi  _et al._ (2022) K. Hamaguchi, N. Nagata, and M. E. Ramirez-Quezada, (2022), arXiv:2204.02413 [hep-ph] .
  * Coffey  _et al._ (2022) J. Coffey, D. McKeen, D. E. Morrissey, and N. Raj, Phys. Rev. D 106, 115019 (2022), arXiv:2207.02221 [hep-ph] .
  * Alvarez  _et al._ (2023) G. Alvarez, A. Joglekar, M. Phoroutan-Mehr, and H.-B. Yu, (2023), arXiv:2301.08767 [hep-ph] .
  * Fujiwara  _et al._ (2024) M. Fujiwara, K. Hamaguchi, N. Nagata, and M. E. Ramirez-Quezada, Phys. Lett. 848, 138341 (2024), arXiv:2309.02633 [hep-ph] .
  * Bertoni  _et al._ (2013) B. Bertoni, A. E. Nelson, and S. Reddy, Phys. Rev. D88, 123505 (2013), arXiv:1309.1721 [hep-ph] .
  * Bell  _et al._ (2023) N. F. Bell, G. Busoni, S. Robles, and M. Virgato, (2023), arXiv:2312.11892 [hep-ph] .
  * Kouvaris (2008) C. Kouvaris, Phys. Rev. D 77, 023006 (2008), arXiv:0708.2362 [astro-ph] .
  * de Lavallaz and Fairbairn (2010) A. de Lavallaz and M. Fairbairn, Phys. Rev. D81, 123521 (2010), arXiv:1004.0629 [astro-ph.GA] .
  * Acevedo  _et al._ (2020) J. F. Acevedo, J. Bramante, R. K. Leane, and N. Raj, JCAP 03, 038 (2020), arXiv:1911.06334 [hep-ph] .
  * Davoudiasl  _et al._ (2010) H. Davoudiasl, D. E. Morrissey, K. Sigurdson, and S. Tulin, Phys. Rev. Lett. 105, 211304 (2010), arXiv:1008.2399 [hep-ph] .
  * Davoudiasl  _et al._ (2011) H. Davoudiasl, D. E. Morrissey, K. Sigurdson, and S. Tulin, Phys. Rev. D 84, 096008 (2011), arXiv:1106.4320 [hep-ph] .
  * Jin and Gao (2018) M. Jin and Y. Gao, Phys. Rev. D 98, 075026 (2018), arXiv:1808.10644 [hep-ph] .
  * Herrero  _et al._ (2019) A. Herrero, M. A. Pérez-García, J. Silk, and C. Albertus, Phys. Rev. D 100, 103019 (2019), arXiv:1905.00893 [hep-ph] .
  * Bai  _et al._ (2020) Y. Bai, J. Berger, M. Korwar, and N. Orlofsky, JHEP 10, 210 (2020), arXiv:2007.03703 [hep-ph] .
  * Leane and Smirnov (2021) R. K. Leane and J. Smirnov, Phys. Rev. Lett. 126, 161101 (2021), arXiv:2010.00015 [hep-ph] .
  * Gresham  _et al._ (2023) M. I. Gresham, V. S. H. Lee, and K. M. Zurek, JCAP 02, 048 (2023), arXiv:2209.03963 [astro-ph.HE] .
  * McKeen  _et al._ (2021a) D. McKeen, M. Pospelov, and N. Raj, Phys. Rev. D 103, 115002 (2021a), arXiv:2012.09865 [hep-ph] .
  * McKeen  _et al._ (2021b) D. McKeen, M. Pospelov, and N. Raj, Phys. Rev. Lett. 127, 061805 (2021b), arXiv:2105.09951 [hep-ph] .
  * Goldman  _et al._ (2022a) I. Goldman, R. N. Mohapatra, S. Nussinov, and Y. Zhang, Eur. Phys. J. C 82, 945 (2022a), arXiv:2203.08473 [hep-ph] .
  * Goldman  _et al._ (2022b) I. Goldman, R. N. Mohapatra, S. Nussinov, and Y. Zhang, Phys. Rev. Lett. 129, 061103 (2022b), arXiv:2208.03771 [hep-ph] .
  * Berezhiani  _et al._ (2021) Z. Berezhiani, R. Biondi, M. Mannarelli, and F. Tonelli, Eur. Phys. J. C 81, 1036 (2021), arXiv:2012.15233 [astro-ph.HE] .
  * Davoudiasl (2023) H. Davoudiasl, Phys. Rev. D 108, 015023 (2023), arXiv:2304.06071 [hep-ph] .


# An Intelligent Transaction Migration Scheme for RAFT-based Private
Blockchain in Internet of Things Applications

Lu Hou , Xiaojun Xu, Kan Zheng Xianbin Wang This work was supported by the
National Natural Science Foundation of China (No. 62001052).L. Hou, X. Xu, and
K. Zheng are with the Intelligent Computing and Communications (IC2) Lab, Key
Lab of Universal Wireless Communications, Ministry of Education, Beijing
University of Posts & Telecommunications, Beijing, China, 100088.X. Wang is
with the Department of Electrical and Computer Engineering, University of
Western Ontario, London, ON N6A 5B9, Canada.Corresponding author: Kan
Zheng.Contact email<EMAIL_ADDRESS>
###### Abstract

The integration of multi-access edge computing (MEC) and RAFT consensus makes
it feasible to deploy blockchain on trustful base stations and gateways to
provide efficient and tamper-proof edge data services for Internet of Things
(IoT) applications. However, reducing the latency of storing data on
blockchain remains a challenge, especially when an anomaly-triggered data flow
in a certain area exceeds the block generation speed. This letter proposes an
intelligent transaction migration scheme for RAFT-based private blockchain in
IoT applications to migrate transactions in busy areas to idle regions
intelligently. Simulation results show that the proposed scheme can apparently
reduce the latency in high data flow circumstances.

###### Index Terms:

IoT, MEC, blockchain, RAFT, transaction migration

##  I Introduction

Blockchain is regarded as the emerging technology that provides transparent
and secure data storage for the Internet of Things (IoT) applications [1].
With the ability of multi-access edge computing (MEC), the gateways and base
stations can participate in blockchain networks and provide tamper-resistant
data services over the edge [2]. However, due to the limited capabilities of
gateways and base stations, the ever-increasing IoT data pose a tremendous
challenge in meeting the requirements of low latency [3]. When an anomaly is
detected in some area, the IoT system needs to increase the measurement
frequency of terminals to obtain high-resolution sensing data for better
decisions. It takes more time for these data to be committed into blockchain
[4].

A lot of existed works have studied how to minimize the latency of blockchain-
based IoT applications. Most of them adopt proof-of-work (PoW) consensus, and
use MEC offloading to reduce the workload [5]. However, a large number of
resources are wasted in PoW consensus to resist malicious nodes. Rovira-
Sugranes _et. al_ [6] optimized the latency by controlling the measurement
rates of sensors in a direct-acyclic-graph (DAG)-based blockchain network.
Nevertheless, DAG is not appropriate for distributed IoT applications since it
makes no guarantee of consistency. As a promising consensus algorithm for
distributed system, RAFT is suitable for blockchain network in IoT
applications because block can be finalized by the unique leader to avoid
heavy computational burden and ensure consistency. Although RAFT cannot
prevent malicious node, an authentication center can be established to manage
the identification and authorization of IoT devices [7]. A few works focus on
addressing issues in blockchain network with RAFT consensus. Yu _et. al_ [8]
analyzed the latency and reliability of RAFT consensus in industrial IoT,
while Xu _et. al_ [9] studied the performance of RAFT when malicious jamming
occurs in the wireless IoT environment. To the best of our knowledge, none of
the literature has addressed the issue of how to reduce latency for IoT
applications in RAFT-based private blockchain network.

Therefore, this letter proposes an intelligent transaction migration scheme by
which leaders can choose to migrate transactions to other clusters optimally.
The latency is defined as the sum of transactions migration latency, block
generation latency and block consensus latency. The optimization problem is
formulated as a Markov decision process (MDP). A deep deterministic policy
gradient (DDPG) based transaction migration scheme with action refinement is
proposed by which the optimal policy is generated by a neural network [10,
11]. The policy ensures that when the sensing data in one area increases, the
incoming transactions can be migrated to other areas that are not in intensive
conditions optimally. Simulations are conducted to demonstrate that the
proposed scheme can reduce a lot of time for packing transactions into the
blockchain.

##  II System Model

Considering a RAFT-based blockchain system over MEC for IoT sensing
application, as shown in Fig. 1. Assume there are $K$ clusters in total, each
of which maintains an independent ledger and can be indexed by
$k={1,2,\dots,K}$. Assume base stations are elected as leaders because they
are more reliable than gateways in this scenario. Therefore, the term Leader
$k$ represents the base station in the $k$-th cluster. Let $V_{k}$ denotes the
number of gateways in cluster $k$. The data sensed by end-devices are sent to
leaders as transactions and wait to be committed on ledgers. Let
$\widehat{\tau}$ denote the epoch when leader starts to generate new block.
Assume new blocks are generated in fixed intervals and each gateway follows
the leader’s committing instructions and replicates the new block directly.

Leader $k$ maintains a decision queue where transactions are waited to be
processed at decision epoch $\tau$. Assume end-devices sense the environment
in fixed intervals. Then, the number of arriving transactions $M_{k}$ is
constant. At one $\tau$, Leader $k$ can take actions on $N$ transactions at
most. The actions can be either to move transactions to the local packing
queue or to migrate transactions to other clusters. Transactions in the
packing queue are encapsulated in a new block at next $\widehat{\tau}$. Let
$\widehat{\tau}=F\tau,F\in\mathbb{Z}^{+}$, which means block generation is
triggered after $F$ decision epochs. Assume the time interval between two
decision epochs is fixed and is denoted as $\Delta\tau$. Let
$\mathbf{a}_{k}(\tau)=\\{a_{k,1}(\tau),a_{k,2}(\tau),\dots,a_{k,N}(\tau)\\},\tau=1,2,\dots,$
denote the decision sequence of Leader $k$ at $\tau$. Let
$a_{k,n}=1,2,\dots,K,1\leq n\leq N$, where $a_{k,n}=k^{\prime}$ means the
transaction $n$ needs to be migrated from Leader $k$ to Leader $k^{\prime}$.
Typically, $a_{k,n}=k$ represents that the transaction is not migrated.

###  II-A Transactions migration latency

The transactions migration latency can be computed as follow,

$t_{k}^{\text{MIG}}=\frac{|I_{k}(\tau)|D}{G},$ (1)

where $I_{k}(\tau)=\\{n|a_{k^{\prime},n}(\tau)=k,k^{\prime}\neq k\\}$ and
$|\cdot|$ represents the count of elements in set. $D$ is the size of a
transaction and $G$ is the transmission rate of the fiber link between two
leaders. The transactions are directly put into the packing queue after being
migrated. No more migration is allowed to avoid high latency.

###  II-B Block generation latency

At $\widehat{\tau}$, each leader generates a new block that contains all
transactions in the packing queue. A Merkle tree is established by repeated
hash operations on all transactions. For $N$ transactions,
$N+2^{\mathcal{N}}-1$ times of hash operations are required [12], where
$\mathcal{N}\in\mathbb{Z}$ and $2^{\mathcal{N}-1}<N\leq 2^{\mathcal{N}}$. Let
$\eta_{k}$ stands for the number of central processing unit (CPU) cycles per
unit time that Leader $k$ possesses. Thus, the expected computing latency of
block generation can be derived as follow,

$t_{k}^{\text{COM}}=\frac{(N+2^{\mathcal{N}}-1)\xi}{\eta_{k}},$ (2)

where $\xi$ stands for the average CPU cycles needed for one hash operation.

###  II-C Block consensus latency

When a block is generated, the leader needs to send the AppendEntries message
that contains the new block to all other followers to reach consensus. The
transmission rate of the wireless link between Leader $k$ and Follower $v$ is
defined as follow,

$r_{k,v}=\frac{B}{V_{k}}\log(1+\gamma_{k,v}),1\leq v\leq V_{k}$ (3)

where $B$ is the total bandwidth, and it is equally allocated to $V_{k}$
followers. Considering only the large-scale fading, the signal-to-noise-ratio
(SNR) of the wireless link from sender $k$ to receiver $v$ can be computed as
follow,

$\gamma_{k,v}=\frac{Pd_{k,v}^{-\beta}\Psi}{BN_{0}},$ (4)

where $P$ is the total transmission power that is equally divided for $V_{k}$
followers. $N_{0}$ stands for the average power spectral density of white
noise, and $d_{k,v}$ is the distance between Leader $k$ and Follower $v$.
$\beta$ stands for the path loss ratio, and the
$\Psi\sim\mathcal{LN}(0,\sigma^{2})$ denotes the shadow fading which follows
log-normal distribution.

To reach consensus on the new block, the leader has to receive confirmation
messages from at least $\zeta_{k}+1$ followers (including the leader itself),
where $\zeta_{k}=\lfloor V_{k}/2\rfloor$. Therefore, the consensus latency is
determined by the transmission latency caused by the leader sending
AppendEntries message to and receiving confirmation from the $\zeta_{k}$-th
follower, i.e.,

Figure 1: Illustration of RAFT-based private blockchain with transaction
migration for IoT applications.

$t_{k}^{\text{CON}}=\frac{C}{r_{k,\zeta_{k}}}+\frac{E}{r_{\zeta_{k},k}},$ (5)

where $C$ denotes the size of an AppendEntries message and $E$ represents the
size of the confirmation message.

###  II-D Objective

Our objective is to minimize the long-term latency of all leaders by an
optimal migration policy $\pi^{*}$. Giving consideration to fairness, the
long-term latency is summed by the maximal latency of all clusters. Therefore,
the objective problem is defined as follow, π ∑_τ{max_k(t_k^COM \+ t_k^CON \+
t_k^MIG)},

##  III DDPG based transaction migration scheme

The optimization problem eq. (II-D) is formulated as an MDP model which
consists of state set $\mathcal{S}$, action set $\mathcal{A}$, reward function
$R:\mathcal{S}\times\mathcal{A}\rightarrow\mathbb{R}$, and transition
probability $T:\mathcal{S}\times\mathcal{A}\times\mathcal{S}\rightarrow[0,1]$.
A reinforcement learning method is adopted to find the optimal policy
$\pi^{*}$ by a training procedure. Besides, an action refinement is applied to
choose the best discrete action from the original output of a neural network.

###  III-A MDP model formulation

####  III-A1 State set

System state $S_{\tau}$ is defined as
$S_{\tau}=\\{\mathbf{D}_{\tau},\mathbf{P}_{\tau},\Gamma_{\tau},\iota\\}$,
where $\mathbf{D}_{\tau}=\\{D_{1,\tau},D_{2,\tau},\dots,D_{K,\tau}\\}$
represents the number of transactions in decision queue of each leader, and
$\mathbf{P}_{\tau}=\\{P_{1,\tau},P_{2,\tau},\dots,P_{K,\tau}\\}$ stands for
the status of packing queue, where $P_{k,\tau}$ includes transactions that
come from local decision queue at previous epoch and that are migrated from
other clusters, i.e., $P_{k,\tau}=N-\bigr{|}\\{a_{k,n}|a_{k,n}\neq
k,n\in[1,N]\\}\bigr{|}+|I_{k}(\tau-1)|$.
$\Gamma_{\tau}=\\{\gamma_{1,1,\tau},\dots,\gamma_{k,V_{k},\tau},\dots,\gamma_{K,V_{K},\tau}\\}$
denotes the SNRs of all wireless links. An indicator $\iota=\tau\mod F$ is
used to indicate the number of decision epochs after a packing epoch. If
$\iota=F$, a new block is generated. All possible states form the state set
$\mathcal{S}$.

####  III-A2 Action set

An action $\mathbf{A}_{K\times N}(\tau)$ is given as a matrix, where each row
stands for a decision sequence by the leader, i.e.,
$\mathbf{A}(\tau)=\\{\mathbf{a}_{k}(\tau)|k\in[1,K]\\}$. All possible actions
form up an action set $\mathcal{A}$.

Without extra explanations, the symbol $\tau$ is ignored for simplicity,
i.e.,$S_{\tau}\coloneqq S$, $\mathbf{D}_{\tau}\coloneqq\mathbf{D}$,
$\mathbf{P}_{\tau}\coloneqq\mathbf{P}$, $\Gamma_{\tau}\coloneqq\Gamma$ and
$\mathbf{A}_{K\times N}(\tau)\coloneqq\mathbf{A}$.

####  III-A3 State transition function

With an action $\mathbf{A}$ at a given state $S$, the following state can be
determined via the state transition function. According to different value of
indicator $\iota$ in $S$, the transition function is defined as follows,

  * •

When $\iota\neq F$, transition function is given as follow,

$\displaystyle S_{\tau+1}$
$\displaystyle=f(S,\mathbf{A})=\\{D_{1}+M_{1}-N,\dots,$ (6) $\displaystyle
D_{K}+M_{K}-N,P_{1,\tau+1},\dots,P_{K,\tau+1},\iota+1\\},$

  * •

When $\iota=F$, leaders need to collect all transactions in the packing queue
to generate new blocks. Therefore, the transition function is given as follow,

$\displaystyle S_{\tau+1}=f(S,\mathbf{A})=$ (7)
$\displaystyle\\{D_{1}+M_{1}-N,\dots,D_{K}+M_{K}-N,0,\dots,0,\iota=0\\},$

####  III-A4 Reward function

The instant reward is defined as the maximal latency of all leaders at $\tau$.
To keep consistency with the DDPG algorithm, the value of the reward function
is set to be negative to make the larger reward stand for the lower latency.
The reward function is given as follow,

$R(\tau)=\begin{cases}0,&\iota\neq F\\\ -\max_{k\in
K}\\{t_{k}^{\text{COM}}+t_{k}^{\text{CON}}+t_{k}^{\text{MIG}}\\},&\iota=F\end{cases}$
(8)

When $\iota\neq F$, no block is generated. Therefore, the reward is set to be
zero. Only at $\widehat{\tau}$ need our model to calculate latency.

TABLE I: Information of neural network structure Network Type | Layer Type |  Number of Cells | Activation | Loss
---|---|---|---|---
| Input Layer |  $2K+1$ $+\sum_{k}V_{k}$ | - |
Hidden Layer 1 | 8 | ReLU
Actor | Hidden Layer 2 | 16 | ReLU | $Q$
| Hidden Layer 3 | 8 | ReLU |
| Output Layer | $K*N$ | $K*\text{Sigmoid}$ |
| Input Layer |  $2K+1$ $+\sum_{k}V_{k}$ $+K*N$ | - |
Hidden Layer 1 | 8 | ReLU | Mean
Critic | Hidden Layer 2 | 16 | ReLU | Squared
| Hidden Layer 3 | 8 | ReLU | Error
| Output Layer | 1 | - |

###  III-B Transaction migration scheme

Due to the large size of action space, the DDPG is adopted to make
approximations on both action-value function $Q$ and policy $\pi$ [10]. The
DDPG model includes an actor network and a critic network, each of which has a
corresponding target network. Since the DDPG generates actions in continuous
space, action refinement is adopted in our algorithm to choose the proper
discrete action from the original continuous action, according to [13].

####  III-B1 Actor and critic networks

Neural networks are used in DDPG to approximate both $Q$ and $\pi$. Let
$\hat{\pi}(S|\phi)$ denotes actor network with parameter $\phi$ and
$\hat{Q}(S,\mathbf{A}|\theta)$ denotes the critic network with parameter
$\theta$. All layers in both networks are fully connected. The structures of
target networks are equivalent to $\hat{\pi}$ and $\hat{Q}$, while the
parameters are updated gradually via exponential moving average (EMA). Let
$\phi^{\prime}$ and $\theta^{\prime}$ denote the parameters of target actor
network and target critic network. Besides, let $\kappa$ denotes the rate of
the EMA update on parameters. Then, the update function of target networks can
be given as follow,

$\phi^{\prime}=\kappa\phi^{\prime}+(1-\kappa)\phi,\theta^{\prime}=\kappa\theta^{\prime}+(1-\kappa)\theta.$
(9)

For simplicity, define short notations for actor and critic networks as
$\pi\coloneqq\hat{\pi}(S|\phi)$, $Q\coloneqq\hat{Q}(S,\mathbf{A}|\theta)$, and
for target networks as
$\pi^{\prime}\coloneqq\hat{\pi}(S^{\prime}|\phi^{\prime})$,
$Q^{\prime}\coloneqq\hat{Q}(S^{\prime},\mathbf{A}^{\prime}|\theta^{\prime})$.

The loss $J(\phi)$ of the actor network is defined as the estimated value of
$Q$ that is computed by critic network, while the loss of the critic network
is defined by the mean squared error of $Q$ and the target value $y$ which is
given as follow,

$y=R+\lambda Q^{\prime},$ (10)

where $\lambda$ is the discount factor. Then, the loss of critic network can
be computed as follow,

$\mathcal{L}(\theta)=\mathbb{E}\\{(Q-y)^{2}\\}$ (11)

The main objective of DDPG is to find the optimal set of parameters
$\theta^{*}$ to maximize $Q$ which is calculated by the long-term sum of
rewards, i.e., eq. (II-D).

Input: Initialize parameters $\phi,\theta\sim\mathcal{G}$, where $\mathcal{G}$
is a Gaussian distribution. Initialize a starting state
$S=S_{0}=\\{0,0,\dots,0\\}$. Set experience deque $\mathcal{E}=\varnothing$,
an index $\chi$ that indicate the epoch of training start and the total
iteration times $\tau_{\max}$.

Output: $\pi^{*}$

1 while  _$\tau\neq\tau_{\max}$_ do

2 Sample a continuous action with random exploration, i.e.,
$\mathbf{A}=\pi(S|\phi)+g$, where $g\sim\mathcal{G}$;

3 Refine actions according to eq. (13);

4 Each leader execute action $\mathbf{A}$ and observe the instant reward $R$ ;

5 Environment transit to state $S_{\tau+1}=f(S,\mathbf{A})$ ;

6 Push the observation sequence $(S,\mathbf{A},R,S_{\tau+1})$ into the
experience deque $\mathcal{E}$ ;

7 Set $\tau\leftarrow\tau+1$ and $S\leftarrow S_{\tau+1}$;

8 if  _$\tau >\chi$_ then

9 Randomly sample a mini-batch $e\sim\mathcal{E}$ ;

10 Obtain target action batch by the target actor network via
$\mathbf{A}^{\prime}=\pi^{\prime}(S)$, where $S\sim e$ ;

11 Acquire the target action value batch by the target critic networks via
$Q^{\prime}(S,\mathbf{A}^{\prime})$ ;

12 Compute the value $y$ according to eq. (10);

13 Calculate the gradient of the critic network as $\nabla\mathcal{L}$;

14 Update parameters $\theta$ of the critic network towards the direction of
gradient descent, i.e., $\theta\leftarrow\theta-\alpha_{1}\nabla\mathcal{L}$,
where $\alpha_{1}$ is the learning rate of critic network;

15 Get raw action batch by the actor network via $\mathbf{A}=\pi(S)$, where
$S\sim e$;

16 Calculate the gradient of the actor network as follow,  $\nabla
J=\frac{\partial Q}{\partial\phi}=\frac{\partial
Q}{\partial\mathbf{A}}\frac{\partial\pi}{\partial\phi}$ (12) Update parameters
$\phi$ towards the direction of gradient ascent, i.e.,
$\phi\leftarrow\phi+\alpha_{2}\nabla J$, where $\alpha_{2}$ is the learning
rate of the actor network;

17 Update the parameters of the target networks according to eq. (9);

18

19 end if

20

21 end while

Algorithm 1 DDPG-based transaction migration training algorithm with action
refinement

The detailed structure of both the actor network and the critic network is
given in Tab. I.

TABLE II: Parameter configuration Parameter | Value | Parameter | Value
---|---|---|---
$K$ | $4$ | $V$ | $4$
$B$ | $20\text{\,}\mathrm{MHz}$ | $N_{0}$ |  $150\text{\,}\mathrm{dBm}$$\mathrm{/}\mathrm{Hz}$
$P$ | $24\text{\,}\mathrm{dBm}$ | $\beta$ | $3.4$
$\xi$ | $300$ | $\eta$ | $1\text{\,}\mathrm{GHz}$

####  III-B2 Action refinement

The output of the actor network is the continuous features of actions, which
cannot be directly executed by leaders. By only using the most approaching
discrete feature of action may not get the maximal $Q$ value [13]. Therefore,
an action refinement, which can choose the discrete action to reach the
maximal action-value function, is given as follows,

$\mathbf{A}^{*}=\arg\max_{\hat{\mathbf{A}}\in\mathbb{A}}\hat{Q}(S,\hat{\mathbf{A}}),$
(13)

where
$\mathbb{A}=\\{\tilde{\mathbf{A}}\bigr{|}\lVert\tilde{\mathbf{A}}-\pi\rVert_{2}\leq\delta\\}$.
By using the proper value of $\delta$, the balance between precision and
efficiency can be reached.

####  III-B3 Training procedure

The training process of the transaction migration algorithm is given in
Algorithm 1. After enough training, the actor network can generate optimal
transactions migration policies for each leader of the cluster.

##  IV Performance evaluation and analysis

###  IV-A Simulation setup

A simulation is conducted to simulate the situation where the sensing
frequency rises in some areas when an anomaly occurs, and descends back to
normal after the anomaly disappears, in an IoT application. There are $4$
areas in total and the anomaly occurs in Area 2. The number of incoming
transactions in other areas stay to be $6$, while $M_{2}$ varies from $4$ to
$9$ and decays back to $4$. Key parameters of our simulations are listed in
Tab. II. Each leader needs to receive $\zeta_{k}=2$ responses from gateways to
reach consensus by RAFT. Besides, the learning rates of both the actor network
and the critic network are both set to be $10^{-5}$.

###  IV-B Numerical results and analysis

Figure 2: Illustration of convergence of actor and critic networks under
different $M_{2}$.

Each simulation runs for $8\times 10^{6}$ epochs. The actor loss $J(\phi)$
rises as the iteration time increases, while the critic loss
$\mathcal{L}(\theta)$ is decreasing, as shown in Fig. 2. This is because actor
loss $J(\phi)$ stands for the average action value of each batch. Thus, the
higher $J(\phi)$ becomes, the higher rewards can be got. On the other hand,
the critic loss $\mathcal{L}(\theta)$ represents the difference between the
estimated action-value function and the target action-value function.
Therefore, $\mathcal{L}(\theta)$ needs to decrease during the training
process.

Figure 3: Illustration of latencies of all schemes under different $M_{2}$.

The proposed scheme is compared to four baselines, i.e., random scheme and
three static schemes include non-migration, 30% migration and 50% migration
schemes. As the name implies, each baseline scheme defines how many
transactions are migrated. The random scheme provides the average performance
for all the possible schemes since each action of this scheme can be selected
with equal probability after training enough times. As shown in Fig. 3, the
three static baselines perform bad similarly. It proves that static schemes
cannot help improve the performance, even if they can migrate transactions
from busy areas. Latency that is reached by the proposed scheme stays the
lowest, because the proposed scheme can act intelligently with regard to the
changing states. The proposed scheme balances the transactions among clusters
optimally to reduce the latency by at most 58% and 29%, compared to the static
schemes and random scheme.

As shown in Fig. 4, migration ratio of Leader $2$ is the lowest at initial
stage, because the workload is smaller than other three leaders. Therefore,
the proposed scheme tends to migrate transactions from other leaders to reduce
the maximal latency. When an anomaly occurs, the number of transactions rises
in Area 2, causing Leader $2$ to migrate transactions to other leaders as many
as possible. Meanwhile, migration ratios of other leaders fall down to a low
level. This is because our objective is to minimize the latency of the whole
system rather than a particular cluster. Therefore, as the number of arriving
transactions increases, the proposed algorithm tends to migrate transactions
from the leader with high burden to the leaders that are idle. After the
anomaly disappears, the migration ratios start to get back to normal.

Figure 4: Illustration of migration ratios of all leaders under different
$M_{2}$.

##  V Conclusion

In this letter, a reinforcement-learning-based scheme that can migrate
transactions intelligently between clusters has been proposed for RAFT-based
private blockchain in IoT application. The scheme can minimize the latency of
storing IoT data on chain by balancing the workload among all clusters. DDPG
algorithm with an action refinement is proposed to obtain optimal transaction
migration policy on consideration of dynamic status of the environment.
Simulation results show that the proposed scheme can reduce at most $58\%$
latency when the system undergoes an anomaly in some sensing area, compared to
baselines.

## References

  * [1] C. H. Liu, Q. Lin, and S. Wen, “Blockchain-enabled data collection and sharing for industrial IoT with deep reinforcement learning,” _IEEE Transactions on Industrial Informatics_ , vol. 15, no. 6, pp. 3516–3526, 2019\.
  * [2] K. Gai, J. Guo, L. Zhu, and S. Yu, “Blockchain meets cloud computing: a survey,” _IEEE Communications Surveys and Tutorials_ , vol. 22, no. 3, pp. 2009–2030, 2020.
  * [3] X. Wang, C. Wang, X. Li, V. C. M. Leung, and T. Taleb, “Federated deep reinforcement learning for internet of things with decentralized cooperative edge caching,” _IEEE Internet of Things Journal_ , vol. 7, no. 10, pp. 9441–9455, 2020.
  * [4] P. Yuan, K. Zheng, X. Xiong, K. Zhang, and L. Lei, “Performance modeling and analysis of a hyperledger-based system using gspn,” _Computer Communications_ , vol. 153, pp. 117–124, 2020.
  * [5] M. Liu, F. R. Yu, Y. Teng, V. C. M. Leung, and M. Song, “Computation offloading and content caching in wireless blockchain networks with mobile edge computing,” _IEEE Transactions on Vehicular Technology_ , vol. 67, no. 11, pp. 11 008–11 021, 2018.
  * [6] A. Rovira-Sugranes and A. Razi, “Optimizing the age of information for blockchain technology with applications to IoT sensors,” _IEEE Communications Letters_ , vol. 24, no. 1, pp. 183–187, 2020.
  * [7] C. Benzaïd and T. Taleb, “AI for beyond 5G networks: A cyber-security defense or offense enabler?” _IEEE Network_ , vol. 34, no. 6, pp. 140–147, 2020.
  * [8] D. Yu, W. Li, H. Xu, and L. Zhang, “Low reliable and low latency communications for mission critical distributed industrial Internet of things,” _IEEE Communications Letters_ , vol. 25, no. 1, pp. 313–317, 2021\.
  * [9] H. Xu, L. Zhang, Y. Liu, and B. Cao, “RAFT based wireless blockchain networks in the presence of malicious jamming,” _IEEE Wireless Communications Letters_ , vol. 9, no. 6, pp. 817–821, 2020.
  * [10] T. P. Lillicrap, J. J. Hunt, A. Pritzel, N. Heess, T. Erez, Y. Tassa, D. Silver, and D. Wierstra, “Continuous control with deep reinforcement learning,” _arXiv e-prints:1509.02971_ , pp. 1–10, Sep. 2015\.
  * [11] L. Lei, Y. Tan, K. Zheng, S. Liu, K. Zhang, and X. Shen, “Deep reinforcement learning for autonomous internet of things: Model, applications and challenges,” _IEEE Communications Surveys & Tutorials_, vol. 22, no. 3, pp. 1722–1760, 2020.
  * [12] L. Hou, K. Zheng, Z. Liu, X. Xu, and T. Wu, “Design and prototype implementation of a blockchain-enabled lora system with edge computing,” _IEEE Internet of Things Journal_ , vol. 8, no. 4, pp. 2419–2430, 2021.
  * [13] G. Dulac-Arnold, R. Evans, H. van Hasselt, P. Sunehag, T. Lillicrap, J. Hunt, T. Mann, T. Weber, T. Degris, and B. Coppin, “Deep reinforcement learning in large discrete action spaces,” _arXiv e-prints:1512.07679_ , pp. 1–11, Dec. 2015.