# Evolution of Urca Pairs in the Crusts of Highly Magnetized Neutron Stars

Michael A. Famiano<EMAIL_ADDRESS>Western Michigan University,
Kalamazoo, MI 49008-5252 USA  National Astronomical Observatory of Japan,
2-21-1 Osawa, Mitaka, Tokyo 181-8588 Japan  Joint Institute for Nuclear
Astrophysics - Center for the Evolution of the Elements, USA  Grant Mathews
<EMAIL_ADDRESS>Joint Institute for Nuclear Astrophysics - Center for the
Evolution of the Elements, USA  Center for Astrophysics, Department of Physics
and Astronomy, University of Notre Dame, Notre Dame, IN 46556, USA  A. Baha
Balantekin<EMAIL_ADDRESS>National Astronomical Observatory of Japan,
2-21-1 Osawa, Mitaka, Tokyo 181-8588 Japan  Department of Physics, University
of Wisconsin-Madison, Madison, Wisconsin 53706 USA  Toshitaka Kajino
<EMAIL_ADDRESS>School of Physics, Beihang University, 37 Xueyuan Road,
Haidian-qu, Beijing 100083, China  National Astronomical Observatory of Japan,
2-21-1 Osawa, Mitaka, Tokyo 181-8588 Japan  Graduate School of Science, The
University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033 Japan  Motohiko
Kusakabe<EMAIL_ADDRESS>School of Physics, Beihang University, 37
Xueyuan Road, Haidian-qu, Beijing 100083, China  Kanji Mori
<EMAIL_ADDRESS>Research Institute of Stellar Explosive Phenomena,
Fukuoka University,
8-19-1 Nanakuma, Jonan-ku, Fukuoka-shi, Fukuoka 814-0180, Japan

###### Abstract

We report on the effects of strong magnetic fields on neutrino emission in the
modified Urca process. We show that the effect of Landau levels on the various
Urca pairs affects the neutrino emission spectrum and leads to an angular
asymmetry in the neutrino emission. For low magnetic fields the Landau levels
have almost no effect on the cooling. However, as the field strength
increases, the electron chemical potential increases resulting in a lower
density at which Urca pairs can exist. For intermediate field strength there
is an interesting interference between the Landau level distribution and the
Fermi distribution. For high enough field strength, the entire electron energy
spectrum is eventually confined to single Landau level producing dramatic
spikes in the emission spectrum.

PPISN – PISN – black holes – massive stars – nuclear physics

##  1 Introduction

It is by now widely accepted that soft gamma-ray repeaters (SGRs) and
anomalous X-ray pulsars (AXPs) correspond to a class of neutron stars known as
magnetars. These objects are warm, isolated, and slowly rotating neutron stars
of age $\sim 10^{5}$ yr with unusually strong surface magnetic fields. Indeed,
both pulsars and magnetars have strong magnetic fields at their surface that
can be as large as $10^{12}$ to $10^{16}$ G (Kouveliotou et al., 1998; Turolla
et al., 2015). Moreover, the interior field can be as large as the
equipartition limit of order $10^{18}$ G (Lai et al., 1991; Chanmugam, 1992).

Among possible explanations for the cooling of magnetars, decay processes
leading to neutrino or anti-neutrino emission such as the direct Urca (DU)
process ($n\rightarrow p+e^{-}+{\bar{\nu}}_{e}$, $p+e^{-}\rightarrow
n+\nu_{e}$), the modified Urca (MU) process ($n+N\rightarrow
p+e^{-}+N^{\prime}+{\bar{\nu}}_{e}$, $p+e^{-}+N\rightarrow
n+N^{\prime}+{\nu}_{e}$ ) (Haensel & Gnedin, 1994; Yakovlev & Levenfish, 1995;
Yakovlev et al., 2001), or the neutrino-pair emission process
($N_{1}+N_{2}\rightarrow N_{1}^{\prime}+N_{2}^{\prime}+\nu+{\bar{\nu}}$) are
possible.

Early in the cooling process the DU process is a viable candidate to explain
the rapid cooling of NSs (Boguta, 1981; Lattimer et al., 1991; Maruyama et
al., 2022). However, in this paper we are more concerned with the MU process.
The DU process takes place at high temperature and density and for certain
ratios of constituents. However, under less extreme conditions when the DU
process is diminished, the MU process can continue to occur.

In particular, the Urca cycles of electron capture and $\beta$ decay on pair
nuclei at a depth of about 150 m in neutron stars were shown to occur in
Schatz et al. (2014) although this mechanism had been previously discussed in
the context of white dwarfs (Tsuruta & Cameron, 1970), Type Ia supernovae
(Paczyński, 1972; Woosley & Weaver, 1986) and electron-degenerate supernovae
(Jones et al., 2013). The Urca cycle operates to cool the outer neutron star
crust by emitting neutrinos while also thermally decoupling the surface layers
from the deeper crust. This cooling eliminates the possibility that interior
heating produces the unexpectedly short recurrence times of energetic
thermonuclear bursts on neutron stars. The Urca cycles also indicate that the
ignition scenario of superbursts by 12C+12C fusion reaction, whose reaction
rate is severely limited in recent theoretical studies (Mori et al., 2019),
would require another heat source because of higher neutrino emissivity for
cooling.

In previous studies, however, the effects of possible strong magnetic fields
on the modified Urca process have not been considered. Here we show that the
appearance of Landau levels for electrons experiencing strong magnetic fields
significantly alters the operation of the various Urca pairs proposed by
Schatz et al. (2014) and affects the neutrino emission spectrum leading to an
angular asymmetry in the neutrino emissivity. For low magnetic fields the
Landau levels have almost no effect on the cooling. However, as the field
strength increases, the electron chemical potential increases resulting in a
lower density at which Urca pairs can exist. For intermediate field strength
there is an interesting interference between the Landau level distribution and
the Fermi distribution. For high enough field strength, eventually the entire
electron energy spectrum is confined to single Landau level producing dramatic
spikes in the emission spectrum.

This paper is organized as follows: The ingredients of the model for the MU
are summarized in Section 2. The results are presented in Section 3. Our
discussion and conclusions are in Section 4.

##  2 The Model

###  2.1 Weak Interaction Rates in External Fields

In a homogeneous plasma at a given temperature and density, the electron
chemical potential will change in the presence of a strong magnetic field.
This will ultimately change the effectiveness of a particular Urca pair for a
given environment. If a dipole field is assumed, then the electron chemical
potential, which depends on the magnetic field, will in turn depend on the
angular position with respect to the magnetic pole for a constant density and
temperature within the crust. The electron number density with the electron
transverse momentum components constrained to Landau levels is (Famiano et
al., 2020; Grasso & Rubinstein, 2001; Kawasaki & Kusakabe, 2012):

$\displaystyle n_{e}=n_{-}-n_{+}$
$\displaystyle=\frac{1}{4\pi^{2}}\int_{0}^{\infty}d^{3}p{\left(\left[{\exp{\left(\frac{E-\mu}{T}\right)}+1}\right]^{-1}-\left[{\exp{\left(\frac{E+\mu}{T}\right)}+1}\right]^{-1}\right)}$
$\displaystyle=\int_{0}^{\infty}dp_{x}dp_{y}dp_{z}{\left(\left[{\exp{\left(\frac{E-\mu}{T}\right)}+1}\right]^{-1}-\left[{\exp{\left(\frac{E+\mu}{T}\right)}+1}\right]^{-1}\right)}$
$\displaystyle=\frac{eB}{2\pi^{2}}\sum\limits_{n=0}^{\infty}g_{n}\int_{0}^{\infty}dp_{z}\left(\left[{\exp{\left(\frac{\sqrt{p_{z}^{2}+m_{e}^{2}+2neB}-\mu}{T}\right)}+1}\right]^{-1}\right.$
$\displaystyle-\left.\left[{\exp{\left(\frac{\sqrt{p_{z}^{2}+m_{e}^{2}+2neB}+\mu}{T}\right)}+1}\right]^{-1}\right)$

where $g_{n}$ is the degeneracy of individual Landau levels $n$:

$g_{n}=\left\\{\begin{array}[]{cl}1,&n=0\\\ 2,&n>0\end{array}\right..$ (2)

Natural units are used in this manuscript ($k=\hbar=c=1)$.

Weak interaction rates are computed as in Famiano et al. (2020) and Arcones et
al. (2010). In the presence of an external field, the electron energy density
is reconfigured by the presence of the field such that the components of the
electron momentum perpendicular to the external magnetic field,
$p_{\perp}^{2}=p_{x}^{2}+p_{y}^{2}$, are placed into individual levels,
$p_{\perp,n}^{2}=neB$. This quantization results in a shift in the electron
chemical potential, $\mu_{e}$, and ultimately the weak interaction rates (Luo
et al., 2020; Grasso & Rubinstein, 2001; Fassio-Canuto, 1969). At low fields,
$B\lesssim B_{c}$, where $B_{c}\sim{m_{e}^{2}}/{e}=4.4\times 10^{13}$ G, the
Fermi distribution of the electrons is very similar to the distribution in
which B=0. In this region, the spacing between individual Landau levels is
small, and the phase space distributions in Equation (2.1) between the
magnetized plasma and the non-magnetized plasma are similar (Luo et al., 2020;
Grasso & Rubinstein, 2001; Fassio-Canuto, 1969):

$dn\propto\frac{d^{3}p}{(2\pi)^{3}}=\sum\limits_{n=0}^{\infty}\left(2-\delta_{n0}\right)\frac{eB}{2\pi^{2}}dp_{z}.$
(3)

From this, the Fermi-Dirac distribution for the $n^{th}$ Landau level is
rewritten:

$f_{FD}(E,\mu_{e})=\frac{1}{\exp{\left[\frac{\sqrt{E^{2}+2neB}-\mu_{e}}{T}\right]}+1}.$
(4)

Following the prescription of Arcones et al. (2010) and Famiano et al. (2020),
the weak interaction rates are approximated as:

$\displaystyle\Gamma_{\beta^{-}}$ $\displaystyle=$
$\displaystyle\kappa\frac{eB}{2}\sum\limits_{n=0}^{N_{max}}\left(2-\delta_{n0}\right)\int\limits_{\omega_{\beta}}^{Q}\frac{E(Q-E)^{2}}{\sqrt{E^{2}-m_{e}^{2}-2neB}}\left(1-f_{FD}(E,\mu_{e})\right)\left(1-f_{FD}(Q-E,-\mu_{\nu})\right)dE,$
(5) $\displaystyle\Gamma_{\beta^{+}}$ $\displaystyle=$
$\displaystyle\kappa\frac{eB}{2}\sum\limits_{n=0}^{N_{max}}\left(2-\delta_{n0}\right)\int\limits_{\omega_{\beta}}^{-Q}\frac{E(-Q-E)^{2}}{\sqrt{E^{2}-m_{e}^{2}-2neB}}\left(1-f_{FD}(E,-\mu_{e})\right)\left(1-f_{FD}(-Q-E,-\mu_{\nu})\right)dE,$
(6) $\displaystyle\Gamma_{EC}$ $\displaystyle=$
$\displaystyle\kappa\frac{eB}{2}\sum\limits_{n=0}^{N_{max}}\left(2-\delta_{n0}\right)\int\limits_{\omega_{EC}}^{\infty}\frac{E(E-Q)^{2}}{\sqrt{E^{2}-m_{e}^{2}-2neB}}f_{FD}(E,\mu_{e})\left(1-f_{FD}(E-Q,\mu_{\nu})\right)dE,$
(7) $\displaystyle\Gamma_{PC}$ $\displaystyle=$
$\displaystyle\kappa\frac{eB}{2}\sum\limits_{n=0}^{N_{max}}\left(2-\delta_{n0}\right)\int\limits_{\omega_{PC}}^{\infty}\frac{E(E+Q)^{2}}{\sqrt{E^{2}-m_{e}^{2}-2neB}}f_{FD}(E,-\mu_{e})\left(1-f_{FD}(E+Q,-\mu_{\nu})\right)dE,$
(8)

in which the following are defined:

$\displaystyle\omega_{EC/PC}$ $\displaystyle\equiv$
$\displaystyle\mbox{max}\left[\pm Q,m_{e}\right],$ (9)
$\displaystyle\omega_{\beta}$ $\displaystyle\equiv$
$\displaystyle\sqrt{m_{e}^{2}+2neB},$ $\displaystyle N_{max}$
$\displaystyle\leq$ $\displaystyle\frac{Q^{2}-m_{e}^{2}}{2eB},$
$\displaystyle\kappa$ $\displaystyle\equiv$ $\displaystyle\frac{B\ln
2}{Km_{e}^{5}},$ $\displaystyle B$ $\displaystyle\equiv$ $\displaystyle
1+3g_{A}^{2}=\left\\{\begin{array}[]{ll}5.76,&\text{nucleons},\\\
4.6,&\text{nuclei},\end{array}\right.$ (12) $\displaystyle K$
$\displaystyle\equiv$ $\displaystyle\frac{2\pi^{3}\hbar^{7}\ln
2}{G_{V}^{2}m_{e}^{5}}=6144\mbox{ s}$

and $Q$ is the nuclear mass difference between the parent and daughter
nucleus.

In this evaluation, we recognize that the above rates are semi-classical
approximations in which nuclear structure and sums over excited states are
ignored. We take this approach to examine the overall gross effects of the
external field. The rates evaluated above can be adapted to individual sums
over individual transitions from parent to daughter states if desired. In this
work, we simplify the above to examine ratios of transition rates in an
external field to those without a field.

In a charge-neutral plasma at temperature $T$ and electron charge density
$\rho Y_{e}/N_{A}$, the electron chemical potential is calculated from Eq.
(2.1). The chemical potential is then used in Eqs. (5) – (8) to evaluate the
modified rates.

###  2.2 Urca Pairs

For an Urca pair to exist (Schatz et al., 2014), a pair of nuclei, ${}^{A}Z$
and ${}^{A}(Z+1)$ must be linked by weak interactions

${}^{A}Z\rightarrow^{A}(Z+1)+e^{-}+\bar{\nu}_{e}$ (13)
${}^{A}(Z+1)+e^{-}\rightarrow^{A}Z+\nu_{e}~{}~{}.$

The electron energy distribution in the plasma must also be non-degenerate at
energies less than the $\beta^{-}$ decay Q value and non-zero at energies
greater than the electron-capture Q value. In other words, the electron phase
space must be simultaneously available for both $\beta^{-}$ decays and
electron capture. For a plasma in which this condition is met $\mu_{e}\approx
Q_{\beta^{-}}=-Q_{EC}$. The finite plasma temperature results in the
availability of electron states at energies below Q and electrons occupying
states at energies above Q. Under these conditions, the Urca pair will undergo
captures and decays, resulting in enhanced cooling via neutrino emission.
Captures may occur to low-lying states of the daughter nucleus (Schatz et al.,
2014).

In the absence of a magnetic field, the electron chemical potential is
constrained by the temperature, $T$, and the charge density $\rho Y_{e}$ of
the plasma. That is, for a specific temperature, there is only a one-to-one
relationship between electron chemical potential and charge density. In fact,
there exists a range of charge densities that define an Urca pair because of
the finite plasma temperature. The Urca pair is constrained to the region
where $Q-kT\lesssim\mu\lesssim Q+kT$. Thus, for a specific temperature, $T$,
and electron chemical potential, $\mu$, a range of densities for which an Urca
pair exists can be defined, $\rho Y_{e}(T,\mu-kT)\lesssim\rho
Y_{e}\lesssim\rho Y_{e}(T,\mu+kT)$. It has been found (Schatz et al., 2014;
Deibel et al., 2016) that this density range results in a thin layer in which
a particular pair can exist.

However, this constraining relationship is broken by the introduction of an
external magnetic field, $\mu=\mu(T,\rho Y_{e},B)$. This may have multiple
effects because:

  * •

The electron chemical potential depends on the external field, the density at
which an Urca pair forms for a particular temperature changes, resulting in a
changed emissivity.

  * •

Weak rates may change at various field strengths, changing the neutrino
emissivity.

  * •

For a specific T, $\rho Y_{e}$, and B, new Urca pairs may result from changes
in the chemical potential. In addition, existing Urca pairs may by prohibited
as the field changes.

  * •

EC rates on exited state nuclei will change with the magnetic field strength.
Concurrent changes in the $\beta$-decay rates of the daughter nucleus may
result in similar rates between exited states and the existance of Urca pairs
which involve formerly inaccessible excited states in nuclei.

  * •

For a non-uniform magnetic field on the surface of a highly-magnetized neutron
star, the location of Urca pairs in the ocean and crust of a neutron star may
depend on the location on the surface of the star.

For a specific Urca pair, the emissivity and luminosity can be calculated
following the method of Deibel et al. (2016): Our model for the neutron star
and the emissivity in the modified Urca process are directly dependent on the
weak interaction rates (Tsuruta & Cameron, 1970; Deibel et al., 2016):

$\epsilon_{\pm}\approx m_{e}\Gamma_{\pm}.$ (14)

In this work, we define the quantity $\epsilon_{22}$:

$\epsilon_{22}\equiv\frac{\left(\epsilon_{-}+\epsilon_{+}\right)}{10^{22}}$
(15)

The geometric thickness of an Urca layer can be shown to be quite thin. In
this case, the luminosity (in units of 1032 erg s-1 as a function of polar
angle, assuming a relativistic correction is:

$L_{32}(\theta)\equiv\varepsilon\times 2\pi
R^{2}\sin(\theta)\Delta\theta\Delta R/(10^{32}\mbox{erg s}^{-1})$ (16)

The radial thickness of each zone is calculated using the formulation of
Schatz et al. (2014) and is found to be $\sim$ 1 m.

##  3 Results

The $\beta^{-}$ decay rate ratio $\lambda(B)/\lambda(B=0)$ is shown as a
function of the magnetic field in Figure 1(a) for $T_{9}$=0.51, $Y_{e}$=0.5,
and $\rho$= 4$\times 10^{10}$ g cm-3. The oscillations in the decay rate
occurs with a change in the field as fewer Landau levels contribute to the
electron energy spectrum and as Landau levels shift across the spectrum.

The decay rate ratio as a function of density is shown in Figure 1(b) for
$T_{9}$=0.51 and B=1014 G. While the oscillatory behavior of Figure 1(a) can
be explained by Landau levels shifting into or out of the electron energy
spectrum as the field changes, in Figure 1(b), the oscillatory behavior can be
explained by the shift in electron chemical potential as the density
increases. The change in chemical potential results in the electron energy
spectrum shifting with respect to the existing Landau levels. The same
behavior is shown for a field of 1015 G in Figure 1(c).

Figure 1: (a) Ratio of $\beta^{-}$ decay rates with and without a magnetic
field as a function of magnetic field strength

for a temperature and $\rho Y_{e}$ resulting in an electron chemical potential
approximately equal to a Q value of 13.3 MeV. (b) Rate ratio as a function of
density for $Y_{e}$=0.5, $T_{9}$=0.51, and B=1014 G. (c) Rate ratio as a
function of density for $Y_{e}$=0.5, $T_{9}$=0.51, and B=1015 G.

Figure 2 shows the ratio of electron capture rates $\lambda(B)/\lambda(B=0)$
as a function of polar angle about the axis of the neutron star for two
different values of field (as indicated at the pole). The field is assumed to
be a dipole field. Electron capture rates are computed at the stellar surface.
For each case, $Q_{EC}$ = 5 MeV. For a weaker field, more Landau levels are
present in the electron phase space, and as the field changes across the
surface of the star a larger number of Landau levels are shifted out of the
electron energy spectrum resulting in the larger number of fluctuations in the
EC rate. However, for a larger field, fewer Landau levels are included in the
electron energy spectrum, and as the field shifts across the surface of the
star, fewer Landau levels shift into or out of the electron energy spectrum.
Thus, there are fewer oscillations in the EC rate across the surface of the
star.

Figure 2: Ratios of $\beta^{-}$ decay rates $\lambda(B\neq 0)/\lambda(B=0)$ as
a function of polar angle, $\theta$ as measured from a magnetic pole of the
star for $Q_{\beta^{-}}=$5 MeV. The top row shows the field dependence at a
constant $\rho Y_{e}$ = 109 g cm-3 for (a) B=1014 G, (b) B=1015 G, and (c)
B=1016 G. The bottom row shows the sensitivity to density at a constant B=1015
G for $\rho Y_{e}/10^{9}$ of (d) 0.3 g cm-3, (e) 0.6 g cm-3, and (f) 2 g cm-3.

In order for efficient Urca pair cooling to occur, EC and $\beta^{-}$ decay
rates must be similar. We thus examine these rates as a function of magnetic
field for several Urca pairs. In Figure 3, rates are shown as a function of
magnetic field for two presumed Urca pairs Schatz et al. (2014). In each
figure, the black line corresponds to $\beta^{-}$ decay rates while the red
line corresponds to EC rates. Optimal Urca pair cooling occurs if the rates
are similar. However, as the field increases, there can be oscillations in the
rates, reducing the efficiency of the associated pair. For example, consider
the 29Mg$\leftrightarrow^{29}$Na pair in Figure 3 at the temperature, density,
and electron fraction indicated. At a field of B$\approx 10^{15.8}$, the EC
rate exceeds the $\beta^{-}$ rate by about five orders of magnitude. However,
at a field strength slightly lower than this, the rates intersect on the
graph. At very high fields, the rates diverge significantly, as only one
Landau level contributes to the electron energy spectrum. A similar comparison
can be made for the 33Al$\leftrightarrow^{33}$Mg pair in the same figure.

Figure 3: Examples of EC and $\beta^{-}$ decay rates as a function of magnetic
field for 29Mg$\leftrightarrow^{29}$Na (left) and 33Al$\leftrightarrow^{33}$Mg
(right). The red line corresponds to the EC rate, while black lines correspond
to the $\beta^{-}$ rate.

The introduction of magnetic fields can also create Urca pairs where they
would not have otherwise existed. This is shown in Figure 4, for two pairs at
densities and temperatures not conducive to the formation of Urca pairs for
the nuclei shown at low fields. However, as the field increases, the rates can
match and possibly form Urca pairs. The additional degree of freedom from the
magnetic field extends the range of possible conditions at which Urca pairs
can form within the NS crust.

Figure 4: Evolution of Urca pair formation for 29Mg$\leftrightarrow^{29}$Na
(left) and 32Si$\leftrightarrow^{32}$Al (right) at the indicated temperatures
and densities.

Finally, in addition to beta decay, Urca pairs can form for EC daughter
excited states. For excited states in the EC daughter nucleus, the EC rate
drops. The daughter nucleus de-excites almost immediately and decays via
$\beta^{-}$ decay back to the original nucleus. However, because the
$\beta^{-}$ rate exceeds the EC rate for excited states, the cooling
efficiency is lower. The rate oscillations, however, may open Urca pair
transitions to excited states in the EC daughter nucleus. This is shown in
Figure 5. EC and $\beta^{-}$ rates to the ground state of 31Mg are shown as a
function of magnetic field, as well as to the first two excited states of
31Mg. While Urca pairs including excited states may not be possible at low
field strength, higher fields may allow for Urca pairs between excited states.
This is particularly interesting because EC to excited states may open up
transition rates that are more favored.

Figure 5: EC and $\beta^{-}$ rates for the 31Al$\leftrightarrow^{31}$Mg Urca
pair for transitions to the ground state in 31Mg (top), the first excited
state (bottom left), and the second excited state (bottom right). The red
bands in the bottom left panel indicate fields where Urca pairing may be more
efficient.

As an example, Figure 6 shows the electron chemical potential as a function of
polar angle for a layer of constant density and temperature within the crust.
For all panels in this figure, a typical temperature of $T_{9}=0.51$ is
adopted. For the top row of panels, $\rho Y_{e}=4.70\times 10^{10}$ g cm-3,
and for the bottom row $\rho Y_{e}=3.73\times 10^{10}$ g cm-3.

Figure 6:  Prevalence of Urca pairs for two different values of $\rho Y_{e}$
at various magnetic fields. Top row: $\rho Y_{e}=1.7\times 10^{10}$ g cm-3,
Bottom row: $\rho Y_{e}=1.3\times 10^{10}$ g cm-3. Values are shown for
magnetic fields of 1015 G (left), 1015.5 G (center), and 1016 G. Shaded
regions are regions where the indicated Urca pair dominates. For the left
column, the dominant Urca pair is 29Mg$\leftrightarrow^{29}$Na and
103Sr$\leftrightarrow^{103}$Rb (top) and 55Ti$\leftrightarrow^{55}$Ca,55Ca
(bottom).

The shaded regions in these figures indicate those for which the chemical
potential is within $kT$ of the EC Q-values of the indicated reactions,
$Q_{EC}-kT\leq\mu_{e}\leq Q_{EC}+kT$. With this evaluation, an Urca pair is
not consistent across the entire surface of the NS at a given temperature and
density. At high enough magnetic fields, an Urca pair may dominate a
particular angular band. This may also have the effect of making additional
Urca pairs possible in various angular regions.

Figure 7:  Left: Electron chemical potential as a function of $\rho Y_{e}$ and
polar angle for a neutron star with a dipole field of 1015.75 G (top), 1016 G
(middle), and 1016.5 G bottom. Units of $\rho Y_{e}$ are 1010 g cm-3. Right:
The electron density $\rho Y_{e}$ and polar angle for the same magnetic fields
at which various crustal Urca pairs are present. These correspond to the lines
in the left column.

Because of the shift in electron chemical potential with magnetic field, the
location or Urca pairs within the neutron star crust varies with field and
latitude (polar angle on the stellar surface). As an example, consider Figure
7, which shows the electron chemical potential as a function of polar angle
for various surface magnetic fields (at the poles), latitudes, and densities
within a NS crust. Because the chemical potential changes with both field and
density, the physical location of Urca pairs may change. This will also result
in a differentiation in the neutrino luminosity of various stars. For example,
for a very large field of 1016.5 G as indicated in the figure, the density for
which an Urca pair occurs is highest at the poles of the star, resulting in a
larger polar neutrino luminosity, whereas, the stellar equator will form Urca
pairs at a lower density. However, for the 63Cr$\leftrightarrow^{63}$V Urca
pair, it can be seen that the minimum luminosity occurs at $\theta\approx$ 1.2
rad. The magnetic field can result in a shift in the locations of Urca pairs
as a function of latitude as seen in this figure. It can also be seen that
magnetic fields will enable the existence of new pairs.

For a constant density and temperature, changes in magnetic field can result
in a change in the electron chemical potential (Equation 2.1). This changes
the overall phase space for electrons in $\beta^{-}$ decay and electron
capture. If an Urca pair is defined as a $\beta^{-}$-EC pair for which
$\mu_{e}\sim Q$, then the density at which an Urca pair can exist also depends
on the environmental magnetic field.

For example, the crusts and oceans of neutron stars have been modeled for
various dipole fields defined by the field at the NS polar region. The density
at which various Urca pairs are viable for various polar fields is shown in
Figure 8 as a function of polar angle on the NS surface. For an assumed dipole
field, the field is dependent on the NS polar angle at the surface. A 12 km
diameter star with M=1.4 M⊙ was considered. A constant temperature and $Y_{e}$
of $T_{9}=0.51$ and $Y_{e}$=0.41 are assumed in each case. For these fixed
parameters and a fixed dipole field, the density at which $\mu_{e}\sim Q$ is
found. This is plotted in the left column of Figure 8 for fields of
$B(\theta=0)$ of 1014 G, 1015 G, and 1016 G. Because the field is not constant
across the NS surface, the electron chemical potential is not constant for
constant density.

Using the densities computed in Figure 8 (left panels), the neutrino
emissivity as a function of polar angle in the NS ocean is plotted in the same
figure (center). Here, the emissivity is normalized to the mass fraction of
the relevant nuclei in these panels, $\varepsilon/X$. The emissivity is
computed using the approximation of Deibel et al. (2016). Because we are
interested in bulk behaviors of the neutrino emissivity and luminosity, we
adopt the approximate weak rates of Arcones et al. (2010) using the phase
space extracted from Equation 2.1 (Famiano et al., 2020). However, given the
extraction of the $ft$ values from the rates presumed in Deibel et al. (2016),
the rate ratios given by the phase space differences with and without the
magnetic fields, $\lambda(B\neq 0)/\lambda(B=0)$ are expected to be
independent of any nuclear structure effects in this evaluation.

Figure 8: Left panels: The density at which an Urca pair is viable for various
ocean Urca pairs. Center panels: The emissivity, $\varepsilon_{22}$, for
various Urca pairs normalized to the mass fraction of the parent nucleus.
Right panels: The luminosity,$L_{34}$, per angular bin for a bin size of
$\pi/200$ rad for each Urca pair. The top, middle, and bottom rows correspond
to a field B of 1014 G, 1015 G, and 1016 G respectively.

The luminosity, $L_{32}$ is shown on the right side of Figure 8 assuming an
axisymmetric field for polar angle increments of $\Delta\theta=\pi/200$
radians. The radial thickness of each zone is calculated using the formulation
of Schatz et al. (2014). For this calculation, a temperature of T9=0.51, an
electron fraction of $Y_{e}=0.41$, a crust radius of 12 km, and a local
gravity of $g=1.85\times 10^{14}$ cm s-2 are assumed. The smaller overall
solid angle near the poles results in an overall reduction of the total
luminosity, though the emissivity may be larger in this region. The value of
the neutrino emissivity, $\varepsilon$, is proportional to the weak
interaction rates and is computed following the prescription of (Deibel et
al., 2016).

There are multiple effects contributing to the behavior of the neutrino
luminosity as a function of field. At the densities necessary for an Urca pair
to exist, only the high(low)-energy tail of the electron energy spectrum is
relevant for $\beta^{-}$(EC) decay. Also, a change in the local magnetic field
will shift the overall electron chemical potential as shown in Figure 6. In
this case, shifting the electron chemical potential in one direction or the
other will result in a shift in the electron phase space, resulting in $\beta$
decays or electron captures being impossible. As the magnetic field increases,
the overlap between the Landau level distribution, the Fermi distribution, and
the electron momentum space can interfere constructively or destructively,
resulting in dramatic shifts in the optimum density at which an Urca pair can
exist and changing the overall emissivities. This will be discussed below.

In this figure one can see that, for very high fields ($\sqrt{eB}\gtrsim$ Q),
the entire electron energy spectrum is confined to a single Landau level. As
the field increases near the poles, the electron chemical potential decreases,
resulting in a lower density at which an Urca pair can exist. This can result
in a reduced emissivity. In addition, as the Landau level spacing increases,
and the tail of the lowest Landau level ($E=m_{e}$) is more prominent in the
electron energy spectrum. As the field decreases, and the Landau levels move
closer together (but with a level spacing still greater than the Q value), the
tail of the lowest Landau level becomes less prominent in the electron
spectrum, resulting in a decrease in the overall rates.

For a very low field, $\sqrt{eB}\ll Q$, the effects of the Landau level
spacing on the overall electron spectrum are minimal. Here, any change in
field along the surface of the NS has little or no effect on the electron
chemical potential, and the optimal density for an Urca pair does not change
significantly.

The “intermediate field” regime ($\sqrt{eB}\sim Q$) is particularly
interesting as the interference between the Landau level distribution and the
Fermi distribution becomes prominent. In this regime, a small shift in the
magnetic field can result in a shift in the optimum density of an Urca pair
through a shift in the electron chemical potential. This contributes to a
shift in the emissivity. Because the location of Landau levels strongly
affects the availability of electrons to decay or capture, the effect can be
magnified.

Figure 9: $\beta^{-}$ Decay energy spectrum for various magnetic fields (G)
for the 23Ne decay. Red dashed lines correspond to the spectrum for B=0, and
the solid black lines correspond to the spectrum for the fields indicated in
each panel. Each plot in this figure is calculated at T9=0.51, $Y_{e}$=0.41,
and a density at which $\mu_{e}=Q_{\beta}$.

This can be seen in Figure 9, which shows the electron phase space
distribution for 23Ne decay in the 23Na$\leftrightarrow^{23}$Ne Urca pair. In
each panel in this figure, the optimum density for this Urca pair to exist is
computed (i.e., $\mu_{e}=Q_{\beta}$), and the resulting electron phase space
spectrum (the integrand in the classical decay rate integral) is shown.

For a low field of 1014 G, the Landau level spacing is slightly less than the
energy range of available electrons. As the field changes, the Landau level
spacing changes, as do the positions of individual Landau levels. One sees
that as the field increases above 1014 G, Landau levels can change location
with respect to the edge of the Fermi distribution. At 1014.5 G, a Landau
level exists at the right edge of the Fermi distribution with a less-prominent
contribution from a level near 4.4 MeV. Most of the energy spectrum is
dominated by the tail of the lower-energy Landau level. At a field of 1014.7
G, the Landau level spacing is larger than the available phase space created
by the Fermi distribution and the electron momentum distribution. For this
reason, much of the available electron energy spectrum is dominated by the
tail of a low-energy Landau level. As the field increases towards 1015 G, the
degeneracy of available Landau levels increases as electrons are placed in
only the lowest few Landau levels, and the occupancy of the tails of the
distributions becomes more important in the electron energy spectrum. When the
field is high enough such that only the lowest Landau level is occupied, the
tail of the distribution becomes quite prominent, and the electron energy
spectrum is dominated by the single Landau level.

##  4 Conclusions

We explored the dependence and evolution of Urca pairs in crusts and oceans of
magnetized neutron stars. Because the electron chemical potential depends on
the environmental magnetic field, the presence or absence of Urca pairs must
also depend on the external field. Urca pairs which may exist at a certain
temperature and density, $\rho Y_{e}$, at zero magnetic field may not exist
for a non-zero field. Conversely, Urca pairs which do not exist at specific
temperatures and densities may appear in the presence of an external field.
The presence of a magnetic field may also make excited states available in
EC-$\beta^{-}$ pairs. In this case, two nuclei which may not transition to
low-lying states, may undergo transitions to excited states in the presence of
an external field.

We also explored the evolution of Urca pairs along the surface of a magnetized
neutron star. While we have assumed a dipole field for simplicity, this is
sufficient to convey the concept that the variations in the field on the
surface of a NS may change Urca pair locations within the crust/ocean of the
star. Here, we have calculated the density at which an Urca pair may exist in
a NS crust/ocean for a constant crust temperature of $T_{9}=0.51$. Of course,
we note that the temperature of the crust varies with depth, so we acknowledge
the simplification in this evaluation. This variation in density will result
in a change in the neutrino emissivity and subsequent luminosity as a function
of polar angle. We find that the emissivity (normalized to the mass fraction
of the Urca pair studied) subsequently depends on the NS latitude, resulting
in changes in the luminosity as a function of latitude.

However, because the density at which an Urca pair may exist in a NS is not
constant, this ultimately affects the actual presence of an Urca pair at a
location in a NS. This is because the density and the mass fraction changes
with depth in the NS crust. At one location on the NS surface, an Urca pair
may exist at one density while it exists at a different density at another
location in the NS. However, the mass fractions of the Urca pair also vary at
different densities/radii. For this reason, the emissivity is presented as
normalized to the mass fraction, and a true computation of the emissivity must
be multiplied by the mass fraction for the radius at which a specific density
exists. Figure 8 can be used to scale Urca pair emissivity as a function of
latitude for a known mass fraction, X. Thus, the emissivity can be expressed
as a functional, $\epsilon=\epsilon(T,\rho
Y_{e},B,X)=\epsilon(T,\rho(r,\theta)Y_{e}(r),B(\theta))=\epsilon(T,r,\theta,B_{0})$,
where $B_{0}$ is the magnetic field at the NS pole.

It may very well be possible that the mass fraction in the region at which an
Urca pair may be viable is zero, while it is non-zero in another viable
region. Thus, an Urca pair may exist in one part of the NS, but not in
another. We have shown that the addition of a magnetic field to NS luminosity
calculations adds another layer of complexity to the total emissivity
calculation, and ultimately may result in uneven NS cooling along the surface
of the star.

There may be several ramifications of this result which will be explored in
subsequent papers. First, we find that the bulk luminosity of a NS is
dependent on its magnetic field. This can ultimately change the cooling curve
of the star. Perhaps variations in NS luminosities are a result of the surface
field. Likewise, limits can be placed on NS surface fields from observations
of their cooling.

Second, uneven neutrino emissivity on the NS surface can result in uneven
cooling and heating in the crust. This can have affect the thermal equation of
state on the surface and may be a way of explaining NS crust quakes or
possibly even NS kicks.

Certainly, an exhaustive treatment of the overall complexity of this problem
is beyond the scope of the present work. More precise work is needed. In
particular, a thorough evaluation of the presence or absence of Urca pairs as
a function of density, temperature, and magnetic field is needed. From this,
luminosity maps can be produced for NSs with various field configurations. In
order to produce these, realistic models of ocean/crust abundances can be
developed. While these will be explored in future work, our current results
introduce the possibility that magnetized neutron stars may have uneven
cooling as a result of variations in the interior magnetic field, and the
overall bulk luminosity changes with the field configuration and magnitude.

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T.K. is supported in part by Grants-in-Aid for Scientific Research of JSPS
(17K05459, 20K03958). A.B.B. is supported in part by the U.S. National Science
Foundation Grants No. PHY-2020275 and PHY-2108339. M.A.F. is supported by
National Science Foundation Grant No. PHY-1712832 and by NASA Grant No.
80NSSC20K0498. K.M. is supported by Research Institute of Stellar Explosive
Phenomena at Fukuoka University and JSPS KAKENHI Grant Number JP21K20369.
M.A.F., G.J.M. and A.B.B. acknowledge support from the NAOJ Visiting Professor
program. Work at the University of Notre Dame (G.J.M.) supported by DOE
nuclear theory grant DE-FG02-95-ER40934.