# Anisotropic neutron star crust, solar system mountains, and gravitational
waves

J.A. Morales<EMAIL_ADDRESS>Center for the Exploration of Energy and Matter
and Department of Physics, Indiana University, Bloomington, IN 47405, USA  Max
Planck Institute for Graavitational Physics (Albert Einstein Institute),
Callinstrasse 38, 30167, Hannover, Germany  C.J. Horowitz<EMAIL_ADDRESS>Facility for Rare Isotope Beams, Michigan State University, East Lansing,
Michigan 48824, USA  Center for the Exploration of Energy and Matter and
Department of Physics, Indiana University, Bloomington, IN 47405, USA

###### Abstract

“Mountains” or non-axisymmetric deformations of rotating neutron stars (NS)
efficiently radiate gravitational waves (GW). We consider analogies between NS
mountains and surface features of solar system bodies. Both NS and moons such
as Europa or Enceladus have thin crusts over deep oceans while Mercury has a
thin crust over a large metallic core. Thin sheets may wrinkle in universal
ways. Europa has linear features, Enceladus has “Tiger” stripes, and Mercury
has lobate scarps. NS may have analogous features. The innermost inner core of
the Earth is anisotropic with a shear modulus that depends on direction. If NS
crust material is also anisotropic this will produce an ellipticity, when the
crust is stressed, that grows with spin frequency. This yields a breaking
index (log derivative of spin down rate) very different from $n=5$ and could
explain the maximum spin observed for neutron stars and a possible minimum
ellipticity of millisecond pulsars.

gravitational waves – stars: neutron

The opening of the gravitational wave (GW) sky is an historic time. We have
observed GW from black hole [1] and neutron star [2] mergers. Galileo opened
the electromagnetic sky and its extraordinary riches. The GW sky, no doubt,
contains additional very exciting signals. Galileo observed mountains on the
Moon. Ongoing searches for continuous GW from “mountains” (large scale
deformations) on rotating neutron stars have not yet detected signals.
Targeted searches have focused on known pulsars, with known spin frequency and
spin down parameters [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18,
19, 20, 21, 22, 23]. On the other hand, directed searches have focused on
locations in the sky that are known or suspected to harbor a neutron star,
without prior knowledge of neither the frequency nor any spin-down parameter
[24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34]. Finally, all-sky searches have
focused on searching for unknown sources at unknown locations [35, 36, 37, 38,
39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49]. These various types of searches
are improving. Next generation GW observatories Cosmic Explorer and Einstein
Telescope should extend these searches to hundreds to thousands of times more
neutron stars [50, 51].

Neutron stars, like many solar system bodies, have solid crusts. “Mountains”,
or non-axisymmetric deformations of the crust, radiate gravitational waves as
the star rotates [52]. The amplitude $h_{0}$ of gravitational wave radiation
from a star a distance $d$ away, rotating with angular frequency $\omega$ and
moment of inertia $I$ is [53],

$h_{0}=16\pi G\frac{I}{d}\omega^{2}\epsilon\,.$ (1)

Here the important unknown is the shape of the star or ellipticity $\epsilon$
defined as the fractional difference in the star’s principle moments of
inertia,

$\epsilon=(I_{1}-I_{2})/I_{3}\,.$ (2)

Large scale deformations in the crust, or mountains, can give rise to a non-
zero $\epsilon$.

An important first step is to calculate the maximum ellipticity that the crust
can support. This involves simulating the breaking strain of neutron star
crust material and then determining the maximum ellipticity the crust material
can support against the star’s gravity. Molecular dynamics simulations,
including the effects of impurities, defects, and grain boundaries, find that
the breaking strain of the crust is large, of order 0.1, because the crust is
under great pressure that prevents the formation of voids or fractures [54,
55, 56].

Given this breaking strain, Ushomirsky et al. [52] have calculated the maximum
ellipticity in an intuitive formalism. They assume the crust can be strained
near its breaking strain everywhere and write the maximum deformation the
crust can support as a simple integral of the crust breaking stress divided by
the local gravitational acceleration. This yields a maximum ellipticitiy
$\epsilon_{max}$ of a few $\times$10-6. Note that Gittins et al., using a
simplified external force to deform the star, claim a smaller $\epsilon_{max}$
[57]. However, using Gittins et al. formalism with an improved external force,
we find a larger $\epsilon_{max}$ consistent with Ushmirsky et al. [58]. We
therefore assume $\epsilon_{max}\approx{\rm few}\times 10^{-6}$.

This value for $\epsilon_{max}$ can be compared to the smallest observed upper
limit, for a rapidly spinning nearby pulsar, $\epsilon_{min}\approx{\rm
few}\times 10^{-9}$ [59]. This gives a dynamic range of
$\epsilon_{max}/\epsilon_{min}\approx 1000$. We can detect mountains 1000
times smaller than the maximum crust mountain.

Unfortunately, we do not know the actual size of neutron star mountains and
$\epsilon$ for particular stars. Electromagnetic observations of surface
features are very limited. For example rotational phase resolved X-ray
spectroscopy has mapped the shape of hot spots on some pulsars [60]. However,
this thermal information does not directly provide elevations or mass
distributions. Furthermore, mountain building mechanisms may be complex and
depend on poorly known material properties. For example, viscoelastic creep,
how a stressed elastic medium relaxes with time, may be important for the
lifetime of neutron star mountains [55].

For a given postulated mechanism, for example temperature dependent electron
capture on accreting stars [52], one can estimate the resulting ellipticity.
However, neutron star crusts may be very rich physical systems that can
involve many possible deformation mechanisms. We consider analogies between
neutron star mountains and surface features of solar system bodies for two
reasons. First, solar system observations may suggest particular mountain
building mechanisms that could produce interesting $\epsilon$ values for
neutron stars and lead to detectable gravitational wave radiation. Second, the
great diversity of solar system bodies suggests that neutron star crusts may
also be diverse. Although the analogy between neutron stars and solar system
bodies is incomplete, we have unique observations of solar system surface
features. This provides “ground truth” for complex mountain building physics.

We consider a large range of solar system planets and moons starting from very
general considerations and proceed to more specialized observations and
mountain building mechanisms. Perhaps the most basic observation is that solar
system moons are extraordinarily diverse. This was revealed with the first
images of the Galilean moons of Jupiter from the Pioneer and Voyager
spacecraft. “The satellite surfaces display dramatic differences including
extensive active volcanism on Io, complex tectonism on Ganymede and possibly
Europa, and flattened remnants of enormous impact features on Callisto” [61].
Not only are these four moons very different from each other, none is similar
to our Moon. If neutron stars are also diverse, this could be promising for
gravitational wave searches. Indeed, we know of several different classes of
neutron stars such as pulsars, millisecond pulsars, or magnetars. Even if some
stars are symmetric with small ellipticities, others may be very different and
have larger ellipticities.

Many solar system bodies have large scale asymmetries. For example the near
and far sides of the Moon are very different. Mars is strikingly asymmetric.
Not only is the low smooth northern hemisphere quite different from the high
cratered southern hemisphere but great volcanoes and high plateaus make the
East very different from the West [62]. Iapetus, a moon of Saturn, is
extremely asymmetric with a very dark leading hemisphere and an
extraordinarily bright trailing hemisphere [63]. These several bodies, with
large observed asymmetries, provide at least moral support for there also
being asymmetric neutron stars.

Mimas, another moon of Saturn, has a very large crater Herschel [64]. This
single feature, by itself, creates a significant ellipticity. Single
catastrophic events on neutron stars could likewise produce large asymmetries.
For example, an event that melts a significant fraction of the crust could
leave a large “scar” when the crust re-freezes and produce a non-zero
ellipticity.

Leptodermous kosmos is a possible Greek translation of thin-skinned worlds.
Neutron stars have a thin crust, approximately 1 km thick, over a deep liquid
core. There are a number of thin-skinned moons in the solar system. Both
Europa and yet another moon of Saturn Enceladus have thin ice crusts over deep
oceans. These moons have characteristic linear surface features. Indeed the
lines on Enceladus look like “Tiger stripes”. Neutron stars, with their thin
crusts, may have analogous linear surface features.

Accretion can spin up the equatorial bulge of a neutron star and put the crust
under tension while EM or GW radiation can spin down the bulge and put the
crust under compression. Thin sheets may wrinkle in universal ways. Examples
of wrinkling under tension include hanging drapes [65], stretched thin sheets
[66], or a water drop on a thin sheet [67]. Compressional examples include
wrinkling from thermal contraction mismatch [68] or growth induced wrinkling
in leafs and flowers [69].

The planet Mercury has a thin silicate crust over a large metallic core.
Lobate scarps on Mercury are bow shaped ridges that can be hundreds of
kilometers long. These are the most prominent tectonic features on the planet
with a few hundred to several thousand meters of vertical relief [70, 71, 72].
The formation of these features is thought to involve the thermal contraction
of the core leading to compressional wrinkling of the thin crust [73]. Neutron
stars that are significantly spun down may have lobate scarp like wrinkles and
these could contribute to a nonzero ellipticity.

Recent observations of seismic waves reverberating through the Earth’s center
find an anisotropic innermost inner core [74]. Here the sound velocity of
innermost inner core material is observed to depend on direction. Don Anderson
wrote “Crystals are anisotropic and tend to be oriented by sedimentation,
freezing, recrystallization, deformation, and flow. Therefore we expect the
solid portions of the earth to be anisotropic to the propagation of seismic
waves and material properties [75].”

We postulate that neutron star crust may also be anisotropic. This provides a
new way to break axial symmetry and generate a non-zero $\epsilon$. Single
crystals are anisotropic. Neutron star crust is believed to form a body
centered cubic (bcc) lattice. A bcc lattice has a small shear modulus for
compressing one axis of the lattice (and expanding the other two axis by half
as much so as to conserve the volume). In addition there is an $\approx$ 8
times larger shear modulus for distorting the square lattice into a rhombus
[76]. Thus a single bcc crystal has a large anisotropy and the velocity of
sound depends strongly on direction [77].

In addition, complex nuclear pasta phases are expected over an approximately
100 m region between the crust and the core [78, 79]. This region can be
important because it is the densest part of the crust and may contain half of
the crust’s mass. Some pasta shapes, such as Lasagna, are strongly anisotropic
[80].

However, one typically assumes macroscopic regions of a neutron star involve
large numbers of micro-crystals (or domains) and each domain has a random
orientation. As a result almost all calculations assume an angle-averaged
shear modulus [81], see also [80], where the velocity of sound is independent
of direction.

We now consider that the micro-crystals may be partially aligned due to re-
crystallization or some other mechanism. For example, material on an accreting
neutron star is constantly being both crystallized as new material is added
and melted as material is buried to higher densities and dissolves into the
core. This may create one or more regions, that are not negligibly small
compared to the size of the star, where crystals are at least partially
aligned. Each region is assumed to have a single orientation determined for
example by the random orientation of a first seed crystal. Alternatively,
pasta may form (partially) aligned with respect to the magnetic field with
spaghetti forming along B or B in the plane of lasagna sheets [80]. The shear
modulus will be anisotropic by an amount that depends on the amount of
alignment of the micro-crystals.

We obtain a first estimate of the ellipticity produced by an anisotropic crust
with a simple two dimensional calculation. This order of magnitude result will
suffice, given the large uncertainty in the anisotropy of the crust. We
replace a hollow sphere by a hollow cylinder that is assumed thin in the z
direction (out of the plane in Fig. 1). We consider a thin disc and treat the
anisotropy as a first order perturbation to the corresponding axially
symmetric constitutive relationships [82]. A thick disk is expected to yield
qualitatively similar results, however the calculation is somewhat more
involved.

Figure 1: Cut through the equatorial plane of a rotating star. The crust
extends from $R_{0}$ to $R$ and is slightly anisotropic in the $X$ direction.

Figure 1 shows the equatorial plane of a rotating neutron star. Elastic
perturbations of the stress tensor $\sigma_{ij}$ are related to the strain
tensor $\epsilon_{ij}$ by the elasticity of the medium,

$\sigma_{xx}=\frac{E}{1-\nu^{2}}(1+\langle\phi\rangle)\epsilon_{xx}+\frac{\nu
E}{1-\nu^{2}}\epsilon_{yy},$ (3)
$\sigma_{yy}=\frac{E}{1-\nu^{2}}\epsilon_{yy}+\frac{\nu
E}{1-\nu^{2}}\epsilon_{xx},$ (4)

and $\sigma_{xy}=E\,\epsilon_{xy}/(1+\nu)$, with $E$ the Young’s modulus and
$\nu$ the Poisson ratio. The degree of alignment of micro-crystals in the
crust is described by the small parameter $\langle\phi\rangle$. If
$\langle\phi\rangle=0$ the medium is isotropic. As an example, we consider a
partially aligned medium with the symmetries of the Lasagna phase of nuclear
pasta [80]. The $X$ axis in Fig. 1 is normal to the Lasagna planes. We assume
this direction arose from spontaneous symmetry breaking, for example the
random oriantation of a seed micro-crystal. We start with a symmetric medium
$\langle\phi\rangle=0$ and then calculate first corrections from
$\langle\phi\rangle\neq 0$.

We note the shear modulus is $\mu=E/[2(1+\nu)]$ and bulk modulus is
$K=E/[3(1-2\nu)]$. We define $\langle\phi\rangle$ to describe the degree of
alignment with respect to the shear (or Young’s) modulus. The larger bulk
modulus (since $\nu$ is near 0.5) is dominated by the isotropic electron
pressure. This symmetric pressure will tend to reduce (but not eliminate) the
ellipticity of the star, see discussion of Eq. 12 below.

We assume the crust froze while the star was rotating with initial angular
frequency $\omega_{0}$. If the star is then spun up or spun down to a new
angular frequency $\omega$, stresses will be induced according to the equation
of motion,

$\frac{\partial\sigma_{rr}}{\partial
r}+\frac{1}{r}(\sigma_{rr}-\sigma_{\theta\theta})=-\rho
r(\omega^{2}-\omega_{0}^{2}),$ (5)

with $\rho$ the average crust density. The radial stress is [82],

$\sigma_{rr}(r)=\frac{3+\nu}{8}\rho(\omega^{2}-\omega_{0}^{2})[R^{2}+R_{0}^{2}-r^{2}-\frac{R^{2}R_{0}^{2}}{r^{2}}]\,,$
(6)

and satisfies boundary conditions $\sigma_{rr}(R_{0})=\sigma_{rr}(R)=0$ at the
inner $R_{0}$ and outer $R$ radii of the crust. The angular stress
$\sigma_{\theta\theta}$ is [82],

$\sigma_{\theta\theta}(r)=\frac{3+\nu}{8}\rho(\omega^{2}-\omega_{0}^{2})[R^{2}+R_{0}^{2}-\frac{1+3\nu}{3+\nu}r^{2}+\frac{R^{2}R_{0}^{2}}{r^{2}}]\,,$
(7)

and does not vanish at $r=R_{0}$ or $R$.

We now consider $\langle\phi\rangle\neq 0$. We rewrite Eqs. 3,4 in polar
coordinates, invert to obtain strain $\epsilon_{ij}$ as a function of stress,
and expand to lowest order in $\langle\phi\rangle$. We provide a first
estimate of the change in strain $\delta\epsilon_{ij}$ with
$\langle\phi\rangle$ by using the unperturbed stresses from Eqs 6 and 7 to
get,

$\begin{bmatrix}\delta\epsilon_{rr}\\\ \delta\epsilon_{\theta\theta}\\\
\end{bmatrix}\approx-\frac{\langle\phi\rangle}{E}\begin{bmatrix}\frac{(C^{2}-S^{2}\nu)^{2}}{1-\nu^{2}}&\frac{C^{2}S^{2}(1+\nu)}{1-\nu}\\\
\frac{C^{2}S^{2}(1+\nu)}{1-\nu}&\frac{(C^{2}\nu-S^{2})^{2}}{1-\nu^{2}}\\\
\end{bmatrix}\begin{bmatrix}\sigma_{rr}\\\ \sigma_{\theta\theta}\\\
\end{bmatrix}$ (8)

with $S=\sin\theta$ and $C=\cos\theta$. We assume a thin crust $R-R_{0}\ll R$
where $\sigma_{rr}\ll\sigma_{\theta\theta}$ and $\sigma_{\theta\theta}$ is
approximately independent of $r$. This gives $\delta\epsilon_{rr}$ and
$\delta\epsilon_{\theta\theta}$ that are also $\approx$ independent of $r$.
The radial displacement is written $u_{r}=u_{r}^{0}+\delta u_{r}$ where
$u_{r}^{0}$ is the displacement in the isotropic case $\langle\phi\rangle=0$.
Likewise $u_{\theta}=u_{\theta}^{0}+\delta u_{\theta}$. The perturbation
$\delta u_{r}\approx(r-R_{0})\delta\epsilon_{rr}(\theta)$ follows from
integrating $\partial\delta u_{r}/\partial r=\delta\epsilon_{rr}$. The angular
displacement follows by integrating
$\delta\epsilon_{\theta\theta}=(\partial\delta
u_{\theta}/\partial\theta+\delta u_{r})/r$ to get $\delta
u_{\theta}\approx\int
d\theta^{\prime}[r\delta\epsilon_{\theta\theta}-(r-R_{0})\delta\epsilon_{rr}]+C_{\theta}$
or,

$\delta u_{\theta}(\theta)\approx
R\int^{\theta}_{0}d\theta^{\prime}\delta\epsilon_{\theta\theta}+C_{\theta}\,,$
(9)

assuming $r\gg r-R_{0}$. Here the integration constant $C_{\theta}$ is
independent of $\theta$ and does not contribute to moments of inertia.

We now calculate the difference in moments of inertia in Eq. 2. For simplicity
we work in two dimensions and actually calculate moments of inertia of a
deformed hoop. This provides an order of magnitude estimate for the
ellipticity. A mass $\rho({\bf r})$ that was initially at ${\bf r}$ is now at
${\bf r}^{\prime}=(r+u^{0}_{r}+\delta u_{r})\hat{r}+(u_{\theta}^{0}+\delta
u_{\theta})\hat{\theta}+u_{z}\hat{z}$. The difference in moments of inertia
is,

$I_{x}-I_{y}=\int d^{3}r\rho(r)[({\bf r}^{\prime}\cdot\hat{y})^{2}-({\bf
r}^{\prime}\cdot\hat{x})^{2}]\,.$ (10)

Working to first order in the small displacement $\delta u_{\theta}(\theta)$,
this becomes $I_{x}-I_{y}\approx 4\int d^{3}r\rho(r)r\delta
u_{\theta}(\theta)\sin\theta\cos\theta$. We write this as $I_{x}-I_{y}\approx
m_{cr}R^{2}A$ where $m_{cr}=\int d^{3}r\rho(r)$ is the mass of the crust and
the important angular integral is

$A=\frac{4}{2\pi}\int_{0}^{2\pi}d\theta\sin\theta\cos\theta\int_{0}^{\theta}d\theta^{\prime}\delta\epsilon_{\theta\theta}(\theta^{\prime})\,.$
(11)

Finally, dividing by the moment of inertia $I\approx\frac{2}{5}MR^{2}$, of a
star of mass $M$, gives the ellipticity
$\epsilon\approx\frac{5}{2}\frac{m_{cr}}{M}A$. Using
$\sigma_{\theta\theta}\approx\rho R^{2}(\omega^{2}-\omega_{0}^{2})$ from Eq.
7, $\delta\epsilon_{\theta\theta}$ from Eq. 8, and evaluating the integral
gives

$A=\frac{5-2\nu+\nu^{2}}{8E(1-\nu^{2})}\rho
R^{2}\langle\phi\rangle(\omega^{2}-\omega_{0}^{2})\,.$ (12)

Before deriving our final result, we first discuss how a thick disk, where the
strain is in the xy plane, might differ from the thin disk (with the stress in
the xy plane) that we have calculated. We expect a thick disk to have a
similar result to Eq. 12 except that the Young’s modulus $E$ would be replaced
by the (larger) bulk modulus $K$. As a conservative approximation we replace
either $E$ or $K$ with $\rho R^{2}\omega_{K}^{2}$ where $\omega_{K}$ is the
Keplarian angular frequency for breakup. This gives our main result for the
ellipticity of a star with anisotropic material in its crust,

$\epsilon\approx\frac{m_{cr}}{M}\langle\phi\rangle\frac{\Omega^{2}-\Omega_{0}^{2}}{\Omega_{K}^{2}}\,.$
(13)

Note that we have rewritten the angular frequencies $\omega$, $\omega_{0}$,
and $\omega_{K}$ in terms of rotational frequencies $\Omega=\omega/(2\pi)$
etc. Here $m_{cr}/M\approx 10^{-2}$ and $\Omega_{K}\approx 1400$ Hz depending
on the equation of state. We see that $\epsilon$ is a strong function of the
rotational frequency $\Omega$ and the initial frequency $\Omega_{0}$ (when the
crust froze).

Unfortunately, we do not know the degree of anisotropy of the neutron star
crust $\langle\phi\rangle$. Even a small average can lead to a significant
$\epsilon$ and produce observable gravitational waves. As an example we
consider the innermost inner core of the Earth because we have no neutron star
observations. There is a few percent anisotropy in the Earth that extends over
the innermost inner core [74]. This region has a radius of 300 km and contains
about $3\times 10^{-4}$ of the Earth’s mass. If we ignore anisotropies in the
rest of the Earth, this corresponds to $\langle\phi\rangle\approx 10^{-5}$
when averaged over the Earth’s total mass.

Of course this value is not directly relevant for neutron stars. Nevertheless,
if material in a neutron star has an anisotropy of $\langle\phi\rangle\approx
10^{-5}$ when averaged over the crust mass, this would yield $\epsilon\approx
10^{-7}(\Omega^{2}-\Omega_{0}^{2})/\Omega_{K}^{2}$ or $\epsilon\approx
10^{-8}$ for a rapidly rotating object with
$(\Omega^{2}-\Omega_{0}^{2})/\Omega_{K}^{2}\approx 0.1$. Gravitational waves
from a nearby star could be detectable for this $\epsilon$ value.

The breaking index $n=d\ln\dot{\omega}/d\ln\omega$ describes how the spin down
rate $\dot{\omega}$ depends on rotational frequency. For simplicity we neglect
spin down from electromagnetic radiation. Spin down from gravitational wave
radiation only with a frequency independent $\epsilon$ has $n=5$. However the
strong spin dependence of $\epsilon$ in Eq. 13 leads to
$n=5+4\Omega^{2}/(\Omega^{2}-\Omega_{0}^{2})$. This is very different from 5
as shown in Fig. 2. In the limit $\Omega\gg\Omega_{0}$, $n=9$ and $n$ changes
rapidly for $\Omega$ near $\Omega_{0}$. Neutron stars that have been spun up
since crust formation have $n>9$ while stars that have spun down since crust
formation have $n<5$. Finally, $n=0$ at $\Omega=\sqrt{5/9}\,\Omega_{0}$. For
constant $\epsilon$, $|\dot{\omega}|$ decreases with decreasing $\Omega$.
However as $\Omega$ decreases $\epsilon$ increases (given $\Omega<\Omega_{0}$)
and this increases $|\dot{\omega}|$. At $\Omega=\sqrt{5/9}\,\Omega_{0}$ the
two effects cancel and $n=0$.

Figure 2: Breaking index $n$ (solid black curve) and ellipticity $\epsilon$
(dashed red curve) vs rotational frequency $\Omega$ assuming the crust froze
while the star was rotating at $\Omega_{0}=300$ Hz.

Torque from gravitational wave radiation could balance the spin up from
accretion and limit neutron star spins [52]. Using $\epsilon$ in Eq. 13, this
torque $N_{gw}$ rises very rapidly with $\Omega$ so
$N_{gw}\propto\Omega^{5}(\Omega^{2}-\Omega_{0}^{2})^{2}\langle\phi\rangle^{2}$.
If $N_{gw}$ balances the accretion torque $N_{a}\approx\dot{M}(GMR)^{1/2}$
then the equilibrium spin,

$\Omega_{eq}\approx 300\ {\rm Hz}\,(\frac{\dot{M}}{10^{-8}{\rm
M_{\odot}\,yr}^{-1}})^{1/9}(\frac{10^{-4}}{\langle\phi\rangle})^{2/9}\,,$ (14)

could agree with observed values. Note that $\Omega_{eq}$ only depends very
weakly on the accretion rate $\dot{M}$ or $\langle\phi\rangle$. Here we assume
$\Omega_{eq}\gg\Omega_{0}$, $M=1.4M_{\odot}$, and $R\approx 10$ km. Because
our ellipticity rises strongly with $\Omega$, this torque balance can be
achieved with modest $\langle\phi\rangle\approx 10^{-4}$. Furthermore, our
mechanism, with a somewhat smaller $\langle\phi\rangle\approx 10^{-6}$, could
explain a possible minimum ellipticity $\epsilon\approx 10^{-9}$ suggested by
an observed minimum spin down rate for millisecond pulsars [83].

In conclusion “mountains” or non-axisymmetric deformations of rotating neutron
stars (NS) efficiently radiate gravitational waves (GW). There are many
ongoing searches for continuous GW from such stars. Present detectors are
sensitive, in the best cases, to mountains that are 1000 times smaller than
the maximum mountain that the crust can support. Unfortunately, we do not know
the size of NS mountains. We consider analogies between NS mountains and
surface features of solar system (SS) bodies. Here SS observations can provide
“ground truth” for complex mountain building physics. Both NS and moons such
as Europa or Enceladus have thin crusts over deep oceans while Mercury has a
thin crust over a large metallic core. Thin sheets may wrinkle in universal
ways. Europa has linear features, Enceladus has “Tiger” stripes, and Mercury
has lobate scarps. NS may have analogous features. The innermost inner core of
the Earth is anisotropic with a shear modulus that depends on direction. If NS
crust material is also anisotropic this could produce a significant
ellipticity that grows rapidly with increasing rotational frequency.
Gravitational wave emission torques from this ellipticity may limit the spin
rate of neutron stars.

###### Acknowledgements.

Matt Caplan, Cole Miller, Jing Ming, and Ruedi Widmer-Schnidrig are thanked
for helpful comments. This work is partially supported by the US Department of
Energy grant DE-FG02-87ER40365 and National Science Foundation grant
PHY-2116686.

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