# A tight universal relation between the shape eccentricity and the moment of
inertia for rotating neutron stars

Yong Gao Department of Astronomy, School of Physics, Peking University,
Beijing 100871, China  Kavli Institute for Astronomy and Astrophysics, Peking
University, Beijing 100871, China  Lijing Shao Kavli Institute for Astronomy
and Astrophysics, Peking University, Beijing 100871, China  National
Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, China
Jan Steinhoff Max Planck Institute for Gravitational Physics (Albert Einstein
Institute), Am Mühlenberg 1, Potsdam 14476, Germany

###### Abstract

Universal relations that are insensitive to the equation of state (EoS) are
useful in reducing the parameter space when measuring global quantities of
neutron stars (NSs). In this paper, we reveal a new universal relation that
connects the eccentricity to the radius and moment of inertia of rotating NSs.
We demonstrate that the universality of this relation holds for both
conventional NSs and bare quark stars (QSs) in the slow rotation
approximation, albeit with different relations. The maximum relative deviation
is approximately $1\%$ for conventional NSs and $0.1\%$ for QSs. Additionally,
we show that the universality still exists for fast-rotating NSs if we use the
dimensionless spin to characterize their rotation. The new universal relation
will be a valuable tool to reduce the number of parameters used to describe
the shape and multipoles of rotating NSs, and it may also be used to infer the
eccentricity or moment of inertia of NSs in future X-ray observations.

dense matter — methods: numerical — stars: rotation

††journal: ApJ††software: RNS (Stergioulas & Friedman, 1995)

##  1 Introduction

Neutron stars (NSs) are the densest stars in the universe, offering a unique
laboratory to study supranuclear matter and gravity in the strong-field
regime. Currently, the equation of state (EoS) for the cores of NSs is still
poorly understood. Many EoS models with varying compositions and states of
dense matter have been developed, leading to significantly different
predictions of global properties for NSs (Lattimer & Prakash, 2001).
Therefore, observed global properties of NSs, such as the mass and the radius,
can be used to constrain EoS models.

Despite the fact that the global properties of NSs depend sensitively on the
EoS models, there exist EoS-insensitive relations that connect various
quantities of NSs. These relations are said to be universal because they are
insensitive to EoS models to a high degree of accuracy. For instance, a
universal relation connecting the frequency and damping time of the
quadrupolar $f$ mode to the mass and moment of inertia of NSs was discovered
by Lau et al. (2010). Yagi & Yunes (2013a) found the famous I-Love-Q relation
for slowly rotating NSs, which links the mass, the moment of inertia, the
tidal Love number, and the spin-induced quadrupole moment. Universal relations
for NSs are of great significance in both astrophysics and fundamental
physics. By providing EoS-insensitive connections between different
quantities, these relations allow us to extract global properties of NSs with
higher accuracy, and help us study the inverse problem of determining the EoS.
The universal relations are also a probe for non-perfect fluid inside NSs. For
instance, it has been shown that anisotropic pressure (Yagi & Yunes, 2015),
strong magnetic fields (Haskell et al., 2014), and ultra-high elasticity (Lau
et al., 2017, 2019) can affect the global structure of NSs and potentially
break the I-Love-Q universal relation. Moreover, universal relations can break
the degeneracy between gravity theories and the uncertainties in EoS, making
NSs the ideal laboratories to test gravity (Shao & Yagi, 2022). We refer
readers to Yagi & Yunes (2013b), Doneva & Pappas (2018), and references
therein for reviews.

The exploration of the universal relations for rotating NSs has garnered lots
of attention since the discovery of the I-Love-Q relation. The calculations of
the I-Q relation were quickly extended to fast rotation by Doneva et al.
(2013), it was shown that the universality of the relation is lost and becomes
increasingly EoS-dependent as the spin frequency increases. However, Pappas &
Apostolatos (2014) and Chakrabarti et al. (2014) demonstrated that the I-Q
relation remains universal if dimensionless quantities are used to
characterize the spin amplitude instead of the spin frequency $f$. Pappas &
Apostolatos (2014) and Yagi et al. (2014) discovered that the first four
multipole moments of rotating NSs are universal to some extent. This relation
allows for a more accurate description of the spacetime geometry around a NS
with fewer parameters. Additionally, Luk & Lin (2018) found another universal
relation connecting the radius and orbital frequency of the innermost stable
circular orbit (ISCO) to the mass and spin frequency of rotating NSs.

Apart from multipoles and the ISCO, the oblate shape is also important. In the
canonical pulse-profile modelling of X-ray pulsars, photons are emitted from
an oblate surface and assumed to propagate in a Schwarzschild background,
which is the so-called “Oblate $+$ Schwarzschild” (O $+$ S) approximation. To
give an analytical formula to describe the rotation-induced oblateness,
Morsink et al. (2007) and AlGendy & Morsink (2014) parametrized the oblate
shape withe the spin frequency and compactness of NSs. Recently, Silva et al.
(2021b) developed a more accurate fitting formula compared to Morsink et al.
(2007); AlGendy & Morsink (2014), which better describes the large deformation
of the surface for very rapid rotation. These fitting formulas can capture the
shape of NSs at a wide range of spin frequencies, compactnesses, and EoSs
(i.e., universal to some extent). In slow rotation, Bauböck et al. (2013) also
explored the universal relation of rotating NSs. They showed that both the
moment of inertia and the surface eccentricity can be approximately
represented by a single parameter, the compactness. Frieben & Rezzolla
(2012)uncovered quasi-universal relations relating surface distortion to spin
frequency and magnetic field, which can be used to calculate surface
distortion up to significant levels of rotation and magnetization.

In this paper, we discover a new universal relation between the surface
eccentricity and the moment of inertia for rotating NSs. The paper is
structured as follows. In Sec. 2, we provide a definition of multipoles and
eccentricity in the slow rotation approximation and present the universal
relation for both conventional NSs and QSs. In Sec. 3, we investigate the
universal relation for fast rotating NSs. Discussion of possible applications
and connections of the new universal relation to early work is shown in Sec.
4. Throughout the paper, we use geometric units with $G=c=1$.

##  2 A new universal relation in the slow-rotation approximation

###  2.1 Multipole moments and shape parameters

To study the universal relation, we first give an overview of the structures
and shape parameters of slowly rotating NSs. Following Hartle (1967) and
Hartle & Thorne (1968), we construct these stars by solving the Einstein
equations perturbatively in a slow-rotation expansion to quadratic order in
the spin. At the zeroth order in spin, we obtain the mass $M$ and the radius
$R$ of the non-rotating background. At the first order in spin, we extract the
angular momentum $J$, from which we can define the moment of inertia $I$ and
the dimensionless spin $\chi$ as $I\equiv{J}/{\Omega}$ and
$\chi\equiv{J}/{M^{2}}$, where $\Omega$ is the angular frequency of the
rotating star. Universal relations usually connect dimensionless quantities.
The dimensionless moment of inertia $\bar{I}$ is usually defined as
${\bar{I}}\equiv{I}/{M^{3}}$. At the second order in spin, the star is
deformed into an oblate shape, and we get the spin-induced quadrupole moment
$Q\equiv-{J^{2}}/{M}-{8}KM^{3}/5$. The parameter $K$ depends on the EoS of NSs
and equals to zero for Kerr black holes according to the no hair theorem. The
dimensionless quadrupole moment is defined as
$\bar{Q}\equiv-{Q}/{M^{3}\chi^{2}}$. The I-Q relation connects the
dimensionless quantities $\bar{I}$ and $\bar{Q}$. The exterior spacetime of a
slowly rotating NS can be fully described up to the quadratic order in spin by
the mass $M$, the angular momentum $J$, and the quadrupole moment $Q$ (Hartle
& Thorne, 1968).

Observationally, some observation of a rotating NS depends on the geometry of
its surface. We use the eccentricity $e_{\rm s}$ to describe the oblate shape
of a NS,

${e}_{\rm s}\equiv\sqrt{\left(\frac{R_{\mathrm{\rm eq}}}{R_{\mathrm{\rm
p}}}\right)^{2}-1}\,,$ (1)

where $R_{\rm eq}$ and $R_{\rm p}$ are the equatorial and polar radii in a
specific coordinate. In the Hartle-Thorne coordinate, the isodensity surface
at radial coordinate $r$ in the non-rotating star is displaced to

$r\to r+\xi_{0}(r)+\xi_{2}(r)P_{2}(\cos\theta)\,,$ (2)

in the rotating configuration, where $\xi_{0}$ and $\xi_{2}$ are spherical and
quadrupole displacements respectively, and $P_{2}(\cos\theta)$ is the Legendre
polynomial. Combining Eqs. (1–2), we get the surface eccentricity in the
Hartle-Thorne coordinate as

$e_{\rm HT}=\left[-3\left(\xi_{2}(R)/R\right)\right]^{1/2}\,.$ (3)

Equation (2) describes the isodensity surface in a particular coordinate
system. By embedding the isodensity surface into a three-dimensional flat
space (denoted by $r^{*}$, $\theta^{*}$, $\phi^{*}$), Hartle & Thorne (1968)
found an invariant parametrization of the oblate surface. To the second order
of the spin, the desired surface is a spheroid with

$\displaystyle r^{*}\left(\theta^{*}\right)$ $\displaystyle=r+\xi_{0}(r)$
$\displaystyle+\Big{\\{}\xi_{2}(r)+r\left[v_{2}(r)-h_{2}(r)\right]\Big{\\}}P_{2}\left(\cos\theta^{*}\right)\,.$
(4)

Here $v_{2}$ and $h_{2}$ are metric functions at the second order in spin. The
eccentricity of the stellar surface embedded in flat space is then given by

$e_{*}=\big{\\{}-3\big{[}v_{2}(R)-h_{2}(R)+\xi_{2}(R)/R\big{]}\big{\\}}^{1/2}\,,$
(5)

where the superscript “$*$” denotes the eccentricity observed in the flat
space.

###  2.2 A universal relation for the eccentricity of NSs

The universal relation that we discovered connects the quantity $e_{\rm
s}/R\Omega$ and the dimensionless moment of inertia $\bar{I}$. Samiliar to the
I-Q relation (Yagi & Yunes, 2013a) and the three hair relation for the
multipole moments (Yagi et al., 2014), the normalization factors $M$ and $R$
are quantities of the non-rotating background in the slow-rotation
approximation. For convenience, we define a dimensionless radius,
$\hat{R}\equiv{R\Omega}$. We have verified that the universal relation exists
for both the eccentricity in the Hartle-Thorne coordinate, $e_{\rm HT}$, and
the eccentricity of the embedding surface, $e_{*}$. In the following, we use
$e_{\rm*}$ to illustrate the results.

Figure 1: The mass-radius relation for selected EoS models of NSs (solid) and
QSs (dashed). The $1$-$\sigma$ regions of the mass measurements of PSR
J0348$+$0432 (Antoniadis et al., 2013) and PSR J0740$+$6620 (Fonseca et al.,
2021) are illustrated.

We first study the universal relation for conventional NSs. Our selection of
realistic EoSs includes BSK21 (Goriely et al., 2010), AU (Wiringa et al.,
1988), HLPS (Hebeler et al., 2013), PAL1 (Prakash et al., 1988), APR (Akmal et
al., 1998), SLy4 (Douchin & Haensel, 2001), MS0 (Mueller & Serot, 1996), and
ENG (Engvik et al., 1994). As shown in Fig. 1, these models cover a wide range
in the mass-radius diagram of static NSs, and all of them have maximal NS mass
larger than $2\,M_{\odot}$. Although the very stiff EoSs MS0 and PAL1 have
been ruled out by the tidal deformability from GW170817 (Abbott et al., 2017),
we include them to demonstrate that the universality exists for a large family
of EoSs.

Our universal relation is described with great accuracy by

$\frac{e_{*}}{\hat{R}}=\sum_{k=0}^{3}a_{k}\left(\ln\bar{I}\right)^{k}\,,$ (6)

where $a_{k}$’s are fitting coefficients with $a_{0}=-0.855572$,
$a_{1}=2.185502$, $a_{2}=-0.428061$, $a_{3}=0.051177$. Note that we use the
dimensionless moment of inertia $\bar{I}\leq 100$, which corresponds to
$M\gtrsim 0.5\,M_{\odot}$ for selected models. We define the relative
deviation to Eq. (6) as

$\Delta=\frac{e_{*}/\hat{R}-(e_{*}/\hat{R})_{\rm fit}}{(e_{*}/\hat{R})_{\rm
fit}}\,.$ (7)

As shown in Fig. 2, the relative deviation to the universal relation is
smaller than $1\%$ for selected models of EoSs. NSs with $M\gtrsim
1\,M_{\odot}$ are more relevant for astrophysics, and in this case, the
dimensionless moment of inertia $\bar{I}\lesssim 30$ for selected EoS models,
and the universal relation takes a simpler form,

$\frac{e_{*}}{\hat{R}}=0.11418+1.04115\ln\bar{I}\,.$ (8)

The relative deviation to this relation is less than $\sim 1\%$.

Figure 2: The $e_{\rm*}/\hat{R}$-$\bar{I}$ universal relation for slowly-
rotating models. The upper panel shows the fitted universal relations for both
NSs and QSs. The middle (lower) panel presents the relative deviation for NSs
(QSs).

For QSs, we use the phenomenological MIT bag model to describe the quark
matter. This model assumes a nearly equal number of $u$, $d$, $s$ quarks and a
small fraction of electrons confined within a bag of vacuum energy density $B$
(Farhi & Jaffe, 1984; Witten, 1984). We account for the mass of the $s$ quark,
$m_{\rm s}$, and include the quark-gluon interaction to the lowest order in
$\alpha_{c}=g^{2}/4\pi$. To investigate the universal relation for QSs, we
employ six different EoSs with varying combinations of $m_{\rm s}$,
$\alpha_{\rm c}$, and $B$ in Table 1. The resulting mass-radius relation is
displayed in Fig. 1.

Table 1: Parameters for QSs in the MIT bag model. Model | $B\,(\rm MeV\,fm^{-3})$ | $m_{\rm s}\,(\rm MeV)$ | $\alpha_{\rm c}$
---|---|---|---
SQM1 | 80 | 100 | 0
SQM2 | 80 | 50 | 0.1
SQM3 | 70 | 150 | 0
SQM4 | 70 | 50 | 0.3
SQM5 | 60 | 0 | 0
SQM6 | 60 | 100 | 0.4

The relation between $e_{*}/\hat{R}$ and $\bar{I}$ is different from that of
NSs, but it is still universal and can be well fitted by

$\frac{e_{*}}{\hat{R}}=\sum_{k=0}^{4}b_{k}\left(\ln\bar{I}\right)^{k}\,,$ (9)

with coefficients $b_{0}=-1.499749$, $b_{1}=2.911859$, $b_{2}=-0.749237$,
$b_{3}=0.137057$, and $b_{4}=-0.007801$. Interestingly, the deviation from the
universal relation for QSs is much smaller than that for NSs, with relative
deviation less than $\sim 0.1\%$, as shown in the lower panel of Fig. 2. For
QSs in our study, the condition for $M\gtrsim 1\,M_{\odot}$ corresponds to
$\bar{I}\lesssim 20$. Within this range, the universal relation can be
approximated by a simpler fitting formula,

$\frac{e_{*}}{\hat{R}}=-0.043372+1.210546\ln(\bar{I}-0.470579)\,,$ (10)

The relative deviation from this fitting formula is less than $\sim 0.3\%$.

Compared to the parametrization in Bauböck et al. (2013), the new universal
relation that we propose incorporates an extra parameter, namely the moment of
inertia, in addition to the parameters $R$, $M$, and $\Omega$. But the new
universal relation is much tighter than that of Bauböck et al. (2013).

##  3 Universal relation for fast rotating NSs

Fast rotation is relevant for sub-millisecond pulsars, nascent NSs after
supernovae, and NSs formed in binary NS mergers. Rapid rotation causes NSs to
develop a more obvious oblate shape. In this section, we explore the universal
relation for rapidly rotating NSs using the RNS code developed by Stergioulas
& Friedman (1995).

The RNS code uses a quasi-isotropic coordinate system to represent the line
element of the stationary axisymmetric spacetime,

$\displaystyle{\rm d}s^{2}=$ $\displaystyle-e^{2\nu}{\rm
d}t^{2}+B^{2}r^{2}\sin^{2}\theta e^{-2\nu}({\rm d}\phi-\omega{\rm d}t)^{2}$
$\displaystyle+e^{2(\xi-\nu)}\left({\rm d}r^{2}+r^{2}{\rm
d}\theta^{2}\right),$ (11)

where $\nu$, $B$, $\omega$, and $\xi$ are metric functions that depend on $r$
and $\theta$. Assuming a perfect fluid and uniform rotation, we obtain the
stellar structure and spacetime metric. The conserved angular momentum $J$ can
be computed from a volume integration over the matter field. The moment of
inertia $I$ and the dimensionless spin have the same definition as before. The
quadrupole moment $Q$ can be obtained from the asymptotic expansion of the
metric functions. The surface eccentricity is formally given by Eq. (1), with
the eccentricity $e_{i}$, equatorial radius $R_{\rm eq}^{i}$, and polar radius
$R_{\rm q}^{i}$ defined in the quasi-isotropic coordinate. Note that, unlike
in the slow-rotation approximation, the normalization factor $M$ is the mass
for the rotating configuration, and $\hat{R}\equiv R_{\rm eq}^{i}\Omega$.

Figure 3: The $e_{i}/\hat{R}$-$\bar{I}$ universal relation for fast-rotating
models with the dimensionless spin $\chi=0.6$. The upper panel shows the
fitted universal relations for both NSs and QSs. The middle (lower) panel
presents the relative deviation for NSs (QSs).

To study universal relations for rapidly rotating NSs, it is necessary to use
a suitable parameter to characterize their spin amplitude. As demonstrated by
Doneva et al. (2013), if one uses the spin frequency $f$ as the parameter, the
I-Q relation for fast rotating NSs is lost. Similarly, the universal relation
that we discovered also breaks down for fixed spin frequencies. However,
Pappas & Apostolatos (2014) and Chakrabarti et al. (2014) found that the I-Q
relation is still universal for fast rotating NSs if one chooses dimensionless
spin parameters such as $\chi$, $Mf$, and $Rf$, instead of the dimensionful
$f$. Inspired by their work, we use $\chi$ to characterize the spin amplitude
and find that the $e_{i}/\hat{R}$-$\bar{I}$ relation for both conventional NSs
and strange QSs is still universal.

According to Lo & Lin (2011), the maximum value of the dimensionless spin
parameter $\chi$ for NSs rotating at the Keplerian frequency is about 0.7 for
various EoS models. This limit is nearly independent of the mass of the NS if
the mass is larger than $1\,M_{\odot}$. However, for QSs in the MIT bag model,
the spin parameter can be larger than unity and does not have a universal
upper limit. Its value also depends strongly on the bag constant and the mass
of the star. Therefore, we generate data points in the regime $0.2\leq\chi\leq
0.6$ for NSs and $0.2\leq\chi\leq 0.8$ for QSs.

Figure 4: The surface of the fitting formula Eq. (12) for NSs (blue) and QSs
(red). Models of NSs and QSs fall on two well-defined surfaces.

In Fig. 3, we show the universal relation for both NSs and QSs with
$\chi=0.6$. The relative errors are on the order of $0.3\%$. More explicitly,
the $e_{\rm i}/\hat{R}$-$\bar{I}$ relation with a dependence on $\chi$ can be
fitted by

${\frac{e_{\rm
i}}{\hat{R}}=\sum_{i,j}\mathcal{A}_{ij}\chi^{i}\log^{j}\bar{I}\,,}$ (12)

where the numerical coefficients $\mathcal{A}_{ij}$ for NSs and QSs are given
in Table 2. The maximum relative errors of the fitting formula are on the
order of $1\%$ for NSs and $0.3\%$ for QSs. In Fig. 4, we present the surfaces
described by the fitting formula Eq. (12). For a given $\chi$ and $\bar{I}$,
the value of $e_{\rm i}/\hat{R}$ for QSs is always larger than that for NSs.
At the maximum mass limit, the two surfaces become closest.

Table 2: Numerical coefficients for the two-parameter fitting formula Eq. (12). $i=$ | $0$ | $1$ | $2$ | $3$
---|---|---|---|---
Coefficients for neutron stars
$\mathcal{A}_{i0}$ | $2.149039$ | $-0.146205$ | $-1.867329$ | $5.65915$
$\mathcal{A}_{i1}$ | $1.23962$ | $2.21818$ | $0.942935$ | $-2.87271$
$\mathcal{A}_{i2}$ | $-0.165024$ | $-2.63504$ | $2.28603$ | $-2.53576$
$\mathcal{A}_{i3}$ | $0.158013$ | $1.04597$ | $-1.80829$ | $2.23285$
Coefficients for quark stars
$\mathcal{A}_{i0}$ | 0.98273 | $-5.06930$ | 12.9114 | 5.65915
$\mathcal{A}_{i1}$ | 4.29319 | 16.3498 | $-48.8296$ | 30.4711
$\mathcal{A}_{i2}$ | $-3.12737$ | $-12.9528$ | 50.1446 | $-37.8440$
$\mathcal{A}_{i3}$ | 1.30675 | 3.27539 | $-16.5473$ | 13.8336

In our study, we have defined the eccentricity in the quasi-isotropic
coordinate system, where the radial coordinate $r$ corresponds to the
isotropic Schwarzschild coordinate in the limit of zero spin. However, Morsink
et al. (2007), AlGendy & Morsink (2014), and Silva et al. (2021b) define the
eccentricity differently. We know that circles centered about the symmetric
axis have circumference $2\pi\bar{r}$, where $\bar{r}$ is related to $r$ and
$\theta$ by

${\bar{r}=e^{-\nu(r,\theta)}B(r,\theta)r\sin\theta=r_{c}(r,\theta)\sin\theta\,.}$
(13)

Here $r_{c}$ corresponds to the Schwarzschild coordinate in the limit of zero
spin. By using $r_{c}$, the equatorial and polar radii can be defined as

${R_{\rm eq}^{c}=r_{c}(R_{\rm eq}^{i},\theta=\frac{\pi}{2})\,,\quad R_{\rm
p}^{c}=r_{c}(R_{\rm p}^{i},\theta=0)\,,}$ (14)

where $R_{\rm eq}^{c}$ is the circumferential radius of the star in the
equatorial plane. Then the surface eccentricity $e_{c}$ can be obtained from
Eq. (1).

We have found that the universality still holds for the eccentricity $e_{c}$,
but its relation differs from that of $e_{i}$. Specifically, for given EoS
families and a fixed value of $\bar{I}$, the value of $e_{i}/\hat{R}$ is
larger than $e_{c}/\hat{R}$. To compare the universal relations for these two
eccentricities, we have included an example in Fig. 5. The figure shows the
$e_{i}/\hat{R}$-$\bar{I}$ and $e_{c}/\hat{R}$-$\bar{I}$ relations for NSs and
QSs at $\chi=0.5$, which highlights the differences of the universal relations
for these two eccentricities.

Figure 5: The $e_{i}/\hat{R}$-$\bar{I}$ and $e_{c}/\hat{R}$-$\bar{I}$
relations for NSs (blue) and QSs (red). Here we take the dimensionless spin
$\chi=0.5$.

##  4 Discussion

The universal relation between $e_{\rm s}/\hat{R}$ and $\bar{I}$ adds a new
tight universal relation to the known ones. Here, $e_{\rm s}$ is the surface
eccentricity formally defined in Eq. (1), including $e_{\rm HT}$, $e_{*}$,
$e_{i}$, and $e_{c}$ in our work. It is important to note that these different
definitions of eccentricity lead to different universal relations. Since the
I-Q relation (Yagi & Yunes, 2013a; Chakrabarti et al., 2014) connects
$\bar{I}$ to $\bar{Q}$, and the three-hair relation (Pappas & Apostolatos,
2014; Yagi et al., 2014) connects $\bar{Q}$ to two other higher-order
multipoles, the relation between $e_{\rm s}/\hat{R}$ and these normalized
multipole moments is also universal. Combined with the I-Love relation (Yagi &
Yunes, 2013a), we have a universal relation between $e_{\rm s}/\hat{R}$ and
the dimensionless tidal Love number. Moreover, the universal relation between
the $f$-mode oscillation and $\bar{I}$ (Lau et al., 2010) helps us connect
$e_{\rm s}/\hat{R}$ to the frequency and damping time of the quadrupolar $f$
mode.

Universal relations are a powerful tool to reduce modelling uncertainties and
inferring NS parameters. As we discussed before, the eccentricity of rotating
NSs is an observable and is an important input for X-ray modelling. The
oblateness induced by rotation at frequencies above $300\,\rm Hz$ produces a
geometric effect that has imprints in the pulse profile of X-ray pulsars
(Morsink et al., 2007). For some emission configurations, the oblateness
effect can rival the Doppler effect. Besides, the effects of oblateness need
to be taken into account when measuring the radii of NSs from rotationally
broadened atomic lines (Baubock et al., 2013).

The new universal relation we obtained can be used to help infer NS
properties. For example, if future X-ray observations can measure the
eccentricity and radius of pulsars, one can use the universal relation to
forecast the moment of inertia of NSs with similar masses, and use the I-Love
relation to test gravity (Silva et al., 2021a). Conversely, if the moment of
inertia is obtained through observations of, say, the Double Pulsar system
(Lattimer & Schutz, 2005; Hu et al., 2020; Kramer et al., 2021) or GWs from a
binary NS inspiral (Lau et al., 2010), our universal relation can be employed
to improve the inference of the radii of NSs through X-ray observations.

For very rapid rotation, deviations from a simple ellipse become potentially
important. Currently, telescopes like NICER observe only slowly rotating NSs,
for which approximating the shape as an ellipse is accurate enough. However,
if NICER or similar telescopes were to observe highly rapidly rotating
pulsars, the full shape function for the surface would be required. Previous
studies, such as Morsink et al. (2007), AlGendy & Morsink (2014), and Silva et
al. (2021b), have parameterized the oblate shape of NSs using the spin
frequency and compactness as parameters. In comparison to these works, our
newly discovered universal relation incorporates an additional parameter, the
moment of inertia, and only focuses on the eccentricity of the star. In the
future, it’s necessary to extend our framework to accurately fit the shape of
the star at all latitudes, especially when interpreting the observations of
rapidly rotating NSs.

We thank the anonymous referee for helpful comments, and Zexin Hu and Enping
Zhou for useful discussions. This work was supported by the National SKA
Program of China (2020SKA0120300), the National Natural Science Foundation of
China (11975027, 11991053), the Max Planck Partner Group Program funded by the
Max Planck Society, and the High-Performance Computing Platform of Peking
University.

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