# Tidal love number of a static spherically symmetric anisotropic compact star

Shyam Das<EMAIL_ADDRESS>Department of Physics, P. D. Women’s
College, Jalpaiguri 735101, West Bengal, India.  Bikram Keshari Parida
<EMAIL_ADDRESS>Department of Physics, Pondicherry University,
Kalapet, Puducherry 605014, India.  Ranjan Sharma<EMAIL_ADDRESS>Department of Physics, Cooch Behar Panchanan Barma University, Cooch Behar
736101, West Bengal, India.

###### Abstract

Tidal deformability of a coalescing neutron star subjected to an external
tidal field plays an important role in our probe for the structure and
properties of compact stars. In particular, the tidal love number provides
valuable information about the external gravitational field responsible for
deforming the star. In this article, we compute the tidal love number of a
particular class of anisotropic stars and analyze the impacts of anisotropy
and compactness on the tidal love number.

Neutron star, Gravity, Anisotropy,

###### pacs:

##  I  Introduction

Compact objects provide extreme conditions in terms of gravity and density and
thus are unique astrophysical laboratories for studying general relativity and
super nuclear-density matter. In general, compact objects exist in binaries
comprising of either two neutron stars (NS-NS binaries) or a black hole (BH)
and a neutron star (NS) (BH-NS binaries). Such binaries radiate away energy.
The merger of these objects generates huge gravitational waves which has been
experimentally verified in the recent past.

Compact stars provide perfect places for investigating the nature of particle
interactions at very high densities in a natural way [1]. Neutron stars (NS)
are compact objects of very high energy density having approximate masses
$1.5M_{\odot}$ and radii $10^{5}$ times smaller than the Sun’s radius.
Therefore, they are perfect natural systems to study nuclear matter properties
at high densities. In fact, density inside the core of a NS can be as high as
several times the density that is reached inside a heavy atomic nuclei [2].
Despite attempts of many decades, we still lack a proper understanding of the
thermodynamical behaviour inside a compact star. The extreme conditions at the
interior of a compact star comprising matter of uncertain composition have
prompted many investigators to study its gross macroscopic properties within
the framework of General Relativity. In order to understand the microscopic
properties, the macroscopic properties such as NS masses and radii have been
used as important tools to constrain its EOS.

In this article, we explore the possibility of introducing tidal deformation
as one of the astrophysically observable macroscopic properties which can be
used to study the interior of a NS [3]. Tidal effects are finite-size effects
arising on extended bodies when they are immersed in an external gravitational
field. Like any other extended object, NS are tidally deformed under the
influence of an external tidal field. The tidal deformability measures the
star’s quadrupole deformation in response to a companion perturbating star
[4]. The induced quadrupole moment of the neutron star affects the binding
energy of the system and increases the rate of emission of gravitational waves
[5, 6, 7]. The tidal deformability has a very important role in the
observation of coalescing NS with gravitational waves, and it has been used to
probe the internal structure of NS. The Tidal Love Number characterizes how
easy or difficult it would be to deform a NS away from sphericity [8], [1].
The tidal love number can be computed by following the standard methods
available in the literature [9, 10, 11, 12, 13].

Tidal properties of a NS has been observed to have a direct bearing on the
emitted gravitational wave signal. The Advanced LIGO [14] and Advanced Virgo
[15] gravitational-wave detectors have made their first observation of a
binary NS inspiral [16], an event known as GW170817. The LIGO observations
lead to the experimental insight of the Tidal Love Number [16]. Later, another
signal emitted during a neutron star binary coalescence, known as GW190425,
was detected. The latter signal was much weaker than GW170817 as it was
originated from a much greater distance [17]. Nevertheless, these observations
has helped investigators to constraint many physical properties of NS such
maximum masses and radii [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28].

For ready references of the calculation on relativistic tidal love number
(TLN), we refer to the following citations [13, 29, 30, 31]. The algorithm has
also been extended to slowly rotating extended compact objects as can be found
in references [32, 33, 31, 34, 35, 36]. As stated earlier, TLN defines the
tidal deformability of the star present in an external tidal field such as the
companion binary [9]. In gravitational wave astronomy, it is an important
parameter which affects the late-spiral GW from coalescence and thereby
provide important information about the nature of the merging objects [37].
Most importantly, TLN can hold us to constrain the EOS of the NS [37, 27].
Note that TLN of a black hole is zero [30, 29, 38, 39, 40]. For relatively
less compact objects, the dominant contribution to the tidal deformability
comes from the ‘even-parity’ quadrupole term $l$, which starts to impact the
phase of the GW signal emitted in a binary at the fifth post-Newtonian (5PN)
order [41]. The leading order (6PN) term of even-parity tidal deformability
has also been calculated [42]. The odd-parity (or gravitomagnetic or mass-
current) tidal deformability was calculated independently in ref [29] & [30].
The choice of fluid properties also affects the odd-parity tidal deformability
[31], as shown in ref [43]. The pioneer in this field was Yagi [44] who for
the first time estimated the impact of odd-parity tidal deformability on the
gravitational waves phase evolution and then extended the work by analyzing
the signal from GW170817 [45].

In our work, we develop a method to estimate the TLN for a spherically
symmetric and anisotropic relativistic star which is in static equilibrium. In
a compact object, pressures may be different in radial, and transverse
directions and the difference of radial pressure ($p_{r}$) and tangential
pressure ($p_{t}$) is defined as pressure anisotropy. Incorporating anisotropy
into the matter distribution of compact objects, several anisotropic models
have been developed and investigated which include the works of Maurya and
Gupta [46], Maurya et al [47], Pandya et al [48], Murad [49], Mafa Takisa et
al [50, 51], Sunzu et al [52, 53], Matondo and Maharaj [54], Karmakar et al
[55], Abreu et al [56], Ivanov [57], Herrera et al [58], Mak and Harko [59],
Sharma and Mukherjee [60], Harko and Mak [61], Herrera et al [62], Maharaj and
Maartens [63], Gokhroo and Mehra [64], Chaisi and Maharaj [65, 66], Thomas and
co-workers [67, 68], Das et al [69], Thirukkanesh and Maharaj [70], fully
covarient framework by Raposo et al [71], amongst others. Ruderman [72] and
Canuto [73] have shown that anisotropy may develop inside highly dense compact
stellar objects due to a variety of factors. Kippenhahn and Weigert [74]
revealed that in relativistic stars anisotropy might occur due to the
existence of a solid core or type $3A$ superfluid. Strong magnetic fields can
also generate an anisotropic pressure inside a self-gravitating body [75].
Anisotropy may also develop due to the slow rotation of fluids [76]. A mixture
of perfect and a null fluid may also be represented by an effective
anisotropic fluid model [77]. Local anisotropy may occur in astrophysical
objects for various reasons such as viscosity, phase transition [78], pion
condensation [79] and the presence of strong electromagnetic field [80]. The
factors contributing to the pressure anisotropy have also been discussed by
Dev and Gleiser [81, 82] and Gleiser and Dev [83]. Ivanov [84] pointed out
that influences of shear, electromagnetic field etc. on self-bound systems can
be absorbed if the system is considered to be anisotropic. Self-bound systems
composed of scalar fields, the so-called ‘boson stars’ are naturally
anisotropy [85]. Same is true for wormholes [86] and gravastars [87, 88] as
well. The shearing motion of the fluid can be considered as one of the reasons
for the presence of anisotropy in a self-gravitating body [89]. Bowers and
Liang [90] have extensively discussed the underlying causes of pressure
anisotropy in the stellar interior and analyzed the effects of anisotropic
stress on the equilibrium configuration of relativistic stars. The point we
would like to stress is that in the studies of relativistic compact stars, we
find it worthwhile to consider anisotropic stress rather than an isotropic
fluid distribution.

The paper has been organized as follows: In Section II, significance of tidal
love number has been discussed. Section III deals with the finding a
physically acceptable model which can be used to calculate the tidal love
number. In the section IV, using the model, the tidal love number $k_{2}$ for
different NS has been estimated. $k_{2}$ has also been calculated for a given
compactness $\mathcal{C}$ but different anisotropies $\alpha$. In Section V,
some concluding remarks have been made.

##  II  Tidal love number

We consider a static spherically symmetric neutron star (NS) immersed in an
external tidal field. In response to the tidal field, by developing a
multipolar structure, the star will be deformed by the tidal force. This kind
of situation occurs in coalescing binary systems where each component is
tidally deformed by the gravitational field of its companion. The Tidal Love
Number(TLN) characterizes the deformability of the NS away from sphericity
[91]. For mathematical simplicity, in our calculation, we shall restrict
ourselves to quadrupole moments $\mathcal{Q}_{ij}$ only. This is reasonable if
the two binary neutron stars remain sufficiently far away from each other. In
such a situation, the quadrupole moment ($l=2$) dominates over the multiple
moments. $\mathcal{Q}_{ij}$ can be related with the external tidal field
$\mathcal{E}_{ij}$ as [92]

$\displaystyle\mathcal{Q}_{ij}=-\Lambda\mathcal{E}_{ij},$ (1)

where, $\Lambda$ is the tidal deformability of the neutron star and it is
related to the tidal love number $k_{2}$ as [92],

$\displaystyle k_{2}=\dfrac{3}{2}\Lambda\,R^{-5}.$ (2)

The Tidal Love Number is dimensionless. The quadrupole fields
$\mathcal{Q}_{ij}$ and $\mathcal{E}_{ij}$ can be expanded in tensor spherical
harmonics $\mathcal{Y}_{ij}^{lm}$ as:

$\displaystyle\mathcal{E}_{ij}$
$\displaystyle=\sum_{m=-2}^{2}\mathcal{E}_{m}\mathcal{Y}_{ij}^{2m}=\mathcal{E}_{0}\mathcal{Y}_{ij}^{20}=\mathcal{E}\mathcal{Y}_{ij}^{20}\,,$
(3) $\displaystyle\mathcal{Q}_{ij}$
$\displaystyle=\sum_{m=-2}^{2}\mathcal{Q}_{m}\mathcal{Y}_{ij}^{2m}=\mathcal{Q}_{0}\mathcal{Y}_{ij}^{20}=\mathcal{Q}\mathcal{Y}_{ij}^{20}.$
(4)

In the second equality, the coordinate system was so oriented that the term
became symmetric in $\phi$. The only component that is non-vanishing is the
$m=0$ component. We can rewrite equation (1) as,

$\displaystyle\mathcal{Q}=-\Lambda\mathcal{E}.$ (5)

Now the background metric ${}^{(0)}g_{\mu\nu}(x^{\nu})$ corresponding to the
neutron star, with a small perturbation $h_{\mu\nu}(x^{\nu})$ due to external
tidal field, gets modified as,

$\displaystyle
g_{\mu\nu}\left(x^{\nu}\right)=^{(0)}g_{\mu\nu}\left(x^{\nu}\right)+h_{\mu\nu}\left(x^{\nu}\right).$
(6)

We write the background geometry of the spherical static star in the standard
form,

${}^{\left(0\right)}ds^{2}$
$\displaystyle=^{\left(0\right)}g_{\mu\nu}dx^{\mu}dx^{\nu}$
$\displaystyle=-e^{2\nu(r)}dt^{2}+e^{2\lambda(r)}dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right).$ (7)

For the linearized metric perturbation $h_{\mu\nu}$, using the method as in
Ref. [93] and [94], we restrict ourselves to static $l=2,\,m=0$ even parity
perturbation. With these assumptions, the perturbed metric becomes,

$\displaystyle\leavevmode\resizebox{195.12767pt}{}{$h_{\mu\nu}=\operatorname{diag}\left[H_{0}(r)e^{2\nu},H_{2}(r)e^{2\lambda},r^{2}K(r),r^{2}\sin^{2}\theta
K(r)\right]Y_{2m}(\theta,\phi)$}.$ (8)

For the spherically static metric (7), the stress-energy tensor is given as
[95, 96, 97, 98],

${}^{(0)}T_{\chi}^{\xi}=\left(\rho+p_{t}\right)u^{\xi}u_{\chi}+p_{t}g_{\chi}^{\xi}+\left(p_{r}-p_{t}\right)\eta^{\xi}\eta_{\chi},$
(9)

with, $u^{\xi}u_{\xi}=-1$, $\eta^{\xi}\eta_{\xi}=1$ & $\eta^{\xi}u_{\xi}=0$.

Furthermore, the energy-momentum tensor is perturbed by a perturbation tensor
$\delta T_{\chi}^{\xi}$ which is defined as,

$\displaystyle T_{\chi}^{\xi}=^{(0)}T_{\chi}^{\xi}+\delta T_{\chi}^{\xi}.$
(10)

The non-zero components of $T_{\chi}^{\xi}$ are:

$\displaystyle T_{t}^{t}$ $\displaystyle=-\dfrac{d\rho}{dp_{r}}\,\,\delta
p_{r}\,\,Y(\theta,\phi)-\rho(r),$ (11) $\displaystyle T_{r}^{r}$
$\displaystyle=\delta p_{r}(r)Y(\theta,\phi)+p_{r}(r),$ (12) $\displaystyle
T_{\theta}^{\theta}$ $\displaystyle=\frac{dp_{t}}{dp_{r}}\delta
p_{r}(r)Y(\theta,\phi)+p_{t}(r),$ (13) $\displaystyle T_{\phi}^{\phi}$
$\displaystyle=\frac{dp_{t}}{dp_{r}}\delta p_{r}(r)Y(\theta,\phi)+p_{t}(r).$
(14)

With these perturbed quantities, we can write down the perturbed Einstein
Field Equations as

$\displaystyle G_{\chi}^{\xi}=8\pi T_{\chi}^{\xi}.$ (15)

Where, the Einstein tensor $G_{\chi}^{\xi}$ is calculated using the metric
$g_{\chi\xi}$.

###  II.1 Derivation of master equation and expression for tidal love number

Using the background field equations
${}^{(0)}G_{\chi}^{\xi}=8\pi^{(0)}T_{\chi}^{\xi}$, we obtain the following
results:

${}^{(0)}G_{t}^{t}$ $\displaystyle=8\pi^{(0)}T_{t}^{t},$
$\displaystyle\Rightarrow\lambda^{\prime}(r)$ $\displaystyle=\frac{8\pi
r^{2}e^{2\lambda(r)}\rho(r)-e^{2\lambda(r)}+1}{2r},$ (16) ${}^{(0)}G_{r}^{r}$
$\displaystyle=8\pi^{(0)}T_{r}^{r},$ $\displaystyle\Rightarrow\nu^{\prime}(r)$
$\displaystyle=\frac{8\pi
r^{2}p_{r}(r)e^{2\lambda(r)}+e^{2\lambda(r)}-1}{2r}.$ (17)

Note that $\nabla_{\xi}^{(0)}T_{\chi}^{\xi}=0$. Choosing $\xi=r$, we obtain,

$\displaystyle
p_{r}^{\prime}(r)=\frac{-rp_{r}(r)\nu^{\prime}(r)-2p_{r}(r)+2p_{t}(r)-r\rho(r)\nu^{\prime}(r)}{r}.$
(18)

For the perturbed metric, using Einstein equations (15), we get the following
results,

$\displaystyle G_{\theta}^{\theta}-G_{\phi}^{\phi}=0\Rightarrow
H_{0}(r)=H_{2}(r)=H(r),$ (19) $\displaystyle G_{r}^{\theta}=0\Rightarrow
K^{\prime}=H^{\prime}+2H\nu^{\prime},$ (20) $\displaystyle
G_{\theta}^{\theta}+G_{\phi}^{\phi}=8\pi(T_{\theta}^{\theta}+T_{\phi}^{\phi}),$
$\displaystyle\Rightarrow\delta
p_{r}=\frac{H(r)e^{-2\lambda(r)}\left(\lambda^{\prime}(r)+\nu^{\prime}(r)\right)}{8\pi\frac{dp_{t}}{dp_{r}}r}.$
(21)

Now, using the identity,

$\displaystyle\leavevmode\resizebox{195.12767pt}{}{$\dfrac{\partial^{2}Y(\theta,\phi)}{\partial\theta^{2}}+cot(\theta)\dfrac{\partial
Y(\theta,\phi)}{\partial\theta}+\csc^{2}(\theta)\dfrac{\partial^{2}Y(\theta,\phi)}{\partial\phi^{2}}=-6Y(\theta,\phi)$},$

eqn (16), (17), (18), (19), (20) & (21), we obtain the master equation for
$H(r)$ as,

$\displaystyle-\frac{1}{e^{-2\lambda(r)}Y(\theta,\phi)}\left[G_{t}^{t}-G_{r}^{r}\right]=-\frac{8\pi}{e^{-2\lambda(r)}Y(\theta,\phi)}\left[T_{t}^{t}-T_{r}^{r}\right]$
$\displaystyle\Rightarrow
H^{\prime\prime}(r)+\mathcal{R}H^{\prime}(r)+\mathcal{S}H(r)=0,$ (22)

Where,

$\displaystyle\mathcal{R}=-\left(\frac{-e^{2\lambda(r)}-1}{r}-4\pi
re^{2\lambda(r)}(p_{r}(r)-\rho(r))\right),$ (23)

$\displaystyle\mathcal{S}$
$\displaystyle=-\left(\frac{4e^{2\lambda(r)}+e^{4\lambda(r)}+1}{r^{2}}+64\pi^{2}r^{2}p_{r}(r)^{2}e^{4\lambda(r)}+16\pi
e^{2\lambda(r)}\left(p_{r}(r)\left(e^{2\lambda(r)}-2\right)-p_{t}(r)-\rho(r)\right)+\right.$
$\displaystyle\left.\frac{-4\pi\frac{d\rho}{dp_{r}}e^{2\lambda(r)}(p_{r}(r)+\rho(r))-4\pi
e^{2\lambda(r)}(p_{r}(r)+\rho(r))}{\frac{dp_{t}}{dp_{r}}}\right).$ (24)

The exterior region of the static spherically symmetric star will be described
by Schwarzschild metric and hence by setting, $\rho=0,\,p_{r}=0,\,p_{t}=0$ and
$e^{2\lambda}=1/(1-2M/r)$, the master equation (22) takes the form,

$\displaystyle\leavevmode\resizebox{203.80193pt}{}{$-H^{\prime\prime}(r)-\frac{2(M-r)H^{\prime}(r)}{r(2M-r)}+\frac{2H(r)\left(2M^{2}-6Mr+3r^{2}\right)}{r^{2}(r-2M)^{2}}=0$}.$
(25)

The solution to this second-order differential equation (25) is obtained as,

$\displaystyle H(r)=$
$\displaystyle\frac{c_{2}\left(-2M\left(2M^{3}+4M^{2}r-9Mr^{2}+3r^{3}\right)-3r^{2}(r-2M)^{2}\log\left(\frac{r}{M}-2\right)+3r^{2}(r-2M)^{2}\log\left(\frac{r}{M}\right)\right)}{2M^{2}r(2M-r)}$
$\displaystyle+\frac{3c_{1}r(2M-r)}{M^{2}}.$ (26)

Where, $c_{1}$ and $c_{2}$ are integration constants. In order to get the
expression for these constants, let us make a series expansion of the equation
(26),

$\displaystyle
H(r)=-\frac{3c_{1}r^{2}}{M^{2}}+\frac{6c_{1}r}{M}-\frac{c_{2}\left(8M^{3}\right)}{5r^{3}}+\mathcal{O}\left(\left(\frac{1}{r}\right)^{4}\right).$
(27)

Now in the star’s local asymptotic rest frame, at large $r$ the metric
coefficient $g_{tt}$ is given by [99, 92, 100],

$\displaystyle\frac{\left(1-g_{tt}\right)}{2}=$
$\displaystyle-\frac{M}{r}-\frac{3\mathcal{Q}_{ij}}{2r^{3}}\left(n^{i}n^{j}-\frac{1}{3}\delta^{ij}\right)+\mathcal{O}(\frac{1}{r^{3}})$
$\displaystyle+\frac{1}{2}\mathcal{E}_{ij}x^{i}x^{j}+\mathcal{O}(r^{3}),$ (28)

Where, $n^{i}=x^{i}/r$. Matching the asymptotic solution using equation (27)
together with the expansion of equation (28) and using equation (1), we have,

$\displaystyle c_{1}=-\frac{M^{2}\mathcal{E}}{3},\quad
c_{2}=\frac{15\mathcal{Q}}{8M^{3}}.$ (29)

Finally, the expression for tidal love number $k_{2}$ can be obtained by using
equations (2), (29) and (26) and also using the expression for $H(r)$ and its
derivatives at the star’s surface $r=R$ as,

$\displaystyle
k_{2}=[8(1-2\mathcal{C})^{2}\mathcal{C}^{5}(2\mathcal{C}(\mathit{y}-1)-\mathit{y}+2)]/X,$
(30)

Where,

$\displaystyle X$
$\displaystyle=(5(2\mathcal{C}(\mathcal{C}(2\mathcal{C}(\mathcal{C}(2\mathcal{C}(\mathit{y}+1)+3\mathit{y}-2)-11\mathit{y}+13)+3(5\mathit{y}-8))-3\mathit{y}+6)$
$\displaystyle\left.\left.+3(1-2\mathcal{C})^{2}(2\mathcal{C}(\mathit{y}-1)-\mathit{y}+2)\log\left(\frac{1}{\mathcal{C}}-2\right)-3(1-2\mathcal{C})^{2}(2\mathcal{C}(\mathit{y}-1)-\mathit{y}+2)\log\left(\frac{1}{\mathcal{C}}\right)\right)\right),$
(31)

Note that $\mathcal{C}=\frac{M}{R}$ and $\mathit{y}$ depend on $r,\,H(r)$ and
it’s derivatives evaluated at $R$ in the form,

$\displaystyle\mathit{y}=\dfrac{rH^{\prime}(r)}{H(r)}_{R}.$ (32)

To calculate the tidal love number $k_{2}$ for a particular compact star, we
need to specify a model which we can be utilized to calculate $y$ and
subsequently $k_{2}$ for a particular NS of given mass $M$ and radius $R$.

##  III  CHOOSING A PHYSICALLY ACCEPTABLE MODEL

###  III.1 Einstein field equations:

To describe the interior of a static and spherically symmetric relativistic
star, we write the line element in coordinates $(x^{a})=(t,r,\theta,\phi)$ as

$ds^{2}=-e^{2\nu(r)}dt^{2}+e^{2\lambda(r)}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}{\theta}d\phi^{2}).$
(33)

We also assume an anisotropic matter distribution for which the energy-
momentum tensor is assumed in the form,

$T_{ij}=\mbox{diag}(-\rho,~{}p_{r},~{}p_{t},~{}p_{t}).$ (34)

The energy density $\rho$, the radial pressure $p_{r}$ and the tangential
pressure $p_{t}$ are measured relative to the comoving fluid velocity
$u^{i}=e^{-\nu}\delta^{i}_{0}.$ For the line element (33), the independent set
of Einstein field equations are then obtained as,

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\frac{1}{r^{2}}\left[r(1-e^{-2\lambda})\right]^{\prime},$ (35)
$\displaystyle p_{r}$ $\displaystyle=$
$\displaystyle-\frac{1}{r^{2}}\left(1-e^{-2\lambda}\right)+\frac{2\nu^{\prime}}{r}e^{-2\lambda},$
(36) $\displaystyle p_{t}$ $\displaystyle=$ $\displaystyle
e^{-2\lambda}\left(\nu^{\prime\prime}+\nu^{\prime
2}+\frac{\nu^{\prime}}{r}-\nu^{\prime}\lambda^{\prime}-\frac{\lambda^{\prime}}{r}\right),$
(37)

where primes (′) denote differentiation with respect to $r$. In the field
equations (35)-(37), we have assumed $8\pi G=1=c$. The system of equations
determines the behaviour of the gravitational field of an anisotropic
imperfect fluid sphere. The mass contained within a radius $r$ of the sphere
is defined as,

$m(r)=\frac{1}{2}\int_{0}^{r}\omega^{2}\rho(\omega)d\omega.$ (38)

We define, $\Delta=p_{t}-p_{r}$ as the measure of anisotropy. The anisotropic
stress will be directed outward (repulsive) when $p_{t}>p_{r}$ (i.e.,
$\Delta>0$) and inwards when $p_{t}<p_{r}$ (i.e., $\Delta<0$).

###  III.2  A particular anisotropic model

To calculate the tidal love number, we choose a particular model which is an
anisotropic generalization of the Korkina and Orlyanskii solution III obtained
earlier by [101]. To examine the physical acceptability of the solution, we
first write the variables which are obtained as,

$\displaystyle e^{2\nu}$ $\displaystyle=$ $\displaystyle
A^{2}(1+aCr^{2})^{2},$ (39) $\displaystyle e^{2\lambda}$ $\displaystyle=$
$\displaystyle\Big{[}1-BCr^{2}(1+3aCr^{2})^{-2/3}-\alpha
Cr^{2}(1+aCr^{2})^{-1}$ (40)
$\displaystyle\times(1+3aCr^{2})^{-2/3}\Big{]}^{-1},$ $\displaystyle\Delta$
$\displaystyle=$ $\displaystyle\frac{\alpha
C^{2}ar^{2}(1+3aCr^{2})^{1/3}}{(1+aCr^{2})^{3}}.$ (41)

The line element (33) then takes the form,

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-A^{2}(1+aCr^{2})^{2}dt^{2}$ (42)
$\displaystyle+\Big{[}1-BCr^{2}(1+3aCr^{2})^{-2/3}-\alpha
Cr^{2}(1+aCr^{2})^{-1}$
$\displaystyle\times(1+3aCr^{2})^{-2/3}\Big{]}^{-1}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}{\theta}d\phi^{2}).$

For an isotropic sphere ($\alpha=0$), if we set $B=0$ and $C=1$, the metric
(42) reduces to,

$ds^{2}=-A^{2}(1+ar^{2})^{2}dt^{2}+dr^{2}+r^{2}(d\theta^{2}+\sin^{2}{\theta}d\phi^{2}).$
(43)

which is the Korkina and Orlyanskii solution III[102]. In other words, the
solution (42) obtained by [101] is an anisotropic generalization of the
solution of Korkina and Orlyanskii[102].

###  III.3 Physical acceptability of the solution

We now examine the physical acceptability of our solution:

  1. i.

In this model, we have
$(e^{2\nu(r)})^{\prime}_{r=0}=(e^{2\lambda(r)})^{\prime}_{r=0}=0$ and
$e^{2\nu(0)}=A^{2},~{}e^{2\lambda(0)}=1$; these imply that the metric is
regular at the centre $r=0$.

  2. ii.

Since $\rho(0)=3\,C(B+\alpha)$ and $\displaystyle
p_{r}(0)=p_{t}(0)=C(4a-B-\alpha)$, the energy density, radial pressure and
tangential pressure will be non-negative at the centre if we choose the
parameters satisfying the condition $a>\frac{B+\alpha}{4}$.

  3. iii.

The interior solution (33) should be matched to the exterior Schwarzschild
metric

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\left(1-\frac{2M}{r}\right)dt^{2}+\left(1-\frac{2M}{r}\right)^{-1}dr^{2}$
(44) $\displaystyle+r^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2}),$

across the boundary boundary of the star $r=R$, where $M$ is the total mass of
the sphere which can be obtained directly from Eq.(38) as

$\displaystyle
M=m(s)=\frac{CR^{3}(B+aBCR^{2}+\alpha)}{2(1+2aCR^{2})(1+3aCR^{2})^{\frac{2}{3}}}.$

Matching of the line elements (33) and (44) at the boundary $r=R$ yields,

$\displaystyle\left(1-\frac{2M}{R}\right)$ $\displaystyle=$ $\displaystyle
1-\frac{\alpha CR^{2}}{(1+aCR^{2})(1+3aCR^{2})^{\frac{2}{3}}}$ (45)
$\displaystyle-\frac{BCR^{2}}{(1+3aCR^{2})^{\frac{2}{3}}},$
$\displaystyle\left(1-\frac{2M}{R}\right)$ $\displaystyle=$ $\displaystyle
A^{2}(1+aCR^{2})^{2},$ (46)

Making use of the junction conditions, the constants $A,\,B,\,C$ are
determined as

$\displaystyle A$ $\displaystyle=$
$\displaystyle\frac{(5M-2R)}{2\sqrt{R(R-2M)}},$ (47) $\displaystyle C$
$\displaystyle=$ $\displaystyle\frac{M}{aR^{2}(2R-5M)},$ (48) $\displaystyle
B$ $\displaystyle=$
$\displaystyle-\frac{(5M-2R)[2^{\frac{8}{3}}a(\frac{M-R}{5M-2R})^{\frac{2}{3}}(2M-R)+R\alpha]}{2R(2M-R)}.$
(49)

  4. iv.

The gradient of density, radial pressure and tangential pressure are
respectively obtained as,

$\displaystyle\frac{d\rho}{dr}$ $\displaystyle=$
$\displaystyle\frac{(2aC^{2}r(-10B(1+aCr^{2})^{4}+(-15+aCr^{2}(-53+aCr^{2}(-49+5aCr^{2})))\alpha))}{((1+aCr^{2})^{3}(1+3aCr^{2})^{\frac{8}{3}})},$
(50) $\displaystyle\frac{dp_{r}}{dr}$ $\displaystyle=$
$\displaystyle-\frac{1}{((1+aCr^{2})^{3}(1+3aCr^{2})^{\frac{5}{3}}))}\times(((2aC^{2}r(-2B(1+aCr^{2})(-1+5a^{2}C^{2}r^{4})+a(4(1+3aCr^{2})^{\frac{2}{3}}$
(51)
$\displaystyle+Cr^{2}(a(16(1+3aCr^{2})^{2/3}+Cr^{2}(12a(1+3aCr^{2})^{2/3}-25\alpha))-8\alpha))+\alpha))),$
$\displaystyle\frac{dp_{t}}{dr}$ $\displaystyle=$
$\displaystyle\frac{1}{(1+aCr^{2})^{4}(1+3aCr^{2})^{\frac{5}{3}}}\times(4aC^{2}r(B(1+aCr^{2})^{2}(-1+5a^{2}C^{2}r^{4})+a(-2(1+3aCr^{2})^{\frac{2}{3}}$
(52)
$\displaystyle+Cr^{2}(6\alpha+a(-10(1+3aCr^{2})^{\frac{2}{3}}+Cr^{2}(17\alpha+a(-14(1+3aCr^{2})^{\frac{2}{3}}$
$\displaystyle+Cr^{2}(-6a(1+3aCr^{2})^{\frac{2}{3}}+5\alpha)))))))).$

The decreasing nature of these quantities is shown graphically.

  5. v.

Within a stellar interior, it is expected that the speed of sound should be
less than the speed of light i.e., $\displaystyle
0\leq\frac{dp_{r}}{d\rho}\leq 1$ and $\displaystyle
0\leq\frac{dp_{t}}{d\rho}\leq 1$.
In this model, we have,

$\displaystyle\frac{dp_{r}}{d\rho}$ $\displaystyle=$
$\displaystyle-\frac{1}{(-10B(1+aCr^{2})^{4}+(-15+aCr^{2}(-53+aCr^{2}(-49+5aCr^{2})))\alpha))}\times\Big{[}(((1+3aCr^{2})$
(53)
$\displaystyle(-2B(1+aCr^{2})(-1+5a^{2}C^{2}r^{4})+a(4(1+3aCr^{2})^{\frac{2}{3}}+Cr^{2}(a(16(1+3aCr^{2})^{\frac{2}{3}}+$
$\displaystyle
Cr^{2}(12a(1+3aCr^{2})^{\frac{2}{3}}-25\alpha))-8\alpha))+\alpha))\Big{]},$
$\displaystyle\frac{dp_{t}}{d\rho}$ $\displaystyle=$
$\displaystyle\frac{1}{(10B(1+aCr^{2})^{5}+15\alpha+aCr^{2}(68+aCr^{2}(102+aCr^{2}(44-5aCr^{2})))\alpha))}\times\Big{[}-((2(1+3aCr^{2})$
(54)
$\displaystyle(B(1+aCr^{2})^{2}(-1+5a^{2}C^{2}r^{4})+a(-2(1+3aCr^{2})^{\frac{2}{3}}+Cr^{2}(6\alpha+a(-10(1+3aCr^{2})^{\frac{2}{3}}$
$\displaystyle+Cr^{2}(17\alpha+a(-14(1+3aCr^{2})^{\frac{2}{3}}+Cr^{2}(-6a(1+3aCr^{2})^{\frac{2}{3}}+5\alpha))))))))\Big{]}.$

By choosing the model parameters appropriately, it can be shown that this
requirement is also satisfied in this model.

  6. vi.

Fulfilment of the energy conditions for an anisotropic fluid i.e.,
$\rho+p_{r}+2p_{t}\geq 0$, $\rho+p_{r}\geq 0$ & $\rho+p_{t}\geq 0$ can also be
shown graphically to be satisfied in this model.

###  III.4 Physical behaviour of the model

Physical quantities in this model are obtained as,

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\frac{(BC(1+aCr^{2})^{2}(3+5aCr^{2})+C(3+aCr^{2}(6-aCr^{2}))\alpha)}{((1+aCr^{2})^{2}(1+3aCr^{2})}^{\frac{5}{3}},$
(55) $\displaystyle p_{r}$ $\displaystyle=$
$\displaystyle-\frac{((C(B(1+aCr^{2})(1+5aCr^{2})+\alpha+a(-4(1+3aCr^{2})^{\frac{2}{3}}+Cr^{2}(-4a(1+3aCr^{2})^{\frac{2}{3}}+5\alpha))))}{((1+aCr^{2})^{2}(1+3aCr^{2})^{\frac{2}{3}}))},$
(56) $\displaystyle p_{t}$ $\displaystyle=$
$\displaystyle\dfrac{1}{((1+aCr^{2})^{3}(1+3aCr^{2})^{\frac{2}{3}})}\times\Big{[}(C(-B(1+aCr^{2})^{2}(1+5aCr^{2})+a(4(1+3aCr^{2})^{\frac{2}{3}}+Cr^{2}(2a(4(1+3aCr^{2})^{\frac{2}{3}}$
(57)
$\displaystyle+Cr^{2}(2a(1+3aCr^{2})^{\frac{2}{3}}-\alpha))-5\alpha))-\alpha))\Big{]},$
$\displaystyle\Delta$ $\displaystyle=$
$\displaystyle\frac{(aC^{2}r^{2}(1+3aCr^{2})^{\frac{1}{3}}\alpha)}{(1+aCr^{2})^{3}},$
(58) $\displaystyle m(r)$ $\displaystyle=$
$\displaystyle\frac{Cr^{3}(B+aBCr^{2}+\alpha)}{2(1+aCr^{2})(1+3aCr^{2})^{\frac{2}{3}}}.$
(59)

The simple elementary functional forms of the physical quantities help us to
make a detailed study of the physical behaviour of the star. Most importantly,
the solution contains an ‘anisotropic switch’ $\alpha$, which allows us to
investigate the impact of anisotropy effectively. We analyze the physical
behaviour of the model by using the values of masses and radii of observed
pulsars as input parameters. We consider the data available from the pulsar
$4U1608-52$ whose estimated mass and radius are $M=1.57~{}M_{\odot}$ and
$R=9.8~{}km$, respectively [103]. For these values, we determine two sets of
constants. For an isotropic case ($\alpha=0$), we obtain the constants
$B=2.48125,~{}C=0.00300606,~{}A=0.56353$ and assuming the star to be composed
of an anisotropic fluid distribution (we assume $\alpha=1$), the constants are
calculated as $A=0.56353,~{}B=1.70527,~{}C=0.0030606$. Note that the parameter
$a$ remains free in this model which we set as $a=1$ in both the cases. Making
use of these values, we show graphically the nature of all the physically
meaningful quantities in Fig. (1)-(8). The plots clearly show that all the
quantities comply with the requirements of a realistic star. In particular,
the figures highlight the effect of anisotropy on the gross physical behaviour
of the compact star.

Figure 1: Density profiles. Figure 2: Radial and tangential pressure profiles.
Figure 3: Radial & transverse component of sound speed. Figure 4: Fulfillment
of energy condition. Figure 5: Radial variation of anisotropy. Figure 6:
Radial variation of adiabatic index. Figure 7: Radial variation of mass.
Figure 8: Equation of state. Figure 9: Forces in equilibrium.

##  IV Numerical calculation of tidal love number

Using the method employed in [104], we can calculate the numerical value of
$k_{2}$ for a particular neutron star. To this end, we rewrite the master
equation (22) using the equation (32) as,

$\displaystyle ry^{\prime}+y^{2}+(r\mathcal{R}-1)y+r^{2}\mathcal{S}=0.$ (60)

From equation (30), at $\mathcal{C}=0$ we expect $k_{2}=0$. This implies that
$y(0)=2$. Moreover, at the horizon formation limit $\mathcal{C}=0.5$, the
tidal love number $k_{2}$ vanishes for all values of $y$.

In order to solve the differential equation (60), we use the initial condition
$y(0)=2$ in addition to the expression for $\mathcal{R},\,\mathcal{S}$ using
equations (23) & (24), respectively. Using the initial condition and equations
(40), (55), (56) and (57) for a particular NS, equation (60) can be solved and
subsequently using equation (30), the tidal love number $k_{2}$ can be
calculated. One can also find the analytical expression for $y(r)$ in terms of
compactness factor $\mathcal{C}$ and anisotropy $\alpha$. Employing this
technique, we plot the relation between $k_{2}$ and compactness factor
$\mathcal{C}$ for different values of $\alpha$ as shown in the figure 10. We
note that $k_{2}$ increases gradually with increasing $\mathcal{C}$ up to a
certain value and then decreases with further increase of $\mathcal{C}$. For a
NS having compactness $\mathcal{C}<3.4$ & $\alpha\leq 2$, the above scheme can
be used to calculate the tidal love number in this model.

Figure 10: $k_{2}$ is plotted against $\mathcal{C}$ for different $\alpha$.

For $\alpha>2$, a discontinuity arises in the plot of $k_{2}$ vs
$\mathcal{C}$. To address this problem, we set a maximum limit on
$\mathcal{C}$ for a particular $\alpha>2$, using the ‘physical acceptability’
conditions discussed earlier. One can check that for all range of values of
$\mathcal{C}<0.34$,
$\rho(r=0),\rho(r=R),p_{r}(r=0),p_{r}(r=R),p_{t}(r=0),p_{t}(r=R)\geq 0$. The
energy condition $\rho+p_{r}+2p_{t}\geq 0$. Therefore, the maximum limit on
$\mathcal{C}$ can be calculated from the condition
$0\leq\dfrac{dp_{r}}{d\rho}\leq 1$ and $0\leq\dfrac{dp_{t}}{d\rho}\leq 1$. For
different values of $\alpha$, $V_{r}^{2}(R)=\dfrac{dp_{r}}{d\rho}$ and
$V_{t}^{2}(R)=\dfrac{dp_{t}}{d\rho}$ are plotted against $\mathcal{C}$ in
figure 11. It then becomes easy to evaluate the maximum limit on $\mathcal{C}$
for different $\alpha$ from the plot. For example, in figure 11, we note that
for $0\leq\alpha\leq 2$, the range of compactness is $0\leq\mathcal{C}\leq
0.4$. For $0\leq\alpha\leq 2$, figure 10 shows that $0\leq\mathcal{C}\leq
0.34$. In table 1, the maximum value of $\mathcal{C}$ for different $\alpha$
is given.

Figure 11: $V_{r}^{2}(R)$ & $V_{t}^{2}(R)$ are plotted against $\mathcal{C}$
for different $\alpha$.

For $\alpha>2$, variation of the tidal love number $k_{2}$ against
$\mathcal{C}$ is shown in figure 12. In table 1, the numerical values of the
tidal love number $k_{2}$ is shown for different $\alpha$.

Figure 12: $k_{2}$ is plotted against $\mathcal{C}$ for different $\alpha$. Only physically allowed range of $\alpha$ & $\mathcal{C}$ are considered. Table 1: The maximum limit of $\mathcal{C}$ and its corresponding $k_{2}$ for $\alpha>2$. Anisotropy $\alpha$ | Compactness $\mathcal{C}$ | Tidal love number $k_{2}$
---|---|---
2.5 | 0.31405 | 0.144954
3 | 0.2288 | 0.173846
3.5 | 0.1800 | 0.221919
4 | 0.1384 | 0.269166

Figure 13: (Top) $k_{2}$ is plotted against mass $M$ for different $\alpha$ at
$R=10\,km$, (Bottom) $k_{2}$ is plotted against mass $M$ for different radius
$R$ at $\alpha=2$.

Figure 14: (Top) $k_{2}$ is plotted against radius $R$ for different $\alpha$
at mass $M=1.5\,M_{\odot}$, (Bottom) $k_{2}$ is plotted against $R$ for
different mass $M$ at $\alpha=2$.

In the fig.13, (Top) $k_{2}$ is plotted against mass $M$ for different
$\alpha$ at $R=10\,km$, (Bottom) $k_{2}$ is plotted against mass $M$ for
different radius $R$ at $\alpha=2$. In the fig.14, (Top) $k_{2}$ is plotted
against radius $R$ for different $\alpha$ at mass $M=1.5\,M_{\odot}$ and
(Bottom) $k_{2}$ is plotted against $R$ for different mass $M$ at $\alpha=2$.

Figure 15:  $y$ is plotted against $\alpha$ for different compact objects. Only physically allowed range of $\alpha$ is considered. Figure 16: $k_{2}$ is plotted against $\alpha$ for different compact objects. Only physically allowed range of $\alpha$ is considered. Table 2:  $y$ and $k_{2}$ for $\alpha=0$ and the corresponding maximum value of $\alpha$are evaluated for different compact objects. The observational values of masses and radii of the compact stars have been considered from reference [103]. Name | M($M_{\odot}$) |  $R$ (km) | $y$ at $\alpha=0$ |  $k_{2}$ at $\alpha=0$ | Max. $\alpha$ |  $k_{2}$ at max. $\alpha$
---|---|---|---|---|---|---
SAX J1748 .9-2021 | $1.81^{+0.25}_{-0.37}$ | $11.7^{+1.7}_{-1.7}$ | 1.88994 | 0.0176463 | 3.006 | 0.174571
4U 1608-52 | $1.57^{+0.30}_{-0.29}$ | $9.8^{+1.8}_{-1.8}$ | 1.93653 | 0.0162621 | 2.9341 | 0.167513
4U 1820-30 | $1.46^{+0.21}_{-0.21}$ | $11.1^{+1.8}_{-1.8}$ | 1.79387 | 0.0221002 | 3.3467 | 0.207726
4U 1724-207 | $1.81^{+0.25}_{-0.37}$ | $12.2^{+1.4}_{-1.4}$ | 1.85006 | 0.0190864 | 3.093 | 0.182933
EXO 1745-268 | $1.65^{+0.21}_{-0.31}$ | $10.5^{+1.6}_{-1.6}$ | 1.90908 | 0.0170474 | 2.973 | 0.171127
KS 1731-260 | $1.61^{+0.35}_{-0.37}$ | $10^{+2.2}_{-2.2}$ | 1.94436 | 0.0160516 | 2.924 | 0.166511
J0437-4715 | $1.44^{+0.07}_{-0.07}$ | $13.6^{+0.9}_{-0.8}$ | 1.78635 | 0.0243844 | 3.78 | 0.248741
M13 | $1.38^{+0.08}_{-0.23}$ | $9.95^{+0.24}_{-0.27}$ | 1.81066 | 0.0209619 | 3.2355 | 0.196986

In the figure 15, variation of $y$ with respect to $\alpha$ is plotted for
different compact stars. In this case, the range of $\alpha$ is assumed to be
$0\leq\alpha<4$. In figure 16, $k_{2}$ is plotted against $\alpha$. Physically
acceptable range of $\alpha$ has been calculated numerically. Obviously, the
maximum value of $\alpha$ is not the same for different class of compact
stars. In table 2, the maximum value of $\alpha$ is calculated for different
neutron stars and the corresponding tidal love number $k_{2}$ is also shown.
In figure 16, we note that the tidal love number increases monotonically with
increasing $\alpha$ for stars having different compactness. In table 2,
$k_{2}$ is calculated for $\alpha=0$ for different compact objects.

##  V  Discussion

In this paper, we have presented a technique to measure the tidal love number
of a compact object when subjected to an external tidal field. Possible role
of anisotropy vis-a-vis matter distribution of the star on the tidal love
number has been investigated. It remains to be seen whether such impact, if
any, can be observationally realized. Moreover, effects of other factors such
as EOS, electromagnetic field etc on the tidal love number needs further
investigation and will be taken up elsewhere.

## Acknowledgement

SD and RS gratefully acknowledge support from the Inter-University Centre for
Astronomy and Astrophysics (IUCAA), Pune, India, under its Visiting Research
Associateship Programme.

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