# ANISOTROPIC COMPACT STARS IN THE MIMETIC GRAVITATIONAL THEORY

G. G. L. Nashed  Centre for theoretical physics, the British University in
Egypt, 11837 - P.O. Box 43, Egypt G. G. L. Nashed<EMAIL_ADDRESS>
###### Abstract

In this paper, we consider the mimetic gravitational theory to derive a novel
category of anisotropic star models. To end and to put the resulting
differential equations into a closed system, the form of the metric potential
$g_{rr}$ as used by Tolman (Tolman, 1939) is assumed as well as a linear form
of the equation-of-state. The resulting energy-momentum components, energy-
density, and radial and tangential pressures contain five constants; three of
these are determined through the junction condition, matching the interior
with the exterior Schwarzschild solution the fourth is constrained by the
vanishing of the radial pressure on the boundary and the fifth is constrained
by a real compact star. The physical acceptability of our model is tested
using the data of the pulsar 4U 1820-30. The stability of this model is
evaluated using the Tolman-Oppenheimer-Volkoff equation and the adiabatic
index and it is shown to be stable. Finally, our model is challenged with
other compact stars demonstrating that it is consistent with those stars.

Mimetic gravitational theory – spherically symmetric interior solution – TOV –
stability –Adiabatic index.

04.50.Kd, 04.25.Nx, 04.40.Nr

##  1 Introduction

Inflation has achieved remarkable success in solving some of the most
important cosmic puzzles, such as the problem of flatness, the problem of the
horizon, and the problems of fine-tuning (Guth, 1981; Albrecht & Steinhardt,
1982; Linde, 1982). Furthermore, it has also been declared that if the pre-
inflationary density fluctuations are allowed to grow very largely over a
short time ($\sim 10^{-33}$ sec.), a clear and systematic explanation of the
construction of the large-scale and anisotropic cosmic microwave background
structures is supplied (Mukhanov & Chibisov, 1981; Guth & Pi, 1982; Hawking,
1982; Starobinsky, 1982).

In Einstein’s general relativity (GR), the scalar field which is coined as
”inflaton” is responsible for executing inflation (Mukhanov & Chibisov, 1981).
In the GR realm, many studies have been carried out that have confirmed the
consistency of the scalar with observations (Hossain et al., 2014; Martin et
al., 2014a, b; Geng et al., 2015; Huang et al., 2016). However, GR still faces
many challenges in confronting the accelerated expansion of our universe,
among others, which is supported by observational data (Riess et al., 1998,
2004; Perlmutter et al., 1999; Ade et al., 2016). Modified gravitational
theories are considered as being reliable in identifying cosmic problems such
as the flat spin curves of spiral galaxies or the late cosmic acceleration,
without the need for dark matter and dark energy (Capozziello & De Laurentis,
2011; Nojiri & Odintsov, 2011; Nashed, 2007; Nojiri et al., 2017; De Felice &
Tsujikawa, 2010). Modified gravitational theories lead to scale-dependent
interactions and preserve the results of Einstein’s theory at scales of the
solar system (Capozziello & De Laurentis, 2011).

Another modified gravitational theory that has aimed to investigate dark
matter is the mimetic dark matter model that was built by Chamseddine and
Mukhanov (Chamseddine & Mukhanov, 2013). They constructed a model whose
concept was based on the isolation of the scalar degree-of-freedom of the
metric. This model was able to predict dark matter as well as dark energy
(Chamseddine et al., 2014) and many cosmological solutions (Chamseddine et
al., 2014) that resolve singularities (Chamseddine & Mukhanov, 2017a, b) and
construct a ghost-free massive gravity model (Chamseddine & Mukhanov, 2018).

The construction of the mimetic theory can be investigated as a specific form
of conformal transformation in which the new metrics, as well as the old ones
are degenerate. The use of non-singular conformal transformation produces an
increase in the number of degrees-of-freedom, and thus; the longitudinal
sector of gravity turns out to be dynamic (Deruelle & Rua, 2014; Domnech et
al., 2015; Firouzjahi et al., 2018; Shen et al., 2019; Gorji et al., 2020).
Generally, conformal transformation means a relation between
${\textit{g}_{\alpha\beta}}$, which is considered the physical one, to
$\bar{{\textit{g}}}_{\alpha\beta}$, which is the auxiliary metric, and
$\zeta$, which is the scalar field, through the following transformation:

${\textit{g}}_{\alpha,\beta}=-\left(\bar{{\textit{g}}}_{\mu\nu}\partial^{\mu}\zeta\partial^{\nu}\zeta\right)\bar{{\textit{g}}}_{\alpha,\beta}\,.$
(1)

The above transformation gives the following constraint (Gorji et al., 2020)

${\textit{g}}^{\alpha\beta}\partial_{\alpha}\zeta\partial_{\beta}\zeta=-1\,.$
(2)

Therefore, $\partial_{\alpha}\zeta$ is a space such that we consider the
positive sign and a time–like such that we consider the negative sign. The
famous mimetic theory is that with a negative sign, and in this study, the
positive sign is dealt with an extension to the mimetic gravitational theory
(MGT). Equation (1) tells us that we cannot write
$\bar{{\textit{g}}}_{\alpha\beta}$ in terms of ${\textit{g}}_{\alpha\beta}$
since it is non-invertible (Deruelle & Rua, 2014). The extra degree of freedom
linked to the transformation (1) accounts for the longitudinal sector of
gravity. Starting with the Einstein-Hilbert action that contains the physical
metric $g_{\alpha\beta}$ and carries out the transformation (1)
${\textit{g}}^{\alpha\beta}$ and $\zeta$ which are dynamic scalar fields
(Chaichian et al., 2014).

Generally, when studying compact stars a spherically symmetric spacetime that
has isotropy can be supposed. The isotropy and homogeneity of astrophysical
compact stellar objects cannot considered as the general physical
characteristics of these stellar objects; even though they could supply us
with some solvable features. This is because the fluid pressure can be
considered to have two different components that provide the anisotropy,
$\Delta=p_{t}-p_{r}$. Such a case is called inhomogeneous where the source of
the inhomogeneity results from the tangential, $p_{t}$, and the radial,
$p_{r}$, pressures therefore, the distribution of matter is not in isotropic
form. The idea of anisotropy was first considered by (Ruderman, 1972) and then
by many researchers (Canuto, 1974; Bowers & Liang, 1974; Herrera & Santos,
1997). Currently, the source of anisotropy can be thought of as being due to
different factors:
$\bullet$ Superfluid 3A, $\bullet$ a core of the region where the density is
very high, $\bullet$ a mixture of fluids of different types,
$\bullet$ different condensate states, $\bullet$ relativistic particles in the
compact stars $\bullet$ phase transition,
$\bullet$ rotational motion, $\bullet$ and the existence of the magnetic
field, among others (Ivanov, 2002; Varela et al., 2010; Rahaman et al., 2012;
Kalam et al., 2012; Deb et al., 2018; Shee et al., 2016; El Hanafy & Nashed,
2016; Maurya et al., 2016, 2018; Deb et al., 2017).
This study aims to derive an interior spherically symmetric solution in the
frame of MGT and to test if this solution represents a true compact star.

The structure of this research is as follows: In Section 2, the cornerstone of
MGT is given and its field equations are derived. In Section 3, the field
equations of MGT are applied to a spherically symmetric line element that has
two unknown functions. To close the system of differential equations, we
assume the metric potential $g_{rr}$ to have the form given by Tolman (Tolman,
1939), and assume a linear form of the equation-of-state (EoS) between the
density and radial pressures. In Section 4, the necessary requirements that
any true compact star should satisfy are stated. In Section 5, the physical
properties of our model are given, showing that the metric potential and the
energy-momentum components are well defined in the center of the star. Because
our model has five constants of integration, we made a junction condition with
an exterior solution, a Schwarzschild solution, and assumed that the radial
pressure vanishes at the boundary of the star. From these constraints and the
use of the radial EoS four constants are determined leaving the fifth one to
be constrained by the comparison with the true star. In Section 6, the mass
and radius of the compact star 4U 1820-30, are used showing that our model
satisfies all the conditions required for any true compact star. In Section 7
the Tolman-Oppenheimer-Volkoff (TOV) equation and the adiabatic index of our
model are derived and the model’s willingness to be stable is shown. Also, in
Section 7, other compact stars are studied and their relevant constants are
derived, with these results being summarized in Tables (1) and (2). In Section
8, the results of the present study are summarized.

##  2 Brief summary of the mimetic gravitational theory

The term “mimetic dark matter” was proposed in the literature by Mukhanov and
Chamseddine (Chamseddine et al., 2014) however, such a term was already known
in the theories of papers written at earlier times (Lim et al., 2010; Gao et
al., 2011; Capozziello et al., 2010; Sebastiani et al., 2017). The Lagrangian
of the mimetic theory in four-dimensions takes the following form:

${\mathcal{L}}:=\frac{1}{2\chi}\int
d^{4}x\sqrt{-g(\bar{g}_{\mu\nu},\zeta)}R(\bar{g}_{\mu\nu},\zeta)-\int
d^{4}x\sqrt{-g(\bar{g}_{\mu\nu},\zeta)}{\mathcal{L}_{m}}\,,$ (3)

where $\chi$ is the gravitational constant, $\chi=\frac{8\pi G}{c^{4}}$, $G$
is the Newtonian constant, $c$ is the speed of light,
$g(\bar{g}_{\mu\nu},\zeta)$ is the determinant of the metric tensor, $\zeta$
is the scalar field, $R$ is the Ricci scalar, and ${\mathcal{L}_{m}}$ is the
Lagrangian of matter. Using Eq. (3) the field equations of the MGT can be
obtained as follows:

$G_{\mu}{}^{\nu}-(G-T)\partial_{\mu}\zeta\partial^{\nu}\zeta=\chi
T_{\mu}{}^{\nu}\,,\qquad\qquad\nabla_{\mu}\left[(G-T)\partial^{\mu}\zeta\right]=0\,,$
(4)

where, $G_{\mu\nu}$ and $T_{\mu\nu}$ are the Einstein and stress energy
tensors and $G$ and $T$ are their traces with $G=-R$. The stress-energy
tensor, $T_{\mu}{}^{\nu}$, is the energy-momentum tensor for fluids whose
configuration has anisotropy and it is represented by:

$\displaystyle
T_{\mu}{}^{\nu}{}=(p_{t}+\rho)u_{\mu}u^{\nu}+p_{t}\delta_{\mu}{}^{\nu}+(p_{r}-p_{t})\xi_{\mu}\xi^{\nu}\,,$
(5)

where $u_{\mu}$ is the time-like vector defined as $u^{\mu}=[1,0,0,0]$ and
$\xi_{\mu}$ is the unit radial vector with its space-like property, defined by
$\xi^{\mu}=[0,1,0,0]$ such that $u^{\mu}u_{\mu}=-1$ and
$\xi^{\mu}\xi_{\mu}=1$.

This study aims to apply the field equations (4) to a spherically symmetric
spacetime.

##  3 Interior solution in mimetic theory

In this section, the field equations of MGT, Eq. (4), are to be applied to a
spherically symmetric spacetime. For this purpose we use the following metric:

$ds^{2}=w(r)dt^{2}-w_{1}(r)dr^{2}-r^{2}\left(d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2}\right)\,,$ (6)

where $w(r)$ and $w_{1}(r)$ are unknown functions. By applying the field
equations (4) to the spacetime (6), we obtain the following non-linear
differential equations:

$\displaystyle{\frac{w^{\prime}_{1}r+w_{1}{}^{2}-w_{1}}{w_{1}{}^{2}{r}^{2}}}=8\pi\rho\,,$
$\displaystyle\frac{1}{2w_{1}{}^{3}{r}^{2}w^{2}}\Bigg{(}2\,w^{\prime}r\,w\,w_{1}{}^{2}-2\,w_{1}{}^{3}w^{2}+2\,w^{2}w_{1}{}^{2}+\zeta^{\prime
2}\Big{[}\pi(16\,{r}^{2}w^{2}w_{1}{}^{2}\rho-16\,{r}^{2}w^{2}w_{1}{}^{2}p_{r}-32\,{r}^{2}w^{2}w_{1}{}^{2}p_{t})-4\,w^{\prime}_{1}rw^{2}$
$\displaystyle-4\,w^{2}w_{1}{}^{2}+4\,w^{2}w_{1}+4\,w^{\prime}r\,ww_{1}-{r}^{2}w^{\prime}w^{\prime}_{1}w+2\,{r}^{2}w^{\prime\prime}ww_{1}-{r}^{2}w^{\prime
2}w_{1}\Big{]}\Bigg{)}=-8\pi\,p_{r}\,,$
$\displaystyle\frac{2\,w^{\prime}_{1}\,w^{2}-2\,w^{\prime}ww_{1}+rw^{\prime}w^{\prime}_{1}w-2\,rw^{\prime\prime}\,w\,w_{1}+rw^{\prime
2}\,w_{1}}{4rw^{2}w_{1}{}^{2}}=-8\pi\,p_{t}\,.$ (7)

where $w\equiv w(r)$, $w_{1}\equiv w_{1}(r)$, $\zeta\equiv\zeta(r)$, and ′ is
the ordinary derivative, i.e., $w^{\prime}\equiv\frac{dw}{dr}$. The above
differential system consists of three independent equations in six unknown
functions; $w$, $w_{1}$, $\rho$, $p_{r}$, $p_{t}$, and $\zeta$. Therefore,
three extra conditions are required to solve the aforementioned system. One of
these conditions is to assume the metric potential $g_{rr}$ to have the form:

$w_{1}=1+s_{1}r^{2}+s_{2}r^{4}\,,$ (8)

where $s_{1}$ and $s_{2}$ are constants to be determined from the matching
conditions. The metric potential (8) was proposed by Tolman (1939) to model
realistic compact stellar objects. Interestingly, the same metric component is
used to describe relativistic anisotropic stellar objects with a prescribed
linear EoS of the form:

$p_{r}=s_{3}\rho+s_{4}\,,$ (9)

where $s_{3}$ and $s_{4}$ are constants. Finally, to satisfy Eq. (2) we assume
the scalar field in the form:

$\zeta=\int\frac{1}{\sqrt{w_{1}}}dr\,.$ (10)

Using Eq. (8) in the first equation of (3) we obtain the form of the density;
then, using this form with Eq. (10) in the remainder of Eqs. (3) and (9) we
obtain the form of $w$ as follows:

$w=s_{5}\,\left(1+s_{1}{r}^{2}+s_{2}{r}^{4}\right)^{s_{3}}{e^{\displaystyle\frac{\,{r}^{2}\left(16\,\pi\,s_{2}\,s_{4}\,{r}^{4}+24\,\pi\,s_{1}\,s_{4}\,{r}^{2}+3\,s_{2}\,s_{3}\,{r}^{2}+3\,s_{2}{r}^{2}+6\,\pi\,s_{3}\,s_{1}+48\,s_{4}+6\,s_{1}\right)}{12}}}\,.$
(11)

Consequently, the physical quantities are obtained as

$\displaystyle\rho(r)=\frac{s_{2}{}^{2}{r}^{6}+2\,s_{1}s_{2}{r}^{4}+\left(5\,s_{2}+s_{1}{}^{2}\right){r}^{2}+3\,s_{1}}{8\pi\,\left(1+s_{1}{r}^{2}+s_{2}{r}^{4}\right)^{2}}\,,$
$\displaystyle
p_{r}(r)=\frac{1}{8\pi\,\left(1+s_{1}{r}^{2}+s_{2}{r}^{4}\right)^{2}}\Big{[}8\,\pi\,s_{2}^{2}\,s_{4}\,{r}^{8}+\left(s_{2}^{2}\,s_{3}+16\,\pi\,s_{1}\,s_{2}\,s_{4}\right){r}^{6}+\left(8\,\,\pi\,s_{1}^{2}\,s_{4}+16\,\pi\,s_{2}\,s_{4}+2\,s_{1}\,s_{2}\,s_{3}\right){r}^{4}$
$\displaystyle+\left(5\,s_{2}\,s_{3}+s_{1}{}^{2}\,s_{3}+16\,\pi\,s_{1}\,s_{4}\right){r}^{2}+3\,s_{1}\,s_{3}+8\,\pi\,s_{4}\Big{]}\,,$

where we give the form of the tangential pressure and the anisotropic force in
Appendix A. It is important to mention that the anisotropic force is defined
as $\frac{2\Delta}{r}$ and it is attractive if $p_{r}-p_{t}>0$ and will be
repulsive if $p_{r}-p_{t}<0$. The mass contained within a radius $r$ of the
sphere is defined as:

$m(r)={\int_{0}}^{r}\rho(\eta)\eta^{2}d\eta\,.$ (13)

Using Eq. (3) in Eq. (13), we get

$m(r)=\frac{{r}^{3}\left(s_{1}+s_{2}{r}^{2}\right)}{16\pi\,\left(1+s_{1}{r}^{2}+s_{2}{r}^{4}\right)}\,.$
(14)

The compactness parameter of a spherically symmetric source with radius $r$
takes the form (Singh et al., 2019)

$\displaystyle
u(r)=\frac{{r}^{2}\left(s_{1}+s_{2}{r}^{2}\right)}{16\pi\,\left(1+s_{1}{r}^{2}+s_{2}{r}^{4}\right)}\,.$
(15)

In the next section, we will state the physical requirements that any viable
stellar structure must satisfy and ascertain whether the model (3) with the
form of tangential pressure given in Appendix A satisfies them.

##  4 Requirements for a physically consistent stellar model

Any physical viable stellar model must satisfy the following conditions
throughout the stellar configurations:
$\diamond$ The gravitational metric potentials, $w(r)$ and $w_{1}(r)$, and the
components of the energy-momentum tensor $\rho$, $p_{r}$ , $p_{t}$ must be
well-defined at the center of the star, as well as being regular and free from
singularity throughout the interior of the star.
$\diamond$ The density, $\rho$, must be positive in the interior of the
stellar model, i.e., $\rho\geq 0$. The value of the density at the center of
the star should be positive, finite and monotonically decreasing toward the
boundary.
$\diamond$ The radial and the tangential pressures must be positive inside the
configuration of the fluid i.e., $p_{r}\geq 0$, $p_{t}\geq 0$. Also, the
derivative of the density and the pressures must be negative, i.e.,
$\frac{d\rho}{dr}<0$, $\frac{dp_{r}}{dr}<0$ and $\frac{dp_{t}}{dr}<0$. The
radial pressure, $p_{r}$, must vanish at the boundary of the stellar model
$r=\mathcal{R}$, however, the tangential pressure, $p_{t}$, need not be zero
at the boundary. Finally, at the center of the star the pressures should be
equal meaning that the anisotropy vanishes, i.e., $\Delta(r=0)=0$.
$\diamond$ Any anisotropic fluid sphere must fulfill the following energy
condition inequalities:
(i) Null energy condition (NEC): $p_{t}+\rho>0$, $\rho>0$.
(ii) Strong energy condition (SEC): $p_{r}+\rho>0$, $p_{t}+\rho>0$, $\rho-
p_{r}-2p_{t}>0$.
(iii) Weak energy condition (WEC): $p_{r}+\rho>0$, $\rho>0$.
(iv) Dominant energy condition (DEC): $\rho\geq|p_{r}|$ and $\rho\geq|p_{t}|$
.
$\diamond$ The interior metric potentials must join smoothly with the
Schwarzschild exterior metric at the boundary.
$\diamond$ For a stable configuration, the adiabatic index must be greater
than $\frac{4}{3}$.
$\diamond$ According to Herrera, the stability of the anisotropic stars should
satisfy $0>v_{r}{}^{2}-v_{t}{}^{2}>-1$ where $v_{r}$ and $v_{t}$ are the
radial and transverse speeds respectively (Herrera, 1992).
$\diamond$ To obtain a realistic model, the causality condition must be
satisfied meaning that the speed of sound must be less than 1 (providing the
speed of light $c=1$) in the interior of the star, i.e.,
$1\geq\frac{dp_{r}}{dr}\geq 0$, $1\geq\frac{dp_{t}}{dr}\geq 0$.

We are now ready to analyze the aforementioned necessary physical conditions
to test whether our model satisfies them.

##  5 Physical properties of the model

Let us test the model (3) and ( A), see if it is consistent with realistic
stellar structures. For this aim, the following topics need to be discussed:

###  5.1 The non-singular model

i- The metric potentials of this model satisfy:

$w(0)=s_{5}\,\qquad\textrm{and}\qquad w_{1}(0)=1\,.$ (16)

Equation (16) implies that the gravitational metric potentials are finite at
the center of the configuration of the stellar model. Furthermore, the
derivatives of these potentials are finite at the center, i.e.,
$w^{\prime}_{r=0}=w^{\prime}_{1}{{}_{r=0}}=0$. The aforementioned conditions
ensure that the metric is regular at the center and behaves well throughout
the interior of the stellar model.
ii-Density, radial, and tangential pressures; at the center, have the form:

$\rho(0)=\frac{3s_{1}}{8\pi}\,,\qquad\qquad
p_{r}(0)=p_{t}(0)=\frac{3s_{1}s_{3}+8\pi s_{4}}{8\pi}.$ (17)

Eq. (17) shows that the density is always positive if $s_{1}>0$, and the
anisotropy is vanishing at the center. The radial and tangential pressures
have positive values if $s_{4}+\frac{3s_{1}s_{3}}{8\pi}>0$; otherwise, they
become negative. Furthermore, the Zeldovich condition (Zeldovich & Novikov,
1971) states that the radial pressure must be less than or equal to the
density at the center i.e., $\frac{p_{r}(0)}{\rho(0)}\leq 1$. Using the
Zeldovich condition in Eq. (17), we obtain:

$s_{1}\leq\frac{3s_{1}s_{3}+8\pi s_{4}}{3}.$ (18)

iii-The derivative of the energy density, radial, and tangential pressures of
the model are, respectively:

$\displaystyle\rho^{\prime}=-\frac{r\left(12\,{s_{2}}^{2}{r}^{4}+3\,{r}^{6}{s_{2}}^{2}s_{1}+{r}^{8}{s_{2}}^{3}+13\,s_{1}{r}^{2}s_{2}+3\,{r}^{4}s_{2}{s_{1}}^{2}-5\,s_{2}+5\,{s_{1}}^{2}+{r}^{2}{s_{1}}^{3}\right)}{4\pi\,\left(1+s_{1}{r}^{2}+s_{2}{r}^{4}\right)^{3}}\,,$
(19) $\displaystyle p^{\prime}_{r}=s_{3}\rho^{\prime}\,,$

where $\rho^{\prime}=\frac{d\rho}{dr}$, $p^{\prime}_{r}=\frac{dp_{r}}{dr}$ and
the form of $p^{\prime}_{t}=\frac{dp_{t}}{dr}$ is given in Appendix B. Eqs.
(19) and ( B) show that the gradients of the density, radial, and tangential
pressures are negative as will be shown in their plotting .
iv-The radial and tangential velocities of sound (c = 1) are obtained as:

$\displaystyle v_{r}{}^{2}=\frac{dp_{r}}{d\rho}=s_{3}\,,$ (20)

where the form of tangential speed is given in Appendix C.

###  5.2 Matching conditions

We assume that the exterior spacetime of a non-rotating star is empty and is
described by the exterior Schwarzschild solution that is given by the
following form:

$\displaystyle
ds^{2}=-\left(1-\frac{2M}{r}\right)dt^{2}+\left(1-\frac{2M}{r}\right)^{-1}dr^{2}+r^{2}d\Omega^{2},$
(21)

where $M$ is the total mass and $r>2M$. It is necessary to match the interior
spacetime metric (8) and (11) with the exterior Schwarzschild spacetime metric
(21) at the boundary of the star where $r=\mathcal{R}$. The continuity of the
metric functions across the boundary $r=\mathcal{R}$ gives the conditions

$\displaystyle
w(r=\mathcal{R})=\left(1-\frac{2M}{\mathcal{R}}\right),\qquad\qquad
w_{1}(r=\mathcal{R})=\left(1-\frac{2M}{\mathcal{R}}\right)^{-1}.$ (22)

Furthermore, we use the fact that the radial pressure approaches zero at a
finite value of the radial parameter $r$ r that coincides with the radius of
the star $\mathcal{R}$. Therefore, the radius of the star can be obtained
using the physical condition $p_{r}(r=\mathcal{R})=0$. From the above
conditions, we obtain the constraints on the constants $s_{1}$, $s_{2}$ and
$s_{5}$. Using the above constraints, we obtain the constants $s_{1}$, $s_{2}$
and $s_{5}$ as follows:

$\displaystyle
s_{1}=-\frac{8\,s_{3}\,{M}^{2}-5\,s_{3}\,\mathcal{R}M-4\,s_{4}\,\pi\,{\mathcal{R}}^{4}}{{\mathcal{R}}^{2}s_{3}\left(\mathcal{R}-2\,M\right)^{2}}\,,\qquad
s_{2}=\frac{-3\,s_{3}\,\mathcal{R}M+4\,s_{3}\,{M}^{2}-4\,s_{4}\,\pi\,{\mathcal{R}}^{4}}{{\mathcal{R}}^{4}s_{3}\,\left(\mathcal{R}-2\,M\right)^{2}}\,,$
$\displaystyle
s_{5}=\frac{\left(\mathcal{R}-2\,M\right)}{\mathcal{R}}e^{\frac{\left(36\,{M}^{2}-21\,\mathcal{R}M\right){s_{3}}^{2}-12\,\ln\left({\frac{\mathcal{R}}{\mathcal{R}-2\,M}}\right)^{\left(\mathcal{R}-2M\right)^{2}{s_{3}}^{2}}+\left(36\,{M}^{2}-64\,{M}^{2}{\mathcal{R}}^{2}s_{4}\,\pi+120\,Ms_{4}\,\pi\,{\mathcal{R}}^{3}-21\,\mathcal{R}M-60\,s_{4}\,\pi\,{\mathcal{R}}^{4}\right)s_{3}-32\,{\mathcal{R}}^{6}{s_{4}}^{2}{\pi}^{2}-12\,s_{4}\,\pi\,{\mathcal{R}}^{4}}{12\left(\mathcal{R}-2\,M\right)^{2}{s_{3}}}}\,.$

The constant $s_{4}$, is determined from the fact that
$p_{r}(r=\mathcal{R})=0\Rightarrow s_{4}=-s_{3}\rho(r=\mathcal{R})$ and the
constant $s_{3}$ remains arbitrary so that its value will be adjusted with the
real compact star.

##  6 Matching the model with the realistic compact stars

Let us now consider the previous physical conditions of the model derived to
test it using the masses and radii of the observed pulsars. To support our
model, the pulsar 4U 1820-30 whose estimated mass and radius are $M=1.46\pm
0.21M_{\circledcirc}$ and $\mathcal{R}=11.1\pm 1.8$ km, respectively, will be
studied (Gangopadhyay et al., 2013; Das et al., 2021; Roupas & Nashed, 2020).
The maximal values $M=1.67M_{\circledcirc}$ and $\mathcal{R}=12.9$km can be
used as the input parameters. The boundary conditions are adopted to determine
the constants $s_{1}=0.01294823993$, $s_{2}=-0.00005537504986$, and
$s_{5}=0.5917241379\,e^{-0.7529881502-1.480957408\,s_{3}}$.

Adopting these constants, the physical quantities can be plotted. The regular
behavior of these can be assumed as being a first requirement to fit a
realistic star model. Figure 1 0(a) represents the behavior of metric
potentials for 4U 1820-30 as well as the junction conditions. As Figure 1
shows, the metric potentials assume the values $w(0)=0.1328969066$ and
$w_{1}(0)=1$ for $r=0$ and $s_{3}=0.5$. This means that they are both finite
and positive at the center.

(a) Metric potentials (8) and (11)

(b) Junction condition of the temporal component with Schwarzschild exterior
solution

(c) Junction condition of the spatial component with Schwarzschild exterior
solution

Figure 1: Schematic plots: 0(a) the metric potentials (8) and (11); 0(b) the
junction condition at the surface boundary of star between the metric
potential w(r) and the temporal potential of Schwarzschild; 0(c) the junction
condition at the surface boundary of star between the metric potential
$w_{1}(r)$ and the spatial potential of Schwarzschild.

The metric potentials of solution (3) are plotted in Figure 1 0(a); Figure 1
0(b) represents the junction condition of the temporal metric potential (11)
and the temporal metric potential of the Schwarzschild solution. Figure 1 0(c)
represents the junction condition of the spatial metric potential (8) and the
spatial metric potential of the Schwarzschild solution.

(a) Density

(b) Radial pressure

(c) Tangential pressure1

Figure 2: Plot of the radial coordinate $r$ in km vs. the density, radial and
tangential pressures of (3) using the constants constrained from 4U 1820-30.

Figure 2 shows that density, radial and tangential pressures are positive as
required for realistic stellar configuration. Moreover, as Figure 2 1(a)
shows, the density is high at the center, $\rho(r=0)=0.001546366234$ and
decreases with the distance from it. Figure 2 1(b) shows that the radial
pressure tends to be zero at the boundary; which is relevant for a realistic
model. Figure 2 1(c) shows that the tangential pressure has a positive value
and that it has a high value at the center, decrease away from the center.

(a) Anisotropy and Anisotropic force

(b) The gradient of energy-density, radial and tangential pressure

(c) Equation of state

Figure 3: Schematic plots: 2(a) of the radial coordinate $r$ in km vs. the
anisotropy and anisotropic force of solution (3) using the constants
constrained from 4U 1820-30; 2(b) the radial coordinate $r$ in km vs. the
gradient of density, radial and tangential pressures of solution (3) using the
constants constrained from 4U 1820-30; 2(c) the radial and tangential equation
of states using the constants constrained from 4U 1820-30.

Figure 3 2(a) shows that the anisotropy, $\Delta(r)=p_{t}-p_{r}$ and the
anisotropic force. As Figure 3 2(a) shows, the anisotropy vanishs at the
center and decreases at the surface of the star. In particular, Fig. 3 2(a)
shows that the anisotropic force $\frac{\Delta}{r}$ is negative. This means
that it possesses an inward gravitational since $p_{r}-p_{t}>0$. Figure 3 2(b)
shows that the gradients of density, radial and tangential pressures are
negative confirming the decreasing of density, radial and transverse pressures
through the stellar configuration. Also Fig. 3 2(c) shows that the radial and
tangential EoSs are positive within the star configuration.

(a) Weak energy conditions

(b) Null energy conditions

(c) Strong energy condition

(d) Dominant energy condition

Figure 4: Schematic plots: 3(a) the weak, 3(b) null, 3(c) strong and 3(d)
dominant energy conditions of solution (3) using the constants constrained
from 4U 1820-30.

Figures 4 3(a), 4 3(b), 4 3(c) and 4 3(d) show the positive values of the WEC,
NEC, SEC and DEC. Therefore, all the energy conditions are satisfied
throughout the stellar configuration as required for a physically meaningful
stellar model.

Figure 5 4(a) represents the velocity of the sound, radial and transverse,
speeds; this should be less than the speed of the light, i.e.,
$\frac{dpr}{d\rho}<1$ and $\frac{dpt}{d\rho}<1$. This condition is known as
the causality condition. The mass function given by Eq. (14) is plotted in
Fig. 54(b), showing that it is a monotonically increasing function of the
radial coordinate and $M(r=0)=0$. Furthermore, Figure 54(b) shows the behavior
of the compactness parameter of star which is increasing. The radial variation
of the surface redshift is plotted in Figure 54(c). Böhmer and Harko (Böhmer &
Harko, 2006) constrained the surface red-shift to be $Z\leq 5$. The surface
redshift of this model is calculated according to $4U1820-30$ and is found to
be $\approx 0.008$.

(a) Radial and tangential speed of sound

(b) Mass function

(c) Redshift

Figure 5: Schematic plots: 4(a) of the radial coordinate $r$ in km vs, the
radial, tangential speeds of solution (3) and the difference between the
radial and tangential speeds of sound; 4(b) the mass and compactness vs the
radial coordinate $r$ in km 4(c) the surface red-shift vs the radius $r$ using
the constants constrained from 4U 1820-30.

##  7 Stability of the model

In this section we are going to discuss the stability issue using two
different techniques; the TOV equations and the adiabatic index.

###  7.1 Equilibrium analysis through TOV equation

In this subsection, we are going to discuss the stability of any stellar
model. For this goal, we assume hydrostatic equilibrium through the TOV
equation. Using the TOV equation (Tolman, 1939; Oppenheimer & Volkoff, 1939)
as that presented in (Ponce de Leon, 1993), we obtain the following form:

$\displaystyle\frac{2[p_{t}-p_{r}]}{r}-\frac{M_{g}(r)[\rho(r)+p_{r}]\sqrt{w}}{r\sqrt{w_{1}}}-\frac{dp_{r}}{r}=0,$
(24)

with $M_{g}(r)$ being the gravitational mass at radius $r$, as defined by the
Tolman-Whittaker mass formula which gives:

$\displaystyle
M_{g}(r)=4\pi{\int_{0}}^{r}\Big{(}{T_{t}}^{t}-{T_{r}}^{r}-{T_{\theta}}^{\theta}-{T_{\phi}}^{\phi}\Big{)}r^{2}\sqrt{ww_{1}}dr=\frac{rw^{\prime}\sqrt{w_{1}}}{2w}\,,$
(25)

Inserting Eq. (25) into (24), we obtain:

$\displaystyle\frac{2(p_{t}-p_{r})}{r}-\frac{dp_{r}}{dr}-\frac{w^{\prime}[\rho+p_{r}]}{2\sqrt{w}}=F_{g}+F_{a}+F_{h}=0\,,$
(26)

where $F_{g}=-\frac{w^{\prime}[\rho+p_{r}]}{2\sqrt{w}}$,
$F_{a}=\frac{2(p_{t}-p_{r})}{r}$, and $F_{h}=-\frac{dp_{r}}{dr}$ are the
gravitational, anisotropic and hydrostatic forces, respectively. The behavior
of the TOV equation for model (3) is shown in Figure 6 in which the three
different forces are plotted.

(a) Different forces

(b) Adiabatic index

Figure 6: Schematic plots: 5(a) the gravitational, the anisotropic and the
hydrostatic forces vs. the radius $r$; 5(b) the adiabatic index vs. the radius
$r$ using the constants constrained from 4U 1820-30.

This shows that hydrostatic force is positive and is dominated by the
gravitational and anisotropic forces, which are negative to maintain the
system in static equilibrium.

###  7.2 Adiabatic index

The stable equilibrium configuration of a spherically symmetric system can be
studied using the adiabatic index, which is a basic ingredient of the
stability criterion. Let us consider an adiabatic perturbation, the adiabatic
index $\Gamma$, is defined as (Chandrasekhar, 1964; Nashed, 2002; Merafina &
Ruffini, 1989; Chan et al., 1993):

$\displaystyle\Gamma=\left(\frac{\rho+p}{p}\right)\left(\frac{dp}{d\rho}\right)\,.$
(27)

A Newtonian isotropic sphere is in stable equilibrium if the adiabatic index
$\Gamma>\frac{4}{3}$ as reported in Heintzmann and Hillebrandth (Heintzmann &
Hillebrandt, 1975). For $\Gamma=\frac{4}{3}$, the isotropic sphere is in
neutral equilibrium. Based on some works carried out by Chan et al. (Chan et
al., 1993), the following condition is required for the stability of a
relativistic anisotropic sphere $\Gamma>\gamma$ where:

$\displaystyle\gamma=\frac{4}{3}-\left\\{\frac{4(p_{r}-p_{t})}{3|p^{\prime}_{r}|}\right\\}_{max}\,.$
(28)

Using Eq. (28), we obtain:

$\displaystyle\Gamma=\frac{4}{3}-\frac{2}{3}\Big{[}(c_{2}r^{2}+1)^{2}(4k_{1}{}^{2}k^{8}+6k^{4}-8r^{2}k^{2}+3r^{4})\Big{]}\times\Big{[}2(c_{2}r^{2}+1)^{2}k^{8}k_{1}{}^{2}+16k^{6}c_{2}-18c_{2}{}^{2}r^{4}k^{4}-36k^{4}c_{2}r^{2}+6k^{4}$
$\displaystyle+16c_{2}r^{4}k^{2}-16k^{2}r^{2}+16c_{2}{}^{2}r^{6}k^{2}+2c_{2}r^{6}+9r^{4}-3c_{2}{}^{2}r^{8}+4c_{2}{}^{2}k^{8}\Big{]}^{-1}.$
(29)

From Eq. (27), we obtain the adiabatic index of solution (3) in the form:

$\displaystyle\Gamma_{r}=s_{3}\,\Bigg{(}8\,s_{4}\,\pi\,{r}^{8}{s_{2}}^{2}+16\,{r}^{6}s_{4}\,\pi\,s_{1}s_{2}+{r}^{6}s_{3}\,{s_{2}}^{2}+8\,{r}^{4}s_{4}\,\pi\,{s_{1}}^{2}+2\,{r}^{4}s_{2}s_{3}\,s_{1}+16\,s_{4}\,\pi\,{r}^{4}s_{2}+{r}^{2}s_{3}\,{s_{1}}^{2}+16\,s_{4}\,\pi\,{r}^{2}s_{1}$
$\displaystyle+5\,s_{3}\,{r}^{2}s_{2}+3\,s_{3}\,s_{1}+8\,s_{4}\,\pi+{s_{2}}^{2}{r}^{6}+2\,s_{1}{r}^{4}s_{2}+5\,s_{2}{r}^{2}+{r}^{2}{s_{1}}^{2}+3\,s_{1}\Bigg{)}\times\Bigg{\\{}8\,s_{4}\,\pi\,{r}^{8}{s_{2}}^{2}+16\,{r}^{6}s_{4}\,\pi\,s_{1}s_{2}+{r}^{6}s_{3}\,{s_{2}}^{2}$
$\displaystyle+8\,{r}^{4}s_{4}\,\pi\,{s_{1}}^{2}+2\,{r}^{4}s_{2}s_{3}\,s_{1}+16\,s_{4}\,\pi\,{r}^{4}s_{2}+{r}^{2}s_{3}\,{s_{1}}^{2}+16\,s_{4}\,\pi\,{r}^{2}s_{1}+5\,s_{3}\,{r}^{2}s_{2}+3\,s_{3}\,s_{1}+8\,s_{4}\,\pi\Bigg{\\}}^{-1}\,.$
(30)

In Fig. 65(b) $\Gamma_{r}$ and $\Gamma_{t}$ are reported. As can be seen from
these plots, the value of $\Gamma_{t}$ is greater than that of $\Gamma_{r}$
throughout the stellar interior, and hence, the stability condition is
fulfilled.

###  7.3 Stability in the static state

For stable compact stars in terms of the mass-central mass–radius, and
relations for the energy density, Harrison, Zeldovich, and Novikov (Harrison
et al., 1965; Zeldovich & Novikov, 1971) claimed that the gradient of the
central density, with respect to the mass, must be positive, i.e.,
$\frac{\partial M}{\partial\rho_{r_{0}}}>0$. If this condition is satisfied,
then we have stable configurations. To be more specific, the stable stable or
unstable region is satisfied for constant mass i.e. $\frac{\partial
M}{\partial\rho_{r_{0}}}=0$ (Singh et al., 2019). Let us apply this procedure
to our solution (3). For solution (3), the central density has the form:

$\displaystyle\rho_{{}_{r_{0}}}=\frac{3s_{1}}{8\pi}\Rightarrow
s_{1}=\frac{8\pi\rho_{r_{0}}}{3},$ $\displaystyle
M(\rho_{r_{0}})=\frac{R^{3}(8\,\pi\,\rho_{r_{0}}+3s_{2}R^{2})}{16\,\pi\,(3+8\pi
R^{2}\rho_{r_{0}}+3s_{2}R^{4})}\,.$ (31)

With Eq. (7.3) we have:

$\displaystyle\frac{\partial M}{\partial\rho_{r_{0}}}=\frac{3R^{3}}{2(3+8\pi
R^{2}\rho_{r_{0}}+3s_{2}R^{4})^{2}}\,.$ (32)

From Eq. (32)), the solution (3) has a stable configuration since
$\frac{\partial M}{\partial\rho_{r_{0}}}>0$ (Singh et al., 2019). The
behaviors of (7.3) and (32) are shown in Fig. 7. It follows from these figures
that the mass and the gradient-of-mass decrease as the energy density become
larger. The above discussion of solution (3) shows that we have a good model.

(a) Mass as a function of central density

(b) differentiation of mass w.r.t. the central density

Figure 7: Schematic plots: 6(a) the gravitational mass as a function of the
central density; 6(b) the differentiation of gravitational mass w.r.t. the
central density using the constrained from 4U 1820-30. Table 1: Values of
model parameters (Özel et al., 2016)

Pulsar | Mass ($M_{\odot}$) | Radius (km) | $s_{1}$ | $s_{2}$ | $s_{3}$ | $s_{4}$ | $s_{5}$
---|---|---|---|---|---|---|---
4U 1724-207 | $1.81^{+0.25}_{-0.37}$ | $12.2^{+1.4}_{-1.4}$ |  $\approx$0.013 |  $\approx$-0.45$\times 10^{-4}$ | 0.5 |  $\approx$-0.18$\times 10^{-12}$ |  $\approx$0.10
4U 1820-30 | $1.46^{+0.21}_{-0.21}$ | $11.1^{+1.8}_{-1.8}$ |  $\approx$0.011 |  $\approx$-0.41$\times 10^{-4}$ | 0.5 |  $\approx$-0.76$\times 10^{-13}$ |  $\approx$0.12
SAX J1748.9-2021 | $1.81^{+0.25}_{-0.37}$ | $11.7^{+1.7}_{-1.7}$ |  $\approx$0.014 |  $\approx$-0.5$\times 10^{-4}$ | 0.5 |  $\approx$-0.59$\times 10^{-13}$ |  $\approx$0.11
EXO 1745-268 | $1.65^{+0.21}_{-0.31}$ | $10.5^{+1.6}_{-1.6}$ |  $\approx$0.017 |  $\approx$-0.75$\times 10^{-4}$ | 0.5 |  $\approx$-0.59$\times 10^{-13}$ |  $\approx$0.14
4U 1608-52 | $1.57^{+0.30}_{-0.29}$ | $9.8^{+1.8}_{-1.8}$ |  $\approx$0.02 |  $\approx$-0.99$\times 10^{-4}$ | 0.5 |  $\approx$-0.33$\times 10^{-12}$ |  $\approx$0.17
KS 1731-260 | $1.61^{+0.35}_{-0.37}$ | $10.0^{+2.2}_{-2.2}$ |  $\approx$0.018 |  $\approx$-0.8$\times 10^{-4}$ | 0.5 |  $\approx$-0.27$\times 10^{-12}$ |  $\approx$0.13

Table 2: Values of physical quantities

Pulsar | $\rho|_{{}_{{}_{0}}}$ | $\rho|_{{}_{{}_{R}}}$ | $\frac{dp_{r}}{d\rho}|_{{}_{{}_{0}}}$ | $\frac{dp_{r}}{d\rho}|_{{}_{{}_{R}}}$ | $\frac{dp_{t}}{d\rho}|_{{}_{{}_{0}}}$ | $\frac{dp_{t}}{d\rho}|_{{}_{{}_{R}}}$ | $(\rho-p_{r}-2p_{t})|_{{}_{{}_{0}}}$ | $(\rho-p_{r}-2p_{t})|_{{}_{{}_{R}}}$ | $z|_{{}_{{}_{R}}}$ |
---|---|---|---|---|---|---|---|---|---|---
4U 1724-207 | 0.15$\times 10^{-2}$ | .36$\times 10^{-12}$ | .5 | .5 | .56 | 2.13 | .76$\times 10^{-3}$ | .63$\times 10^{-3}$ | .009 |
4U 1820-30 | 0.13$\times 10^{-2}$ | 0 | .5 | .5 | .59 | 1.87 | .63$\times 10^{-4}$ | .6$\times 10^{-4}$ | .0077 |
SAX J1748.9-2021 | 0.16$\times 10^{-2}$ | 1.18$\times 10^{-13}$ | .5 | .5 | .56 | 2.16 | .81$\times 10^{-3}$ | .66$\times 10^{-3}$ | .0092 |
EXO 1745-268 | 0.20$\times 10^{-2}$ | 1.33$\times 10^{-8}$ | .5 | .5 | .56 | 2.16 | .1$\times 10^{-2}$ | .8$\times 10^{-3}$ | .0092 |
4U 1608-52 | .24$\times 10^{-2}$ | 6.53$\times 10^{-14}$ | .5 | .5 | .55 | 2.26 | .12$\times 10^{-2}$ | .92$\times 10^{-3}$ | .0096 |
KS 1731-260 | 0.22$\times 10^{-2}$ | 2.2$\times 10^{-13}$ | .5 | .5 | .55 | 2.25 | .11$\times 10^{-2}$ | .83$\times 10^{-3}$ | .0096 |

In addition to 4U 1820-30, a similar analysis can be developed for other
pulsars. In Tables I and II, we report the results for other observed systems.

##  8 Discussion and conclusions

In this research, we studied anisotropic spherically symmetric spacetime in
the frame of MGT. We obtained a system of three differential equations in six
unknowns. To put this system in a closed form, we used the formula of the
radial component of the metric potential given by Tolman (Tolman, 1939), ], a
linear form of the radial EoS, and the constraint given by the mimetic field
and the metric potential $g_{rr}$ given by Eq. (2). The solution of this
system involves five constants, three of them are fixed using the junction
condition, matching the interior solution with the Schwarzschild exterior, as
well as the vanishing of the radial pressure at the boundary. The fourth
constant is determined from the radial EoS, leaving the fifth constant to be
determined from the study of a real compact star. The main features of this
study can be summarized as follows:
$\star$ To demonstrate that our solution is compatible with a real compact
star, we used the stellar $4U1820-30$, which has mass
$1.46^{+0.21}_{-0.21}M_{\odot}$, and radius $11.1^{+1.8}_{-1.8}$km (Roupas &
Nashed, 2020; Das et al., 2021). The use of the mass and the radius fixed the
two constants $s_{1}=0.01049285312$ and $s_{2}=-.4062000611\times 10^{-4}$.
Furthermore, the use of the vanishing of the radial pressure on the boundary
yields $s_{5}=.1195529130$. Finally, using the linear radial EoS through the
assumption of $s_{3}=0.5$ we obtained the constant $s_{4}=.7572649088\times
10^{-13}$.
$\star$ As shown in Figure 1, the metric potentials have no singularity either
at the center of the star or at the boundary. Furthermore, in Figure 1 we show
how the junction condition is made between the interior and exterior
Schwarzschild solution.
$\star$ Figure 2, shows that the density, radial, and tangential pressures are
positive and decrease toward the center of the star.
$\star$ Figure 3, shows that the anisotropic force has a negative sign,
meaning that it is attractive since $p_{t}-p_{r}<0$. Also, figure 3, 3 shows
that the gradient of the density, radial, and tangential pressures are
negative; this is a necessary condition for any real star configuration.
Finally, Figure 3 shows that the radial and tangential EoS’s are not constant.

$\star$ The importance of energy conditions in the frame of modified
gravitational theories were discussed in detail in (Capozziello et al., 2015,
2014). In the present study, we tested the impact of extra force which comes
from the mimetic field and get the matter content more realistic, as in the
case of all these stars, the energy conditions, i.e., NEC, WEC, DEC, SEC were
all well satisfied which support that the matter destitution is normal matter
(not any kind of exotic matter). In figure 4, 4 shows that our model satisfies
all the energy conditions. Furthermore, Figure 5 shows that the radial and
tangential speeds are ¡1, as required for any realistic star. Also in figure 5
the mass and compactness of our model are positive, and the red-shift is less
than five i.e., $Z<5$.
$\star$ Figure 6, shows that the resulting model is stable because its
adiabatic index is greater than $4/3$.
$\star$ Finally, we extended our study to other stars, as shown in Tables I
and II, which confirms that our model is verified for these compact stars.
$\star$ It is interesting to analyze the solutions presented in this study in
the frame of more compact objects like neutron stars, to make contact with
events like the $GW190814$. These solutions have been discussed in details in
(Astashenok et al., 2021) in the frame of $f(R)$ gravitational theory. In our
case, we must be more careful because our approach applies to inhomogeneous
solutions. However, pulsars with spin less than $3ms$ or even the product of
the merging of two neutron stars if it is a neutron star, can initially be
quite inhomogeneous, thus during the ring-down, our solution could be
relevant. We hope to discuss this topic in future work since such a study
would require the implementation of a numerical recipe appropriately tailored
to our solutions.

Appendix A

The form of tangential pressure and anisotropic force

$\displaystyle
p_{t}(r)=\frac{1}{32\pi\,\left(1+s_{1}{r}^{2}+s_{2}{r}^{4}\right)^{3}}\Big{[}144\,s_{4}\,\pi\,{r}^{4}s_{2}+128\,s_{4}\,\pi\,{r}^{2}s_{1}+208\,s_{4}\,\pi\,{r}^{8}{s_{2}}^{2}+64\,{s_{4}}^{2}{\pi}^{2}{r}^{18}{s_{2}}^{4}+176\,{r}^{4}s_{4}\,\pi\,{s_{1}}^{2}+26\,{r}^{4}s_{3}\,s_{1}s_{2}$
$\displaystyle+384\,{r}^{6}s_{4}\,\pi\,s_{1}s_{2}+40\,s_{3}\,{r}^{2}s_{2}+8\,s_{1}{r}^{4}s_{2}+12\,s_{3}\,s_{1}+32\,s_{4}\,\pi+5\,{s_{2}}^{2}{r}^{6}+3\,{r}^{2}{s_{1}}^{2}+6\,{r}^{6}s_{3}\,{s_{2}}^{2}+8\,{r}^{2}s_{3}\,{s_{1}}^{2}+64\,{s_{2}}^{3}{r}^{14}s_{4}\,\pi\,s_{1}s_{3}$
$\displaystyle+96\,{s_{2}}^{2}{r}^{12}s_{4}\,\pi\,{s_{1}}^{2}s_{3}+64\,{r}^{10}s_{4}\,\pi\,s_{2}{s_{1}}^{3}s_{3}+304\,{r}^{10}\pi\,s_{4}\,{s_{2}}^{2}s_{1}s_{3}+272\,{r}^{8}s_{4}\,\pi\,s_{2}{s_{1}}^{2}s_{3}+288\,{r}^{6}s_{4}\,\pi\,s_{1}s_{2}s_{3}+2\,{s_{2}}^{4}{r}^{14}s_{3}$
$\displaystyle+{s_{2}}^{4}{r}^{14}{s_{3}}^{2}+2\,{r}^{6}{s_{1}}^{4}s_{3}+{r}^{6}{s_{1}}^{4}{s_{3}}^{2}+14\,{r}^{6}s_{2}{s_{1}}^{2}+4\,{r}^{8}{s_{1}}^{3}s_{2}+16\,{r}^{8}{s_{2}}^{2}s_{1}+10\,{r}^{4}{s_{1}}^{3}s_{3}+6\,{r}^{4}{s_{1}}^{3}{s_{3}}^{2}+64\,{r}^{2}{\pi}^{2}{s_{4}}^{2}+9\,{r}^{2}{s_{1}}^{2}{s_{3}}^{2}$
$\displaystyle+25\,{r}^{6}{s_{2}}^{2}{s_{3}}^{2}+6\,{r}^{10}{s_{2}}^{2}{s_{1}}^{2}+10\,{r}^{10}{s_{2}}^{3}{s_{3}}^{2}+16\,{r}^{10}{s_{2}}^{3}s_{3}+4\,{s_{2}}^{3}{r}^{12}s_{1}+256\,{\pi}^{2}{s_{4}}^{2}{s_{2}}^{3}{r}^{16}s_{1}+16\,\pi\,s_{4}\,{s_{2}}^{4}{r}^{16}s_{3}$
$\displaystyle+384\,{s_{2}}^{2}{r}^{14}{s_{4}}^{2}{\pi}^{2}{s_{1}}^{2}+64\,{s_{2}}^{3}{r}^{14}s_{4}\,\pi\,s_{1}+256\,s_{2}{r}^{12}{s_{4}}^{2}{\pi}^{2}{s_{1}}^{3}+96\,{s_{2}}^{2}{r}^{12}s_{4}\,\pi\,{s_{1}}^{2}+768\,{s_{2}}^{2}{r}^{12}s_{1}{\pi}^{2}{s_{4}}^{2}+112\,{s_{2}}^{3}{r}^{12}s_{4}\,\pi\,s_{3}$
$\displaystyle+64\,{r}^{10}s_{4}\,\pi\,s_{2}{s_{1}}^{3}+768\,{r}^{10}s_{2}{s_{1}}^{2}{\pi}^{2}{s_{4}}^{2}+320\,{r}^{10}\pi\,s_{4}\,{s_{2}}^{2}s_{1}+16\,{r}^{8}s_{4}\,\pi\,{s_{1}}^{4}s_{3}+304\,{r}^{8}s_{4}\,\pi\,s_{2}{s_{1}}^{2}+768\,{r}^{8}s_{2}s_{1}{\pi}^{2}{s_{4}}^{2}$
$\displaystyle+176\,{r}^{8}s_{4}\,\pi\,{s_{2}}^{2}s_{3}+80\,{r}^{6}\pi\,s_{4}\,{s_{1}}^{3}s_{3}+112\,{r}^{4}s_{4}\,\pi\,{s_{1}}^{2}s_{3}+80\,{r}^{4}s_{4}\,\pi\,s_{2}s_{3}+48\,{r}^{2}s_{4}\,\pi\,s_{1}s_{3}+16\,\pi\,s_{4}\,{s_{2}}^{4}{r}^{16}+256\,{s_{2}}^{3}{r}^{14}{s_{4}}^{2}{\pi}^{2}$
$\displaystyle+8\,{s_{2}}^{3}{r}^{12}s_{1}s_{3}+4\,{s_{2}}^{3}{r}^{12}s_{1}{s_{3}}^{2}+112\,{s_{2}}^{3}{r}^{12}s_{4}\,\pi+64\,{r}^{10}{s_{4}}^{2}{\pi}^{2}{s_{1}}^{4}+12\,{r}^{10}{s_{2}}^{2}{s_{1}}^{2}s_{3}+6\,{r}^{10}{s_{2}}^{2}{s_{1}}^{2}{s_{3}}^{2}+384\,{r}^{10}{s_{4}}^{2}{\pi}^{2}{s_{2}}^{2}$
$\displaystyle+16\,{r}^{8}s_{4}\,\pi\,{s_{1}}^{4}+8\,{r}^{8}{s_{1}}^{3}s_{3}\,s_{2}+4\,{r}^{8}{s_{1}}^{3}s_{2}{s_{3}}^{2}+256\,{r}^{8}{s_{4}}^{2}{\pi}^{2}{s_{1}}^{3}+42\,{r}^{8}{s_{2}}^{2}s_{1}s_{3}+26\,{r}^{8}{s_{2}}^{2}s_{1}{s_{3}}^{2}+96\,{r}^{6}\pi\,s_{4}\,{s_{1}}^{3}+36\,{r}^{6}{s_{1}}^{2}s_{3}\,s_{2}$
$\displaystyle+22\,{r}^{6}{s_{1}}^{2}s_{2}{s_{3}}^{2}+384\,{r}^{6}{s_{4}}^{2}{\pi}^{2}{s_{1}}^{2}+256\,{r}^{6}{s_{4}}^{2}{\pi}^{2}s_{2}+30\,{r}^{4}s_{1}s_{2}{s_{3}}^{2}+256\,{r}^{4}s_{1}{\pi}^{2}{s_{4}}^{2}+{s_{2}}^{4}{r}^{14}+6\,{r}^{10}{s_{2}}^{3}+{r}^{6}{s_{1}}^{4}+4\,{r}^{4}{s_{1}}^{3}\Big{]}\,,$
$\displaystyle\Delta=\frac{{r}^{2}}{32\pi\,\left(1+s_{1}{r}^{2}+s_{2}{r}^{4}\right)^{3}}\Big{[}3\,{s_{1}}^{2}+16\,{r}^{6}s_{4}\,\pi\,{s_{1}}^{4}+8\,{r}^{6}{s_{1}}^{3}s_{3}\,s_{2}+4\,{r}^{6}{s_{1}}^{3}s_{2}{s_{3}}^{2}+256\,{r}^{6}{s_{4}}^{2}{\pi}^{2}{s_{1}}^{3}+26\,{r}^{6}s_{1}{s_{2}}^{2}{s_{3}}^{2}$
$\displaystyle+30\,{r}^{6}s_{1}{s_{2}}^{2}s_{3}+64\,{r}^{4}s_{4}\,\pi\,{s_{1}}^{3}+22\,{r}^{4}{s_{1}}^{2}s_{2}{s_{3}}^{2}+24\,{r}^{4}{s_{1}}^{2}s_{3}\,s_{2}+384\,{r}^{4}{s_{4}}^{2}{\pi}^{2}{s_{1}}^{2}+256\,{r}^{4}{s_{4}}^{2}{\pi}^{2}s_{2}+80\,{r}^{2}s_{4}\,\pi\,{s_{1}}^{2}+30\,{r}^{2}s_{1}s_{2}{s_{3}}^{2}$
$\displaystyle-14\,{r}^{2}s_{2}s_{3}\,s_{1}+256\,{r}^{2}s_{1}{s_{4}}^{2}{\pi}^{2}+48\,{r}^{2}s_{4}\,\pi\,s_{2}+64\,{s_{4}}^{2}{\pi}^{2}{r}^{16}{s_{2}}^{4}+16\,s_{4}\,\pi\,{s_{2}}^{4}{r}^{14}+256\,{s_{2}}^{3}{r}^{12}{s_{4}}^{2}{\pi}^{2}+8\,{s_{2}}^{3}{r}^{10}s_{1}s_{3}$
$\displaystyle+4\,{s_{2}}^{3}{r}^{10}s_{1}{s_{3}}^{2}+80\,{s_{2}}^{3}{r}^{10}s_{4}\,\pi+64\,{r}^{8}{s_{4}}^{2}{\pi}^{2}{s_{1}}^{4}+12\,{r}^{8}{s_{2}}^{2}{s_{1}}^{2}s_{3}+6\,{r}^{8}{s_{2}}^{2}{s_{1}}^{2}{s_{3}}^{2}+384\,{r}^{8}{s_{4}}^{2}{\pi}^{2}{s_{2}}^{2}+112\,s_{4}\,\pi\,{r}^{6}{s_{2}}^{2}$
$\displaystyle+48\,s_{4}\,\pi\,s_{1}s_{3}+5\,{s_{2}}^{2}{r}^{4}+8\,s_{1}{r}^{2}s_{2}+4\,{r}^{6}s_{2}{s_{1}}^{3}+16\,{r}^{6}{s_{2}}^{2}s_{1}+2\,{r}^{4}{s_{1}}^{4}s_{3}+{r}^{4}{s_{1}}^{4}{s_{3}}^{2}+14\,{r}^{4}s_{2}{s_{1}}^{2}-18\,{r}^{4}s_{3}\,{s_{2}}^{2}+25\,{r}^{4}{s_{2}}^{2}{s_{3}}^{2}$
$\displaystyle+6\,{r}^{2}{s_{1}}^{3}s_{3}+6\,{r}^{2}{s_{1}}^{3}{s_{3}}^{2}+{s_{2}}^{4}{r}^{12}{s_{3}}^{2}+4\,{s_{2}}^{3}{r}^{10}s_{1}+2\,{s_{2}}^{4}{r}^{12}s_{3}+12\,{r}^{8}{s_{2}}^{3}s_{3}+10\,{r}^{8}{s_{2}}^{3}{s_{3}}^{2}+6\,{r}^{8}{s_{2}}^{2}{s_{1}}^{2}+9\,{s_{1}}^{2}{s_{3}}^{2}+{r}^{4}{s_{1}}^{4}$
$\displaystyle+4\,{r}^{2}{s_{1}}^{3}+{s_{2}}^{4}{r}^{12}+6\,{r}^{8}{s_{2}}^{3}+64\,{s_{4}}^{2}{\pi}^{2}+20\,s_{3}\,s_{2}-8\,s_{3}\,{s_{1}}^{2}+32\,s_{4}\,\pi\,s_{1}+224\,{r}^{8}s_{4}\,\pi\,{s_{2}}^{2}s_{1}+16\,{r}^{6}s_{4}\,\pi\,{s_{1}}^{4}s_{3}+208\,{r}^{6}s_{4}\,\pi\,s_{2}{s_{1}}^{2}$
$\displaystyle+768\,{r}^{6}s_{1}s_{2}{s_{4}}^{2}{\pi}^{2}+176\,{r}^{6}s_{4}\,\pi\,{s_{2}}^{2}s_{3}+80\,{r}^{4}s_{4}\,\pi\,{s_{1}}^{3}s_{3}+192\,{r}^{4}s_{4}\,\pi\,s_{1}s_{2}+112\,{r}^{2}s_{4}\,\pi\,{s_{1}}^{2}s_{3}+80\,{r}^{2}s_{4}\,\pi\,s_{2}s_{3}$
$\displaystyle+256\,{s_{4}}^{2}{\pi}^{2}{s_{2}}^{3}{r}^{14}s_{1}+16\,s_{4}\,\pi\,{s_{2}}^{4}{r}^{14}s_{3}+384\,{s_{2}}^{2}{r}^{12}{s_{4}}^{2}{\pi}^{2}{s_{1}}^{2}+64\,{s_{2}}^{3}{r}^{12}s_{4}\,\pi\,s_{1}+256\,s_{2}{r}^{10}{s_{4}}^{2}{\pi}^{2}{s_{1}}^{3}+96\,{s_{2}}^{2}{r}^{10}s_{4}\,\pi\,{s_{1}}^{2}$
$\displaystyle+768\,{s_{2}}^{2}{r}^{10}s_{1}{s_{4}}^{2}{\pi}^{2}+112\,{s_{2}}^{3}{r}^{10}s_{4}\,\pi\,s_{3}+64\,{r}^{8}s_{4}\,\pi\,s_{2}{s_{1}}^{3}+768\,{r}^{8}s_{2}{s_{1}}^{2}{s_{4}}^{2}{\pi}^{2}+64\,{s_{2}}^{3}{r}^{12}s_{4}\,\pi\,s_{1}s_{3}+96\,{s_{2}}^{2}{r}^{10}s_{4}\,\pi\,{s_{1}}^{2}s_{3}$
$\displaystyle+64\,{r}^{8}s_{4}\,\pi\,s_{2}{s_{1}}^{3}s_{3}+304\,{r}^{8}s_{4}\,\pi\,{s_{2}}^{2}s_{1}s_{3}+272\,{r}^{6}s_{4}\,\pi\,s_{2}{s_{1}}^{2}s_{3}+288\,{r}^{4}s_{4}\,\pi\,s_{1}s_{2}s_{3}\Big{]}\,.\hskip
142.26378pt({\mathbf{A}})$

Appendix B

The gradient of the tangential pressure

$\displaystyle
p^{\prime}_{t}=\frac{r}{16\pi\,\left(1+s_{1}{r}^{2}+s_{2}{r}^{4}\right)^{4}}\Big{[}-28\,s_{3}\,{s_{1}}^{2}+40\,s_{3}\,s_{2}+64\,{s_{4}}^{2}{\pi}^{2}+9\,{s_{1}}^{2}{s_{3}}^{2}+15\,{s_{2}}^{2}{r}^{4}+{s_{2}}^{4}{r}^{12}+15\,{r}^{8}{s_{2}}^{3}-{r}^{4}{s_{1}}^{4}+2\,{r}^{2}{s_{1}}^{3}$
$\displaystyle+3072\,{s_{4}}^{2}{\pi}^{2}{s_{2}}^{3}{r}^{14}s_{1}+128\,s_{4}\,\pi\,{s_{2}}^{4}{r}^{14}s_{3}+4224\,{s_{2}}^{2}{r}^{12}{s_{4}}^{2}{\pi}^{2}{s_{1}}^{2}+464\,{s_{2}}^{3}{r}^{12}s_{4}\,\pi\,s_{1}+2560\,s_{2}{r}^{10}{s_{4}}^{2}{\pi}^{2}{s_{1}}^{3}+608\,{s_{2}}^{2}{r}^{10}s_{4}\,\pi\,{s_{1}}^{2}$
$\displaystyle+3840\,{s_{2}}^{2}{r}^{10}s_{1}{s_{4}}^{2}{\pi}^{2}+320\,{s_{2}}^{3}{r}^{10}s_{4}\,\pi\,s_{3}+336\,{r}^{8}s_{4}\,\pi\,s_{2}{s_{1}}^{3}+3456\,{r}^{8}s_{2}{s_{1}}^{2}{s_{4}}^{2}{\pi}^{2}+656\,{r}^{8}s_{4}\,\pi\,{s_{2}}^{2}s_{1}+64\,{r}^{6}s_{4}\,\pi\,{s_{1}}^{4}s_{3}$
$\displaystyle+512\,{r}^{6}s_{4}\,\pi\,s_{2}{s_{1}}^{2}+2048\,{r}^{6}s_{2}s_{1}{s_{4}}^{2}{\pi}^{2}+384\,{r}^{6}s_{4}\,\pi\,{s_{2}}^{2}s_{3}+128\,{r}^{4}s_{4}\,\pi\,{s_{1}}^{3}s_{3}+368\,{r}^{4}s_{4}\,\pi\,s_{1}s_{2}+128\,{r}^{2}s_{4}\,\pi\,{s_{1}}^{2}s_{3}$
$\displaystyle+160\,{r}^{2}s_{4}\,\pi\,s_{2}s_{3}+1664\,{s_{4}}^{2}{\pi}^{2}{s_{2}}^{3}{r}^{16}{s_{1}}^{2}+896\,{s_{4}}^{2}{\pi}^{2}{s_{2}}^{4}{r}^{18}s_{1}+32\,s_{4}\,\pi\,{s_{2}}^{5}{r}^{18}s_{3}+1536\,{s_{2}}^{2}{r}^{14}{s_{4}}^{2}{\pi}^{2}{s_{1}}^{3}+256\,{s_{2}}^{3}{r}^{14}s_{4}\,\pi\,{s_{1}}^{2}$
$\displaystyle+144\,{s_{2}}^{4}{r}^{16}s_{4}\,\pi\,s_{1}+704\,s_{2}{r}^{12}{s_{4}}^{2}{\pi}^{2}{s_{1}}^{4}+224\,{s_{2}}^{2}{r}^{12}s_{4}\,\pi\,{s_{1}}^{3}+96\,{r}^{10}s_{4}\,\pi\,s_{2}{s_{1}}^{4}+16\,{r}^{8}s_{4}\,\pi\,{s_{1}}^{5}s_{3}+48\,s_{4}\,\pi\,s_{1}s_{3}+96\,s_{4}\,\pi\,{r}^{2}s_{2}$
$\displaystyle+256\,s_{4}\,\pi\,{r}^{6}{s_{2}}^{2}+832\,{s_{4}}^{2}{\pi}^{2}{r}^{16}{s_{2}}^{4}+128\,s_{4}\,\pi\,{s_{2}}^{4}{r}^{14}+1408\,{s_{2}}^{3}{r}^{12}{s_{4}}^{2}{\pi}^{2}-4\,{s_{2}}^{3}{r}^{10}s_{1}s_{3}-8\,{s_{2}}^{3}{r}^{10}s_{1}{s_{3}}^{2}+256\,{s_{2}}^{3}{r}^{10}s_{4}\,\pi$
$\displaystyle+576\,{r}^{8}{s_{4}}^{2}{\pi}^{2}{s_{1}}^{4}-6\,{r}^{8}{s_{2}}^{2}{s_{1}}^{2}s_{3}-10\,{r}^{8}{s_{2}}^{2}{s_{1}}^{2}{s_{3}}^{2}+1152\,{r}^{8}{s_{4}}^{2}{\pi}^{2}{s_{2}}^{2}+64\,{r}^{6}s_{4}\,\pi\,{s_{1}}^{4}-8\,{r}^{6}{s_{1}}^{3}s_{3}\,s_{2}-8\,{r}^{6}{s_{1}}^{3}s_{2}{s_{3}}^{2}$
$\displaystyle+576\,{r}^{8}{s_{4}}^{2}{\pi}^{2}{s_{1}}^{4}+1024\,{r}^{6}{s_{4}}^{2}{\pi}^{2}{s_{1}}^{3}-16\,{r}^{6}{s_{2}}^{2}s_{1}{s_{3}}^{2}+64\,{r}^{6}{s_{2}}^{2}s_{1}s_{3}+112\,{r}^{4}s_{4}\,\pi\,{s_{1}}^{3}-9\,{r}^{4}{s_{1}}^{2}s_{2}{s_{3}}^{2}+42\,{r}^{4}{s_{1}}^{2}s_{3}\,s_{2}$
$\displaystyle+896\,{r}^{4}{s_{4}}^{2}{\pi}^{2}{s_{1}}^{2}+448\,{r}^{4}{s_{4}}^{2}{\pi}^{2}s_{2}+96\,{r}^{2}s_{4}\,\pi\,{s_{1}}^{2}+60\,{r}^{2}s_{1}s_{2}{s_{3}}^{2}-100\,{r}^{2}s_{2}s_{3}\,s_{1}+384\,{r}^{2}s_{1}{s_{4}}^{2}{\pi}^{2}+192\,{s_{4}}^{2}{\pi}^{2}{r}^{20}{s_{2}}^{5}$
$\displaystyle+32\,s_{4}\,\pi\,{s_{2}}^{5}{r}^{18}+12\,{s_{2}}^{3}{r}^{12}{s_{1}}^{2}s_{3}+8\,{s_{2}}^{4}{r}^{14}s_{1}s_{3}+6\,{s_{2}}^{3}{r}^{12}{s_{1}}^{2}{s_{3}}^{2}+4\,{s_{2}}^{4}{r}^{14}s_{1}{s_{3}}^{2}+128\,{r}^{10}{s_{4}}^{2}{\pi}^{2}{s_{1}}^{5}+8\,{r}^{10}{s_{2}}^{2}{s_{1}}^{3}s_{3}$
$\displaystyle+4\,{r}^{10}{s_{2}}^{2}{s_{1}}^{3}{s_{3}}^{2}+16\,{r}^{8}s_{4}\,\pi\,{s_{1}}^{5}+2\,{r}^{8}{s_{1}}^{4}s_{3}\,s_{2}+{r}^{8}{s_{1}}^{4}s_{2}{s_{3}}^{2}+3\,{s_{1}}^{2}+32\,s_{4}\,\pi\,s_{1}+{s_{2}}^{5}{r}^{16}+16\,s_{1}{r}^{2}s_{2}+32\,{r}^{6}{s_{2}}^{2}s_{1}-4\,{r}^{4}{s_{1}}^{4}s_{3}$
$\displaystyle-3\,{r}^{4}{s_{1}}^{4}{s_{3}}^{2}+19\,{r}^{4}s_{2}{s_{1}}^{2}-182\,{r}^{4}s_{3}\,{s_{2}}^{2}+75\,{r}^{4}{s_{2}}^{2}{s_{3}}^{2}+4\,{r}^{2}{s_{1}}^{3}s_{3}-6\,{r}^{2}{s_{1}}^{3}{s_{3}}^{2}-2\,{s_{2}}^{4}{r}^{12}s_{3}-3\,{s_{2}}^{4}{r}^{12}{s_{3}}^{2}+4\,{s_{2}}^{3}{r}^{10}s_{1}$
$\displaystyle+4\,{r}^{8}{s_{2}}^{2}{s_{1}}^{2}-25\,{r}^{8}{s_{2}}^{3}{s_{3}}^{2}+62\,{r}^{8}{s_{2}}^{3}s_{3}+144\,s_{4}\,\pi\,{s_{2}}^{4}{r}^{16}s_{3}\,s_{1}+96\,{r}^{10}s_{4}\,\pi\,{s_{1}}^{4}s_{3}\,s_{2}+256\,{s_{2}}^{3}{r}^{14}s_{4}\,\pi\,{s_{1}}^{2}s_{3}$
$\displaystyle+224\,{s_{2}}^{2}{r}^{12}s_{4}\,\pi\,{s_{1}}^{3}s_{3}+480\,{s_{2}}^{3}{r}^{12}s_{4}\,\pi\,s_{1}s_{3}+640\,{s_{2}}^{2}{r}^{10}s_{4}\,\pi\,{s_{1}}^{2}s_{3}+352\,{r}^{8}s_{4}\,\pi\,s_{2}{s_{1}}^{3}s_{3}+832\,{r}^{8}s_{4}\,\pi\,{s_{2}}^{2}s_{1}s_{3}$
$\displaystyle+640\,{r}^{6}s_{4}\,\pi\,s_{2}{s_{1}}^{2}s_{3}+544\,{r}^{4}s_{4}\,\pi\,s_{1}s_{2}s_{3}+{r}^{8}{s_{1}}^{4}s_{2}+4\,{r}^{10}{s_{1}}^{3}{s_{2}}^{2}+2\,{s_{2}}^{5}{r}^{16}s_{3}+{s_{2}}^{5}{r}^{16}{s_{3}}^{2}+6\,{s_{2}}^{3}{r}^{12}{s_{1}}^{2}+4\,{s_{2}}^{4}{r}^{14}s_{1}\Big{]}\,,$
$\displaystyle\hskip 483.69684pt({\mathbf{B}})$

Appendix C

The tangential sound speed

$\displaystyle
v_{r}{}^{2}=\frac{dp_{t}}{d\rho}=-\frac{1}{4\left(1+s_{1}{r}^{2}+s_{2}{r}^{4}\right)\left(12\,{s_{2}}^{2}{r}^{4}+3\,{r}^{6}{s_{2}}^{2}s_{1}+{r}^{8}{s_{2}}^{3}+13\,s_{1}{r}^{2}s_{2}+3\,{r}^{4}s_{2}{s_{1}}^{2}-5\,s_{2}+5\,{s_{1}}^{2}+{r}^{2}{s_{1}}^{3}\right)}$
$\displaystyle\times\Big{[}-28\,s_{3}\,{s_{1}}^{2}+40\,s_{3}\,s_{2}+64\,{s_{4}}^{2}{\pi}^{2}+9\,{s_{1}}^{2}{s_{3}}^{2}+15\,{s_{2}}^{2}{r}^{4}+{s_{2}}^{4}{r}^{12}+15\,{r}^{8}{s_{2}}^{3}-{r}^{4}{s_{1}}^{4}+2\,{r}^{2}{s_{1}}^{3}+3072\,{s_{4}}^{2}{\pi}^{2}{s_{2}}^{3}{r}^{14}s_{1}$
$\displaystyle+128\,s_{4}\,\pi\,{s_{2}}^{4}{r}^{14}s_{3}+4224\,{s_{2}}^{2}{r}^{12}{s_{4}}^{2}{\pi}^{2}{s_{1}}^{2}+464\,{s_{2}}^{3}{r}^{12}s_{4}\,\pi\,s_{1}+2560\,s_{2}{r}^{10}{s_{4}}^{2}{\pi}^{2}{s_{1}}^{3}+608\,{s_{2}}^{2}{r}^{10}s_{4}\,\pi\,{s_{1}}^{2}+3840\,{s_{2}}^{2}{r}^{10}s_{1}{s_{4}}^{2}{\pi}^{2}$
$\displaystyle+320\,{s_{2}}^{3}{r}^{10}s_{4}\,\pi\,s_{3}+336\,{r}^{8}s_{4}\,\pi\,s_{2}{s_{1}}^{3}+3456\,{r}^{8}s_{2}{s_{1}}^{2}{s_{4}}^{2}{\pi}^{2}+656\,{r}^{8}s_{4}\,\pi\,{s_{2}}^{2}s_{1}+64\,{r}^{6}s_{4}\,\pi\,{s_{1}}^{4}s_{3}+512\,{r}^{6}s_{4}\,\pi\,s_{2}{s_{1}}^{2}$
$\displaystyle+2048\,{r}^{6}s_{2}s_{1}{s_{4}}^{2}{\pi}^{2}+384\,{r}^{6}s_{4}\,\pi\,{s_{2}}^{2}s_{3}+128\,{r}^{4}s_{4}\,\pi\,{s_{1}}^{3}s_{3}+368\,{r}^{4}s_{4}\,\pi\,s_{1}s_{2}+128\,{r}^{2}s_{4}\,\pi\,{s_{1}}^{2}s_{3}+160\,{r}^{2}s_{4}\,\pi\,s_{2}s_{3}$
$\displaystyle+1664\,{s_{4}}^{2}{\pi}^{2}{s_{2}}^{3}{r}^{16}{s_{1}}^{2}+896\,{s_{4}}^{2}{\pi}^{2}{s_{2}}^{4}{r}^{18}s_{1}+32\,s_{4}\,\pi\,{s_{2}}^{5}{r}^{18}s_{3}+1536\,{s_{2}}^{2}{r}^{14}{s_{4}}^{2}{\pi}^{2}{s_{1}}^{3}+256\,{s_{2}}^{3}{r}^{14}s_{4}\,\pi\,{s_{1}}^{2}+144\,{s_{2}}^{4}{r}^{16}s_{4}\,\pi\,s_{1}$
$\displaystyle+704\,s_{2}{r}^{12}{s_{4}}^{2}{\pi}^{2}{s_{1}}^{4}+224\,{s_{2}}^{2}{r}^{12}s_{4}\,\pi\,{s_{1}}^{3}+96\,{r}^{10}s_{4}\,\pi\,s_{2}{s_{1}}^{4}+16\,{r}^{8}s_{4}\,\pi\,{s_{1}}^{5}s_{3}+48\,s_{4}\,\pi\,s_{1}s_{3}+96\,s_{4}\,\pi\,{r}^{2}s_{2}+256\,s_{4}\,\pi\,{r}^{6}{s_{2}}^{2}$
$\displaystyle+832\,{s_{4}}^{2}{\pi}^{2}{r}^{16}{s_{2}}^{4}+128\,s_{4}\,\pi\,{s_{2}}^{4}{r}^{14}+1408\,{s_{2}}^{3}{r}^{12}{s_{4}}^{2}{\pi}^{2}-4\,{s_{2}}^{3}{r}^{10}s_{1}s_{3}-8\,{s_{2}}^{3}{r}^{10}s_{1}{s_{3}}^{2}3+256\,{s_{2}}^{3}{r}^{10}s_{4}\,\pi+576\,{r}^{8}{s_{4}}^{2}{\pi}^{2}{s_{1}}^{4}$
$\displaystyle-6\,{r}^{8}{s_{2}}^{2}{s_{1}}^{2}s_{3}-10\,{r}^{8}{s_{2}}^{2}{s_{1}}^{2}{s_{3}}^{2}+1152\,{r}^{8}{s_{4}}^{2}{\pi}^{2}{s_{2}}^{2}+64\,{r}^{6}s_{4}\,\pi\,{s_{1}}^{4}-8\,{r}^{6}{s_{1}}^{3}s_{3}\,s_{2}-8\,{r}^{6}{s_{1}}^{3}s_{2}{s_{3}}^{2}+1024\,{r}^{6}{s_{4}}^{2}{\pi}^{2}{s_{1}}^{3}$
$\displaystyle-16\,{r}^{6}{s_{2}}^{2}s_{1}{s_{3}}^{2}+64\,{r}^{6}{s_{2}}^{2}s_{1}s_{3}+112\,{r}^{4}s_{4}\,\pi\,{s_{1}}^{3}-9\,{r}^{4}{s_{1}}^{2}s_{2}{s_{3}}^{2}+42\,{r}^{4}{s_{1}}^{2}s_{3}\,s_{2}+896\,{r}^{4}{s_{4}}^{2}{\pi}^{2}{s_{1}}^{2}+448\,{r}^{4}{s_{4}}^{2}{\pi}^{2}s_{2}$
$\displaystyle+96\,{r}^{2}s_{4}\,\pi\,{s_{1}}^{2}+60\,{r}^{2}s_{1}s_{2}{s_{3}}^{2}-100\,{r}^{2}s_{2}s_{3}\,s_{1}+384\,{r}^{2}s_{1}{s_{4}}^{2}{\pi}^{2}+192\,{s_{4}}^{2}{\pi}^{2}{r}^{20}{s_{2}}^{5}+32\,s_{4}\,\pi\,{s_{2}}^{5}{r}^{18}+12\,{s_{2}}^{3}{r}^{12}{s_{1}}^{2}s_{3}$
$\displaystyle+8\,{s_{2}}^{4}{r}^{14}s_{1}s_{3}+6\,{s_{2}}^{3}{r}^{12}{s_{1}}^{2}{s_{3}}^{2}+4\,{s_{2}}^{4}{r}^{14}s_{1}{s_{3}}^{2}+128\,{r}^{10}{s_{4}}^{2}{\pi}^{2}{s_{1}}^{5}+8\,{r}^{10}{s_{2}}^{2}{s_{1}}^{3}s_{3}+4\,{r}^{10}{s_{2}}^{2}{s_{1}}^{3}{s_{3}}^{2}+16\,{r}^{8}s_{4}\,\pi\,{s_{1}}^{5}$
$\displaystyle+2\,{r}^{8}{s_{1}}^{4}s_{3}\,s_{2}+{r}^{8}{s_{1}}^{4}s_{2}{s_{3}}^{2}+3\,{s_{1}}^{2}+32\,s_{4}\,\pi\,s_{1}+{s_{2}}^{5}{r}^{16}+16\,s_{1}{r}^{2}s_{2}+32\,{r}^{6}{s_{2}}^{2}s_{1}-4\,{r}^{4}{s_{1}}^{4}s_{3}-3\,{r}^{4}{s_{1}}^{4}{s_{3}}^{2}+19\,{r}^{4}s_{2}{s_{1}}^{2}$
$\displaystyle-182\,{r}^{4}s_{3}\,{s_{2}}^{2}+75\,{r}^{4}{s_{2}}^{2}{s_{3}}^{2}+4\,{r}^{2}{s_{1}}^{3}s_{3}-6\,{r}^{2}{s_{1}}^{3}{s_{3}}^{2}-2\,{s_{2}}^{4}{r}^{12}s_{3}-3\,{s_{2}}^{4}{r}^{12}{s_{3}}^{2}+4\,{s_{2}}^{3}{r}^{10}s_{1}+4\,{r}^{8}{s_{2}}^{2}{s_{1}}^{2}-25\,{r}^{8}{s_{2}}^{3}{s_{3}}^{2}$
$\displaystyle+62\,{r}^{8}{s_{2}}^{3}s_{3}+144\,s_{4}\,\pi\,{s_{2}}^{4}{r}^{16}s_{3}\,s_{1}+96\,{r}^{10}s_{4}\,\pi\,{s_{1}}^{4}s_{3}\,s_{2}+256\,{s_{2}}^{3}{r}^{14}s_{4}\,\pi\,{s_{1}}^{2}s_{3}+224\,{s_{2}}^{2}{r}^{12}s_{4}\,\pi\,{s_{1}}^{3}s_{3}+480\,{s_{2}}^{3}{r}^{12}s_{4}\,\pi\,s_{1}s_{3}$
$\displaystyle+640\,{s_{2}}^{2}{r}^{10}s_{4}\,\pi\,{s_{1}}^{2}s_{3}+352\,{r}^{8}s_{4}\,\pi\,s_{2}{s_{1}}^{3}s_{3}+832\,{r}^{8}s_{4}\,\pi\,{s_{2}}^{2}s_{1}s_{3}+640\,{r}^{6}s_{4}\,\pi\,s_{2}{s_{1}}^{2}s_{3}+544\,{r}^{4}s_{4}\,\pi\,s_{1}s_{2}s_{3}$
$\displaystyle+{r}^{8}{s_{1}}^{4}s_{2}+4\,{r}^{10}{s_{1}}^{3}{s_{2}}^{2}+2\,{s_{2}}^{5}{r}^{16}s_{3}+{s_{2}}^{5}{r}^{16}{s_{3}}^{2}+6\,{s_{2}}^{3}{r}^{12}{s_{1}}^{2}+4\,{s_{2}}^{4}{r}^{14}s_{1}\Big{]}\,.\hskip
142.26378pt({\mathbf{C}})$

Appendix D

The adiabatic tangential index

$\displaystyle\Gamma_{t}=-\Bigg{(}-28\,s_{3}\,{s_{1}}^{2}+40\,s_{3}\,s_{2}+64\,{s_{4}}^{2}{\pi}^{2}+9\,{s_{1}}^{2}{s_{3}}^{2}+15\,{s_{2}}^{2}{r}^{4}+{s_{2}}^{4}{r}^{12}+15\,{r}^{8}{s_{2}}^{3}-{r}^{4}{s_{1}}^{4}+2\,{r}^{2}{s_{1}}^{3}+3072\,{s_{4}}^{2}{\pi}^{2}{s_{2}}^{3}{r}^{14}s_{1}$
$\displaystyle+128\,s_{4}\,\pi\,{s_{2}}^{4}{r}^{14}s_{3}+4224\,{s_{2}}^{2}{r}^{12}{s_{4}}^{2}{\pi}^{2}{s_{1}}^{2}+464\,{s_{2}}^{3}{r}^{12}s_{4}\,\pi\,s_{1}+2560\,s_{2}{r}^{10}{s_{4}}^{2}{\pi}^{2}{s_{1}}^{3}+608\,{s_{2}}^{2}{r}^{10}s_{4}\,\pi\,{s_{1}}^{2}+3840\,{s_{2}}^{2}{r}^{10}s_{1}{s_{4}}^{2}{\pi}^{2}$
$\displaystyle+320\,{s_{2}}^{3}{r}^{10}s_{4}\,\pi\,s_{3}+336\,{r}^{8}s_{4}\,\pi\,s_{2}{s_{1}}^{3}+3456\,{r}^{8}s_{2}{s_{1}}^{2}{s_{4}}^{2}{\pi}^{2}+656\,{r}^{8}s_{4}\,\pi\,{s_{2}}^{2}s_{1}+64\,{r}^{6}s_{4}\,\pi\,{s_{1}}^{4}s_{3}+512\,{r}^{6}s_{4}\,\pi\,s_{2}{s_{1}}^{2}$
$\displaystyle+2048\,{r}^{6}s_{2}s_{1}{s_{4}}^{2}{\pi}^{2}+384\,{r}^{6}s_{4}\,\pi\,{s_{2}}^{2}s_{3}+128\,{r}^{4}s_{4}\,\pi\,{s_{1}}^{3}s_{3}+368\,{r}^{4}s_{4}\,\pi\,s_{1}s_{2}+128\,{r}^{2}s_{4}\,\pi\,{s_{1}}^{2}s_{3}+160\,{r}^{2}s_{4}\,\pi\,s_{2}s_{3}$
$\displaystyle+1664\,{s_{4}}^{2}{\pi}^{2}{s_{2}}^{3}{r}^{16}{s_{1}}^{2}+896\,{s_{4}}^{2}{\pi}^{2}{s_{2}}^{4}{r}^{18}s_{1}+32\,s_{4}\,\pi\,{s_{2}}^{5}{r}^{18}s_{3}+1536\,{s_{2}}^{2}{r}^{14}{s_{4}}^{2}{\pi}^{2}{s_{1}}^{3}+256\,{s_{2}}^{3}{r}^{14}s_{4}\,\pi\,{s_{1}}^{2}+144\,{s_{2}}^{4}{r}^{16}s_{4}\,\pi\,s_{1}$
$\displaystyle+704\,s_{2}{r}^{12}{s_{4}}^{2}{\pi}^{2}{s_{1}}^{4}+224\,{s_{2}}^{2}{r}^{12}s_{4}\,\pi\,{s_{1}}^{3}+96\,{r}^{10}s_{4}\,\pi\,s_{2}{s_{1}}^{4}+16\,{r}^{8}s_{4}\,\pi\,{s_{1}}^{5}s_{3}+48\,s_{4}\,\pi\,s_{1}s_{3}+96\,s_{4}\,\pi\,{r}^{2}s_{2}+256\,s_{4}\,\pi\,{r}^{6}{s_{2}}^{2}$
$\displaystyle+832\,{s_{4}}^{2}{\pi}^{2}{r}^{16}{s_{2}}^{4}+128\,s_{4}\,\pi\,{s_{2}}^{4}{r}^{14}+1408\,{s_{2}}^{3}{r}^{12}{s_{4}}^{2}{\pi}^{2}-4\,{s_{2}}^{3}{r}^{10}s_{1}s_{3}-8\,{s_{2}}^{3}{r}^{10}s_{1}{s_{3}}^{2}+256\,{s_{2}}^{3}{r}^{10}s_{4}\,\pi+576\,{r}^{8}{s_{4}}^{2}{\pi}^{2}{s_{1}}^{4}$
$\displaystyle-6\,{r}^{8}{s_{2}}^{2}{s_{1}}^{2}s_{3}-10\,{r}^{8}{s_{2}}^{2}{s_{1}}^{2}{s_{3}}^{2}+1152\,{r}^{8}{s_{4}}^{2}{\pi}^{2}{s_{2}}^{2}+64\,{r}^{6}s_{4}\,\pi\,{s_{1}}^{4}-8\,{r}^{6}{s_{1}}^{3}s_{3}\,s_{2}-8\,{r}^{6}{s_{1}}^{3}s_{2}{s_{3}}^{2}+1024\,{r}^{6}{s_{4}}^{2}{\pi}^{2}{s_{1}}^{3}$
$\displaystyle-16\,{r}^{6}{s_{2}}^{2}s_{1}{s_{3}}^{2}+64\,{r}^{6}{s_{2}}^{2}s_{1}s_{3}+112\,{r}^{4}s_{4}\,\pi\,{s_{1}}^{3}-9\,{r}^{4}{s_{1}}^{2}s_{2}{s_{3}}^{2}+42\,{r}^{4}{s_{1}}^{2}s_{3}\,s_{2}+896\,{r}^{4}{s_{4}}^{2}{\pi}^{2}{s_{1}}^{2}+448\,{r}^{4}{s_{4}}^{2}{\pi}^{2}s_{2}$
$\displaystyle+96\,{r}^{2}s_{4}\,\pi\,{s_{1}}^{2}+60\,{r}^{2}s_{1}s_{2}{s_{3}}^{2}-100\,{r}^{2}s_{2}s_{3}\,s_{1}+384\,{r}^{2}s_{1}{s_{4}}^{2}{\pi}^{2}+192\,{s_{4}}^{2}{\pi}^{2}{r}^{20}{s_{2}}^{5}+32\,s_{4}\,\pi\,{s_{2}}^{5}{r}^{18}+12\,{s_{2}}^{3}{r}^{12}{s_{1}}^{2}s_{3}$
$\displaystyle+8\,{s_{2}}^{4}{r}^{14}s_{1}s_{3}+6\,{s_{2}}^{3}{r}^{12}{s_{1}}^{2}{s_{3}}^{2}+4\,{s_{2}}^{4}{r}^{14}s_{1}{s_{3}}^{2}+128\,{r}^{10}{s_{4}}^{2}{\pi}^{2}{s_{1}}^{5}+8\,{r}^{10}{s_{2}}^{2}{s_{1}}^{3}s_{3}+4\,{r}^{10}{s_{2}}^{2}{s_{1}}^{3}{s_{3}}^{2}+16\,{r}^{8}s_{4}\,\pi\,{s_{1}}^{5}$
$\displaystyle+2\,{r}^{8}{s_{1}}^{4}s_{3}\,s_{2}+{r}^{8}{s_{1}}^{4}s_{2}{s_{3}}^{2}+3\,{s_{1}}^{2}+32\,s_{4}\,\pi\,s_{1}+{s_{2}}^{5}{r}^{16}+16\,s_{1}{r}^{2}s_{2}-2\,{s_{2}}^{4}{r}^{12}s_{3}-3\,{s_{2}}^{4}{r}^{12}{s_{3}}^{2}+4\,{s_{2}}^{3}{r}^{10}s_{1}+4\,{r}^{8}{s_{2}}^{2}{s_{1}}^{2}$
$\displaystyle-25\,{r}^{8}{s_{2}}^{3}{s_{3}}^{2}+62\,{r}^{8}{s_{2}}^{3}s_{3}+32\,{r}^{6}{s_{2}}^{2}s_{1}-4\,{r}^{4}{s_{1}}^{4}s_{3}-3\,{r}^{4}{s_{1}}^{4}{s_{3}}^{2}+19\,{r}^{4}s_{2}{s_{1}}^{2}-182\,{r}^{4}s_{3}\,{s_{2}}^{2}+75\,{r}^{4}{s_{2}}^{2}{s_{3}}^{2}+4\,{r}^{2}{s_{1}}^{3}s_{3}$
$\displaystyle-6\,{r}^{2}{s_{1}}^{3}{s_{3}}^{2}+144\,s_{4}\,\pi\,{s_{2}}^{4}{r}^{16}s_{3}\,s_{1}+96\,{r}^{10}s_{4}\,\pi\,{s_{1}}^{4}s_{3}\,s_{2}+256\,{s_{2}}^{3}{r}^{14}s_{4}\,\pi\,{s_{1}}^{2}s_{3}+224\,{s_{2}}^{2}{r}^{12}s_{4}\,\pi\,{s_{1}}^{3}s_{3}+480\,{s_{2}}^{3}{r}^{12}s_{4}\,\pi\,s_{1}s_{3}$
$\displaystyle+640\,{s_{2}}^{2}{r}^{10}s_{4}\,\pi\,{s_{1}}^{2}s_{3}+352\,{r}^{8}s_{4}\,\pi\,s_{2}{s_{1}}^{3}s_{3}+832\,{r}^{8}s_{4}\,\pi\,{s_{2}}^{2}s_{1}s_{3}+640\,{r}^{6}s_{4}\,\pi\,s_{2}{s_{1}}^{2}s_{3}+544\,{r}^{4}s_{4}\,\pi\,s_{1}s_{2}s_{3}+4\,{s_{2}}^{4}{r}^{14}s_{1}$
$\displaystyle+{r}^{8}{s_{1}}^{4}s_{2}+2\,{s_{2}}^{5}{r}^{16}s_{3}+{s_{2}}^{5}{r}^{16}{s_{3}}^{2}+6\,{s_{2}}^{3}{r}^{12}{s_{1}}^{2}+4\,{r}^{10}{s_{2}}^{2}{s_{1}}^{3}\Bigg{)}\Bigg{(}12\,s_{1}+20\,s_{2}{r}^{2}+32\,s_{4}\,\pi+12\,s_{3}\,s_{1}+29\,{s_{2}}^{2}{r}^{6}+{s_{2}}^{4}{r}^{14}$
$\displaystyle+10\,{r}^{10}{s_{2}}^{3}+{r}^{6}{s_{1}}^{4}+8\,{r}^{4}{s_{1}}^{3}+19\,{r}^{2}{s_{1}}^{2}+256\,{s_{4}}^{2}{\pi}^{2}{s_{2}}^{3}{r}^{16}s_{1}+16\,s_{4}\,\pi\,{s_{2}}^{4}{r}^{16}s_{3}+384\,{s_{2}}^{2}{r}^{14}{s_{4}}^{2}{\pi}^{2}{s_{1}}^{2}+64\,{s_{2}}^{3}{r}^{14}s_{4}\,\pi\,s_{1}$
$\displaystyle+256\,s_{2}{r}^{12}{s_{4}}^{2}{\pi}^{2}{s_{1}}^{3}+96\,{s_{2}}^{2}{r}^{12}s_{4}\,\pi\,{s_{1}}^{2}+768\,{s_{2}}^{2}{r}^{12}s_{1}{s_{4}}^{2}{\pi}^{2}+112\,{s_{2}}^{3}{r}^{12}s_{4}\,\pi\,s_{3}+64\,{r}^{10}s_{4}\,\pi\,s_{2}{s_{1}}^{3}+768\,{r}^{10}s_{2}{s_{1}}^{2}{s_{4}}^{2}{\pi}^{2}$
$\displaystyle+320\,{r}^{10}s_{4}\,\pi\,{s_{2}}^{2}s_{1}+16\,{r}^{8}s_{4}\,\pi\,{s_{1}}^{4}s_{3}+304\,{r}^{8}s_{4}\,\pi\,s_{2}{s_{1}}^{2}+768\,{r}^{8}s_{2}s_{1}{s_{4}}^{2}{\pi}^{2}+176\,{r}^{8}s_{4}\,\pi\,{s_{2}}^{2}s_{3}+80\,{r}^{6}s_{4}\,\pi\,{s_{1}}^{3}s_{3}$
$\displaystyle+384\,{r}^{6}s_{4}\,\pi\,s_{1}s_{2}+112\,{r}^{4}s_{4}\,\pi\,{s_{1}}^{2}s_{3}+80\,{r}^{4}s_{4}\,\pi\,s_{2}s_{3}+48\,{r}^{2}s_{4}\,\pi\,s_{1}s_{3}+144\,s_{4}\,\pi\,{r}^{4}s_{2}+128\,s_{4}\,\pi\,{r}^{2}s_{1}+208\,s_{4}\,\pi\,{r}^{8}{s_{2}}^{2}$
$\displaystyle+64\,{s_{4}}^{2}{\pi}^{2}{r}^{18}{s_{2}}^{4}+16\,s_{4}\,\pi\,{s_{2}}^{4}{r}^{16}+256\,{s_{2}}^{3}{r}^{14}{s_{4}}^{2}{\pi}^{2}+8\,{s_{2}}^{3}{r}^{12}s_{1}s_{3}+4\,{s_{2}}^{3}{r}^{12}s_{1}{s_{3}}^{2}+112\,{s_{2}}^{3}{r}^{12}s_{4}\,\pi+64\,{r}^{10}{s_{4}}^{2}{\pi}^{2}{s_{1}}^{4}$
$\displaystyle+12\,{r}^{10}{s_{2}}^{2}{s_{1}}^{2}s_{3}+6\,{r}^{10}{s_{2}}^{2}{s_{1}}^{2}{s_{3}}^{2}+384\,{r}^{10}{s_{4}}^{2}{\pi}^{2}{s_{2}}^{2}+16\,{r}^{8}s_{4}\,\pi\,{s_{1}}^{4}+8\,{r}^{8}{s_{1}}^{3}s_{3}\,s_{2}+4\,{r}^{8}{s_{1}}^{3}s_{2}{s_{3}}^{2}+256\,{r}^{8}{s_{4}}^{2}{\pi}^{2}{s_{1}}^{3}$
$\displaystyle+26\,{r}^{8}{s_{2}}^{2}s_{1}{s_{3}}^{2}+42\,{r}^{8}{s_{2}}^{2}s_{1}s_{3}+96\,{r}^{6}s_{4}\,\pi\,{s_{1}}^{3}+22\,{r}^{6}{s_{1}}^{2}s_{2}{s_{3}}^{2}+36\,{r}^{6}{s_{1}}^{2}s_{3}\,s_{2}+384\,{r}^{6}{s_{4}}^{2}{\pi}^{2}{s_{1}}^{2}+256\,{r}^{6}{s_{4}}^{2}{\pi}^{2}s_{2}$
$\displaystyle+176\,{r}^{4}s_{4}\,\pi\,{s_{1}}^{2}+30\,{r}^{4}s_{1}s_{2}{s_{3}}^{2}+26\,{r}^{4}s_{2}s_{3}\,s_{1}+256\,{r}^{4}s_{1}{s_{4}}^{2}{\pi}^{2}+40\,s_{3}\,{r}^{2}s_{2}+48\,s_{1}{r}^{4}s_{2}+2\,{s_{2}}^{4}{r}^{14}s_{3}+{s_{2}}^{4}{r}^{14}{s_{3}}^{2}+4\,{s_{2}}^{3}{r}^{12}s_{1}$
$\displaystyle+6\,{r}^{10}{s_{2}}^{2}{s_{1}}^{2}+10\,{r}^{10}{s_{2}}^{3}{s_{3}}^{2}+16\,{r}^{10}{s_{2}}^{3}s_{3}+4\,{r}^{8}{s_{1}}^{3}s_{2}+28\,{r}^{8}{s_{2}}^{2}s_{1}+2\,{r}^{6}{s_{1}}^{4}s_{3}+{r}^{6}{s_{1}}^{4}{s_{3}}^{2}+26\,{r}^{6}s_{2}{s_{1}}^{2}+6\,{r}^{6}s_{3}\,{s_{2}}^{2}$
$\displaystyle+25\,{r}^{6}{s_{2}}^{2}{s_{3}}^{2}+10\,{r}^{4}{s_{1}}^{3}s_{3}+6\,{r}^{4}{s_{1}}^{3}{s_{3}}^{2}+8\,{r}^{2}s_{3}\,{s_{1}}^{2}+9\,{r}^{2}{s_{1}}^{2}{s_{3}}^{2}+64\,{r}^{2}{s_{4}}^{2}{\pi}^{2}+64\,{s_{2}}^{3}{r}^{14}s_{4}\,\pi\,s_{1}s_{3}+96\,{s_{2}}^{2}{r}^{12}s_{4}\,\pi\,{s_{1}}^{2}s_{3}$
$\displaystyle+64\,{r}^{10}s_{4}\,\pi\,s_{2}{s_{1}}^{3}s_{3}+304\,{r}^{10}s_{4}\,\pi\,{s_{2}}^{2}s_{1}s_{3}+272\,{r}^{8}s_{4}\,\pi\,s_{2}{s_{1}}^{2}s_{3}+288\,{r}^{6}s_{4}\,\pi\,s_{1}s_{2}s_{3}\Bigg{)}\Bigg{\\{}4\Bigg{(}12\,{s_{2}}^{2}{r}^{4}+3\,{r}^{6}{s_{2}}^{2}s_{1}+{r}^{8}{s_{2}}^{3}$
$\displaystyle+13\,s_{1}{r}^{2}s_{2}+3\,{r}^{4}s_{2}{s_{1}}^{2}-5\,s_{2}+5\,{s_{1}}^{2}+{r}^{2}{s_{1}}^{3}\Bigg{)}\Bigg{(}32\,s_{4}\,\pi+12\,s_{3}\,s_{1}+5\,{s_{2}}^{2}{r}^{6}+{s_{2}}^{4}{r}^{14}+6\,{r}^{10}{s_{2}}^{3}+{r}^{6}{s_{1}}^{4}+4\,{r}^{4}{s_{1}}^{3}+3\,{r}^{2}{s_{1}}^{2}$
$\displaystyle+256\,{s_{4}}^{2}{\pi}^{2}{s_{2}}^{3}{r}^{16}s_{1}+16\,s_{4}\,\pi\,{s_{2}}^{4}{r}^{16}s_{3}+384\,{s_{2}}^{2}{r}^{14}{s_{4}}^{2}{\pi}^{2}{s_{1}}^{2}+64\,{s_{2}}^{3}{r}^{14}s_{4}\,\pi\,s_{1}+256\,s_{2}{r}^{12}{s_{4}}^{2}{\pi}^{2}{s_{1}}^{3}+96\,{s_{2}}^{2}{r}^{12}s_{4}\,\pi\,{s_{1}}^{2}$
$\displaystyle+768\,{s_{2}}^{2}{r}^{12}s_{1}{s_{4}}^{2}{\pi}^{2}+112\,{s_{2}}^{3}{r}^{12}s_{4}\,\pi\,s_{3}+64\,{r}^{10}s_{4}\,\pi\,s_{2}{s_{1}}^{3}+768\,{r}^{10}s_{2}{s_{1}}^{2}{s_{4}}^{2}{\pi}^{2}+320\,{r}^{10}s_{4}\,\pi\,{s_{2}}^{2}s_{1}+16\,{r}^{8}s_{4}\,\pi\,{s_{1}}^{4}s_{3}$
$\displaystyle+304\,{r}^{8}s_{4}\,\pi\,s_{2}{s_{1}}^{2}+768\,{r}^{8}s_{2}s_{1}{s_{4}}^{2}{\pi}^{2}+176\,{r}^{8}s_{4}\,\pi\,{s_{2}}^{2}s_{3}+80\,{r}^{6}s_{4}\,\pi\,{s_{1}}^{3}s_{3}+384\,{r}^{6}s_{4}\,\pi\,s_{1}s_{2}+112\,{r}^{4}s_{4}\,\pi\,{s_{1}}^{2}s_{3}+80\,{r}^{4}s_{4}\,\pi\,s_{2}s_{3}$
$\displaystyle+48\,{r}^{2}s_{4}\,\pi\,s_{1}s_{3}+144\,s_{4}\,\pi\,{r}^{4}s_{2}+128\,s_{4}\,\pi\,{r}^{2}s_{1}+208\,s_{4}\,\pi\,{r}^{8}{s_{2}}^{2}+64\,{s_{4}}^{2}{\pi}^{2}{r}^{18}{s_{2}}^{4}+16\,s_{4}\,\pi\,{s_{2}}^{4}{r}^{16}+256\,{s_{2}}^{3}{r}^{14}{s_{4}}^{2}{\pi}^{2}$
$\displaystyle+8\,{s_{2}}^{3}{r}^{12}s_{1}s_{3}+4\,{s_{2}}^{3}{r}^{12}s_{1}{s_{3}}^{2}+112\,{s_{2}}^{3}{r}^{12}s_{4}\,\pi+64\,{r}^{10}{s_{4}}^{2}{\pi}^{2}{s_{1}}^{4}+12\,{r}^{10}{s_{2}}^{2}{s_{1}}^{2}s_{3}+6\,{r}^{10}{s_{2}}^{2}{s_{1}}^{2}{s_{3}}^{2}+384\,{r}^{10}{s_{4}}^{2}{\pi}^{2}{s_{2}}^{2}$
$\displaystyle+16\,{r}^{8}s_{4}\,\pi\,{s_{1}}^{4}+8\,{r}^{8}{s_{1}}^{3}s_{3}\,s_{2}+4\,{r}^{8}{s_{1}}^{3}s_{2}{s_{3}}^{2}+256\,{r}^{8}{s_{4}}^{2}{\pi}^{2}{s_{1}}^{3}+26\,{r}^{8}{s_{2}}^{2}s_{1}{s_{3}}^{2}+42\,{r}^{8}{s_{2}}^{2}s_{1}s_{3}+96\,{r}^{6}s_{4}\,\pi\,{s_{1}}^{3}+22\,{r}^{6}{s_{1}}^{2}s_{2}{s_{3}}^{2}$
$\displaystyle+36\,{r}^{6}{s_{1}}^{2}s_{3}\,s_{2}+384\,{r}^{6}{s_{4}}^{2}{\pi}^{2}{s_{1}}^{2}+256\,{r}^{6}{s_{4}}^{2}{\pi}^{2}s_{2}+176\,{r}^{4}s_{4}\,\pi\,{s_{1}}^{2}+30\,{r}^{4}s_{1}s_{2}{s_{3}}^{2}+26\,{r}^{4}s_{2}s_{3}\,s_{1}+256\,{r}^{4}s_{1}{s_{4}}^{2}{\pi}^{2}$
$\displaystyle+40\,s_{3}\,{r}^{2}s_{2}+8\,s_{1}{r}^{4}s_{2}+2\,{s_{2}}^{4}{r}^{14}s_{3}+{s_{2}}^{4}{r}^{14}{s_{3}}^{2}+4\,{s_{2}}^{3}{r}^{12}s_{1}+6\,{r}^{10}{s_{2}}^{2}{s_{1}}^{2}+10\,{r}^{10}{s_{2}}^{3}{s_{3}}^{2}+16\,{r}^{10}{s_{2}}^{3}s_{3}+4\,{r}^{8}{s_{1}}^{3}s_{2}$
$\displaystyle+16\,{r}^{8}{s_{2}}^{2}s_{1}+2\,{r}^{6}{s_{1}}^{4}s_{3}+{r}^{6}{s_{1}}^{4}{s_{3}}^{2}+14\,{r}^{6}s_{2}{s_{1}}^{2}+6\,{r}^{6}s_{3}\,{s_{2}}^{2}+25\,{r}^{6}{s_{2}}^{2}{s_{3}}^{2}+10\,{r}^{4}{s_{1}}^{3}s_{3}+6\,{r}^{4}{s_{1}}^{3}{s_{3}}^{2}+8\,{r}^{2}s_{3}\,{s_{1}}^{2}$
$\displaystyle+9\,{r}^{2}{s_{1}}^{2}{s_{3}}^{2}+64\,{r}^{2}{s_{4}}^{2}{\pi}^{2}+64\,{s_{2}}^{3}{r}^{14}s_{4}\,\pi\,s_{1}s_{3}+96\,{s_{2}}^{2}{r}^{12}s_{4}\,\pi\,{s_{1}}^{2}s_{3}+64\,{r}^{10}s_{4}\,\pi\,s_{2}{s_{1}}^{3}s_{3}+304\,{r}^{10}s_{4}\,\pi\,{s_{2}}^{2}s_{1}s_{3}$
$\displaystyle+272\,{r}^{8}s_{4}\,\pi\,s_{2}{s_{1}}^{2}s_{3}+288\,{r}^{6}s_{4}\,\pi\,s_{1}s_{2}s_{3}\Bigg{)}\left(1+s_{1}{r}^{2}+s_{2}{r}^{4}\right)\Bigg{\\}}^{-1}\,.\hskip
199.16928pt({\mathbf{D}})$

Data availability

The data underlying this article will be shared on any reasonable request to
the corresponding author.

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