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ORIGINAL RESEARCH article

Front. Astron. Space Sci., 17 April 2023
Sec. Stellar and Solar Physics
This article is part of the Research Topic Current Challenges of Compact Objects View all 5 articles

Analysis of class I complexity induced spherical polytropic models for compact objects

Adnan MalikAdnan Malik1S. A. Mardan
S. A. Mardan2*Tayyaba NazTayyaba Naz3Shiraz KhanShiraz Khan2
  • 1School of Mathematical Sciences, Zhejiang Normal University, Jinhua, Zhejiang, China
  • 2Department of Mathematics, University of Management and Technology, Lahore, Pakistan
  • 3National University of Computer and Emerging Sciences, Lahore, Pakistan

In this research, we present a comprehensive framework that uses a complexity factor to analyze class I generalized relativistic polytropes. We establish class I generalized Lane–Emden equations using the Karmarkar condition under both isothermal and non-isothermal regimes. Our approach considers a spherically symmetric fluid distribution for two cases of the generalized polytropic equation of state: 1) the mass density case μo and 2) the energy density case μ. To obtain numerical solutions for both cases, we solve two sets of differential equations that incorporate the complexity factor. Finally, we conduct a graphical analysis of these solutions.

1 Introduction

Physical factors interact to create complex characteristics in large-scale objects such as stars, galaxies, and their clusters, making their analysis challenging. A system is composed of elements arranged in a specific manner, and any disruption can cause complications. Complexity refers to how these elements work together to form a complex system. In astronomy, the complexity factor (CF) has become crucial to understanding various features of compact objects. Herrera (2018) proposed a new definition of complexity in general relativity for static self-gravitating objects by considering the orthogonal splitting of the curvature tensor into scalars, known as structural scalars. This definition includes physical factors such as anisotropic pressure and inhomogeneous energy density in terms of active gravitational mass. Abbas and Nazar (2018) applied this idea of CF in the framework of the f(R) theory for anisotropic fluid, calculating the effect of the f(R) term on the CF and obtaining exact solutions to the alternated field equations. Sharif and Butt (2018) and Sharif and Butt (2019) studied the impact of electric charge on the cylindrical static system with CF. Khan et al. (2019), Khan et al. (2021a), and Khan et al. (2021b) investigated uncharged and charged generalized polytropes (GPs) for spherical and cylindrical anisotropic inner fluid distribution using CF.

Modeling and describing compact objects are crucial topics in the relativistic discipline. It is not new to consider four-dimensional spacetime being embedded with higher-dimensional space. Schlafli (1871) introduced the embedding problem, and Eiesland (1925) proposed the necessary condition for the n-dimensional spacetime to be embedded in higher-dimensional space, which is that the Gaussian curvature must be zero, referred to as the Christoffel curvature tensor. Karmakar (1948) thoroughly examined this condition and developed an equation for class I embedding called the class I Karmarkar condition. However, Pandey and Sharma (1982) corrected the insufficiency of this equation. Maurya et al. (2015) solved the Einstein–Maxwell field equations by considering the charged ordinary baryonic matter and analyzed three sets of solutions (I, II, and III) of stellar models using the spherically symmetric metric of embedded class I. Singh and Pant (2016) and Singh et al. (2017) derived exact results for anisotropic fluid distribution using the Karmarkar class I condition and described several well-behaved models for various neutron stars. Ramos et al. (2021) found a class I interior solution for spherically symmetric inner fluid distribution using the polytropic equation of state (PEoS) and developed a compatible Lane–Emden equation with the Karmarkar condition under both isothermal and non-isothermal regimes. Malik et al. (2022a) investigated the class I interior solution for spherically symmetric inner fluid distribution in the f(R) theory of gravity. Some interesting work related to stellar structures in modified theories of gravities has been done by other researchers (Malik et al., 2022b; Malik et al., 2022c; Malik et al., 2022d; Malik, 2022; Shamir, 2022; Malik et al., 2023).

PEoS has been widely used in astrophysics and cosmology due to its simplicity. In the study of star structure, PEoS has been applied to investigate physical models of white dwarfs through Newtonian polytropes. Several researchers have analyzed different neutron stars in general relativity using PEoS. The idea of Newtonian polytropes was introduced by Chandrasekhar (1939), who used principles of thermodynamics to determine the mass and density limit of white dwarf stars. Tooper (1964) applied PEoS to analyze the solution of field equations for compressible fluid spheres in view of the general theory of relativity. He also studied models of massive hot stars through the numerical solutions of the Lane–Emden equation (LEe) (Tooper, 1965). Kaplan and Lupanov (1965) studied the relativistic effects in the context of the theory of polytropes and derived a relation between the mass of a sphere and its central density in general relativity. Managhan and Roxburgh (1965) used an approximation technique to examine the structure of turning polytropes by matching two solutions at the interface. Kaufmann (1967) investigated the solution of a single integro-differential equation (DE) under spherical static symmetry, which depended on the value of different polytropic indices. Occhionero (1967) studied the structures of turning polytropes for polytropic index n⩾2 and showed that a second-order approximation for the internal core of these structures, with suitable parameters, was more accurate than a first-order approximation. Finally, Kovetz (1968) removed some ambiguities in the theory of slowly turning polytropes defined by Chandrasekhar (1939).

PEoS has proven to be a useful tool for studying the structure of stars in astrophysics and cosmology due to its simplicity. Many researchers have explored different aspects of polytropic models for stars and spheres. For example, Chandrasekhar (1939) introduced the concept of Newtonian polytropes and determined the mass and density limit of white dwarf stars based on thermodynamics principles. Tooper (1964) and Tooper, (1965) used PEoS to analyze compressible fluid spheres in the context of general relativity, while Kaplan and Lupanov (1965) studied the relativistic effects of polytropes. Managhan and Roxburgh (1965) investigated the structure of turning polytropes using an approximation technique, and Kaufmann (1967) explored the solution of a single integro-differential equation under spherical static symmetry. Horedt’s research (Horedt, 1973; Horedt, 1987) investigated the behavior of slowly rotating cylinders, polytropic rings, and radially symmetric polytropes in dimensions higher than three. Sharma (1981) used Pade’s approximation to obtain analytic solutions for fundamental field equations in the context of stationary spheres, while Singh and Singh (1983) formulated models for relativistic polytropes that take into account rotation, tides, and deformations. Pandey et al. (1991) used relativistic PEoS to explore various parameters in static spherically symmetric structures, and Hendry (1993) developed uncomplicated polytropic models to describe the Sun’s interior using power-law relationships.

Herrera and Barreto (2004); and Herrera and Barreto (2013a) proposed a comprehensive approach to modeling different types of relativistic stars using PEoS. They introduced the Tolman mass to explain certain features of these models, particularly for static dissipative fluid spheres. Other studies, such as those by Herrera et al. (2014) and Herrera et al. (2016), applied PEoS to investigate the properties of spherical static fluids under conformally flat conditions and used the cracking method to analyze the relationship between energy density and mass. These investigations also explored various physical models and presented numerical results regarding spherical compact stars.

The generalized polytropic equation of state (GPEoS) offers greater freedom to explore the universe and astronomical objects at a deeper level. This equation of state consists of two equations, enabling us to study these objects in more detail.

i) PEoS, which discusses the dark matter of the universe, is

Pr=Kμo1+1n,(1)

where γ, K, n, and Pr are polytropic exponent, polytropic index, polytropic constant, and radial pressure, respectively.

ii) The linear equation of state, which discusses the dark energy of the universe, is defined as

Pr=α1μo,(2)

where α1 is a constant of proportionality.

The combination of Eqs 1, 2 defines the GPEoS (Azam et al., 2016) as

Pr=Kμoγ+α1μo=Kμo1+1n+1μo,(3)

replacing μo with μ; then, Eq. 3 is taken as

Pr=Kμ1+1n+α1μ.(4)
Azam et al. (2016) and Azam and Mardan (2017) used GPEoS to examine the impact of charge on generalized polytropes (GPs) while considering both spherical and cylindrical symmetries. Mardan et al. (2018) and Mardan et al. (2019) focused on the gravitational consequences of massive compact objects (COs) through GPs with spherical symmetry and explored various mathematical models of COs with radiation factors using GPEoS for different values of polytropic index n. They found these models physically plausible and well-behaved. In addition, Mardan et al. (2020a) and Mardan et al. (2020b) introduced novel classes of mathematical models and investigated the radius–mass relationship of compact stars using spherical symmetry and GPEoS.

The outline of this document is as follows: in Section 2, a spherically static anisotropic fluid distribution will be used to develop the basic equations and the Tolman–Oppenheimer–Volkoff (TOV) equation. In Section 3, the mass function for a self-gravitating source will be studied with the help of the Weyl tensor. The study of CF, which is defined through orthogonal splitting of the curvature tensor, will be covered in Section 4. In Section 5, relativistic GPs for the two cases, mass density and energy density, will be discussed. We will discuss the Karmarkar condition to develop the class I GPs in Section 6. A graphical solution will be given for class I GPs with CF in Section 7. A summary of this work will be given in Section 8.

2 Basic equations

Consider a static, spherically symmetric metric for an anisotropic fluid distribution, as

ds2=eνdt2eλdr2r2dθ2+sin2θdϕ2,(5)

where ν and λ are functions of “r”. The coordinates are x0 = t, x1 = r, x2 = θ, x3 = ϕ. The field equation Gνμ=8πTνμ must be satisfied by Eq. 5. The energy–momentum tensor defines the matter content for anisotropic fluid distribution as

Tμν=PrPsμsν+μ+PuμuνPgμν,(6)

where P denotes the tangential pressure.

sμ=0,eλ2,0,0,(7)

Eq. 7 represents four vectors, and four velocity uμ is given by

uμ=eν2,0,0,0,(8)

with sμuμ = 0, sμsμ = −1. Using Eqs 58, we have

8πμ=λr+1r2eλ1r2,(9)
8πPr=νr1r2eλ1r2,(10)
8πP=2λνr+ν2λν+2νeλ4,(11)

where the prime denotes the differentiation with respect to “r.” We take Schwarzschild spacetime at the exterior of the fluid distribution as

ds2=2Mrdt21+2Mr11dr2rdθ2+sin2θdϕ2.(12)

For smooth matching of the two metrics, Eq. 5 and 12, the first and second basic forms must be continuous (the Darmois condition) on the boundary r = rΣ = constant. Matching of this type gives the following results:

12MrΣ=eνΣ,(13)
12MrΣ=eλΣ,(14)
PΣ=0.(15)

Using Eqs 911, the TOV equation can be read as

Pr=2PPrrν2μ+Pr,(16)

and

ν=2m+4πPrr3rr2m,(17)

so,

Pr=2PPrrm+4πPrr3rr2mμ+Pr,(18)

m (mass function) is defined as

2mr=R2323=1eλ,(19)

otherwise,

m=4π0rr̄2μdr̄.(20)

It is better to write the energy–momentum tensor as

Tνμ=Δνμ+μuμuνPhνμ,(21)

with

Δνμ=Δsμsν+13hνμ;P=Pr+2P3.Δ=PrP;hνμ=δνμuμuν.(22)

3 Mass function through the Weyl tensor

The Weyl tensor Cαβμρ, Ricci scalar R, and Ricci tensor Rαβ can be used to illustrate the Riemann tensor as

Rαβμρ=Cαβμρ+12Rβgαμρ12Rαβδμρ+12Rαμδβρ12Rμμgαβ16Rδβρgαμgαβδμρ.(23)

The electric part (Eαβ = Cαγβδuγuδ) of the Weyl tensor can be written as

Cμνκλ=gμναβgκλγδημναβηκλγδuαuγEβδ,(24)

with gμναβ = gμαgνα and ημναβ being the Levi-Civita tensor and its magnetic part dissipating in a spherically symmetric case. Note that Eαβ can also be expressed as

Eαβ=E13hαβ+sαsβ,(25)

with

E=21eλr2+ν2λν2+ννλreλ4,(26)

satisfying

Eαγuγ=0,Eαγ=Eαγ,Eαα=0(27)

Using Eqs 911, 19, 23, and 25, we have

m=r3E3+4π3r3PPr+μ,(28)

and we have

E4π=1r30rr̄3μdr̄+PrP.(29)

Using Eq. 2928, we have

mr=4π3r3μ4π30rr̄3μdr̄.(30)

4 Vanishing complexity factor through orthogonal splitting of the Riemann tensor

We will now discuss the structural scalars that result from the orthogonal splitting of the curvature tensor (Bel, 1961). These scalars contribute to the definition of the CF (Herrera, 2018), and the subsequent tensors (Herrera et al., 2009; Herrera et al., 2011) represent the outcome of this splitting.

Yαβ=Rαγβδuγuδ,(31)
Zαβ=*Rαγβδuγuδ=12ηαγϵμRβδϵμuγuδ,(32)
Xαβ=*Rαγβδ*uγuδ=12ηαγϵμRϵμβδ*uγuδ,(33)

where * shows the dual tensor, i.e., Rαβγδ*=12ηϵμγδRαβϵμ. Equation 23 may be expressed as

Rβδαγ=Cβδαγ+28πTβδδαγ+8πT13δβδδαγδβδδαγ.(34)

Using Eq. 18 in Eq. 34 gives the splitting of the Riemann tensor as

Rβδαγ=RIβδαγ+βIIβδαγ+RIIIβδαγ,(35)

where

RIβδαγ=16πμuαγuβδδ28πPhβδδαγ+8μ3P13δβδδαγδαγβδδ,(36)
RIIβδαγ=16πΔβδδαγ,(37)
RIIIβδαγ=4uαγuβEδϵμαγϵβδνEμν=0,(38)

with

ϵαγβ=uμημαγβ,ϵαγβuβ=0.(39)

We can find the explicit expressions for the three tensors, Yαβ, Zαβ, and Xαβ, as

Yαβ=hαβ3P+μ4π3+4πΔαβ+Eαβ,(40)
Zαβ=0,(41)

and

Xαβ=8π3μhαβ+4πΔαβEαβ.(42)

Scalars XT, XTF, YT, and YTF, are defined through the tensors Xαβ and Yαβ (Bel, 1961) as

XT=8πμ,(43)
XTF=4πΔαβE,(44)
XTF=4πr30rr3̄μdr̄,(45)
YT=4π3Pr+μ+2Δ,(46)
YTF=E+4πΔ,(47)

using Eq. 29

YTF=8πΔ4πr30rr̄3μdr̄.(48)

From Eq. 45 and 48, we get

8πΔ=XTF+YTF.(49)

Many factors, including pressure anisotropy, charge, heat dissipation, viscosity, and density inhomogeneity, are responsible for the complexity of a system. Any system, in general, lacking these factors, with the exception of isotropic pressure and energy density, should be regarded as the simplest system with vanishing complexity. In addition, the complexity of the system for fluid distribution is brought about by inhomogeneous density and anisotropic pressure. The “complexity factor” is related to structure scalar YTF, since Eq. 49, which defines it, involves these factors. Therefore, when we apply the condition YTF = 0 on Eq. 48, it gives

Δ=12r30rr̄3μdr̄.(50)

5 Relativistic spherical generalized polytropes

In this section, we talk about the mass density and energy density of GPEoS for anisotropic fluids in both isothermal and non-isothermal regimes (Azam et al., 2016).

5.1 Non-isothermal regime

5.1.1 Case 1

In this case, mass density is used to study GPEoS.

Pr=α1μo+Kμoγ=α1μo+Kμo1+1n,(51)

taking γ ≠ 1 and energy density μ connected with total mass density μo (Herrera et al., 2014) as

Pr=1nμμo.(52)

The following assumptions are made:

α=Prcμc,rA=ξ,αn+1A2=4πμc.(53)
μocψo=μo,4πμcvξ=mrA3.(54)

The dimensionless form of TOV Eq. 18 is

4πPrc2ψon1+nαα1+αα1nψo+1αnα1+α1n+1×ξ3ψonα1+ψoαα1+αα1nαα1n+v/αξ8πPrcvαA2ξ2Δξ+1αPrcψon1ψo×1+nψoαα1nα1α+1αnα1n=0,

where prime shows differentiation w. r. t. ξ. From Eq. 19 and 9, we have

m=4πr2μ,(55)

or, using Eqs 53 and 54, we get

dvdξ=ξ2ψonnψoαα1+αα1nαn1α1n+1.(56)

ξ = ξn defines the boundary such that ψo(ξo) = 0, and the following assumptions are made at the boundary:

vξ=0=0andψoξ=0=1.(57)

Equations 55 and 56 give the generalized spherical LEe

4πψo1+nPrc2αα1+nαα1ψon×ξ3ψonα1+ψoαα1+αα1nαα1n+v/αξαA2ξ8πPrcv4πnPrc2ψon1ψoαn1×α1+α1n+1n+1ψoαα1+αα1n×ξ3ψonα1+ψoαα1+αα1n+αα1n+v/αξαA2ξ8πPrcv+4πPrc2ψonαn1×1+α1+α1nψo1+nαα1+αα1n×ξ3ψonα1+αα1n+αα1ψoαα1n+v/αξ2αA2ξ8πPrcv+4πPrc2ψon1+n×αα1+αα1nψoαn1α1+α1n+1×ξ3ψonα1+αα1+αα1nψoαα1n+v×8πξ2Prcψonnψoαα1+αα1nαn1α1n+1αA2/αξαA2ξ8πPrcv2+4πξPrc2ψo2n11+nψoαα1+αα1nαn1α1+α1n+1ξψo1+nψo×αα1+αα1n+α1n1αn+ψon+3×αα1+αα1nψo+1αnα1n+3+1/ααA2ξ8πvPrc+2Δξ2+Prcψon1ψo1+n×αα1+αα1nψoα1nαn11α1+n+Prcψon2ψo21+nαα1n+αα1ψo+α1n1αn1α+1α1+nPrcαα1+αα1nψon1ψo2=0.

5.1.2 Case 2

It is easy to see GPEoS with energy density (Azam et al., 2016) as

Pr=Kμ1+1n+α1μ,(58)

total energy density μ is used in place of mass density μo in Eq. 52 according to the relation (Herrera and Barreto, 2013b) as

μ=μo1Kμo1/nn,(59)

considering

ψn=μμc.(60)

The TOV equation is obtained as

1αξ8πPrcvαA2ξ4πPrc2ψnψαα1+1+α1×vξ3ψnα1αψα12Δξ+1αPrcψn1ψn+1αα1ψ+α1n=0,

and from Eq. 55, we have

dvdξ=ξ2ψn.(61)

Equation 61 gives the generalized LEe

1αξ8πPrcvαA2ξ4πnPrc2ψn1ψψαα1+1+α1×vξ3ψnα1+αα1ψ1αξ8πPrcvαA2ξ×4πPrc2αα1ψnψvξ3ψnα1αψα1+1αξ28πPrcvαA2ξ4πPrc2ψnψαα1+1+α1×vξ3ψnψαα1+α1+1αξαA2ξ8πPrcv2×4πPrc2ψnαα1ψ+1+α18πξ2PrcψnαA2×vξ3ψnψα1αα11α8πPrcvαA2ξ×4πξPrc2ψ2n1ψαα1+α1+1×ψ3αα1ψ+3α1+1+ξ1+nαα1ψ+α1nψ+2Δξ2+1αψn1ψPrcnα1+1+nψαα1+1αPrcn1ψn2ψ2n+1ψαα1+α1n+1α1+nPrcαα1ψn1ψ2=0.

5.2 Isothermal regime

In this regime, only the energy density case (μ) will be discussed because mass density (μo) and energy density (μ) become the same in both cases for the isothermal regime (γ = 1). In this regime, ψ is defined as

eψ=μμc.(62)

Introducing dimensionless variables, we get

α=Prcμc,rA=ξ,B2α=4πμcvξ4πμc=mrB3,(63)

so, Eq. 55 changes to

dvdξ=eψξ2,(64)

and the TOV equation becomes

1αξαB2ξ8πPrcv4πα+1Prc2e2ψαξ3+eψv2ΔξPrceψψ=0.(65)

From Eq. 64 and Eq. 65, we have the second-order generalized LEe

Bαξ2αB2ξ8πPrcv24πα+1ξ3Prc2e2ψ×αB2ξ2αξψ+α+1+8πPrcv2αξψ2α1+4πα+1Prc2e2ψveψ8πPrcvξψ+1αB2ξξψ+2+8πξ3Prc+2αΔαB2ξ8πPrcv2+αξ2Prceψψ2ψαB2ξ8πPrcv2+32π2αα+1ξ6Prc3e3ψ=0.

6 Class I spherical generalized polytropes

It is sometimes helpful to merge four-dimensional spacetime with higher dimensions when studying cosmological phenomena (Karmakar, 1948), and one such merging is the Karmarkar condition (Maurya et al., 2015), read as

R1010R2323=R1212R0303+R1202R1303,(66)

with R2323 ≠ 0, it gives

2ν=eλλeλ1λ2,eλ1.(67)

6.1 Case 1

From Eqs 5154, 67, we have

Δ=3mrmm4πr3ψn+1Prcα1μc4πα1μcr3ψn16πmr3.(68)

Equation 68 in dimensionless form is

Δ=14vξ3μcvξ3ψonα1α1+ααα1nψoαα1n3vξ3ψonnψoαα1+αα1n+1αn1+α1n.(69)

Equations 69 and 55 together provide the class I spherical generalized TOV equation.

12αAPrc18πPrcvαA2ξξ8πPrcψon1+n×αα1+αα1nψoαn1α1+α1n+1×ξ3ψonα1+ψoαα1+αα1nαψoαn+v+2ψoψon1ψo1+nα+αα1nα1α1nαn11ξ4vvξ3ψonα1+α+αα1nα1ψoαα1n×3vξ3ψonnψoαα1+αα1n+1αnα1n+1=0,

so, the class I spherical generalized LEe is

12αAPrc1ψo22n1nnα1α1ψo2+1ξ2A2αξ8πPrcvnα1α+αα1ψo×8πPrcn+3v+n+1ξvA2n+3αξ+1ξψon2n+1nα1α+αα1ξψo2nα1×n+3α1+1ψo+21nαξα1ψo+1ξ2A2αξ8πPrcvξA2αnα13+nα1+1+1+nnα1α+αα1ξn+3ψo+2ξψo8πPrcnα1nα1+α1+1v8πPrcvnα1×3+nα1+1+1+nnα1α+αα1ξ×n+3ψo+2ξψoψon+1v2×ξ2nαα1+α1+nα1α+αα1ψo×nnα1α+αα1ψonα1nα1+1vψo2n+1A2αξ8πPrcv28πPrcξψo2n1n+1×nα1α+αα12αξA2+8πPrcξv2vψo3+nα1α+αα12n+12nα1α+αα1ξ×A2αξ8πPrcvψonα12n+1α1+1×αξA2+8πPrcξv2vψo2nα1×2n+12n+1α1+1nα1α+αα1ξ×A2αξ8πPrcvψo+1nαα11+α1+nα1×αξA2+8πPrcξv2vψo+2n1nα2α1×1+α1+nα1ξA2αξ8πPrcvψo1vξψo2n12nnα1α+αα12ψo3+2nα1α+αα1nn+1nα1α+αα1ξψonα12nα1+1ψo2+nα12nα1α1×nα1+12n+12nα1+1nα1α+αα1ξψo×ψo+2nnα12α1nα1+1ξψo+1ξ54v1A2αξ8πPrcv22πPrcξ3ψon12n+1×nα1α+αα1A2αξ4πPrcv+ξvψo2+nα1α+αα1ξA2αξ8πPrcvψo1+n212×1nαα1+nα1++1A2αξ4πPrcv+ξvψo+n1nαnα1+α1+1ξA2αξ8πPrcvψo+33vξ4=0.(70)

6.2 Case 2

Using Eqs 58, 60, 67,

Δ=3mrmm4πr3ψn+1Prcα1μc4πα1μcr3ψn16πmr3,(71)

the dimensionless form of Eq. 71 is

Δ=μc3vξ3ψnξ3ψnα1αψα1+v4ξ3v.(72)

The class I spherical generalized TOV equation for the energy density case is

1ξ8πPrcvαA2ξ8πPrcψnψ1+αα1+α1×vψnψα1αψα1+2ψn1ψn+1×αα1ψ+α1n+1ξ4vξ3ψn3v×ξ3ψnα1αψα1+v=0.(73)

Then, using Eq. 61 and Eq. 73, the second-order class I spherical generalized LEe is

ψn1ξ2αA2ξ8πPrcvξαA23α1+ξn+1αα1×2ψξ+3ψ1+8πα1+1Prcv+8πPrcv3α1n+1ξαα13ψ+2ξψ+1+1ξ2αα1ψ8πξPrcvαA2ξ8πPrcv3+1ξψn2α1ξψ+ψ3α1+2n+1ξαα1ψ+1+1v2α1n1nψ2+1ξ54v1αA2ξ8πPrcv2×2πξ3Prcψn1ξψαA2ξ8πPrcvn+1ψ×αα1+α1+1n2ψ2αα1+α1+1×αA2ξ4πPrcξv+v+3+1αA2ξ8πPrcv2×8πξPrcψ2n1αα1ψ22n+1ξαα1ψ×αA2ξ8πPrcv+2α1+1αA2ξ+8πPrc×ξv2v+ψ2α1+12n+1ξαα1ψ×αA2ξ8πPrcv+α1α1+1αA2ξ+8πPrc×ξv2v+2α1α1+1nξψαA2ξ8πPrcv+αα12ψ3αA2ξ+8πPrcξv2v1v2ξ2ψ2nvα1αψα1+1v×ξψ2n12ψα1αψα1+ξψ2n+1×α1αψ2α1n3vξ4=0.(74)

In the isothermal regime (γ = 1), the procedure is now repeated for only the energy density case. In order to accomplish this, we take Eqs 58, 62, and 67 and obtain

Δ=eψ3mrmmeψ4πPrcr316πmr3,(75)

the dimensionless form of Eq. 75 is

Δ=Prceψ3νξννeψαξ34ανξ3,(76)

for (γ = 1), the class I spherical generalized TOV equation is

12BPrce2ψ8πPrceψαeψ+eψαξ3+eψvαξαB2ξ8πPrcv2eψψeψ3vξ3eψeψvαξ3αξ4v=0,(77)

and the class I spherical generalized LEe is

1αξvαB2ξ8πPrcvBPrceψα3B4ξ11+v×16παξ9Prc4πα+2PrcvαB2ξ+12e3ψv2×αB2ξ8πPrcv2+2αξ6eψα2B4ξ2ξψ1+4πB2ξPrcv2αα+3ξψ+αα+6+1+64π2×α+2Prc2v2ξψ1+ξ3e2ψvα2B4ξ2×ξψ2αξψ3α12αξψ3α4+8παB2ξPrcvξ4αξψ4αξψ2+5α+1ψ+4α+6+64π2Prc2v22αξψξψ1ξψ2α3=0.(78)

The following conditions should be satisfied:

μ>0,Prμ,Pμ.(79)

For case 1, (γ ≠ 1), conditions Eq. 79) take the form

αα1+nαα1n+1+nα11nαψonμc>0,α1ψ0+αψoα1αα1+nαα1n+1+nα11nα1,3μov2ξ+μoξ5ψo2nα1+nαα1+αα1+nαα1ψoαα1+nαα1n+1+nα11nαξ2vψon1nα5+3α1μc+3+5nnα1α1μc+4α1μoc,(80)

and these conditions (Eq. 79) for case 2 (γ ≠ 1) are taken as

μ>0,α1+ψαα11,3vξψn+αξ5ψn+1v+ξ24α15+α4α1ψ0.(81)

Now, for the isothermal regime (γ = 1), these conditions will be

μceψ>0,α1,eψα5ξ3+e2ψαξ6v+3v0.(82)

7 Vanishing complexity factor and class I relativistic spherical generalized polytropes

7.1 Case no. 1

Vanishing complexity factor YTF = 0 is integrated with Eqs 53 and 54, and the result is

1ψo4πξ22ξψoΔ+6Δψo+μcnξψonψon+1ψo×αα1+αα1nαn1α1n+1=0.(83)

A system of first-order DEs is formed by Eq. 5670 and 83. For constant values of α = .5 and α1 = .5, this system is numerically solved. Figures 13 show the patterns of v, ψo, and Δ for various n values.

FIGURE 1
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FIGURE 1. v(ξ) curves.

FIGURE 2
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FIGURE 2. ψo(ξ) curves.

FIGURE 3
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FIGURE 3. Δ(ξ) curves.

7.2 Case no. 2

CF in this case will be expressed as

4πξ22ξΔ+6Δμcnξψn1ψ=0,(84)

for both α1 = .5 and α = .5, Figures 46 illustrate how v, ψ, and Δ behave for various n values. The system of ordinary DEs (61, 73, 71) is solved to achieve these characteristics numerically.

FIGURE 4
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FIGURE 4. v(ξ) curves.

FIGURE 5
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FIGURE 5. ψ(ξ) curves.

FIGURE 6
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FIGURE 6. Δ(ξ) curves.

Vanishing CF YTF = 0 with class I GPs (γ = 1) will be regarded as

4πξ22ξΔ+6Δμcξeψψ+6sinhψ.(85)

The variables that are used in case 2(γ ≠ 1) constitute a set of first-order DEs (62, 77, 85). This set of DEs was also numerically solved for different values of parameters. Graphs of Figures 46 are used to illustrate the actions of v, ψ, and Δ.

8 Summary

Dark energy and dark matter are vital aspects of the universe which are explained by the GPEoS. It describes different scenarios of these two special features of astrophysics and gives information about the early and late universe (Babichev et al., 2004; Mukhopadhyay et al., 2008; Chavanis, 2012; Chavanis, 2014a; Chavanis, 2014b). Different self-gravitating system have been discussed through LEe over a long period of time (Lane, 1870; Chandrasekhar, 1939). The significance of this equation is that it helps to study the gravitational collapse of systems, stability of relativistic stellar objects, and density and pressure profiles of dark matter (Abellan et al., 2019; Bhatti and Tariq, 2019; Wojnar, 2019). In recent years, the concepts of GPEoS (Azam et al., 2016; Azam and Mardan, 2017; Mardan et al., 2018; Khan et al., 2019; Mardan et al., 2019; Mardan et al., 2020a; Mardan et al., 2020b; Khan et al., 2021a; Khan et al., 2021b), complexity factor (Abbas and Nazar, 2018; Herrera, 2018; Sharif and Butt, 2018; Khan et al., 2019; Sharif and Butt, 2019; Khan et al., 2021a; Khan et al., 2021b), and the Karmarkar condition (Karmakar, 1948; Maurya et al., 2015; Singh and Pant, 2016; Singh et al., 2017; Ramos et al., 2021) have all been widely used to explain various physical aspects and characteristics of self-gravitating compact objects. These three theories about self-gravitating stellar structures have been combined in the current study to explore a few of their characteristics (v, ψo, ψ, and Δ) under isothermal and non-isothermal regimes. For this reason, a generalized framework is built to create a modified version of the class I spherical LEe. The class I spherical TOV equation is constructed using field equations. Structure scalars are generated by means of the curvature tensor, the Weyl tensor and mass function are built, and CF is defined using these scalars. Class I spherical generalized LEes are developed through GPEoS for two cases: 1) mass density and 2) energy density in both non-isothermal and isothermal regimes. These LEes give us class I spherical GPs to study some features of self-gravitating stellar structures. Additionally, the energy conditions for each case have been determined. Vanishing CF with three pairs of LEes (55, 70), (61,73), and (64,77) generate three sets of DEs. The numerical solutions to these sets of DEs are presented graphically.

The numerical solution of the set of DEs (55, 70, 83) of case (1) is explained by the curves in Figures 13. The value of v for different values of n has been shown in Figure 1, which shows that the value of v is zero at the center of the spherical self-gravitating object, and it increases for higher values of n along the increasing direction of radius. It can also be seen that this object is more compact for n = .5 (curvea), and its compactness decreases for higher values of n(curved). The curves in Figure 2 show the behavior of ψo, which has its highest value at the center and it steadily diminishes as the radius increases. For n = .5, it is zero at the boundary surface. The curves in Figures 1, 2 are all smooth and exhibit normal behavior for various parameter values. The curves of Figure 3 express the response of variable Δ. It can be observed that these curves exhibit the same pattern as shown by the curves of Figure 2, except curve (a) of variable Δ, which shows some abnormal behavior at n = .5.

In case (2), Figure 4 shows the pattern of the variable v for different values of ξ. It has zero value at the center and gradually increases with the increase of ξ, and it attains maximum value at the boundary surface for the maximum value of n.

Figure 5 explains the behavior of variable ψ for different values of n. It attains maximum values at the center, which continuously decrease toward the boundary. It can be observed from Figure 6 that the measure of anisotropy has smaller values at the center of a self-gravitating object, and it attains maximum values at the middle of the radius of the object. It then starts decreasing until it reaches the minimum again at the boundary of the object.

Figures 79 illustrate the results of variables v, ψ, and Δ through the solutions of the set of DEs (62, 77, 85), for the isothermal regime. Figure 7 shows the exponential increase in variable v from the center to the boundary as the value of α decreases. Meanwhile, variable ψ in Figure 8 shows an exponential decrease as α decreases. The variable Δ in Figure 9 exhibits some abnormal behavior for smaller values of α. It can be seen from Figure 9 that Δ has the maximum value at the center and has a very small value at the boundary (curvea), and with a decrease in the value of α, it has higher values at the boundary (curvebandc), while at α = .0005, Δ changes its orientation and becomes minimum at the center (curved).

FIGURE 7
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FIGURE 7. v(ξ) curves.

FIGURE 8
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FIGURE 8. v(ξ) curves.

FIGURE 9
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FIGURE 9. Δ(ξ) curves.

In the presence of anisotropic pressure, we have proposed the basic framework for the solutions of class I spherical generalized relativistic LEes with CF. We undertook this task by showing the perceptible presence of anisotropy in cosmological objects and its influence on the structure of such objects. Another factor is that fluid systems can be represented by the solutions of LEe with a number of applications in astrophysics and cosmology. The major goals of this study are to build a modified version of the class I spherical generalized LEe in relation to CF under isothermal and non-isothermal regimes and to numerically solve the systems of DEs. Some possible extensions to this work are the development of more generalized frameworks in modified theories of gravity such as f(R) and f(R, T) (Manzor and Shahid, 2021; Mumtaz et al., 2022).

Data availability statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding authors.

Author contributions

All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.

Funding

This work was financially supported by Zhejiang Normal University for Postdoctoral project (No. YS304023912).

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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Keywords: complexity, compact objects, spherical polytropic models, spherical symmetry, anisotropy

Citation: Malik A, Mardan SA, Naz T and Khan S (2023) Analysis of class I complexity induced spherical polytropic models for compact objects. Front. Astron. Space Sci. 10:1182772. doi: 10.3389/fspas.2023.1182772

Received: 09 March 2023; Accepted: 22 March 2023;
Published: 17 April 2023.

Edited by:

Sehrish Iftikhar, Lahore College for Women University, Pakistan

Reviewed by:

Akram Ali, King Khalid University, Saudi Arabia
Shamaila Rani, COMSATS University Islamabad, Pakistan

Copyright © 2023 Malik, Mardan, Naz and Khan. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: S. A. Mardan, syedalimardanazmi@yahoo.com

Disclaimer: All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.