# The role of an equation of state in the dynamical (in)stability of a radiating star

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## Abstract

The influence of an equation of state on the dynamical (in)stability of a sphere undergoing dissipative collapse is investigated for various forms of matter distributions. Employing a perturbative scheme we study the collapse of an initially static star described by the interior Schwarzschild solution. As the star starts to collapse it dissipates energy in the form of a radial heat flux to the exterior spacetime described by the Vaidya solution. By imposing a linear equation of state of the form \(p_r = \gamma \mu \) on the perturbed radial pressure and density we obtain the complete gravitational behaviour of the collapsing star. We analyse the stability of the collapsing star in both the Newtonian as well as the post-Newtonian approximations.

## 1 Introduction

The phenomenon of gravitational collapse has become a widely investigated subject due to its many interesting applications in astrophysics. The first solution for the non-adiabatic collapse of a spherically symmetric matter distribution in the form of a dust cloud and having a Schwarzschild exterior was produced by Oppenheimer and Snyder [1]. Taking into consideration that a radiating collapsing mass distribution has outgoing energy, its exterior spacetime is no longer a vacuum but contains null radiation. In this regard, Vaidya [2] obtained an exact solution to the Einstein field equations which describe the exterior field of a radiating spherically symmetric fluid. Santos [3] derived the junction conditions for a collapsing spherically symmetric shear-free non-adiabatic fluid with heat flow. This pioneering work allowed for the matching of the interior and exterior spacetimes of a collapsing star, and paved the way for the study of dissipative gravitational collapse. For some interesting developments on the subject of dissipative collapse, refer to [4, 5, 6, 7, 8, 9, 10].

When a system initially in static equilibrium experiences a disturbance or a perturbation, the stability of that system is affected. Dynamical stability is the property of a system to retain its stable state under a perturbation. The problem of stability is an important one in the study of self-gravitating objects because a static stellar model is of little significance if it proves to be highly unstable under collapse. Also, depending on the extent of instability of a system in collapse, the resulting patterns of evolution will vary. Hence the stability problem has become a subject of much investigation.

Chandrasekhar [11] was the first to examine the dynamical instability of a spherically symmetric mass with isotropic pressure. With the aid of the adiabatic index \(\varGamma \), he showed that for a system to remain stable under collapse, \(\varGamma \) must be greater than \(\frac{4}{3}\). Following this, much has been written in the literature about dynamical instability. Herrera et al. [12] studied the instability range for a non-adiabatic sphere and showed that relativistic corrections as a result of heat flow decreases the unstable range of \(\varGamma \) and renders the fluid less unstable. Chan et al. [13] investigated the stability criteria by deviating from the perfect fluid condition in two ways : they considered radiation in the free-streaming approximation, and they assumed local anisotropy of the fluid. Herrera et al. [14] also examined the dynamical instability of expansion-free, locally anisotropic spherical stellar bodies.

In this work, we consider a spherically symmetric static configuration undergoing radiative collapse. We assume shear-free and isotropic conditions. We impose a linear equation of state of the form \(p_r = \gamma \mu \) on the perturbed radial pressure and energy density, and we obtain the complete gravitational behaviour of the collapsing star. The stability ranges have been explored through the adiabatic index \(\varGamma \) for the Newtonian and post-Newtonian regimes.

The structure of this paper is as follows: In Sect. 2 we introduce the field equations describing the geometry and matter content for a star undergoing shear-free gravitational collapse. In Sect. 3 we present the exterior spacetime and the junction conditions necessary for the smooth matching of the interior spacetime with Vaidya’s exterior solution across the boundary. In Sect. 4 the perturbative scheme is described and the field equations for the static and perturbed configurations are given. In Sect. 5 we present the temporal equation employed in the perturbative scheme that begins with an initially static star that is perturbed so that the perturbations decay exponentially with time. Sect. 6 presents respectively a radiating and a static model . In Sect. 7 we discuss dissipative collapse, express the perturbed quantities in terms of two unspecified quantities and introduce an equation of state which allows us to present the perturbed quantities in terms of *r* only . We explore the stability of our collapsing model in the Newtonian and post-Newtonian approximations in Sect. 8, and our results are discussed in Sect. 9.

## 2 Interior spacetime

*t*and

*r*respectively.

## 3 Exterior spacetime and junction conditions

*m*(

*v*) represents the Newtonian mass of the gravitating body as measured by an observer at infinity.

*t*and

*v*.

## 4 The perturbed equations

We assume that the fluid is initially in static equilibrium, hence the fluid is described by quantities that are not time-dependent, ie they are expressed in terms of the radial coordinate only. We then assert that the static system is perturbed, undergoing slow shear-free collapse and producing pure radiation. We denote the quantities such as energy density, radial pressure and tangential pressure of the static system by a zero subscript and those of the perturbed fluid by an overhead bar. We further assume that the metric functions *A*(*r*, *t*) and *B*(*r*, *t*) have the same time dependence in their perturbations. This assumption would imply that the perturbed material functions also have the same time dependence.

*r*inside \(\varSigma \) for the static and perturbed configurations are respectively given by

## 5 The temporal equation of the perturbations

*t*increases. The solution (39) has been used by many authors investigating gravitational collapse. This solution formed the basis of the work of Chan et al. [13] in which they investigated the dynamical instability for radiating anisotropic collapse. More recently, Govender et al. [16] used the solution in their investigation on the role of shear in dissipative collapse.

## 6 A radiating model

*R*are constants and \(\lambda = -\beta _{\varSigma } + \sqrt{\alpha _{\varSigma }+\beta ^2_{\varSigma }}\). Then (25) and (26) can be written as

## 7 Dissipative collapse

*a*(

*r*) and

*b*(

*r*). Following Chan et al. [13] we adopt the following form for

*b*(

*r*)

*f*(

*r*), \(f^{\prime }(r) = 2r > 0\). Furthermore, \({\dot{T}} < 0\) since the fluid is collapsing. The heat flow is positive so it follows that \(\xi < 0\). For dissipative collapse \(\xi \ne 0\). We observe that for \(\xi = 0\) the heat flow vanishes.

*a*(

*r*) we impose an equation of state of the form

*a*(

*r*) which has the solution

## 8 Stability

## 9 Discussion

In this work we investigated the effect of an equation of state on the dynamical stability or instability of a spherically symmetric star undergoing dissipative collapse. We adopted a perturbative approach in which the star starts collapsing from an initial static configuration in the infinite past \((t = - \infty )\). The radial perturbation in the gravitational potential was determined by specifying a linear equation of state of the form \(p_r = \gamma \mu \). The temporal evolution of the model was determined from the junction conditions. We showed that the temporal behaviour of the model is intrinsically linked to the equation of state parameter, \(\gamma \). We studied the stability of our model by investigating the stability index and its dependence on \(\gamma \). We were in a position to clearly show the variation of the stability index with the equation of state parameter throughout the stellar fluid. More importantly, our investigation revealed that the stability depends on the nature of the matter making up the stellar interior. Our results are new and contribute to recent findings by Naidu et al. [18] and Bogadi et al. [19] in which they found that the equation of state affects the end state of collapse and temperature profiles of the radiating body.

## Notes

### Acknowledgements

The authors thank the unnamed referee for insightful remarks and references which greatly enhanced the quality of this work. This work was initiated at the “Developing Research Streams in the Department of Mathematics” workshop, Drakensberg KwaZulu Natal (2017). Financial support through the Incentive Funding for Rated Researchers from the National Research Foundation, Grant number 103420 is acknowledged.

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