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We here present a relativistic model for a spherically symmetric anisotropic fluid to study the various factors of physical and thermal phenomenon during the evolution of a collapsing star dissipating energy in the form of radial heat flow. We also proposed a table of some new parametric class of solutions which will be useful for constructing the new compact star models. The constructed algorithm obeys all the relevant requirements of a realistic model and matched with Vaidya exterior metric over the boundary. At the initial stage the interior solutions represent a static configuration of perfect fluid which then gradually starts evolving into radiating collapse. The apparent luminosity as observed by the distant observer at rest at infinity and the effective surface temperature are zero in remote past at the instant when collapse begins and at the stage when collapsing configuration reaches the horizon of the black hole.

In modern Astrophysics and Cosmology, a detailed description of gravitational collapse of massive stars and the modeling of the structure of compact objects such as Neutron star, Quasar, Supernovae, Black hole etc. under various conditions is the most interesting phenomena. The final outcome of the gravitational collapse is an important open issue in relativistic astrophysics (Joshi and Malafarina [

In order to construct the new realistic models, it is desirable to solve the Einstein's field equations but due to non linear character of the equations it is a very difficult task; various efforts have been made in this direction. The maiden exact solution of spherical gravitational collapse was due to Oppenheimer and Snyder [

Herrera and Santos [

Bowers and Liang [

The main objective of this work is to present a simple anisotropic collapsing radiating fluid model and discuss all the relevant thermal and physical conditions by taking Tewari and Charan [

The interior space-time of a shear-free spherically symmetric fluid with the coordinates

The energy-momentum tensor for the matter distribution with anisotropy in pressure is

where

Assuming comoving coordinates, we have

Since the interior fluid is radiating energy in the form of heat therefore the exterior space-time of a collapsing radiating star is described by Vaidya’s outgoing exterior metric [

where v is the retarded time and

The junction conditions for radiating star matching two line elements (1) and (3) at the boundary continuously across a spherically symmetric time-like hyper surface

where

Some other physical and thermal features of the collapsing matter are the surface luminosity and the boundary redshift

The total luminosity for an observer at rest at infinity is

In order to solve the Non-trivial Einstein’s field equations which are generated by (1) and (2), we choose a separable form of the metric coefficients given in (1) into functions of r and t coordinates as

where

Here the quantities with the suffix 0 corresponds to the static star model with metric components

In the absence of non-adiabatic dissipative forces the Equation (5),

where

Tewari [

Here we observed that the function

To find a new parametric class of exact solutions of pressure anisotropy equation which is created by the Equa- tions (11) and (12), Tewari and Charan [

Making an adhoc relationship between the variables in (21), the above mentioned authors obtained the following solution

where

where n is real if

For different values of n or l Equations (22) and (23) give a variety of solutions and they are categorized as isotropic pressure and homogeneous density, isotropic pressure and inhomogeneous density while some inhomogeneous density and anisotropic pressure. For

To construct a new relativistic model, in the present study we assume

The junction condition

Here from (30) and (31), we are seeing that at the centre radial and transverse pressures are equal and anisotropy vanishes there.

A physically reasonable solution should satisfy certain energy conditions and they are:

The central values of both the pressures, energy density and gravitational potential should be non-zero positive definite and the solution should have monotonically decreasing expressions for the pressures and density with the increase of r. Thus in view of these conditions, we write the bounds of model parameters

Now using (10)-(13), (19), (26) and (27) the expressions for

We can see the physical parameters

The fluid collapse rate

where by using (18), (26) and (27) we have

The constant

The total energy entrapped inside the surface

where

Using (4) and (27), we get the physical radius of the collapsing radiating star as

Using (7)-(9), (18), (26), (27) and (38) the expressions for the surface luminosity, the boundary redshift on

To obtain the black hole formation time, the surface redshift goes to infinity, for this the term in the parentheses of (44) goes to zero and we have

To investigate the temperature inside and on the boundary surface, we utilize temperature gradient law (Israel et al. [

where

The effective surface temperature observed by external observer can be calculated from the expression (Schwarzschild [

where for photons the constant

The arbitrary function

Temperature distribution throughout the interior of the collapsing radiating star is given by

It follows that the surface temperature of the collapsing star tends to zero at the beginning of the collapse

We here presented a new radiating fluid model collapsing in the influence of its own gravity using Tewari and Charan [

S.N. | n | l | |||
---|---|---|---|---|---|

1 | 2 | ||||

2 | 1 | ||||

3 | 0 | ||||

4 | |||||

5 | 0 | −2 | |||

6 | 1 | −2 | 0 | ||

7 | |||||

8 | −2 | −8 | |||

9 | −3 | −8 | |||

10 | −4 | ||||

11 | |||||

12 | −5 | ||||

13 | |||||

14 | −6 | ||||

15 | |||||

16 | −7 | ||||

17 | |||||

18 | −8 | ||||

19 |

20 | −9 | ||||
---|---|---|---|---|---|

21 | |||||

22 | −10 | ||||

23 | |||||

24 | −11 | ||||

25 |

and tangential pressures and the flux density throughout the fluid sphere. We observed that the function

as observed by the distant observer at rest at infinity is zero in remote past at the instance when the collapse begins and at the stage of black hole formation.

Particularly we have constructed a radiating star model by taking suitable parameters

The time of black hole formation is observed as 0.62996 S. We observed that the model is well behaved for the chosen constraints and there are a number of such sets for which the solution is well behaved. Thus the parametric class of solution and constructed table is very fruitful for further study of radiating and static compact star’s modeling. The surface temperature of the collapsing radiating star tends to zero at the beginning of the

collapse

Authors express their gratitude and thanks to the anonymous referee for his rigorous review and valuable suggestions.

B. C. Tewari,Kali Charan,Jyoti Rani, (2016) Spherical Gravitational Collapse of Anisotropic Radiating Fluid Sphere. International Journal of Astronomy and Astrophysics,06,155-165. doi: 10.4236/ijaa.2016.62013