A Study of Stellar Model with Kramer’s Opacity by Using Runge Kutta Method with Programming C

In this paper, we have made an investigation on a stellar model with Kramer’s Opacity and negligible abundance of heavy elements. We have determined the structure of a star with mass 2.5M , i.e. the physical variables like pressure, density, temperature and luminosity at different interior points of the star. We have discussed about some equations of structure, mechanism of energy production in a star and energy transports in stellar interior in a star and then we have solved radiative envelope and convective core by the matching or fitting point method and Runge-Kutta method by C Programming language. In future, it will help us to know about the characteristics of new stars.


Introduction
A star is a dense mass that generates its light and heat by nuclear reactions, specifically by the fusion of hydrogen and helium under conditions of enormous temperature and density. Stars are like our sun. The star is powered by hydrogen fusion. The fusion only takes place at core of the star where it is dense enough.
The "life" of a star is the time during which it slowly burns up its hydrogen fuel, and evolves only slowly in the process. The star is in force balance between pressure and gravity. It is also in energy balance between production by fusion reactions, transport by photon radiation, and loss from the surface by the (usually) visible radiation by which can detect the star. The "birth" of a star refers to the process by which it is formed from diffuse clouds of cold gas that are Schwarzschild (1955) have shown that the maximal possible mass of the star is 60M and minimum mass of star is 0.01M [2]. The chemical element of star is hydrogen, helium and other heavier elements [3]. If hydrogen, helium and other elements are denoted by X, Y and Z respectively, then For the sun X = 0.73, Y = 0.25 and Z = 0.02.

HR Diagram
HR diagram is diagram where the absolute magnitudes and luminosities of stars are plotted against heir surface temperatures or colors. In 1905, Danish astronomer Einar Hertzsprung, and independently American astronomer Henry Norris Russell developed the technique of plotting absolute magnitude for a star versus its spectral type to look for families of stellar type. These diagrams, called the Hertzsprung-Russell or HR diagrams, plot luminosity in solar units on the Y axis and stellar temperature on the X axis [2], as shown in Figure 1.

Energy Production in Stars
A normal main sequence star derives energy from its nuclear source. Enormous amount of energy are continually radiated at a steady rate over long spars of time; for example the sun radiates approximately 10 41 ergs per year. Those thermonuclear reactions do produce energy. That a star can derive energy from thermonuclear reaction is understood from the following example  [5]. That means four hydrogen atoms combine to give one helium atom with the production of two positrons (β + ), two neutrinos (ν) and radiation (γ). Energy production mainly in two ways: 1) Proton-Proton chain (PP chain); 2) Carbon-Nitrogen chain (CN chain).

Hydrostatic Equilibrium of Star
Consider a cylinder of mass dm located at a distance r from centre of the star with height dr and surface area A at the top and bottom. Also denote . p t F and . p b F to be the pressure forces at the top and bottom of the cylinder respectively If F g < 0 is the gravitational forces on the cylinder then from Newton's second Defining the change in pressure force dF p across the cylinder by The gravitational force on the small mass dm is given by From the definition of pressure as the force per unit area we have Putting (3) and (4) in (2) ( ) Assuming the density of the cylinder is ρ, then its mass is d d m rA r = , now Assuming the star is static the acceleration term will be zero which then leads This is the condition of hydrostatic equilibrium.

Mass Conservation
Consider a spherically symmetric shell of mass dM r with thickness dr and r is the distance from the centre of the star. The local density is of the shell is ρ. The shell's mass is then given by In the limit where 0 r δ → , which is the mass conservation equation.

Energy Conservation
Consider a spherical symmetric star in which energy transport is radial in which time variations are very important. Let L r is the rate of energy flow a across of sphere of radius r and L r+dr for radius r + dr Now, the volume of the shell = This is the equation of energy conservation.

Energy Transport in Stellar Interior
Energy transport in stellar interiors occurs by three mechanisms, i.e. radiation, convection and conduction [6] [7].

Radiation
Photons carry energy but constantly interact with electrons and ions. Each interaction causes the photon, on average, to lose energy to the plasma ⇒ increase in gas temperature.

Convection
Energy is carried by macroscopic mass motion (rising gas) though there is no net mass flux. If the density of an element of gas is less than that of its surroundings, it rises ⇒ Schwarzschild criterion for convection.

Conduction
Energy is carried by mobile electrons, which collide with ions and other electrons, but still make progress through the star. The diffusive nature of this process makes it describable in a way similar to radiative transport.

Radiative Energy Transport
If the condition of the occurrence of convection is failed then radiative transfer occurs. The energy carried by radiation per square meter per second i.e. flux F rad can be expressed in term of the temperature gradient and a coefficient of radiative conductively λ rad as follows where -ve sign indicates that heat flows down the temperature gradient.
Assuming all energy is transported by radiation. We will now drop the suffix rad, Astronomers prefer to work with an inverse of the conductivity known as opacity which opacity κ defined as Putting (12) in (10) we have We know flux and luminosity equation is This equation is known as the equation of radiative transfer.

Convective Energy Transport
Let * 1 ρ and * 1 P be the density and pressure inside the blob in its original position, the corresponding quantities outside being ρ 1 and P 1 . In its displaced position, let * 2 ρ and * 2 P be the density and pressure inside the blob white corresponding quantities outside be ρ 2 and P 2 .
Before the perturbation, * ρ ρ > . Therefore mass motion will occur if * 2 2 ρ ρ < . Now we have from the above equations The equilibrium is stable if And the equilibrium is unstable if Expanding left side of the above inequalities in Taylor series and neglecting higher order terms we have Taking log and differentiating we have For stability condition we have Therefore mass motion will occur when 1958) has shown that the temperature gradient for the convection is well represented by which is known as convective energy equation [2].

Schwarzschild Method and Variable
When one is searching for the numerical solution to a physical problem, it is convenient to re-express the problem in terms of a set of dimensionless variables whose range is known and conveniently limited. This is exactly what the Schwarzschild variables accomplish. Define the following set of dimensionless International Journal of Astronomy and Astrophysics ( ) Note that the first three variables are the fractional radius, mass and luminosity, respectively and after three variables represented the pressure, temperature and density. In addition, let us assume that the opacity and energy generation rate can be approximately by Putting (17), (18), (20) and (19) in (6), we have ( ) Again, putting (17), (18) and (22) in (7), we have Now putting (17) putting (17), (19) and (20) in (16) If the star has a convective core, then all the energy is produced in a region where the structure is essentially specified by the adiabatic gradient and so the energy conservation Equation (29) is redundant. This means that the D is unspecified and the problem will be solved by determining C alone. Such a model is known as a Cowling model.

Solution of the Model
Since the model star is likely to have a small convective core with a radiative envelope, in principle we have two solutions, one is the envelope and another is

Polytropic Core Solution
Eliminating q from (27) and (28) we have Now introducing the polytropic variables η and θ , defined by which is the Lane-Emden equation with index 3 2 n = .
The general solution of (34) is For small η this is a rapidly convergent series.
We take Introducing Schwarzschild homology variables defined by [2] So as to good approximation  From Figure 2 it is found that this happens for . This is the correct value of C for this model star.
Then from Equation (6) the values of the parameters that point are found to be. The radiative solution for the pack 0.168 1 x ≤ < for 7 9.46e C − = is given is

Core Solution of the Model
From the Table 1   1.2323  Table 2.  The packet solution and the core solution together give the complete internal structure of the star. The complete structures are shown in Table 3.

Conclusion
In this paper we have assumed a non-rotating and non-magnetic star with mass 2.5M . The structure of the star with Kramer's opacity with negligible abundances heavy element i.e. the pressure, temperature, mass, luminosity and density at various interior points is determined numerically and non-dimensional result of the radiative envelope is shown in Table 1 and convective core in Table   2. However, the complete structure is shown in Table 3. We also determined the actual radius 1.5011 R R = and total luminosity 6.4957 L L = . And our calculated results are in good agreement with the recent published results book Bohm-Vitense (W. Brunish). If the mass varies and composition is fixed, then T eff and R are found to vary but L is increased quite sharp. Again if hydrogen and heavy elements are increased, then R is increased but L and T eff are decreased.
For the increase in M, the position of the star in the HR diagram [2] is slightly shifted toward the upper end of the main sequence. If the mass is constant, then the decrease in the hydrogen content of the star increases luminosity and effective temperature. But as time goes on in the main sequence lifetime of a star, its