A Study of Stellar Model with Cramer's Opacity by using Runge Kutta Method with Programming C

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 we 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 present in its galaxy. A cloud collapses to form a number of stars when it is disturbed so that its gravity overcomes its motion and pressure. The “death” of a star occurs when its fusion fuel, first hydrogen and then heavier nuclei, has run out. This can be very violent if the star is very massive, ending in things like a black hole and/or a supernova, perhaps leaving a neutron star behind. If the star is not very massive, like the Sun or even smaller, it ends by ejecting part of its atmosphere and then settling down to a cold, dense white dwarf. Harm and Schwarzschild (1955) has shown that the maximal possible mass of the star is 60MÅ and minimum mass of star is 0.01MÅ. The chemical element of star is hydrogen, helium and other heavier elements. If hydrogen, helium and other element were denoted by X, Y and Z, respectively. Then X+Y+Z=1. For the sun X=0.73, Y=0.25 and Z=0.02.


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 we 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 present in its galaxy. A cloud collapses to form a number of stars when it is disturbed so that its gravity overcomes its motion and pressure. The "death" of a star occurs when its fusion fuel, first hydrogen and then heavier nuclei, has run out. This can be very violent if the star is very massive, ending in things like a black hole and/or a supernova, perhaps leaving a neutron star behind. If the star is not very massive, like the Sun or even smaller, it ends by ejecting part of its atmosphere and then settling down to a cold, dense white dwarf. Harm and Schwarzschild (1955) has shown that the maximal possible mass of the star is 60MÅ and minimum mass of star is 0.01MÅ. The chemical element of star is hydrogen, helium and other heavier elements. If hydrogen, helium and other element were denoted by X, Y and Z, respectively. Then X+Y+Z=1. For the sun X=0.73, Y=0.25 and Z=0.02.

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, 4 1H1=2He4 + 2b+ + 2ν + γ. That means four hydrogen atoms combine to give one helium atom with the production of two positrons (b+), two neutrinos (ν) and radiation (γ). Energy production mainly in two ways (i) Proton-Proton chain (PP chain) (ii) 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 as shown in Figure 1. Also denote F p.t and F p.b to be the pressure forces at the top and bottom of the cylinder respectively If Fg<0 is the gravitational forces on the cylinder then from Newton's second law we have 2 . .
Defining the change in pressure force dFp across the cylinder by Then gives radial in which time variations are very important [1]. Let Lr 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=4pr 2 dr. If r is the density, then mass of the shell is illustrated in Figure 3.
The energy released in the shell can be written as 4pr 2 rdre, where e is defined as the energy released per unit mass per unit time. The conservation of energy leads that

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

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 [2].

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 Frad can be expressed in terms of the temperature gradient and a coefficient of radiative conductively lrad as follows From the definition of pressure as the force per unit area we have Putting eqns. (3) and (4) in eqn. (2) Assuming the density of the cylinder is r, then its mass is dm=rAdr, Now eqn. (5) becomes Assuming the star is static the acceleration term will be zero which then leads to This is the condition of hydrostatic equilibrium.

Mass Conservation
Consider a spherically symmetric shell of mass dMr with thickness dr and r is the distance from the centre of the star. The local density is of the shell is r (Figure 2). The shell's mass is then given by dM=d Vr (r). Since δV=4πr 2 δr. Then we have M=4πr 2 δrρ(r) 2 ( ) 4 ( ) dM r r r dr π ρ ⇒ = (7) In the limit where d r ® 0 which is the mass conservation equation. Now, the volume of the shell = 4pr 2 dr. If r is the density, then mass of the shell = 4 pr2rdr. The energy released in the shell can be written as 4 p r2rdre, where e is defined as the energy released per unit mass per unit time. The conservation of energy leads that  Where -ve sign indicates that heat flows down the temperature gradient. Assuming that all energy is transported by radiation. We will now drop the suffix rad, Astronomers prefers to work with an inverse of the conductivity known as opacity which opacity Where a is the radiation density constant, c is the speed of light.
From eqn. (12) we have Putting eqns. (13) in eqn. (11) 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 r1 and P1. In its displaced position, let * 2 ρ and * 2 P be the density and pressure inside the blob white corresponding quantities outside be r 2 and P2. Before the perturbation, ρ ρ < . Now we have from the above equations 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 transport equation.

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 [3]. Define the following set of dimensionless variables 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 eqns. (17), (18), (20) and (19) in eqn. (6), we have Again, putting eqns. (17), (18) and (22) in eqn. (7), we have GM dp Again, putting eqns. (17), (18) and (22) in eqn. (7), we have 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 [4].

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 the core. The two solutions must match at the interface.

a)
Polytropic core solution b) Envelope solution

Polytropic core solution
Eliminating q from eqns. (27) and (28) we have This gives the core solution in the U-V plane.

Envelope solution of the matching point
The envelope of the model star is radiative equilibrium. This structure is determined by equations (27), (28) and (30). The equation (30) contains an unknown parameter C. Our aim is to determine the correct value of C and obtain the envelope solution for the value of the parameter. In order to do this we have to solve the envelope solutions for different trial value of C and find which value of C the solution just matches the core solution at the interface. However the solution is not straightforward. Because of the existence of singularity at the surface, integration cannot be started right from the surface (x=l). To avoid this difficulty we have to look for series expansion of the variables about the singular point. The envelope solutions we have calculated numerically, however since the equations are singular at the surface, p=t=0. We have chosen the series expansion of the variables near the singular point in the following way [1,3]. Let x d x dp qp dp qp dp qp Here the singular point is x=0, since x=1 i.e., x=0. Now the series expansion of variables about x=0 can be easily done. By Fuchs theorem, a convergent development of the solution in a power series about the singular point having a finite number of terms is possible. We therefore take And q ≈1.
These relations determine the values of the parameters at any point near the surface. With these values as the boundary values the envelope equations can easily be solved numerically for given of C. C is an unknown constant whose value for a start of given mass depends on its luminosity and radius. For solar type stars C is of the order of 10 -6 . We shall treat C as a free parameter and consider of values of close to 10 -6 . We take a point x=0.99 very near to the surface. Appropriate for convection, by the fourth order Runge-Kutta method for a number of trial values of C. Some of these calculations, namely for C=1.56e -6 ,C=5.6e -7 C=9.46e -7 . Together with the convective track, equation (36), are drawn in the (U-V) plane (Figure 3) at the junction between the convective core and the radiative envelope both (U, V) and their derivatives must be continuous. So the curve for the correct radiative solution must touch the convective curve at the interface. Form Figure  4 it is found that this happens for C=9.46e -7 . This is the correct value of C for our model star.
Then from equation (6) the values of the parameters that point are found to be. Taking these values as the boundary values we have integrated the equations for the radiative envelop numerically inwards up to where 0.168 1 x ≤ < For this value of C the matching point is at x f =0 .168 The radiative solution for the envelop is 0.168 1 x ≤ < for c=9.46e -7 is given in Table 1 Radiative structure of the model star M=2.5, X=0.90, Y=0.09, Z=0.01 (solar Unit).

Core Solution of the Model
From the Table 1 we find that Since all the energy is produced in the core. With these values as our boundary conditions we have to solve the core equations, namely equations (27), (28), (29) and (31) inwards numerically. In order to do this we need the correct value of D. This can be done by integrating the equation. Total luminosity, Since p and t are continuous at xf

Conclusion
In this paper we have assumed a non-rotating and non-magnetic star with mass 2.5M Å . 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 point are determined numerically and non-dimensional result of the radiative envelope are 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 R=1.5011 R Å and total luminosity L=6.4957L Å . And our   calculated results are in good agreement with the recent published results book Bohm-Vitense (W. Brunish). If the mass varies and composition fixed, then Teff and Rare found to varies but L is increase quite sharp. Again if hydrogen and heavy elements are increase, then R is increase but decrease L and Teff. For an increase in M the position of the star in the HR diagram is slightly shifted to toward the upper end of the main sequence. If the mass is constant then a 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 hydrogen content gradually diminishes giving rise to the helium content. That means, as a main sequence star ages its position in the HR diagram slowly moves along the main sequence toward the hot end.