The distance variation due to mass transfer and mass loss in ( 13 . 6 + 8 ) M and ( 13 + 10 ) M binary star systems

E-mail: mohammed_plasma@Scbaghdad.edu.iq Abstract

Most of the stars in our universe are born in binary systems.The binary star system can be classified into two categories:-Close binary stars and wide binary stars.In close binaries the initial orbital period is less than few years while in wide binaries the inverse is true.The course of evolution of the binary can be affected with the tidal force between the stars of the binary system [1].
The known four types of binary star systems are visual binaries which its image can be distinguished, Spectroscopic binaries which also can be distinguished by the displacement of the wavelength of their spectral lines, and Eclipsing binaries which can be recognized by the periodic drops in brightness as the couple stars periodically eclipse each other.The two assumed binary system under studying will be visual binary star systems.Nearly 85% of all stars in the universe are binaries.The physical parameters of binary star systems are about mass, distance between each other, luminosity, orbital period, mass transfer and etc. [2].

The orbital period and the separation of the binary stars
The binary star system is about couple stars orbit each other under their mutual gravitational attraction.The distance between the two stars is called the separation (a) and when the star completes one cycle period, it is called the orbital period (P).The total mass of the binary star can be calculated as the following equation:- Where M 1 is mass of the primary star, M 2 is mass of the secondary star.[1].The importance of the binary star system appeared clearly because the mass of the binary star can be calculated if the calculation of their orbit is done.And also indirectly the mass-luminosity relation could be extracted.[3] There is another classification of the binary star depending on the distance between the binary star as follow:-

1-Detached binaries
They are binary stars where each component is within its Roche lobe.The stars have no maximum influence on each other .Most binaries fall in this category

2-Semidetached binary stars
They are binary stars which one of the components fills its Roche lobe but the other doesn't.The mass transfer in this case will commence and dominate the evolution stage.The matter will be transferred from the donor star to the other.

3-Contact binary stars
They are binary stars in which both components of the double stars will fill their Roche lobe.The common envelope will be formed in the upper atmosphere of the binary system.The two stars will probably merge [3].The above three types of binary stars have been shown as in Fig. 1[4].

Fig.1: The three types of the binary star systems depending on their Roche lobe [4].
The orbital period of the binary stars differs from system to system depending on the separation between them, e.g. it could be few days or also hundreds of thousands of years.[3] The binary star interactions play an important role to determine the state of its compact component.The component object may be White dwarf (WD), Neutron star (NS) or Black hole (BH) depending on the evolutionary stage and mass of the star.For example, star with mass less than 2.3M  can evolve to White dwarfs.
A star in binary star system with mass larger than 12M  can form a Neutron star in evolutionary stage.A Black holes compact can be existed when the Neutron star core collapses inwardly with mass between (25-40)M  .[5] Both the rotational and gravitational energy per unit mass near the surface of stars generate the surface called equipotential surface .Near the stellar centers these surfaces are usually spherical.These surfaces of the binary stars denoted as Roche surfaces after the French mathematician E.Roche who described them in 1873.[6] The surface will form a common envelope in the binary star.The Roche lobe is the region around star in a binary system in which the gravity of that star dominates as in Fig. 2
Usually the size of the Roche lobe in one component of the binary stars can be calculated by using Eggleton formula [1] (2) [7].
Where R L is the radius of Roche lobe, q is the mass ratio=M 2 /M 1 [7].Sometimes the R L is called effective radius [6] Paczynski also calculated the effective Roche lobe radius R L of the donor star in binary system as follow:- (3) [8].
Where M 2 is the mass of the donor star, M is the total mass, and a is the separation of the binary star [8].

The mass transfer phenomenon
In evolution course of the binary star system, if one of the stars expands and fills its Roche lobe then matter will flow from it to the companion star.The orbital parameters of the binary star would be changed clearly due to the mass transfer process.The total orbital angular momentum of the binary star J can be stated as:-( 4) [1] Where M is the total mass of the binary system.Then (5) [1] in the present work the conservative case has been adopted i.e. no matter or angular momentum leaves the binary star system, then both and will be zeroed.This leads to (6) [1] notice that because of the mass conservation [1] It is important to say that at the early case of binary evolution, the mass transfer between the components and the orbital momentum loss play a great dynamical role [9].Finally the most general form of the Eeq.( 5) has been given by:-(7) [9] where β is the fraction of the ejected matter which leaves the system [9].

Calculations and discussion 1-(13.6+8) M Binary star system
First the mass of the primary star has been taken into account in the calculations.The reduction in mass of the primary is due to two main processes: the mass loss and the mass transfer.
Initially the data of the binary system parameters have been gathering from Table1.Then according to the mass dynamics, the life of the binary star under studying can be divided into two stages: Mass loss and Mass transfer.Table1 explains the various phases of binary star system initially and finally .In Table (2),the calculation of and have been evaluated.and have been computed according to the mass status ,if the mass phase is mass loss then can be computed by simple formula and also can be computed by:-,if the mass phase is mass transfer then and can be computed by and Eq.( 6) respectively. 3Mass loss, 4 Mass transfer The sign(-) of means that the primary star loses the mass to the secondary star.The primary star would shed matter into secondary until the secondary star would fill its Roche lobe and reversely it would shed matter into the primary star that means will be positive after 23.015x10 6 yr.Of course the separation of the two stars will be affected due to these dynamics changes.First will increase due to mass loss of the primary star of the binary system, then it will increase more and more.reaches its maximum value at 14.7841x10 6 yr in mass transfer phase.As increases will increases and vice versa.At 14.7841Myr (1Myr=10 6 yr), the configuration of the binary star system has critical case leading to Supernova.This is clear that in Fig. 4 as increases, will increase.In Fig. 5, the peak of clearly has been seen in 14.1354Myr (Critical case), Sharp drops.And after 14.1354Myr the nearly will be steady and constant with proceeding time.the mass transfer stage which affects on the , in mass transfer stage will increase with time.After finishing of mass transfer will decrease sufficiently.as in Fig. 6 would be nearly study and constant after 22.9777Myr.

2-(13+10)M  Binary star system
Similar to the previous mentioned binary star system ,the data of the (13+10)M  binary star system has been gathered from Table3.There are also two stage of mass process:-mass loss and mass transfer.The results in Table4 shows that and in system (13+10)M  have been calculated as the same method in Table2 in system(13.6+8)M .First the in MT phase increases with time progression.In 16.1359Myr it will decrease and sequentially will increase again until its value vanished in 23.8042Myr and 23.8204Myr where the system will be converted to Neutron star.also will increase with time specially in mass transfer period.Then it will decrease, and in 18.3771Myr will increase.When the secondary star sheds the matter into primary star, the values will increase.In Fig. 7 the relation between and nearly study proportionally especially in mass transfer phase.
In Fig. 8 the maximum value of versus time is in 15.8036Myr and has value 2.6931 R  /yr and then it will be inear proportion.In Fig. 9, has maximum value at early evolution of the binary star when the primary star fills its Roche lobe and shedding materials into the secondary star at 15.7821Myr which is represented by peak hump as shown in Fig. 9. Then it will be semilinear with time after 16.1359Myr.In Fig. 9, has maximum value at early evolution of the binary star when the primary star fills its Roche lobe and shedding materials into the secondary star at 15.7821Myr which is represented by peak hump as shown in Fig. 9 then it will be semilinear with time after 16.1359Myr.and can be different from one binary star system to another due to their masses and separations.

Fig. 6 Fig. 4 :Fig. 5 :
Fig.6 shows initially at early time, suffers from dramatic changes, starts with minmum value and increased with time until

Table 2 : The results of the mass transfer process and the distance vartiation in the binary star system
(13.6+8)M  .

Table 4 : The results of the mass transfer process and the distance vartiation in the binary star system (13+10)M  . Fig.6: The relation between and time in binary star system (13.6+8) M 
3Mass loss,4Mass transfer