A New Limit to the Core Mass in Stars with M ≥ 2M_Θ

A New Limit to the Core Mass in Stars with M 2M

M 2M

Nouara.Tinakiche1,2

1Departement of Physics, Faculty of Science, U.M.B.B, Boumerdes, ALGERIA

2Faculty of physics, U.S.T.H.B, Algiers-ALGERIA

Abstract–According to the studies of (Schönberg & Chandrasekhar 1942; Henrich & Chandrasekhar 1941)[6,4], it exists an upper limit to the mass of the isothermal core for the stars situated on the post main sequence MS on the HR diagram with a mass M 2M . In the present work, and using another

called circum-nuclear shell situated above the core. This shell feeds the core with the nuclear reaction products and contribuate in increasing the mass of this core. Henrich & Chandrasekhar (1941) [4]and Schönberg & Chandrasekhar

approach that I find more rigorous than the calculus

(1942) [6] calculated the greatest mass

M iso

done in the other works, I demonstrate the existence of an other value to this upper limit and I establish in function of this upper limit M iso the formulae of the luminosity produced by these stars.

supported by this isothermal core. In the present work, I find an other upper limit to this mass and in function of this mass I establish the formulae of the luminosity produced by these stars in the frame of the following approach:

1. INTRODUCTION

2. CALCULUS OF THE MASS M iso

   WHICH

   For stars of masses greater than

   2M

   and classified

   CORRESPONDS TO THE MAXIMUM PRESSURE IN THE CORE

   within the post main sequence on the HR diagram, the

   interior region or the core is under the rule of the gravitational contractions in the phase of hydrogen rarefaction. Because of the lack of the hydrogen, the luminosity produced by the core is null ( L =0) and therfore the core is isothermal. In this phase, the gravitational energy generated from these contractions in the core heats the upper layers, and this increasing in temperature in these layers allows the nuclear reactions to take place in the so

   In this calculus, the core is assumed to be a sphere of gas with a quasi constant density. Using the equations of the hydrostatic equilibrium and the equation of the mass in a star, one can find the equation relating

   between the quantities dP dM r, M r and r , where P is the pressure, M r is the mass of the

   star and r is the radial coordinate.

   The hydrostatic equilibrium of a spherical star of a quasi

   and,

   G 6.67 108 erg cm

   g 2

   is the gravitational

   constant density and a mass M r is expressed as in [5],

   constant.

   dP

   dr

   GM r r2

   (1)

   However,

   4 r 2 dr dM

   where M r 4 r3

   3

   Combining the relations (1) and (2), one finds:

   (2)

   4 r 3

   dP

   dM r

   GM r r

   (3)

   the density of the core

   iso is considered quasi-constant,

   One can rewrite 4

   form,

   r 3 dP

   dM r in the following

   the radius of the core

   and M iso are so that,

   1

   Riso varies when

   M iso varies.

   Riso

   3M 3

   dP d (4

   r 3 P) 3P

   R iso

   (8)

   4 r 3

   dM (r)

   dM (r)

   (4)

   iso

   4

   iso

   If one inserts this last relation in the relation (3) and integrates over the whole isothermal core of mass M iso ,

   and M

   iso

   R 4 R3

   iso 3 iso

   iso

   one obtains,

   For each mass

   M iso correspond Riso , Tiso ,

   iso

   and

   iso

   which are being now functions of rewritten as,

   M iso .

   Piso

   can be

   3P

   M iso d 4

   r 3 P

   M iso

   1 3

   dM r dM r

   dM (r)

   P kTiso iso G 4

   4 2

   3 M 3

   (9)

   0 0

   M iso GM r

   iso

   iso mH

   iso

   5

   3

   iso

   0

   dM r

   r

   (5)

   Hence the derivative of by,

   Piso with respect to M iso

   is given

   The right term of the relation (5) represents the gravitational 1

   energy of the core, its equal to

   • 3GM 2 5R if

     dPiso kTiso

     4G 4

     3 13 M 2 3 diso

     iso iso

     dM m 15 3

     iso iso dM

     M r=0 when r=0. Using the following state equation relative to a perfect isothermal gas,

     iso iso H

     iso

     P kT mH ,where T is the temperature,

     H

     1

     T =Tiso=cte, is the molecular weight, m is the atomic

     • kiso

     iso mH

     dTiso dM iso

     kTiso iso

     m

     2

     H iso

     diso

     dM iso

     mass of the hydrogen, and k is the Boltzman constant, the

     2G 4 3 4 1

     relation (5) becomes then,

     3M

     M

     3GM 2

     15 3

     3 3

     iso

     (10)

     4R3 P

   • iso kT

   iso

   (6)

   iso iso

   iso mH

   iso

   5Riso

   Assuming that

   iso

   is quasi-constant in the core and

   where Riso ,

   Piso , M iso , iso and Tiso

   are respectively

   doesnt vary appreciably with the small variation of the mass M iso , one can neglect the derivative diso dMiso and

   the radius, the pressure, the mass, the molecular weight and the temperature of the isothermal core. Then, the pressure of this core is given by,

   the relation (10) becomes,

   3 M kT

   1 GM 2

   dP kT

   4G 4 3 1

   2 d

   P iso iso iso

   (7)

   iso iso

   1

   3 M

   3 iso

   iso

   4R3 m

   5 R

   iso

   iso

   iso

   iso H

   iso

   dM iso

   iso mH

   15 3

   dM iso

   In order to find the maximum value of the pressure in the

   k dT

   2G 4 3 4 1

   1

   M iso iso

   M

   3

   core, one proceeds by the variation of the mass iso . Since

   iso mH

   dM iso

   15 3

   iso

   1

   3

   iso

   (11)

   4R 2 dR

   • iso iso iso

   dP dM

   iso

   iso 4 3

   To evaluate

   iso

   iso , we need to calculate the

   3 Riso

   derivatives

   diso dMiso

   and

   dTiso dMiso appearing in

   the relation (11).

   Where,

   M iso

   iso

   4 3

   1. Calculus of the derivative diso dMiso

      3 Riso

      diso dMiso

      can be rewritten under the following form,

      then, d

      3iso dR

      d d dR

      iso

      iso

      iso R

      iso

      iso iso iso

      (12)

      iso

      dM iso dRiso dM iso

      we obtain,

      we set

      diso iso iso

      where

      iso

      is the density of

      diso

      3iso

      (17)

      the core after the variation of the mass

      M iso from

      M iso to

      dRiso

      Riso

      M iso dM iso , this leads to a radius variation from

      Riso

      The relation (12) becomes,

      to Riso dRiso

      (the density

      iso is still considered quasi-

      d 3 dR

      constant in the core which is supposed to conserve its

      iso iso iso

      (18)

      spherical form and has the new mass

      M iso dM

      iso ).

      dM iso

      Riso

      dM iso

      Therefore iso is given by,

      From the relation (16), the drivative dRiso dMiso

      to,

      is equal

      M iso dM iso

      (13)

      3

      iso

      4

      3

      Riso

      • dRiso

        dRiso dM

        1

        4R2

        Since

        Riso

        dRiso , one can do the following

        iso

        iso

        iso

        approximation,

        and the relation (18) becomes:

        M dM

        diso

        dM iso

        3iso

        Riso

        1

        iso

        4R2

        iso

        iso iso

        (14)

        iso 4 3

        we obtain,

        3 Riso

        M dM

        diso

        dM

        3

        4R3

        (19)

        iso iso

        (15)

        iso

        iso

        iso

        4 3 4 3

        3 Riso

        3 Riso

        as Riso is given by the relation (8), diso dMiso

        is finally

        given by,

        and since,

        dM 4R dR

        2

        iso iso iso iso

        (16)

        diso

        dM iso

        iso

        M iso

        (20)

        the relation (15) becomes,

        we insert this expression of

        diso dMiso

        in the relation

        (11) and obtain the following relation for the derivative of

        the pressure,

        we replace the expression of

        dTiso dMiso in the relation

        1

        dP kT 4G 4 3 1

        2

        (21) and we obtain the following expression for the

        iso iso

        3 M

        3 iso

        derivative dPiso dMiso ,

        dM iso

        iso mH

        15 3

        1

        iso

        iso M iso

        • kiso

      dTiso

      2G 4 3 3 1

      4

      1

      iso mH

      dM iso

      15 3

      iso 1

      M 3

      dPiso

      kTiso

      4G 4 3 13

      2 iso

      3

      (21)

      iso

      dM iso

      iso mH

      4

      1

      15 3

      iso M iso M

      iso

   2. Calculus of the derivative dTiso dMiso :

   7G 4 3 3 1

   The relation (3) can be rewritten as,

   15 3

   M

   iso

   1

   3

   iso

   4R3

   dPiso

   dTiso

   GMiso r

   (28)

   dT

   iso

   iso

   therefore,

   dM iso

   Riso

   (22)

   The value of by,

   M iso

   for which

   dPiso dMiso 0

   is given

   dT GM dT 1 3 32

   iso iso iso

   (23)

   3 15 3 2 1 2 kT

   dM 4R 4 dP

   M

   iso

   iso

   iso

   iso

   iso

   4 11

   Gm

   13

   From the state equation used above which

   3

   H iso iso

   3

   2

   is given by P kT

   mH

   , we can write,

   1 2 kT

   0.778

   iso

   (29)

   Gm 13

   k H iso iso

   dP iso dT

   (24)

   so,

   iso

   iso mH

   iso

   The maximum value of pressure corresponding to this mass is then given by,

   8 kTiso iso

   dTiso

   iso mH

   (25)

   Piso max

   11 m

   dPiso

   iso k

   1

   8 4 3

   iso H

   M

   4 2

   After replacing the expression of

   15 3

   G 3

   3

   iso

   (30)

   dTiso dPiso

   in the relation (23), we find,

   dTiso dM iso

   GMiso

   4R

   4

   iso

   iso mH

   iso k

   (26)

   C. Calculus of the pressure at the interface between the envelope and the core Piso

   env

   Integrating the equation of the hydrostatic equilibrium over

   Since the radius

   Riso

   depends on

   M iso

   and its given by

   the envelope and by taking the pressure null on the star surface, one obtains,

   Riso

   3M

   iso

   4

   iso

   13 , the relation (26) becomes,

   4

   dT G 4 3

   1

   m 3

   M GM

   iso

   iso H iso

   (27)

   Piso

   dM r

   4

   r

   (31)

   dM iso

   4 3 k

   1

   M

   3

   iso

   env 4

   M iso

   where

   Piso

   is the pressure at the interface between the

   81 1 kT 4

   env

   Piso

   iso

   (38)

   isothermal core and the envelope, M is the whole mass of the star. Approximately, one finds,

   env

   4 G3 M 2

   inv

   mH

   iso

   Piso G M 2 M 2

   (32)

   Confronting now the two pressures

   Penv

   and

   Piso max to

   env

   8 r 4

   iso

   find the relation between the mass of the whole star M

   and

   where,

   the mass of the isothermal core of the same star

   M iso which

   4 R

   4

   r (33)

   2

   corresponds to the maximum of pressure Piso max in the core, we find,

   P = Piso

   Since M Miso the relation (32) becomes,

   iso max

   env

   8 kT

   81 1 kT 4

   G M 2

   iso iso =

   iso

   (39)

   Piso (34)

   11 m

   4 G3 M 2 m

   env 4 R4

   iso H

   env H

   Using the state equation of a perfect gas,

   From the relation (29) we deduce the expression of

   T

   m piso

   kTiso GmH iso ,

   k

   iso env H env env iso

   env

   iso

   iso

   (35)

   1

   2

   kTiso M 3 3

   where Tenv ,

   env

   and

   env

   are respectively the

   Gm

   iso iso

   1

   temperature, the molecular weight in the envelope

   (supposed to be constant in the envelope), and the density at

   H iso

   15

   3 3

   the interface between the isothermal core and the envelope.

   11 4

   env

   The density iso

   is assumed to be approximately equal to,

   iso env

   M

   4 R3

   (36)

   Then from the relation (39), we obtain,

   3

   M 24

   15 2 2

   where M and R are respectively the mass and the radius of

   iso

   env

   the whole star.

   M 1215 11

   iso

   From the relations (34), (35), and (36) one can deduce the following expression of the radius of the star R ,

   2

   0.261 env

   iso

   (40)

   1 GM m

   Therefore the allowed values of M iso are given by,

   R env H

   (37)

   3 Tiso k

   2

   This expression of R

   P

   is inserted into the relation (34) and

   M 0.261 env M

   iso

   iso

   (41)

   one finds the expression of the pressure

   iso env ,

   which verify the condition

   P Piso

   iso max env .

   This is the result obtained in this present work. The calculus of this upper mass by L. R. Henrich and S. Chandrasekhar

   Riso 3M

   iso

   4 13

   where is the mean density of

   [2,3,4] and Schönberg & Chandrasekhar [6] gives the following result,

   the star, its given by M

   4 R3

   3

   M iso 0.35env

   2

   M

   iso

   which is given by

   Then,

   Miso 0.35M

   for env iso 1.

   M R dL

   L 1 iso

   (46)

   It appears that the two approaches to estimate the value of

   M 3

   dr r R

   M iso dont diverge and although they give different values the results arent very far from each other.

   If the energy transport in the envelope is radiative, the luminosity Lr is so that in [7], page 89,

   4acT 3

   2 dT

   L r 4

   r

   (47)

   1. The calculus of the luminosity produced by the envelope

      Since the luminosity produced in the isothermal core is null

      3

      dr rad

      Lr,

      0 r Riso

      0, we can assume the luminosity

      where a is the radiative constant, c is the light velocity,

      produced by such stars to be qual to,

      is the opacity, and dT

      rad

      is the radiative temperature

      dr

      R

      L 4

      r 2 iso

      r dr

      (42)

      gradient. From the relation (47), we can deduce the

      R iso

      rR

      env

      expression of dL dr and replace its expression in the relation (46). Then,

      where r

      is the energy produced by a mass unit in the

      envelope,

      iso env

      is the density of the envelope assumed to be

      quasi constant as mentioned above,

      dLr 4ac 8

      rT 3 dT

      dr 3

      dr

      iso env

      cte

      (43)

      rad

      R and Riso are respectively the radius of the star and the

      4ac 4

      3

      r 2 3T

      2 dT

      dr

      d 2T

      T

      3

      dr 2

      radius of the isothermal core.

      rad

      rad

      If r

      can be supposed constant and approximated such

      that in [7], page 89,

      2 iso

      rad

      r 1 dL

      4R dr

      (44)

      we can rewrite dT

      dr in the form,

      (48)

      env r R

      Replacing the expression of r into (42), the luminosity of such stars is then given by,

      dT dT dM

      dr dM dr

      1 R3

      R3 dL

      rad

      L

      iso

      (45)

      R 2 3

      R

      3 dr

      r R

      M and d 2T

      dr 2

      in the form,

      iso is related to the mass of the isothermal core iso by

      rad

      Riso 3M

      iso

      4 13 .

      which can be approximated to be equal to,

      d 2T

      d 2T dM

      dT d 2 M

      dr 2

      drdM dr

      dM

      dr 2

      This relation provide us with an estimation of the luminosity

      rad

      produced by a star from the POST MS region with a mass

      M 2M in which the envelope is considered radiative.

      From the relations (2) and (27), we find,

      4 4 3

      As expected, the luminosity of these stars is in function of

      M iso .

      dT

      4

      3 mH 2

      dr

      = G k 1 r

   2. Choice of the parameters:

   and,

   rad

   3

   M 3

   (49)

   The molecular weight

   To calculate the luminosity of the star given by the relation (52), we need to know the value of the molecular weight

   . To get the values of this parameter, we use the

   formulae given by [1], page 119:

   4

   d 2T

   G 4 3

   mH 3 4

   dr 2

   1

   1

   4 3 k

   rad

   1

   8 r 4 r 2 2 1

   2

   1

   3 4

   0.00309 4 M

   M 3

   M 3

   (50)

   M

   where M and M

   are respectively the mass of the star and

   Replacing these two last relations into (48), we find,

   dLr 4ac 4

   the mass of the sun. is a constant relating between the

   total pressure PTOT , the radiative pressure PR ,and the gas

   G 4 3 mH 13

   pressure P of the star. They are defined by (see [1], page

   dr 3

   4 3

   k

   116),

   GAS

   32 2 r 3T 3 48 2 r 4T 2 32 2 r 3T 3

   1

   M 3

   64 3 r 6T 3 2

   PR 1 PTOT

   The opacity

   and PGAS

   PTOT

   4

   3M 3

   (51)

   For the calculus of the luminosity, we use the opacity formulae given by (see [1], page 119),

   Then the expression of the luminosity given by (46) becomes:

   4 c G M 1

   4

   M R

   4ac

   G 4 3 m 1 L

   L 1 iso

   H 3

   M 3 3

   4 3 k

   where is the constant mentioned above, c the light

   32 2 r 3T 3 48 2 r 4T 2 32 2 r 3T 3

   velocity , G the gravitational constant, M is the mass of

   1

   M 3

   64 3 r 6T 3 2

   (52)

   the star, and L its luminosity. In this work is calculated using the observational values of the luminosity L .

   4

   3M 3

3. DISCUSSION OF THE RESULTS:

   On the Table 1 and the Table 2, its shown the results of the luminosity calculated for several stars from the POST MS zone which their masses exceed 2M . The values of the luminosity L of the present work arent far from those obtained from the bolometric measures. The differences between the observational and the calculated values of the luminosity are very reasonable with respect to the approximations done in this work. So this work can be

   considered as a very appreciable approach to find the theoretical expression of the luminosity which fit the best

   the observation and in the same time takes into account the upper limit of the mass of the isothermal core.

4. CONCLUSION

The present work is a contribution to the study of the internal structure of the stars with M 2M . According to the results obtained in the frame of this work ,one can deduce that it consists a good approach to establish the most accurate theoretical expression for the luminosity produced by this category of stars taking into account the upper limit

of the isothermal core mass M iso

Tableau 1:The calculated luminosities produced by stars with isothermal core and radiative envelope.

These results are obtained for iso env so for Miso 0.261M , Teff is the effective temperature References. (1) Eddington & Chandrasekhar 1988, page 145 [1]; (2) Eddington & Chandrasekhar 1988, page 182[1].

Stars Mass 1

Radius Temperature Density Molecular Luminosity Reference

SZ.TAU SU.CYG RT.AUR T.VUL

6.60M

6.80M

6.90M

7.70M

0.380

0.390

0.390

0.420

13.51011

12.0 1011

13.91011

16.7 1011

5850

6450

5950

5750

1.273103

1.868103

1.219103

7.850104

2.091

2.107

2.092

2.121

3.0761037

2.5571037

3.7881037

7.7821037

(2)

(2)

(2)

(2)

POLARIS 780M

0.420

19.61011

5250

4.919104

2.108

1.1531038

(2)

Tableau 2:The comparison between the observational luminosities and those calculated in the present work.

References. (1) Eddington & Chandrasekhar 1988, page 145[1]; (2) Eddington & Chandrasekhar 1988, page 182[1]

Stars The Luminosity calculated Bolometric Luminosity

using M iso of the present work

LMEASURED erg s1

LCALCULATED L

Reference

LCALCULATED

erg s1

MEASURED

REFERENCES

1. Eddington, A.S., & Chandrasekhar, S. 1988, The internal constitution of the stars, (Cambridge University Press)

2. Henrich, L. R., & Chandrasekhar S. 1929, M. N., 89, 739.

3. Henrich, L. R., & Chandrasekhar S. 1936, M. N., 96, 179.

4. Henrich, L. R., & Chandrasekhar S. 1941, M. N., 94, 525.

5. Schatzman, E., & Praderie, F. 1990, les étoiles, (savoirs actuels interedition

   /Editition du CNRS, Paris) page 130

6. Schönberg, M., & Chandrasekhar S. 1942, ApJ, 96, Vol. 2, 161.

7. Unno, W., Osaki, Y., Ando, H., Saio, H., Shibahashi, H. 1989, Nonradial oscillations of stars, (2nd ed.; university of Tokyo press)