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 Total Downloads : 102
 Authors : Nouara Tinakiche
 Paper ID : IJERTV3IS040168
 Volume & Issue : Volume 03, Issue 04 (April 2014)
 Published (First Online): 29042014
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
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, AlgiersALGERIA
Abstract–According to the studies of (SchÃ¶nberg & 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 circumnuclear 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 SchÃ¶nberg & 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:

INTRODUCTION

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 quasiconstant,
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 quasiconstant 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

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


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 SchÃ¶nberg & 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)

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

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


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.

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].
M 
g 
R 
cm 
Teff 
c 
g cm3 
Mass 
L erg s1 

RR LYR. 
3.70M 
0.260 
4.321011 
7800 
2.180102 
2.127 
4.7931035 
(2) 

CAPELLA 
4.18M 
0.283 
9.551011 
5200 
2.270103 
2.110 
4.0451036 
(1) 

B 

Cephei 
5.10M 
0.330 
1.52 1011 
19000 
6.895101 
2.125 
1.0531035 (2) 

SU.CAS 
5.30M 
0.330 
9.20 1011 
6350 
3.232103 
2.084 
7.1941036 (2) 
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
RR LYR. 
4.7931035 
5.030 1035 
0.953 
(2) 

CAPELLA B 
4.045 1036 
4.800 1035 
8.427 
(1) 

1.0531035 
2.215 1036 
0.047 
(2) 

Cephei 

SU.CAS 
7.194 1036 
1.012 1036 
7.109 
(2) 

SZ.TAU 
3.076 1037 
1.5611036 
19.109 
(2) 

SU.CYG 
2.557 1037 
1.809 1036 
14.135 
(2) 

RT.AUR 
3.788 1037 
1.776 1036 
20.329 
(2) 

T.VUL 
7.782 1037 
2.215 1036 
35.133 
(2) 

POLARIS 
1.1531038 
2.096 1036 
55.009 
(2) 
REFERENCES

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

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

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

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

Schatzman, E., & Praderie, F. 1990, les Ã©toiles, (savoirs actuels interedition
/Editition du CNRS, Paris) page 130

SchÃ¶nberg, M., & Chandrasekhar S. 1942, ApJ, 96, Vol. 2, 161.

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