A&A 394, 965969 (2002)
DOI: 10.1051/00046361:20021233
A more powerful evolution model for rotating stars
Y. R. Zeng
Yunnan Astronomical Observatory and National
Astronomical Observatories, PO Box 110, 650011 Kunming, PR China
Received 30 May 2002 / Accepted 1 August 2002
Abstract
A new onedimensional rotating stellar evolution model was
developed which, distinct from its previous counterparts based on
a conservative rotation case or a "shellular rotation'' case,
made no special assumptions on the distribution of angular
velocity inside a rotating star. The results of the evolutionary
calculations are presented and compared with the results of
previous studies.
Key words: stars: evolution  stars: rotation 
stars: HertzsprungRussell (HR) and CM diagrams
The study of rotating stars has attracted much attention during
the last decades as a consequence of the observational findings
such as the He enrichments
in the fast rotators among Ostars (Herrero et al. 1992), the N/C and ^{13}C/^{12}C enrichments on the Red Giant Branch (see e.g. Charbonnel
1994, 1995), the He and Nenrichments in OBA supergiants and in SN 1987A
(Walborn 1976; Fransson et al. 1989; Venn 1995A,b; Venn et al. 1996) and the
problem of the blue to red supergiants ratio in galaxies (Langer & Maeder
1995) etc.
In a rotating star, the centrifugal forces reduce the effective
gravity according to the latitude and introduce deviations from
sphericity. Thus, the structure equations of a rotating star should
be two dimensional.The original method devised by Kippenhahn,
MeyerHofmeister & Thomas (1970) (subsequently referred to as KMT)
and applied in most subsequent works (MeyerHofmeister 1972; Endal
& Sofia 1976 (hereafter ES), 1978; Pinsonneault et al. 1989, 1990;
Chaboyer et al. 1995a,b; Sills et al. 2000; Heger et al. 2000) to keep
the problem one dimensional is based on a socalled conservative
rotation case, i.e., the angular velocity is constant on
cylinders. In such a rotation case, the total potential is
conservative and the structural variables P, T,
are
constant on a equipotential surface. Meynet & Maeder
(1997) (subsequently referred to as MM) argue that not a
conservative rotation case but a "shellular rotation'' case,
i.e., the angular velocity is constant on an isobar (an isobar is a
constant pressure surface) is the real rotation case in an evolved
star. In a "shellular rotation'' case, they find a

(1) 
where
,
instead of ,
is
constant on an isobar and it can denote the property of
on the whole isobar. Furthermore, the temperature T and the
chemical composition
are supposed to be constant on an
isobar. Thus, if the structural equations are written on isobars,
the problem can be kept one dimensional.
The distribution of angular velocity inside a real star is still a
mystery, therefore, it is important to develop an one dimensional
method which is not based on any special assumptions on the
distribution of angular velocity. I propose a method based on
Meynet & Maeder (1997) to deal with this problem.
The new structural model is introduced in Sect. 2, and the results of
evolution calculations are presented in Sect. 3.
In a Roche model, a constant total potential surface is given by
the equation:

(2) 
where ,
and V is the gravitational potential, r is the radius,
the angular velocity and
the colatitude.
The constant
surfaces, the constant surfaces, the constant
surfaces, the constant Tsurfaces, and the isobars do not correspond to each other if no
special assumption about the distribution of angular velocity is
made. However, some parameters ,
,
and
can always be found to make
,
,
and
constant on an isobar, furthermore, if
and
denote the value of density and temperature
of a point P (
on an isobar,
and
denote the properties of density and temperature on the
whole isobar, different values of
and
will be given for
different points P. Thus, the problem can be kept one
dimensional, if the structural equations are written on isobars.
The four parameters ,
,
and ,
indicating the relationship between the constant
,
,
,
T surfaces and the isobars, accordingly, are
not independent of one another. Once ,
the distribution
of angular velocity, is decided, the other three parameters are
decided, too. Consequently, there is only one independent
parameter need to be determined by advanced theories or by
comparing the results of evolution model with the observational
data. The relationships of
and
with will be shown in the Appendix and how these parameters are decided
in the "shellular rotation'' case will be shown in Sect. 3.
An isobar in a rotating star is assumed to be an ellipsoid of
resolution with the major axis a and the minor axis b, and the
volume inside an isobar is given by

(3) 
For every isobar, an assumed equivalent sphere with the radius
can be defined by:

(4) 
The structural equations are written at the point P (
where an isobar intersects its equivalent sphere (see Fig. 1).
(1) Conservation of mass
As from Eq. (4), the mass between the two isobaric surfaces is given by

(5) 
which gives the equation of conservation of mass as

(6) 
(2) The hydrostatic equilibrium
The hydrostatic equilibrium implies that

(7) 
where
is the effective gravitational
acceleration. In the spherical coordinates, its components are given by

(8) 

(9) 
where
is the centrifugal acceleration,
the gravitational acceleration. The values of
and
are given by

(10) 

(11) 
The radial component of the vectorial Eq. (7) is

(12) 
where

(13) 
From Eqs. (11), (12) and (6), the equation of hydrostatic equilibrium
is obtained as

(14) 
where

(15) 
and

(16) 
(3) Conservation of the energy
The net energy outflow from a shell between the two isobaric surfaces Pand
is equal to

(17) 
where
is the rate of energy production in the shell. Writing
in its nuclear, internal and neutrino components,
one obtains the equation for conservation of the energy as

(18) 
(4) Energy transport in the region of radiative equilibrium
The temperature gradient in the region of radiative equilibrium is defined as

(19) 
Using (12) and (19), one obtains

(20) 
where n is the unit normal of the isobar at point P.
The radiative
energy flux at the point P on the isobaric surface is given by

(21) 
Hence, one obtains the luminosity as

(22) 
where
is the area of the isobaric surface. Comparing (22)
with (20), one obtains

(23) 
where

(24) 
Using (14) and (19), one obtains the equation of energy transport in the
region of radiative equilibrium as

(25) 
(5) Energy transport in the convective region
The equation of energy transported in the convective region of stellar
interior can be written as

(26) 
where

(27) 
In the practical calculations, the nonspherical stratification of
rotating stars can be replaced by the spherical stratification of
equivalent sphere. Thus, the calculations of the structure Eqs. (6), (14), (18), (25) and (26) are based on the equivalent sphere. However, the two factors
and ,
defined by (15) and (24), which are related to the characteristic of rotation, should be calculated on the isobar.
When the radius of the equivalent sphere
and the angular velocity
are given,the volume
inside the isobaric surface can be
written from (4) as

(28) 
From Eq. (2), one obtains

(29) 
can be supposed to be 1.
Then, the major and minor axis of the isobar can be obtained as

(30) 

(31) 

(32) 
where

(33) 
Using the value b and the relation

(34) 
the value of the colatitude
can be obtained.
The factor
can be calculated by using (15) when the values of
and
are known.
The factor
can be calculated by using (24) when the surface
area
of the isobar is given by

(35) 
The calculation of the case without rotation is performed by using
a modified version of the stellar structure and evolution program
developed by Kippenhahn et al. (1967). This version has included the
recent opacities (cf. Iglesias & Rogers 1996, complemented at
lower temperature with the lowtemperature rosseland opacities by
Alexander & Ferguson 1994) and energy generating rates
(cf.
Maeder 1983; Maeder & Meynet 1987, 1989). The case with rotation
is calculated using the rotating evolution code which is modified
from the previous nonrotating program, and have included the
equations represented in Sect. 2.
To compare the new results with previous ones, the "shellular
rotation'' case (compared with MM) has been calculated.
MM assumed that the angular velocity ,
the pressure P and
the temperature T and a
are constant on
isobars. Therefore, in the new method,

(36) 

(37) 
where

Figure 2:
Upper ZAMS in the theoretical HR diagram for different
rotation rates. 
Open with DEXTER 
The comparison can only be made on the ZAMS because in the new
method, the mixing of chemical species and angular momentum by
circulation currents and turbulence has not been included while it
has been considered in MM's method, and solid body rotation is
assumed on the ZAMS, therefore:

(38) 

(39) 
Some characteristics of the models being calculated (X=0.7,
Z=0.02.) are given in Table 1 as a function of the initial mass
and angular velocity, and Fig. 2 presents the ZAMS obtained in the
theoretical HR diagram. These results do not differ from MM's very
much.
A new onedimensional rotating stellar evolution model was
developed. Distinct from its previous counterparts based on a
conservative rotation case or a "shellular rotation'' case, it
made no special assumptions on the distribution of angular
velocity inside a rotating star.
One independent parameter
has been introduced in the
new method; to determine its value is another complicated task.
This and other improvements, such as including the mixing of
chemical species and angular momentum by circulation currents and
turbulent, will be presented in the future.
Acknowledgements
I wish to express thanks to R.Q. Huang for his most fruitful advice.
Following MM,
and under the condition of no special assumption about
,
one can get:
is parallel to
,
thus, the effective
gravity can be expressed as

(A.3) 
where:
,
or

(A.4) 
where
,
hence:

(A.5) 
and:

(A.6) 
The hydrostatic equation then becomes

(A.7) 
or

(A.8) 
From this equation, one can deduce that
is
constant on an isobar.
Using (A.6), one can get
and
Thus

(A.11) 
and

(A.12) 
Hence

(A.13) 

(A.14) 
Furthermore, once
is given,
can be decided from
other ways and (A5) and (A14), which decide
and respectively, can be more applicable.
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