R. Q. Huang
National Astronomical Observatories / Yunnan Observatory, the Chinese Academy of Sciences, Kunming 650011, PR China
Received 1 September 2003 / Accepted 16 March 2004
Abstract
A method to treat rotational mixing in stars is
proposed. The evolution of a 8
star with rotation is studied, and the
effects of rotation on the interior structure and evolution of
the star are discussed.
Key words: stars: evolution - stars: interiors - stars: rotation
To construct a model for a rotating star, we have to take into account two effects of rotation: centrifugal force and that of meridian circulation. Due to the first effect, the structure of a star changes from sphericaly symmetric to non-sphericaly symmetric, and the stellar model becomes two-dimensional. Owing to the second effect, a radial flow of mass occurs in the star. As a result of radial mass flow, the transport of angular momentum and chemical composition takes place in the star. The sum of the two effects makes the structure and evolution of a rotating star differ from that of a non-rotating star.
The effect of centrifugal force and the changes in the stellar model have been studied by many investigators (Kippenhahn & Thomas 1970; Endal & Sofia 1976; Pinsonneault et al. 1989, 1990, 1991; Charboyer et al. 1995a,b; Langer 1998; Heger & Langer 1998; Meynet & Maeder 1997; Maeder & Meynet 2000). Among these investigations the study of Kippenhahn & Thomas (1970) has attracted special attention because they introduced a method to simplify the two-dimensional model with conservative rotation to a one-dimensional model, and gave the structure equations. Without a detailed study of the properties of stellar rotation, we make a simple assumption that the method developed by Kippenhahn & Thomas (1970) can be used in our model. The effect of meridian circulation and the radial mass flow in a star have also been studied in many investigators (Endal & Sofia 1976; Tassoul 1978; Schatzman et al. 1981; Langer 1991, 1992; Chaboyer & Zahn 1992; Zahn 1992; Heger et al. 2000; Maeder & Meynet 2000). The calculations of meridian circulation and the radial mass flow are quite complicated. Here we propose a simple method to treat the transport of angular momentum and the chemical composition of a star.
In Sect. 2, we discuss the rotational mixing and the model for a rotating star. In Sect. 3, the evolution a 8 star with rotation is studied, and the differences between the results for the evolution with and without the effects of rotation are discussed in detail.
The effect of rotation causes an outward mass flow along the
rotational axis and an inward mass flow along the equatorial
plane. Such an outward and inward mass flow in a star is called
meridian circulation (Kippenhahn & Weigert 1990; Maeder & Meynet
2000). As a result of the meridian circulation and the shear
turbulence, a transport of angular momentum and chemical
composition occurs in the star. We can treat this transport
process approximately as an advection process. The meridian
circulation is caused by the effect of rotation, and the strength
of rotation of a mass layer in the star can be expressed by the
difference between the gravity g_{i} and the mean effective
gravity
of this layer. The shear turbulence is caused
by the differential rotation, so that the rate of the change in
the composition of the element can be expressed as
= | (1) |
(2) |
At the outer and inner boundary, reflecting conditions are used:
(3) |
(4) |
Another result of the meridian circulation and the shear turbulence is the transport of angular momentum in the star.
If J_{i} is the angular momentum of the ith layer, the rate of
the change in angular momentum can be expressed as:
(5) |
At the outer and inner boundary reflecting conditions are used:
(6) |
(7) |
(8) |
The stellar model for rotating stars is based on the method developed
by Kippenhahn & Thomas (1970). The
structure equations are the following:
(9) |
(10) |
(11) |
(12) |
(13) |
(14) |
The equipotentials are defined by the equation of
(15) |
Figure 1: The evolutionary tracks in the HR diagram of a 8 star for the sequences with and without rotation. The solid and dotted curves correspond to the tracks for the STD sequence and the ROT sequence, respectively. | |
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The evolution of a 8 star is studied. Two evolutionary sequences corresponding to the cases with and without the effects of rotation are calculated. The sequence denoted as STD corresponds to the evolution without the effects of rotation, while the sequence denoted as ROT corresponds to that with the effects of rotation. The initial chemical composition of X=0.70, Z=0.02 is adopted for both sequences. The effects of mass loss due to stellar winds and convective overshooting (0.22l over the Schwartzschield border) are considered for both sequences. The initial rotating velocity for the 8 star has a value of 150.49 km s^{-1}. The parameters C_{1} and C_{2} in Eq. (1) have the values of 0.02 and 0.1, while the parameter C_{3} in Eq. (5) has the value of 0.05.
Figure 2: The time dependent variation of the border of the burning core of the 8 star. The solid and dotted curves have the same meanings as in Fig. 1. | |
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Figure 3: he evolution of the ratio as a function of the mass fraction in the star. X(c) is the hydrogen composition at the center. | |
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Table 1: The parameters at different evolutionary points a, b, c and d of the STD and ROT sequences.
The evolutionary tracks in the HR diagram for the 8 star of two sequences are illustrated in Fig. 1. The solid curve corresponds to the track for the STD sequence, while the dotted curve corresponds to that of the ROT sequence. From the difference between the solid and the dotted curves in Fig. 1 one finds that rotation causes the evolutionary track in the HR diagram to shift towards lower luminosity.
Figure 2 shows the time dependent variation at the border of the burning core of a 8 star. The solid and the dotted curves correspond to the sequences STD and ROT respectively. From Fig. 2 one finds that the ends of the H- and He-burning phases are at later times for the ROT sequence than for the STD sequence. This means that the effect of rotation causes the star to evolve more slowly. In addition, Fig. 2 shows that the border of the H-burning core of the ROT sequence is slightly smaller than that of the STD sequence. This explains the fact that the evolutionary track in the HR diagram of the star with rotation shifts towards lower luminosity in Fig. 1, because the luminosity of the star is dependent on the size of the burning core.
In Table 1 we list the ages, masses, effective temperatures, luminosities, the central hydrogen and helium compositions, the compositions of helium and heavy elements on the surface, and the rotating velocities at the critical evolutionary points of the sequences. From the compositions of helium and heavy elements on the surface of the ROT sequence at the points b, c and d, we find the effect of the rotational mixing. However, it is quite small.
Figure 3 illustrates the time-dependent variation in the rotating velocity of the ROT sequence. From Fig. 3 we find that the rotating velocity of the star decreases progressively during the central hydrogen burning phase, and reduces rapidly to very small values during the H-shell burning phase. This shows that the effects of rotation have an influence on the structure and evolution of the star mainly in the main sequence phase.
Acknowledgements
This work is supported by the Chinese National Science Foundation No. 10073020.