A&A 394, 965-969 (2002)
DOI: 10.1051/0004-6361:20021233

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

A new one-dimensional 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: Hertzsprung-Russell (HR) and C-M diagrams

1 Introduction

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 O-stars (Herrero et al. 1992), the N/C and 13C/12C enrichments on the Red Giant Branch (see e.g. Charbonnel 1994, 1995), the He- and N-enrichments 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, Meyer-Hofmeister & Thomas (1970) (subsequently referred to as KMT) and applied in most subsequent works (Meyer-Hofmeister 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 so-called 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, $\rho ,...$ 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

\begin{displaymath}\overline{\rho }=\frac{\rho \left( 1-r^2\sin ^2\theta \omega ...
...ft\langle g^{-1}r^2\sin ^2\theta \right\rangle \omega \alpha }
\end{displaymath} (1)

where $\alpha =\frac{\rm d\omega }{\rm d\psi}$, instead of $\rho $, is constant on an isobar and it can denote the property of $\rho $ on the whole isobar. Furthermore, the temperature T and the chemical composition $\mu$ 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.

2 A new rotating stellar evolution model

In a Roche model, a constant total potential surface is given by the equation:

\begin{displaymath}\Psi =\Phi +\frac 12\omega ^2r^2\sin ^2\theta =\rm constant
\end{displaymath} (2)

where $\Phi =-V$, and V is the gravitational potential, r is the radius, $%
\omega $ the angular velocity and $\theta $ the colatitude.

The constant $\Psi _{\rm P}$ surfaces, the constant $\omega $surfaces, the constant $\rho $ 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 $f_\Psi $, $f _\omega $, $f_{\rm d}$ and $f_{\rm T}$ can always be found to make $f_\Psi \Psi $, $f_\omega \omega
$, $f _{\rm d}\rho $ and $f_{\rm T}T$ constant on an isobar, furthermore, if $\rho _{\rm P}$ and $T_{\rm P} $ denote the value of density and temperature of a point P ( $r_{\rm P}~\theta _{\rm P})$ on an isobar, $\left\langle
f_{\rm d}\right\rangle \rho _{\rm P}\ $ and $ \left\langle f_{\rm T}\right\rangle
T_{\rm P} $ denote the properties of density and temperature on the whole isobar, different values of $\left\langle f_{\rm d}\right\rangle \
$ and $\left\langle f_{\rm T}\right\rangle $ 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 $f_\Psi $, $f _\omega $, $f_{\rm d}$ and $f_{\rm T}$, indicating the relationship between the constant $\Psi _{\rm P}$, $\omega $, $\rho $, T surfaces and the isobars, accordingly, are not independent of one another. Once $f _\omega $, 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 $f_\Psi $ and $f_{\rm d}$ with $f _\omega $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

\begin{displaymath}V_{\rm P}=\frac{4\pi }3a^2b.
\end{displaymath} (3)

For every isobar, an assumed equivalent sphere with the radius $r_{\rm P}$ can be defined by:

\begin{displaymath}r_{\rm p}=\left(\frac{3V_{\rm p}}{4\pi }\right)^{\frac 13}\cdot
\end{displaymath} (4)

The structural equations are written at the point P ($r_{\rm P}$ $\theta
_{\rm P})$ where an isobar intersects its equivalent sphere (see Fig. 1).

\par\includegraphics[width=8.8cm,clip]{f1.eps}\end{figure} Figure 1: The isobaric surface.
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2.1 The equations for the stellar interior

(1) Conservation of mass

As from Eq. (4), the mass between the two isobaric surfaces is given by

\begin{displaymath}{\rm d}M_{\rm P}=\left\langle f_{\rm d}\right\rangle \rho _{\...
f_{\rm d}\right\rangle \rho _{\rm P}{\rm d}r_{\rm P}
\end{displaymath} (5)

which gives the equation of conservation of mass as

\begin{displaymath}\frac{{\rm d}r_{\rm P}}{{\rm d}M_{\rm P}}=\frac 1{4\pi r_{\rm P}^2\left\langle f_{\rm d}\right\rangle \rho _{\rm P}}\cdot
\end{displaymath} (6)

(2) The hydrostatic equilibrium

The hydrostatic equilibrium implies that

\begin{displaymath}\overrightarrow{\nabla }P=-\left\langle f_{\rm d}\right\rangle \rho _{\rm P}
\overrightarrow{g_{\rm eff}}
\end{displaymath} (7)

where $\overrightarrow{g_{\rm eff}}$ is the effective gravitational acceleration. In the spherical coordinates, its components are given by

\begin{displaymath}g_{\rm eff,r}=-g_{\rm r}+a_{\rm n}\sin \theta _{\rm P}
\end{displaymath} (8)

\begin{displaymath}g_{\rm eff,\theta }=a_{\rm n}\cos \theta _{\rm P}
\end{displaymath} (9)

where $\overrightarrow{a_{\rm n}}$ is the centrifugal acceleration, $%
\overrightarrow{g_{\rm r}}$ the gravitational acceleration. The values of $%
\overrightarrow{a_{\rm n}} $ and $\overrightarrow{g_{\rm r}}$ are given by

\begin{displaymath}a_{\rm n}=\omega ^2r_{\rm P}\sin \theta _{\rm P}
\end{displaymath} (10)

\begin{displaymath}g_{\rm r}=\frac{GM_{\rm P}}{r_{\rm P}^2}\cdot
\end{displaymath} (11)

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

\begin{displaymath}\frac{{\rm d}P}{{\rm d}r_{\rm P}}=-\left\langle f_{\rm d}\right\rangle \rho _{\rm P}g_{\rm eff}\cos \alpha
\end{displaymath} (12)


\begin{displaymath}\cos \alpha =\frac{g_{\rm r}^2+g_{\rm eff}^2-a_{\rm n}^2}{2g_{\rm r}g_{\rm eff}}\cdot
\end{displaymath} (13)

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

\begin{displaymath}\frac{{\rm d}P}{{\rm d}M_{\rm P}}=-\frac{GM_{\rm P}}{4\pi r_{\rm P}^4}f_{\rm P}
\end{displaymath} (14)


\begin{displaymath}f_{\rm P} =\frac 12\frac{g_{\rm r}^2+g_{\rm eff}^2-a_{\rm n}^2}{g_{\rm r}^2}=\frac 12\left( 1+\xi \right)
\end{displaymath} (15)


\begin{displaymath}\xi =\frac{g_{\rm eff}^2-a_{\rm n}^2}{g_{\rm r}^2}\cdot
\end{displaymath} (16)

(3) Conservation of the energy

The net energy outflow from a shell between the two isobaric surfaces Pand $P +{\rm d}P $ is equal to

\begin{displaymath}{\rm d}L_{\rm P} =\varepsilon {\rm d}M_{\rm P}
\end{displaymath} (17)

where $\varepsilon $ is the rate of energy production in the shell. Writing $%
\varepsilon $ in its nuclear, internal and neutrino components, one obtains the equation for conservation of the energy as

\begin{displaymath}\frac{{\rm d}L_{\rm P} }{{\rm d}M_{\rm P} }=\varepsilon _{\rm n}+\varepsilon _{\rm g}-\varepsilon _\nu.
\end{displaymath} (18)

(4) Energy transport in the region of radiative equilibrium

The temperature gradient in the region of radiative equilibrium is defined as

\begin{displaymath}\nabla _{\rm R}=\left( \frac{{\rm d}\ln T}{{\rm d}\ln P}\right) _{\rm R}=\frac PT\frac{{\rm d}T}{{\rm d}P} \cdot
\end{displaymath} (19)

Using (12) and (19), one obtains

\begin{displaymath}\frac{{\rm d{}T}}{{\rm d}n}=-\frac{\rho Tg_{\rm eff}}P\nabla _{\rm R}
\end{displaymath} (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

\begin{displaymath}\pi F=-\frac{4acT^3}{3\kappa \rho }\frac{{\rm d}T}{{\rm d}n}\cdot
\end{displaymath} (21)

Hence, one obtains the luminosity as

\begin{displaymath}L_{\rm P}=-S_{\rm P}\frac{4acT^3}{3\kappa \rho }\frac{{\rm d}T}{{\rm d}n}
\end{displaymath} (22)

where $S_{\rm P}$ is the area of the isobaric surface. Comparing (22) with (20), one obtains

\begin{displaymath}\nabla _{\rm R}=\frac{3\kappa L_{\rm P}P}{4acT^4S_{\rm P}g_{\...
...langle f_{\rm T}\right\rangle T_{\rm P})^4GM_{\rm P}}f_{\rm R}
\end{displaymath} (23)


\begin{displaymath}f_{\rm R}=\frac{g_{\rm r}}{g_{\rm eff}}\frac{4\pi r_{\rm p}^2}{S_{\rm P}}\cdot
\end{displaymath} (24)

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

\begin{displaymath}\frac{{\rm d}(\left\langle f_{\rm T}\right\rangle T_{\rm P})}...
...ght\rangle T_{\rm P})f_{\rm P}}{4\pi r^4\rm P}\nabla _{\rm R}.
\end{displaymath} (25)

(5) Energy transport in the convective region

The equation of energy transported in the convective region of stellar interior can be written as

\begin{displaymath}\frac{{\rm d}(\left\langle f_{\rm T}\right\rangle T_{\rm P})}...
...}\right\rangle T_{\rm P})f_{\rm P}}{4\pi r^4P}\nabla _{\rm ad}
\end{displaymath} (26)


\begin{displaymath}\nabla _{\rm ad}=\frac{\delta P}{C_{\rm P}(\left\langle f_{\rm T}\right\rangle T_{\rm P})\rho _{\rm P}}\cdot
\end{displaymath} (27)

2.2 Calculation of the quantities f $_\mathsfsl{P} $ and f $_\mathsfsl{R}$

In the practical calculations, the non-spherical 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 $f_{\rm P} $ and $f_{\rm R}$, 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 $r_{\rm P}$ and the angular velocity $%
\omega $ are given,the volume $V_{\rm P} $ inside the isobaric surface can be written from (4) as

\begin{displaymath}V_{\rm P}=\frac{4\pi }3r_{\rm P}^3.
\end{displaymath} (28)

From Eq. (2), one obtains

\begin{displaymath}f_{\Psi a}\left[\frac{GM_{\rm p}}a+\frac 12(f_{\omega a}\omega )^2a^2\right]=f_{\Psi b}\left(\frac{GM_{\rm p}}b\right)
\end{displaymath} (29)

$f_{\omega a}$ can be supposed to be 1.

Then, the major and minor axis of the isobar can be obtained as

\begin{displaymath}a=\frac b{f-\eta }
\end{displaymath} (30)

\begin{displaymath}b=r_{\rm p}(f-\eta )^{\frac 23}
\end{displaymath} (31)

\begin{displaymath}f=\frac{f_{\Psi b}}{f_{\Psi a}}
\end{displaymath} (32)


\begin{displaymath}\eta =\frac 12\frac{\omega ^2r_{\rm p}^3}{GM_{\rm p}}\cdot
\end{displaymath} (33)

Using the value b and the relation

\begin{displaymath}f_{\Psi}\left[\frac{GM_{\rm p}}{r_{\rm p}}+\frac 12(f_\omega ...
_{\rm p}\right]=f_{\Psi b}\left(\frac{GM_{\rm p}}b\right)
\end{displaymath} (34)

the value of the colatitude $\theta _{\rm p}$ can be obtained.

The factor $f_{\rm p}$ can be calculated by using (15) when the values of $\omega
, $ $r_{\rm p}$ and $\theta _{\rm p}$ are known.

The factor $f_{\rm R}$ can be calculated by using (24) when the surface area $S_{\rm P}$ of the isobar is given by

\begin{displaymath}S_{\rm p}=\frac{4\pi }3(2a^2+b^2).
\end{displaymath} (35)

3 Evolutionary calculations

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 low-temperature 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 non-rotating 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 $\omega $, the pressure P and the temperature T and a $\overline{\rho} $ are constant on isobars. Therefore, in the new method,

\begin{displaymath}f=f_\omega =\left\langle f_{\rm T}\right\rangle =1
\end{displaymath} (36)

\begin{displaymath}\left\langle f_{\rm d}\right\rangle =\frac{\rho\left( 1-r^2\s...
g^{-1}r^2\sin ^2\theta \right\rangle \omega \alpha )}
\end{displaymath} (37)

where $\alpha =\frac{\rm d\omega}{\rm d\psi}\cdot$

\par\includegraphics[angle=270,width=11.2cm,clip]{27462.ps}\end{figure} Figure 2: Upper ZAMS in the theoretical HR diagram for different rotation rates.
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Table 1: Effect of rotation on the ZAMS.
Initial $\omega_0$ $\omega_0$/ $\omega_{\rm crit}$ $V_{\rm eq}$ $\log L/L_{\odot}$ $\log T_{\rm eff}$ $\frac{R_{\rm P}\left( \omega \right)
}{R_{\rm e}\left( \omega \right) }$ $\frac{R_{\rm e}\left( \omega \right)
}{R_{\rm e}\left( \omega =0\right) }$ $E_{\rm h}$
mass   [10-4 s-1] [km s-1]         [1037 erg]
  $M_{\odot}$ $\omega_{\rm crit }
10^{-3}~[\rm s^{-1}]$      
$M_{\odot}$ 0.00 0.00 0.000 3.595 4.376 1.0000 1.000 0.7743
$M_{\odot}$ 0.84 53.13% 227.6 3.585 4.367 0.9626 1.029 0.7563
$M_{\odot}$ 1.54 97.41% 529.2 3.561 4.330 0.7978 1.189 0.7154
  20 $M_{\odot}$ $\omega_{\rm crit }
10^{-3}~[\rm s^{-1}]$      
20 $M_{\odot}$ 0.00 0.00 0.000 4.628 4.538 1.0000 1.000 3.763
20 $M_{\odot}$ 0.64 53.60% 270.9 4.618 4.529 0.9628 1.029 3.673
20 $M_{\odot}$ 1.54 97.57% 629.3 4.594 4.490 0.7977 1.196 3.476
  40 $M_{\odot}$ $\omega_{\rm crit }
10^{-4}~[\rm s^{-1}]$      
40 $M_{\odot}$ 0.00 0.00 0.000 5.354 4.635 1.0000 1.000 10.01
40 $M_{\odot}$ 0.495 55.49% 309.9 5.345 4.625 0.9639 1.032 9.814
40 $M_{\odot}$ 0.88 98.65% 722.6 5.326 4.582 0.7974 1.232 9.382
  60 $M_{\odot}$ $\omega_{\rm crit }
10^{-4}~[\rm s^{-1}]$      
60 $M_{\odot}$ 0.00 0.00 0.000 5.713 4.671 1.0000 1.000 15.25
60 $M_{\odot}$ 0.413 58.58% 331.4 5.705 4.661 0.9647 1.038 14.99
60 $M_{\odot}$ 0.701 99.43% 765.7 5.691 4.611 0.7981 1.285 14.50

Symbols: $\omega_0$ and $\omega_{\rm crit}$ denote the initial angular velocity and the breaking equatorial angular velocity at the surface on the ZAMS, respectively; $V_{\rm eq}$ denotes the equatorial velocity at the surface; $\frac{R_{\rm P}\left( \omega \right)
}{R_{\rm e}\left( \omega \right) }$ and $\frac{R_{\rm e}\left( \omega \right)
}{R_{\rm e}\left( \omega =0\right) }$denote the oblateness (polar radius over the equatorial radius) and the ratios between the equatorial radius obtained with rotation and that obtained without rotation, respectively; and $E_{\rm h}$ denote the energy production of core H-burning.

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:

\begin{displaymath}\alpha =\frac{\rm d\omega}{\rm d\psi}=1
\end{displaymath} (38)

\begin{displaymath}\left\langle f_{\rm d}\right\rangle =1.
\end{displaymath} (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.

4 Conclusion

A new one-dimensional 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 $f _\omega $ 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.

I wish to express thanks to R.Q. Huang for his most fruitful advice.

Appendix A:

Following MM,

$\displaystyle {\overrightarrow{\nabla }P}$=$\displaystyle -\rho \overrightarrow{g}_{\rm eff}$
=$\displaystyle -\rho\left(\overrightarrow{\nabla }\Psi _{\rm P}-r^2\sin ^2\theta \omega
\overrightarrow{\nabla }\omega \right)$ (A.1)

and under the condition of no special assumption about $\omega $ , one can get:
$\displaystyle {\overrightarrow{\nabla }P}$=$\displaystyle -\rho \overrightarrow{g}_{\rm eff}$
=$\displaystyle -\rho \left[\frac{\overrightarrow{\nabla }\Psi
_{\rm P}}{\overrig...
...w{\nabla }
(f_\omega \omega )}\overrightarrow{\nabla }(f_\omega \omega )\right]$
=$\displaystyle -\rho [\frac 1{\left\langle f_\Psi \right\rangle
...left\langle f_\omega \right\rangle
}\overrightarrow{\nabla }(f_\omega \omega )]$ (A.2)

$\overrightarrow{\nabla }(f_\Psi \Psi _{\rm P})$ is parallel to $\overrightarrow{\nabla }(f_\omega \omega)$, thus, the effective gravity can be expressed as

\begin{displaymath}g=\left[ \frac 1{\left\langle f_\Psi \right\rangle }-r^2\sin
}\right] \frac{{\rm d}(f_\Psi \Psi _{\rm P})}{{\rm d}n}
\end{displaymath} (A.3)

where: $\alpha =\frac{{\rm d}(f_\omega \omega )}{{\rm d}(f_\Psi \Psi _{\rm P})}$, or

\begin{displaymath}g=\left[\frac \beta {\left\langle f_\Psi \right\rangle }-r^2\...
}\right]\frac{{\rm d}(f_\omega \omega )}{{\rm d}n}
\end{displaymath} (A.4)

where $\beta =\frac{{\rm d}(f_\Psi \Psi _{\rm P})}{{\rm d}(f_\omega \omega )}$, hence:

\begin{displaymath}\left\langle f_\Psi \right\rangle =\frac \beta
{g\frac{{\rm d...
...^2\theta \omega \frac
1{\left\langle f_\omega \right\rangle }}
\end{displaymath} (A.5)


 \begin{displaymath}\frac{{\rm d}n}{{\rm d}(f_\Psi \Psi _{\rm P})}=\frac{\frac 1{...
...ga \alpha \frac
1{\left\langle f_\omega \right\rangle }}g\cdot
\end{displaymath} (A.6)

The hydrostatic equation then becomes

\begin{displaymath}\frac{{\rm d}P}{{\rm d}n}=-\rho \left[ \frac 1{\left\langle f...
...rangle }\right] \frac{{\rm d}(f_\Psi \Psi
_{\rm P})}{{\rm d}n}
\end{displaymath} (A.7)


\begin{displaymath}\frac{{\rm d}P}{{\rm d}(f_\Psi \Psi _{\rm P})}=-\rho \left[ \...
...lpha \frac
1{\left\langle f_\omega \right\rangle }\right]\cdot
\end{displaymath} (A.8)

From this equation, one can deduce that
$-\rho \left[ \frac
1{\left\langle f_\Psi \right\rangle }-r^2\sin ^2\theta \omega
\alpha \frac 1{\left\langle f_\omega \right\rangle }\right]$ is constant on an isobar.

Using (A.6), one can get

$\displaystyle {\rm d}M_{\rm P}$=$\displaystyle \int_{\rm isobar}\rho {\rm d}n{\rm d}\sigma$
=$\displaystyle {\rm d}(f_\Psi \Psi
_{\rm P})\int_{\rm isobar}\rho \frac{{\rm d}n}{{\rm d}(f_\Psi \Psi _{\rm P})}{\rm d}\sigma$
=$\displaystyle {\rm d}(f_\Psi \Psi _{\rm P})\int_{\rm isobar}\rho \frac{ \frac 1...
...eta \omega \alpha \frac
1{\left\langle f_\omega \right\rangle }}g{\rm d}\sigma.$ (A.9)

$\displaystyle {\rm d}V_{\rm P}$=$\displaystyle \int_{\rm isobar}{\rm d}n{\rm d}\sigma$
=$\displaystyle {\rm d}(f_\Psi \Psi
_{\rm P})\int_{\rm isobar}\frac{{\rm d}n}{ {\rm d}(f_\Psi \Psi _{\rm P})}{\rm d}\sigma$
=$\displaystyle {\rm d}(f_\Psi \Psi _{\rm P})\int_{\rm isobar}\frac{\frac 1{\left...
...eta \omega \alpha \frac
1{\left\langle f_\omega \right\rangle }}g{\rm d}\sigma.$ (A.10)


\begin{displaymath}\frac{{\rm d}(f_\Psi \Psi _{\rm P})}{{\rm d}M_{\rm P}}=\frac ...
...ht\rangle }\right]
\left\langle g^{-1}\right\rangle S_{\rm P}}
\end{displaymath} (A.11)


\begin{displaymath}\frac{{\rm d}V_{\rm P}}{{\rm d}(f_\Psi \Psi _{\rm P})}=\left[...
...\alpha{\left\langle f_\omega \right\rangle }\right] S_{\rm P}.
\end{displaymath} (A.12)


\begin{displaymath}\overline{\rho }=\frac{\rho \left[ \frac 1{\left\langle f_\Ps...
...le \frac
\alpha{\left\langle f_\omega \right\rangle }\right] }
\end{displaymath} (A.13)

\begin{displaymath}\left\langle f_{\rm d}\right\rangle =\frac{\rho \left[ \frac
\alpha{\left\langle f_\omega \right\rangle }\right] }\cdot
\end{displaymath} (A.14)

Furthermore, once $f _\omega $ is given, $f_{\rm T}$ can be decided from other ways and (A5) and (A14), which decide $f_\psi$ and $f_{\rm d}$respectively, can be more applicable.


Copyright ESO 2002