## Field Amplification by WindingUp

Differential rotation stretches the frozen-in field lines and after several rotation periods the field lines are wound-up with a strong toroidal component. The field lines become very close to each other, meaning that the field has been amplified. Some energy of the differential rotation is converted into magnetic energy. Let us assume 545 that the initial field is weak, i.e., the average Alfven frequency rnA (13.17) is small compared to the rotation frequency Q, where B and Q are the mean...

## Mass Conservation and Continuity Equation

In spherical symmetry, the change of the mass Mr (t) in a sphere of radius r can be written as dMr (r, t ) 4nr2 q dr - 4 nr2 gvdt. (1.7) The first term on the right represents the change of mass due to a variation of radius r at a given time t, the second term expresses the flux of mass out of the sphere of constant r due to an outward motion with velocity v > 0. The differential dMr(r, t) can also be written as Comparing with (1.7), we make the identifications, 4nr2q and -4nr2qv . (1.9) The...

## From Maclaurin Spheroids to the Roche Models

The stability of rotating configurations has been studied since long (see review in 315 ), for example with Maclaurin spheroids, where the density q is supposed constant or with the Roche model, which assumes an infinite central condensation. The complex reality lies between these two extreme cases. In the case of the Maclaurin spheroids, the equilibrium configurations flatten for high rotation. For extremely high angular momentum, it tends toward an infinitely thin circular disk. The maximum...

## Ac 2Hp CR1 lT g

Near the stellar surface, as T decreases rapidly, the cutoff frequency rises fast. This fast growth constitutes the limit of acoustic waves near the surface. In the atmosphere of the Sun, ac 0.03 s-1, which corresponds to a period P 1 v 2n a 200 s. Let us now examine the behavior of the other key frequencies which determine the nature of the waves. - The Brunt-Vaisala Nad (Sect. 5.1). Figure 16.1 shows its behavior in a solar model. At the very center, the gravity...

## The Initial Cloud Structure and its Evolution

In the above derivation of the Jeans mass, we have assumed constant q and T. While constant T is a valuable assumption, this is not the case for density. We examine in more detail the structure of contracting clouds, which is critical for fragmentation and also for the accretion rates (Sect. 20.3). The gravitational collapse of clouds is non-homologous, i.e., central regions collapse faster than external ones. Non-homology starts during the isothermal phase. As a result of contraction and...

## Overshoot Semiconvection Thermohaline Convection Rotation and Solberg Hoiland Criterion

The devil is often hidden in details or in what is looking at first as a detail and then reveals itself as a point of prime importance. Convection theory is full of such details able to critically change the results. We may mention the problem of overshooting, i.e., where is the exact edge of a convective zone, the role of the gradients, the heat losses by convective fluid elements, semiconvection and thermohaline mixing, the effects of rotation on convection, convection at sonic velocities,...

## FP

The simplifications (4.6)-(4.8) are not severe 408 . First, the equations describing the hydrostatic equilibrium and the conservation of mass are strictly valid (in the Roche model approximation) in the case of a shellular rotation law, provided that q is considered as the dependent variable for density. Moreover the strong horizontal turbulence homogenizes the chemical composition and reduces the q and T contrasts on isobars. The above equations are used in models of rotating stars in...

## Transport by Gravity Waves

A stone thrown into the water generates waves which propagate making nice circles on the water surface away from the impact point. These are surface gravity waves, because the water displaced by the stone is recalled back by gravity. In stars, gravity waves are typically produced by the convective pistons injecting energy into a radiatively stable adjacent region. Due to the piston motions, the fluid elements at some depth are slightly moved up or down into regions of different densities....

## The Rosseland Mean Opacity

Some care must be brought to the definition of the mean opacity k in (3.17). In particular, it is to be noted that the mean opacity is not simply the integral of the monochromatic opacity, i.e., k JO Kx dX. An appropriate definition has to be adopted to achieve consistency with the definition of the integrated flux over the frequencies or over the wavelengths, The radiation pressure is Prad 4nI0 (3c), with I0 B(T) (cf. Appendix C.1). Thus, (3.15) becomes The second form is obtained if instead...

## Kudritzkis Wind Momentum Luminosity Relation

There is an important relation between the mechanical wind momentum Mv of the mass outflow and the luminosity L of hot stars predicted by the theory of radiative winds. It was found and studied in a series of works by Kudritzki and coworkers 297, 298, 481 . Such a relation is established for O-type stars in the Galaxy and in the Magellanic Clouds, as well as for A- and B-type supergiants in the Galaxy 298 (Fig. 14.2). The WLR provides a new way to obtain absolute luminosities from spectroscopic...

## Gravitational Settling

Let us consider a binary mixture, where the component i has a very low abundance, so that its contribution to the total density is negligible. This is the so-called test atom approximation. Let us also consider initial conditions with constant P, T and q and suppose that there is an acceleration ai acting on particles i only. These particles will reach a state of hydrostatic equilibrium characterized by the equality of the force acting on them by volume unity and the gradient of partial...

## The Mechanical Equilibrium of Stars

If the stars would not be in equilibrium during most of their life, stable conditions permitting life would not have been present on the Earth. Mechanical equilibrium is a necessary condition for stable luminosity and temperature over long periods of time. It is a fundamental property of stars, implying the exact balance between the gravity force which attracts the matter toward the center and the force due to the thermal pressure, which resists gravity. Any departure from this equilibrium will...

## The CNO Cycles

When T6 > 17 for a standard composition, the H burning occurs mainly through the CNO cycles. CNO elements must be initially present, their sum XC + XN + XO does not change when the cycles operate, as easily verified from Table 25.2. However, the ratios of CNO elements, like XN XC, are modified by the cycles. There is a basic cycle, the CN cycle (see Fig. 25.1), to which two ON loops are added, the relative importance of the ON loops increases for higher T. In addition, there is a rare loop...

## A K

This shows that for a star of a given gravity, the effective temperature is limited. 3.6.2.1 The Upper Bound of Stellar Mass First Approximation The outward-directed radiation forces put an upper limit to the stellar masses. The Eddington luminosity grows linearly with mass. For a hydrogen mass fraction X 0.70, one gets Fig. 3.6 The steeper line represents schematically the mass-luminosity relation, while the other line represents the Eddington luminosity as a...

## Evolution in the HeBurning Phase and Dredgeup

Let us further follow the evolution in the giant phase (Fig. 26.1) for stars which do not experience the He flash (cf. Sect. 25.3.2), i.e., for stellar masses above about 2.3M0. D-E, Red giant branch, first dredge-up the low Teff resulting from envelope expansion produces very large opacities, which in turn favor convection. The external convective envelope becomes very deep and may cover 90 of the stellar radius (Fig. 5.8). The deep convective envelope reaches the layers processed by the CNO...

## Internal Properties Tracks in the HR Diagram

During the MS phase, the fusion of hydrogen into helium progressively modifies the H and He profiles in the deep interior and thus the distribution of the mean molecular weight x, while the composition of the outer layers does not change, in principle. The average 1 is thus increased and a higher luminosity is resulting. In this inhomogeneous body, the growth of the central L favors a global expansion of the outer layers, the associate cooling increases the opacity and makes Teff to further...

## Magnetic Buoyancy

The simplest case of magnetic buoyancy, originally considered by Parker 459 , is that of a horizontal tube of magnetic flux lying at some depth in the solar envelope, where the average surrounding field is negligible with respect to that inside the tube. The tube is assumed in pressure equilibrium with the surroundings, which implies motions slower than the sound velocity. The tube is also in thermal equilibrium due to the rapid heat transfer by convection or radiation. The presence of an...

## Core Collapse and Explosion

In general, one good reason is better than many ones. For core collapse, there are four good reasons for its occurrence 1. As evolution proceeds from H to Si burning, there is less and less nuclear energy available (Fig. 9.1), thus core contraction must inexorably go on to compensate for the energy lost. 2. As central T and q increase, the v emissions (Sect. 9.5) remove more energy (Fig. 28.6), which can only be provided by core contraction. 3. At T 1010 K, q 1010 g cm3 1 MeV), the energetic...

## Polar Radius as a Function of Rotation

In first approximation, one may consider that the polar radii are independent of rotation and use values such as given by Fig. 25.7. In reality the polar radii Rp(a) have a slight dependence on m, which results from the small changes of internal structure brought about by centrifugal force. The rate of change is given by the models of internal structure with rotation. While the equatorial radius strongly inflates, the polar radius decreases by a few percent in general (Fig. 2.7), mostly as a...

## CpTo 1 Vad Z

Instead of dQ q -3 dr r for a sphere. We can thus replace the geometry factor of 3 by r0 D in the density variations and subsequent equations and one gets for Z in (3.90) with all other conclusions being the same. This means that for a perfect gas, for D C r0, we have Z < 0 and then A > 0. Thus the nuclear burning in very thin shell is unstable, one sees from (3.95) that for a perfect gas A > 0 for D r0 < 5 12. From (3.84) and (3.94), one gets which means that a change of density in the...

## Mass Loss Effects in the HR Diagram

Figure 27.4 illustrates the effects of mass loss for a 30 M0 star with a simple parametrization. The mass reduction makes the star less luminous, however it is overluminous for its actual mass, the MS band is more extended as the core mass fraction is larger (Fig. 24.3). In the expression of the MS lifetime tH qcM L, the quantities (qc x M) and L are reduced by mass loss so that on the whole tH does not change very much, increasing for current M rates, by about 5-10 . Figure 27.4 illustrates...

## The NeNa and MgAl Cycles

There are two cycles of reactions not significant for energy production, but which change some isotopic ratios, working above T6 > 25. These are the NeNa and MgAl cycles (Fig. 25.4). The NeNa chain starts from 20Ne. The 20Ne abundance is high enough not to be modified by the creation or destruction of the other much less-abundant isotopes (Appendix A.3). For these isotopes, the NeNA chain has the following effects for different T6 T (106 K) 17 - 21Ne its abundance first increases with T up to...

## Stellar Surface and Gravity

The stellar surface is an equipotential W const., otherwise there would be mountains on the star and matter flowing from higher to lower levels. The total potential at a level r and at colatitude ( 0 at the pole) in a star of constant angular velocity Q can be written as GMr 1 T T T W(r, ) --j- Q2 r2 sin2 . (2.9) One assumes in the Roche model that the gravitational potential 0 GMr r of the mass Mr inside radius r is not distorted by rotation. The inner layers are considered as spherical, which...

## The Schnberg Chandrasekhar Limit

There is a maximum mass fraction permitted for an isothermal core of perfect gas. As P increases toward the center and T is constant, the density must provide the whole pressure and this becomes no longer possible. Above the limit, the isothermal core cannot sustain the pressure and collapses. This can be shown from the Virial theorem (1.54) applied to an isothermal core, with parameters labeled with an index 1. Let us call P1 the pressure at the core surface. With account for (1.48) the Virial...

## Complements on Radiative Transfer and Thermodynamics

Here, we define some basic properties of the radiation field in stars. Let us consider a medium with a radiation beam at frequency v in a given direction s (Fig. C.1). The energy dUv transmitted by a surface element da in a direction making an angle 6 with the normal to da, over the length ds centered on a solid angle dQ, during the time dt and in the frequency interval dv, is dUv Iv da cos 6 dvdQ dt. (C.1) This defines the intensity Iv of a radiation beam. Density of radiation energy The...

## Transport Processes Diffusion and Advection

There does not seem to by anything in the Universe which stays for ever where it has been put once. This also applies to the elements synthesized in the stellar interiors. Classical evolution models assume that an element in a radiative zone, e.g., in the solar center, always stays exactly at the same place, i.e., for the next 10 billions years in the case of the Sun. Such an assumption ignores microscopic diffusion and the fact that stars, especially rotating stars, are subject to many...

## Rrn1

Which has many properties, in particular r(n + 1) nr(n), r(L5) yfn 2 and r(2.5) (3 4) n. This gives ' Fx 2(f) - and F3 2(f ) 3 ef . (7.138) (2mekT)3 2 Me mu 2 ef (7.139) If one eliminates f between these two equations, one gets the law of perfect gas for the electrons P q kT (p mu). 7.7.2.2 Case of Very Strong Degeneracy The integration is performed up to f, where the cells in the phase space are occupied. This allows us to express F1 2 and F3 2 in the parametric equations (7.135) and (7.136)...

## From AGB to the White Dwarfs

G-H, Upward the AGB phase we briefly examine these stages the complex physics of AGB stars is studied in Sect. 26.6. In point G, the He core is exhausted the star is at the base of the so-called asymptotic giant branch (AGB), as also seen for the Sun (Fig. 25.11). A fraction of about 10 of the total mass has been removed by stellar winds. From the He-burning phase, the evolution depends much on the mass domain considered, the limits of the domains being influenced by rotation and overshooting....

## Stellar Convection

In most systems, whether stars, Earth or even humans, a heat excess may bring disorders. In stars, a heat excess (with respect to what radiation can transfer) drives turbulent chaotic convective motions. Convection, i.e., the turbulent turnover of matter in a medium heated from below, is a basic mechanism of energy transport in stars together with radiative transfer. In addition, it produces fast mixing of the chemical elements, generally leading to the chemical homogeneity of the convective...

## E2

According to (18.23), with e in g cm-3. The factor 1 3 between the variations of T and e corresponds to r3 4 3 for an adiabatic photon gas (Sect. 7.5). A choice of values e 0, T0, n leads to a certain distribution of the elements (e0, To, n) --distribution. (28.23) Which triplet of values (e0, T0, n) gives the observed distribution of elements in the domain of atomic masses A corresponding to the considered onion skin layer(s) is searched. Remarkably, the values of e0 and T0 are slightly larger...

## Magnetic Braking of Rotating Stars

The convective dynamo operating in the external layers of solar-type stars creates a magnetic field, as seen above. As a consequence, the field may force the outer plasma resulting from the weak stellar winds to co-rotate with the star up to a large distance. This produces a loss of angular momentum from solar-type and lower mass stars, with many consequences for stellar evolution. There are also direct effects on the distributions of rotational velocities as a function of ages and masses, in...

## Cgrav dt dMr

The equation for Lr becomes (24.6) accounting also for ev. The two equations (24.7) and (24.8) are unmodified (Sect. 5.3). If the convective turnover time is not much shorter than the evolutionary timescale, time-dependent convection has to be applied (Sect. 6.3). The numerical solution of these equations requires great care in the process of discretization 404 . 24.1.3 Boundary Conditions at the Center and Surface Let us consider the Lagrangian form of hydrostatic equations. At the center,...

## Continuity Equation Atomic Diffusion and Motion

Let us consider a mixture of several species of particles i with mass fractions X, assumed to be small. This is the test particle approximation. The total density is q. The elements i receive a net momentum which makes them move with an average total velocity < u, > < v, > +u, where < v, > is the average diffusion velocity of particles i with respect to the rest of the elements. The continuity equation applied to elements i writes djtX)+ V (QX, < ui > ) 0 (10.5) The gas globally...

## Photoionization or Bound Free Transitions

A bound electron in an atom is ejected by an incident photon and becomes a free electron (Fig. 8.1). Such a bound-free (bf) absorption process occurs only for photon energies higher than the ionization potential of the considered level n, where R 2n2mee4 (ch3) 1.09737 x 105 cm-1 is the Rydberg constant. This expression also defines an absorption edge, i.e., the lowest frequency v* for bound-free absorption, Fig. 8.1 Schematic illustration of the bound-free and free-free absorptions Fig. 8.1...

## Nuclear Cross Sections

Let us examine more closely the effects determining the nuclear cross-sections. For energies below 50 MeV, a typical reaction, which can occur through several channels, can be decomposed in three steps a + X C* Y + b . (9.12) 1. The interaction of a and X the wave associated to the incident particle interacts with the diffusion center of the target nucleus X. 2. The penetration of the barrier of electrostatic potential V (r) of the target nucleus, which leads to an unstable compound nucleus C*....

## Hydrostatic Equilibrium for Solid Body Rotation

We first consider the angular velocity Q as constant throughout the star. Let us assume hydrostatic equilibrium and ignore viscous terms. The Navier-Stokes equation (1.2) becomes with account of the centrifugal acceleration according to (B.24) and following remarks. r sin is the distance to the rotation axis (Fig. 2.1). The above expression of the centrifugal force gives a projection Q2 sin along vector r and a projection Q2mcos along vector . The quantity 0 is the gravitational potential,...

## Baroclinic Instabilities

For stars with solid or cylindrical rotation, the centrifugal force can be derived from a potential (Sect. 2.1.2). If so, the equipotentials and isobars coincide, the star is said barotropic q, P and T are constant on equipotentials. For other cases of differential rotation, all quantities other than P vary with colat-itude on an isobar. The star is said baroclinic. The relations between the fluctuations of the various quantities on isobars have been found in Sect. 11.2.2, they depend on the...

## Equation of the Surface for Shellular Rotation

In the case of shellular rotation, the isobars are defined by expression (2.27), which is identical to the expression of the equipotentials for solid body rotation. We may search the equation of the equipotential, in particular for the stellar surface. An equipotential is defined by the condition that a displacement ds on it neither requires nor produces energy, The effective gravity is given by (2.31) and the above product becomes dr rdtf + r2sin2 tfQ dr + r2sin2 tf rrdtf 0 . (2.49) For...

## Electron Conduction in Nondegenerate

In a non-degenerate gas, the ions of mass Amu and electrons of mass me have average quadratic velocities (C.63), V i (AmJ > Vv2e ( ) (8.30) The electrons are faster than the ions in a ratio (Amu me)1 2 43 VA. Thus, the electron gas is made of fast particles with respect to the slower ion gas. If there is a T gradient, the electrons move faster from regions of higher T toward regions of lower T than the opposite. The ions also move faster in the same direction, but with a smaller velocity...

## The MHD Equations in Astrophysics

The magnetohydrodynamic (MHD) equations are an ensemble formed with the equations of electromagnetism (see Appendix B.2) and the equations of the fluid mechanics. Some properties are specific to astrophysics 480 A. Maeder, Physics, Formation and Evolution of Rotating Stars, Astronomy and 311 Astrophysics Library, DOI Springer-Verlag Berlin Heidelberg 2009 - In a plasma, the particles experience the Lorentz force Fl qE +1 qv x B or Fl qE + (l c) jdV x B , (13.1) where q is the charge (q > 0...

## Info

Where udisp is the dispersion velocity in the cluster, M the stellar mass and R the stellar radius. The last term corresponds to the so-called gravitational focusing. In order to get a collision time < 105 yr, one needs a concentration n > 108 stars pc 3, which is extremely high. The mean observed concentration in the Orion Nebula is n 103 stars pc 3, the maximum value is n 2 x 104 stars pc 3 in the core of the Nebula. These two values of the concentrations give, respectively, collision...

## Plasma Bremsstrahlung Recombination Neutrinos

Plasma-Neutrinos An electromagnetic wave entering a sufficiently dense plasma generates collective oscillations of the electrons. In turn, these oscillations of a charged medium generate the emission of electromagnetic waves. A plasmon is the quantum of energy associated to these waves. They propagate in various directions and their sum has to be taken. According to (9.51), associated to each plasmon there is probability of emission of veve. The energy loss by plasma-neutrino dominates at high...

## Toward the Onion Skin Model

Figures 28.1 and 28.2 show the evolution of the internal abundances of the elements in a rotating and a non-rotating star with an initial mass of 20 M0, with an overshooting of 0.1HP 251 . The figures show the models at the end of the main phases of nuclear burning. There are major changes in the center and limited changes in the He- and H-rich outer layers. Between the ends of the He- and C-burning phases, 12C is destroyed and 20Ne and 24Mg are produced, a small amount of 16O is also turned to...

## Complements on Mechanics and Electromagnetism

B.1 Equations of Motion and Continuity We derive the general equations of motion and continuity. Although their form is very simple for spherical stars in hydrostatic equilibrium, the present general forms are useful when these simplifications do not apply. B.1.1 Equations of Continuity and of Motion Let us consider a volume element dxdydz with coordinates x, y, z. Let vx be the component of the velocity at a point x through the face dzdy and vx + - dx the velocity at point x + dx through the...

## The Ferraro Law of Isorotation

The law of isorotation was found by Ferraro 183,184 . This law or theorem says the following The magnetic field of a star can only remain steady if it is symmetrical about the axis of rotation and each line of force lies wholly in a surface which is symmetrical about the axis and rotates with uniform angular velocity. To derive this law, we separate the magnetic field into poloidal and toroidal components 558 in spherical coordinates (r, ft, p), Several simplifications are made. The fluid is...

## Equation of Radiative Transfer

Let us consider a radiation beam at frequency v in a given direction s in stellar medium as illustrated in Fig. C.1. The energy dUv transmitted by an element of the medium in a direction perpendicular to the surface element do centered on a solid angle dQ, during the time dt and in the frequency interval dv is Here, the surface element is perpendicular to the direction of the radiation beam, so that cos 1 in Eq. (C.1). The above expression defines the specific intensity Iv in erg s-1 cm-2...

## Properties of the Isobars

In the case of shellular rotation, the centrifugal force cannot be derived from a potential and thus (2.3) does not apply. Let us consider the surface of constant W (2.9), W 0- 1Q2 r2 sin2 const. (2.27) As in Sect. 1.2.1, the gravitational potential is defined by d0 dr GMr r2 and 0 -GMr r in the Roche approximation. The components of the gradient of W are in polar coordinates (r, ) dW d0 9 9 9 9 dQ - Q2 r sin2 - r sin2 Q , (2.28) 1 S 1 d0 - Q2 r sin cos - r2sin2 tfQ1 , (2.29) r d r d r d The...

## Isochrones and Age Determinations

Isochrones (or time-lines) in the logL vs. log Teff diagram are obtained by connecting the points of the same ages on the tracks of various masses. The lines obtained in this way are the time lines or isochrones. In practice, a lot of intermediate tracks have to be calculated or interpolated (great care has to be given in the interpolations ). The isochrones in logL vs. log Teff have to be expressed in observables quantities, as for Fig. 25.18 Isochrones in the My vs. (B-V) diagram derived from...

## Stationary Disks

During a part of their existence, disks may be considered as stationary, with constant rates of mass accretion. In such a state, some of their properties are independent of the viscosity 229 . Let us start from the conservation equations as above and impose a steady state. Mass conservation in the disk yields for the inward mass flow at level r where a positive radial velocity is directed outward. Equation (19.10) for mass conservation becomes after integration avr Q avr3 - + C, or - va -aQv +...

## The Blue Loops

The blue loops described in the HR diagram (Fig. 26.1) by stars in the range of 3-12 M0 allow the stars to spend a fraction of their He-burning lifetime as Cepheids. We can easily understand which effects favor or inhibit the blue loops by using an interesting result by Kippenhahn and Weigert 285 The blue extension of the loops mainly depends on the potential of the core c Mc Rc. There is a critical value of the potential such that c > crit(M) red giant. (26.11) crit(M) increases with mass,...

## The Evolution of the

The evolution of the Sun in the HR diagram is illustrated in Fig. 25.11. It also shows all the various evolutionary phases discussed in other chapters. From the emergence of the Sun at the end of the protostellar stage (Chap. 19) characterized by the dynamical timescale 106 yr), the pre-main sequence proceeds at the Kelvin-Helmholtz timescale (tKH 3 x 107 yr), first descending along the Hayashi branch and then joining the MS after a small hook due to the settling of the CN cycle to equilibrium....

## Evolution on the Birthline

Let us follow the evolution of an accreting star on the birthline with an accretion rate of 10 5 M0 yr-1. The initial age is equal to about (M < M > ) for an average initial rate < M > . Apart from this choice of the initial age, the evolution on the birthline is independent of the initial conditions. Up to 1.2M0 for brown dwarfs and low-mass stars between 0.01 and 0.4M0, the luminosity is so low and the Kelvin-Helmholtz time so long that even the moderate thermostatic support of D...

## Helium Burning

To build elements heavier than 4He, it would seem normal to first consider proton or a captures by helium nuclei however the absence of stable nuclei of atomic mass A 5 and 8 makes this kind of building problematic. In the early 1950s, nothing was supposed to halt the collapse at the end of H burning until supernova explosion intervenes at T > 109 K. It was the merit of Salpeter 506 to find that a Saha-like equilibrium 2 a 8Be allows a tiny concentration of 8Be to exist (typically X (8Be) 1.4...

## Differences in Structure

High- and low-mass stars have very different internal structures (Fig. 25.5). Above about 1.2 M0, the CNO cycles dominate the energy production (Fig. 25.2). The nuclear energy generation rate e strongly depends on T, this dependence is expressed by eT or v (Fig. 25.3). This means that the luminosity L is rapidly built near the center. In turn, the thermal gradient Vrad (5.32) which depends on the ratio Lr Mr is large. In stars with masses above 1.2 M0, Vrad is larger than the adiabatic gradient...

## DPm2 Zt SPm1

Where i 1,2,3,4 in all equations except the two in Ri. The dots represent the non-explicitly written derivatives. In each line one has the corresponding derivatives with respect to (P, T, r, L). In the last line, there are no derivatives with respect to rm and Lm, because in the center Srm 0 and SLm 0. The above expressions represent a total of 2 + 4(m - 2)+4 equations for SPj, Srj, STj, SLj with j 1, ,m - 1 and STm, SPm, (24.37) i.e., a total of 2 + 4(m - 1) unknowns. The system is thus...

## Equation of Diffusion

We consider here a diffusion process due to the motion of particles with a diffusion velocity vi depending only on the abundance gradient according to expression (10.14). This expression satisfies condition (10.4), because of XX 1 and of the relation between the concentration ni and the partial density q Xi is with Ai the atomic mass expressed in atomic mass units mu. Let us consider a one-dimensional problem, where particles i may diffuse along the radial direction r. Thus, (10.14) writes The...

## Q Jo Ju Ub

Q( 0*17 0Q2(r)Ps(cos ft) sin3 ftdft ( ( Ps(cos ft) sin3 ftdft () nsin3ftdft s 0 s( V onsin3ftdft J. This defines Is its denominator is equal to 4 3 thus Is - sin2 ft Ps(cos ft) sin ft dft . (B.73) Let us express sin2 ft in terms of P2(cos ft) one has P2(cosft) 1 - 2 sin2 ft thus sin2 ft 2 1 -P2(cosft) . (B.74) Is - - 1 - P2 (cos ft) Ps(cosft) sin ft dft - P0(cos ft)Ps(cos ft) - P2 (cos ft)Ps(cos ft) sin ft dft 20 2 J P0 (x)Ps (x) dx + 2 y P2 (x)Ps (x) dx where Ss,i is the Kronecker symbol....

## Q dTjQ

The terms containing U have disappeared from dq. As independent variables in stellar structure, one usually takes (P, T) rather than (q, T) and thus we can eliminate q by using the equation of state in its general form (7.63) with a _ (dnp) , 8 _ - (dn,) , 9 _ (dm) , om dln P ) T , d ln T J , * d ln j pj K J Let us first consider the case of a constant mean molecular weight .If is constant, the last term in (3.60) is absent, where we have used the expressions of a and 5. One can write since one...

## Equations of Stellar Structure for Shellular Rotation

Let us consider the interesting case of shellular rotation, where Q is constant on isobars (i.e., surface of constant pressure), but varies according to the radial coordinate of the isobars. Rotation is shellular because differential rotation in radiative regions produces anisotropic turbulence 632 , much stronger in the horizontal direction than in the vertical one due to stable stratification (Sect. 12.1). Since one has the relation VP geff, the words constant in the horizontal direction mean...

## Rotation Driven Instabilities

Rotation, and especially differential rotation, generates a number of instabilities. We know this on the Earth with for example the west winds, the jet streams, the many effects of the Coriolis force. On the fast rotating surface of Jupiter, we observe turbulent waves at the interface of differentially rotating zones, the Red Spot is a long-living hurricane generated by rotation, etc. The various rotational instabilities produce some mixing of the elements and transport angular momentum in...

## Radial Pulsations of Stars

Variable stars have always fascinated mankind, showing non-immutable objects on the celestial sphere thought to be the domain of gods. Indeed, oscillatory phenomena are frequent in natural systems the level of the sea shows tidal oscillations and waves, the blowing winds produce oscillating noise, the clouds form waves on the side of mountains, some geysers are periodic, etc. Stars do not escape to this rule of Nature. What makes the beautifully self-controlled stellar nuclear reactors...

## The Opacity Limited Fragmentation

Fragmentation goes on as long as the energy from the cloud contraction is radiated away. When this is no longer the case, the process stops. Let us estimate the size of the smallest fragment in the opacity-limited fragmentation. The gravitational energy GM2 R of a cloud of mass M, radius R and average density q is liberated in a time of the order of 1 - Gg . Thus, the gravitational power produced is . GM2 i 3 2 G3 M5 grav GRr (G Q)1 ( j . (18.47) The radiated power is at most that of the black...