Stellar oscillations and stellar convection in the presence of an Urca shell

Abstract

The problem of damping of stellar oscillations in presence of a Urca shell is solved analytically in a plane symmetrical approximation. Low-amplitude oscillations are considered. Oscillatory pressure perturbations induce beta reactions of the electron capture and decay in the thin layer around the Urca shell, leading to damping of oscillations. Owing to the non-linear dependence of beta reaction rates on the pulsation amplitude in degenerate matter, even a low-amplitude oscillation damping follows a power law. It is shown that in the presence of the Urca shell the energy losses owing to neutrino emission and the entropy increase resulting from non-equilibrium beta reactions are much smaller than the rate of decrease of the energy of pulsations by the excitation of short-wavelength acoustic waves. The dissipation of the vibrational energy by the last process is the main source of heating of matter. Convective motion in the presence of an Urca shell is considered, and equations generalizing the mean free path model of the convection are derived.

convection, elementary particles, stars: oscillations

1 Introduction

Supernova explosions of type Ia are believed to happen because of carbon ignition in the highly degenerate matter of a medium mass star, leading to a rapid carbon burning, and total disruption of the star (Arnett 1969). Paczynski (1972) suggested that cooling of the matter in a convective region may be enhanced in the presence of Urca shells, first considered by Tsuruta & Cameron (1970). The Urca shell appears when the matter contains an isotope with a threshold Fermi energy for electron capture, corresponding to a density less then the central one. For example, the elements ²¹Ne, ²³Na, ²⁵Mg and ⁵⁵Mn have threshold Fermi energies for the electron capture, with the addition of the electron rest mass, 5.7, 4.4, 3.76 and 2.50 MeV, respectively, which corresponds to densities 3.44, 1.68, 1.17 and 0.42 in units of 10⁹ g cm⁻³, for a number of electrons per one electron In conditions such that the central density in the pre-supernova may be close to 10¹⁰ g cm⁻³ (see e.g Iben 1982), the presence of such an isotope leads to the existence of a jump in the composition at a density corresponding to a threshold energy. During the convective motion, the matter in eddies around this density periodically crosses the boundary. That implies continuous beta capture and beta decay in the matter of these eddies.

The physical processes accompanying the eddy motion around the Urca shell are rather complicated. Different interpretations had been suggested, with different conclusions about the direction of influence: stabilizing or destabilizing the carbon burning in the convective degenerate core (Bruenn 1973; Paczynski 1973; Ergma & Paczynski 1974; Couch & Arnett 1975; Lazareff 1975; Iben 1978, 1982; Barkat & Wheeler 1990; Mochkovitch 1996; Stein, Barkat & Wheeler 1999). In the last section of the paper we analyse physical processes in the convective Urca shell and formulate an approximate quantitative approach to the solution of this problem.

The main part of this paper is devoted to revisiting the related problem of stellar oscillations in a degenerate star in the presence of Urca shells, considered before by Aparicio & Isern (1993). Attention is given to derivation of a relation that properly describes the damping of oscillations owing to the presence of an Urca shell. The time dependence of the pulsation amplitude is obtained in the absence of any excitation mechanisms. Such conditions occur when stellar oscillations are exited by some occasional event, like a star quake, or a surface nuclear explosion in a nova white dwarf star. Stellar collapse with the formation of a neutron star is accompanied by an excitation of strong pulsations. These pulsations take place under different conditions, in which neutrinos may be trapped inside, temperature effects are important, and beta processes occur in the bulk of the matter, not confined by a thin shell. Damping of such oscillations had been considered by Finzi & Wolf (1968) and Sawyer (1980).

2 Plane-parallel layer with a phase transition

In order to reduce the problem to one having an analytical solution, let us consider a plane-parallel layer in a constant gravitational field with an acceleration g, with a phase transition at the pressure P*. Taking a polytropic equation of state with at and at we obtain the following relations for the static equilibrium:

 

(1)

Here the pressure is continuous, but the density ρ has a jump at P* owing to the jump of the constant K, M is the mass of 1 cm² of the slab and m is the mass of 1 cm² of the slab under the layer with Lagrangian coordinate x, which is related to the density as

 

(2)

with for a chemically uniform slab. The main parameters of the slab at a given M with a phase transition at are determined as follows:

 

(3)

Here ρ₀ and P₀ are the density and the pressure at the bottom of the slab, and x₀ and x* are the total thickness and the thickness of the inner denser phase layer of the slab, The phase transition in the slab happens only if its specific mass

3 Linear oscillations of the slab

The neutron star oscillations in the presence of beta processes had been considered by Finzi (1965) for fully degenerate matter, and by Finzi & Wolf (1968) for the case of arbitrary degeneracy. It was shown that a rapid damping of oscillations happens because of the electron capture and emission by nucleons in non-equilibrium conditions, when the pressure depends not only on the density and the temperature, but also on the composition of the matter, described by a kinetic equation. This dynamical behaviour is similar to the action of the ‘second’ or ‘bulk’ viscosity (Landau & Lifshits 1988), but the non-linear dependence of beta reaction rates, even for small perturbations, makes the situation more complicated than in the case of classical hydrodynamics. In the papers of Finzi (1965) and Finzi & Wolf (1968), the action of the beta processes was considered in the main body of the star, without any account being taken of Urca shells and a jump of the composition.

In a static star the presence of an Urca shell may be taken into account as a change in the equation of state at a given pressure, similar to a phase transition. Stellar stability in the presence of the phase transition was investigated by Ramsey (1950), Seidov (1967) and Bisnovatyi-Kogan, Blinnikov & Shnol (1975), using the variational approach. The linear oscillations of a star with a phase transition have been studied by Grinfeld (1982) and Haensel, Zdunik & Schaeffer (1989). Non-linear oscillations in the model of a non-compressible fluid with a phase transition have been considered by Bisnovatyi-Kogan & Seidov (1984). Damping of oscillations in the presence of the phase transition, owing to the dissipation in the shock for an ideal, infinitely rapid transition, was considered by Bisnovatyi-Kogan & Seidov (1984). The case of a phase transition resulting from pionization (Migdal, Chernoutsan & Mishustin 1979) may be treated in the frame of the classical bulk viscosity (Haensel et al. 1989).

Equations describing the oscillations of the slab in Lagrangian coordinates t, m are written as

 

(4)

For the perturbations , and we obtain the equations

 

(5)

Neglecting damping of the oscillations, we search the solution in the exponent form Using the adiabatic relation between the perturbations we obtain the following equation for the pressure perturbation:

 

(6)

Taking into account the relations (1) and (3), and introducing a non-dimensional coordinate we obtain the equation for linear oscillations of the slab in the form

 

(7)

The solution of this equation is a combination of two linear independent solutions in the form of the Bessel functions,

 

(8)

where for two phases

 

(9)

The frequency of the oscillations ω and relations between the constants A₁, B₁, A₂ and B₂ are obtained from boundary conditions and from relations for the phase jump.

3.1 The case without a phase transition

The solution for this case was obtained by Bisnovatyi-Kogan, Popov & Samokhin (1976). It is described by one solution with two constants A and B. From equations (5) and (8) we obtain the solution for the velocity in the form

 

(10)

At the outer boundary the pressure perturbation is zero, and because of a divergency of Y(z) at we get Taking into account the inner boundary condition at we obtain a dispersion equation in the form

 

(11)

where ξ_i, are the roots of a Bessel function, It follows from equation (11) that the slab is stable to plane-parallel oscillations when its equilibrium state exists at Note that owing to the expansion of Bessel functions in the form

 

(12)

we obtain from equations (9) and (10) the finite value of the surface velocity of oscillations.

3.2 Conditions on the phase jump

Conditions on the phase jump are obtained from the mass, momentum and energy conservation laws, written as (Landau & Lifshits 1988)

 

(13)

 

(14)

 

(15)

Here ẋ is a velocity of a boundary between two phases. We consider a phase transition with a zero transition heat, which is characteristic of equilibrium transitions in strongly degenerate matter. For the case of a non-compressible fluid we have and obtain the conditions used by Bisnovatyi-Kogan & Seidov (1984). We write equations (13)–(15) in the following form:

 

(16)

Taking into account the second conditions on the jump in (16) may be written as

 

(17)

where β is not uniquely determined. The irreversible (positive) heat released in the jump during compression is determined as

 

(18)

where at the adiabatic compression In the case of the contraction, the condition of a non-decreasing entropy puts a restriction with an adiabatic jump at For the case of the expansion, the heat production during the jump is equal to allowing the interval . Bisnovatyi-Kogan & Seidov (1984) considered the contraction with and the adiabatic expansion with for the case of an incompressible fluid with a phase transition. In the case of a medium with a uniform pressure, a matter region with a mixture of two phases may exist (Landau & Lifshits 1964), but in a gravitational field with a pressure gradient for an ideal phase transition there is a sharp boundary between these two phases.

The situation becomes much simpler for linear oscillations, when neglecting the quadratic terms and for a jump with small perturbations in the pressure and the velocity we obtain

 

(19)

The last condition in (19) corresponds to the entropy conservation and adiabaticity of the motion in the linear approximation.

3.3 The case of a frozen phase transition

The velocity in the inner phase is written as in equation (10) with A₂, B₂, η₂ instead of A, B, η, and for the outer phase we obtain a relation

 

(20)

In the case of a ‘frozen’ phase transition, the boundary between phases does not change, so and It follows from equation (18) that the frozen phase jump is uniquely determined during oscillations, and remains adiabatic. As in the case of one phase layer, the outer boundary condition implies The velocity is zero at the inner boundary, so that

 

(21)

Conditions of a continuity of the pressure and the velocity perturbations on the phase jump give relations

 

(22)

 

(23)

Here η_1*, η_2* are determined in (9) at Introducing the dimensionless frequency

 

(24)

we obtain from (21)−(23) the following dispersion equation:

 

(25)

In the limiting case when the boundary between the phases is on the inner boundary of the layer, and we obtain from (25) the equation

 

(26)

Taking account of the relation for the Bessel functions ,

 

(27)

the expression in the square brackets in equation (26) is reduced to the Wronskian of the Bessel equation, which has no roots. So the dispersion equation is reduced to similar to (11), which corresponds to the uniform slab of the outer lighter phase of the matter. At when the level between the phases is moving to the outer boundary, we have, from equation (25) using equation (12), the dispersion equation corresponding to the oscillations of the slab of the inner phase of the matter. In Fig. 1 the frequencies Ω_i for 2, 3, 4 are given as functions of z* for and The values at and correspond to the slab without a phase transition, and are determined by the first four roots of and J₃(Ω_i), relatively.

Figure 1.

Non-dimensional frequencies Ω of oscillation of the polytropic slab with as functions of z* for the basic mode (1, 1a), and modes with one (2, 2a), two (3, 3a) and three (4, 4a) nodes. Upper and lower curves correspond to the frozen and equilibrium cases, respectively. The frequencies of these two cases coincide when one of the node coincides with the phase transition.

3.4 The case of an equilibrium phase transition

The equilibrium (ideal) phase transition corresponds to the case in which the boundary between the phases is always situated at , and the rate of the phase transition is infinite. Here the phase boundary level is changing with the rate

 

(28)

That gives the condition for a phase jump in the form

 

(29)

The conditions of the finiteness of the pressure, and of zero velocity at the inner boundary, imply conditions and equation (21), as in the previous case. The condition of the pressure continuity at the phase jump gives equation (22), and equation (29) is written in the form

 

(30)

The dispersion equation following from equations (21), (22) and (30) is written in the form

 

(31)

In the limiting case we obtain, taking account of equation (27) and the equality the dispersion equation in the form

 

(32)

At we get from equation (31), using equation (12), the dispersion equation as in the frozen case. Consider the limiting case of an incompressible fluid, where the oscillations are possible only in the presence of a phase transition. This case corresponds to the limit leading for finite Ω in equation (24) to the limits for all modes, except the first one ω₁, becoming a phase mode with at Using the expansions (12) for this mode, and relation we obtain from equation (31)

 

(33)

This phase mode is the only one, remaining in the spectrum of oscillations of the incompressible slab. The frequencies of oscillations of the slab Ω_i with the ideal phase transition are given in Fig. 1 for the same 2, 3, 4, The frequencies of the slab oscillations with frozen and equilibrium phase transitions coincide when the node of the oscillation mode is situated on the phase boundary. In the case of a spherical star, the non-dimensional frequency of the mode connected with the phase transition, tends to the universal value, not depending on γ: when a phase transition happens in the centre. This value is the same as the frequency of the phase mode of the incompressible fluid with a phase transition in the centre (Grinfeld 1982; Bisnovatyi-Kogan & Seidov 1984). In contrast to the case of a spherical star, the phase frequency of the slab, as follows from equation (32), depends on γ, and in the limit for this dependence is written as

To investigate the dependence of the oscillation modes with an ideal phase transition on γ, it is convenient to introduce and to rewrite the equation (32) in the form

 

(34)

with the first root at All other roots tend to infinity at The dependence (γ) from equation (34) is presented in Fig. 2.¹ The frequencies of oscillations of a star with a phase transition have been calculated by Haensel et al. (1989). In the case of a non-compressible slab in a constant gravitational field, the phase frequency ω_p determined by equation (33) does not depend on the total mass of the slab. It depends only on the mass over the phase transition layer which depends only on P*.

Figure 2.

Non-dimensional frequencies of oscillation of a polytropic slab as functions of γ for the case of the phase transition at the bottom At the frequency of the basic mode (1) tends to the finite value (33), those of other modes (2,3,4) increase to 8.

4 Equations describing damping of oscillations resulting from an Urca shell in highly degenerate matter

In conditions of high degeneracy, both beta decay and beta capture are taking place in non-equilibrium conditions, when there is a finite difference between the threshold and the electron Fermi energy. It leads to a rise of the entropy and heating (Bisnovatyi-Kogan & Seidov 1970; Nakazawa et al. 1970). Here we do not take into account the influence of heating on the equation of state, which is always taken as for cold degenerate electrons. The finite rate of beta reactions produce an effect, which may be called ‘a non-linear bulk viscosity’ owing to the non-linear dependence of the reaction rate on the pressure perturbation. The bulk viscosity acts always at a finite rate of a change of matter composition during perturbations, which implies a phase shift between the density and the pressure oscillations, and leads to damping of the oscillatory motion. Usually the reaction rate depends linearly on the deviations from the equilibrium, leading to the term in the equation of motion (Landau & Lifshits 1988). In our case the equation of motion contains terms, non-linear to ࢚v_k/࢚x_k.

4.1 Thermodynamic relations and reaction rates

Consider for simplicity the case of the ultrarelativistic electron degeneracy. In most Urca shells this approximation works well. This case corresponds to the polytrope with and constant

 

(35)

where is the average number of nucleons on one electron. Here a two-component mixture is considered, consisting of elements with an atomic weight A and atomic numbers Z and with a beta transition between them, and x_Z and are mass concentrations of these elements, The concentrations of the elements n_Z and are determined by the continuity equations, similar to equations (4), which in the presence of beta reactions are written as

 

(36)

Here and are the probabilities of the electron capture and electron decay. In fully degenerate matter we have (see e.g. Bisnovatyi-Kogan 1989)

 

(37)

Here F₀(x) is a Fermi function of a beta decay which has an analytical presentation, u_Fe and δ are Fermi energy plus rest mass energy of the electrons, and the threshold energy for a beta capture, in units of m_ec²; g_z and are statistical weights of the elements (A,Z) and Ft_1/2 is a non-dimensional value measured in the beta-decay experiments, or estimated theoretically when the experiments are not available. For a small difference we have the simple expressions (Bisnovatyi-Kogan 1989)

 

(38)

The value u_Fe is defined by the following expression:

 

(39)

where

It is convenient to rewrite equation (36) in terms of the mass concentrations and We then obtain, taking account of the continuity equation in (4),

 

(40)

We consider matter that is transparent to neutrinos. The rates of neutrino energy loss per nucleus during the electron capture and during the beta decay are determined as

 

(41)

which for small differences reduces to

 

(42)

Part of the energy of oscillations during damping heats the matter. For the case of strongly degenerate electrons, the finite difference between ɛ_Fe and Δ leads to electron capture or decay, when neutrino losses are accompanied by heating of the matter and increase of the entropy (Bisnovatyi-Kogan 1989). The expressions for an entropy increase during beta decay and capture in fully degenerate matter are written as

 

(43)

The last expressions in the equations (43) are related to small differences when the rate of the entropy increase is equal to one third of the energy loss rate by the neutrino emission.

4.2 Linearized equations for averaged change of a composition

Taking into account, that in the ultrarelativistic approximation the pressure may be written as

 

(44)

and P* is determined by equation (44) at we may write equations (40) and (38) in the form

 

(45)

During linear oscillations the beta reactions take place only in a thin layer of the matter, crossing in its motion the boundary , In this layer the pressure may be represented by an expansion

 

(46)

leading to the linearized equations in the form

 

(47)

Having in mind a linear dependence of the rate of change of the concentrations on their values, and zero equilibrium concentrations of one of the elements in two parts of the slab, we may neglect the beta decay and capture of the elements with zero equilibrium concentrations.

Continuous supply of the mechanical energy for supporting the oscillations leads to large deviations from the static equilibrium case in the presence of a phase transition with a jump in the composition at from to After the long-lasting induced oscillations, the concentration in the perturbed Urca shell should reach a dynamically quasi-stationary state with compatible concentrations of x_Z and in the perturbed layer. Such ‘enhanced’ oscillations will be accompanied by neutrino losses, and by the non-equilibrium heating of the degenerate matter, the energy for which is coming from the kinetic energy supply. The low-amplitude pulsation regime with concentrations in dynamical equilibrium may be treated as linear from the mechanical point of view, but it is highly non-linear with respect to beta reactions, because the concentrations in the thin layer are very different from the static equilibrium ones.

Let us find the rates of the element transformation averaged over the pulsation period for the stage in which deviations from the initial concentrations are small. In this approximation, pulsations may be described by the solution for the frozen phase transition in Subsection 3.3. The action of beta processes leading to damping of oscillations is taken into account by a slow decrease with time of the coefficients A₁, A₂, and B₂, out of which only one coefficient is independent. Consider the time dependence of the perturbed pressure in the form

 

(48)

with ω determined by the dispersion equation (25). For the layer that had the equilibrium beta capture takes place only during the phases when the total pressure exceeds P*, which is determined by the inequality

 

(49)

and the opposite inequality should be fulfilled for the beta decay reaction in the layer with Let us find an equation for the rate of changing of the average compositions in two layers, crossing the boundary Introducing

 

(50)

with and averaging the equations in (47) over the corresponding layers for long-wavelength oscillations with we obtain

 

(51)

Because of the thinness of the oscillating layers, we may take The values averaged over the oscillation period determine the change of the composition over a time much greater than one oscillation period,

 

(52)

The equations for these variables are determined by averaging over the oscillation period of (51):

 

(53)

The equations (52) and (53) are valid only when the composition is close to the equilibrium one. That leads to the time restriction of their applicability,

 

(54)

4.3 Decrease of the pulsation amplitude in the presence of an Urca shell

To derive an equation describing the decrease of the pulsation amplitude, we should take into account the change of the pressure resulting from the change of the electron concentration. Let be a perturbation of the pressure determined by the beta reactions,

 

(55)

Introducing

 

(56)

we write the equation of state in the form The pressure perturbations are defined as

 

(57)

which gives at

 

(58)

In the presence of damping let us represent the velocity perturbation in the form where _a inside and outside the jump are determined by equations (10) and (20). The function , defining the damping of oscillations, is non-zero only in a thin layer around the phase jump of the equilibrium model. The value of _a inside and outside this layer may be written as

 

(59)

where A₂ and B₂ may be taken as linear functions of A₁, determined by the equations (22) and (23), and the dispersion equation (25). Let us define so that The amplitude of the perturbed outside pressure my be written as a function of V,

 

(60)

The equation of motion gives the relation describing damping of oscillations in the layer after a subtraction of the proper oscillations of the slab at the frozen composition and averaging over the motion of the whole slab:

 

(61)

where Taking into account equation (3), we obtain

 

(62)

Multiplying equation (62) by sin ωt cos²ωt, and integrating over the pulsation period 2π/ω, we obtain, using equation (58) with an input of both layers around P* at different times, and an integration by parts on the right-hand side,

 

(63)

Taking account of equation (51), we obtain

 

(64)

Integration gives equal values π/(4ω) for the integrals on both sides, so we obtain an equation

 

(65)

Using the connection between V and _a in equation (60), we finally obtain

 

(66)

The change of the amplitude of pulsations is defined by a solution

 

(67)

Let us note the power-law damping of the oscillations in fully degenerate matter, owing to the non-linear character of the bulk viscosity. A characteristic time of damping τ_d of oscillations owing to the presence of Urca shell is defined by

 

(68)

When the Urca shell is not situated close to the boundary, so that it follows from equation (68) that

 

(69)

and when the phase transition happens close to the boundary, we obtain an estimation, using the expansion (12):

 

(70)

Here If the Urca shell is situated inside the star, the damping of oscillations always happens before the linear approximation breaks, as follows from equations (54) and (69), and the relation Numerical estimations for give and for the Urca shell deep inside, and close to the outer boundary, relatively.

5 Energy balance and damping of oscillations

5.1 Neutrino emission and non-equilibrium heating

Equations (43) describe the increase of the entropy resulting from non-equilibrium beta reactions. Rewriting these equations in a more compact form,

 

(71)

After using the expansion (46), we obtain

 

(72)

Taking account of equations (48)–(51), we obtain, after averaging over the layers around the Urca shell for long-wavelength oscillations, similarly to (51),

 

(73)

Making averaging over the oscillation period, we obtain, similarly to (53),

 

That entropy increase is related only to the layer oscillating around the Urca shell. The total rate of heating owing to the non-equilibrium beta reactions in the oscillating layer is

 

(75)

The rate of the neutrino energy losses L (erg s⁻¹ cm⁻²) is obtained by averaging of losses over the oscillation period in the whole oscillating layer. Using equation (42) we obtain, similarly to equation (75),

 

(76)

A similar dependence of the neutrino energy losses during the oscillations because of the Urca shell had been obtained by Tsuruta & Cameron (1970), who also took equation (76) for the whole rate of loss of the kinetic energy of oscillations. In the approximation of strong degeneracy, the matter will be heated during oscillations with the rate (75), and the neutrino luminosity is determined by (76). The source of both kind of energy fluxes is the vibrational energy of the slab, giving the rate of vibrational energy losses directly connected with the beta reactions as (see also Finzi & Wolf 1968)

 

(77)

5.2 Acoustic wave excitation

Defining and using approximately from equations (8) and (10) for the Urca shell in the middle of the slab, we obtain from equation (65) the equation for decrease of the vibrational energy, connected with the hydrodynamical processes, in the form

 

(78)

Taking account of (44), (45) and (71), we obtain

 

(79)

where ω₀ is approximately defined by equation (11) at It follows from equation (79) that the main source of damping of oscillations in the presence of the Urca shell is connected not with the neutrino emission/non-equilibrium heating, but with dynamical action of the non-equilibrium layer of the slab, where beta reactions take place. This action leads to an excitation of short-wavelength acoustic waves with the length Damping of these acoustic waves is connected with the transport properties and formation of weak shocks near the outer boundary. The amplitude of these waves reaches an equilibrium level when their excitation is balanced by the damping processes. When the wavelength of the excited eigenmode approaches the thickness of the non-equilibrium layer Δx, formed by the oscillations, both mechanisms of damping become comparable. For taking account of relations and we obtain The importance of the dynamical damping of stellar oscillations in the presence of an Urca shell is connected with the non-linearity of weak interaction rates and deviations from the eigenoscillations under the action of weak interactions in this layer, leading to the excitation of acoustic waves.

5.3 Formation of shocks

In the case of an ideal phase transition, formation of shocks is possible during small oscillations near the phase jump, leading to even stronger dissipation of the vibrational energy. From equations (18), (28) and (49) follows the following equation for the vibrational energy losses in the equilibrium phase jump resulting from the shock wave formation:

 

(80)

Note that this kind of dissipation increases with increasing frequency of oscillations. Comparing the two mechanisms of damping (78) and (80), we obtain, taking account of equation (45),

 

(81)

where it is assumed that

6 Urca shell in a convective motion

Barkat & Wheeler (1990) had concluded that non-equilibrium heating is balanced by the change in the convective flow, leading to the net cooling resulting from the convective Urca shell. Nine years, later the same authors (Stein et al. 1999) changed their mind, concluding that ‘convective Urca process can reduce the rate of heating by nuclear reactions but cannot result in a net decrease in entropy, and hence in temperature, for a constant or increasing density’. This conclusion, as well as the opposite one, made by using thermodynamic relations only, seems not to be convincing. Following this line, let us present two plausible scenarios, leading to two opposite conclusions.

A. Owing to the action of a non-linear bulk viscosity, the convection is damped in the vicinity of the Urca shell, decreasing the convective heat flux from the central part of the star. In terms of the general heat balance of the star, this means that cooling become less effective, and nuclear reactions become thermally unstable and lead to a nuclear explosion earlier than without the presence of the Urca shell. Non-equilibrium heating gives additional heating, supporting the earlier nuclear explosion.

B. Owing to the action of a non-linear bulk viscosity, the convection is damped in the vicinity of the Urca shell, decreasing the convective heat flux from the central part of the star. Owing to a local decrease of the heat flux from the core, the average temperature gradient increases, leading finally to an increase of the convective flux soon after the Urca shell enters into the convective zone. If the increase of the convective flux prevails over the non-equilibrium heating in the Urca shell, the general heat balance would be shifted to a larger temperature with more effective cooling, and the boundary of the thermal explosion would be postponed in time, if not eliminated.

In a highly non-linear system, such as a star with nuclear reactions, neutrino losses, degeneracy, convection and many feedback influences, the numerical simulations are needed, because it seems to be impossible to make a conclusion about the direction of the process under the action of an additional Urca shell, based only on thermodynamical grounds.

The main difference between the dissipation of pulsations considered above and dissipation of convective motion is connected with the role of the excitation of the sound waves. Such excitation is of the utmost importance for dissipation of plane parallel oscillations of the slab, which are analogous to radial oscillations of a star, both belonging to the p-mode family of oscillations (Cox 1980). In contrast with them, the convective modes belong to another family of perturbations, related to stellar g modes, in which the local pressure perturbations are much smaller, and could be neglected when the convective velocity is much less than the velocity of sound. In this situation the sound wave dissipation of the convective modes imposed by the Urca shell is much less important than in the case of p modes. The convective Urca shell may only change the convective velocity and dissipation of the convection connected with the wave excitation by the convective motion, which is neglected in the standard mixing length model. Therefore we do not consider the dynamical dissipation of the convective motion in the model of convection in which the Urca shell influence is taken into account.

The equations of stellar evolution in the presence of the Urca shell should take into account the following physical processes.

• (1)

  Loss of the energy owing to neutrino emission in the Urca shell.

• (2)

  Heating of the matter in the convective region around the Urca shell owing to non-equilibrium beta processes.

• (3)

  Decrease of the convective velocity in the layer around the Urca shell owing to energy dissipation connected with the non-equilibrium beta processes. The kinetic energy of the convection is the source of the energy for neutrino losses and non-equilibrium heating of matter.

6.1 Energy equation in presence of the convective Urca shell

In the condition of static equilibrium only the energy and heat transfer equations should be modified. In the energy equation

 

(82)

in addition to the other neutrino cooling processes ɛ, the new term is connected with heating due to the non-equilibrium beta processes around the Urca shell, having in mind the strong degeneracy of electrons in this region. The neutrino emission in the non-equilibrium Urca processes is accompanied by heating at high degeneracy, because the positive term ࢣμ_i dn exceeds the energy carried away by the neutrinos (Bisnovatyi-Kogan & Seidov 1970). The convective motion, consisting of convective vortexes around the Urca shell, is a source of additional neutrino energy losses, and of heating of the matter. This dissipation of the convective energy may be described in the same way as the corresponding dissipation and heating during stellar pulsations. Therefore we use for a description of these processes the formulae from the previous sections. If we accept that the pressure difference in the convective vortex is about one half of the local pressure, we use for the amplitude of pressure pulsations in equations (75) and (78) Taking also approximately we obtain

 

(83)

Taking into account the equality on the phase boundary in equations (39) and (44) we obtain

 

(84)

where

 

(85)

Equation (84) is related to energy losses averaged over the whole slab. In the convective motion the losses are localized in the layer around the Urca shell radius r*,

 

(86)

here l_conv is taken from the mean free path model. The local rate is obtained from equation (84), if we take into account the fact that all of the heating is concentrated inside the layer (86). We then obtain

 

(87)

6.2 Convective flux

The heat flux consists of the radiative and convective parts The convective part should be modified, taking account of damping of the convective motion in the layer around the Urca shell. The convective heat flux density is written in the mean free path model of the convection as (see e.g. Bisnovatyi-Kogan 1989)

 

(88)

Here is the difference between the adiabatic temperature gradient and the actual one, c_p is the heat capacity at a constant pressure, v_conv is the convective velocity, and dr is the length that the convective element crosses before its merging with the background. In the conventional mixing length theory the convective velocity is the result of action of the buoyancy force on the length dr. While neutrino energy losses and non-equilibrium beta heating have the kinetic energy of convective motion as a source, we should also add the corresponding terms explicitly to the equation determining the convective velocity. Taking account of this we write the equation for the convective velocity in a form

 

(89)

Here the difference between the actual density gradient and the adiabatic one Δ࢟ρ is written as

 

(90)

The average value of the convective element path dr is connected with the mixing length l as in the approximation of small density variations accepted in this paper, so we have, as in equation (77), the relation We obtain more complicated relations for the convective energy flux density than in the standard mixing length model, because the convective velocity is not expressed explicitly, but should be found from the third-order algebraic equation. Finally, the temperature gradient and the total radial heat flux L_r are connected by the following system of algebraic equations in the region (86) around the Urca shell:

 

(91)

 

(92)

 

(93)

 

(94)

where is found from equation (84), κ is the Rosseland matter opacity, and the pressure gradient is determined by the equation of static equilibrium of the star. These relations may be applied for a description of the Urca shell convection only for sufficiently rapid convective motion, when the first term in equation (89) considerably exceeds the second one. For a slow convective motion this model cannot be applied, because it was suggested that the change in the initial composition caused by beta processes in the Urca shell is small during the period of damping of oscillations. For the case of a convective Urca shell where the convection is excited by the external source, this condition may be softened and reduced to the demand that the changes of composition are small during one rotation of the convective vortex. The equation (93) has roots only when

 

(95)

The violation of this inequality may result in an abrupt termination of the convection in the layer (86) around the Urca shell.

7 Discussion

The dissipation of pressure oscillations connected with the excitation of sound waves is inherent to any kind of dissipation when the main term is non-linear, as in the processes of the beta capture and beta decay. The linear mechanism, connected with a conventional bulk viscosity, does not change the form of the eigenfunction of oscillations, and so no additional waves are excited. Any non-linear mechanism leads to non-coherent damping, distortion of the eigenfunction and appearance of high-frequency and rapidly decaying harmonics. In the case when the damping is concentrated in the thin layer, it is evident that the excited harmonics should have lengths of the order of the thickness of the perturbed layer. We come then to a paradoxical situation, when an increase of the pulsation amplitude leads to an increase of the wavelength of the excited modes, with less effective dissipation. The wave dissipation could even be stopped when, in the condition of small dissipation, a high level of wave excitation is reached, and the energy losses of the stellar oscillations are compensated by the excitation of these oscillations by a high-frequency ‘noise’. We may expect also that an additional minimum of the dissipation rate would appear at a finite amplitude of pulsations.

When analysing the Urca shell convection in a star, it would be premature to predict the results of the evolutionary calculations taking account of a convective Urca shell according to equations (91)–(94) before such calculations are done. However, two possibilities may be expected. One possibility is connected with the obtaining of a definite result which has a little sensitivity to the input parameters of the problem, such as α_p, Ft_1/2, accepted rates of nuclear reactions, neutrino losses etc. Another possibility could be a high sensitivity of the result to the same input parameters. If the second possibility were realized we could still remain in the situation of an ambiguity, because the set of the input parameters for a pre-supernova model cannot be established with sufficient precision.

Numerical two-dimensional simulations of the Urca shell convection could be useful for clarifying the problem further. Nevertheless, numerical problems connected with a simulation of the convective motion seem to remain severe enough, and therefore we should first obtain some understanding from the simplified one-dimensional model of the convective Urca shell, generalizing the mixing length model, the variant of which is presented in this paper.

Acknowledgments

The author is grateful to R. Canal, who attracted his attention to the existing ambiguities in the problem of the convective Urca shell; S. I. Blinnikov, J. Isern and R. Mochkovich for useful discussions; W. Hillebrandt and E. Müller for the possibility of participating in some workshops on ‘Nuclear Astrophysics’, where this work was initiated; and O. V. Shorokhov and O. D. Toropina for help. This work was partly supported by Russian Basic Research Foundation grant No. 99-02-18180 and a grant of the Ministry of Science and Technology 1.2.6.5.

1

Numerical solution of the dispersion equations (25), (31) and (34) and plotting of Figs 1 and 2 was performed by O. V. Shorokhov.