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Astron. Astrophys. 359, 1124-1138 (2000)

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Appendix A: solution of a magneto-thermal condensation for weakly heterogeneous magnetic fields

In this Appendix, assuming that the magnetic field is weakly heterogeneous and neglecting viscosity and conduction, we show that the idealized system corresponding to Eqs. (1-6) admits self-similar solutions. These solutions derive from the resolution of one single time ordinary differential equation. We use the adimensional variables defined in Sect. 3.1 and look for self-similar solutions as defined in Eqs. (14-15). Partial derivatives are simply related, for any function G of the reduced spatial coordinate [FORMULA]

[EQUATION]

The integration of the x-momentum equation gives the total pressure [FORMULA]

[EQUATION]

where [FORMULA] is the pressure at origin. We define

[EQUATION]

Consequently, the temperature is given by

[EQUATION]

Although exact solutions exist with a magnetic field equal to (see Appendix C).

[EQUATION]

the magnetic field vanishes at the center and contributes to the compression of the gas instead of opposing to the condensation, as would be expected. We then rather search for magnetic fields equal to

[EQUATION]

[FORMULA] is the homogeneous part of the magnetic field and [FORMULA] is the heterogeneous one. Eq. (4), in its one dimensional form, is as follows

[EQUATION]

Considering the expression of the temperature (A.4), of the magnetic field (A.6) and the property (A.1), we obtain

[EQUATION]

In the general case, the three terms of the l.h.s., which have functionally independent spatial variations, lead to three time differential equations. As we will see later two additional constraints can be set on [FORMULA] and [FORMULA]. Thus, with five constraints and four variables there is no solution in the general case. In Appendix B, we find exact solutions in the case [FORMULA] and for a special choice of the loss function. Here, to overcome this difficulty, we assume that:

[EQUATION]

This means that the magnetic field is weakly heterogeneous. We then can neglect the third term of the l.h.s. in Eq. (A.8). Even with this assumption, Eq. (A.8) admits solutions only for special loss functions. As in Paper I, we consider the loss function: [FORMULA], which has been shown to be in reasonable agreement with a more realistic simulation during the strong condensation phase. It is also possible to add a term proportional to a powerlaw of density to this loss function, but special density distributions f are then required (see Appendix B). With this restriction Eq. (A.8) implies

[EQUATION]

[EQUATION]

Integrating once, we obtain

[EQUATION]

[EQUATION]

with [FORMULA], [FORMULA] and [FORMULA]. One has to keep in mind that these equations assume that the magnetic field is weakly heterogeneous (Eq. A.9). We will not consider further Eq. (A.12) because it describes only a uniform component that has no effect on the dynamics.

The evolution of the magnetic field is given by Eq. (5) and Eq. (6). For the x-component, they lead to

[EQUATION]

and [FORMULA] is constant. For the y-component, we have

[EQUATION]

Let us consider the y-component of Eq. (2)

[EQUATION]

This equation integrates to

[EQUATION]

[EQUATION]

and any function M is allowed. From Eqs. (A.15, A.17, A.18) one derives

[EQUATION]

[EQUATION]

where [FORMULA] is a constant and [FORMULA]. These two equations can be integrated and lead to

[EQUATION]

[EQUATION]

Finally, with Eq. (A.13)

[EQUATION]

The system of Eq. (1-6) in its ideal version with cooling only is reduced to a single time dependent equation.

Appendix B: exact solution of a magneto-thermal condensation

In this appendix, we find an exact solution in the idealised limit of Eq. (1-6) in the case where the adiabatic index [FORMULA] is equal to 2. We start as in Appendix A and consider Eq. (A.8). In the case [FORMULA], Eq. (A.21) leads to

[EQUATION]

If [FORMULA], the third term of l.h.s. of Eq. (A.8) is equal to zero. We consider a loss function equal to

[EQUATION]

The thermal equilibrium condition leads to the relation

[EQUATION]

Thus, for [FORMULA], we have

[EQUATION]

If [FORMULA] the gas is linearly thermally stable and unstable in the opposite case. Eq. (A.8) becomes

[EQUATION]

Let us consider a gaussian distribution of density: [FORMULA]. One has

[EQUATION]

We have the identities

[EQUATION]

[EQUATION]

A solution of Eq. (B.5) can thus be found if the time dependent coefficient of the three independent spatial functions are all zero, which yields

[EQUATION]

[EQUATION]

[EQUATION]

With Eq. (A.22)

[EQUATION]

one obtains two time differential equations describing the whole system. These self-similar solutions are exact and no approximation is assumed but for the thermal function. Eq. (B.11) describes the evolution of a gaussian perturbation associated to an homologous velocity field, it does not depend on the central pressure. Eq. (B.10) describes the evolution of the central pressure. It depends on the evolution of the perturbation. Eq. (B.11) allows to study, at least numerically, the threshold dependence with initial velocity ([FORMULA]) and magnetic field ([FORMULA] and [FORMULA]). This will be considered in a forthcoming study.

Appendix C: exact solution of a magnetically assisted thermal condensation

In this appendix, we find an exact solution of the idealised limit of Eq. (1-6) with the magnetic field equal to zero at the origin ([FORMULA]). In this situation the magnetic pressure compresses the gas and enhances the condensation process. We start as in Appendix A and consider Eq. (A.5) with a magnetic field equal to

[EQUATION]

where H is a constant. The evolution of [FORMULA] is given by the y-component of Eq. (2) that can be rewritten with Eq. (C.1)

[EQUATION]

This leads to: [FORMULA], with

[EQUATION]

and

[EQUATION]

The evolution of [FORMULA] is given by Eq. (5) which leads to

[EQUATION]

The condition for the existence of a self-similar solution is that

[EQUATION]

where [FORMULA] is a constant. This leads to:

[EQUATION]

and

[EQUATION]

But, [FORMULA], so H is equal to zero and the sign is +. One has:

[EQUATION]

where cst is a constant coming from the integration, and

[EQUATION]

The density is a decreasing function, and the magnetic pressure an increasing function, so that magnetic pressure enhances the condensation. The time dependent functions obey the following relations

[EQUATION]

[EQUATION]

[EQUATION]

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© European Southern Observatory (ESO) 2000

Online publication: July 13, 2000
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