## 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
The integration of the x-momentum equation gives the total pressure where is the pressure at origin. We define Consequently, the temperature is given by Although exact solutions exist with a magnetic field equal to (see Appendix C). 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 is the homogeneous part of the magnetic field and is the heterogeneous one. Eq. (4), in its one dimensional form, is as follows Considering the expression of the temperature (A.4), of the magnetic field (A.6) and the property (A.1), we obtain 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 and . 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 and for a special choice of the loss function. Here, to overcome this difficulty, we assume that: 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:
, 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 with , and . 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 and is constant. For the y-component, we have Let us consider the y-component of Eq. (2) and any function M is allowed. From Eqs. (A.15, A.17, A.18) one derives where is a constant and . These two equations can be integrated and lead to Finally, with Eq. (A.13) 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 condensationIn this appendix, we find an exact solution in the idealised limit of Eq. (1-6) in the case where the adiabatic index is equal to 2. We start as in Appendix A and consider Eq. (A.8). In the case , Eq. (A.21) leads to If , the third term of l.h.s. of Eq. (A.8) is equal to zero. We consider a loss function equal to The thermal equilibrium condition leads to the relation Thus, for , we have If the gas is linearly thermally stable and unstable in the opposite case. Eq. (A.8) becomes Let us consider a gaussian distribution of density: . One has We have the identities 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 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 () and magnetic field ( and ). This will be considered in a forthcoming study. ## Appendix C: exact solution of a magnetically assisted thermal condensationIn 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 (). 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 where H is a constant. The evolution of is given by the y-component of Eq. (2) that can be rewritten with Eq. (C.1) This leads to: , with and The evolution of is given by Eq. (5) which leads to The condition for the existence of a self-similar solution is that where is a constant. This leads to: and But, , so where 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 © European Southern Observatory (ESO) 2000 Online publication: July 13, 2000 |