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

3. Self-similar solutions of a magneto-thermal condensation

Self-similar solutions of the idealised (all dissipative terms are neglected) system of Eqs. (1-6) can be obtained, with a slight additional approximation. The system is then reduced to a single ordinary differential equation, derived in Appendix A, and numerically solved in Sects. 3.3, 3.4. Exact solutions for special cases are found in Appendix B and C.

3.1. Reduction to a single ordinary differential equation

Self-similar solutions are widely found in the literature in several contexts (Sedov 1959, Barenblatt & Zeldovich 1972, Ferrara & Shchekinov 1996, Shu 1977, Li & Shu 1997, Munier & Feix 1982, Bouquet et al. 1985). They are known to describe intermediate regimes, that do not depend on the initial and boundary conditions. These exact solutions usually offer a deep insight into the physical processes; they mainly provide an analytical description of fully non-linear processes that cannot be described by linear or weakly non-linear analysis.

Let us consider the idealised system of Eq. (1-6). We first normalize these equations ( is the equilibrium gas density in the WNM) and set

We assume that the x-component of the velocity field is

where a is a function of and its derivative against . This field diverges at infinity, but can be relevant locally. The associated density field is

where f is an arbitrary function and . We now consider a magnetic field equal to

where

and assume that the magnetic field is only weakly heterogeneous, or

( is the homogeneous part of the magnetic field and the heterogeneous part). Physically, this assumes that the magnetic energy dominates the kinetic energy. The magnetic field is strong and the field lines are weakly bended by the flow. Finally, the loss function is assumed to write

Thus, at this stage no heating is included and only a phase of the evolution, during which the gas cools, is described. These limitations will be relaxed in the numerical simulations of Sect. 4. Under conditions (14)-(20) the idealized system corresponding to Eqs. (1-6) is shown to reduce to a single time dependent ordinary equation (Appendix A)

with , , and

The transverse velocity is equal to

3.2. Physical interpretation

In the non-magnetized case, studied in Paper I, Eq. (21) is

It admits the following exact solution

provided initially

Solution (26) has a clear physical meaning: the thermal energy is radiatively dissipated and tends to a Dirac function (if ).

Let us now consider the case of an initial density distribution weakly peaking at the origin (for instance a gaussian distribution). The case is unphysical because high densities correspond to low thermal pressures, which leads to an unphysical collapse. For similar reasons, we only consider initial conditions with, and , i.e. high densities (center) correspond to high thermal and magnetic pressure.

a) If the magnetic field is transverse () and the initial transverse velocity is equal to zero (, ), then system of Eqs. (21-23) is as follows

Hence, the magnetic field is proportional to the density and the ratio is constant. The condensation is prevented by the magnetic pressure through the term (in the following, we will simply refer to as the magnetic pressure ). With , we have

Thus, if increases strongly (condensation), so does because . Consequently, increases and tends to become positive, then tends to decrease (re-expansion).

b) If the magnetic field is oblique () and the initial transverse velocity is equal to zero (, )

The ratio is not constant anymore, and a new term appears in the l.h.s. of Eq. (34), which corresponds to a transverse velocity field generated by the magnetic tension. It contributes through the term of Eq. (5) to the evolution of the transverse magnetic component. At , this term is equal to zero, then it grows and if and is large enough, decreases. If becomes comparable to the approximation stated in Eq. (19) is no longer valid.

c) When the transverse velocity is initially not zero (may be a consequence of the previous phase), the evolution may be more complex and will not be further considered here.

3.3. Numerical solution with a transverse and an oblique magnetic field

An analytical solution of Eq. (34) seems to be out of reach and numerical solution is now adressed. Let us consider the variable

Eq. (34) becomes

with

We numerically solve this 4th order differential equation, using a 4th order Runge-Kutta method. For initial values close to the values given in Eq. (27), the non-magnetized case (described by Eq. 25) behaves very differently from the adiabatic one. Thus, we choose initial conditions near these two values ( and with ). We explore two different magnetic cases: (1) case of a purely transverse magnetic field and (2) case of an oblique magnetic field. The results can be seen on Fig. 1. The adiabatic case () and non adiabatic case () without magnetic field are also shown for easier comparison.

Fig. 1. Numerical resolution of Eq. (34) in four different cases. The normalised density at origin 1/a, is plotted as a function of , the a dimensional time. The two upper panels display the non-magnetised solutions of Paper I, the adiabatic and cooling cases respectively. The thermal approximation breaks down for large densities. The two lower panels display solutions with a magnetic field. Third panel: the transverse magnetic field prevents the condensation. Fourth panel: a longitudinal field has been added. Unfortunately, the approximation stated in Eq. (19) is no longer valid after and the computation cannot be continued. The essential difference between the third and fourth panel nevertheless appears clearly in Fig. 2.

While a strong condensation is observed in the absence of a magnetic field (Paper I), the transverse field quickly stops the condensation. The case of the oblique field is subtly different, the contraction is effectively slowed down, but the magnetic pressure which initially increases, vanishes before the contraction stops (see Fig 2). Our weak heterogeneity approximation (Eq. 19) breaks down at this point and the solution cannot be further calculated. But the condensation is expected to proceed further, because the thermal energy has already been radiated away and the magnetic pressure is weak. Whether this magnetic pressure leak mechanism operates, depends on the initial magnitude of the longitudinal magnetic field, hence on the angle between the initial velocity and magnetic fields.

Fig. 2. Evolution of the in the two magnetized cases of Fig. 1 ( and ). In the first case, the magnetic pressure is proportionnal to (). In the second case, the behaviour is more complex. starts to increase but then decreases whereas density still increases (see Fig. 1). Before it becomes equal to zero, the approximation stated in Eq. (19) breaks down.

3.4. Magnetic pressure leak , efficiency as a function of the incidence angle

How does the magnetic pressure evolution depend on the magnetic intensity () and on the angle ? We investigate this issue in the oblique case just considered (, , ) by varying the initial values of , and , in the weak heterogeneity case (). In the purely transverse case, the ratio stays constant, while steadily increases during the compression, preventing compression far beyond the pressure equilibrium point. On the opposite, we saw (Fig. 2) that the magnetic pressure rapidly decreases before the compression stops in the oblique case considered. Using this sudden decrease as condensation criterion we can draw the condensation threshold line, in the plane (Fig. 3) for the selected initial conditions. This qualitative criterion constitutes a first approach towards an understanding of the condensation conditions. Calculations have not been carried on for , as in these cases the approximation of a weakly heterogeneous field is not appropriate. This situation will be covered in the simulation presented in the next section.

Fig. 3. Dependence of with the initial magnetic intensity () (first approach). The points below this curve are expected to lead to a condensation because the magnetic pressure is weak, whereas the points above will not give such condensations because the gas re-expands before the magnetic pressure decreases sufficiently. For small values of the magnetic field our criterion is invalid, the magnetic pressure is low and the condensation is possible independently of .

In the conditions explored, the maximum angle increases with the magnetic field intensity.

We conclude that:

• i) In a one dimensional condensation process with an initial transverse velocity field equal to zero, the magnetic pressure starts to increase and can possibly decrease before re-expansion if the initial flow angle with respect to the magnetic field, , is small enough (see Fig. 2).

• ii) In this case, the higher the magnetic intensity is and the smaller the angle , the faster the decrease of the magnetic pressure will be.

• iii) Whatever the intensity of the magnetic field, the condensation process is still possible if the angle is below a maximum value: . For weak magnetic fields, the magnetic pressure has small effect on the dynamics and the condensation process can occur even for large values of . For large magnetic fields, increases with (see Fig. 3).

© European Southern Observatory (ESO) 2000

Online publication: July 13, 2000
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