Astron. Astrophys. 331, 1099-1102 (1998)

2. Model

In this first paper, we treat the simple case where the magnetic field is in the plane of the ionization front. The jump conditions are: (cf. Bertoldi & Draine 1996; Melrose 1986)

where is the gas density, u is the gas velocity in the frame of the IF, P is the pressure and B is the magnetic field strength. The subscripts 1 and 2 refer to upstream and downstream values, respectively. We define

where is the Alfvén speed. The parameter is the reciprocal of the usual plasma parameter . From Eqs. (1-3)

is then given by

where . For the numerical calculations presented here we take . However, lower values may be appropriate if the IF is proceeded by a dissociation front (Bertoldi & Draine 1996).

2.1. Non-magnetized ionization fronts

If , Eq. (9) becomes

with (in addition to the trivial solution, ) the usual solution

Since must be real, the bounds on which allow real solutions are and giving R-type and D-type IFs respectively. For these bounds on there are two solution branches to . These are

with the positive root solution giving a weak R-type IF and the negative root solution giving a strong R-type IF. Similarly,

the positive root solution gives a strong D-type IF and the negative root solution gives a weak D-type IF. Note that these values will also apply to the case where the magnetic field is perpendicular to the plane of the IF. There may, however, also be `switch-on' type IFs in this case for some initial Mach numbers (cf. Draine & McKee 1993).

2.2. Magnetized ionization fronts

If , Eq. (9) is a cubic that can have either two or zero real positive roots (since is positive at and ). At the boundary between these two regimes there are two coincident roots, at where . Thus the boundary is given parametrically in terms of by solving these simultaneous equations for and . We find

In Fig. 1 is plotted against . The solid lines are where the two roots are coincident; the forbidden range of solutions lies in between. As and with , we recover the standard non-magnetic limits of and for the R-type and D-type IFs respectively (Sect. 2.1). As the magnetic field strength increases, the forbidden region between the R- and D-type classes of solutions decreases significantly, which corresponds to a smaller velocity difference between the two types of IFs.

Fig. 1. plotted against . The forbidden range of velocities decreases as the magnetic field strength increases

In the top panel of Fig. 2 we show the allowed and forbidden regimes for values of and , i.e. is plotted in terms of the fast mode sound speed, . We identify the solutions with as being the D-type IFs and as being R-type IFs.

Fig. 2. Top: The ranges of and that give real solutions to Eq.  9. is plotted in terms of the fast mode sound speed. Bottom: The family of plots of as a function of , for constant (thin lines). The thick line marks the critical solutions between the weak and the strong cases. This line corresponds exactly to the plot in the top panel and can be used to judge the value of beta used for a given () curve

In the bottom panel of Fig. 2 we show the ranges of and that are permitted by the presence of a magnetic field. The two solution branches, separated by the line of critical solutions, give rise to the weak and the strong IFs in the R-type and D-type zones. The standard non-magnetized ionization front solutions (Eqs.  12and   13) are recovered at low and are seen as the D-type curve in the top left hand corner and the R-type curve in the bottom right hand corner.

The plots in the two panels are orthogonal to each other in space. The solid line corresponds exactly between them.

The D-critical solutions are particularly important, as they often occur at the surface of neutral clumps ablated by external ionization. In Fig. 3 we show the effects that the magnetic field will have on various parameters. The velocity ratio, is initially the high standard value but falls off with increasing field strength so that the velocity ratio is close to 1. The IF speed () into the dense material increases as increases. The exit speed of material through the front, relative to the gas immediately ahead of the front decreases slightly, from in the unmagnetized case to as . The downstream magnetic : thermal pressure ratio is always less than .

Fig. 3. Physical parameters for the D-critical IF as functions of .

Weak R-type fronts are found in the early stages of ionization of a uniform gas distribution by a source such as a massive star. R-critical fronts mark the end of this phase. As in the non-magnetized case, strong R-type fronts are over determined (Goldsworthy 1961) and require very special circumstances for their existence. In Fig. 4 some relevant physical parameters of R-critical fronts are displayed. Significant changes occur only for . This is a high value for diffuse neutral clouds but may be expected in molecular clouds (Sect. 1). Weak R-type fronts generally evolve into D-type fronts which are affected by lower values of , so that magnetic fields will play a role in this transition.

Fig. 4. Physical parameters for the R-critical IF as functions of .

© European Southern Observatory (ESO) 1998

Online publication: March 3, 1998
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