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

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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)

[EQUATION]

[EQUATION]

[EQUATION]

where [FORMULA] 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

[EQUATION]

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

[EQUATION]

[FORMULA] is then given by

[EQUATION]

where [FORMULA]. For the numerical calculations presented here we take [FORMULA]. 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 [FORMULA], Eq. (9) becomes

[EQUATION]

with (in addition to the trivial solution, [FORMULA]) the usual solution

[EQUATION]

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

[EQUATION]

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

[EQUATION]

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

[EQUATION]

In Fig. 1 [FORMULA] is plotted against [FORMULA]. The solid lines are where the two roots are coincident; the forbidden range of solutions lies in between. As [FORMULA] and with [FORMULA], we recover the standard non-magnetic limits of [FORMULA] and [FORMULA] 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.

[FIGURE] Fig. 1. [FORMULA] plotted against [FORMULA]. 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 [FORMULA] and [FORMULA], i.e. [FORMULA] is plotted in terms of the fast mode sound speed, [FORMULA]. We identify the solutions with [FORMULA] as being the D-type IFs and [FORMULA] as being R-type IFs.

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

In the bottom panel of Fig. 2 we show the ranges of [FORMULA] and [FORMULA] 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 [FORMULA] 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 [FORMULA] 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, [FORMULA] 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 ([FORMULA]) into the dense material increases as [FORMULA] increases. The exit speed of material through the front, [FORMULA] relative to the gas immediately ahead of the front decreases slightly, from [FORMULA] in the unmagnetized case to [FORMULA] as [FORMULA]. The downstream magnetic : thermal pressure ratio [FORMULA] is always less than [FORMULA].

[FIGURE] Fig. 3. Physical parameters for the D-critical IF as functions of [FORMULA].

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 [FORMULA]. 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 [FORMULA], so that magnetic fields will play a role in this transition.

[FIGURE] Fig. 4. Physical parameters for the R-critical IF as functions of [FORMULA].
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© European Southern Observatory (ESO) 1998

Online publication: March 3, 1998
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