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)
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
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.
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.
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 .
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.
© European Southern Observatory (ESO) 1998
Online publication: March 3, 1998