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Astron. Astrophys. 327, 909-920 (1997)

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Appendix A: density gradients from radiation pressure

It is customary to assume constant pressure in photoionization calculations, i.e. to employ at each point of the nebula a total density which depends on the local temperature and on the ionization fraction of H and He. The dynamical impact of the impinging radiation is not considered and the gas is assumed static. These assumptions can be justified by the fact that, in general, the timescales for photoionized plasmas to reach local thermal and ionization equilibrium are much shorter than the dynamical timescales. From this perspective, photoionization models represent a series of "snapshots" of the nebular ionization and density structure. However, if one adopts unusually large ionization parameters ([FORMULA]) radiation pressure can no longer be ignored, since it exceeds gas pressure, as shown in Sect 2.3. To take account of radiation pressure in a semi-quantitative manner in the one dimensional case, we have developed a density prescription which is consistent with the assumption of hydrostatic equilibrium, but takes into account the force exerted by radiation from an external source.

If a cloud of neutral gas is suddenly irradiated by an external ionizing radiation field, the cloud surface becomes ionized, and an ionization front travels into the cloud. The photoionized layer is heated to [FORMULA]  K, resulting in a substantial overpressure which drives an outward flow of gas whose density decreases (thereby increasing U) with time. However, in the one dimensional case, the radiation pressure of the photon field on the gas will tend to stop this outward flow. Let us consider what happens in one dimension if the gas density profile evolves towards that of a balance between the gas pressure gradient (directed outward) and the force due to the gradual absorption of the incident ionizing photon field (directed inward).

For the simple case of a pure hydrogen slab we derive (Sect. A.1) analytical expressions describing the principles behind the production of a density gradient by radiation pressure. In Sect. A.2, we discuss some of the properties of internal density gradients, while in Sect. A.3 we describe the numerical implementation of this density prescription in MAPPINGS IC, which includes elements other than hydrogen.

A.1. Density behaviour in the hydrostatic case

Let us assume that the line emission originates from gas clouds which are `near' hydrostatic equilibrium. We assume that the emitting region consist of many clouds at various distances [FORMULA] from the central ionizing source. In this scheme, the cloud dimensions are much smaller than [FORMULA], which allows us to approximate each cloud as a slab and to describe its characteristics in the one dimension x. Unlike other works on ionized nebulae and only for the purpose of simplifying the analytical expressions, we adopt the convention that the spatial coordinate x increases outward from the cloud interior, having its origin beyond the position of the "Strömgren boundary" [FORMULA] where hydrogen becomes neutral. Note that all quantities measured at the irradiated surface (i.e. at [FORMULA], cf. equations A.10-A.11) carry 0 as a superscript or a subscript.

The clouds may or may not be of the same density. If their `front' density [FORMULA] (i.e. at their irradiated face) was to differ such that [FORMULA] then the ratio of radiation pressure to gas pressure would be the same for all clouds. On the other hand, if their `front' dentities were all the same, the impact on each cloud of the radiation pressure would be a decreasing function of the distance from the ionizing source.

Let us now concentrate on a single cloud which in our simplified treatment is approximated as an isothermal pure hydrogen ionization-bounded photoionized slab. We have the equation of state :

[EQUATION]

where P is the local gas pressure, n ([FORMULA]) the proton density (function of x) and T the temperature (a constant). Neglecting the presence of other forces, we have :

[EQUATION]

where [FORMULA] is the force produced by the radiation pressure as a result of photoelectric absorption and [FORMULA] ([FORMULA]) is the number density of neutral hydrogen. [FORMULA] is given by :

[EQUATION]

where [FORMULA] is the monochromatic ionizing energy flux impinging on the cloud and [FORMULA] the photoionization cross section from the ground state of hydrogen. The equilibrium ionization balance of hydrogen is described by :

[EQUATION]

where [FORMULA] is the hydrogen radiative recombination coefficient to excited states and [FORMULA] is given by :

[EQUATION]

Combining equations A.1-A.4 and assuming an isothermal temperature distribution, we obtain :

[EQUATION]

A simple solution to this differential equation can be obtained if we neglect the effects of the diffuse radiation and the hardening of the ionizing photon field. In this approximation, the ratio [FORMULA] is independent of x and so is a constant in the integration of eqn. A.6. We should note that for a [FORMULA] power law ionizing photon field, this ratio takes the simple form :

[EQUATION]

where [FORMULA] is the Lyman frequency, and we have made use of the fact that [FORMULA]. In this way, we can integrate eqn. A.6 to obtain the important result that the position-dependent density of the cloud, [FORMULA], decreases outward, namely :

[EQUATION]

where [FORMULA] is the density at the outer boundary ([FORMULA]) of the cloud. The product [FORMULA], the significance of which is discussed below, contains the following constants :

[EQUATION]

The implied divergence of the gas density for [FORMULA] in eqn. A.6 is an artifact of not considering opacity explicitly in equations A.3 and A.5. As power law continua generate an extensive semi-ionized region, the details of the transition between the fully ionized region and the neutral region at [FORMULA] are not described in the above simplified formulation. As described in Sect. A.3, these two limitations are overcome in MAPPINGS IC by properly solving for the transfer across the slab and by taking into account the opacity of all abundant metals and ions.

A.2. Implications of strong density stratification

In a paper by Binette & Raga (1990; BR90), photoionization calculations were presented in which the gaseous condensations had `built-in' density stratification with a density profile of the form :

[EQUATION]

[EQUATION]

where [FORMULA] corresponds to the density at the slab's irradiated surface (i.e. at [FORMULA]). It was found that when the density decreases outward as [FORMULA] (i.e. [FORMULA]), the value taken by the `mean' (i.e. emissivity averaged) ionization parameter ([FORMULA]), as well as the line ratios, does not change with increasing intensity of the ionizing radiation. The authors did not explore in detail how such a stratification might come about. The hydro-static solution presented above, however, shows that radiation pressure, if important ([FORMULA]), can induce such a gradient.

Let us adopt the same definition of the `mean' ionization parameter as BR90 :

[EQUATION]

where [FORMULA] is the mean density, weighted by the local emissivity of the gas (which we take to be proportional to [FORMULA]) :

[EQUATION]

where the inner limit of integration is [FORMULA], the position of the "Strömgren boundary" which is assumed to be abrupt. [FORMULA] is given by :

[EQUATION]

For the case that interests us ([FORMULA]), BR90 found that the mean ionization parameter [FORMULA] is simply given by :

[EQUATION]

and is therefore independent of the ionizing flux q0 and depends only on the normalization constant [FORMULA]. BR90 found that the line ratios also become independent of the strength of the source (see their Fig. 2) except for the density sensitive lines which become collisionally suppressed as [FORMULA] increases linearly with q0. As shown in Fig. 3 of Sect. 2.5, we obtain a similar result when the gradient results from radiation pressure.

Constant [FORMULA] is obtained only in the case of a strong density gradient. BR90 showed that this condition requires [FORMULA], where [FORMULA] is the Strömgen distance at which hydrogen would become neutral in the slab if it were of constant density [FORMULA] throughout (i.e. isochoric). We rewrite this condition (which applies only to the `built-in' density stratification with [FORMULA] of BR90) in terms of q0 or [FORMULA] as follows :

[EQUATION]

In eq. 15, the normalization constant [FORMULA] is a free parameter. To the extent that the above condition is satisfied, clouds of similar [FORMULA] distributed at various [FORMULA] around a point-like ionizing source will have [FORMULA] independent of the distance from the source (see BR90 for a discussion of the limitations of this property). For line ratios, the situation is more complex due to the process of collisional de-excitation which introduces a dependence of the emissivity on the critical density. Only in the low density regime are line ratios of all clouds of similar [FORMULA] strictly constant, provided that [FORMULA] and the species of interest are fully contained within the ionization structure of the slab (between [FORMULA] and [FORMULA]).

A.3. Implementation of radiation pressure in MAPPINGS IC

The treatment presented is a numerical expression of Sect. A.1 generalized to take into account the opacity of heavier elements. Since in a photoionization code the integration of the transfer equations proceeds from the irradiated face inward, let us introduce the spatial coordinate [FORMULA]. The code MAPPINGS IC divides the photoionized slab into N small layers ([FORMULA]) of geometrical depth [FORMULA]. Starting at [FORMULA], the code sequentially solves for the temperature and ionization balance at each of the boundaries i, that is at each [FORMULA] (starting with [FORMULA]). The gas densities [FORMULA] are derived from the equation of state, that is from the local gas pressure, [FORMULA]. The gas pressure at boundary i must balance the cumulative pressure exerted by the outer gas layers [FORMULA] as well as by the volume force exerted within [FORMULA] by absorption of ionizing radiation. We therefore have :

[EQUATION]

where [FORMULA] is the volume force evaluated at [FORMULA] due to absorption of the impinging radiation and [FORMULA] a term representing the inertial and gravitational forces (see below). We have :

[EQUATION]

where [FORMULA] is the number density of ion [FORMULA] of atomic species m within layer [FORMULA], [FORMULA] the corresponding photoionization cross section with threshold [FORMULA] and [FORMULA] the cumulative opacity up to [FORMULA]  :

[EQUATION]

The summations in A.18 and A.19 are carried over all ions of significant abundance and can be generalized to include continuous opacity due to dust extinction.

The implementation of bulk acceleration and gravity will be the subject of a subsequent paper. For completeness, let us define [FORMULA] as :

[EQUATION]

where [FORMULA] is the density in [FORMULA] @ [FORMULA] the acceleration due to the nuclear blackhole and/or stellar masses and [FORMULA] the bulk acceleration of the whole cloud (cf. Mathews 1986). Only the effective acceleration [FORMULA] needs be defined in our calculations. For future work, we define this input parameter as [FORMULA], the ratio of effective acceleration to radiative acceleration at the irradiated face : [FORMULA]. In this work we have set [FORMULA] (i.e. [FORMULA]). This corresponds to the situation in which the photoionized clouds are pushing (without acceleration) against a hot and tenuous medium with the compression being counterbalanced by the drag force. Adopting a formalism similar to Mathews (1986), the relative pressure difference between the irradiated face and the back of the slab, [FORMULA], is :

[EQUATION]

It represents the ratio of drag pressure to external pressure (i.e. the dynamical pressure of the `pushing' clouds whenever [FORMULA]). For the MB slab with [FORMULA] discussed in Sect. 2, we obtain [FORMULA]. In the simple case [FORMULA], the increase in pressure across the slab due to absorption of radiation is given by :

[EQUATION]

A significant gradient in density or gas pressure will only occur if the available radiation pressure is sufficiently high. Therefore, in the case of stratification due to radiation pressure, the condition for constant [FORMULA] discussed in Sect. 2.3 (with [FORMULA]) can be expressed as follows :

[EQUATION]

where [FORMULA] is the gas pressure at the irradiated surface. [This condition differs from eq A.14 above which concerned only the `built-in' density stratification discussed in BR90.]

The above density prescription remains an idealized way of considering radiation pressure and is not intended to describe the full complexity of photoionized regions. Clouds submitted to significant radiation pressure are particularly prone to derimming through the establishment of a lateral flow as shown by Mathews (1986). Mathews & Veilleux (1989) have reviewed the problem of cloud stability under conditions appropriate to the NLR.

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© European Southern Observatory (ESO) 1997

Online publication: April 6, 1998
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