1. INTRODUCTION

Strömgren (1939) studied the problem of the “grey” transfer of ionizing photons in a homogeneous, photoionized region, and found an approximate solution with ionization fraction x = n_HII/n_H(where n_HIIand n_Hare the ionized and total H number densities, respectively) as a function of the spherical radius R which goes to zero at the so-called “Strömgren radius” R_S. This radius can also be obtained from a simple, “Strömgren sphere” analysis, in which one considers the balance between the ionizing photon rate S∗ produced by the star and the total recombination rate inside a fully ionized sphere. Petrosian et al. (1972) used the same approximation as Strömgren (1939) in order to analytically solve the transfer of ionizing photons within a dusty, homogeneous HII region, and found the resulting x vs. R relation. Also, the paper of Franco et al. (1990) has an appendix with an interesting discussion of the effect of dust on the outer radii of HII regions. Models of photodissociation regions (e.g. Hollenbach & Tielens 1999 ; Krumholz et al. 2008 ; Sternberg et al. 2014 ) of course also share many common features with dusty HII region models.

In the present paper, we derive the “Strömgren sphere” analysis for a dusty HII region. This analysis results in simple prescriptions for the outer radius of the nebula as a function of the dust-free Strömgren radius ( RS ) and the dust optical depth λd associated with a distance RS (see § 2).

We then solve the radiative transfer problem to find the ionizaton stratification x(R) with the approximation of Raga (2015) , who studied the case of a dust-free nebula obtaining an improvement on the simpler approximation of Strömgren (1939) . We find that under this improved approximation, the problem of a dusty nebula also has an analytic solution (§ 3), which compares well with exact (i.e., numerical) solutions of the radiative transfer equation.

The astrophysically relevant ranges of the dimensionless parameters of the problem are discussed in § 4, and examples are given of the values of these parameters for different types of HII regions. A discussion of the limits of applicability of homogeneous HII region models is given in § 5. Finally, the results are summarized in § 6.

2. THE “DUSTY STRöMGREN SPHERE” APPROACH

In this section, we extend the “global photoionization balance” condition (which gives in a direct way the well-known “Strömgren radius” of an HII region) to the case of a homogeneous nebula with dust. The dust (assumed to be homogeneously distributed in the volume of the ionized nebula) contributes to the absorption of ionizing photons.

The requirement of a global balance between the rate of stellar ionizing photons S∗ and the total recombination rate within the ionized region (together with the assumption of homogeneous density and temperature distributions and a H ionization fraction x ≈ 1 within the nebula) gives the Strömgren radius:

where n_H is the H number density and α H = 2.6 × 10⁻¹³cm⁻³is the “case B” H recombination coefficient at ≈ 104 K (see, e.g., the book of Dyson & Williams 1980 ).

In order to include the effect of absorption of ionizing photons by dust particles (present within the ionized region), we write the balance between the ionizing photon rate S _∗and the absorptions within the nebula:

where R ₀is the outer radius of the photoionized region, J _ʋis the angular average of the specific intensity, ʋ is the frequency (ʋ₀being the Lyman limit frequency), h is Planck’s constant, n_HI is the neutral H number density, σ _H is the H photionization cross section, and σ _d is the dust absorption cross section per H atom.

We now consider the standard “grey ISM” approximation, namely that σ _H and σ _d are independent of ʋ, assume that the ionizing radiation is dominated by the direct stellar photons (the diffuse ionizing photons being included in an approximate way by considering the “case B” value of α _H ) to obtain:

where

where T_H is the photoionization and T_d the dust absorption optical depths.

Substituting equation (3) and the photoionization balance condition

with

into equation (2) we finally obtain:

where we have considered that the electron density n_eand the ionized H density n_HIIhave values n_e≈ n_HII≈ n_Hwithin the ionized region (i.e., for R ≤ R0).

Clearly, if we set σ_d= 0 (i.e., no dust absorption), equation (7) is the standard Strömgren sphere relation, giving R₀= R_S(with R_Sgiven by equation 1 ). Using the Strömgren radius to adimensionalize equation (7) we then obtain:

where

is the dust optical depth corresponding to the Strömgren radius.

In order to proceed, we now note that in the inner part of the nebula H is mostly ionized, so that n_HI≪ n_H(this is of course not true close to the outer radius of the nebula, where H has the ionized → neutral transition), so that the first term in the right hand side of equation (4) can be neglected. We then have

which can be substituted in equation (8) to obtain:

This is a trascendental equation for (R₀/R_S) as a function of λ_d(see equation 9 ) that gives R₀= R_Sfor λ_d= 0 (i.e., for no dust absorption), and can be inverted numerically for λ_d> 0. The result of this inversion is shown in Figure 1 .

Interestingly, in the Appendix of the paper by Franco et al. ( 1990 , which discusses the effect of dust on the radii of photoionized regions), equation (11) is derived through the simple argument that the dust will attenuate the ionizing photon rate S∗ by a factor of ≈ e^−Td. This attenuated ionizing photon rate is then inserted in equation (1) to obtain the outer radius of the ionized nebula. The derivation above shows that the relation deduced by Franco et al. (1990) is indeed obtained from a proper “Strömgren sphere” analysis provided that the assumption of equation (10) is made. An explicit, approximate form for R₀/R_Scan be obtained by taking the cube root of equation (11) and then expanding the exponential to second order in λ_dR₀/R_S. The resulting quadratic equation can then be inverted to obtain

This approximate equation follows well a numerical inversion of equation (11) for λ_d≤ 3, but has large deviations from the true solution for larger values of λ_d.

It is also possible to drop the assumption of equation (10) by noting that the number of absorptions due to photoionization out to a radius R is:

where T_His the photoionization optical depth (see equation 4 ), from which we obtain

Using this value for TH (instead of setting TH = 0, as we have done for deriving equation 11 ), equation (8) then takes the form

Doing the integral, one obtains

where r₀= R₀/R_S. This equation can be inverted numerically to obtain R₀/R_Sas a function of λ_d. The results of this excersize are shown in Figure 1 , in which we see that for the displayed range of λ_d equation (16) gives very similar results to the ones obtained from equation (11) . The good agreement between equations (11) and (16) is an indirect justification of the approximation (see equation 10 ) used for deriving equation (11) .

In the following section we compare the outer radii R0 for the photoionized region obtained from equations (11) and/or (16) with numerical and analytic solutions of the radiative transfer+ photionization equilibrium problem.

3. THE RADIAL STRATIFICATION OF THE IONIZATION FRACTION

We now write the ionization balance condition ( equation 6 ) in terms of the (spatially dependent) H ionization fraction x = n_HII/n_H= n_e/n_H:

This quadratic equation for x can be inverted to obtain

with

We now use equations (19) and (6) to define

in terms of the dimensionless radius

and where the dimensionless parameter,

is the optical depth over a Strömgren radius R_Sassociated with photoionizations in the neutral gas. The above derivation is identical to the one of Raga (2015) and very similar to the one of Strömgren (1939) , who studied the case of a dust-free HII region. Now, from equation (4) we see that f obeys the differential equation

where x is given as a function of f and r by equations (18-20). This differential equation can be integrated numerically with the boundary condition f(0) = 1, but no exact analytic integral has been found.

In order to obtain an approximate numerical integral, one expands the term in parentheses on the right hand side of equation (18) to first order in A, obtaining

This Taylor series expansion is compared with the exact form of 1 − x(A) (obtained from equation 18 ) in Figure 1 .

We now write A in terms of f and r using equation (20) and insert the result into equations (24) and (23) obtaining the differential equation

where we have used the notation “f _p ” in reference to the paper of Petrosian et al. (1972) . This equation can be directly integrated with the boundary condition f _p (0) = 1 to obtain the approximate solution

This is the approximate analytic solution derived by Petrosian et al. (1972) . It can be easily shown that for λ _d → 0 this solution converges to the f ( r ) = 1 − r ³, dust-free solution of Strömgren (1939) . Also, one sees that this solution gives an infinite optical depth (i.e., f_p = 0, see equation 20 ) at a finite value r_p , which can be obtained by inverting numerically the condition f_p ( r_p ) = 0 (see equation 26 ). The resulting values of rp as a function of λ _d are shown in Figure 1 , in which we see that similar values to the r ₀radii (resulting from global photoionization balance arguments, see equations 11 and 16 ) are obtained.

To obtain a better analytic approximation we follow Raga (2015) in approximating the x(A) dependence (see equation 18 ) with a three-segment interpolation of the form

with A ₁= 0.1, A ₂= 8.402, a = 0.316 and b = 0.5 (Raga 2015 used a b = 0.6 value, but as will be evident below, it is more convenient to choose a slightly different value for this coefficient).

This approximation has a transition from the linear regime to a ∝ A ^1/2power law at a neutral fraction y1 = A 1 = 0.1. The resulting interpolation is compared with the value of 1 − x ( A ) obtained using the exact form for x ( A ) (given by equation 18 ) in Figure 2 .

With the approximate form for 1 − x ( A ) given by equation (27) , equation (23) can be integrated analytically to obtain

for r ≤ r ₁(see equation 26 ),

for r ₁< r ≤ r ₂,

for r ₂< r . In equation (29) the a constant has the value given after equation (27) , and we have considered the b = 1/2 case. It is also possible to obtain exact analytic integrals for b = n /2 values (with integer n ), but for other values of the power law exponent b (see equation 27 ) only approximate integrals can be obtained, resulting in more complex forms of the solution.

In equation (30) , f ₂is the value given by equation (29) when evaluated in r = r ₂. The matching consant E in equation (29) is given by

The switch between equations (28) and (29) occurs at a dimensionless radius r ₁which is obtained from a numerical inversion of the equation

with fp ( r ) given by equation (26) .

The switch between equations (28) and (29) occurs at a dimensionless radius r ₂which is obtained from a numerical inversion of the equation

with f ( r ) given by equation (29).

Approximate explicit forms for r ₁and r ₂can be obtained by expanding the corresponding equations (32 and 33) to first → fourth order around r _1,2= 1, and then finding analytically the roots of the resulting polynomials. We do not give here the results of this exercise, because they are hardly worthwhile.

We can now calculate the radial dependence of the ionization fraction x (see equations 18 and 20) using the “exact” (i.e. numerical) solution of equation (23) and the two approximate analytic forms obtained for f ( r ) (the solution of Petrosian et al. 1972 given by equation 26 and the 3-segment approximation given by equations 28-30).

The three corresponding forms for x ( r ) obtained for the sets of parameters (λ = 100, λ _d = 0.2), (λ = 1000, λ _d = 0.2), (λ = 1000, λ _d = 0.0) and (λ = 105, λ _d = 20.0) are shown in Figure 3 . It is clear that the 3-segment approximation has a transition to neutral gas (i.e., for x → 0) that follows the exact solution in a more accurate way than the solution of Petrosian et al. ( 1972 , which reaches x = 0 at a finite radius).

In Figure 4 , we show the radius R _0.9at which the ionization fraction has a x = 0.9 value as a function of λ _d , for three values of the dimensionless parameter λ (=1000, 100 and 10) obtained from the exact (i. e., numerical) and from the 3-segment approximation (given by equations 28-30). These two solutions give almost indistinguishable results at the resolution of the figure. In the three plots we also show the R ₀vs. λ _d dependence obtained from the “dusty Strömgren sphere” analysis (see equation 16 and Figure 1 ). It is clear that for λ = 100 and 1000, the R _0.9radius closely follows the R ₀vs. λ _d dependence. For the lower, λ = 10 value, not surprisingly we find larger deviations between the R _0.9and R ₀radii.

4. THE VALUES OF THE DIMENSIONLESS PARAMETERS

In § 3 we present solutions of the ionization structure of a homogeneous HII region which are given in terms of two dimensionless parameters:

• λ = n_Hσ_HR_S : the optical depth due to photoionization of H over a distance RS in a completely neutral medium,

• λd = n_Hσ_dR_S : the optical depth over a distance RS due to dust absorption.

Even though λ typically has values ≫ λ _d , this does not necessarily imply that the dust absorptions are negligible, since within the nebula the neutral fraction of H is very low.

Also, in § 2, we presented a “dusty Strömgren sphere” model, in which the only dimensionless parameter that appears is λ _d . This model is applicable for HII regions in the λ ≫ 1 limit.

The dimensionless parameter λ (see equation 22 ) has values

where we have used typical parameters for a galactic HII region. The last term on the right (σ _H /σ_ʋ0) is the ratio between the average photoionization cross section (that has to be taken out of the integral in equation 2) and the Lyman limit cross section. This ratio is ≈ 1 for an HII region photoionized by a stellar source, but can have substantially smaller values for the case of a region photoionized by the “monster” (i.e., the accreting black hole) in the center of an active galaxy (which has a power law ionizing spectrum, see, e.g., Binette et al. 1993 ; Haro-Corzo et al. 2007 ). Therefore, for narrow line regions of AGNs we can have HII regions with substantially lower λ ∼ 10 → 100 values. Conversely, in very high density HII regions such as ultracompact regions within molecular clouds (see, e.g., Kurtz 2005 ), we can have very high densities, resulting in higher values, up to λ ∼ 10⁵. Another way to have higher λ values is to have an HII region photoionized by a cluster with many O stars.

The average galactic dust absorption cross section per H atom at the Lyman limit has a value σ _d ≈ 1.15 × 10⁻²¹cm²(see the discussion of Vasconcelos et al. 2011 ). Therefore, the dimensionless parameter λd (see equation 9 ) has a value

where ξ_d≤ 1 is the fraction of dust that survives within the HII region.

Combining equations ( 34 - 35 ) we obtain

In Figure 3 , we show the ionization stratifications of four different HII regions:

• an HII region photoionized by a hard spectrum (resulting in λ = 100), and with average galactic dust abundance (so that λ _d = 0.2, see equation 35 ),

• an evolved galactic HII region (λ = 10³) with average galactic dust abundance (λ _d = 0.2),

• an evolved galactic HII region (λ = 10³) with no dust (λ _d = 0),

• an ultracompact HII region with density n_H ≈ 10⁶cm⁻³, with λ = 10⁵and λ _d = 20.

It is clear that in the last case the dust absorption has a very strong effect on the HII region, which has an outer radius of ≈ 1/4 of the Strömgren radius of a dust-free nebula.

5. APPLICABILITY OF A HOMOGENEOUS MODEL

The models described in this paper are limited to “Strömgren’s model”, i.e., to the case of a nebula with a homogeneous density and temperature. The way to obtain such a configuration is to “turn on” an O star within a homogeneous, neutral medium, and to wait for the “initial expansion” to take place. In this initial expansion, a fast R-type ionization front travels out and reaches the “initial Strömgren radius” (see, e.g., the lucid discussion in the book of Dyson & Williams 1980 ).

This initial Strömgren sphere does have a homogeneous density distribution, so that a homogeneous model is in principle applicable. However, at the ionization front it has a strong temperature gradient. The feedback of this temperature gradient on the ionization structure (through the rather weak temperature dependence of the H recombination coefficient) is not included in a homogeneous HII region model.

The other possibility is the application of a homogeneous model to the final, pressure equilibrium configuration (in which the HII region has expanded until it has reached pressure balance with the surrounding, neutral medium). In this final configuration, the almost constant temperature ( T ∼ 10⁴K) resulting from the thermal balance guarantees an internal region with a quite homogeneous density. However, in the HII/HI transition region (in which the temperature has a transition from ∼ 10⁴K to the ∼ 10²→ 10³K temperature of the neutral medium), the pressure balance condition implies a strong density gradient. This effect is of course not included in a homogeneous model.

Another effect that will introduce density gradients within the fully ionized region of a hydrostatic, dusty nebula is the radiation pressure on the dust grains. This effect was studied in detail by Draine (2011) . In order to evaluate the importance of the radiation pressure on dust grains let us consider the following, semi-qualitative analytic argument.

Let us assume that we have a low enough value of λd, so that the outer radius of the nebula has a value ∼ R_S (i.e., the Strömgren radius of a dust-free nebula, see equation 1 ). This implies that the dust optical depth is at most of order 1, so that most of the stellar radiation (dominated by the non-ionizing photons) reaches regions close to the outer edge of the nebula. At radii close to R_S , we then write the hydrostatic balance between the pressure gradient and the radiation pressure force as

where we have assumed a “grey” σ d ≈ 10⁻²¹cm²over all of the frequencies of the stellar spectrum. We can then calculate the radial scale ΔR of the pressure variations in the outer regions of the hydrostatic stratification as

where we have set P = 2 n_Hk_T . Clearly, the pressure gradient induced by the radiation pressure force will be important for Δ R / R_S ≈ 1 (or higher). Setting equation (38) equal to 1 and using equation (1) for R_S , we then derive the density

above which the ionized region will develop an important pressure (and therefore density) gradient. In equation (39) we have used the values of S∗ and L∗ appropriate for an O5 dwarf, and a T = 10⁴K temperature for the HII region.

If we insert the value of n_H,r ( equation 39 ) in equation (35) , we obtain λ _d ≈ 3.5. Therefore, for regions with values of λ _d substantially above unity, we expect the constant density models (discussed in this paper) not to be applicable to the final, pressure equilibrium configuration of a nebula. However (as discussed above), even for lower λ _d values, the homogeneous models do not describe appropriately the ionized to neutral transition in a pressure equilibrium nebula.

6. DISCUSSION

We have first presented a “Strömgren sphere” description for calculating the outer radius R ₀of a homogeneous, dusty photoionized region (see § 2). In its most simple form (see equation 11 ), this approach leads to a simple, trascendental equation which can be inverted to obtain R ₀/ R_S (where R_S is the Strömgren radius of a dust-free but otherwise identical region) as a function of the dust optical depth λ _d = n_Hσ_dR_S associated with Rs (see equation 35 ). We also find an approximate, explicit analytic inversion of this trascendental equation, which works well for λ d < 3 (see equation 12 ).

We have also studied a more detailed “Strömgren sphere” approach, which results in a different trascendental R ₀/ R_S vs. λ _d relation (see equation 16 ). However, this relation gives similar results to the simpler equation (11) for the astrophysically relevant λ _d range (see Figure 1 and § 3 ). Our simpler relation for the radius of a dusty Strömgren sphere was previously obtained by Franco et al. (1990) with a more qualitative approach.

We then compare the results of the “Strömgren sphere” approach with solutions of the associated radiative transfer problem (see equation 23 ), from which the ionization fraction x = n_HII / n_H is obtained as a function of radius. We integrate the resulting transfer equation numerically, and we also integrate it analytically using the approximation described by Raga (2015) , who studied the case of a dust-free HII region. This approximation is an improvement on the classical approximation of Strömgren (1939) , which was applied to the case of a dusty HII region by Petrosian et al. (1972) .

The presence of dust has a strong effect in the size of the ionized region for high nebular densities, as is the case in compact, ultra-compact and hypercompact HII regions (see the review of Kurtz 2005 ). Dust has of course been included in the past in detailed photoionization models of dense HII regions (see, e.g., Morisset et al. 2002 and Draine 2011 ).

The analysis presented above gives a clear physical picture of the transfer of ionizing photons in dusty HII regions, and results in relatively simple recipes ( equations 11 , 12 and 16 ) for obtaining the outer radius of such a region. We have also obtained a full analytic solution for the ionization fraction as a function of radius (see § 3), which can be useful for initializing numerical simulations of dusty, photoionized flows.

We end by noting again that homogeneous models (such as the ones described in the present paper) are a strong idealization of the situation found in real HII regions. Very special conditions are indeed necessary for such models to be directly applicable to observed objects (see the discussion of § 5).

We acknowledge support from the CONACyT grants 61547, 101356, 101975 and 167611 and the DGAPA-UNAMgrants IN105312 and IG100214. We acknowledge an anonymous referee for comments which lead to the discussion in § 5.

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