3 The energy implications of temperature fluctuations

To quantify the energy injection rate needed to fuel temperature fluctuations in a nebula, we follow the model proposed by Binette & Luridiana (2000) and Binette et al. (2001). The first of these two papers contains a detailed description of the method, while we refer to the second for an alternative representation of the results. The model depicts temperature fluctuations as a collection of hot spots, arising above a uniform equilibrium temperature floor, and fed by an unknown heating agent. This representation allows one to compute the average thermal properties of the nebula in the temperature-fluctuation regime, but without having to compute real localized fluctuations, as described below.

  
3.1 Definition of temperature fluctuations

The amplitude of temperature fluctuations in a nebula is measured by the parameter t², defined as follows (Peimbert 1967):

$$t^2={{\int_V (T_{\rm e} (\vec{r})-T_{\rm0})^2 N_{\rm e}(\vec{... ...rm0}^2\int_V N_{\rm e}(\vec{r}) N_{\rm i}(\vec{r}) {\rm d}V }}$$ (1)

where $T_{\rm e}(\vec{r})$ is the local electron temperature, $N_{\rm e}(\vec{r})$ and $N_{\rm i}(\vec{r})$ are the local values of the electron and ionic density respectively, V is the observed volume, and the average temperature $T_{\rm0}$ is given by:

$$T_{\rm0}={{\int_V T_{\rm e}(\vec{r}) N_{\rm e}(\vec{r}) N_{\r... ...{\int_V N_{\rm e}(\vec{r}) N_{\rm i}(\vec{r}) {\rm d}V }}\cdot$$ (2)

Since the temperature of a model H II region is not spatially constant, temperature fluctuations appear and evolve following the evolution of the ionization field even in the simplest case of a static, pure photoionization nebula; in particular, Pérez (1997) showed that large temperature fluctuations arise during the WR phase, as a consequence of the hardening of the spectrum. Following the definition by Ferland (1996), we will refer to such "structural'' temperature fluctuations as $t^2_{\rm str}$. Typical $t^2_{\rm str}$ values for photoionization models of chemically and spatially homogeneous model nebulae are in the range $0.00\le t^2_{\rm str}\le 0.02$. The best-fit photoionization model by Luridiana et al. (1999) yields a value $t^2_{\rm str}\sim 0.009$.

On the observational side, González-Delgado et al. (1994) find for knot A in NGC 2363 the value $t^2_{\rm obs}=0.064$by comparing the Paschen temperature, $T_{\rm e}$(Pa), to the [ O III] temperature, obtained from the $\lambda 4363/\lambda 5007$ ratio. Even if in this case the formal error on $t^2_{\rm obs}$ is quite large (we calculated $\sigma (t^2_{\rm obs})=0.045$, mainly due to the large error on $T_{\rm e}$(Pa)), the differences between the model predictions and the observations seem to imply that an extra-heating mechanism, other than photoionization, is at work in most objects, producing an additional $t^2_{\rm extra}$ such that $t^2_{\rm obs} = t^2_{\rm str} + t^2_{\rm extra}$. In the case of NGC 2363, we find $t^2_{\rm extra} = 0.055\pm 0.045$. Note that such value still leaves the door open to the possibility, small but not negligible, that $t^2_{\rm extra} = 0.00$; however, we assume this not to be the case. In this respect, it is interesting to note that $t^2_{\rm obs}=0.098$ in knot B of the same region (González-Delgado et al. 1994). Given the presumed age and metallicity of this region (Luridiana et al. 1999), and assuming the same filling factor as in knot A, we can roughly estimate $t^2_{\rm str} = 0.02$ (see, e.g., Pérez 1997), yielding $t^2_{\rm extra} = t^2_{\rm obs} - t^2_{\rm str} = 0.078$, with an associated error of about $\sigma (t^2_{\rm obs})=0.019$.

More generally, even though the errors might be compatible with $t^2_{\rm extra}=0$ in individual cases of ionized regions with $t^2_{\rm obs}>0$, this is certainly not the case when large samples are considered. As an example, galactic H II regions and planetary nebulae collectively show higher values, typically around 0.04 (e.g., Peimbert et al. 1995; Peimbert 1995, and references therein).

The presence of temperature fluctuations inside NGC 2363 bears important cosmological consequences, as it affects the determination of the chemical abundances, lowering the extrapolated primordial helium abundance $Y_{\rm p}$(e.g., Peimbert et al. 2001a, and references therein). In our case, $t^2_{\rm obs}=0.064$ implies a downward change in the helium abundance of NGC 2363 of order 2%, the exact figure depending on which He lines the abundance determination is based on (see also González-Delgado et al. 1994). It is important to note, however, that temperature fluctuations are not the only factor affecting a proper chemical abundance determination, other being, e.g, the ionization structure and the collisional excitation of the hydrogen lines (Stasinska & Izotov 2001; Peimbert et al. 2001b).

3.2 The hot-spot scheme

The overall energy balance in a nebula is described by the following equation:

 

H = G (3)

where H and G are the heating and cooling rates respectively. In a nebula in ionization equilibrium, with no extra-heating sources, the heating is provided by photoionization, and the cooling takes place mainly through collisionally excited line emission and free-free radiation. A local equilibrium temperature $T_{\rm eq}$ is implicitly defined by Eq. (3), such that the heating terms counterbalance the cooling terms. This generic model is the reference $t^2_{\rm extra}=0$ model.

In the hot-spot model, a particular temperature profile of a hypothetic nebula is postulated, consisting of a collection of randomly-generated hot spots arising above the $T_{\rm eq}$ floor. Following the definitions of Eqs. (1) and (2), the $t^2_{\rm extra}$ and $T_{\rm0}$ values can be computed, giving $t^2_{\rm extra}>0$ and $T_{\rm0}>T_{\rm eq}$. The recombination rates of this hypothetic model depend on the assumed temperature structure, being in general different from those of the reference $t^2_{\rm extra}=0$ model. It is possible to account for such variation by introducing a new "global'' temperature, $T_{\rm rec}$, such that the intensity of a recombination line is given by:

$$I_{\rm rec} \propto T_{\rm rec}^\alpha,$$ (4)

where $\alpha$ is a representative average of the H I recombination exponent $\alpha(T_{\rm e})$ in the appropriate temperature range.

Analogously, the intensity of a collisionally excited line in the temperature-fluctuation regime can be calculated by means of a "global'' collisional temperature $T_{\rm col}$, such that the collisional excitation and de-excitation rates are proportional to $T_{\rm col}^\beta\, \exp\,[-\Delta E_{ij}/kT_{\rm col}]$and $T_{\rm col}^\beta$ respectively, with the $\beta$ and $\Delta E_{ij}$values appropriate for each considered line.

Through the definition of these corrected temperatures, the new line emissivities can be computed in a straightforward way. In the hot-spot model, the relationship between $t^2_{\rm extra}$ and the derived temperatures is put in analytical form and generalized, so that the effects of fluctuations on the output quantities can be easily computed. For each postulated $t^2_{\rm extra}$ value, a plain photoionization model is calculated, in which the line emissivities are corrected for the temperature-fluctuation effects, and the global energy balance is consequently modified. This energetic change can be expressed by means of the quantity $\Gamma _{\rm heat}$, defined as follows:

$$\Gamma_{\rm heat}=\frac{L_{\rm fluc}-L_{\rm eq}}{L_{\rm eq}+Q_{\rm eq}},$$ (5)

where $L_{\rm fluc}$ and $L_{\rm eq}$ are the energies radiated by the nebula through line emission with and without temperature fluctuations respectively, and $Q_{\rm eq}$ is the energy radiated through processes other than line emission. Thus, ${L_{\rm eq}+Q_{\rm eq}}$ is the equilibrium cooling rate, and $L_{\rm fluc}-L_{\rm eq}$ gives the extra-luminosity radiated because of temperature fluctuations, so that $\Gamma _{\rm heat}$ gives a measure of the excess energy provided by the putative heating mechanism driving the fluctuations, and dispersed through collisional-line enhancement.

The relationship between $t^2_{\rm extra}$ and $\Gamma _{\rm heat}$depends on the properties of both the ionizing source and the gaseous nebula. In Fig. 1 we show the dependence of $\Gamma _{\rm heat}$on $t^2_{\rm extra}$ for the Luridiana et al. (1999) model of NGC 2363, which we computed using a modified version of the photoionization-shock code MAPPINGS IC (Ferruit et al. 1997). As a general rule, a tailored model should be computed for each considered case, and the calibrations obtained for simpler models should only be used as rough guidelines when estimating the energy implications of temperature fluctuations in a given object. We can illustrate this point comparing our calibration to the one by Binette & Luridiana (2000) at a representative temperature. We first introduce an equivalent effective temperature for the cluster, defined, according to the method by Mas-Hesse & Kunth (1991), as the temperature of an early-type star with the same Q(He⁰)/Q(H⁰) ratio as the synthetic stellar energy distribution (SED). Using the calibration by Panagia (1973), we found an equivalent effective temperature $T_{\rm eff} = 43\,700\; {\rm K}$. In Fig. 1 we reproduce the calibration computed by Binette & Luridiana (2000) for the case of a constant-density, Z=0.004 nebula ionized by an unblanketed LTE atmosphere of $T=45\,000$ K (Hummer & Mihalas 1970). Although the two curves show the same qualitative behavior, they rapidly diverge for increasing $t^2_{\rm extra}$ values.

Figure 1: $\Gamma _{\rm heat}$ as a function of $t^2_{\rm extra}$ for the model of NGC 2363 (solid line) and for a simpler model (dashed line, see text). The $t^2_{\rm extra}$ value inferred from observations of NGC 2363 is also shown.