A&A 387, 295-300 (2002)
DOI: 10.1051/0004-6361:20020331

Electron temperature fluctuations in 30 Doradus

A. C. Krabbe1 - M. V. F. Copetti1,2


1 - Laboratório de Análise Numérica e Astrofísica, Departamento de Matemática, Universidade Federal de Santa Maria, 97119-900 Santa Maria, RS, Brazil
2 - Physics Department, University of Cincinnati, Cincinnati OH 45221-0011, USA

Received 13 December 2001 / Accepted 18 February 2002

Abstract
We present an observational study of the spatial variation of the electron temperature in the 30 Doradus Nebula. We used the [O III] $(\lambda 4959 + \lambda 5007)/\lambda 4363$ ratio to estimate the electron temperature at 135 positions in the nebula across three different directions. We analysed long-slit spectrophotometric data of high signal-to-noise in the range of 4100 to 5030 Å obtained with the Cassegrain spectrograph attached to the 1.60 m telescope of the Laboratório Nacional de Astrofísica, Brazil. No large-scale electron temperature gradient was detected in 30 Doradus. The electron temperature estimates obtained are fairly homogeneous with a mean value of $10~270 \pm 140~(3\sigma)$ K. The compatibility between the present estimates with optical and radio temperature determinations found in the literature for other positions or for the entire nebula corroborates this conclusion. Temperature fluctuations of small amplitude were observed with a variance relative to the mean of $t_{\rm s}^{2}= 0.0025$ or equivalently with a dispersion of only 5%. The areas with lower surface brightness seem to present slightly higher electron temperatures. This would indicate that the bright arcs of 30 Doradus, which correspond to the densest regions, would have lower electron temperatures than the most diffuse areas.

Key words: ISM: H II regions - ISM: individual objects: 30 Doradus Nebula


1 Introduction

Traditionally, the abundances in planetary nebulae and H II regions of chemical elements other than hydrogen and helium have been obtained from their collisionally excited emission lines. Since the emissivities of these lines are exponentially dependent on the electron temperature, $T_{\rm e}$, an accurate determination of this property is a key step in this process. However, considerable differences are found between the temperature estimates based on distinct methods. These discrepancies have been attributed to internal spatial temperature fluctuations in the nebulae (Peimbert 1967).

The recent developments in astronomical instruments have made possible the determination of the abundances of CNO elements based on their recombination lines. These lines are 103-104 times fainter than the strong forbidden lines. So, the use of this method is limited to a few objects with very high surface brightness. On the other hand, it has the advantage of being only weakly dependent on the electron temperature. Huge discrepancies between abundance determinations from collisionally excited and recombination lines have been reported in the literature, which cast doubt on the accuracy of the abundance determinations in gaseous nebulae. For example, Liu et al. (1995) determined abundances of C, N and O in the planetary nebula NGC 7009 by recombination lines and found that they are about 5 times higher than those derived from forbidden lines. These discrepancies have also been explained by the presence of fluctuations of electron temperature. However, the magnitudes of the temperature fluctuations needed are considerably higher than those predicted by standard photoionisation models and the physical mechanisms that could possibly explain the large temperature fluctuations presumed are unknown. So, the subject of spatial variations of electron temperature in H II regions and planetary nebulae has gained renewed interest.

In the present paper, we report the results of a study on the spatial variation of the electron temperature in the 30 Doradus Nebula based on point-to-point measurements of the [O III] $(\lambda 4959 + \lambda 5007)/\lambda 4363$ line ratio obtained from long-slit spectrophotometric observations of high signal-to-noise ratio.

2 Observations and data reduction

The observations were performed during the nights of September 12/13 1994 and April 14/15 2001 at the Laboratório Nacional de Astrofísica (LNA), Brazil, with the Cassegrain spectrograph attached to the 1.6 m telescope. An EEV CCD of 800 $\times$ 1024 pixels and a SITe CCD of 1024 $\times$ 1024 pixels were used in 1994 and 2001, respectively. A grid of 1200 grooves mm-1 was used. The spatial scale was 0.90$\arcsec$ pxl-1 for the EEV CCD and 1.0$\arcsec$ pxl-1 for the SITe CCD. The slit used had an entrance of $2.5\arcsec\times~360\arcsec$ on the plane of the sky. The spectra obtained covered the wavelength range of 4100 to 5030 Å with a dispersion of 0.87 Å pxl-1 and a resolution of 2.8 Å, measured as the full-width-at-half-maximum (FWHM) of the emission lines of comparison lamps. Exposures of dome flat-fields and several measurements of bias were made during each night. For flux calibration, the spectrophotometric standard stars LTT 7379, EG 274, HR 9087 and HR 1544 were observed during the 1994 run and HR 3454 and HR 4963 during the 2001 run. Spectra of a He-Ar-Ne lamp were taken before and after each object exposure for wavelength calibration.

Nine two-dimensional spectra of the 30 Doradus Nebula were obtained at three different slit positions. The exposure times were limited to 1200 s. to minimize the effects of cosmic rays and to avoid saturation of the brightest emission lines. Table 1 lists the number and time of the exposures,

 

 
Table 1: Journal of observations.

Date		 Exp. time (s) $ \rm\alpha$(2000) $\rm\delta(2000)$ PA

9.12.1994		 3$\times$1200 $\rm 5^{\rm {h}}38^{\rm {m}}42\fs 4$ $-69\degr 06\arcmin 01\arcsec$ $26\degr$
9.12.1994 2$\times$1200+600 $\rm 5^{\rm {h}}38^{\rm {m}}38\fs 2$ $-69\degr 05\arcmin 42\arcsec$ $58\degr$
4.14.2001 3$\times$1200 $\rm 5^{\rm {h}}38^{\rm {m}}41\fs 5$ $-69\degr 05\arcmin 14\arcsec$ $90\degr$


the equatorial coordinates of the slit center and its position angle, PA, measured from north through east.

The data reduction was made using the IRAF software. We have followed the standard procedures for bias correction, flat-fielding, cosmic ray cleaning and wavelength and flux calibrations. In order to increase the signal-to-noise ratio, a rebinning of 5 (EEV) and 4.5 (SITe) CCD rows along the spatial direction was performed, leading to a final spatial scale of 4.5$\arcsec$ pxl-1. The two-dimensional spectra were divided into series of one-dimensional spectra, each one corresponding to an aperture of $2.5\arcsec \times 4.5\arcsec$.

The line intensities were measured by integrating the flux over a linear local continuum between two given limits.These measurements were made with the splot routine of the IRAF package. All the line intensities were normalized to $\rm H\beta$. The error estimates were calculated by $\sigma^2 = \sigma_{{\rm cont}}^2 + \sigma_{{\rm line}}^2$, where $\sigma_{{\rm cont}}$ and $\sigma_{{\rm line}}$ are the continuum rms and the Poisson error of the line respectively. The effect of interstellar extinction was corrected by comparing the $\rm H\gamma/H\beta$ ratios measured in each aperture with the theoretical ones by Hummer & Storey (1987) for an electron temperature of 10 000 K and a density of 100 cm-3. The reddening law of Howarth (1983) appropriate for the Large Magellanic Cloud was used.

3 Determination of the electron temperature

The electron temperatures were calculated from the [O III] $(\lambda 4959 + \lambda 5007)/\lambda 4363$ intensity ratios by resolving numerically the equations of equilibrium for the five-level atom using the temden routine of the nebular package of IRAF. The values of energy levels, transition probabilities and collision strengths used were from Bowen (1960), Wiese et al. (1996) and Lennon & Burke (1994) respectively. In this method there is a dependence of electron temperature $T_{\rm e}$ on the electron density $N_{\rm e}$ assumed. Notwithstanding, this dependence is very weak and errors in $T_{\rm e}$ due to variations of the $N_{\rm e}$ are practically negligible. In our study we adopted an electron density of 300 cm-3as a representative value. Electron densities in the range of 75 to 800 cm-3have been measured in 30 Doradus (Feast 1961; Boeshaar et al. 1980). For densities in this range the errors in the electron temperature estimates would be below $0.2\%$.

4 Results and discussions

In Fig. 1 a sample of spectra from areas with different surface brightness is shown.

  \begin{figure} \par\includegraphics*[angle=-90,width=17cm,clip]{ms2197f1.eps}\end{figure} Figure 1: A sample of spectra in the range of 4275 to 4460 Å from areas with different surface brightnesses along PA = 90$\degr$. The corresponding positions are marked in Fig. 2. To emphasize the variation of the [O III] $(\lambda 4959 + \lambda 5007)/\lambda 4363$ ratio, the flux scale was normalized to the peak of [O III] $\lambda 5007$.
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The spatial profiles of the H$\beta $ flux, the [O III] $(\lambda 4959 + \lambda 5007)/\lambda 4363$ ratio and the electron temperature are shown in Figs. 4-2.
  \begin{figure} \par\includegraphics*[width=8.8cm,clip]{ms2197f2.eps}\end{figure} Figure 2: Spatial profiles of the H$\beta $ flux (in units of 10-13 ergs cm-2 s-1), the [O III] $(\lambda 4959 + \lambda 5007)/\lambda 4363$ ratio, R, and the electron temperature along PA = 90$\degr$. The labels indicate the positions corresponding to the spectra shown in Fig. 1.
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  \begin{figure} \par\includegraphics*[width=8.8cm,clip]{ms2197f3.eps}\end{figure} Figure 3: Same as Fig. 2, but for PA = 58$\degr$.
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  \begin{figure} \par\includegraphics*[width=8.8cm,clip]{ms2197f4.eps}\end{figure} Figure 4: Same as Fig. 2, but for PA = 26$\degr$.
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The position along the slit is positive to the west (PA = 90$\degr$) or northwest (PA = 26$\degr$ and 58$\degr$) of the reference points whose equatorial coordinates are shown in Table 1. Table 2 presents some statistics of the data,
 

 
Table 2: [O III] ratio and electron temperature statistics.

[O III] $(\lambda 4959 + \lambda 5007)/\lambda 4363$   $T_{\rm e}\ ({\rm K})$

Position angle ($^{\circ}$)   Position angle ($^{\circ}$)
  26 58 90 26+58+90   26 58 90 26+58+90

number of data		 37 39 59 135   37 39 59 135
minimum 97 $\pm$ 14 115 $\pm$ 20 123 $\pm$ 9 97 $\pm$ 14   9766 -131+142 9854 -156+171 9707 -97+103 9707 -97+103
first quartile 125 141 190 155   10275 10195 9930 10003
median 156 173 203 190   10977 10598 10094 10296
third quartile 191 196 214 209   11838 11349 10291 10998
maximum 226 $\pm$ 11 219 $\pm$ 12 231 $\pm$ 8 231 $\pm$ 8   13011 -666+869 12207 -670+906 11918 -321+368 13011 -666+869
weighted mean 179 183 208 195   10598 10497 10015 10267
standard deviation 33.4 28.3 12.1 26.5   728 605 196 543


including the number of different apertures, the median, the first and third quartiles (the limits between which 50% of the values lie), the minimum and maximum, and the mean, standard deviation and mean error weighted by the flux in H$\beta $.

The electron temperature estimates obtained are relatively homogeneous with fluctuations of very small amplitude along the three directions observed. For the entire set of 135 apertures observed, we found a weighted mean electron temperature of $10~270 \pm 140~(3\sigma)$ K. The temperature estimates present a small dispersion around the mean value of only 6%, measured as the weighted standard deviation. The individual mean temperatures for each of the three different slit positions are in agreement with the general average within an error of 3%.

No large-scale electron temperature gradient was found in 30 Doradus. Of course local observations, which are in fact integrations along the line of sight, tend to smooth out small spatial scale fluctuations of any line ratio. Nevertheless, point-to-point measurements are still able to detect global internal gradients. So, any significative large-scale systematic variation of the electron temperature should be revealed by our high signal-to-noise observations, as was the case of the studies of the planetary nebulae NGC 6720 (Garnett & Dinerstein 2001) and NGC 4361 (Liu 1998) and of the Orion Nebula (Walter et al. 1992).

As can be seen in Figs. 2-4, the areas with lower surface brightness in H$\beta $ tend to be associated with lower values for the [O III] $(\lambda 4959 + \lambda 5007)/\lambda 4363$ ratio and consequently with higher electron temperatures. If this apparent pattern is an artifact of the line flux estimation it will be certainly due to errors in the intensities of the [O III] $\lambda 4363$ line since the other two lines are very strong and easily measured. In fact, Rola & Pelat (1994) have demonstrated that weak emission lines tend to be overestimated. To investigate this possibility we have recalculated the line intensities of [O III] $\lambda 4363$ and H$\gamma$ with methods other than the initial one. Using the single line profile option of splot we have performed integrations by Gaussian fitting of each line individually with the central positions and line widths as free parameters and with an eye estimation of the continuum. With the deblending command of splot we have simultaneously fitted a linear function to the continuum in the range of 4275 to 4460 Å and Gaussian profiles with a single line width and free central wavelengths to the lines H$\gamma$, [O III] $\lambda 4363$ and the HeI $\lambda 4388$. We have calculated the intensities for each spectrum individually and for their sums. We have also co-added each three contiguous apertures in order to increase the signal-to-noise. All these tests have confirmed the trend of lower temperatures at brighter spots, indicating that in the bright arcs of 30 Doradus, which correspond to the densest areas, we find electron temperatures lower than in the most diffuse regions.

Interestingly, a similar anti-correlation between [O III] electron temperature and density can be seen in the data obtained for the Orion Nebula (Walter et al. 1992) and for the planetary nebula NGC 6720 (Garnett & Dinerstein 2001; Guerrero et al. 1997), the Orion Nebula presenting an outwards radial increase in temperature and decrease in density and NGC 6720 showing the opposite behaviour. These facts indicate that the density structure may play an important role in the production of the temperature fluctuations.

4.1 Comparison with other authors

We have recalculated the electron temperatures from measurements of the [O III] ratio found in the literature (Faulkner & Aller 1965; Peimbert & Torres-Peimbert 1974; Dufour et al. 1982; Mathis et al. 1985; Rosa & Mathis 1987) for 37 areas in 30 Doradus using the same atomic parameters and electron density adopted for our data. These estimates, with a mean value of 10 580 K and a standard deviation of 880 K, are in good agreement with those obtained from our own observations even for regions not in common. In particular, the data from Rosa & Mathis (1987) obtained at 10 positions in the outer regions of 30 Doradus are entirely compatible with those from the core of the nebula which is another indication that the electron temperature in the O III ionization zone does not present striking large-scale spatial variations.

We have also compared our results with estimations of electron temperature based line-to-continuum ratios of radio recombination lines. Peck et al. (1997) have made interferometric observations of 30 Doradus in the H90$\alpha$, H92$\alpha$ and H109$\alpha$ lines smoothed to a resolution of 15$\arcsec$$\times$15$\arcsec$. No significant temperature variations were found across the nebula. Although large differences between the optical and radio temperature estimates have been reported in the literature, the average global electron temperature of $9200 \pm 1000$ obtained from single dish observations of 30 Doradus (Shaver et al. 1983; Mezger et al. 1970; McGee et al. 1974) and from the data of Peck et al. (1997) integrated over the entire nebula is only 10% lower than our mean value. So, we verify than the mean optical and radio temperatures are consistent with each other within the error estimates. This is another indication that the fluctuations of electron temperature on a large spatial scale are not strong in 30 Doradus since these two methods provide mean values of electron temperature with different weights for different regions, the collisionally excited lines being heavily weighted toward the hottest locations.

4.2 Magnitude of the electron temperature fluctuations

Following Peimbert (1967), the magnitude of the temperature fluctuations are usually quantified by the parameter t2 defined as

 \begin{displaymath}t^2 = \frac{ \int{(T_{\rm e}- T_0)^2 N_{\rm i} N_{\rm e} {\rm d}V }} { T_0^2 \int{N_{\rm i} N_{\rm e} {\rm d}V }}, \end{displaymath} (1)

with

\begin{displaymath}T_0 = \frac{ \int{T_0 N_{\rm i} N_{\rm e} {\rm d}V }} { \int{N_{\rm i} N_{\rm e} {\rm d}V }}, \end{displaymath} (2)

where $N_{\rm i}$ is the density of the ion used to measure the temperature and the integrations are calculated over the observed volume V of the nebula. Essentially, T0 and t2 are the mean and the relative variance of the temperature distribution weighted by the square of the local density. Although t2 cannot be directly measured, it can be estimated through the effects of the temperature fluctuations on the values of temperature or abundance obtained from different methods. By this method, values of $t^2 \approx 0.02{-}0.10$ have been attributed to planetary nebulae (Dinerstein et al. 1985; Liu & Danziger 1993; Peimbert et al. 1995), galactic (Esteban et al. 1998, 1999a,b) and extragalactic (Luridiana et al. 1999; Gonzalez-Delgado et al. 1994) H II regions. However, these high values for t2 are not reproduced by the standard photoionization models, which predict $t^2 \leq 0.02$ (Gruenwald & Viegas 1995; Kingdon & Ferland 1995).

An independent estimation of t2 can be obtained through point-to-point determinations of the electron temperature across the nebula. We may rewrite Eq. (1) and find a lower limit for t2 as

 \begin{displaymath}t^2=\frac{ \int{\langle(T_{\rm e}-T_0)^2\rangle_{\bf\Omega} E... ...{\rm d}\Omega}} { T_0^2 \int{ E_{\bf\Omega} {\rm d}\Omega }}, \end{displaymath} (3)

where: $\langle . \rangle_{\bf\Omega}$ stands for the mean value weighted by the square density along the direction $\bf\Omega$; $E_{\bf\Omega} = \int{N_{\rm i} N_{\rm e} {\rm d}l}$ is the emission measure; d$\Omega$ and dl are respectively the elements of solid angle and distance along the line of sight. A discrete approximation for the right hand side of expression 3 is given by (see Liu 1998)

 \begin{displaymath}t_{\rm s}^2=\frac{ \sum_{i}{(T_{\rm e}^i-T_0)^2 F_i({\rm H}\beta)}} { T_0^2 \sum_{i}{F_i({\rm H}\beta)}}, \end{displaymath} (4)

where $T_{\rm e}^i$ and $F_i({\rm H}\beta)$ are the electron temperature and the H$\beta $ flux obtained for the aperture i. Since part of the observed variance, $t_{\rm s}^2{\rm (obs)}$, is due exclusively to errors in the measurements, the final estimation of $t_{\rm s}^2$ should be corrected by $t_{\rm s}^2 = t_{\rm s}^2{\rm (obs)} - t^{2}_{\rm er},~ $ $t^{2}_{\rm er}$ being the relative mean quadratic error of the electron temperature measurements.

As the temperature measured at any point is a mean value along the line of sight, any small-scale temperature fluctuation would be smoothed out by the present observations. For this reason, it is clear that $t_{\rm s}^2$ can only give a lower limit to t2. So, 2D mapping of the electron temperature in the nebula can possibly confirm the existence of large temperature fluctuations, but never disprove it. Theoretically, such small-scale temperature fluctuations are not expected and the temperature structure predicted follows a large-scale radial gradient. If indeed most of the temperature variation is along the radial direction, $t_{\rm s}^2$ will be a bias but still useful estimator of t2. Liu (1998) has expected the diference between them to be smaller than a factor of 2. From the numerical simulations of point-to-point observations of Gruenwald & Viegas (1995), we have found that $0.1 \leq t_{\rm s}^2/t^2 \leq 1$, with a median value of 0.7.

From our measurements of [O III] electron temperature in 135 areas of the 30 Doradus Nebula we have obtained a value of $t_{\rm s}^{2}= 0.0025$ for the variance of the projected temperature distribution, which corresponds to a dispersion amplitude of only 5%. A similar value of $t_{\rm s}^2 = 0.002$ was found by Liu (1998) for the planetary nebula NGC 4361.

The small difference between the mean [O III] and radio electron temperatures also indicates a low amplitude for the temperature fluctuations in this nebula. From the expressions by Peimbert (1967) relating the [O III] and radio electron temperature estimates to T0 and t2, we derive t2 = 0.02, assuming the same values for T0and t2 in the O++ and H+ zones.

The low amplitude for the large-scale temperature fluctuations found in the 30 Doradus Nebula are consistent with the results of photoionisation models (Kingdon & Ferland 1995; Gruenwald & Viegas 1995) and are below the levels needed to explain the discrepancies between abundances derived from forbidden and recombination lines.

5 Conclusions

We present an observational study on the variation of the electron temperature in the 30 Doradus nebula based on long-slit spectrophotometry of high signal-to-noise ratio in the range of 4100 to 5030 Å. Electron temperatures were derived from the [O III] $(\lambda 4959 + \lambda 5007)/\lambda 4363$ ratios measured at 135 locations along three different directions. The main results are the following:

1.
The electron temperature estimates obtained are fairly homogeneous. No large-scale electron temperature gradient has been detected in 30 Doradus. The compatibility between the present estimates with optical and radio temperature determinations found in the literature for other positions or for the entire nebula corroborates this conclusion.

2.
An emission-weighted mean electron temperature of $10~270 \pm 140~(3\sigma)$ K was found.

3.
Temperature fluctuations of small amplitude have been observed. The temperature distribution across the nebula presents a variance relative to the mean of $t_{\rm s}^{2}= 0.0025$ or equivalently a dispersion of only 5%. Temperature fluctuations of this magnitude are compatible with the prediction of photoionization models but are too small to be invoked as the cause of the discrepancy between the abundances derived from forbidden and recombination lines. However, we must emphasize that the existence of small-scale temperature fluctuations cannot be ruled out by the present observations.

4.
The areas with lower surface brightness tend to present slightly higher electron temperatures. This would indicate that the bright arcs of 30 Doradus, which correspond to the densest regions, would have smaller electron temperatures than the most diffuse areas.

Acknowledgements
This work was partially supported by the Brazilian institutions CAPES, CNPQ and FAPERGS. We thank the anonymous referee for helpful comments and suggestions.

References

 

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