Free Access
Volume 503, Number 2, August IV 2009
Page(s) L13 - L16
Section Letters
Published online 28 July 2009

Online Material

Appendix A: The FUV luminosity

For the modeling in Sect. 4, the FUV luminosity $L_{\rm FUV}$ is required. Assuming the protostar to emit a blackbody spectrum, this quantity depends on the bolometric luminosity $L_{\rm bol}$ and the effective temperature $T_{\rm eff}$. While $L_{\rm bol}$ of the embedded protostar can be determined relatively well from photometry in the IR and is assumed to be given in the following, only rough estimations of $T_{\rm eff}$ are available, since photons are absorbed or redistributed to longer wavelengths by the high dust and gas column density toward the source.

\end{figure} Figure A.1:

Luminosity in the FUV band depending on $T_{\rm eff}$ for $L_{\rm bol}=L_\odot $. The spectral classification is indicated by red circles.

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The Stefan-Boltzmann law requires

\begin{displaymath}L_{\rm bol}=4\pi R^2 \sigma T_{\rm eff}^4\ ,\ {\rm hence}\ R=...}{T_{\rm eff}} \right)^2 \sqrt{\frac{L_{\rm bol}}{L_\odot}}
\end{displaymath} (A.1)

with the source radius R, the Stefan-Boltzmann constant $\sigma $, and the solar temperature, radius, and luminosity ($T_\odot$, $R_\odot$, and $L_\odot$). The FUV band is limited by the Ly$\alpha$ edge (13.6 eV, $\lambda_{\rm min}=912$ Å) at short wavelengths and the average dust working function at long wavelengths (6 eV, $\lambda_{\rm max}=2067$ Å). For temperatures between $2.4\times 10^4$ K and $5.6\times 10^4$ K, the peak of the blackbody intensity $B_\lambda(T_{\rm eff})$ is within the FUV band (Wien's displacement law, $\lambda_{\rm max} [\textrm{\AA}] = 5.1 \times 10^7 / (T [K])$). The FUV luminosity $L_{\rm FUV}$ is obtained from integrating $B_\lambda(T_{\rm eff})$ between $\lambda_{\rm min}$ and  $\lambda_{\rm max}$ by
                         $\displaystyle L_{\rm FUV}$ = $\displaystyle 4 \pi R^2 \int_{\lambda_{\rm max}}^{\lambda_{\rm min}} \pi B_\lambda(T_{\rm eff}) ~ {\rm d}\lambda$ (A.2)
  = $\displaystyle L_{\rm bol} \times \frac{60\sigma}{\pi^3} \frac{R_\odot^2 T_\odot...
...odot} \times \int_{x_{\rm a}}^{x_{\rm b}} \frac{x^3}{{\rm e}^{x}-1} ~ {\rm d}x,$ (A.3)

with $x_{\rm a}=hc/kT_{\rm eff} \lambda_{\rm min}$ and $x_{\rm b}=hc/kT_{\rm eff} \lambda_{\rm max}$. For $x_{\rm a}=0$ and $x_{\rm b} \rightarrow \infty$, the integral in Eq. (A.3) is $\pi^4/15$, and Stefan-Boltzmanns law is recovered. In Fig. A.1, the FUV luminosity depending on $T_{\rm eff}$ is given for $L_{\rm bol}=L_\odot $. At a temperature of $2.7\times 10^4$ K, where $L_{\rm FUV}$ peaks, the ratio $L_{\rm FUV} / L_{\rm bol}$ is 0.55. Considering temperatures below this peak, the FUV luminosity is within a factor of 3 for $T_{\rm eff} > 1.2\times 10^4$ K and a factor of 10 for $T_{\rm eff} > 9\times 10^3$ K. We note that this is valid independently of $L_{\rm bol}$, since $L_{\rm FUV} \propto L_{\rm bol}$.

How does this temperature dependence affect the results of the models in Sect. 4? In the absence of any attenuation, $L_{\rm FUV} > 6 \times 10^{35}$ erg s-1 is required to provide the necessary FUV field of $3 \times 10^{3}$ ISRF at position B for heating. Assuming the bolometric luminosity to be correct, the temperature needs to be higher than 6800 K. For a temperature of $1.5 \times 10^4$ K instead of $3\times 10^4$ K, the FUV luminosity decreases by a factor of 2 and the required column density for attenuation (Sect. 4) reduces to $\tau=3.5$. We conclude that the modeling results are not affected by $T_{\rm eff}$ as long as the temperature exceeds about 104 K.

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