EDP Sciences
Free Access
Volume 503, Number 2, August IV 2009
Page(s) L13 - L16
Section Letters
DOI https://doi.org/10.1051/0004-6361/200912620
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.

Open with DEXTER

The Stefan-Boltzmann law requires

\begin{displaymath}L_{\rm bol}=4\pi R^2 \sigma T_{\rm eff}^4\ ,\ {\rm hence}\ R=...
...dot}{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.

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.