Recent studies on high redshift galaxies and star formation obscured by dust have shown the importance of the reddening in the high-z universe. Therefore, another effect that must be taken into account is dust extinction, produced inside the galaxies themselves.

The five reddening laws presently implemented in hyperz are:

  1. Allen (1976) for the Milky Way (MW);
  2. Seaton (1979) fit by Fitzpatrick (1986) for the MW;
  3. Fitzpatrick (1986) for the Large Magellanic Cloud (LMC);
  4. Prévot et al. (1984) and Bouchet et al. (1985) for the Small Magellanic Cloud (SMC);
  5. Calzetti et al. (2000) for starburst galaxies.
The different laws normalized to $k(B)-k(V)=1$, are presented in Figure 4. The input value is $A_V$, corresponding to a dust-screen model, with
\begin{displaymath}f_{\rm obs}(\lambda)=f_{\rm int}(\lambda) 10^{-0.4 A_{\lambda}}\:,\end{displaymath}

where $f_{\rm obs}$ and $f_{\rm int}$ are the observed and the intrinsic fluxes, respectively. The extinction at a wavelength $\lambda$ is related to the colour excess $E(B-V)$ and to the reddening curve $k(\lambda)$ by

\begin{displaymath}A_{\lambda} = k(\lambda) E(B-V) = {k(\lambda) A_V \over R_V} \:,\end{displaymath}

with $R_V=3.1$ except for the Small Magellanic Cloud ($R_V=2.72\pm0.21$) and Calzetti's law ($R_V=4.05\pm 0.80$).

\begin{figure}\centerline {\psfig{,width=0.89\textwidth,angle=270}}\end{figure}
Figure:Extinction curves $k(\lambda)$ for the different reddening laws implemented in hyperz.
In detail, Allen's law for the MW is computed from the values of $k(\lambda)/R_V$ tabulated in Table 3. At shorter and longer wavelengths we extrapolate the slope from the last available points. For Seaton's law, we adopt the fit presented by Fitzpatrick (1986), that follows the same relation found by the same author for the LMC:
$\displaystyle \frac{E(\lambda -V)}{E(B-V)}$ $\textstyle =$ $\displaystyle C_1 + C_2 \lambda^{-1} +\frac{C_3}{\left[\lambda^{-1}-\frac{(\lambda_0^{-1})^2}{\lambda^{-1}}\right] + \gamma^2}$  
  $\textstyle +$ $\displaystyle C_4\left[0.539(\lambda^{-1}-5.9)^2+ 0.0564 (\lambda^{-1}-5.9)^3 \right]$  

with $\lambda$ in $\mu m$. The two laws, for the MW and the LMC, are different in the coefficient values:

\begin{displaymath}\begin{array}{l\vert cccccc}&\lambda_0^{-1} [\mu {\rm m}^{-......MC} & 4.608 & 0.994 & -0.69 & 0.89 & 2.55 & 0.50\end{array}\end{displaymath}

whereas $C_4 = 0$ for $\lambda^{-1} < 5.9\,\mu {\rm m}^{-1}$ in any case. Because the validity of these laws is limited to $\lambda^{-1}<8.3\,\mu{\rm m}^{-1}$, i.e. $\lambda>1200$Å, we extrapolated the slope at shorter wavelengths from the 1100 - 1200Å range. At wavelengths longer than 3650 and 3330Å respectively for the two quoted laws, we adopt the points from Allen, to avoid $k(\lambda)$ being too flat in the red and near-IR regions.


Table:Left: Values of $k(\lambda)/R_V$ for the MW reddening law from Allen (1976). Right: Values of $E(\lambda-V)/E(B-V)$ for the SMC reddening law from Prévot et al. (1984) and Bouchet et al. (1985).
$\lambda$[Å] $k(\lambda)/R_V$ $\lambda$[Å] $k(\lambda)/R_V$
1000. 4.20 3650. 1.58
1110. 3.70 4000. 1.45
1250. 3.30 4400. 1.32
1430. 3.00 5000. 1.13
1670. 2.70 5530. 1.00
2000. 2.80 6700. 0.74
2220. 2.90 9000. 0.46
2500. 2.30 10000. 0.38
2850. 1.97 20000. 0.11
3330. 1.69 100000. 0.00
$\lambda$[Å] $\frac{E(\lambda-V)}{E(B-V)}$ $\lambda$[Å] $\frac{E(\lambda-V)}{E(B-V)}$ $\lambda$[Å] $\frac{E(\lambda-V)}{E(B-V)}$
1275. 13.54 1810. 7.17 2778. 3.15
1330. 12.52 1860. 6.90 2890. 3.00
1385. 11.51 1910. 6.76 2995. 2.65
1435. 10.80 2000. 6.38 3105. 2.29
1490. 9.84 2115. 5.85 3704. 1.81
1545. 9.28 2220. 5.30 4255. 1.00
1595. 9.06 2335. 4.53 5291. 0.00
1647. 8.49 2445. 4.24 12500. -2.02
1700. 8.01 2550. 3.91 16500. -2.36
1755. 7.71 2665. 3.49 22000. -2.47
The points used to compute the law for the SMC are taken from Prévot et al. (1984) and Bouchet et al. (1985) and are listed in Table 3. The peculiarity of this law is the lack of the graphite 2175Å bump. The non detection of this bump is probably related to the large underabundance of carbon in the SMC. As for Allen's law, we made an extrapolation from the slope computed from the available points.

The most used attenuation curve in high-redshift studies is the law derived by Calzetti et al. (2000). They derived a law as a purely empirical result from a sample of near starburst (SB) galaxies. Again, the most prominent feature of the MW law is absent: this characteristic suggests that starburst galaxies contain SMC-like dust grains, being the 2175Å bump an excellent probe of the type of dust in a galaxy. From Calzetti et al. (2000) we have:

\begin{displaymath}k(\lambda) = \left\{\begin{array}{ll}2.659\left(-2.156+\fra...... {\rm m} \le \lambda \le 2.20\,\mu {\rm m}\end{array}\right. \end{displaymath}

with $R_V=4.05$. Below the validity wavelength range, we obtain the slope of the reddening law by interpolating $k(\lambda)$ at 1100 and 1200Å. In a similar way, we compute the slope at $\lambda >22000$Å from the values at 21900 and 22000Å.

In particular, the application of this law is suggested for the central star-forming regions of galaxies, and then for high-z galaxies. For this reason, Calzetti's law is used to correct the value of the SFR at high-redshift, as seen in the previous chapter. It is evident from Figure 4 that the UV region of the spectrum will be more affected by dust. As a consequence, taking extinction into account is of paramount importance for galaxies at $z\mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfil$\displaystyle ..., when the rest frame UV emission is shifted to optical wavelengths.

Similarly, but in the opposite way, an extinction curve and a value of the colour excess can be used to deredden the magnitudes of objects seen through the dust of the Milky Way. see next section with parameters description

micol bolzonella