The HI 21 cm Line

Atomic hydrogen has been measured towards Sgr B2 via its 21 cm line by Garwood & Dickey (1989). The ground state of HI is split by the interaction of the proton and electron spins and this causes a hyperfine transition with a wavelength of 21.1061 cm (e.g. Kulkarni & Heiles, 1988). Garwood & Dickey (1989) observed the 21 cm line in absorption close to the Sgr B2 (M) source using the VLA. The final spectral resolution they obtained after smoothing was 5.16 km s$^{-1}$. They fitted the observed spectrum with Gaussian components. However, examination of the values quoted in their paper indicates that one of the velocity components was given incorrectly. This was the 0 km s$^{-1}$ component which was given a velocity of $+11.39$ km s$^{-1}$ which does not reproduce their data. Changing this component to $+1.1$ km s$^{-1}$ significantly improves the fit with the data, indicating that the decimal point was incorrectly placed in the paper. In order to get a better determination of the Gaussian parameters, the spectrum from Garwood & Dickey (1989) was scanned and re-fitted by C. Vastel. These fit parameters are given in Table 5.3. The main difference between the new fit and the original published values is that there is a better definition of the velocity components in the range $-50$ to $-100$ km s$^{-1}$. The two fits are compared to the data in Figure 5.8.

Figure 5.8: The approximate shape of the HI absorption spectrum of Garwood & Dickey (1989) is shown by the solid black line. The Gaussian fit to the original spectrum is shown in blue (with the $+11$ km s$^{-1}$ component replaced as described in the text). A fit to the scanned spectrum by C. Vastel is shown in red.

The HI column densities can be calculated for each component in the line of sight from the fitted optical depths and line widths. The optical depth across an absorption line depends on the column density in the lower state, $N_{j}$, the absorption cross section, $s(\nu)$, and the normalised line shape function, $\phi(\Delta\nu)$,

$$\tau(\nu)=N_{j} s(\nu) \phi(\Delta\nu)$$ (E.7)

where the absorption cross section per particle is given by (Spitzer, 1978),

$$s(\nu)=\frac{g_i}{g_j} \frac{c^{2} A_{ij}}{8 \pi \nu_{ji}^{2}} [1-\exp{({-h\nu_{ji}/kT})}]$$ (E.8)

where $\nu_{ji}$ is the frequency of the transition, $A_{ij}$ is the Einstein coefficient for spontaneous emission and $g_i$ and $g_j$ are the statistical weights for the upper ($i$) and lower ($j$) levels. The factor $[1-\exp{({-h\nu_{ji}/kT})}]$ is a correction for stimulated emission and reflects the fact that the total energy absorbed depends on the difference between stimulated absorption and stimulated emission.

The line shape function can be expressed in terms of the relative velocity from line centre (defined by $v=c\Delta\nu/\nu_{ji}$),

$$P(v)=\frac{\nu_{ji} \phi(\Delta\nu)}{c}$$ (E.9)

and if the frequency of the transition is converted to wavelength, the final optical depth across the line is given by,

$$\tau(v)=N_{j} \frac{g_i}{g_j} \frac{\lambda_{ji}^{3} A_{ij}}{8 \pi} P(v) [1-\exp{({-hc/\lambda_{ji}kT})}]$$ (E.10)

Integrating across the line and assuming a Maxwellian velocity distribution gives the optical depth at line centre (Spitzer, 1978),

$$\tau_{0}=N_{j} \frac{g_i}{g_j} \frac{\lambda_{ji}^{3} A_{ij}... ...2\sqrt{\ln2}} [1-\exp{({-hc/\lambda_{ji}kT})}] \times 10^{-17}$$ (E.11)

where $\Delta{v}$ is the line width in km s$^{-1}$, the transition wavelength is in $\mu$m and the column density is in cm$^{-2}$.

The relative populations of each level are determined by the Boltzmann relation,

$$\frac{N_i}{N_j}=\frac{g_i}{g_j}\exp{\left(\frac{-h c}{\lambda_{ij} k T}\right)}$$ (E.12)

where $g_i$ is the statistical weight for level $i$. The temperature that determines the populations of the levels for HI is the spin temperature, $T_{\mathrm{s}}$, and this is determined by the excitation mechanism. In cold clouds where collisional excitation dominates, $T_{\mathrm{s}}$ should be equal to the gas kinetic temperature, $T_{\mathrm{K}}$ (Kulkarni & Heiles, 1988). For the HI 21 cm line $hc/\lambda_{ij}k\sim0.07$ K, which is always less than the spin temperature. In this case the correction for stimulated emission can be approximated to by $hc/\lambda_{ij}T_{\mathrm{s}}$ giving,

$$N(\mathrm{HI})=\frac{32 \pi k}{c h \lambda_{ij}^2 A_{ij} g_{... ...j}} \frac{\sqrt{\pi}}{2\sqrt{\ln{2}}} T_{s} \Delta{v} \tau_{0}$$ (E.13)

where $A_{ij}=2.869\times10^{-15}$ s$^{-1}$ (e.g. Kulkarni & Heiles, 1988), $g_{i}=3$, $g_{j}=1$ and $\lambda_{ij}=21.1061$ cm (e.g. Kulkarni & Heiles, 1988). The spin temperature in each absorbing component along the line of sight to Sgr B2 has been derived by from observations of HI 21 cm absorption by Cohen (1977). The values obtained were in the range 40-160 K with a mean $\approx100$ K. This range agrees with values found in the local spiral arm of $\sim$100-140 K (Normandeau, 1999; Wendker & Wrigge, 1996). However, the Sgr B2 observations were carried out with a large beam size of $13\hbox{$^\prime$}$ and so included absorbing clouds outside of the narrow VLA beam used by Garwood & Dickey (1989). Therefore the individual spin temperatures may not be directly applicable to the components seen with the VLA. A value of 150 K was adopted for all the components along the line of sight and this probably represents an upper limit. Table 5.3 shows the calculated HI column densities. Decreasing the spin temperature for some components would decrease their derived column densities.

Table 5.35.3: Gaussian fit parameters (central velocity, FWHM and optical depth) for the HI absorption spectrum of Garwood & Dickey (1989). This fit was carried out by C. Vastel (see Vastel et al., 2002). Column 4 shows the calculated column density for atomic hydrogen using a spin temperature of 150 K. Columns 5 and 6 show the fitted optical depths and column densities for CII (from Vastel et al., 2002). The last column shows the column density of OI derived from the CII column density using the O/C ratio in the local ISM.

Velocity	$\Delta{v}$(HI)	$\tau_{0}\mathrm{(HI)}$	$N(\mathrm{HI})$	$\tau_{0}\mathrm{(CII)}$	$N(\mathrm{CII})$	$N(\mathrm{OI})$
(km s$^{-1}$)	(km s$^{-1}$)		(10$^{20}$ cm$^{-2}$)		(10$^{17}$ cm$^{-2}$)	(10$^{17}$ cm$^{-2}$)
$-108.0$	7.0	0.14	2.8	0.1	1.3	2.9
$-92.0$	14.0	0.13	5.3	0.08	1.5	3.4
$-77.0$	14.0	0.19	7.7	0.1	1.8	4.1
$-60.5$	7.0	0.30	6.1	0.1	0.9	2.0
$-51.9$	8.0	0.55	12.8	0.1	1.0	2.3
$-44.0$	8.0	1.10	25.6	0.6	6.2	14.0
$-21.5$	15.0	0.50	21.8	0.3	5.9	13.3
$-3.5$	11.5	1.40	46.8	0.5	7.5	16.9
$+5.5$	12.0	1.20	41.9	0.7	10.9	24.5
$+15.0$	8.5	1.25	30.9
$+30.0$	11.0	0.29	9.3
$+52.8$	17.0	1.10	54.4
$+67.0$	15.0	2.00	87.2