MEASURING TURBULENCE IN THE INTERSTELLAR MEDIUM BY COMPARING N(H i; Lyα) AND N(H i; 21 cm)

A fundamental aspect of understanding the structure of the interstellar medium (ISM) involves the origin of the small-scale structure that is observed. A number of formulations provide a framework for analyzing such structure and explaining its creation and distribution. These formulations include those of Houlahan & Scalo (1992), who developed a hierarchical tree structure, describing clouds as a series of partitions. Falgarone et al. (1991) explored fractal structure and showed that this description is applicable to molecular clouds. Vogelaar & Wakker (1994) used fractal structure in an attempt to describe the structure of high-velocity clouds (HVCs). Fractal structure may arise naturally from the density statistics of interstellar turbulence. More recently, Lazarian & Pogosyan (2000, 2004, 2006, 2008) developed techniques for comparing observations of velocity and density structure in the ISM with statistics derived from theories of turbulence. Kowal et al. (2007) used these ideas to construct three-dimensional (3D) magnetohydrodynamic (MHD) models of the ISM that predict the spectrum of density fluctuations, while Burkhart et al. (2009) studied how statistical measures can be used to connect these models to observations. In this paper, we look at the column density distribution of neutral hydrogen, trying to compare measurements made at different angular resolutions, which allows us to apply one of the measures discussed by Burkhart et al. (2009).

Galactic H i column densities can be measured using 21 cm radio observations or by using Lyα absorption-line spectra. 21 cm observations are made with single-dish or interferometer radio telescopes. Single-dish telescopes have a large beam size, typically 36' for all-sky surveys and 9'–21' for more targeted observations. Interferometers produce beams of less than 2', although for Galactic gas it is necessary to combine this with single-dish observations to derive an accurate total column density. With Lyα observations of ultraviolet bright background targets, however, one measures N(H i) in a very small (subarcsecond) area centered on the background target. If the background target is a galactic disk star, the Lyα absorption is caused only by the H i in front of the star, whereas the 21 cm observations measure H i both in front of and behind the star. When observing active galactic nuclei (AGNs), both Lyα and 21 cm observations measure all H i in the line of sight. Some halo stars may also be sufficiently high above the galactic plane to lie above most of the H i. Thus, valid comparisons between N(H i; Lyα) and N(H i; 21 cm) can only be made in the directions of AGNs or distant halo stars.

Precise measurements of N(H i) are also necessary to derive abundances of heavy elements in the ISM. Usually, 21 cm data are used to derive the H i reference column density, which is necessary whenever absorption lines show different components. However, the metal-line absorption is produced only by the ions in the small beam toward the AGN, while the 21 cm data average N(H i) across the radio telescope beam. Thus, there is a question as to whether it is correct to use the 21 cm data to derive N(H i). Do Lyα measurements and 21 cm measurements in fact give consistent results? If they do not, what is the reason for this difference?

Hobbs et al. (1982) were the first to directly compare N(H i) measured using Lyα absorption lines (from Copernicus data) to N(H i; 21 cm), measured using the 140-ft Green Bank Telescope (GBT; 21' beam). Although they did not present errors on their measurements, the ratios they found varied from about 0.9 to 3.9, with the outliers for the stars closest to the plane. For the three stars above most of the Galactic H i layer (z > 2 kpc), the ratios were given as 0.9, 0.9, and 1.1.

This was followed by a study by Lockman et al. (1986a), who used International Ultraviolet Explorer (IUE) spectra to measure N(H i; Lyα) toward 45 stars, by fitting the profiles of the damping wings. The resulting errors in N(H i) were estimated to be about 0.1 dex (25%). These values were again compared to 140-ft GBT 21 cm data, which were corrected for stray radiation. They also used a spin temperature of 75 K to correct the column densities for optical depth effects, though in most cases this correction is small (a few percent). For the stars closest to the plane (z < 1 kpc), the ratio N(H i; Lyα)/N(H i; 21 cm) tends to be <1, because not all of the H i in the sightline is seen in absorption. For the six stars at z > 1.5 kpc, the average ratio was about 1, to within the errors, although the typical error in each ratio was about 0.3.

Savage et al. (2000) revisited this comparison in the directions of 14 QSOs, using data from the G130H grating in the Faint Object Spectrograph (FOS) on the Hubble Space Telescope (HST). Unlike the Copernicus and IUE data, the FOS spectra had low resolution (230 km s⁻¹). Savage et al. (2000) compared these measurements to 21 cm data from the Green Bank (GB) 140-ft telescope. For 10 of the QSO spectra, it was possible to correct for geocoronal emission, and for these Savage et al. (2000) found that the ratio N(H i; Lyα)/N(H i; 21 cm) had values in the range 0.62–0.91 with errors of about 0.1.

Savage et al. (2000) suggested two possible origins for differences between the column densities measured from Lyα absorption and 21 cm emission. First, differences might arise from a combination of systematic and random errors in the Lyα and 21 cm observations. In the Lyα observations, systematic errors can be produced by uncertain continuum placement, geocoronal H i emission removal, the spectrograph background and scattered light correction, interfering QSO and intergalactic medium (IGM) absorption, and detector fixed pattern noise. For the FOS dataset of Savage et al. (2000), all of these effects were present. Using data at higher spectral resolution removes all of the trouble associated with geocoronal H i, IGM absorption, and fixed pattern noise, while it greatly reduces the other problems. In the 21 cm data, systematic errors can be created by the absolute calibration of the radio telescope, baseline fitting, and the stray-radiation correction. Alternatively, the differences in Lyα and 21 cm column densities could be caused by the structure of the ISM. If there are small bright spots embedded in a smoother background, the 21 cm data will include these, but a random sightline to an AGN has a high probability of missing the brighter spots. To study such effects requires a large sample of sightlines.

In this paper, we analyze 59 sightlines using higher resolution and higher signal-to-noise ratio (S/N) Lyα spectra than used by Hobbs et al. (1982), Lockman et al. (1986b), and Savage et al. (2000). We compared all these measurements to the column densities found in the Leiden–Argentina–Bonn (LAB) survey (Kalberla et al. 2005) as well as to the Lockman & Savage (1995) GB 140-ft data that are available in many directions. In addition, we obtained new 21 cm data with the GBT toward 35 AGNs. In Section 2, we describe the Lyα and 21 cm datasets that we used as well as the method used to derive N(H i; Lyα). During our analysis, we discovered the presence of a spurious, broad underlying component in the LAB and 140-ft spectra. We show this in Section 3. After removing this component, we find the results that are presented in Section 4 and discussed in Section 5.

2.1. HST Data

We used HST observations of AGNs that were observed at sufficient spectral resolution to (mostly) resolve the interstellar and intergalactic absorption lines. This includes observations using the G140M grating and E140M echelle in the Space Telescope Imaging Spectrograph (STIS). G140M spectra cover a wavelength interval about 55 Å wide at 30 km s⁻¹ resolution, while E140M spectra range from 1140 to 1710 Å with a resolution of 6.5 km s⁻¹. The calibrated HST data were retrieved from the Multimission Archive at STScI (MAST) server. We do not look at the many targets observed with the STIS−G140L grating, since the interstellar lines near 1200 and 1206 Å in the damping profile are not fully resolved in such low-resolution (300 km s⁻¹) data, nor are low-redshift intergalactic Lyα or high-redshift intergalactic metal lines. When unresolved, such lines change the apparent continuum, making it more difficult to obtain reliable results. We list the sample of 59 AGNs in Table 1. The locations of these targets on the sky are shown in Figure 1.

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Table 1. Observational Data

Object	Lon.	Lat.	Type	ObsID	Grating	λmin	λmax	Texp	GBT
	(°)	(°)				(Å)	(Å)	(ks)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
3C 249.1	130.39	38.55	QSO	O6E1/24-30	E140M	1140	1729	68.8	Y
3C 273.0	289.95	64.36	QSO	O5D3/01-02	E140M	1140	1709	18.7	Y
3C 351.0	90.08	36.38	QSO	O579/01-04	E140M	1140	1709	77.0	Y
H 1821+643	94.00	27.42	QSO	O5E7/03-05	E140M	1140	1709	50.9	Y
HE 0226−4110	253.94	−65.77	QSO	O6E1/07-11	E140M	1143	1729	43.8
HE 0340−2703	222.68	−52.12	QSO	O8EI/03	G140M	1194	1250	4.9
HE 1029−1401	259.33	36.52	QSO	O4EC/05	G140M	1194	1248	4.1
HE 1228+0131	291.26	63.66	QSO	O56A/01-02	E140M	1140	1729	27.2
HS 0624+6907	145.71	23.35	QSO	O6E1/12-16	E140M	1140	1729	62.0	Y
HS 1543+5921	92.40	46.36	QSO	O8MR/01	G140M	1194	1250	25.3
HS 1700+6416	94.40	36.16	QSO	O4SI/01-05	E140M	1140	1729	99.4
MCG +10−16−111	144.21	55.08	Sey	O5EW/02	G140M	1194	1249	19.5	Y
MRC 2251−178	46.20	−61.33	QSO	O4EC/03	G140M	1194	1249	6.0	Y
Mrk 110	165.01	44.36	Sey	O4N3/52	G140M	1194	1249	2.2	Y
Mrk 205	125.45	41.67	Sey	O62Q/03-05	E140M	1140	1729	62.1	Y
Mrk 279	115.04	46.86	Sey	O6JM/01	E140M	1140	1709	13.2
				O8K1/01-05	E140M	1140	1709	52.8
Mrk 335	108.76	−41.42	Sey	O8N5/04-05	E140M	1140	1729	15.2	Y
Mrk 478	59.24	65.03	Sey	O4EC/14	G140M	1194	1249	7.6	Y
Mrk 509	35.97	−29.86	Sey	O6AP/01	E140M	1140	1709	7.6	Y
Mrk 771	269.44	81.74	Sey	O4N3/05	G140M	1194	1249	2.0	Y
				O4EC/07	G140M	1194	1249	5.8
Mrk 876	98.27	40.38	Sey	O8NN/01-02	E140M	1140	1729	29.2	Y
Mrk 926	64.09	−58.76	Sey	O4EC/12	G140M	1194	1249	3.9
Mrk 1044	179.69	−60.48	Sey	O8K4/01	G140M	1194	1250	2.4	Y
Mrk 1383	349.22	55.13	Sey	O8PG/01-02	E140M	1140	1729	19.2	Y
Mrk 1513	63.67	−29.07	Sey	O4EC/10	G140M	1194	1249	7.3	Y
NGC 985	180.84	−59.49	Sey	O4EC/11	G140M	1194	1249	3.7	Y
NGC 1705	261.08	−38.74	Sey	O58N/01-02	E140M	1140	1709	17.1
NGC 3516	133.24	42.40	Sey	O57B/02	E140M	1140	1729	5.5
NGC 3783	287.46	22.95	Sey	O57B/01	E140M	1140	1729	5.4
				O63M/01-17	E140M	1140	1729	87.0
				O63M/53	E140M	1140	1729	4.9
NGC 4051	148.88	70.09	Sey	O5F0/01	E140M	1140	1729	10.3
NGC 4151	155.08	75.06	Sey	O578/01	E140M	1140	1729	5.5
				O5KT/02,53,54	E140M	1140	1729	6.6
				O61L/01	E140M	1140	1729	9.4
				O6JB/01	E140M	1140	1729	7.6
NGC 5548	31.96	70.50	Sey	O4LL/01	E140M	1140	1729	4.8	Y
				O6JD/01-02	E140M	1140	1729	15.3
				O6KW/01-04	E140M	1140	1729	52.2
NGC 7469	83.10	−45.47	Sey	O6BN/01	E140M	1140	1729	13.0	Y
				O8N5/01-02	E140M	1140	1729	22.8
PG 0804+761	138.28	31.03	QSO	O4N3/01	G140M	1194	1249	2.4	Y
				O4EC/06	G140M	1194	1249	4.9	Y
PG 0953+414	179.79	51.71	QSO	O4X0/01-02	E140M	1140	1729	25.5	Y
PG 1001+291	200.08	53.21	QSO	O6E1/17-23	E140M	1140	1729	48.4	Y
PG 1004+130	225.12	49.12	QSO	O5EW/03	G140M	1194	1249	5.2
PG 1049−005	252.28	49.88	QSO	O4N3/03	G140M	1194	1249	1.5
PG 1103−006	256.66	52.30	QSO	O4N3/04	G140M	1194	1249	1.4
PG 1116+215	223.36	68.21	QSO	O5A3/01-02	E140M	1139	1709	6.6
				O5E7/01-02	E140M	1140	1709	19.9
PG 1149−110	280.47	48.89	Sey	O5EW/05	G140M	1194	1250	8.1
PG 1211+143	267.55	74.31	Sey	O61Y/01-08	E140M	1140	1729	42.5	Y
PG 1216+069	281.07	68.14	QSO	O6E1/31-39	E140M	1140	1729	69.8	Y
PG 1259+593	120.56	58.05	QSO	O63G/05-11	E140M	1143	1729	95.8
PG 1302−102	308.59	52.16	QSO	O5BU/01,02,61	E140M	1140	1729	22.1	Y
PG 1341+258	28.71	78.15	QSO	O5EW/01	G140M	1194	1250	6.9	Y
PG 1351+640	111.89	52.02	Sey	O4EC/54	G140M	1194	1248	14.7	Y
PG 1444+407	69.90	62.72	QSO	O6E1/01-06	E140M	1140	1729	48.6	Y
PHL1811	47.47	−44.82	QSO	O8D9/01-04	E140M	1140	1729	33.9	Y
PKS 0312−77	293.44	−37.55	QSO	O65T/01,02,13	E140M	1140	1729	8.4
PKS 0405−12	204.93	−41.76	QSO	O55S/01,02	E140M	1139	1729	27.2	Y
PKS 2005−489	350.37	−32.60	BLLac	O4EC/09	G140M	1194	1249	6.1
PKS 2155−304	17.73	−52.25	BLLac	O5BY/01-02	E140M	1139	1729	28.5
RX J0100.4−5113	299.48	−65.84	Sey	O8P8/02	G140M	1194	1250	2.3
RX J1830.3+7312	104.04	27.40	Sey	O5EW/09	G140M	1194	1249	5.8	Y
Ton S180	139.00	−85.07	Sey	O4EC/02	G140M	1194	1249	4.1	Y
Ton S210	224.97	−83.16	QSO	O6L0/01-02	E140M	1140	1709	12.3
UGC 12163	92.14	−25.34	Sey	O5IT/05	E140M	1140	1709	10.3
VIIZw118	151.36	25.99	Sey	O4EC/13	G140M	1194	1249	9.5	Y

Notes. Column 1: object name. Columns 2 and 3: Galactic longitude and latitude. Column 4: object type; QSO is a quasar and Sey is a Seyfert galaxy. Column 5: observational program identification datasets with exposure identifications. Column 6: grating used for observation; either HST STIS−G140M or HST STIS−E140M. Columns 7 and 8: minimum and maximum wavelengths of observational wavelength spanned. Column 9: total exposure time. Column 10: "Y" means there is a spectrum taken with the GBT.

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2.2. Fitting Method

To measure N(H i), the continuum-reconstruction method was used, which works as follows. First, one assumes an H i column density and a linewidth, which yields an optical depth (τ) profile. Multiplying the observed spectrum with exp(+τ) then gives a "reconstructed" continuum, which is assumed to be smooth as well as continuous with the parts of the spectrum where the continuum is observed directly. The H i column density is then varied until this is actually the case. We now describe this in more detail.

Before applying this method, we first increased the S/N of the spectra by binning the E140M data by 15 pixels (to 48 km s⁻¹ or 7 resolution elements) and the G140M data by 3 pixels (to 40 km s⁻¹ or 1.3 resolution elements). Then we defined a continuum (C) by selecting regions free of absorption and emission in a window about 50 Å wide around the Lyα line and fitting a Legendre polynomial of order up to four through these selected points, using the method of Sembach & Savage (1992). To ensure the continuity and smoothness of the reconstructed continuum, some of these regions are placed so that the final fit is made through the reconstructed continuum in regions where there is Lyα absorption.

Next, we created a reconstructed continuum starting with the N(H i) value obtained from the 21 cm brightness temperature. The velocity of the H i profile was also determined from the 21 cm H i emission data toward the sightline (see Section 2.3). Using the velocity and column density, we multiplied the observed flux (F) by exp(+τ), where τ is the optical depth of the Lyα line, including its damping wings. We note that the width of the central Gaussian part of the absorption profile is not important for the shape of the damping wings.

Finally, we made a fit to the continuum by changing the column density value until the fitted continuum (1) looked smooth and (2) minimized the difference between the fit and the reconstruction. The minimization is done by calculating a reduced χ² as

$$\begin{eqnarray*} &&\chi ^2 = {1\over n} \Sigma \left({ C - F\ e^\tau \over \delta F\ e^\tau }\right)^2, \end{eqnarray*}$$

where C is the fit to the reconstructed continuum, F is the observed flux, δF is the error in the flux, and n is the number of independent pixels. This calculation is only done in selected regions of the spectrum, which are different from those used to define the continuum. The selected regions are those that (1) are within about 10 Å of the central wavelength of Lyα (1215.67 Å), (2) are free of intergalactic or interstellar absorption lines, (3) have the optical depth of the Lyα line <1.5, and (4) have an S/N (F/δF)>2. The range of N(H i) values at which χ² = χ²_min + 1 determines the error in each N(H i) result.

Some sightlines contain high-velocity H i components. In such cases, we applied a procedure to determine a systematic error in N(H i) for the low-velocity components. To do this, we systematically varied the column density of one component, while fitting the other. Specifically, for sightlines with an HVC having N(H i) between ∼2 × 10¹⁹ and 10²⁰ cm^-2, we fixed N(H i; HVC) at the nominal 21 cm value, as well as at nominal × 1.5 and nominal/1.5, and then fitted the column density of the low-velocity component for each of these three HVC component values. The resulting range of N(H i) values for the low-velocity component then gave an estimate of the uncertainty associated with having the high-velocity component present. For sightlines containing high- and low-velocity components with similar strengths, the components were varied by 20% instead of 50%, since the result is better constrained in this case. We then executed the procedure of fixing one component and fitting the other for both components separately, while defining a region to calculate χ² on only one side of the Lyα absorption line. In most of these cases, we decided in the end that the systematic error in the final values was too large to use the results in the analysis of column density ratios.

2.3. 21 cm Data

We used three sources of 21 cm data. For 35 of the targets that have Lyα spectra we obtained new data with the GBT at an angular resolution of 91 (see below for more details on these observations). In 167 directions (42 overlapping with the Lyα sample), we also used the spectra of Lockman & Savage (1995), which were taken with the GB 140-ft (21' beam). For each of the directions in the merged Lyα+GBT+140' list, we interpolated between the grid points of the LAB survey (Kalberla et al. 2005), a whole-sky survey done at 1.3 km s⁻¹ velocity resolution with a 36' beam on a 05 × 05 grid. For each object, the 21 cm column density, N(H i; 21 cm), was obtained by integrating the brightness temperature over a specified velocity interval and converting this to a column density using

$$\begin{eqnarray*} &&N(\rm{H}\,{\mathsc i}) =\int 1.823 \times 10^{18}\ T_s\ \ln \left({T_s \over T_s-T_{\rm B} } \right)\ dv. \end{eqnarray*}$$

We analyze the highest T_B half of each spectrum with T_s = 135 K and the faintest T_B half with T_s = 5000 K and then add the two values.

This is justified by the fact that Heiles & Troland (2003) found that about 50% of the H i in the disk is "cold neutral medium" (CNM), with spin temperatures ranging from 30 K to several 100 K, with a median value of 70 K. The other half of the H i is "warm neutral medium" (WNM) with T_s > 500 K. Our value of 135 K for the CNM is an average. We use the different possible values for T_s to derive a systematic error (see below). For column densities below 10²⁰ cm^-2, the differences between assuming an optically thin cloud and T_s = 70 K or T_s = 135 K are <1.5%. Only above column densities of about 5 × 10²⁰ cm^-2 (log N(H i)>20.7) does the difference become >5%. In our sample, this only happens for sightlines toward which we compare 21 cm column densities derived with different radio telescopes, but not for sightlines where we measure Lyα.

The total error on the column density was estimated by adding in quadrature four sources of error, one statistical and three systematic. First, for each spectrum the rms in the brightness temperature was calculated and converted to a corresponding error in column density for the velocity interval that the column density was integrated over. Second, a systematic error of 1 × 10¹⁸ cm^-2 due to baseline placement was assumed for LAB and GBT data. For the older GB spectra, this was set to 2 × 10¹⁸ cm^-2. In one case (3C 273.0), strong continuum emission in the beam results in a noticeably worse instrumental baseline. Therefore, we set the baseline error to 5 × 10¹⁸ cm^-2. Similar effects cause the baseline error for NGC 985 to be set to 2 × 10¹⁸ cm^-2. The third source of error is the stray-radiation correction. Blagrave et al. (2010) and A. T. Boothroyd et al. (2011, in preparation) study the stray-radiation effects for the GBT in detail and estimate that the stray-radiation correction adds another 4 K km s⁻¹ of uncertainty to an H i profile, equivalent to 7 × 10¹⁸ cm^-2 in N(H i). However, this uncertainty only applies to the low-velocity emission near 0 km s⁻¹. Profile components in the H i spectrum that have velocities above about 50 km s⁻¹ are not affected as strongly, as there is much less bright emission elsewhere in the sky at those velocities. Therefore, for intermediate-velocity components (v ∼ 50 km s⁻¹) we used a systematic error associated with the stray-radiation correction of 1 × 10¹⁸ cm^-2, while for high-velocity components (v > 90 km s⁻¹) we set this error to zero. The final source of error is the assumed value for the spin temperature. We calculated N(H i) using the formula above with T_s set to 70 K, with T_s = 135 K, and using the optically thin assumption (reducing the T terms to T_B), and we took the rms variation between those three values as the corresponding error. The combined error is at most 0.04 dex (10%) for sightlines with log N(H i) < 20.8, the highest value we find for a direction where Lyα is measured, but for the great majority of sightlines the combined systematic error is <5% and usually it is <2%.

Finally, we calculated the brightness-temperature weighted average velocity of the profile, which is used to center the absorption-line model (see below). The velocity intervals were defined by visually determining the extent of the emission in each of the 21 cm spectra toward a target. In most cases, the chosen interval was then set to be the same at each angular resolution, but in a few cases small adaptations were necessary. To compare the 21 cm emission to the Lyα absorption, the full extent of the H i emission was used, but high- and intermediate-velocity components were measured separately when comparing column densities between different 21 cm observations.

2.4. GBT Data

In this section, we discuss the comparison between the column densities measured using Lyα absorption and 21 cm emission. A straightforward comparison between the values of N(H i) measured from 21 cm emission with those measured using various radio telescopes reveals significant differences between the two. In particular, for our set of low-column-density directions, the values found with the GB 140-ft and from the LAB survey are on average about 10% larger than those derived from Lyα. On the other hand, no significant difference is found when comparing GBT and Lyα column densities. Below we will show why we think that these differences arise from the presence of a broad, spurious component in the LAB and 140-ft spectra. The amplitude and FWHM of this component are 0.048 K and 167 km s⁻¹ for LAB spectra, 0.023 K and 134 km s⁻¹ for 140-ft data with 2 km s⁻¹ channels, and 0.12 K and 70 km s⁻¹ for 140-ft data with 1 km s⁻¹ channels, corresponding to column densities of 15.3 × 10¹⁸, 5.9 × 10¹⁸, and 15.8 × 10¹⁸ cm^-2.

We discovered this after our referee pointed us to a Web site commonly used to obtain column densities from the LAB data (http://heasarc.gsfc.nasa.gov/cgi-bin/Tools/w3nh/w3nh.pl). Although this helped us to figure out the problems, we want to point out that we have concluded that this Web site gives incorrect answers. In particular, it gives column densities that are derived after interpolating between the pixels of a gridded all-sky map smoothed (by default) to a 1° beam. Thus, a varying number of original LAB spectra are used to derive a column density. When using half a degree as the smoothing radius there are directions where the Web site says that there are no data on positions where an actual observation was done. Furthermore, in most cases the column density that it gives for a position where an actual spectrum was taken differs from the column density derived from that actual spectrum.

However, comparing our numbers with those given by this Web site led us to the realization that integrating LAB spectra from −400 to 400 km s⁻¹ gives different results than integrating over the line core, i.e., the velocity range where emission appears to be present. We had done the latter, in order to increase the S/N by avoiding integrating over a 300 km s⁻¹ window containing only noise, as well as to be able to separate out high-velocity emission. For the LAB spectra, the two different integrals differ by a constant offset of ∼6 × 10¹⁸ cm^-2, as shown in Figure 2. Since the highest H i column density in the set of sightlines with Lyα data is only about 7 × 10²⁰ cm^-2, we added 90 sightlines with higher column density to make this comparison. These 90 sightlines were chosen at latitudes b = 0° to b = 30°, at longitudes 170°, 180°, and 190°. This extends the comparison to N(H i) ∼ 7 × 10²¹ cm^-2.

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The origin of the offset seen in Figure 2 becomes clear if we examine the parts of the 21 cm spectra outside the bright line cores. Figure 3 shows the resulting averages of the non-signal regions of each spectrum. That is, for each direction we selected the channels outside the velocity limits given in Table 2. These limits are based on visually inspecting the spectra and selecting the velocity range where emission appears present. We did this for the GBT, 140-ft, and LAB spectra in the same set of directions. Panel (a) gives the residual spectrum for the combined set of all LAB directions (i.e., the those with Lyα, GBT, and 140-ft, as well as the higher column densities directions used to make Figure 2). Panels (d), (e), and (f) show the residuals for the two kinds of 140-ft spectra (with 2 km s⁻¹ and 1 km s⁻¹ channel spacing) and for the GBT spectra. Further, in panels (b) and (c), we show the residual LAB spectra for the directions corresponding to the 140-ft and the GBT sightlines. Gaussians were fit to the resulting features in order to determine the total emission in this broad component. Note that the GBT data show no significant broad component, although we show the formal fit.

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The broad residual component in the LAB spectra has an FWHM of 167 km s⁻¹ (dispersion 71 km s⁻¹) and a column density of 1.5 × 10¹⁹ cm^-2. We note that a similar broad underlying component in the LAB spectra was originally found by Kalberla et al. (1998), who measured a component with dispersion 60 km s⁻¹ (FWHM 140 km s⁻¹) and total column density 1.4 × 10¹⁹ cm^-2. Kalberla et al. (1998) interpreted this as evidence for a halo component in the Galactic H i, with scale height 4.4 kpc. However, we will now show that the overwhelming preponderance of evidence points to the conclusion that this broad underlying component is an artifact. This is based on the following four arguments.

• 1.  
  There is no residual component in GBT spectra. Since theoretically the GBT has very stable baselines and is less affected by stray radiation than the LAB dataset, we weigh the absence of a residual in the GBT spectra as an argument in favor of the conclusion that the residual seen in the LAB spectra is an artifact.
• 2.  
  The residual emission seen in LAB and 140-ft spectra differs. Where the LAB spectra produce a residual feature that is a Gaussian with T = 0.048 K and FWHM = 167 km s⁻¹, the 140-ft data with 2 km s⁻¹ channels give a Gaussian with T = 0.023 K, FWHM = 134 km s⁻¹. Notably, 140-ft with 1 km s⁻¹ channel spacing give T = 0.12 K, FWHM = 70 km s⁻¹. Thus, the width of the Gaussian differs for the two sets of similarly calibrated but differently setup 140-ft data, even though the same velocity ranges in the same spectra were used to create these residuals. Further, both residuals differ from that in the LAB data. This again suggests that the residual emission is an artifact. Still, it is possible that the calibration of the LAB spectra was better and the broad component is only properly picked up in that dataset. On the other hand, the difference in residual between the two kinds of 140-ft data suggests that it may be caused by a (small) baseline fitting error.
• 3.  
  The broad component produces no metal-line absorption. If there is a broad H i component in the Galaxy, this should result in a detectable optical depth for many ionic absorption lines in the ultraviolet. For instance, for solar metallicity gas with typical dust content, a component with FWHM 170 km s⁻¹ will produce a peak optical depth of >2 in the C ii λλ1334.532, 1036.337, O i λ1302.169, and Si ii λ1260.422 lines. This would yield a dark line out to ±150 km s⁻¹ in every sightline. As can be seen from the spectra toward 100 AGNs presented by Wakker et al. (2003), this is clearly not observed. A typical example for a direction without any known 21 cm emitting high-velocity gas is shown in Figure 4. In fact, in all directions the extent of the strong metal lines is rather similar to the visual extent of the H i emission. Figure 4 also includes the absorption profile that would be associated with the high-dispersion gas if it had one-tenth solar metallicity. This could be the case if the high-dispersion gas consisted of infalling gas. However, in that case there would still be clear extended wings, which are also not observed. Thus, if this high-dispersion component were Galactic gas, there would be unmistakable evidence for it in these spectra, unless its metallicity is much less than 0.1 times solar.
• 4.  
  The broad component cannot originate as an average of a population of small clouds. The only way in which a broad component might be absent in absorption-line spectra is if it consists of many small clumps, which are missed in all of the 100 or so observed AGN sightlines. Since 100 sightlines were sampled to have a 50–50 probability of missing the small clouds in all sightlines requires that they have an area covering fraction of <0.7%, since 0.993¹⁰⁰ = 0.5. To have one such cloud in every 05 beam thus requires clouds with diameters that are less than 3'. As the observed brightness temperature at 36' resolution is 0.048 K, this implies an actual brightness temperature of 7 K, which for clouds with linewidth 15 km s⁻¹ would imply a column density of 2 × 10²⁰. When observed with a 10' beam, these clouds would have apparent brightness temperatures of 0.6 K and apparent column densities of 10¹⁹ cm^-2. The population should have a cloud-to-cloud dispersion of 70 km s⁻¹and be widespread at low and high latitudes.

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A population of small clouds has indeed been discovered (Lockman 2002; Ford et al. 2008, 2010). These studies find 400 and 255 small clouds in two 720 deg² regions (0.45 deg⁻²), with the clouds having typical peak brightness temperature 0.5 K, typical velocity width 13 km s⁻¹, and typical size 20' (i.e., many are unresolved). However, although the brightness temperature of these clouds is similar to what is required, they are much too large, the population strongly is confined to low latitudes, and the cloud-to-cloud dispersion is only 16 km s⁻¹. Thus, these clouds do not fit the requirements, and the observations show that a population of clouds with the required characteristics to mimic the broad component in the LAB spectra does not exist.

Combining points (1) through (4) leads to the conclusion that the residual emission in the LAB spectra is an artifact. The residual could be due to a final imperfection in the stray-radiation correction. It might also be caused by a small error in the polynomials used to fit the baselines. Such an error would be difficult to discern since the largest visible deviation is ∼0.02 K or less than one-third the rms noise in a single spectrum. The fact that the spectra from different telescopes have different residuals argues for the second option. The fact that no significant residual is seen in the GBT spectra argues that the first option might also play a role.

Based on these considerations, we decided to correct the LAB and the 140-ft spectra, subtracting the Gaussians shown in Figure 3. This eliminates the discrepancies between the column densities derived from Lyα and those derived from 21 cm observations. As we show in the next section, the average ratio N(H i; Lyα)/N(H i; 21 cm) then becomes 0.96–1.00, with a dispersion of ∼0.10, not significantly different from 1. Without this correction, the average ratio is 0.90 ± 0.10 for the LAB data and the LAB spectra are in tension with the Lyα column densities, with the GBT spectra, with the non-detection of metal-line absorption, and with the observed population of small clouds. However, after applying the corrections, all tension between the many different observations disappears. In the remainder of the paper, we will thus only use corrected LAB and 140-ft column densities. We leave the GBT column densities uncorrected because there is no strong evidence that a correction is needed.

We also note the following. In Section 2.4, we had said that for the standard field S8 comparing the column density derived from the GBT data to that derived from the LAB survey gave a ratio of 0.991. The actual LAB column density toward S8 is 1.805 × 10²¹ cm^-2. Subtracting the spurious 1.5 × 10¹⁹ cm^-2 from this gives a column density of 1.790 × 10²¹ cm^-2. The ratio of these two values is 0.992. Thus, after correcting the LAB data, they give exactly the same column density as the GBT data in this direction.

4.1. 21 cm and UV Data

In Figure 5, we show the 21 cm and UV spectra of all AGN targets toward which we measured N(H i; Lyα). The 21 cm spectrum that is displayed is the one obtained with the smallest telescope beam. In the panels with the UV spectra, we show the observed flux as well as the fitted continua and resulting models. The H i column densities derived from the 21 cm and Lyα absorption are shown for comparison. Table 2 presents a more detailed summary of the results, giving the UV column densities, as well as the 21 cm column densities obtained with different telescopes (see Section 2).

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The spectra of targets observed with the STIS−G140M grating sometimes show a rise on the short-wavelength side of the spectrum, below 1200 Å. We have been unable to determine the origin of this spectral slope, but it may be due to a calibration error in G140M observations. The targets that show this effect are Mrk 1044, Mrk 1513, PG 0804+761, PG 1049−005, PG 1341+258, PKS 2005−489, RX J0100.4−5113, RX J1830.3+7312, Ton S180, and VII Zw118, as can be seen in Figure 5. Where possible, we avoided fitting the continuum using pixels below 1200 Å.

4.2. Data Quality

4.3. The 21 cm to Lyα Column Density Ratio

In Figure 6, we plot all combinations of log N(H i) derived using Lyα, LAB, GB 140-ft, and GBT against each other. In Figure 7, we show the ratios of these column densities. These figures show that although the values derived from the 21 cm and Lyα data are generally similar, they are not identical. Most of the low-column-density points in the scatter plots for 21 cm only data are values for the HVCs. Since the scatter clearly is larger for these clouds (red and blue points) than for the low-velocity gas (black points), they evidently have small-scale structure on scales of 9'–36'. The low-velocity components are possibly a mixture of clouds at different distances, each of which may have a column density similar to that of an HVC, as well as small-scale structure. However, all of these clouds get blended together in both the telescope beam and in the velocity space of the sightline.

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In the ratio panels, we also give the average ratio and its rms for all targets, and for targets with quality 3/4 only. Using LAB, GB, and GBT data, on average, the ratios for all qualities are 0.96 ± 0.11, 0.95 ± 0.09, and 1.00 ± 0.07, respectively. Using only the reliable (qualities three and four) data, the 21 cm telescopes give average ratios of 0.95 ± 0.09, 0.92 ± 0.08, and 0.98 ± 0.06, respectively. See Section 5.1 for further discussion of these ratios. Clearly, on average the ratio is slightly less than 1 in most cases, although not by much.

There are 22 sightlines for which we rated the Lyα measurement as high quality (three or four). For these datasets, the ratio N(H i; Lyα)/N(H i; LAB) ranges from 0.78 to 1.24, with the most extreme low ratio for toward PKS 2155−304, and the most extreme high toward PG 1049−005. For 16 sightlines with Q = 3 or 4 and GB 140-ft data, the range is 0.77–1.16 (also for PKS 2155−304 and PG 1049−005). Nineteen sightlines with GBT data fall in the high-quality category, and the ratio of column densities ranges from 0.88 to 1.12 (Mrk 478 and PG 1444+407).

4.4. The 21 cm to 21 cm Column Density Ratio

4.5. Comparison with FOS Results

5.1. Does N(H i; Lyα)/N(H i; 21 cm) Differ Significantly from 1?

5.2. Comparing the Observed Distribution of Column Densities to Models

In Figure 8, we compare the predictions from a number of different models with the observed distribution of column density ratios (shown as the black histogram). As we describe below, we used three different approaches to make these predictions: (1) modeling the structure using a simple hierarchical approach, (2) assuming that the distribution is log-normal, and (3) using the results of a 3D MHD simulation.

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5.2.1. Modeling the Structure Using a Simple Hierarchical Approach

One way to represent the distribution of small-scale structure in column density measurements is a hierarchical model. This means that higher column density regions are enclosed by and cover some fraction of the area of lower column density regions. We can construct such a model by starting with assuming that a patch has some column density, N, which covers some fraction A of the area inside the 21 cm beam. Next, we assume that there are one or more other patches with a total area that is a fraction β times smaller, and which have a column density that is a factor α larger. We then construct an indefinite number of patches in this fashion. In the end, each patch is meant to represent the total area covered by all cloud structures at a certain column density within a region, regardless of where the structure is located. This model assumes that as area decreases, column density decreases, thus α must be greater than 1 and β must be less than 1. In fact, we find that their product also needs to be <1.

We can now derive a mathematical prediction for the distribution of column densities that can be compared to the data. We start with deriving $\bar{N}$, which is the area-weighted column density average in a large region. Since $\bar{N}$ is the total column density divided by the total area, $\bar{N}$ for an area divided into k patches is

$$\begin{eqnarray*} &&\bar{N} = { A N\ +\ \beta A\, \alpha N\ +\ \beta ^2 A\, \alpha ^2 N\ +\ \cdots \beta ^k A\, \alpha ^k N \over A\ +\ \beta A\ +\ \beta ^2 A\ +\ \cdots +\ \beta ^k A} = { {1\over 1-\alpha \beta } \over {1\over 1-\beta } }, \end{eqnarray*}$$

where A is the area of the first patch, having column density N. Thus, A can be divided out, and by extracting N and inverting the formula to derive N/$\bar{N}$, and plugging in the result of the infinite-series summation, we find

$$\begin{eqnarray*} &&r_0 = {N\over \bar{N}} = {1 - \alpha \beta \over 1 - \beta }. \end{eqnarray*}$$

Here, r₀ is the ratio of the column density in the first patch to the average over the whole target. Each subsequent mth patch will have area A_m = Aβ^m, and column density ratio r_m = r₀ α^m.

To derive the distribution f(r), we need to calculate the relative fraction of the total area covered by patch m:

$$\begin{eqnarray*} &&f(r_m) = { A_m \over \Sigma _{m=0}^{m=\infty }\ A_n} = \beta ^m\ (1-\beta). \end{eqnarray*}$$

Thus, given α and β, the probability of a beam hitting a patch can be calculated, as well as the column density ratio corresponding to that patch. The probability is the fraction of area that a single patch covers (i.e., f(r_m)). The column density ratio in that patch is given by r_m. This probability can be interpreted as the number of times one would expect to observe a beam hitting the respective patch, given a number of pencil-beam measurements taken.

In Figure 8, we display the hierarchical model (blue curves) for α = 1.14 and β = 0.44. These values were determined such that the modeled hierarchical distribution has similar peak and dispersion as the observed distributions for N(H i; Lyα)/N(H i; LAB) and N(H i; Lyα)/N(H i; GBT). This combination of parameters implies that 56% of the area is covered by pencil beams having a column density that is 89% of the average, 25% is covered by pencil beams with column density that is 101% of the average, while 19% of the beams are 116% of the average or higher. Thus, the hierarchical model predicts a high peak near r ∼ 0.90, and a small fraction of sightlines with ratio >1. Observationally, for the comparison of Lyα to LAB column densities, we find 49% (18 of 37) of the beams have r between 0.8 and 0.95, while 41% (15 of 37) have r between 0.95 and 1.08, and 4 (11%) have r above 1.08. Thus, the column densities inside a 36' LAB beam have a distribution that is not too dissimilar from the hierarchical model. In contrast, comparing Lyα to GBT column densities yields 36% (10 of 28) with r = 0.8–0.95, and 43% (12 of 28) having r = 0.95–1.08, with 6 (22%) at r > 1.08. In this case, the predicted peak at lower ratio is not observed. In general, the hierarchical model underpredicts the number of high ratios.

5.2.2. Assuming that the Distribution is Log-normal

It is likely that the structure of Galactic H i is determined by turbulence (see, e.g., Kowal et al. 2007; Burkhart et al. 2009). The resulting 3D structure is then effectively determined by a multiplicative random walk. That is, a parcel of gas will be compressed and will expand proportionally to its current density. Intuitively, this should produce a column density distribution that is log-normal, i.e., the log of the density has a Gaussian distribution around some mean. In most circumstances, if the 3D turbulence produces a log-normal distribution of volume densities, the two-dimensional column density projection will also be log-normal.

We note that converting a log-normal distribution to a linear scale results in the mode being at a value below the average. Taking a random sampling of sightlines through the gas gives a distribution with the mode at some density. Averaging this same parcel of gas by observing it with a large beam will thus result in a value slightly larger than the mode. The offset is determined by the width of the distribution. Using N_m for the mode of the distribution, we can write

$$\begin{eqnarray*} &&f(N/N_m) = \exp \left(-{ \left(\ln \,N/N_m\over b\right)^2 } \right), \end{eqnarray*}$$

where f(N/N_m) is the log-normal distribution of the column densities and b is a parameter characterizing the width of the distribution. The column densities are normalized to the modal value, N_m. The average value of N observed in the area of integration, $\bar{N}$ is then found to be

$$\begin{eqnarray*} &&{\bar{N}\over N_m} = { \int N/N_m\ f(N/N_m) {\it dN} \over \int f(N/N_m) {\it dN} }. \end{eqnarray*}$$

To compare this to N(H i; Lyα)/N(H i; 21 cm), we have to invert this, since N(H i; Lyα) corresponds to N, the column density in each pencil beam inside the area of averaging, while N(H i; 21 cm) corresponds to $\bar{N}$. Then the integral works out to be

$$\begin{eqnarray*} &&{N_m \over \bar{N}} = \exp { -3 b^2\over 4 }. \end{eqnarray*}$$

Thus, taking for instance b = 0.155, the ratio of mode to average is 0.98, while for b = 0.25 it is 0.95.

We have too few sightlines to determine whether the mode of the distribution of ratios differs significantly from 1. However, we can match the dispersion of the log-normal distribution to the observations. For the Lyα/LAB, Lyα/140-ft, and Lyα/GBT ratios this requires b = 0.155, 0.126, and 0.101, respectively. For GBT/LAB, 140-ft/LAB, and GBT/140-ft the required b = 0.085, 0.094, and 0.043. The green curves in Figure 8 show two log-normal distributions. For the Lyα to LAB and 140-ft comparisons, we used b = 0.155, while for comparing Lyα/GBT we used b = 0.100, the values for which the dispersions match. In the case of comparing 21 cm to 21 cm data, we used b = 0.085, which matches the GBT/LAB dispersion. Clearly, a log-normal distribution resembles the observed distributions. It is unclear whether it is significant that the distribution of Lyα/LAB ratios is wider than that of Lyα/GBT ratios, but it might in principle be possible to attribute this to the fact that the GBT beam samples a smaller region, so that the relative fluctuations are smaller.

5.2.3. Using the Results of a 3D MHD Simulation

Kowal et al. (2007) presented a set of simulations of turbulence. We used the model datacubes that they produced to extract column densities by projecting these cubes onto one of their faces. We added to this some cubes made with the same software, but which were not included in their paper. These cubes can be projected onto one axis and converted to a table of 65,536 column densities. We then plotted the distribution of these column densities, normalizing by the average of all column densities ($\bar{N}$) in the simulation. Kowal et al. (2007) used the gas and magnetic Mach numbers to parameterize their simulations. In fact, there is a distribution of Mach numbers throughout their cubes, and the numbers represent the average Mach numbers.

We extracted the predicted column density distributions for each of 16 simulations and compared them to the data. In Figure 8, the orange and red curves give the theoretical distributions of ratios, N/$\bar{N}$, for the two best models, which are the cases with sonic and Alfvénic Mach numbers of (M_S, M_A) = (0.65, 0.61) and (M_S, M_A) = (0.90, 2.00), respectively. For models with M_S below 0.5 the width of the distribution is much narrower than observed, while if M_S is larger than 1.4 the predicted distribution is much wider than observed (and the mode lies near a ratio of 0.7 when M_S > 2). In all models, the influence of the Alfvénic Mach number is modest, changing only the details of the distribution but not the width, although the models with higher M_A fit better than those with lower M_A. It is also clear that these instances of the MHD models predict distributions that are quite similar to a log-normal distribution with b ∼ 0.155.

The simulations were also used to compare the GBT versus LAB telescope pairs. In these cases, N is the column density in an area that is (10/36)² ∼ (1/13)th times the full size of the simulation. Since none of the 21 cm telescopes actually make a pencil-beam measurement, it is incorrect to use each pencil-beam measurement as N in these instances. When comparing, e.g., GB 140-ft to LAB data, the number of 140-ft beams inside the LAB beam is only (36/21)² ∼ 3. This means that in such cases, there is no clear difference between N and $\bar{N}$, making N/$\bar{N}$ a trivial comparison between 21 cm telescopes of similar beam size. Thus, we only compare the GBT to the LAB data (Figure 8(d)). Using our approach, the predicted width of the distribution is clearly narrower than was the case for the comparison between Lyα and 21 cm, as is predicted by the simulations.

Our approach of comparing column densities between observations made with very different resolutions suggests a simple way of characterizing the small-scale structure of the ISM. The predictions from different 3D turbulent MHD models are clearly sensitive to the sonic Mach number that is used. The observed distribution of ratios fits very well with those predictions. The Cosmic Origins Spectrograph instrument on HST is expected to provide the possibility of measuring N(H i; Lyα) toward several 100 additional sightlines, strongly improving the statistics and thus the constraints on the model parameters. Another possibility is to use high-resolution H i data, such as those that will become available when the GALFA (see Peek & Heiles 2008) or GASKAP surveys are complete (GASKAP is a survey of the Galactic Plane to be executed with the ASKAP telescope, building of which is in progress). These surveys have 3' and 1' resolution and thus provide a large dynamic range in resolution.

We derive the column density of neutral H i from the Lyα line using data from the STIS spectrograph on HST and from the 21 cm line using the LAB survey, GB 140-ft, and GBT observations. Using these column densities, we compare the ratio of N(H i; Lyα) to N(H i; 21 cm) and N(H i; 21 cm) to N(H i; 21 cm) in order to analyze the structure of the ISM. Based on the results, we conclude the following.

• 1.  
  For 59 AGNs surveyed, 36 yield reliable Lyα column densities. There are 163 GB 140-ft and 35 GBT sightlines for which N(H i; 21 cm) is derived. For each of the unique sightlines, we also measured N(H i) using LAB data.
• 2.  
  We conclude that the published LAB data, as well as our old (from the late 1980s) GB 140-ft data, suffer from a problem that results in an excess column density of ∼1.5 × 10¹⁹ cm^-2. This problem is revealed by extracting the spectral regions outside the range where signal appears to be present. There is no residual emission in the combined GBT spectrum, in contrast to the LAB and 140-ft data. The residual can be fitted by a Gaussian, which for the LAB dataset has v = −22 km s⁻¹, T = 0.048 K, and FWHM = 167 km s⁻¹. The parameters of the residual are different for the 1 km s⁻¹ and 2 km s⁻¹ channel spacing 140-ft spectra.
• 3.  
  We conclude that the H i spectra from the LAB survey need to be corrected for the presence of a broad underlying Gaussian. Without such a correction, the LAB data conflict with measurements of N(H i) made with the GBT and made using Lyα absorption, as well as with UV absorption-line studies, and with the properties of the recently found population of small H i clouds. With such a correction, all tension between measures of N(H i) at different resolutions disappears.
• 4.  
  Using data from the FOS on HST, Savage et al. (2000) had found that on average N(H i; Lyα)/N(H i; 21 cm) was 0.81 ± 0.09 for 12 sightlines. Using our new data, the same set of sightlines yields Lyα column densities that are on average ∼0.06 dex higher (compared to the typical error of 0.2–0.3 dex in the FOS results). We also find that a correction is needed to the 140-ft data, which corresponds to 0.05 dex in these 12 sightlines. As a result of these corrections, the average ratio N(H i; Lyα)/N(H i; 21 cm) for these sightlines is now found as 1.02 ± 0.08.
• 5.  
  After applying the corrections to the LAB and 140-ft observations, we find that the ratios between N(H i; Lyα) and N(H i; 21 cm) are on average 0.96 ± 0.11 (Lyα/LAB), 0.95 ± 0.09 (Lyα/140-ft), and 1.00 ± 0.07 (Lyα/GBT). A statistical test shows that these averages do not differ from 1 in a statistically significant way.
• 6.  
  A hierarchical model for the ISM matches the observed column density distribution for N(H i; Lyα)/N(H i; 21 cm) ratios adequately well, although it underpredicts the number of high ratios.
• 7.  
  A log-normal model matches column density ratio distribution moderately well, with a different width parameter for different cases.
• 8.  
  Using a 3D MHD model from Kowal et al. (2007), we can match most features of the column density distributions when choosing cases with M_S = 0.65–0.90. These distributions are similar to a simple log-normal distribution, as is expected for turbulence.
• 9.  
  We conclude that by comparing H i column densities observed at very different resolutions it becomes possible to characterize the small-scale structure of the ISM. Although the number of sightlines in our sample is small, the distribution of column density ratios approximately follows a log-normal distribution, which is also similar to the predictions of 3D MHD modeling, using a sonic Mach number in the range 0.65–0.90.

B.P.W. acknowledges support from NSF grant AST-0607154 and NASA-ADP grant NNX07AH42G. The Green Bank Telescope is part of the National Radio Astronomy Observatory, a facility of the NSF operated under cooperative agreement by Associated Universities, Inc. The Lyα data in this paper were obtained with the NASA ESA Hubble Space Telescope, at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. Spectra were retrieved from the Multimission Archive (MAST) at STScI. We thank UW graduate students Alex Hill and Blakesley Burkhart for extracting the column densities from the simulation of Kowal et al. (2007) and for providing additional models. J.M.B. acknowledges Steve Schmitt and Erin Conrad for mathematical discussions. We thank the anonymous referee for insisting that we investigate possible errors in the 21 cm surveys.

3C 249.1. The quality factor of the column density determination is only 2, because the spectrum is relatively noisy and the 21 cm spectrum shows the presence of several IVCs. There is an unidentified line near 1210 Å, which cannot be intergalactic Lyα, but which also does not fit into any known system of absorbers toward this sightline.

3C 273.0. Intrinsic O vi emission at 1195.32 Å and 1201.91 Å creates a moderately broad peak in the continuum at wavelengths shorter than Galactic Lyα. Combined with the intermediate-velocity gas at v ∼ 25 km s⁻¹, this leads us to assign quality two to this sightline. A third-order polynomial fits the right side of the O vi wing well and creates a good fit overall from 1197 Å to 1247 Å.

3C 351.0. The spectrum of this AGN is flat near Galactic Lyα, and the 21 cm spectrum is simple, resulting in quality factor 4 for this spectrum.

H 1821+643. The continuum of this target is flat and does not contain features. However, there are weak H i components at high negative velocities (<−100 km s⁻¹), lowering the confidence in the resulting value of N(H i) and leading us to give quality three to this target.

HE 0226−4110. The spectrum of this AGN is flat near Galactic Lyα, and the 21 cm spectrum is simple, resulting in quality factor 4 for this spectrum. Where we find a value of 20.09 ± 0.04 for the H i column density, Savage et al. (2007) reported 20.12 ± 0.03 based on an earlier analysis of the same data.

HE 0340−2703. Uncertain continuum placement due to the presence of multiple strong IGM lines results in a low quality fit for this QSO continuum. In addition, on the short-wavelength side there appears to be an (unidentified) intrinsic emission line, making the continuum even more uncertain. We therefore assign Q = 1 to this target.

HE 1029−1401. The spectrum of this AGN is flat near Galactic Lyα, and the 21 cm spectrum is simple, resulting in quality factor 4 for this spectrum.

HE 1228+0131. The noisy continuum of this spectrum makes it is impossible to obtain a good fit. This is further complicated by the lack of knowledge about the intrinsic continuum of the QSO, which is higher at λ < 1215 Å than at λ>1215 Å. Intrinsic S iv λ1062.664 (redshifted to 1186 Å) and intrinsic N ii λ1083.994 (redshifted to 1211 Å) emission may be present, and there is also a steep continuum drop across Galactic Lyα. These problems make the continuum too uncertain to fit, and we do not use this sightline in our analysis.

HS 0624+6907. The spectrum of this AGN is flat near Galactic Lyα, but the 21 cm spectrum is not simple, resulting in quality factor 2 for this spectrum.

HS 1543+5921. A complicated continuum combined with low S/N makes a good fit for the continuum almost impossible. In particular, Lyα absorption in SBS 1543+593 at 2800 km s⁻¹ blends with Galactic Lyα. We do not use this spectrum in our analysis.

HS 1700+6416. The continuum is too difficult to obtain a good fit due to multiple IGM lines and moderate to high noise. We do not use this spectrum in our analysis.

MCG +10−16−111. The 21 cm spectrum shows two components of similar strength. As explained in Section 2.2, we fix each one in turn and fit the other, in order to determine a systematic error. As can be seen in Table 2, the sum of the two stays more or less constant. As we cannot reliably compare the 21 cm and Lyα column densities, we assign Q = 1 to this sightline.

MRC 2251−178. Intrinsic Lyα emission peaks at 1296 Å, creating a moderately strong emission wing for measuring the Galactic Lyα absorption. A third-order polynomial provides a good fit from 1195 Å to 1241 Å. Since the 21 cm spectrum is simple and the upward slope is minor, we assign Q = 3 to this sightline.

Mrk 110. Intrinsic Lyα emission peaks at 1259 Å, creating an upward slope in the continuum. A third-order polynomial fits this wing adequately and provides an adequate fit for the rest of the continuum in the wavelength range plotted, resulting in Q = 3.

Mrk 205. The 21 cm spectrum contains multiple H i components including absorption originating in NGC 4319 at 1289 km s⁻¹ and an HVC at v = −204 km s⁻¹. The HVC is relatively strong, resulting in a 0.04 dex systematic error in the value of N(H i; Lyα) for the Galactic emission. Extra curvature is present in the continuum and a second-order polynomial was used to handle this. On balance, however, the result is reliable and we assign Q = 3.

Mrk 279. Many factors contribute to a low-quality (Q = 1) fit. Multiple H i components are present in the 21 cm spectrum, but only one component at v = −40 km s⁻¹ is used for the fitting. Strong Lyα emission is also present at 1252.69 Å creating a large wing in the continuum. A fourth-order polynomial is used to handle these features and fits the continuum adequately well.

Mrk 335. Intrinsic Lyα emission peaks at 1247.02 Å, creating a large wing beginning around 1208 Å and producing significant curvature in the continuum. A fourth-order polynomial fits this continuum moderately well.

Mrk 478. An order three polynomial is needed to handle the curvature in the continuum, but the fitting lines provide a good fit over the range of 1201 Å to 1234 Å. The curvature is caused by intrinsic Fe iii-1122 emission, centered at 1209.02 Å.

Mrk 509. The 21 cm spectrum contains multiple components, including an IVC at v = 61 km s⁻¹. All components were included in the measurement of the 21 cm and Lyα H i column density. Strong intrinsic Lyα emission at 1256 Å creates a large wing. A fourth-order polynomial fits this wing and the continuum well from 1180 Å to 1230 Å.

Mrk 771. The continuum is flat across the Galactic Lyα line. Although the 21 cm spectrum shows two strong components, their separation in velocity is low enough that the final fit to the Lyα absorption results in Q = 4.

Mrk 876. The 21 cm spectrum shows multiple H i components, including an HVC at v = −130 km s⁻¹, thus a systematic error in the N(H i) value is present due to the HVC's impact. Intrinsic O vi emission at 1165 Å and 1171 Å also causes the continuum to slope downward at the short-wavelength side of Lyα, but an order two polynomial still provides a good fit. These complications lead us to assign quality factor 2 to this measurement.

Mrk 926. A third-order polynomial provides a good fit to the continuum. However, multiple factors contribute to a systematic error in the value of N(H i). One factor is strong intrinsic Lyα emission at 1273 Å that creates a broad wing in the continuum. Another factor is the fact that the G140M appears to show an extra upturn to the continuum below wavelengths of 1200 Å (not easily visible in Figure 5). Combined with a relatively noisy spectrum, we decided to assign Q = 1 to this sightline.

Mrk 1044. Strong intrinsic Lyα emission at 1235.86 Å adds a large wing to the continuum. A fourth-order polynomial fits this wing well and fits the continuum adequately from 1203 Å to 1223 Å. The strong curvature across Lyα results in Q = 2 for this target.

Mrk 1383. An order two polynomial is needed to handle the slight curvature in the continuum, but the fit is still good enough to derive a result with quality factor 4.

Mrk 1513. The continuum is flat, except near 1197 Å where it shows an upturn, although there are no known intrinsic emission features near this wavelength. The tail end of intrinsic Lyα emission also causes the continuum to rise at wavelengths above 1235 Å. The net result of both issues is that the continuum fit is not as reliable as it might seem, resulting in quality three for this measurement.

NGC 985. Intrinsic Lyα emission peaks at 1268 Å, creating a moderate upward slope near Galactic Lyα. A third-order polynomial provides a good fit from 1195 Å to 1240 Å.

NGC 1705. The continuum is too complicated to make a good fit due to intrinsic N v emission at 1241.42 Å and 1245.41 Å as well as intrinsic Lyα emission from the galaxy, which has a redshift of only 628 km s⁻¹. Therefore, we do not derive N(H i; Lyα) for this sightline.

NGC 3516. The continuum is too difficult to obtain a good fit due to moderate to high noise and strong intrinsic Lyα emission at 1225 Å. Therefore, we do not derive N(H i; Lyα) for this sightline.

NGC 3783. The continuum is too difficult to obtain a good fit due to the strong Lyα emission at 1227.50 Å, as well as the presence of multiple strong H i components at intermediate and high velocity. Therefore, we do not derive N(H i; Lyα) for this sightline.

NGC 4051. Strong Lyα emission at 1218.51 Å and N v emission at 1241.71 Å and 1245.71 Å creates a continuum too difficult to obtain a good fit.

NGC 4151. The continuum is too difficult to make a good fit due to strong Lyα emission at 1219.70 Å and N v emission at 1242.93 Å and 1246.93 Å.

NGC 5548. Strong Lyα emission at 1236.55 Å adds a large wing to the continuum. A fourth-order polynomial provides an adequate fit. Since the 21 cm profile is simple, the measurement is given a final quality of three.

NGC 7469. Lyα emission at 1235.51 Å creates a large rise in the continuum, but this is fit adequately well by a fourth-order polynomial over the range of 1197 Å to 1230 Å. This results in a quality three measurement.

PG 0804+761. The continuum is flat from 1201 Å to 1234 Å but is pushed above the fitting line at 1196 Å and below at 1245 Å. The final fit is given quality three.

PG 0953+414. Intrinsic C iii emission is present at 1204.92 Å  which adds curvature that creates a hill in the continuum over a range from 1188 Å to 1236 Å. A fourth-order polynomial fits this hill well over a range of 1205 Å to 1248 Å, but the curvature leads us to assign Q = 3 to this measurement.

PG 1001+291. The quality of this continuum is diminished by multiple factors, including a moderate S/N and the continuum resting above the fitting line at 1183 Å.

PG 1004+130. A high level of noise in the continuum makes it difficult to obtain a good quality fit and a reliable value of N(H i) for this QSO.

PG 1049−005. The continuum is flat, but the high level of noise greatly reduces the quality of the fit. As a result, we assign Q = 1 to this sightline.

PG 1103−006. Low S/N makes obtaining a reliable fit impossible for this target.

PG 1116+215. The intrinsic O vi emission lines are redshifted to 1214.06 Å and 1220.75 Å, which results in a large bump in the continuum across the Galactic Lyα line. This can be modeled by using a fourth-order polynomial, which fits the continuum adequately well from 1180 Å to 1248 Å. The 21 cm spectrum shows two components of similar strength at −39 and −5 km s⁻¹. These factors combine to yield Q = 1 for the resulting Lyα column density.

PG 1149−110. Intrinsic Lyα emission peaks at 1275.24 Å and a low S/N makes a good fit for the continuum too difficult to obtain.

PG 1211+143. Intrinsic Fe iii λ1122.52 emission is redshifted to 1212.77 Å, which pushes the continuum slightly upward on the short-wavelength side of Galactic Lyα. This lowers the quality of the fit, although a fourth-order polynomial is used and fits the continuum adequately well from 1180 Å to 1248 Å. The final fit quality is assigned a value of 3.

PG 1259+593. Multiple H i components are present in the continuum including an HVC at v = −127 km s⁻¹ and an IVC at v = −52 km s⁻¹. A two-sided fit is used, as described in Section 2.2.

PG 1302−102. The continuum is flat, but the low S/N visibly diminishes the quality of the fit.

PG 1341+258. The continuum is slightly above the fitting line from 1223 Å to 1230 Å, but a linear fit still provides a high-quality (Q = 4) fit.

PG 1351+640. The continuum contains multiple absorption features, lowering the quality of the fit. The 21 cm spectrum shows multiple H i components including an HVC at v = −156 km s⁻¹. Thus, a systematic error is introduced into the value for N(H i) due to the HVC's impact on the continuum. Intrinsic Fe iii emission is also present at 1221.53 Å. Although a fourth-order polynomial is used to deal with these features and provides an adequate fit for the continuum, the uncertainty associated with the HVC's column density is such that we assign a final quality factor of 1.

PG 1444+407. A flat continuum gives an acceptable fit. However, it is a little above the fitting line from 1180 Å to 1200 Å and from 1233 Å to 1237 Å, thus reducing the quality of this fit. Combined with the multiple 21 cm components, the final column density value is quality two.

PHL 1811. A flat continuum is a good fit despite the presence of intrinsic O vi emission at 1230.06 Å and 1236.84 Å, which pushes the continuum slightly above the fitting line from 1224 Å to 1236 Å. Combined with the simplicity of the 21 cm profile, the final quality for this sightline is four.

PKS 0312−77. The 21 cm spectrum contains multiple H i components, including the Magellanic Bridge at v = 191 km s⁻¹ and v = 166 km s⁻¹. A two-sided fit is used as described in Section 2.2. Lehner et al. (2008) give Lyα-derived column densities of 20.78 ± 0.06 and 20.12 ± 0.30 for components at 5 and 210 km s⁻¹, respectively. The combined value is 20.86. From the LAB data, we find a column densities of 20.83 and 20.23 for these two components. Varying the Magellanic Bridge component between 20.14 and 20.31 results in values for the low-velocity column density of 20.67 ± 0.18 to 20.71 ± 0.17. A combined fit yields values between 20.79 ± 0.14 and 20.87 ± 0.11, depending on the precise selections. Thus, we find a column density for the Magellanic Bridge component that is 0.1 dex higher than that of Lehner et al., but within their error. We also find that Lehner et al. underestimated the error for the low-velocity gas by a factor of ∼3.

PKS 0405−12. The continuum is flat but a few extra features reduce the quality of the fit. There may be Ne viii emission at 1211.06 Å and intrinsic O iv emission at 1238.75 Å. N iii emission at 1200.42 Å pushes the continuum up slightly, and a dip from 1243 Å to 1255 Å pulls the continuum down, creating a twist in the continuum that causes the fitting line to slope upward. In the end, we only assign Q = 2 to this sightline.

PKS 2005−489. This spectrum is flat across Galactic Lyα, and the sightline has a simple 21 cm profile. The derived Lyα column density is quality four.

PKS 2155−304. This spectrum is flat across Galactic Lyα, and the sightline has a simple 21 cm profile. The derived Lyα column density is quality four.

RX J0100.4−5113. The continuum placement is too uncertain for this QSO to obtain a good fit or a reliable value for N(H i). The continuum is further complicated by the presence of an HVC at v = 92 km s⁻¹.

RX J1830.3+7312. The moderate curvature of the continuum reduces the quality of the fit to Q = 3, but a third-order polynomial still provides a good fit for this QSO.

Ton S180. The fit is of lower quality due to significant curvature in the continuum. This results from the intrinsic Lyα line centered at 1291 Å, and a rise toward the lower wavelength edge that is seen in many G140M spectra, which is probably due to a calibration problem.

Ton S210. The fit is of lower quality due to multiple factors. One is the moderate S/N. Another is the uncertainty of the continuum, which rises toward the lower wavelengths due to the wing of the intrinsic O vi emission, centered at about 1155 Å.

UGC 12163. This spectrum is relatively noisy, and the wing of the intrinsic Lyα line, centered at 1245 Å, extends above the Galactic Lyα absorption, making a determination of the continuum almost impossible.

VII Zw 118. A flat continuum is present, but the slope combined with a continuum dip between 1195 Å to 1205 Å reduces the quality of the fit.