THE ORIGIN AND DISTRIBUTION OF COLD GAS IN THE HALO OF A MILKY-WAY-MASS GALAXY

We first study the H i distribution and kinematics at z = 0 viewed from different angles. We construct position–position–velocity cubes from the H i density and velocity arrays for three projections: the unrotated frame,¹ edge-on, and face-on. We do this by first creating a velocity range with a 20 km s⁻¹ resolution that includes all of the velocities seen in the simulation and then assigning the simulation cells to a specific bin depending on their line-of-sight velocity (which depends on the projection). In each velocity bin, we compute a column density by adding along the line of sight, which results in an H i density map for positions (x, y) at different velocities. We then take the first and second moments along the velocity axis to make H i column density and velocity maps. We show these maps in Figure 1 with a field of view of 108 × 108 kpc². In addition to the maps, we calculate the amount of H i as a function of projected distance from the center of the galaxy. The different columns in Figure 1 correspond to the three viewing angles. The plots in the first two rows are made using the highest-resolution extraction box (10 levels of refinement; 1 cell size = 0.27 kpc), while the plots in the bottom row were produced from eight levels of refinement (1 cell size = 1.09 kpc), which provides a larger field of view (480 kpc on a side).

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The first row corresponds to H i column density maps, which include column densities above 10¹⁴ cm⁻². The disk of the simulated galaxy spans about 15–20 kpc in radius, and the halo population exhibits a wide array of morphological features that tend to be arranged in complexes. We see compact clouds, elongated features resembling filaments, and interconnected structures. Most of these structures are located in the vicinity of the disk and tend to have higher column densities than the gas seen at larger distances. The features seen at the edges of the figures are part of bigger complexes (see Figure 4 for a bigger field of view at z = 0). The edge-on view depicts a small warp to the right of the disk, which interestingly points in the direction of stripped gas material that is located below the disk. This is suggestive of the warp being caused by the satellite as it approached the disk. The face-on projection shows that many of these clouds were once part of coherent structures that fragmented as they approached the disk.

The velocity maps presented in the second row use the line-of-sight velocity for direct comparison with observations. We only include velocity cells with an H i column density above 10¹⁴ cm⁻² when making the map. We subtract the systemic velocity of the galaxy and plot velocities between −400 and 400 km s⁻¹. The kinematics of the clouds overall follow the rotation of the disk. The big complex seen in the bottom right-hand corner of the unrotated view is at the systemic velocity, indicating that the gas is not moving significantly with respect to the galaxy. The gas in the immediate vicinity of the disk has velocities indicating infall. We also see the warp in the kinematics of the edge-on view, but most of the H i gas in its proximity is near the systemic velocity, not showing a direct connection to the warp. The face-on projection does not show any extreme behavior in the clouds' velocity, since most of the visible gas in that projection is near the systemic velocity. Overall, the three viewing angles show that most of the cold halo gas structures have velocities near the systemic velocity or have velocities that indicate infall. This is consistent with the appearance of gas motions in a movie made from multiple time steps in the simulation, where we see gas accreting onto the disk.

The plots in the third row show how the H i is distributed as a function of projected distance from the center of the galaxy. We created these plots by constructing spherical annuli of 1 kpc in thickness with their origin at the center of the galaxy and calculating the amount of H i in each. We calculate the H i mass out to 240 kpc by using the level 8 refinement extraction box without imposing a column density limit. The distribution is overall declining with distance, with the first 20 kpc corresponding to the gas found in the disk. Note that the H i gas density within the solar neighborhood is consistent with the observed value of 13.2 M_☉ pc⁻² (Flynn et al. 2006). There is at least 10⁵ M_☉ of H i gas in each annulus out to 90 kpc, where there is a peak at around 40–50 kpc, which is due to the complex seen at the edges of the column density maps in the first row. After this, the H i mass distribution continues to decrease to values that are not observable, with isolated spikes that correspond to satellites.

Figure 2 shows the temperature and metallicity maps that serve to further illustrate the properties of the H i at z = 0. Both maps are of the unrotated galaxy and are H i-gas density-weighted along each line of sight. As seen in the temperature map, the galaxy is embedded in a diffuse hot halo of (1–3) × 10⁶ K, whose mean temperature declines slowly with increasing distance from the galactic center and is surrounded by colder fragmented flows with a range of temperatures 8 × 10³–3 × 10⁵ K. The higher than expected H i gas temperatures are due to the weighting by density along the line of sight. We overlay H i contours of 10¹⁴ and 10¹⁸ cm⁻² to show which structures are observable in either H i emission or absorption studies.

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The metallicity map shows that all of the gas in the halo has been enriched to some extent. Most of the cold halo gas has an approximate metallicity of 0.15 Z_☉, in contrast with the gas in the disk that has a metallicity of ∼1 Z_☉. The filaments show a slight level of enrichment (Z = 0.2 Z_☉), as seen in the feature enclosed by Circle A. Another feature in the vicinity of the disk (Circle B) has a metallicity of 0.8 Z_☉, corresponding to stripped gas from a satellite. We can also see an enriched component of around 0.3 Z_☉ perpendicular to the major axis of the disk, which corresponds to traces of the supernova feedback wind (Circle C). As we will discuss later, the metallicity of the gas turns out to be an important indicator of the origin of the neutral gas.

We calculate the H i mass accretion rate at z = 0 as a function of radius from the level 8 extraction box following Peek et al. (2008). We use the following equation:

$$\begin{equation} \dot{M}(R)=\sum _{i=1}^{n(R)} \frac{M_i\textbf {\em V}_i \cdot (-\hat{\mathbf {\textbf {\em r}}}_i)}{dR}, \end{equation} \tag{ 1 }$$

where M_i is the H i mass in the ith cell, $\textbf {\em V}_i$ is the velocity vector of that cell, $\hat{\mathbf {\textbf {\em r}}}_i$ is the radial unit vector, and dR is the thickness of the spherical shell. Figure 3 shows the resulting mass accretion rate as a function of radius. We calculate the mass accretion rate out to 240 kpc but only plot the first 100 kpc since there is no halo H i gas at larger distances. We exclude the first 20 kpc since that is the radius of the disk. As seen in the plot, the mass accretion rate does not linearly decrease with increasing radius. The spike at 63 kpc corresponds to one of the satellites that still has gas at z = 0. Excluding this, there are various peaks at different distances. The two most massive ones are at 20 kpc and 40 kpc, the first one being due to the H i in the immediate vicinity of the disk, while the other one represents the big complex that is moving at the systemic velocity of the galaxy. From this analysis, we estimate an H i mass accretion of ∼0.2 M_☉ yr⁻¹ onto the disk.

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We now examine the origin of the neutral gas in the halo by analyzing a sequence of 117 projection maps taken from a range of redshifts each separated uniformly by 43 Myr, starting at z = 0.5 and ending at z = 0. This time sequence is of the unrotated galaxy, which gives a close to edge-on view with a larger field of view (540 kpc comoving on a side). Figure 4 shows six representative snapshots at z = 0.5, 0.4, 0.3, 0.2, 0.1, and 0.0, showing the H i distribution. These maps show the presence of filamentary flows at all times and also provide some examples of satellites getting stripped as they approach the main galaxy.

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In addition, Figure 5 shows the evolution of the H i halo gas mass in the redshift interval z = 0.5–0.0. We calculate the amount at every five simulation outputs, which corresponds to an interval of 215 Myr. In calculating this number, we do not set a minimum column density and exclude the amount of gas inside the main disk and in the satellite cores, which then leaves a combination of gas from filamentary flows and stripped galaxies. We remove the disk and satellites by excluding the high-column-density gas (>10²⁰ cm⁻²) within a specified distance from the center of the dark matter halo. The solid curve shows that the amount of H i is larger at earlier times, when the disk is actively accreting greater amounts of gas (z > 0.3). At later times, the plot shows a roughly constant amount of H i gas in the halo (∼10⁸ M_☉.) There are some peaks at different times, which could be due to more satellites being stripped or to streams being more active at certain times. We discuss how we studied these two possible origins for the cold halo gas and our findings below.

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3.3.1. Satellites

We identify 20 gas-rich satellites by selecting dark matter subhalos that have both stars and H i gas (above 10¹⁶ cm⁻²). Figure 4 provides examples of what happens to satellites as they approach the main galaxy. In the second snapshot (at z = 0.4) there is a spherical satellite immediately above the disk, which we label S3. This satellite eventually merges with the disk and has no noticeable tidal tails in its H i morphology in this projection. At z = 0.3, we see another satellite approaching the main galaxy from the bottom left corner. This is actually a system of three galaxies (S7, S8, S9), and its H i morphology is visibly disturbed, as evident from the elongated features. This is possibly a product of both tidal interactions within the system and an interaction with the hot medium. The last example is seen at z = 0.1 in the bottom right corner. The circle encloses S13, a satellite that is also disturbed as it enters this hot medium. The long tail seen diagonally extending to the right is likely a product of ram pressure stripping.

We follow the evolution of the settled gas within the dark matter halo of the satellites as they enter our projection map volume. When necessary, we examine the stellar mass distribution maps to locate the exact positions of the satellites. We are able to identify and track the gas inside the satellite cores since it is compact and does not mix easily with the rest of the gas in the halo. We start tracking the gas content from the moment that the satellites appear in the box analyzed until they lose all of their gas, exit the simulation box, merge with the main disk, or until z = 0. The total amount of gas inside the satellites is shown in Figure 5 by the dotted line and is calculated at every five simulation outputs. As seen in the curve, there is more satellite gas at earlier times, showing that there are a greater number of satellites at around z = 0.5, with many of these, such as S3, being gas rich.

Table 1 summarizes our satellite analysis. It lists the satellite IDs, initial and final redshifts, number of years this redshift interval corresponds to, initial and final masses, mass losses, mean mass-loss rates, projected distances of closest approach, and the percentage of mass loss. We see a variety of satellites with a range of initial H i content 10⁴–10⁸ M_☉ that generally make a close encounter with the main galaxy before losing all of their gas or exiting the simulation box. The majority of these galaxies are losing their H i content at different rates. Thirteen satellites lose 50% or more of their original gas content, with nine of these losing 80% or more. The other satellites have lost less than 50% of their initial H i content (5), one kept its gas content, and one actually gained some H i due to an interaction with companion satellites.

Table 1. Mass-loss Rates

Satellite	z0	zf	Time	Mo	Mf	ΔM	$\Delta \dot{M}$	D	Mass
Number			(Myr)	(106 M☉)	(106 M☉)	(106 M☉)	(M☉ yr−1)	(kpc)	Loss
S1	0.495	0.221	2346	0.07	0.01	0.06	2.56 × 10−5	45	86%
S2	0.495	0.300	1579	3.66	0.31	3.35	2.12 × 10−3	66	92%
S3	0.489	0.339	1195	174.00	125.92	48.08	4.02 × 10−2	0	28%
S4	0.477	0.443	256	87.25	46.23	41.02	1.60 × 10−1	31	47%
S5	0.454	0.339	939	30.75	1.31	29.44	3.14 × 10−2	57	96%
S6	0.374	0.282	811	1.92	0.64	1.28	1.58 × 10−3	0	67%
S7	0.363	0.282	725	22.09	12.46	9.63	1.33 × 10−2	0	44%
S8	0.363	0.255	981	20.78	27.28	−6.50	−6.63 × 10−3	0	−31%
S9	0.287	0.238	469	2.20	0.01	2.19	4.67 × 10−3	0	99%
S10	0.334	0	3712	46.25	37.25	9.00	2.42 × 10−3	132	19%
S11	0.247	0	2901	40.80	12.58	28.22	9.73 × 10−3	45	69%
S12	0.221	0.033	2176	8.14	0.13	8.01	3.68 × 10−3	68	98%
S13	0.196	0	2389	53.14	0.71	52.43	2.19 × 10−2	37	99%
S14	0.196	0.172	256	0.10	0.01	0.09	3.52 × 10−4	67	90%
S15	0.196	0	2389	19.22	15.38	3.84	1.61 × 10−3	28	20%
S16	0.196	0.184	128	0.03	0.01	0.02	1.56 × 10−4	81	67%
S17	0.196	0.149	512	0.05	0.05	0	0	282	0
S18	0.180	0.024	1877	0.60	0.01	0.59	3.14 × 10−4	64	98%
S19	0.105	0	1365	17.54	2.42	15.12	1.11 × 10−2	0	86%
S20	0.084	0.056	341	0.03	0.01	0.02	5.87 × 10−5	248	67%

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The satellite H i mass-loss rate is not necessarily correlated with the projected distance of closest approach. The two satellites with largest distances (d > 200 kpc) have mass-loss rates that are negligible. At d < 200 kpc, the mass-loss rate for each satellite varies since it depends on the initial H i mass of the satellite, the three-dimensional location, and the timescale at which it is losing gas. For example, S1 is not very massive in H i and loses gas slowly, which leads to an insignificant rate. On the other hand, S4 is fairly massive and loses its gas on a very short timescale, translating to the highest mass-loss rate. The median mean mass-loss rate for all of the satellites that have lost H i gas is 3.1 × 10⁻³ M_☉ yr⁻¹. If we calculate the median mean mass-loss rate for only the galaxies that have gas at z = 0, we get 9.7 × 10⁻³ M_☉ yr⁻¹. In a Gyr, these numbers would translate to 3.1 × 10⁶ M_☉ and 9.7 × 10⁶ M_☉ of stripped gas, showing that satellite gas provides an important, but not dominant, origin for the cold halo gas.

There is the question, however, of how long the stripped neutral gas will remain in the halo before it gets disrupted or ionized. It is difficult to make a general estimate on this timescale since it depends on unique conditions such as the H i mass, the properties of the medium, and the velocity. We find that the stripped gas in the simulation survives for 2–5 outputs, which corresponds to 80–200 Myr. This agrees with previous detailed studies (e.g., Mayer et al. 2006) showing that stripped gas can survive for 100 Myr.

There are various possible gas-loss mechanisms including tidal interactions, ram pressure stripping, star formation, or supernova feedback. Two satellites, S13 and S19, are prime examples for ram pressure stripping. We have checked the stellar components of the satellites, and they show no evidence of interaction, implying that tidal interactions are minor. In addition to this, we have constructed star formation histories for the five satellites that still have gas at z = 0; none show a recent burst of star formation, indicating that the H i gas was stripped rather than consumed.

3.3.2. Streams

In contrast to our satellite analysis, we cannot track the filaments accurately since they are not tightly bound and consequently do not have a defined structure. They fragment easily and interact with other gas components present in the halo. This limits a precise quantitative assessment of the amount of cold flow gas present at different times. We can, however, qualitatively describe what happens at different redshifts using the movie, snapshots of which are shown in Figure 4. At z = 0.5, the gas in the vicinity of the disk has not settled, and there are three main filaments that are directly feeding the disk, accompanied by various smaller and less dense structures. The main filaments are very diffuse and tend to remain in the same directions (upper left, upper right, and lower center). The small-scale filaments are dominant in the second snapshot, where they are still delivering gas to the disk. Some of these streams are intermittent and appear to be inactive at certain times, but then become a prominent feature again. As time progresses, the filaments tend to fragment, become less dense, and are unable to directly reach the disk as coherent structures. For example, at z = 0.1, there is a dominant filamentary stream to the left of the simulated galaxy with an H i mass of ∼3 × 10⁷M_☉. This stream moves around the galaxy in the counterclockwise direction and collides with a smaller stream that is entering from the bottom of the simulation box, fragmenting the flow and leaving several discrete features concentrated in the bottom right corner of the simulation box at z = 0 with an approximate total mass of 4 × 10⁷M_☉. It is important to note that we do not expect numerical fragmentation since all cells inside filaments satisfy the Truelove criterion.

We look at the metallicity of the gas as a possible way to place a constraint on the amount of cold flow gas at different times. In principle, the satellite gas should have a higher metallicity than the gas due to filamentary streams. Figure 6 shows metallicity maps at different redshifts. These are projected metallicities and are density-weighted along each line of sight. The disk has been removed, and we only plot metallicities for cells that have at least 10¹⁶ cm⁻² in H i column density. We restrict our metallicity range to 0.01–0.50 Z_☉ for better contrast. Many satellites have metallicities above 0.5 Z_☉, which appear white in the metallicity map. As expected, we do see a difference in metallicity between satellite and filamentary stream gas. The best example of this is seen at z = 0.1, where S13 is seen getting stripped in the middle right of the plot and where various cold streams are approaching the disk. The metallicity of the streams is around 0.2 Z_☉, and the stripped gas from the satellites generally has values above 0.3 Z_☉.

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Despite the distinction here, there is also mixing going on, which is mostly due to supernova feedback from the galactic disk interacting with the filaments. At z = 0, the gas in the bottom right corner of the plot originates from filamentary streams contaminated by the supernova wind, as shown by the temperature and metallicity movies. Another example of contamination is seen in the fourth panel (z = 0.3), where the elongated features with higher metallicities above the disk are not associated with satellite gas. Some of these features condensed out of filamentary streams but were contaminated by the higher-metallicity supernova wind. The feature enclosed by the white circle is a product of mixing between cold flow material and a fragmented supernova wind shell.

We conclude from this analysis that all lower metallicity gas (≲0.2 Z_☉) corresponds to cold flows in the simulation, but not all higher metallicity gas is due to satellites. This allows us to place a minimum on the amount of gas that is due to cold flows at different redshifts. The dashed line in Figure 5 represents the total amount of H i at each redshift with metallicities less than 0.2 Z_☉. This is a minimum since the amount of gas originating from cold streams is higher, but some of it is indistinguishable from satellite gas in terms of metallicity. The peaks in this curve roughly correspond to the maxima in the solid curve, showing that active cold flows dominate the H i content at those particular times. The relative amount of this low-metallicity gas with respect to the total amount of H i halo gas at different redshifts ranges between 25% and 75%.

We first compare the simulation with the deep H i observations of M31 carried out by Thilker et al. (2004, hereafter T04). We choose M31 since it is the nearest spiral galaxy of similar size and mass to the Milky Way and several properties of the clouds can be directly compared to the simulation. The wide-field H i observations (94 × 94 kpc²) were conducted with the Green Bank Telescope (GBT) and achieve an rms flux sensitivity of 7.2 mJy beam⁻¹ (1σ), which corresponds to a column density of 2.5 × 10¹⁷ cm⁻² over 18 km s⁻¹. The data include velocities between −827 and 226 km s⁻¹, where −300 km s⁻¹ is the systemic velocity of the system. The authors exclude data with velocities more positive than −160 km s⁻¹ due to Galactic emission. After removing the disk H i emission, they find at least 20 sources in the halo above 2σ within 50 kpc of the disk and with a total H i mass of 3 × 10⁷ M_☉. Follow-up observations carried out at Effelsberg by Westmeier et al. (2008) indicate that there are no clouds above a 3σ mass detection limit of 8 × 10⁴ M_☉ at distances greater than 50 kpc from the disk.

Figure 7 shows the total H i distribution of our simulated galaxy at z = 0, matching the viewing geometry of M31. We have rotated the galaxy accordingly to correspond to M31's inclination (i = 78°) and position angle (P.A. = 35°) and use the same contour levels as T04 [(0.5, 1, 2, 10, 20) × 10¹⁸ cm⁻²]. If we exclude the gas in the disk, we find a total H i mass of 1.1 × 10⁸ M_☉. We detect about 30 discrete sources, and many are arranged in complexes. Some of the structures are compact clouds with masses within the range of 10⁴–10⁶ M_☉, while others have an elongated filamentary morphology. These filamentary features have higher masses, such as the nearly horizontal structure located about 15 kpc above the disk, which has an H i mass of 1.1 × 10⁷ M_☉.

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The morphology, distribution, and kinematics of the clouds are similar to the observations. T04 detect a wide array of morphological features with similar masses to the ones seen in the simulation. The cold halo gas in the simulation does seem to extend beyond 50 kpc from the disk, as is seen in the large complex south of the disk that extends to about 70 kpc. If we limit the H i to only 50 kpc to better compare with observations, we detect about 88% of the value quoted above, which means that the bulk of the gas is located within 50 kpc of the galaxy. One issue of concern we see in this comparison is that our contour levels are closely spaced together, showing that our simulation is overpredicting values for the column density for some of the clouds. This discrepancy might be a consequence of not including interstellar UV in the simulation (Bland-Hawthorn & Maloney 2002; Maloney & Putman 2003). Kinematically, the comparison between the GBT data and the simulation is more complicated since the observations mostly probe redshifted velocities to avoid Galactic emission. In general, T04 find that the discrete features have velocities in between −515 and −172 km s⁻¹, which can be compared to M31's V_sys = −300 km s⁻¹. In contrast, M31's H i gas with stream-like morphology has values within ∼80 km s⁻¹ of the systemic velocity. This is roughly consistent with the simulated H i halo gas, where many of the diffuse streams are close to the systemic velocity and the compact features have a wider range in velocities. We also compare the covering fraction of M31 with the simulation in Section 4.3.

The population of HVCs in the Milky Way has been extensively studied over the past decades. We can compare the distribution, amount, and metallicities of HVCs to the cold halo gas seen in the simulation. As various surveys have shown (e.g., Putman et al. 2002; Kalberla et al. 2005; Peek et al. 2011), the HVCs are grouped into large complexes, which is consistent with the distribution of the gas in the simulation. If we exclude the gas in the Magellanic Stream and place all of the HVCs at a distance of 10 kpc, there is a total of ∼3 × 10⁷ M_☉ in H i (Putman et al. 2012). In addition to this, HVCs in the Milky Way exhibit metallicities at around ∼0.1–0.3 Z_☉ (Wakker 2001), which is consistent with the values of the simulated H i gas.

In addition to the Milky Way and M31, other spirals have been shown to have cold gas in their halos. Another well-studied galaxy is NGC 891 (e.g., Oosterloo et al. 2007), which has been shown to have an extended and massive halo of H i with a mass of 1.2 × 10⁹ M_☉ when both the halo and disk–halo interface region are examined, accounting for about 30% of the total mass of neutral gas found in the galaxy. Its distribution of cold gas is not smooth, showing clouds with H i masses of ∼1 × 10⁶ M_☉ and an extended filament extending to the northwest of the galaxy. If the disk and disk–halo interface region are subtracted from the H i distribution map, the remaining neutral gas is ∼1 × 10⁸ M_☉ in the form of clouds and the extended filament (Sancisi et al. 2008), resembling what is seen in the simulation.

Sancisi et al. (2008) quote that ∼25% of field galaxies have extra-planar features of ≳10⁸ M_☉. This number would go up to 50% if warps and kinematic deviations from the disk are included (Sancisi et al. 2008; Haynes et al. 1998). Ongoing deep H i observations (e.g., HALOGAS; Heald et al. 2011) are further examining the cold halo gas of a larger sample of spirals. We expect that these will only detect part of the features seen in the simulation. First, the column densities being probed in the observations are higher (>10¹⁹ cm⁻²) than the bulk of the simulated clouds. In addition to this, the average column densities in the simulation may be lower since we did not include interstellar UV. Future H i deep surveys will allow us to compare the simulation with a larger sample of galaxies.

We use the level 8 extraction box to calculate the projected covering fraction out to 240 kpc for different column densities at z = 0 (see Figure 8). Here we plot four representative values: 1 × 10¹³, 1 × 10¹⁵, 5 × 10¹⁷, 5 × 10¹⁹ cm⁻². We also include a curve showing the exponential fit of f = 2.1exp (− d/12) done by Richter (2011) to the covering fraction for M31.

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Most of the curves show the covering fraction exponentially decreasing as a function of distance. We calculated covering fractions for intermediate values between 10¹³ and 10¹⁵ cm⁻² and found the exponential drop-off to hold true for $N_{{\rm H\,\mathsc{i}}} > 5 \times 10^{13}$ cm⁻². Below this value (shown by the blue curve), we find covering fractions near unity out to 50 kpc, with the value slowly decreasing afterward to a minimum of 0.85 at 240 kpc. This indicates that the halo of the simulated galaxy is filled with low-density gas. In contrast, the covering fraction for $N_{{\rm H\,\mathsc{i}}} > 1 \times 10^{15}$ cm⁻² is less than 0.2 beyond 50 kpc, indicating that denser absorbers are not a common feature in the extended halo.

The calculated covering fractions agree well with absorption-line work and H i observations. There are several studies (Bowen et al. 2002; Wakker & Savage 2009; Prochaska et al. 2011) that have detected Lyα absorption against background quasars, probing low-density gas in the range 10¹⁵ cm⁻² > $N_{{\rm H\,\mathsc{i}}} > 10^{13}$ cm⁻². All of these studies find covering fractions close to 100% within 300 kpc of a galaxy, which agrees with the blue curve. For higher-density gas, we can compare our results to the covering fractions from the H i data of M31. The exponential fit found by Richter (2011) expresses the covering fraction as a function of distance. If we compare this fit to the cyan curve ($N_{{\rm H\,\mathsc{i}}} > 5 \times 10^{17}$ cm⁻²), which is the lowest contour value in T04, we find similar values for the covering fraction. The observational fit to larger distances is not reliable since there are no detections at d > 60 kpc. The covering fraction of HVCs in the Milky Way above 7 × 10¹⁷ cm⁻² is 0.37 (Murphy et al. 1995), which is roughly consistent with the cyan curve at ∼30 kpc.