The warm Gaseous Disk and the Anisotropic Circumgalactic Medium of the Milky Way

TL;DR

This study models the warm gaseous disk and circumgalactic medium of the Milky Way using absorption line data, revealing the disk's structure, mass, and anisotropic distribution of the CGM, with implications for galactic feedback processes.

Contribution

The paper introduces a 2D disk-CGM model for warm gas in the Milky Way, constraining its density profile, mass, and anisotropic distribution based on Si IV and O VI absorption data.

Findings

The warm gas disk has a scale height of 2.6 kpc and a scale length of 6.1 kpc.

The total warm gas mass in the disk is approximately 4 million solar masses.

The CGM shows higher column density perpendicular to the disk, indicating anisotropic distribution.

Abstract

The warm ($\log~T\approx5$) gas is an important gaseous component in the galaxy baryonic cycle. We built a 2-dimension disk-CGM model to study the warm gas distribution of the Milky Way (MW) using the absorption line surveys of Si IV and O VI. In this model, the disk component of both ions has the same density profile ($n(r, z)=n_0\exp(-|z|/z_0)\exp(-r/r_0)$) with a scale height of $z_0=2.6\pm0.4\rm~kpc$ and a scale length of $r_0=6.1\pm1.2\rm~kpc$. For this disk component, we calculate the warm gas mass of $\log(M/M_\odot)=(7.6\pm0.2)-\log\frac{Z}{Z_\odot}$. The similar disk density profiles and total masses of Si IV and O VI-bearing gas set constraints on the ionization mechanisms. We suggest that the warm gas disk might be dominated by the Galactic fountain mechanism, which ejects and recycles gas to set both the scale height and the scale length of the warm gas disk. The CGM component in our model has a dependence on Galactic latitude with a higher column density along the direction perpendicular to the Galactic plane ($b=90^\circ$) than the column density along the radial direction ($b=0^\circ$). The column density difference between these two directions is $0.82\pm0.32\rm~dex$ at $6.3\sigma$ for both ions. This difference may be due to the enrichment of Galactic feedback to the entire CGM, or an additional interaction layer between the warm gas disk and the CGM; existing data cannot distinguish between these two scenarios. If this higher column density at $b=90^\circ$ is for the entire CGM, the total warm CGM mass is $\log(M/M_\odot)\approx(9.5-9.8)-\log\frac{Z}{0.5Z_\odot}$ within the MW virial radius of $250\rm~kpc$.

Tables

Table 1. Table 1 Sample

Ion	$𝒩_{\mathrm{star}}^{}{}_{}{}^a$			$𝒩_{\mathrm{AGN}}^{}{}_{}{}^a$			$\sigma _p^{}{}_{}{}^b$	Ref.
Si IV	49	13	13		$\dots$		0.30	SW09
		$\dots$		119	11	0	0.13	Zheng19
O VI	73	0	4		$\dots$		0.23	SW09
		$\dots$		93	0	8	0.15	Savage03

Table 2. Table 2 The Disk and CGM model for Si IV and O VI

Model	$\log n_0$	$r_0$	$z_0$	$\log N_{\mathrm{mp}}^{\mathrm{CGM}}$	$\log N_{\mathrm{nd}}^{\mathrm{CGM}}$	red. ${\chi }^2$ (dof)	$\log n_{\odot }^{\mathrm{disk}}$	$\log n_{\odot }{z}_0$	$\log M_{\mathrm{disk}}$
	$\left({\mathrm{cm}}^{-3}\right)$	kpc	kpc	$\left({\mathrm{cm}}^{-2}\right)$	$\left({\mathrm{cm}}^{-2}\right)$		$\left({\mathrm{cm}}^{-3}\right)$	$\left({\mathrm{cm}}^{-2}\right)$	$\left(M_{\odot }\right)$
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
The 2-D disk-CGM models with isotropic CGM of Si IV
$R_E{Z}_E$	$-7.82\pm 0.24$	$4.3\pm 1.2$	$2.5\pm 0.6$	$13.18\pm 0.12$		1.232 (201)	-8.68	13.21	3.78
$R_G{Z}_E$	$-8.21\pm 0.16$	$8.3\pm 1.5$	$2.8\pm 0.7$	$13.14\pm 0.15$		1.182 (201)	-8.67	13.26	3.70
$R_E{Z}_G$	$-7.74\pm 0.23$	$3.7\pm 0.9$	$3.3\pm 0.6$	$13.16\pm 0.11$		1.267 (201)	-8.73	13.22	3.80
$R_G{Z}_G$	$-8.21\pm 0.15$	$7.4\pm 1.1$	$3.5\pm 0.7$	$13.12\pm 0.14$		1.202 (201)	-8.78	13.26	3.66
The 2-D disk-CGM models with isotropic CGM of O VI
$R_E{Z}_E$	$-7.19\pm 0.20$	$5.7\pm 1.8$	$2.3\pm 0.6$	$13.98\pm 0.14$		1.210 (174)	-7.84	14.02	4.39
$R_G{Z}_E$	$-7.51\pm 0.13$	$9.8\pm 1.9$	$2.6\pm 0.7$	$13.93\pm 0.18$		1.196 (174)	-7.83	14.08	4.29
$R_E{Z}_G$	$-7.18\pm 0.20$	$5.4\pm 1.6$	$2.4\pm 0.5$	$14.03\pm 0.09$		1.240 (174)	-7.86	13.95	4.31
$R_G{Z}_G$	$-7.50\pm 0.12$	$8.8\pm 1.4$	$2.7\pm 0.6$	$13.98\pm 0.12$		1.222 (174)	-7.90	14.01	4.16
The 2-D disk-CGM models with anisotropic CGM of Si IV
$R_E{Z}_E$	$-7.93\pm 0.20$	$5.2\pm 1.4$	$2.6\pm 0.6$	$12.62\pm 0.40$	$13.32\pm 0.08$	1.128 (200)	-8.64	13.27	3.86
$R_G{Z}_E$	$-8.25\pm 0.14$	$9.2\pm 1.6$	$2.7\pm 0.7$	$12.67\pm 0.44$	$13.28\pm 0.10$	1.106 (200)	-8.62	13.29	3.74
$R_E{Z}_G$	$-7.89\pm 0.19$	$4.6\pm 1.0$	$3.4\pm 0.6$	$12.46\pm 0.45$	$13.32\pm 0.07$	1.138 (200)	-8.69	13.28	3.86
$R_G{Z}_G$	$-8.27\pm 0.12$	$8.4\pm 1.2$	$3.5\pm 0.7$	$12.48\pm 0.52$	$13.28\pm 0.08$	1.105 (200)	-8.72	13.31	3.70
The 2-D disk-CGM models with anisotropic CGM of O VI
$R_E{Z}_E$	$-7.35\pm 0.15$	$8.0\pm 2.5$	$2.6\pm 0.6$	$13.21\pm 0.58$	$14.19\pm 0.08$	1.104 (173)	-7.82	14.08	4.55
$R_G{Z}_E$	$-7.60\pm 0.09$	$12.2\pm 2.4$	$2.7\pm 0.6$	$13.18\pm 0.66$	$14.16\pm 0.10$	1.108 (173)	-7.81	14.11	4.40
$R_E{Z}_G$	$-7.33\pm 0.15$	$7.2\pm 2.2$	$2.9\pm 0.5$	$13.21\pm 0.50$	$14.22\pm 0.07$	1.121 (173)	-7.84	14.05	4.49
$R_G{Z}_G$	$-7.59\pm 0.09$	$10.7\pm 1.8$	$3.0\pm 0.6$	$13.18\pm 0.57$	$14.19\pm 0.08$	1.119 (173)	-7.84	14.08	4.28

Table 3. Table 3 The Joint Fitting Results of Si IV and O VI

Ion	$\log n_0$	$r_0$	$z_0$	$\Delta \log N^{\mathrm{CGM}}{}^a$	$\log N_{\mathrm{nd}}^{\mathrm{CGM}}$	red. ${\chi }^2$ (dof)	$\log n_{\odot }^{\mathrm{disk}}$	$\log n_{\odot }^{\mathrm{disk}}{z}_0$	$\log M_{\mathrm{disk}}$
	$\left({\mathrm{cm}}^{-3}\right)$	kpc	kpc	$\mathrm{dex}\left({\mathrm{cm}}^{-2}\right)$	$\left({\mathrm{cm}}^{-2}\right)$		$\left({\mathrm{cm}}^{-3}\right)$	$\left({\mathrm{cm}}^{-2}\right)$	$\left(M_{\odot }\right)$
Si IV	$-8.02\pm 0.12$	$6.1\pm 1.2$	$2.6\pm 0.4$	$0.82\pm 0.32$	$13.32\pm 0.07$	1.126 (376)	-8.63	13.28	3.91
O VI	$-7.22\pm 0.12$	$\dots$	$\dots$	$\dots$	$14.17\pm 0.08$	$\dots$	-7.83	14.08	4.46
Si IV	$-8.04\pm 0.13$	$6.1\pm 1.2$	$2.9\pm 0.5$	$0.86\pm 0.43$	$13.30\pm 0.08$	1.123 (374)	-8.64	13.30	3.93
O VI	$-7.21\pm 0.12$	$\dots$	$2.3\pm 0.5$	$0.74\pm 0.32$	$14.20\pm 0.08$	$\dots$	-7.81	14.04	4.42
Si IV	$-7.92\pm 0.16$	$5.2\pm 1.2$	$2.6\pm 0.4$	$0.69\pm 0.32$	$13.32\pm 0.07$	1.117 (374)	-8.64	13.26	3.86
O VI	$-7.36\pm 0.13$	$8.0\pm 2.3$	$\dots$	$0.99\pm 0.49$	$14.19\pm 0.08$	$\dots$	-7.82	14.08	4.56
Si IV	$-7.96\pm 0.16$	$5.5\pm 1.2$	$2.7\pm 0.5$	$0.81\pm 0.32$	$13.32\pm 0.08$	1.118 (374)	-8.63	13.29	3.89
O VI	$-7.32\pm 0.15$	$7.5\pm 2.1$	$2.4\pm 0.5$	$\dots$	$14.18\pm 0.08$	$\dots$	-7.82	14.06	4.50

Table 4. Table 4 The Fitting Results of the North-South Asymmetry

Ion	$\log n_0$ a	$\Delta \log n_0^{\mathrm{NS}}$ b	$r_0$	$z_0^N$	$z_0^S$	$\Delta \log N^{\mathrm{CGM}}$	$\log N_{\mathrm{nd}}^{\mathrm{CGM}}$ a	$\Delta \log N_{\mathrm{CGM}}^{\mathrm{NS}}$ b	red. ${\chi }^2$ (dof)
	$\left({\mathrm{cm}}^{-3}\right)$	$\left({\mathrm{cm}}^{-3}\right)$	kpc	kpc	kpc	$\mathrm{dex}\left({\mathrm{cm}}^{-2}\right)$	$\left({\mathrm{cm}}^{-2}\right)$	$\left({\mathrm{cm}}^{-2}\right)$
Si IV	$-8.01\pm 0.12$	$0.06\pm 0.05$	$6.0\pm 1.1$	$3.4\pm 0.6$	$2.5\pm 0.4$	$0.92\pm 0.46$	$13.21\pm 0.14$	$-0.09\pm 0.15$	1.034 (373)
O VI	$-7.20\pm 0.12$	$\dots$	$\dots$	$\dots$	$\dots$	$\dots$	$13.96\pm 0.16$	$\dots$	$\dots$
Si IV	$-7.97\pm 0.12$	$0.14\pm 0.02$	$6.0\pm 1.0$	$2.9\pm 0.4$	$\dots$	$0.98\pm 0.46$	$13.26\pm 0.08$	$\dots$	1.045 (375)
O VI	$-7.17\pm 0.11$	$\dots$	$\dots$	$\dots$	$\dots$	$\dots$	$14.06\pm 0.11$	$\dots$	$\dots$
Si IV	$-8.03\pm 0.12$	$\dots$	$5.9\pm 1.1$	$3.5\pm 0.5$	$2.3\pm 0.4$	$0.83\pm 0.41$	$13.26\pm 0.08$	$\dots$	1.039 (375)
O VI	$-7.23\pm 0.12$	$\dots$	$\dots$	$\dots$	$\dots$	$\dots$	$14.04\pm 0.11$	$\dots$	$\dots$
Si IV	$-7.95\pm 0.13$	$\dots$	$5.2\pm 1.0$	$2.5\pm 0.4$	$\dots$	$0.56\pm 0.20$	$13.40\pm 0.06$	$0.29\pm 0.09$	1.063 (375)
O VI	$-7.18\pm 0.12$	$\dots$	$\dots$	$\dots$	$\dots$	$\dots$	$14.25\pm 0.06$	$\dots$	$\dots$

Equations

$$N(l,b,d)$$

$$N(l,b)$$

$$n(r,z)=n_0{f}_r(r)f_z(z),$$

$$\log N_{\mathrm{CGM}}(b)=\sqrt{{\log }^2{N}_{\mathrm{mp}}{\cos }^{2}b+{\log }^2{N}_{\mathrm{nd}}{\sin }^{2}b},$$

$${\mathcal{N}}_{\mathrm{total}}^{\mathrm{disk}}=n_0{\int }_0^{R_{\mathrm{vir}}}\mathrm{d}r\exp (-\frac{r}{r_0}){\int }_{-R_{\mathrm{vir}}}^{R_{\mathrm{vir}}}\mathrm{d}z\exp (-\frac{|z|}{z_0}).$$

$$\log (M_{\mathrm{H}}/M_{\odot }{)}_{\mathrm{SiIV}}^{\mathrm{disk}}=(7.6\pm 0.1)-\log \frac{f_{\mathrm{SiIV}}}{0.2}-\log \frac{Z}{Z_{\odot }},$$

$$\log (M_{\mathrm{H}}/M_{\odot }{)}_{\mathrm{OVI}}^{\mathrm{disk}}=(7.6\pm 0.2)-\log \frac{f_{\mathrm{OVI}}}{0.1}-\log \frac{Z}{Z_{\odot }},$$

$$\log (M_{\mathrm{H}}/M_{\odot }{)}_{\mathrm{SiIV}}^{\mathrm{CGM}}$$

$$\log (M_{\mathrm{H}}/M_{\odot }{)}_{\mathrm{OVI}}^{\mathrm{CGM}}$$

$$\frac{M_{\beta }}{M_{\mathrm{u}}}=\frac{1-3\beta }{1-\beta }\frac{R_{\max }-R_{\min }}{R_{\max }^3-R_{\min }^3}\frac{R_{\max }^{3-3\beta }-R_{\min }^{3-3\beta }}{R_{\max }^{1-3\beta }-R_{\min }^{1-3\beta }},$$

Full text

1. Introduction

The gaseous baryons in galaxies can be found in both gaseous disks (the interstellar medium; Dickey & Lockman 1990) and gaseous halos (the circumgalactic medium; CGM; Putman et al. 2012; Tumlinson et al. 2017). The gaseous disks are roughly cospatial with the galaxy stellar disks, whereas the CGM are surrounding the stellar disks. H I 21 cm line surveys or ultraviolet (UV) observations reveal that the gaseous disks can extend beyond the stellar disks (up to $30-50$ kpc along the major axis) with masses from $10^{7}~{}M_{\odot}$ to $\lesssim 10^{10}~{}M_{\odot}$ for low-redshift galaxies ($z\lesssim 0.2$; Oosterloo et al. 2007; Bregman et al. 2018). With statistical assembly of Quasar-galaxy pairs in UV absorption lines, the CGM can be detected out to large radii ($>150\rm~{}kpc$) and contribute a large amount of baryonic materials ($\gtrsim 10^{10}~{}M_{\odot}$) for galaxies at $z\lesssim 0.5$ (Stocke et al., 2013; Werk et al., 2014; Lehner et al., 2015; Keeney et al., 2017).

The existence of these gaseous components is important for galaxy evolution by mediating both accretion and feedback processes (Mo et al., 2010). For $L\gtrsim L^{*}$ galaxies, the CGM is normally volume-filled by warm-hot gas ($\log T\approx 5-7$) together with discrete cool gas clouds ($\log T\approx 4-5$), which is mainly shock-heated by gas accreted from the intergalactic medium (IGM) and altered by feedback processes from host galaxies (Cen & Ostriker, 2006). The existence of the CGM could prevent direct accretion from the IGM; instead, it provides cooling materials from itself by thermal or gravitational instabilities (Kereš et al., 2009). These cooling flows from the CGM supplement the gaseous disk, and sustain the star formation activities in the stellar disk (Lehner & Howk, 2011; Li et al., 2014; Borthakur et al., 2015; Qu & Bregman, 2018). In turn, the stellar and active galactic nuclei (AGN) feedback could enrich the CGM by ejecting gas, energy, and metals (Borthakur et al., 2013; Fielding et al., 2017; Oppenheimer et al., 2018).

The warm ($\log~{}T\approx 5$) gas is of special importance, since it is at transitional temperatures (the peak of the radiative cooling curve). In this temperature range, the gas could be the interaction layer between the cool gas and the warm-hot gas. These interactions are mostly associated with galactic feedback (e.g., outflows) or gas accretion onto the disk (e.g., accretion shocks). Therefore, the distribution of the warm gas is crucial to investigate these fundamental processes. Observationally, the warm gas distribution could be divided into two components: the warm gas disk and the warm CGM. The warm gas disk has been studied for both other galaxies (Boettcher et al., 2016; Zheng et al., 2017; Qu et al., 2019) and the Milky Way (MW; Howk et al. 2002; Finkbeiner 2003; Savage et al. 2003; Wakker et al. 2012). For other galaxies, the warm gas disk is detected in various observations. Direct imaging on nearby edge-on galaxies has detected the warm gas disk at radii of $\approx 1-10\rm~{}kpc$ in X-ray band, UV band, or nebula emission lines (Rand et al., 2008; Li & Wang, 2013; Boettcher et al., 2016; Hodges-Kluck et al., 2016). However, for the warm CGM, these observations are limited at larger radii ($>50$ kpc) due to the current instrument limitations and the low surface brightness of the diffuse ionized gas. An alternative is to use UV absorption lines against the continua of background AGN/stellar objects to detect gas with column densities as low as $\approx 10^{12}~{}\rm cm^{-2}$ (Tumlinson et al. 2017 and reference therein). The warm gas is traced by intermediate-high ionization state UV ions, such as Al III, Si IV, C IV, and O VI (Stocke et al., 2013; Werk et al., 2014; Johnson et al., 2015; Lehner et al., 2015; Zheng et al., 2017; Qu et al., 2019). However, extragalactic absorption-line studies are all limited by the sample of available sightlines for individual galaxies.

The MW is a unique target to study the warm gas distribution with hundreds of sightlines over the sky mapping both the disk and the CGM (Savage et al., 1997; Howk et al., 2002; Wakker et al., 2003; Savage & Wakker, 2009; Lehner & Howk, 2011; Lehner et al., 2012; Wakker et al., 2012; Fox et al., 2014, 2015; Bordoloi et al., 2017; Karim et al., 2018). Previous studies indicated that the warm gas could be discrete kpc-size clouds, which can be detected as intermediate-velocity clouds (IVCs) or high-velocity clouds (HVCs; Sembach et al. 2003; Wakker et al. 2003; Fox et al. 2004; Shull et al. 2009; Wakker et al. 2012; Werk et al. 2019). At large scales, the warm gas distribution in the MW was modeled as a plane-parallel slab with only one dimensional (1-D) variation over the vertical direction (perpendicular to the disk) as an exponential function of $n(z)=n_{0}\exp(-|z|/z_{0})$, where $z_{0}$ is the scale height (Savage & Wakker 2009 and references therein; hereafter SW09). The scale height of the warm gas disk is measured using column densities against UV-bright stars at different $z$-heights and AGNs. In this model, the stellar sightlines are used to estimate the ion density at the mid-plane of the disk. Combining with the sightlines toward AGNs (determining the maximum projected column density), the scale heights of the plane-parallel slab model are obtained for various ions, where no CGM component is considered. However, more recent observations reveal the CGM also contributes to the column densities of the intermediate-high ionization state ion measured in the AGN sightlines (Werk et al., 2014; Johnson et al., 2015; Lehner et al., 2015; Stocke et al., 2017; Zahedy et al., 2019). As pointed out by Zheng et al. (2019, hereafter Zheng19), the MW CGM contributes to a significant amount of column density to the Si IV absorption lines measured with 132 AGN sightlines obtained by the Cosmic Origins Spectrograph (COS; Green et al. 2012) on the Hubble Space Telescope (HST). They demonstrated that the warm gas in the MW, as observed with all-sky AGN sightlines, should be modeled with a two-component model (i.e., a disk component and a CGM component). In their two-component model, the disk component follows the 1-D plane-parallel slab model as SW09 adopted, and the halo component is modeled as a uniform global background.

This paper is built upon the two-component model of Zheng19 to develop a disk-CGM model that accounts for both the radial and vertical density profiles of the disk. This model is applied to Si IV and O VI, which are typical ions tracing the transitional temperature gas (Savage et al., 1997; Wakker et al., 2012). In Section 2, we summarize the data used in this study, which includes column density measurements from MW stellar sightlines by SW09, and from all-sky AGN sightlines by Savage et al. (2003, hereafter Savage03) and Zheng19. We introduce our model in Section 3, and show that the inclusion of the disk radial profile can alleviate the disagreement between the plane-parallel slab model based on the stellar sample (SW09) and the two-component model based on the AGN sample (Zheng19). The anisotropic CGM model is also introduced in Section 3, showing that the MW is likely to have a warm CGM with anisotropic column density distribution. In Section 4, we discuss the implication of this work on the warm gas disk (Section 4.2), the warm CGM of the MW (Section 4.3), and the north-south asymmetry of the MW warm gas absorption features (Section 4.4). The key results are summarized in Section 5.

2. Data

At the temperature of $\approx 10^{5}\rm~{}K$, the transitional gas can be traced by intermediate-high ionization state ions, such as Si IV with ionization potential of 33.5 - 45.1 eV, C IV (47.9 - 64.5 eV), N V (77.5 - 97.9 eV), or O VI (113.9 - 138.1 eV). These ions are detectable in absorption against the continua of background UV-bright stars or AGNs. For observations of the MW warm gas disk and CGM, the stellar sightlines are normally at low Galactic laitutdes ($|b|\lesssim 5^{\circ}$; SW09), whereas the AGNs are at high Galactic latitudes ($|b|\gtrsim 30^{\circ}$; \al@Savage:2003aa, Zheng:2019aa; \al@Savage:2003aa, Zheng:2019aa). The stellar sightlines are employed to measure the mid-plane density of the disk, while the AGN sightlines can trace the large-scale variation of the disk (e.g., scale height of the disk) and the CGM. Therefore, these two samples are equally important to constrain both the gaseous disk shape and the CGM contribution. We only consider the ions that have high $S/N\gtrsim 15$ samples for both disk stars and AGNs of sample sizes $\mathcal{N}\gtrsim 100$.

Based on this criterion, Si IV and O VI are the two ions, which have both stellar sightlines from SW09 and AGN sightlines from Savage03 and Zheng19. In this study, we do not consider C IV, since the current largest AGN sample ($\mathcal{N}\approx 30-40$) from Wakker et al. (2012) does not have sufficient sightlines to obtain a good fitting result. All of the used three samples (\al@Savage:2003aa, Savage:2009aa, Zheng:2019aa; \al@Savage:2003aa, Savage:2009aa, Zheng:2019aa; \al@Savage:2003aa, Savage:2009aa, Zheng:2019aa) have similar velocity ranges ($|v|\lesssim 100~{}\rm km~{}s^{-1}$) for the measurement of the column density, so we only study the low-intermediate velocity gas of the MW (without HVCs $|v|\gtrsim 100~{}\rm km~{}s^{-1}$). Zheng et al. (2015) showed that a significant amount of the CGM is at low-intermediate velocities using a MW-mass hydrodynamic simulation (Joung et al., 2012), so we expect that we could detect both the warm gas disk and CGM using these samples.

For the stellar sample, SW09 summarized the FUSE, IUE, and Copernicus sightlines toward the 109 MW stars, 25 AGNs, and 6 LMC/SMC stars with good measurements ($\sigma_{N}<0.4$ dex) or limits of Si IV and O VI. This sample is mainly based on the Galactic O VI surveys, such as Bowen et al. (2008) for low-latitude disk stars and Zsargó et al. (2003) for halo stars. Compared to the Bowen et al. (2008) sample, SW09 excluded sightlines with large uncertainties of O VI and other transitional ions (e.g., C IV, Si IV; Savage et al. 2001). The excluded sightlines are mainly stellar sightlines within 1 kpc. Since SW09 also included sightlines toward halo stars at $|z|>1$ kpc, this sample is better to constrain the scale height of the disk component. We exclude sightlines that might be contaminated by foreground H II regions as marked out by SW09. Besides the Galactic stellar sample, SW09 also included 6 stars in LMC/SMC, which are not used in our analyses. This is because one needs to assume the radial density profile of the MW CGM to model the ion column densities from stars at the distance of $50-60\rm~{}kpc$, which is highly uncertain. Therefore, we do not implement this variation in our model, and omit the sightlines toward LMC/SMC stars. The final stellar sample used in our analyses is composed of 77 sightlines, 75 of which have good column density ($\log N$) measurements or limits for Si IV, and all of them have good $\log N$ values or limits for O VI (Table 1).

For the AGN sample, we adopt two data sets. We make use of the Si IV measurements from the COS-GAL sample (Zheng19), which is based on the Hubble Spectroscopic Legacy Archive (Peeples et al., 2017). Moreover, we retrieve the O VI measurements from the FUSE observations analyzed by Savage03. We do not include the AGN sample in SW09, since it has a large overlap with the Zheng19 sample (18/25) and the Savage03 sample (22/25). The final AGN sample includes 130 sightlines for Si IV and 101 sightlines for O VI (Table 1).

3. Models and Results

3.1. Previous Models

Previously, the warm gas disk (e.g., traced by Si IV and O VI) of the MW is modeled as an 1-D plane-parallel slab model (Jenkins 1978; Bowen et al. 2008; SW09). The model only has one dimensional variation: the density distribution of the warm gas over the disk height $z$ as an exponential function of $n(z)=n_{0}\exp(-|z|/z_{0})$, where $n_{0}$ is the ion density at the mid-plane and $z_{0}$ is the scale height. The current stellar sightlines are normally close to the Sun with a distance of $d\lesssim 2$ kpc, which implies that the average ion densities traced by these sightlines do not vary significantly at large scales. Therefore, the stellar sightlines mainly determine the average density of the mid-plane around the solar system ($n_{\odot}$). For the AGN sightlines, both the disk and the CGM are detected to show the large-scale variation. Based on AGN sightlines, one could obtain the maximum projected column density along the $z$-direction ($N\sin|b|$) for the disk component, since in the plane-parallel slab model, the CGM contribution is ignored. Combining these two measurements, the scale height $z_{0}$ in the plane-parallel slab model is derived as $N\sin|b|/n_{\odot}$ around the solar neighborhood.

This model works well for the sample dominated by stellar sightlines, such as SW09, which has $\approx 100$ stellar sightlines and $\approx 20$ AGN sightlines. However, this model might have two problems with more and more AGN sightlines obtained by HST/COS. First, for sightlines toward AGNs, the contribution from the MW CGM is not considered, which has been shown as an important component for low redshift galaxies ($z\lesssim 0.5$; e.g., Stocke et al. 2013; Werk et al. 2014; Johnson et al. 2015). Second, the AGN sightlines could trace the large scale variation of the disk in both the vertical and radial directions, so the plane-parallel slab model might lead to divergence from the observations.

The first problem is partially solved in Zheng19 by introducing an isotropic CGM component ($N_{\rm CGM}$) into the plane-parallel slab model; their model is referred as the two-component disk-CGM model hereafter. They applied the two-component disk-CGM model to fit the Si IV column density distribution measured along 130 AGN sightlines across the Galactic sky, and found a significant contribution of the MW CGM of $\log N_{\rm CGM}\approx 13.53$. The Zheng19 analyses provide the first statistical evidence that the MW hosts an extended warm CGM. However, in this model, the disk component ($\log N_{\rm disk}=12.1$) is different from SW09 ($\log N_{\rm disk}=13.4$) by more than one order of magnitude. Therefore, there is still a huge gap between the model dominated by stellar sightlines (the plane-parallel slab model; SW09) and the model dominated by AGN sightlines (the two-component disk-CGM model; Zheng19). In the following, we introduce a 2-D disk-CGM model with a disk radial profile that alleviates the tension between the flat-slab model by SW09 and the two-component disk-CGM model by Zheng19 in studying the warm gas in the MW.

3.2. The 2-D Disk-CGM Model

We improve the previous models by introducing a two-dimension (2-D) disk into the two-component disk-CGM model of Zheng19. In this model, we consider the number density distribution of the disk component, which is a 2-D distribution ($n_{\rm disk}(r,z)$) depending on the radius ($r$) from the Galactic center (GC), and the $z$-height above and below the Galactic plane. For a given sightline at a given distance ($l$, $b$, and $d$), we can calculate the column density contribution from the disk by integrating the 2-D density distribution of the disk component. For the CGM component, we first consider a constant CGM column density over all directions (isotropic $N_{\rm CGM}$; the same as Zheng19). This CGM component is only applied to the AGN sightlines, while the disk component is calculated for both stellar and AGN sightlines. For a given sightline, the model predicted column densities are

[TABLE]

where $d_{\rm max}$ is the maximum distance for the disk component, which is set to be the virial radius of the MW halo ($R_{\rm vir}=250$ kpc).

Here, we emphasize that the decomposition of the disk and the CGM component is phenomenological, since we assume the stellar sightlines do not trace any CGM gas. This is limited by the current sample, which does not have sightlines in the MW halo that trace the radial profile of the MW CGM at large radii, so we cannot calculate the CGM contribution to the column density measurements in stellar sightlines. However, this assumption is also reasonable with the current sample. For stellar sightlines, most stars are close to the disk mid-plane center $|z|\lesssim 3~{}\rm kpc$, which are marginally affected by the CGM component. There are only three stars have $|z|>3~{}\rm kpc$, leading a tiny effect on the fitting results.

For the 2-D disk component, the radial and vertical profiles are assumed to be independent from each other, so the ion number density distribution in the disk is

[TABLE]

where $f_{r}(r)$ and $f_{z}(z)$ are the profile functions in the radial and vertical directions. For $f_{z}(z)$, we adopt the same exponential profile, $f_{z}(z)=\exp(-|z|/z_{0})$, as the plane-parallel slab model (SW09). We also assume the radial profile to be exponential as $f_{r}(r)=\exp(-r/r_{0})$, where $r_{0}$ is the scale length. Both of the radial and vertical exponential profiles are empirical as inferred from the H I disk and the stellar disk (Kalberla & Dedes, 2008; Bovy & Rix, 2013). It is possible that the warm gas disk follows a different density distribution, since the warm gas disk is more extensive and affected by Galactic feedback. Therefore, in our following analyses, we also consider the Gaussian function $f(x)=\exp(-(x/x_{0})^{2})$ for both $f_{r}(r)$ and $f_{z}(z)$ to test whether the shape of the warm gas disk can be distinguished from the observations. In total, there are four phenomenological models for our disk density profiles, named as $\rm R_{E}Z_{E}$ (exponential radial and vertical profiles), $\rm R_{G}Z_{E}$ (Gaussian radial profile and exponential vertical profile), $\rm R_{E}Z_{G}$ (Exponential radial profile and Gaussian vertical profile), and $\rm R_{G}Z_{G}$ (Gaussian radial and vertical profiles). In these models, the solar system is placed at $r_{\odot}=8.5\rm~{}kpc$ (Ghez et al., 2008) and $z_{\odot}=0\rm~{}kpc$.

We apply these models to the column density measurements of Si IV and O VI (\al@Savage:2003aa, Savage:2009aa, Zheng:2019aa; \al@Savage:2003aa, Savage:2009aa, Zheng:2019aa; \al@Savage:2003aa, Savage:2009aa, Zheng:2019aa), and obtain the best parameters using the minimum $\chi^{2}$ estimation. In our fittings, we include the lower or upper limits of $\log N_{\rm SiIV}$ and $\log N_{\rm OVI}$ values, which are typically not considered in previous modelings (e.g., SW09). For these limits, we only calculated the $\chi^{2}$ value when it is opposite to the limits, i.e., higher than the upper limit and lower than the lower limit; otherwise, the $\chi^{2}$ value is fixed to 1 for these limits. The uncertainties of the sightlines are one-sided uncertainty, which we set according to the intrinsic scatters (i.e., the patchiness parameter derived later; Table 1). Therefore, the uncertainty of limits in the stellar sample is set as 0.3 dex, while the AGN sample is set as 0.1 dex.

Previous studies show that the intrinsic column density scatters of the disk and the CGM are the major contributors of the deviation in the fitting (i.e., Bowen et al. 2008; \al@Savage:2009aa, Zheng:2019aa; \al@Savage:2009aa, Zheng:2019aa). The intrinsic scatter is modeled as the patchiness parameter ($\sigma_{\rm p}$), which is an additional uncertainty attached to the measurement uncertainty as $\sigma^{2}_{\rm f}=\sigma^{2}_{\rm p}+\sigma^{2}_{\rm m}$. $\sigma_{\rm f}$ is the final adopted uncertainty in model fittings, and $\sigma_{\rm m}$ is the measurement uncertainty. There are two methods to implement patchiness parameters. The first method is varying the patchiness parameter to obtain the reduced $\chi^{2}=1$, which is adopted in SW09 (and reference herein). The second method is to implement the patchiness uncertainty into the Bayesian model as introduced in Zheng19. These two methods obtain similar results, and we adopt the first method in our analyses. The calculated $\sigma_{\rm p}$ values are shown in Table 1, which leads to reduced $\chi^{2}$ of $0.95-1.05$ due to the significant figures. In SW09, the patchiness parameters of Si IV and O VI are 0.266 and 0.233, respectively. The larger patchiness parameters (0.30 and 0.23) in Table 1 are mainly due to the inclusion of upper or lower limits in our fittings, and the exclusion of AGN sightlines, which normally have smaller scatters. For the AGN samples, the previously $\sigma_{\rm p}$ are 0.18 and 0.25 for Si IV and O VI, respectively (\al@Savage:2003aa, Zheng:2019aa; \al@Savage:2003aa, Zheng:2019aa), which are larger than our values in Table 1 (0.13 and 0.15). The reduction of the patchiness parameter indicates that the radial distribution of the disk affects the AGN sample more significantly.

The fitting results are summarized in Table 2. Overall, the exponential function leads to smaller scale lengths or scale heights than the Gaussian function, because the exponential function has a slower decay with the same characteristic length. With the isotropic CGM, the $\rm R_{G}Z_{E}$ model (indicating Gaussian function for the radial profile and exponential function for the vertical profile) is preferred with the significance of $\lesssim 2\sigma$ (inferred from the difference of total $\chi^{2}\lesssim 4$ ). Similarly, the anisotropic CGM models show that no specific model is preferred, which will be described and discussed in detail in Section 3.3. Therefore, we suggest that the current stellar and AGN samples cannot distinguish the density profiles (exponential or Gaussian) of the warm gas disk, and we set the $\rm R_{E}Z_{E}$ model as the fiducial model.

All of our four models show that both the disk and the CGM components contribute significantly to the observed column densities in AGN sightlines. Using $n_{0}z_{0}$ as the characteristic column density of the disk, the disk component is comparable to the CGM component for both Si IV and O VI (Table 2). Previously, the plane-parallel slab model shows that the disk component has Si IV column density ranging from $\log N=\log n_{0}z_{0}$ = 13.36 to 13.56, and O VI from $\log N$ =14.12 to 14.28 (SW09). These values are all larger than our values of $13.21-13.26$ (Si IV) and $13.95-14.08$ (O VI). The lower values of our disk component are because we take into account the contribution of the MW CGM to ion column density measurements toward AGN sightlines, whereas the plane-parallel slab model assumes no contribution from the CGM. Our fitting results show that the CGM components are $13.12-13.17$ (Si IV) and $13.91-14.02$ (O VI). These values are comparable with the Si IV and O VI column densities measured from transverse AGN sightlines at $R\approx 100\rm~{}kpc$ ($\approx 0.5R_{\rm vir}$) for low redshift $L^{*}$ galaxies ($z\approx 0.5$; Werk et al. 2013; Savage et al. 2014; Johnson et al. 2015). Although the sightlines through the MW CGM have a different geometry from sightlines for external galaxies, this consistency indicates a decreasing column density dependence on the radius of a power law with a slope of about $-1$ (Werk et al., 2013).

For the disk component, although we cannot distinguish between the exponential and Gaussian profiles, the scale height ($z_{0}$) and the scale length ($r_{0}$) can be determined. In the fiducial model ($\rm R_{E}Z_{E}$), the scale heights are $2.6\pm 0.6$ kpc and $2.4\pm 0.6$ kpc for Si IV and O VI, respectively. The scale lengths are $4.2\pm 1.2$ kpc and $5.6\pm 1.7$ for Si IV and O VI, respectively. The scale lengths are first measured in this work for the MW warm gas disk.

The radial profile of the disk component is important to solve the divergence between the plane-parallel slab model (SW09) and the two-component disk-CGM model (Zheng19), which have different relative contributions between disk and CGM components. In Fig. 1, we predict the projected column density distribution ($\log N\sin|b|$) as a function of $|z|$-height and Galactic latitude ($b$) using the $\rm R_{E}Z_{E}$ model with isotropic CGM with best-fit parameters in Table 2.

Before introducing the plots, we define the observed scale height, which is an observable for the warm gas analysis. This parameter is defined as $N\sin|b|/n_{\odot}$, where $N\sin|b|$ is the projected column density observed from the Sun, and $n_{\odot}$ is the mid-plane ion density around the Sun. The observed scale height could be estimated as the $z$-height of the turnover point in the projected column density function of $z$-height. For example, in the left panel of Fig. 1, the observed scale height is about $|z_{\rm obs}|\approx 0.03$ kpc at $|b|=0.1^{\circ}$, while it is $|z_{\rm obs}|\approx 1$ kpc at $|b|=5^{\circ}$. This is different from the scale height $z_{0}$ defined in Equation (2), which is a constant over the entire sky. The scale height $z_{0}$ could be calculated as $N_{\rm r,nd}/n_{\rm r,mp}$, where $N_{\rm r,nd}$ and $n_{\rm r,mp}$ are the column density toward $b=90^{\circ}$ (the normal direction of the disk) and the mid-plane density at any given radius of $r$.

The difference between the scale height ($z_{0}$) and the observed scale height ($z_{\rm obs}$) is mainly due to the radial density distribution of the disk component. Using two AGN sightlines as an example, one sightline is toward $b=90^{\circ}$, while another sightline is toward the anti-Galactic center (anti-GC; $l=180^{\circ}$) direction with any Galactic latitudes. Considering the calculations of the scale height and the observed scale height, the mid-plane densities are the same, since both of densities are around the Sun. However, the projected column densities of the disk component are different: the term $N_{r_{\odot},{\rm nd}}$ is always larger than $N\sin|b|$ at different $b$. This is a result of the disk radial distribution, since $N\sin|b|$ could be approximated as $N_{r_{\odot},{\rm nd}}\exp(-(r-r_{\odot})/r_{0})$, where $r$ is always larger than $r_{\odot}$ for anti-GC sightlines. Therefore, we expected that the observed scale height is always lower than the real scale height for anti-GC sightlines. Also, since a low $b$ leads to a small $\tan b$ value, the low latitude sightlines need a longer path length to reach the same height. Then, the effect of the disk radial distribution is more significant for low latitude sightlines, and lower projected column densities are expected for these sightlines (the left panel in Fig. 1).

The sightlines toward the GC ($l=0^{\circ}$) is more complex, since the disk radial distribution leads to higher density around the GC. However, the stellar sightlines are mainly at low latitudes $|b|\lesssim 5^{\circ}$. In our fiducial $\rm R_{E}Z_{E}$ model, the scale heights of both Si IV and O VI are higher than 2 kpc. Using this scale height and a Galactic latitude of $5^{\circ}$, one expects a radius difference of $z_{0}/\tan|b|>20$ kpc to reach the scale height of the disk. With this radius difference, the final effect on the observed scale height will be a competition between the high-density gas around the GC and the low-density gas at large radii. Our numerical calculation shows that it is possible to have larger observed scale heights around the GC direction (the left panel in Fig. 1). Therefore, the sightlines around the GC direction have higher projected column densities than those toward anti-GC directions due to the high-density gas at the GC. Since the solar system is at $r_{\odot}=8.5$ kpc, this difference is most significant around $|z|=8.5\tan|b|~{}\rm kpc$, which is $\approx 0.7$ kpc for $|b|=5^{\circ}$.

In the middle panel of Fig. 1, we show the predicted projected column density for the AGN samples using the fiducial $\rm R_{E}Z_{E}$ model. Our model predicts that the projected column density has a dependence on both Galactic latitude and Galactic longitude, while the previous models only have dependence on Galactic latitude (SW09 and Zheng19; the right panel of Fig. 1). The $\log N$ dependence on both $l$ and $b$ is due to the radial profile of the disk component, so it is similar to the case in the disk-only model (the left panel of Fig. 1), but for higher Galactic latitudes ($|b|>30^{\circ}$). The projected column densities are generally higher toward the GC direction than the anti-GC direction, and all the values converge at $|b|=90^{\circ}$. The anti-GC sightlines show a more significant dependence on the Galactic latitude $|b|$ (decreasing rapidly), which is due to the radial profile in our model. The Zheng19 model also reproduces this feature, but in a different way. The Galactic latitude dependence in the Zheng19 model is due to the term of $N_{\rm CGM}\sin|b|$, and do not have dependence on the Galactic longitude, so this model does not reproduce the feature that the GC sightlines have higher column density than the anti-GC sightlines (also see Fig. 5 in Zheng19, and Fig. 7 in Wakker et al. 2012).

3.3. The Anisotropy of the CGM Component

In the previous section, we adopt the isotropic CGM assumption from Zheng19. However, this isotropic CGM profile over the entire sky may not best represent the gas density distribution in the CGM. For example, Bordoloi et al. (2011) found that the absorption features are stronger along the minor axis using Mg II absorption lines for external galaxies (also see Lan & Mo 2018). Besides the absorption strength, Martin et al. (2019) found that the non-detections of Mg II are mainly along the minor axis (perpendicular to the disk), which indicates a lower detection rate along the minor axis. Therefore, we consider the azimuthal variation of CGM in our 2-D disk-CGM model of the MW.

For the MW, the azimuthal variation of external galaxies is equivalent to a variation of the CGM column densities as a function of Galactic latitude. We refine our 2-D disk-CGM model by changing CGM density from an isotropic distribution to an anisotropic distribution with a dependence on Galactic latitude. In this model, we define two characteristic CGM column densities: the column density along the disk ($N_{\rm mp}$; denoting the mid-plane) and the column density perpendicular to the disk ($N_{\rm nd}$; denoting the normal direction of the disk). These two directions are similar to the major and minor axis directions for external galaxies. For simplicity, we assume that the CGM column density depends on Galactic latitude $b$ as an elliptical function:

[TABLE]

where $N_{\rm mp}$ and $N_{\rm nd}$ are free parameters in our model. For AGN sightlines, the term $N_{\rm CGM}$ in Equation 2 has a dependence on Galactic latitude ($N_{\rm CGM}(b)$). In this model, we assume the variation of the CGM column density is in the logarithmic scale rather than in the linear scale, i.e., $N_{\rm CGM}(b)=(N_{\rm mp}^{2}\cos^{2}b+N_{\rm nd}^{2}\sin^{2}b)^{1/2}$. In the linear scale variation model, if $N_{\rm nd}$ is much larger than $N_{\rm mp}$ (e.g., a factor of $>3$), the CGM column density will be dominated by $N_{\rm nd}\sin b$, and $N_{\rm mp}$ cannot affect the fitting results. Therefore, the linear scale variation model does not have the ability to trace the large amplitude CGM variation (i.e., $|\log N_{\rm nd}-\log N_{\rm mp}|>0.5$ dex).

The $\chi^{2}$ fitting results show that the anisotropic CGM model is significantly better than the isotropic CGM model (Table 2). The total $\chi^{2}$ values are reduced by $15.6-26.4$ and $15.7-21.2$ for Si IV and O VI, respectively. The mean values of the $\chi^{2}$ difference are $20.8$ and $18.5$, which lead to a $4.6\sigma$ and $4.3\sigma$ significance considering the degree of freedom (dof) is reduced by 1. According to the fitting, the CGM column density is higher along the normal direction of the MW disk ($N_{\rm nd}$) than the direction along the disk ($N_{\rm mp}$) by $0.6-0.9$ dex and $\approx 1.0$ dex for Si IV and O VI, respectively. The differences between $N_{\rm nd}$ and $N_{\rm mp}$ are consistent for different disk density profiles (exponential or Gaussian). This consistency indicates the CGM anisotropy is a real feature rather than an artificial feature due to the choice of the disk density profiles. Therefore, we suggest that the anisotropic model is preferred at least at a level of $4.0\sigma$ for both Si IV and O VI distributions. Combining these two ions together, the significance is about $6.3\sigma$. However, this result does not imply the CGM has an elliptical geometry, and it is even unknown whether this feature is completely due to the CGM, which will be discussed in Section 4.3.

Adopting the anisotropic CGM model does not affect the disk parameters significantly, but one interesting difference is the larger scale length. This is the result of the smaller CGM column density at low Galactic latitudes. In the isotropic CGM model, the CGM column density is dominated by the AGN sample at high Galactic latitudes. This isotropic column density ($\approx N_{\rm nd}$) is higher than the real column density at low Galactic latitudes ($N_{\rm mp}$), and suppresses the extension of the disk component along the radial direction. In the fiducial $\rm R_{E}Z_{E}$ disk model with an anisotropic CGM, the scale lengths are $5.3\pm 1.4$ kpc and $7.8\pm 2.4$ for Si IV and O VI, respectively. These numbers indicate that the warm gas disk is more extensive than the stellar disk ($\approx 2$ kpc; Bovy & Rix 2013) and the H I disk ($\approx 3.5$ kpc; Kalberla & Dedes 2008) at about $2\sigma$.

Column density predictions by the preferred models ($\rm R_{E}Z_{E}$ disk with anisotropic CGM) are compared to the observations in Fig. 2 and Fig. 3 for Si IV and O VI, respectively. We plot the stellar samples in the top panels, which generally follow the model ($\rm R_{E}Z_{E}$ disk with isotropic CGM; Fig. 1) described in Section 3.2. Different from the plane-parallel slab model (SW09), we predict the maximum projected column density has a dependence on Galactic latitude due to the radial profile of the disk (Section 3.2). However, the difference between low and high Galactic latitude sightlines does not show up for the observed scale heights, since there are no high-$|z|$ stellar sightlines at low latitude ($|b|\lesssim 3^{\circ}$), which are expected to follow the purple lines in the top left panels. The observations show the longitude dependence of the projected column density, since the sightlines toward the anti-GC generally have lower projected column density than the sightlines toward the GC.

The middle panels show the global variation of the total column densities predicted for AGN sightlines. It is of interests to find that the predicted global minimum among the AGN sightlines occurs around $|b|=30^{\circ}-50^{\circ}$ around the anti-GC, which is the combination of the disk variation and the anisotropic CGM. The disk component has the minimum around the polar regions ($|b|\approx 90^{\circ}$), where has the shortest path length, while the anisotropic CGM component in our model has the minimum along the disk radial direction ($|b|=0^{\circ}$). Then, considering these two effects together, the minimum of the total column density will be around $|b|\approx 45^{\circ}$. Observationally, this feature was found by Wakker et al. (2012), showing an O VI deficit region at $l=70-280^{\circ}$, $b=-60$ to $-10^{\circ}$. This deficit is more clear in the southern hemisphere, since the southern hemisphere has systematically lower column densities.

The projected column density of the AGN samples is plotted in the lower panels of Fig. 2 and Fig. 3. The northern hemisphere sightlines have systematically higher column densities than the southern hemisphere by $0.1-0.2$ dex. The origin of this north-south asymmetry is beyond the scope of this paper, but we discuss it phenomenologically in Section 4.4. Here, we do not consider this north-south column density asymmetry in our modeling. The predicted tendency stated in Section 3.2 matches with the observations, which show the anti-GC sightlines have lower projected column densities.

We find that the anisotropic model can better reproduce the large $\log N$ variation at lower Galactic latitudes for two reasons (Fig. 2 and Fig. 3). First, the variation of the projected column density at low latitudes is mainly caused by the radial variation of the disk. The low column density along the disk radial direction ($\log N_{\rm mp}$) allows a more extended disk, which increases the column density variation at low Galactic latitudes, since this variation is due to the variation of the disk (the middle panel of Fig. 1). Second, the variation of the CGM column density leads to a steeper decrease of the projected column density at low latitudes around the anti-GC. Since the CGM column density is higher at $b=90^{\circ}$, the total column density is also increased at high latitudes.

In all, we prefer the $\rm R_{E}Z_{E}$ model with anisotropic CGM profile to other models (Fig. 4). We examine the Si IV column density residuals to evaluate the performance of the plane-parallel slab model (SW09; left column), the two-component disk-CGM model (Zheng19; middle column), and our preferred model (right column). The plane-parallel slab model (SW09) fits the stellar sample well, which does not have unaccounted features in the residual (i.e., flat residuals over Galactic longitude), although there is a large scatter. However, there are significant unaccounted structures in the residuals for the AGN sample (low residuals at low Galactic latitudes), although the intrinsic scatter is less than the stellar sample. The two-component disk-CGM model (Zheng19) has comparable residuals to our 2-D disk-CGM model for the AGN sample, but the residuals show a peak around $l=0^{\circ}$. Also, this model predicts a disk component of $\log N=12.1$, which is about one order of magnitude lower than SW09 and this work ($\log N\approx 13.3$). The two-component disk-CGM model does not have distance constraints, so the $\log N$ measurement from stellar sightlines cannot be reproduced in this model (Zheng19). Our new model could reproduce the column density measurements from both the stellar and the AGN sightlines equally well without unaccounted features in the residuals.

4. Discussion

4.1. The Comparison with Zheng19

Using the archival data (\al@Savage:2003aa, Savage:2009aa, Zheng:2019aa; \al@Savage:2003aa, Savage:2009aa, Zheng:2019aa; \al@Savage:2003aa, Savage:2009aa, Zheng:2019aa), we build a 2-D disk-CGM model to fit the Si IV and O VI column density measurements for warm gas moving at $|v|\lesssim 100~{}\rm km~{}s^{-1}$ from both the AGN and stellar sightlines simultaneously. Previously, the plane-parallel slab model is commonly used to study the MW disk (Jenkins 1978; Bowen et al. 2008; SW09). However, the plane-parallel slab model cannot explain a mismatch between the AGN and stellar sample as noted and discussed in Zheng19, which is due to the lack of CGM component in the plane-parallel slab model. Zheng19 introduced an additional isotropic CGM component to the plane-parallel slab model to account for the CGM contribution in the AGN sightlines. Here, we mainly compare our 2-D disk-CGM model with the two-component disk-CGM model with 1-D disk (Zheng19).

For the model setting, the major difference is the inclusion of the disk radial distribution in our 2-D disk-CGM model (Section 3.2), while a minor difference is an improvement from the isotropic CGM model to the Galactic latitude-dependent CGM model (Section 3.3). In Zheng19, the disk component is still the 1-D plane-parallel slab model, which leads to a lower disk component. This is because the plane-parallel slab model has a constant projected column density of AGN over different Galactic latitudes. Therefore, the low Galactic latitude sightlines with low projected column densities (Fig. 2 and Fig. 3) lead to a low value of the disk component in the two-component disk-CGM model (Zheng19). For the CGM component, we find that the column density distribution of MW CGM is likely to be a function of Galactic latitude instead of an isotropic one (Section 3.3). Since the AGN samples are mostly around the high latitude ($|b|\gtrsim 30^{\circ}$), this CGM modification lead to a significant difference ($\approx 0.7-1.0$ dex for both Si IV and O VI) at low Galactic latitudes between our models and the two-component disk-CGM model (Zheng19).

Another difference is the adopted statistical method, where we used the $\chi^{2}$ optimization, while Zheng19 used the Bayesian frame. Assuming the Gaussian distribution for the measurement uncertainty and the uniform prior, these two methods are equivalent in the sense to obtain the minimum of $\chi^{2}$ or the maximum of likelihood. Besides the method to obtain the fitting results, another difference in the statistical method is the choice of the likelihood or the uncertainty distribution. Zheng19 assumed the column density uncertainty follows a normal distribution (the linear scale) rather than a lognormal distribution, while the latter distribution is adopted in our models. Although the real distribution of the uncertainty is unknown, the lognormal distribution is more used for the data with a large variation (e.g., one order of magnitude). It is worth to notice that these two distributions are similar to each other when the uncertainty is small ($\approx 0.02$ for most sightlines in Zheng19).

Zheng19 also used the block bootstrapping to account for the possible unknown large scale structures. From the residual map in Fig. 2 and Fig. 3, we noticed the north-south asymmetry, which is the most prominent variation over the entire sky. This feature is addressed and discussed phenomenologically in Section 4.4. In section 4.5, we introduce several blocking tests for large scale structures, which show consistent results with the unblocked fitting.

Another minor difference is that we obtain the patchiness parameter by reducing the reduced $\chi^{2}$ to 1, while Zheng19 implemented the patchiness parameter in the Bayesian frame. Again, Zheng19 assumed a normal distribution rather than a lognormal distribution for this intrinsic scatter. Therefore, their patchiness parameter estimate is $N_{\rm p}=1.4\times 10^{13}$ in the linear scale. We convert it into the logarithmic scale by $\log_{10}e\times\frac{N_{\rm p}}{N_{\rm SiIV}}\approx 0.179$, where $N_{\rm SiIV}$ is the mean column density of the AGN sample of Si IV. This value is larger than the one in our models (0.13 dex; Table 1), and we suggest that this difference is mainly due to the inclusion of the disk radial profile to better account for the column density scatters as seen in the AGN data.

4.2. The Warm Gas Disk

The warm gas at $(1-5)\times 10^{5}$ K is important for gas assembly and recycling in a galaxy due to their high cooling rates and short lifetimes ($\approx 10$ Myr; Oppenheimer & Schaye 2013). Theoretically, this gas normally trace the interaction layer between cool and hot gases, and the cooling from hotter mediums (Gnat et al., 2010; Kwak et al., 2015). These phenomena are usually associated with galactic outflows (feedback processes), infall gas (gas accretion), and interactions between the disk and the CGM (McQuinn & Werk, 2018; Qu & Bregman, 2018). Therefore, one could obtain unique insights into the disk and the CGM formation by observing the warm gas.

The scale height is a key property of the warm gas disk, since it indicates how extensive the disk is, which is a test for the ionization mechanism and the gas origin (Bowen et al. 2008; SW09; Wakker et al. 2012). For example, the scale height should be larger for ions with higher ionization potentials under collisional ionization equilibrium (CIE). However, as shown in SW09, the O VI disk ($z_{0}=2.6\pm 0.5$ kpc) has slightly lower scale height than both Si IV ($3.2_{-0.6}^{+1.0}$ kpc) and C IV ($3.6_{-0.8}^{+1.0}$ kpc) in the SW09 model. This phenomenon might indicate that the Galactic Si IV and O VI are produced under different ionization mechanisms (SW09).

However, as stated in Section 3.2, we find that the Si IV scale height is reduced from $3.2_{-0.6}^{+1.0}$ kpc (SW09) to $2.6\pm 0.6$ kpc (the $\rm R_{E}Z_{E}$ model with the anisotropic CGM). The O VI scale height ($2.6\pm 0.6$ kpc) is similar to SW09 ($2.6\pm 0.5$ kpc). Therefore, our models do not support that Si IV and O VI have different scale heights. The different behaviors between Si IV and O VI scale heights are because of: the inclusion of the disk radial profile and the anisotropic CGM component, the different samples, and the exclusion of AGN sightlines around the north Galactic polar region in SW09.

Besides the scale heights, we find that the scale lengths are also similar between Si IV and O VI within $1\sigma$. Therefore, we consider whether both of the ions follow the same density profile distributions of the disk component. A joint fitting model is applied to Si IV and O VI samples simultaneously, where we tie the parameters of the O VI model to the Si IV model, including the scale length ($r_{0}$), the scale height ($z_{0}$), and the CGM difference between two axes ($\Delta\log N^{\rm CGM}$). In the model where the three parameters are all tied (Table 3), the difference of total $\chi^{2}$ is 3.51 compared to the best model with all parameters are free (the models in Table 2). Because the best model has a 3 more dof, it is $1\sigma$ better than the most limited model, so the best model is not a significantly better model. We also tie these three parameters in turn to check which is the most dominant factor in the $\chi^{2}$ difference. We find that tying the scale length leads to the highest $\chi^{2}$, but the difference is still insignificant. Therefore, we prefer the most limited model with all three parameters tied, and suggest that there is no significant difference of the density profile between Si IV and O VI adopting the new models.

Our model measures the scale length of the warm gas disk of the MW for the first time. We can further estimate the total mass of the warm gas disk of the MW. First, we obtain the total number of ions ($\mathcal{N}_{\rm total}^{\rm disk}$; for Si IV or O VI) within the warm gas disk by integrating the ion number density over the radial and vertical directions:

[TABLE]

Then, we calculate the masses of Si IV and O VI ions in the warm gas disk for different models (Table 2 and Table 3). For each ion, various models lead to similar ion masses within 0.3 dex. Summarizing our results, we obtain the mass of Si IV is $\log(M/M_{\odot})=3.8\pm 0.1$, while the O VI mass is $\log(M/M_{\odot})=4.4\pm 0.2$. To obtain the total mass of the Si IV or O VI-bearing gases, we assume the warm gas has the solar metallicity, and adopt log(Si/H) and log(O/H) solar abundance values from Asplund et al. (2009). Also, we assume the average ionization fraction of $0.2$ and $0.1$ for Si IV and O VI, respectively (about half of the maximum in CIE or PIE to represent the average ionization fraction; Gnat & Sternberg 2007; Oppenheimer & Schaye 2013). Then, the expected total number of hydrogen atoms is $\mathcal{N}_{\rm H}=\mathcal{N}_{\rm total}^{\rm disk}/f/a$, where $f$ is the ionization fraction of Si IV or O VI, and $a$ is the abundance of silicon or oxygen. Taking the helium mass into account, the total mass of the warm gas disk is $1.3\mathcal{N}_{\rm H}m_{\rm H}$, where $m_{\rm H}$ is the hydrogen atom mass. Finally, the derived total masses of the warm gas disk based on Si IV and O VI are

[TABLE]

which are similar to each other.

The similarities of shapes and masses between Si IV and O VI disks indicate that these two ions might trace the same gases. However, it does not mean that these two ions are cospatial, since the ion ratio (Si IV/O VI) shows large scatters ($\approx 0.5$ dex; SW09). Si IV and O VI occupy the same space at large-scale (Galactic scale) due to the similarities of the disk shapes, but these two ion-bearing gases are clumpy to be non-cospatial at small-scale (single cloud size; kpc size; Werk et al. 2019). The Si IV gas is more clumpy than O VI because it has larger intrinsic scatters (the patchiness parameter; Table 1). The same shapes of the Si IV and the O VI disk profiles from our models indicate that the warm gas disk cannot be in equilibrium. If these ions are in photoionization equilibrium, the Si IV gas should have a larger scale height, while the thermal-supported collisional disk predicts the opposite behavior.

A possible explanation of the same scale heights for Si IV and O VI is that these ions are produced by feedback processes (e.g., the Galactic fountain; Bregman 1980; Melso et al. 2019). In the Galactic fountain, the gas could be IVCs, which are separate clouds (Wakker et al., 2008; Shull et al., 2009; Werk et al., 2019). Then, the Si IV gas is close to the core of H I, while the O VI gas is likely to be the envelope, since Si IV has a lower excitation potential. The scale heights of these two ions are both set by the ejection due to Galactic feedback. It is also explained that the Si IV gas is more clumpy than the O VI gas, since as an envelope, the O VI gas should have a larger volume filling factor.

As a comparison to the neutral gas, the H I disk has a total mass of $7.1\times 10^{9}~{}M_{\odot}$, a scale height of $0.15$ kpc, and a scale length of $3.25$ kpc within 30 kpc (Kalberla & Dedes, 2008; Nakanishi & Sofue, 2016). Besides the thin H I disk component, there is also a more extensive H I disk with a scale height of $1.6_{-0.4}^{+0.6}$ kpc, which contains a mass of $3.2_{-0.9}^{+1.0}\times 10^{8}~{}M_{\odot}$ (named as the H I halo in Marasco & Fraternali 2011). The warm gaseous disk has a larger scale height than the thick H I disk, while the mass is about one order of magnitude lower.

4.3. The Anisotropic CGM

As stated in Section 3.2, the preferred CGM component in our model is anisotropic with a dependence on Galactic latitude. The joint fitting of Si IV and O VI shows that there is an enhancement of $\Delta\log N=0.82\pm 0.32$ for the column density perpendicular to the disk compared to the direction along the disk. It is worth noticing that although the component is named as “CGM”, it does not mean that this enhancement of the column density is completely due to the CGM of the MW. This column density enhancement could be due to the enriched CGM of the MW or the interaction layer between the disk and the CGM (e.g., interface layers around low-intermediate velocity clouds).

In the first scenario, the CGM above the disk is enriched by feedback processes from the disk which ejected (and recycled) materials/metals into the CGM. Also, the escaping ionizing fluxes are more intense in the $z$-direction, which could lead to higher ionization states (Si IV and O VI) by photoionization. Another possibility is that there is an interaction layer between the disk and the halo gas above the disk, such as the Galactic fountain (Bregman, 1980), which can be observed as low-intermediate velocity clouds (Wakker et al., 2008; Werk et al., 2019). If this component cannot be included in the disk component in our modeling, then it has to be attributed to the anisotropic “CGM” component, which might be the case here. Although these two possibilities are both associated with the feedback processes, the difference is the location of the gases, which could affect the estimation of the mass of the MW warm CGM. However, current observations cannot determine the location of these gases. Hereafter, we assume it is the enriched CGM scenario.

To estimate the mass of the warm CGM, we calculate the the average CGM column density over the entire sky, which is $\frac{1}{2}\int_{-\pi/2}^{\pi/2}{\rm d}bN_{\rm CGM}(b)\cos b$. The average CGM column densities are $\log N=12.84$ and $\log N=13.70$ for Si IV and O VI, respectively. The maximum radius of the CGM is fixed as the virial radius of the MW (250 kpc). The total ion mass is $\log(M/M_{\odot})=(5.6\pm 0.2)+2\log(R_{\rm max}/250\rm~{}kpc)$ assuming the uniform density distribution for Si IV, and $\log(M/M_{\odot})=(6.3\pm 0.2)+2\log(R_{\rm max}/250\rm~{}kpc)$ for O VI, where $R_{\rm max}$ is the maximum radius of the CGM. The metallicity of the MW CGM is assumed to be $0.5~{}Z_{\odot}$ (Bregman et al., 2018), and the average ionization fraction is about the half of the peak from CIE or PIE (similar to the disk calculation in Section 4.2). Then, the Si IV and O VI-bearing gases have masses of

[TABLE]

Different from the disk component, we cannot constrain whether Si IV and O VI have similar shapes for the CGM, but if we assume they follow the same density profile, the masses are the same for these two ions. If one wants to estimate the mass for the interaction-layer scenario, one could use $\log N=12.50$ and $\log N=13.36$ instead in the mass estimation, which is a difference of $\approx 0.34$ dex.

The mass of the CGM has a dependence on the radial profile of the density. Although the radial profile of the warm gas cannot be determined using current observations for the MW, we show the effect of this variation as the following. For simplicity, we assume a $\beta$-model of $n(r)=n_{0}r^{-3\beta}$ (a power law model), which is empirical for the MW hot gas (Li & Bregman, 2017) and warm gas in external galaxies (Werk et al., 2013; Johnson et al., 2015). Then, the new CGM mass is

[TABLE]

where $M_{\beta}$ is the mass in the $\beta$-model, while $M_{\rm u}$ is the mass in the uniform density model (Equation 5). $R_{\rm max}$ and $R_{\rm min}$ are the maximum and the minimum radii. With a boundary of 10 kpc and 250 kpc, the ${M_{\beta}}/{M_{\rm u}}$ ratio is 0.24 with $\beta=1/2$ (the theoretical hydrostatic equilibrium solution; Mo et al. 2010), which is a correction of $-0.6$ dex for Equation 6. Generally, a larger $\beta$ leads to a smaller mass of the CGM. Varying $\beta$ from $1/3$ to $2/3$, the mass ratio varies from $0.45$ to $0.12$, and the mass is always lower than the uniform model. Therefore, we suggest that the mass in the Equation 6 is the upper limit if the radial profiles of Si IV and O VI are decreasing at larger radii with the same assumptions of the abundance and the ionization fraction. The suggested mass region is $\log(M/M_{\odot})\approx 8.9-9.5$ with a correction of $-\log\frac{Z}{0.5Z_{\odot}}$.

The mass of the HVCs are not included in the previous discussion, since the Savage03 and Zheng19 samples only measured absorption features at low and intermediate velocities ($|v|\lesssim 100~{}\rm km~{}s^{-1}$). One of the major contributor of the HVCs is the Magellanic Systems (MS), which has a total mass of $\log(M/M_{\odot})\approx 9.3$ for atomic and warm-ionized gases: $\approx 4.9\times 10^{8}~{}M_{\odot}$ in H I; $\approx 1.0\times 10^{9}~{}M_{\odot}$ in the warm gas in MS; and $\approx 5.5\times 10^{8}~{}M_{\odot}$ in the envelope of the MS (Brüns et al., 2005; Fox et al., 2014). Besides the Magellanic systems, other HVCs have a total H I mass of $2.6\times 10^{7}~{}M_{\odot}$ (Wakker, 2004; Putman et al., 2012). Assuming other HVCs have a similar the H I/total warm gas ratio of MS (1:4), the total mass of other HVCs is about $1\times 10^{8}~{}M_{\odot}$ (Lehner et al., 2012). Then, the total HVC mass in the MW is $\log(M/M_{\odot})\approx 9.4$, which is comparable to the derived mass of low- and intermediate-velocity gas in this paper. Therefore, the total mass of the warm-ionized gaseous halo is about $\log(M/M_{\odot})\approx 9.5-9.8$ for the all velocity range.

This derived mass is consistent with the mass of $\log(M/M_{\odot})\gtrsim 9.3$ reported in Zheng19, which only used the Si IV AGN sample to estimate the CGM column density of the MW. Our estimation of the warm CGM mass is comparable to the Andromeda galaxy, which has a total mass of $\log(M/M_{\odot})\approx 9.1-\log(Z/Z_{\odot})$ for the warm gas (up to C IV; Lehner et al. 2015). The warm CGM mass of the MW is consistent with some $L^{*}$ galaxy samples at redshifts of $z\lesssim 0.2$ with $\log(M/M_{\odot})\approx 9.5-10.4$ (Stocke et al., 2013), while there are also samples showing significant differences of $L^{*}$ galaxies at $z\approx 0.2$ (e.g., COS-Halos), which obtained a mass of $\log(M/M_{\odot})\approx 10.8-11.0$ (Werk et al., 2014; Prochaska et al., 2017). However, significantly different masses are derived with different models using the same COS-Halos data, such as $\log(M/M_{\odot})\approx 10.1$ (Stern et al., 2016; Bregman et al., 2018). These uncertainties suggest that the mass estimation of CGM is model-depended, but the local $L^{*}$galaxies (i.e., the MW and the Andromeda) do not favor a warm CGM with a mass comparable to the stellar mass.

4.4. Comments on the North-South Asymmetry

As shown in Fig. 2 and Fig. 3, there is a significant north-south (NS) asymmetry for the observed scale height of AGN samples, which indicates the asymmetry of the warm gas distribution. This asymmetry is similar to the NS asymmetry of the Galactic X-ray background, which shows more soft X-ray emission in the northern hemisphere (Snowden et al., 1997). The physical origin of this asymmetry is unclear, and beyond the scope of this paper. However, it is of interest to determine the origin of the warm gas asymmetry phenomenologically (i.e., the disk or the CGM).

Based on previous results, we assume Si IV and O VI have the same behaviors for both the disk and the CGM: the scale length, the scale height, and the CGM difference between two axes. Considering the NS asymmetry, there are three possible variations between two hemispheres: the scale height of the disk; the disk normalization density; and the CGM column density. Here, we ignore the possible difference of the scale length, which is fixed to the same for both hemispheres. Then, this model can have different scale heights and different disk density normalizations between the north and south disks, and different azimuthal CGM column densities (Table 4). The total $\chi^{2}$ difference is 35.1, which is $5.3\sigma$ with a dof difference of 3. The fitting reveals that the differences of the disk density normalizations and the CGM column density normalizations are close to zero, within the uncertainty. The largest variation is due to the difference in the scale heights.

Quantitatively, we vary these parameters individually to determine the dominant factor (Table 4). The fitting results show that the scale height is the dominant parameter rather than the disk normalization or the CGM normalization with the smallest reduced $\chi^{2}$. Only varying the scale height, this model is $0.8\sigma$ worse compared to the “best” model (with all parameters free) by $\Delta\chi^{2}=1.78$ and the dof difference of 2. Similarly, the disk normalization model is $1.5\sigma$ away from the “best” model. Although the scale height model is preferred, it is not a large statistical difference between these two models with varied disk shapes. Compared to models where the disk is varied, the model with CGM-only differences is less preferred since it is $2.9\sigma$ away from the “best” model.

In the different scale height models, the northern and the southern hemispheres have scale heights of $3.5\pm 0.5\rm~{}kpc$ and $2.3\pm 0.4\rm~{}kpc$, respectively. The difference of the scale heights is about $1.2\rm~{}kpc$, which is at about $2\sigma$. As shown in Fig. 5, a larger scale height leads to a larger scatter at low latitudes, which is favored by the observations. In this model, the variation of the model parameters does not affect the mass estimation in Section 4.2 and 4.3, which are all within $1\sigma$. Therefore, we do not report new values for the masses of both the disk and the CGM.

4.5. The Possible Non-Uniform Structures

It is well known that the warm gas disk and CGM of the MW both have lots of structures, e.g., the Fermi Bubbles (FBs), HVCs, and the Local Bubble. These structures may have a non-uniform contribution to the measured column density of warm gases, which is opposite to our assumption that the density profile of the warm gas can be modeled by smooth functions. Therefore, we adopt the blocking method to test whether these possible non-uniform structures affect our fittings; a similar method has also been used by Zheng19 to study the underlying gaseous structures in the MW halo. For the blocking, we mean to block some part of the sky to obtain a new set of sample and fitting result. The (non-)consistency between blocked and unblocked fitting results show the hints for the effect of the possible structures in the blocked region.

First, we consider a known structure – the FBs. Bordoloi et al. (2017) and Karim et al. (2018) showed the enhancement of HVCs due to the FBs (both the northern and southern bubbles), which are not included in our modeling. Therefore, we block the sky region of $-60^{\circ}<b<60^{\circ}$ and $-30^{\circ}<l<30^{\circ}$ to avoid the AGN sightlines (17 for Si IV and 5 for O VI) througth the FB. The stellar sightlines are not masked out since none of the stars are distant enough to be in the FB. The fitting results are $R_{0}=5.5\pm 1.1\rm~{}kpc$, $z_{0}=2.9\pm 0.5$ kpc, and the CGM difference of $0.80\pm 0.34$ dex. This solution is within $0.5\sigma$ from the fiducial model (Table 3). The mass estimates of the warm gas are all within uncertain of 0.2 dex for both the disk and the CGM. Therefore, we suggest that the FBs do not contribute to the low-intermediate velocity warm gas significantly, although detailed studies on the FBs show evidence for the enhancement of HVCs (Bordoloi et al., 2017; Karim et al., 2018).

Then, we consider possible unknown large scale structures, such as the possible connection with HVCs and IVCs (Sembach et al., 2003). Zheng19 used the block bootstrapping to study it, while we consider this in a simple way. Following Zheng19, the sky is divided into eight regions with $90^{\circ}\times 90^{\circ}$ based on the latitude and the longitude. Each region is blocked out in turn, so we have eight new samples with about 7/8 sightlines of the fiducial sample. We applied the joint model to these new samples. The fittings of the blocked sample lead to the parameter region of $R_{0}\approx 5.2-7.4$ kpc, $z_{0}\approx 2.2-3.0$ kpc, and $\Delta\log N^{\rm CGM}\approx 0.63-1.23$. These results are all within uncertainty ($1\sigma$) of the fiducial model, which indicates that there are no significant contributions from the unknown structures, and our assumption of the smooth profiles is roughly hold at large-scale.

5. Summary

We develop a 2-D disk-CGM model for the MW absorption line samples of Si IV and O VI. The radial density profile of the disk is introduced to determine if it alleviates the tension between the stellar sample and the AGN sample, where a thick warm disk is supported by the stellar sample (SW09), but not by the two-component disk-CGM model of the AGN sample (Zheng19). More details can be found in Section 4.1 for the difference between the new model and the previous studies (e.g., \al@Savage:2009aa, Zheng:2019aa; \al@Savage:2009aa, Zheng:2019aa). Adopting the new model, we obtain the scale heights and the scale lengths for the warm gas disk traced by Si IV and O VI, and estimate the masses in both the gaseous disk and the gaseous halo. Here, we summarize our results:

For the MW, the preferred warm gas distribution has a 2-D disk component ($\rm R_{E}Z_{E}$) with exponential radial and vertical profiles ($n(r,z)=n_{0}\exp(-|z|/z_{0})\exp(-r/r_{0})$) and an anisotropic CGM component (depending on Galactic latitude). The joint fitting of Si IV and O VI shows that these two ions could be modeled by the same density profile, which has a scale length of $r_{0}=6.1\pm 1.2$ kpc and a scale height of $z_{0}=2.6\pm 0.4$ kpc. The same shape of Si IV and O VI might indicate that these two ions are physically associated with each other despite a significant difference in their ionization potentials. This scale length is larger than the H I disk ($\approx 3-4$ kpc) and the stellar disk ($\approx 2$ kpc). In SW09, O VI was found to have a lower scale height than Si IV, which was suggested as an evidence for different ionization mechanisms between Si IV and O VI. However, our fitting shows that there is no significant difference between the Si IV and the O VI scale heights, but this does not mean that these two ions are cospatial. 2. 2.

From our best-fit model (the $\rm R_{E}Z_{E}$ disk and anisotropic CGM), the total mass of the warm gas disk ($\log T\approx 5$) is about $\log(M/M_{\odot})_{\rm SiIV}=(7.6\pm 0.2)-\log\frac{Z}{Z_{\odot}}$. 3. 3.

The CGM component in our model makes a comparable contribution of the column density as the warm gas disk. Our modeling indicates that it has a higher column density in the direction perpendicular to the disk than the direction along the disk at $>4\sigma$ levels for both Si IV and O VI. Combining these two ions, the difference is $0.82\pm 0.32$ dex at about $6.3\sigma$ between the vertical and radial directions. However, some of this difference may be due to an interaction layer close to the disk, which we attribute to the CGM. 4. 4.

The mass of the low-intermediate velocity ($|v|\lesssim 100~{}\rm km~{}s^{-1}$) warm ($\log T\approx 5$) gas in the CGM is estimated to be $\log(M/M_{\odot})\approx 9.8\pm 0.2$ with a uniform density distribution and a metallicity of $0.5~{}Z_{\odot}$. When we adopt a $\beta$-model (power law; $n(r)=n_{0}r^{-3\beta}$) to approximate the density profile to 250 kpc, the total mass will be reduced to $\log(M/M_{\odot})\approx 9.5$ ($\beta=1/3$), $\log(M/M_{\odot})\approx 9.2$ ($\beta=1/2$), and $\log(M/M_{\odot})\approx 8.9$ ($\beta=2/3$). Then, the total mass of the warm CGM is estimated to be $\log(M/M_{\odot})=9.5-9.8$ for the MW, combining with the HVC mass of $\log(M/M_{\odot})=9.4$ for the MW. 5. 5.

The projected column density ($\log N\sin|b|$) of AGN indicates a significant north-south asymmetry. Our models suggest that this asymmetry is more likely due to an asymmetric disk rather than an asymmetric CGM at about $2\sigma$. For the asymmetric disk, the variation of the density or the scale height cannot be distinguished, but the model with varying scale heights shows a smaller reduced $\chi^{2}$ (at $\approx 0.7\sigma$). The northern and the southern hemispheres have scale heights of $3.5\pm 0.5\rm~{}kpc$ and $2.3\pm 0.4\rm~{}kpc$, respectively.

We thank the anonymous referee for her/his constructive comments and suggested improvements. The authors also thank for Yong Zheng, Blair Savage, Josh Peek, Mary Putman, and Jiangtao Li for their thoughtful discussions and helpful comments. We also acknowledge the Hubble Spectroscopic Legacy Archive (HSLA) and the software Astropy, and their developers (Astropy Collaboration et al., 2013; Peeples et al., 2017). Z.Q. and J.N.B. would like to acknowledge support from NASA grants NNX15AM93G and NNX16AF23G.