THE ROTATION OF THE HOT GAS AROUND THE MILKY WAY

To measure the global velocity of the million-degree gas around the Galaxy, one needs to measure Doppler shifts toward a range of sources across the sky in lines sensitive to this temperature. This gas is detected in X-ray emission and absorption, but the emission lines are far too faint for a focused grating observation. X-ray imaging CCDs measure the energies of incoming photons and are thus also low-resolution spectrometers, but their spectral resolution is far too low to measure Doppler shifts of tens of km s⁻¹. The only instruments capable of this accuracy are the Chandra Low/High-energy Transmission Grating (LETG/HETG) Spectrometers and the XMM-Newton Reflection Grating Spectrometer (RGS), and the 21.602 Å O vii line is the only line that probes the relevant temperatures and is detected at z = 0 toward many background continuum sources. The Oviii line at 18.96 Å probes slightly hotter gas and is only detected toward a few objects.

Our initial sample included all archival LETG and RGS data sets where the line has been detected in the literature. The LETG has modestly better spectral resolution at 21.6 Å (R = 400) than the RGS (R = 325), but it only has a third of the effective area at this wavelength (15 cm² for the LETG, 45 cm² for the RGS). In addition, unlike the LETG the RGS is always on (it has a dedicated telescope, whereas the gratings must be moved into the focal plane on Chandra), and thus has accumulated many more spectra. These factors lead to many more detected O vii lines in the RGS, so we only use the LETG data toward several calibration sources as a check on the wavelength solution (see below).

Our analysis sample includes 37 known O vii absorbers at z = 0 with RGS data (Fang et al. 2002, 2006, 2015; Nicastro et al. 2002; McKernan et al. 2004; Williams et al. 2005, 2006, 2007; Yao & Wang 2005; Yao et al. 2009b; Gupta et al. 2012; Table 1). These include active galactic nuclei (AGNs) as well as several X-ray binaries in the Milky Way's halo and Magellanic Clouds. We tried to include all sources known to be outside the disk with reported absorption lines, but we excluded three sources: NGC 3783, PKS 2005–489, and Swift J1753.5–0127. NGC 3783 has an intrinsic oxygen line with a P Cygni profile where we cannot disentangle the Galactic line, PKS 2005–489 has a broad line that suggests blending or a non-Galactic origin, and the line in Swift J1753.5–0127 is only detected in two of four high signal-to-noise ratio (S/N) exposures. We include NGC 5408 X-1 (which is an X-ray binary, not an AGN) and NGC 4051, but these systems have redshifts smaller than 1000 km s⁻¹ so the lines may be intrinsic. NGC 4051 and MCG-6-30-15 also have known outflows; as they have O vii lines attributed to the Galaxy in some prior studies (e.g., Fang et al. 2015) we include them here, but we show below that excluding them does not strongly change our results.

Table 1.  Oxygen Absorption Line Sample

Name	Gal Long.	Gal Lat.	z	$\mathrm{sin}(l)\;\mathrm{cos}(b)$	v0	${v}_{0}-{v}_{0,\mathrm{nc}}$	Corotating	Stationary	Eq. Width	Cont. S/N
	(deg)	(deg)			(km s−1)	(km s−1)	(km s−1)	(km s−1)	(mÅ)
Mkn 421	179.832	65.031	0.0300	0.0013	$-{65}_{-21}^{+21}$	−29	0	0	13.1 ± 0.5	232
3C 273	289.951	64.360	0.1580	−0.4069	${33}_{-45}^{+45}$	−27	21	98	24.6 ± 1.6	55
PKS 2155-304	17.730	−52.245	0.1160	0.1865	$-{39}_{-38}^{+38}$	−36	24	−45	15.4 ± 0.9	95
Mkn 509	35.971	−29.855	0.0340	0.5094	$-{45}_{-50}^{+53}$	−41	0	−122	30.2 ± 2.4	48
NGC 4051	148.883	70.085	0.0023	0.1761	${110}_{-41}^{+46}$	8	−16	−42	40.4 ± 3.6	33
MCG -06-30-15	313.292	27.680	0.0077	−0.6448	${53}_{-54}^{+51}$	−1	26	154	35.1 ± 3.2	38
Ark 564	92.138	−25.337	0.0247	0.9032	$-{57}_{-97}^{+93}$	21	−98	−217	12.3 ± 1.9	50
ESO 141-55	338.183	−26.711	0.0371	−0.3323	$-{154}_{-264}^{+242}$	102	−35	80	21.2 ± 5.1	18
H1426 + 428	77.487	64.899	0.1290	0.4142	${83}_{-145}^{+153}$	−7	−25	−99	16.0 ± 3.9	25
1H 0707-495	260.169	−17.672	0.0405	−0.9388	${45}_{-121}^{+124}$	−21	109	225	24.3 ± 6.1	22
PKS 0558-504	257.962	−28.569	0.1370	−0.8589	${7}_{-104}^{+110}$	8	98	206	16.6 ± 4.2	32
NGC 4593	297.483	57.403	0.0090	−0.4781	$-{63}_{-137}^{+153}$	11	24	115	27.1 ± 7.1	13
Mrk 335	108.763	−41.424	0.0258	0.7100	$-{161}_{-206}^{+158}$	−40	−79	−170	19.3 ± 5.3	22
PG 1211 + 143	267.552	74.315	0.0809	−0.2702	${69}_{-195}^{+188}$	−57	14	65	40.3 ± 11.	7
PG 1244 + 026	300.041	65.214	0.0482	−0.3631	${130}_{-127}^{+286}$	−188	12	87	40.8 ± 11.	10
Mkn 501	64.600	38.860	0.0337	0.7034	${41}_{-173}^{+155}$	−68	−51	−169	25.2 ± 10.	22
1ES 1028 + 511	161.439	54.439	0.3600	0.1853	${40}_{-196}^{+180}$	81	−21	−44	36.4 ± 11.	12
ESO 198-24	271.639	−57.948	0.0455	−0.5305	$-{652}_{-219}^{+231}$	−393	42	127	62.2 ± 19.	7
NGC 2617	229.300	20.939	0.0142	−0.7079	${58}_{-128}^{+124}$	−95	91	170	30.1 ± 9.5	10
PG 1553 + 113	21.909	43.964	0.3600	0.2686	$-{149}_{-175}^{+162}$	45	25	−64	27.6 ± 8.7	13
H1101-232	273.190	33.079	0.1860	−0.8367	${358}_{-152}^{+141}$	2	85	201	45.5 ± 17.	8
3C 390.3	111.438	27.074	0.0561	0.8289	${130}_{-515}^{+242}$	−21	−99	−199	25.0 ± 10.	10
1H 0419-577	266.963	−42.006	0.1040	−0.7421	${783}_{-255}^{+273}$	17	74	178	39.9 ± 16.	6
Fairall 9	295.073	−57.826	0.0470	−0.4825	$-{18}_{-333}^{+291}$	−226	26	116	37.2 ± 15.	6
IRAS 13224-3809	310.189	23.979	0.0660	−0.6982	${63}_{-198}^{+201}$	31	35	167	45.1 ± 19.	7
NGC 5408 X-1	317.149	19.496	0.0017	−0.6414	$-{19}_{-174}^{+216}$	−98	19	154	30.4 ± 14.	7
PDS 456	10.392	11.164	0.1840	0.1770	${57}_{-261}^{+267}$	47	76	−42	60.0 ± 29.	7
Mrk 279	115.042	46.865	0.0305	0.6195	${58}_{-229}^{+278}$	−29	−68	−149	14.2 ± 7.0	15
E1821 + 643	94.003	27.417	0.2970	0.8855	${85}_{-262}^{+241}$	−28	−97	−212	35.7 ± 18.	7
IC 4329A	317.496	30.920	0.0160	−0.5799	$-{93}_{-229}^{+367}$	−73	15	139	38.3 ± 20.	7
PG 0804 + 761	138.279	31.033	0.1000	0.5704	${57}_{-236}^{+219}$	98	−73	−137	28.5 ± 15.	6
NGC 5548	31.960	70.496	0.0172	0.1768	$-{46}_{-185}^{+183}$	−51	7	−42	18.0 ± 10.	13
Mrk 766	190.681	82.270	0.0129	−0.0249	${191}_{-365}^{+240}$	93	0	6	6.0 ± 4.2	27
Local Group and Halo Sources
LMC X-3	273.576	−32.082	-	−0.8457	${35}_{-66}^{+69}$	7	63	203	23.1 ± 3.0	36
4U 1957 + 11	51.308	−9.330	-	0.7702	$-{264}_{-152}^{+145}$	−124	32	−185	20.6 ± 4.1	22
MAXI J0556-032	238.939	−25.183	-	−0.7751	${137}_{-82}^{+85}$	84	39	186	16.0 ± 3.3	33
SMC X-1	300.414	−43.560	-	−0.6251	${117}_{-122}^{+127}$	−3	17	150	21.0 ± 4.9	13

Note. The quantity ${v}_{0}-{v}_{0,\mathrm{nc}}$ is the difference between measurements with and without the heliocentric/Sun-angle corrections. The corotating and stationary columns refer to model velocities. The equivalent width is for a Lorentzian line of fixed width (0.028 Å), and the continuum S/N is per resolution element (∼0.055 Å).

Download table as:  ASCIITypeset image

The data for each target were reprocessed using standard methods in the XMM-Newton Science Analysis Software (SAS v14.0.0) with the appropriate calibration files. This included excluding hot, cold, and "cool" pixels, and data from periods when the background count rate exceeds 3σ from the mean. We applied the (default) empirical correction for the Sun angle of the spacecraft and its heliocentric motion (de Vries et al. 2015). We used the highest precision coordinates available rather than the proposal coordinates, which improves the accuracy of the wavelength scale. For each object, we merged the first-order RGS1 spectra and response matrices into a "stacked" spectrum. Standard processing resamples the data from native bins (about 0.011 Å at 21.6 Å) into a user-specified bin size. We binned the data to 0.02 Å (one resolution element is about 0.055 Å). This resampling causes small but stochastic changes in the bin assignment for some events, leading to variation under the same protocols, which we quantify by processing each object ten times in the same way.

To measure the Doppler shifts, we fit a model consisting of a power law and an absorption line to the spectrum in the 21-22 Å bandpass using XSPEC v12.9.0 (Arnaud 1996). We exclude an instrumental artifact between 21.75 and 21.85 Å in each spectrum, and in several spectra there are one or more bad channels in the bandpass that we also exclude.¹ The parameters of the absorption-line model include the line centroid, the line width, and the line strength, but we fix the line width at the instrumental line-profile width because we do not expect detectable line broadening, an assumption we validated in the brightest sources. The best-fit centroid is converted to a velocity using the best-fit line energy from a stacked spectrum of Capella as a reference, corrected for the radial velocity of the star (described below). We measured the velocity for each of the ten stacked spectra per object, and we report the mean value with its 1σ uncertainty in Table 1, including the resampling uncertainty.

The systematic uncertainty in the wavelength scale limits the accuracy of our measurements, and recent improvements in the calibration of the wavelength scale (de Vries et al. 2015) and multiple high S/N spectra for the objects in our sample are largely what enable this study. Here, we show the accuracy of the wavelength solution for the protocols we adopt and briefly describe the sources of uncertainty and their magnitudes.

We created spectra for the active stars Capella and HR 1099 following the protocols above, then measured the (emission) line centroids for strong, mostly unblended lines, and compared them to their laboratory rest wavelengths (Figure 1). We accounted for the radial velocity of each star ($+29.2$ km s⁻¹ for Capella and −15.3 km s⁻¹ for HR 1099; Karataş et al. 2004). We find no systematic offset in the 5–30 Å bandpass or change over time. The wavelength offsets in O vii λ21.602 Å for stacked spectra are ${\rm{\Delta }}\lambda =0.0\pm 0.8$ mÅ (Capella) and 0.7 ± 1.6 mÅ (HR 1099). These correspond to $v=0\pm 11$ and −7 ± 22 km s⁻¹. To verify that stacking does not introduce artificial offsets, we also measured centroids in each individual exposure (16 for Capella and 14 for HR 1099) and computed the weighted mean. The offsets are consistent with the stacked spectrum: $v=-8\pm 26$ and −3 ± 28 km s⁻¹ for Capella and HR 1099.

Zoom In Zoom Out Reset image size

We also checked the wavelength solution against the LETG and HETG data for these stars. We reduced te data using the Chandra Interactive Analysis of Observations software (CIAO v4.7), and we found good agreement with spectra from the reduced data available through the TGCat project (Huenemoerder et al. 2011). We co-added the ±first-order spectra and stacked all observations. The LETG offsets are v = 1±22 km s⁻¹ (Capella) and $v=10\pm 26$ km s⁻¹ (HR 1099), whereas the HETG Medium Energy Grating offsets are $v=-3\pm 21$ km s⁻¹ (Capella) and $v=-22\pm 38$ km s⁻¹ (HR 1099). These results agree with the RGS and the laboratory wavelength (Figure 1). We also compared velocity measurements between the RGS and LETG in the four brightest quasars (Rasmussen et al. 2003, 2007; Williams et al. 2005; Bregman & Lloyd-Davies 2007; Hagihara et al. 2010; Fang et al. 2015). The LETG and RGS centroids agree to within the 1σ error bar.

These results show that the wavelength scale is sufficiently accurate for our measurement, but it is important to note that there is a substantial systematic scatter that makes individual observations unreliable, and also that there will be systematic differences between our measurements and those reported for the same data using an earlier version of SAS or using incorrect source coordinates.

Different observations of the same object have a systematic scatter of ∼5 mÅ in the wavelength solution around the true mean value. This corresponds to 70 km s⁻¹ at 21.602 Å, and leads to the standard quoted systematic uncertainty² of 100 km s⁻¹. The reason for this scatter is not clear, but about 3 mÅ could be explained by limits in the pointing accuracy of the telescope (de Vries et al. 2015). In any case, the scatter is normally distributed (based on measurements from many exposures of calibration stars such as Capella), and can thus be strongly mitigated with multiple observations of the same source. To reduce the scatter to within 20 km s⁻¹ requires 15–20 independent spectra (assuming equal S/N and a $\sigma =5$ mÅ). The sources in our sample have between 2 and 60 observations, and at the low end the reported 1σ statistical errors (200 km s⁻¹ or more) are much larger than a 70 km s⁻¹ systematic error. For the bright quasars with many observations, we estimate a typical systematic error due to this scatter of 10–15 km s⁻¹.

In addition to the scatter, there are systematic offsets produced by the Sun angle of the telescope and its projected heliocentric motion (possibly an inaccuracy in the star-tracker) that were measured and corrected by de Vries et al. (2015); we refer the reader to their paper for a detailed description. The earliest version of SAS that conained these corrections was v13.0.0, and it was not enabled by default until v14.0.0; all prior absorption-line halo studies using the RGS used an earlier version or did not mention the correction. We measured line centroids with and without these corrections, and found centroid shifts in our sample with magnitudes between 0 and 100 km s⁻¹ (higher for weak lines). To show the effect of not including them, we show the measured centroids without these corrections in Table 1. Finally, if we used the default (proposal) coordinates instead of the SIMBAD coordinates, we measured offsets of ±10 km s⁻¹.

Thus, we would expect our measured centroids to be correlated with prior results but perhaps significantly different relative to the statistical errors. For example, there is a systematic offset of about 60 km s⁻¹ between our measurements and Fang et al. (2015) for the several systems where it can be measured. This offset cannot be entirely explained by the Sun angle and heliocentric motion corrections, but it is consistent with shifts seen relative to prior versions of SAS (de Vries et al. 2015). They do not report the line centroids for active stars, as they only need to measure the centroids to sufficient accuracy to identify halo absorbers. Thus, we do not regard the apparent inconsistency as reflecting an inherent uncertainty in line centroid measurements.

In addition to the wavelength grid, there are a few sources of systematic uncertainty and some fitting choices that affect our final results but where we believe there is a correct choice. We briefly describe these here.

Cool Pixels. There are several "cool" pixels in the vicinity of 21.6 Å that have a lower than expected signal (by about 20%). By default, these pixels are included in the spectrum. We exclude them because they can affect weak absorption features. The typical velocity shift between keeping and excluding them is ${\rm{\Delta }}v=10$ km s⁻¹ in bright sources.

Binning and Resampling. The native RGS binning is about 11 mÅ at 21.6 Å, but the default processing resamples the events onto a user-specified grid, with a default value of 10 mÅ. This resampling is probabilistic with a random element (normally distributed), which means that running the same protocol on a given data set multiple times will result in slightly different spectra. The magnitude of the velocity shift is strongly dependent on the line strength and the continuum S/N, so we reprocessed each data set ten times per reduction protocol set. We then added the standard deviation in the measured velocities in quadrature to the statistical error, since it behaves in essentially the same way. The bin sizes also affect the measured velocities, with a mean line centroid shift of Δv = 7 km s⁻¹ between bins of 10 and 20 mÅ in bright systems (we use the latter).

Stacking. Temporal changes in the instrumental response or changes in the spectral shape of the source can bias the results of stacked spectra. On the other hand, jointly fitting several low-S/N spectra with the continuum shape as a free parameter leads to poorer constraints on the velocity of a line based on spectral bins far from the absorption line. We stack the spectra to improve the continuum S/N, but to determine if this provides systematic bias we measured the line shift between joint fits and stacked spectra in our brightest sources and the calibration sources. We find a typical ${\rm{\Delta }}v=5$ km s⁻¹ because the instrumental response in these regions does not appear to have temporal changes.

Line Profile and Fixed Line Width. For lines with Doppler $b\;\lt$ 200 km s⁻¹, we do not expect to measure reliable line widths (however, see the Doppler b measurements in some of the same lines in Fang et al. 2015). Since the RGS line-spread function (LSF) is best described as a Lorentzian near the core, we fit our spectra using a Lorentzian profile with the width fixed at the instrumental line width. If we use a Gaussian line profile instead but keep the width fixed, the line shift is consistent with zero in most cases but can be up to ${\rm{\Delta }}v=30$ km s⁻¹ in weak lines (statistical errors greater than 200 km s⁻¹). When the line width is a free parameter, we find shifts in the Lorentzian centroids of ${\rm{\Delta }}v=0-15$ km s⁻¹ (the shift magnitude is negatively correlated with S/N) and ${\rm{\Delta }}v=20-50$ km s⁻¹ in the Gaussian case. However, for the Gaussian lines the best-fit line widths tend to be 1.5–2 times the instrumental resolution, and the statistical error also increases. These fits are usually not significantly better than with a fixed line width, so in our view a non-zero line width is not required by the data, and these line widths reflect some small curvature in the continuum.

Bandpass. The fitting bandpass is important because the continuum model needs to fit well even if the result is unphysical. Typically, one fits the local continuum, but in the literature for the O vii line this can vary from a fitting interval less than 1 Å wide to about 5 Å wide. Our choice of fitting bandpass (21–22 Å) is motivated by strong instrumental features below 21 Å and above 22 Å, but if we ignore these features and expand our bandpass by ±1 Å, the typical velocity shift is ${\rm{\Delta }}v=3$ km s⁻¹.

Bad Columns near 21.6 Å. There is asymmetry in the LSF, which is the instrumental response to a δ-function, near the O vii line. This is not the same as the well known asymmetry in the LSF at the 1% level in the line wings (which is not important), but rather due to bad columns that are not included in the cool pixel list (Figure 2). For an arbitrarily strong δ-function line, the offset in the measured centroid from this defect can range from ${\rm{\Delta }}v=0\mbox{--}100$ km s⁻¹ depending on where the true line centroid is. However, if the line is unresolved but has some physical width (so that incident photons would be dispersed over multiple bins even before the LSF spreads them out), the error is strongly mitigated. We used XSPEC simulations to determine the error as a function of Doppler b parameter, and for $b\gt 40$ km s⁻¹ (about the thermal width even without turbulence), the typical error is reduced to 15 km s⁻¹ (Figure 2). We expect the Galactic halo lines to be in this regime. Alternatively, one can ignore the columns, but because of their proximity to the region of interest, this will also bias the results.

Zoom In Zoom Out Reset image size

Overall, the systematic error should allow measurements to better than 30 km s⁻¹ accuracy with sufficient S/N. For the high S/N lines we estimate a typical systematic error of 15-20 km s⁻¹, and for lines weaker than about 4σ, the systematic error is dominated by the statistical error.

The interpretation of the measured Doppler shifts as a rotation signature depends on the validity of our single-component halo model. This depends on the following assumptions: (1) A volume-filling spherical halo is an approximately accurate representation of most of the hot gas around the Galaxy; (2) there is no strong local absorber that we have ignored; (3) The spherical halo has bulk global motion; (4) the assumption of ${{dv}}_{\phi }/{dR}=0$ is reasonable within about 50 kpc; and (5) the Doppler shift measurements are accurate. The fourth and fifth assumptions were addressed above, so here we focus on the first three.

Two basic models are suggested in the literature for the structure of the hot halo: an extended, spherical distribution or an exponential disk with a scale height of a few kpc. When using individual sight lines or small samples (Yao et al. 2009a; Hagihara et al. 2010), the observed O vii and O viii absorption and emission line strengths are consistent with an exponential disk model with a scale height of a few kpc. They are also consistent with a spherical model. However, larger samples of absorption lines (∼40 sight lines; Bregman & Lloyd-Davies 2007; Gupta et al. 2012; Fang et al. 2015) and the all-sky emission-line intensities (≈1000 sight lines Henley & Shelton 2012; Miller & Bregman 2013, 2015) favor a spherical model. Similar analyses on independent observables that also probe the hot gas, such as the pulsar dispersion measure toward the Large Magellanic Cloud and the ram-pressure stripping of Milky Way satellites favor an extended halo (Fang et al. 2013; Salem et al. 2015). Finally, a recent analysis of both Galactic and extragalactic sightlines for L_* galaxies finds that the O vii traces hot gas (Faerman et al. 2016). Thus, we adopted the spherical density profile of Miller & Bregman (2015).

However, these analyses are based on single-component models, and from basic galaxy models we expect at least two X-ray absorbing components: infalling gas that is shock heated to the virial temperature ($T\approx 2\times {10}^{6}$ K for the Milky Way) and forms an extended halo (e.g., White & Frenk 1991), and supernova-driven outflows from the disk (e.g., Shapiro & Field 1976; Hill et al. 2012). In the latter case, we expect an exponential disk of hot gas with a scale height set (for a given Galactocentric radius) by the temperature at the midplane. It is worth noting that for $T=2\times {10}^{6}$ K, the scale height at the Solar circle is larger than the Galactocentric distance, so the distribution of outflowing gas could also be spherical, if not very extended; the true shape depends on how widespread the outflows are in the disk, the actual midplane temperature, and the amount of radiative cooling. Within the galaxy disk itself, supernovae contribute to the hot interstellar medium, much of which is confined within the disk. Since X-ray absorption covers a wider temperature range than emission, we are also weakly sensitive to cooler gas (described in the following subsection).

These components may be kinematically distinct, as is seen in the warm gas by Nicastro et al. (2003), Savage et al. (2003), but at the relatively low resolution of the RGS, we expect them to be blended. This complicates the interpretation of the measured centroids, which are the weighted average of the offsets for each component along that line of sight. At a qualitative level, we expect the gas confined in the galaxy disk to rotate with it (although depending on the distances to the absorbers, there may not be any rotation signature) and the gas in supernova-driven outflows to rotate in the same direction but lagging the disk as it reaches larger heights or radii.

To constrain the column and thus the influence on the line centroids from a disk component, we extended the analysis of Miller & Bregman (2015) to fit a disk+halo model to the same data set they used: 648 O viii emission lines from Henley & Shelton (2012), which are filtered for contamination from solar-wind charge exchange and ignore most of the Galactic plane. We refer the reader to Miller & Bregman (2015) for a more detailed explanation of the modeling procedure, but we summarize our model components here.

The Sun exists in a region known as the Local Bubble, which emits soft X-rays (e.g., Cox & Reynolds 1987; Snowden et al. 1997), and we adopt for the Bubble a temperature of $1.2\times {10}^{6}$ K, a variable path length between 100 and 300 pc, and a density ${n}_{\mathrm{LB}}=4\times {10}^{-3}$ cm⁻³. The nature of the Bubble and the density remain debatable because of the contribution from solar-wind charge exchange (e.g., Kuntz 2009; Welsh & Shelton 2009; Galeazzi et al. 2014; Henley & Shelton 2015; Snowden et al. 2015). We have ignored hot interstellar gas confined within the galaxy disk other than the Local Bubble. Constraints from the intensity of the soft X-ray background imply that our sight lines will incur only a small column from this material, and the emission-line sample excludes the Galactic plane. For the extended halo component, we assume an isothermal ($2\times {10}^{6}$ K) plasma with a density profile described by Equation (1). For the exponential disk component, we assume a form for the density of

$$\begin{eqnarray}&&n={n}_{0,\mathrm{disk}}\;{e}^{-z/h}\;{\mathrm{cm}}^{-3},\end{eqnarray} \tag{ 2 }$$

where h is the scale height. The temperature declines in an analogous way, and we fix the midplane temperature and its scale height to $2.5\times {10}^{6}$ K and 5.6 kpc, respectively (Hagihara et al. 2010) because we do not model the line ratios as a temperature diagnostic. The free parameters are the normalizations, β, and h, which we constrain using the same MCMC method as Miller & Bregman (2015).

We assume that each component is optically thin and compute its contribution to the intensity along each line of sight s:

$$\begin{eqnarray}&&I=\displaystyle \frac{1}{4\pi }\int n{(s)}^{2}\epsilon (T(s)){ds},\end{eqnarray} \tag{ 3 }$$

where $\epsilon (T)$ is the volumetric line emissivity from the APEC thermal plasma code (Foster et al. 2012). Finally, the line intensities from Henley & Shelton (2012) do not account for absorption due to Galactic H i, so to compare our model intensities to theirs we apply photoelectric absorption using the neutral hydrogen column from Kalberla et al. (2007) to the halo and disk components:

$$\begin{eqnarray}&&I(s)={I}_{\mathrm{LB}}(s)+({I}_{\mathrm{halo}}(s)+{I}_{\mathrm{disk}}(s)){e}^{-\sigma {N}_{{\rm{H}}}},\end{eqnarray} \tag{ 4 }$$

where σ is the absorption cross-section and ${I}_{\mathrm{LB}}$ is the intensity from the Local Bubble.

Figure 7 shows the results from the MCMC analysis as marginalized posterior probability distributions for each of the free parameters. The model is not a significant improvement on the pure halo model, and the best-fit parameters for the halo (${n}_{0,\mathrm{halo}}{r}_{c}^{3\beta }=1.43\pm 0.25\times {10}^{-2}$ cm⁻³ kpc${}^{3\beta }$, $\beta \ =0.52\pm 0.03$) are consistent with the results in Miller & Bregman (2015). The disk parameters are poorly constrained and indicate that the exponential disk contributes at most 10% of the total O vii column density. This corroborates the Miller & Bregman (2015) result. In contrast, the Hagihara et al. (2010) disk model (which assumes the halo gas can be described entirely by a disk and is based on one sightline) finds ${n}_{0,\mathrm{disk}}=1.4\times {10}^{-3}$ and $h=2.3\;{\rm{kpc}}$. Yao & Wang (2005), who include some sight lines near the Galactic plane and some outside the plane, find that the O vii column is consistent with ${n}_{0,\mathrm{disk}}=6.4\times {10}^{-3}$ cm⁻³ and a scale height of $h=1.2\;{\rm{kpc}}$, but the authors acknowledge that mixing results from sight lines toward nearby X-ray binaries at low latitudes and those at high latitudes can lead to a strong bias; the hot material confined to the galaxy disk is not part of the exponential disk structure that we have modeled, but it can contribute at low latitudes.

Zoom In Zoom Out Reset image size

Thus, the impact of the disk component on the measured centroids must be small, since 80%–90% of the column density will come from the spherical component. Assuming that the two components are kinematically distinct, for the latter to be stationary and consistent with the measured centroids requires a disk speed much faster than co-rotation. We verified this by modeling a rotating disk and a stationary halo where the disk contributes 10% of the column. Since this is inconsistent with models where the gas originates in the galaxy disk, the data probe motion in the spherical component. For reference, if the Hagihara et al. (2010) disk model is adopted instead of the Miller & Bregman (2015) model or our disk+halo model, the best-fit azimuthal and radial velocities are ${v}_{\phi }\ =151\pm 32$ km s⁻¹ and ${v}_{r}=-15\pm 18$ km s⁻¹.

We have assumed that, to first order, the spherical component moves as a solid body with some ${v}_{\phi }$ and v_r. There may be second-order effects such as the internal flows mentioned above (perhaps due to satellite motions or infalling clouds) or modes in the fluid, which would add scatter to the velocity measurements. However, if the halo is volume filling and in steady state, then large-scale disturbances will tend to dissipate in a sound-crossing time (which could be a long time if the halo is very extended), and the gas (if stationary) will tend toward hydrostatic equilibrium. Multiple major kinematic components are therefore not expected, although we would expect differential rotation. In this case, the measured rotation velocity is some average but still provides useful information. Also, if the material is fresh infall from the cosmic web then it likely accretes along filaments, which lead to a preferential orbital angular momentum axis. Finally, gas ejected from the disk (either cold or hot) could spin up the halo (Marinacci et al. 2011), although perhaps not to the velocity that we infer.

It is possible that there is a layer of warm-hot gas near the Galactic plane (in addition to the Local Bubble) that affects every sight line out of the Galaxy, but to which our models are not sensitive because we ignore data close to the Galactic plane (where dust, supernova remnants, and other features make modeling very difficult). Characterizing the structure and filling factor of the hot interstellar medium has been a major effort by itself (e.g., Yao & Wang 2005; Hagihara et al. 2011; Nicastro et al. 2016) and is beyond the scope of this paper, but if there is such a layer (possibly a disk with a scale height of a few hundred pc) with a high density, the rotation signature in the RGS data could be misattributed to the halo. On the other hand, for plausible path lengths, oxygen column densities, and foreground absorption column densities this layer should also produce emission in excess of the Local Bubble contribution to the soft X-ray background.

To summarize, the halo models that are based on many data points as opposed to a few sightlines favor a spherical halo (especially in emission) as the dominant component. Any contribution to the column density from a kinematically distinct (thick) exponential disk is small in this scenario, so the Doppler shifts support a non-stationary extended halo. Reinterpreting these shifts will be necessary if future data or analyses rule out a spherical halo (or at least a volume-filling one) or a high-resolution X-ray spectrometer resolves the lines into components inconsistent with rotation. Also, we reiterate that the parametric fit strongly depends on the few AGNs with the highest S/N, but the nonparametric tests indicate rotation at some level.

The O vii ion fraction is high between $(3\mbox{--}20)\times {10}^{5}$ K, so the cooler (non-coronal) gas seen in and around the Galaxy in O vi with FUSE (Nicastro et al. 2003; Savage et al. 2003; Wakker et al. 2003; Wang et al. 2005; Shelton et al. 2007; Yao & Wang 2007; Bowen et al. 2008) will also absorb O vii. The O vi lines toward background objects reveal Galactic and high-velocity absorbers (Nicastro et al. 2003; Savage et al. 2003; Sembach et al. 2003). The Galactic absorbers (also seen toward stars) are consistent with an exponential disk of scale height 1–4 kpc (Savage et al. 2003; Bowen et al. 2008), whereas the high-velocity absorbers may come from the halo or outside the Galaxy. If the column densities of these O vi absorbers are high, then their O vii lines could bias our results. Since the disk component will rotate, we are most concerned about contamination from this gas.

Bowen et al. (2008) measured O vi column densities for about 150 sight lines around the Galaxy and found a typical column of ${N}_{{\rm{O}}{\rm{VI}}}\mathrm{sin}(b)\sim 1.6\quad \times \quad {10}^{14}$ cm⁻². For the sight lines in our sample, we expect ${N}_{{\rm{O}}{\rm{VI}}}=(1\mbox{--}4)\times {10}^{14}$ cm⁻². If the O vi absorbers are at $T=3\times {10}^{5}$ K and in collisional ionization equilibrium, the Ovii/O vi ratio is 1.7, which leads to an expected contribution of ${N}_{{\rm{O}}{\rm{VII}}}=(2\mbox{--}7)\times {10}^{14}$ cm⁻². In contrast, the typical O vii column (assuming an optically thin plasma) is ${N}_{{\rm{O}}{\rm{VII}}}\sim (4\mbox{--}5)\times {10}^{15}$ cm⁻² (Miller & Bregman 2013). Hence, the contribution directly from O vi absorbers in the Galaxy is at most 15%. It is likely lower, since Gupta et al. (2012) and Miller & Bregman (2015) argue that the O vii lines are not optically thin. The bias from the O vi absorbers also declines at higher latitude, since the O vii column declines more slowly than the O vi column with increasing b (corroborated in emission by Henley & Shelton 2013). For most of our sight lines we estimate that the local and disk contribution to the O vii column is less than 10%.

Even a 10% contribution is important for measuring the true halo velocity, but compared to the uncertainty in our best-fit velocity parameters, it is small. More importantly, as with the hot disk considered above, it cannot by itself account for the measured velocities if we assume that the million-degree halo is stationary.

We have measured a signature of rotation in the O vii line around the Milky Way using 37 sight lines toward background AGNs and archival RGS data. The parametric fit strongly depends on the brightest AGNs, but nonparametric tests indicate that the O vii absorbers are not stationary, even when removing suspect lines. From larger samples of emission and absorption lines, we believe that an extended halo is more consistent with the data than an exponential (thick) disk, and in this case this halo rotates at some velocity smaller than ${v}_{\mathrm{LSR}}$. Taken at face value, this implies that the million-degree gas has a comparable angular momentum to the galaxy disk. It is possible that both components exist, in which case the RGS lines are blends of kinematically distinct components. We use a large sample of emission lines to constrain the contribution of each component to the column density, and find that an exponential disk accounts for no more than 10% of the O vii column density. Thus, the measured Doppler shifts are dominated by the motion of the gas in the extended halo, which is consistent with prograde rotation. Even if a disk were to dominate (which, for our data, produces an unacceptable ${\chi }^{2}$ value), the best-fit azimuthal velocities imply that it is rotating at nearly the same speed and with a comparable amount of angular momentum to the spherical model.

This conclusion depends on assumptions about the underlying model. Measuring the true velocity and separating the X-ray absorbers into their various components requires a high-resolution X-ray instrument with a large effective area (e.g., ARCUS, Smith et al. 2014), which would also give important information on their line shapes that could constrain the optical depth and Doppler b parameters (Miller et al. 2016), as well as reveal the contribution to the total column from the disk and the coronal halo gas seen in X-ray emission.

The recent work on calibrating the conversion of dispersion angle to a wavelength grid for the RGS (de Vries et al. 2015) and stacking multiple observations of the same object enables a wavelength accuracy of tens of km s⁻¹ for the first time (Figure 1). After investigating a variety of systematic issues, we find that the statistical uncertainty (due to the low S/N in many spectra and the relatively small sample) remains the major source of error. The inferred halo velocity and angular momentum are strongly model dependent (and the uncertainties that we report are large), but the basic conclusion that the hot gas distribution rotates is less so.

Several scenarios could produce a rotating halo that lags behind the disk, depending on the geometry. These include a galactic fountain of cool gas that spins up hot gas (Marinacci et al. 2011), a hot galactic fountain of superbubble ejecta that produces an exponential disk before cooling, or infall from the cosmic web with some preferential direction. Our measurements cannot, by themselves, distinguish between these models (which may not be mutually exclusive), but they are an important kinematic constraint for future halo and galaxy formation models.