Recovering interstellar gas properties with HI spectral lines: A comparison between synthetic spectra and 21-SPONGE
Claire E. Murray, Snezana Stanimirovic, Chang-Goo Kim, Eve C. Ostriker, Robert R. Lindner, Carl Heiles, John M. Dickey, Brian Babler
DDraft version September 24, 2018
Preprint typeset using L A TEX style emulateapj v. 5/2/11
RECOVERING INTERSTELLAR GAS PROPERTIES WITH H i SPECTRAL LINES: A COMPARISONBETWEEN SYNTHETIC SPECTRA AND 21-SPONGE
Claire E. Murray † , Sneˇzana Stanimirovi´c , Chang-Goo Kim , Eve C. Ostriker , Robert R. Lindner , CarlHeiles , John M. Dickey , Brian Babler Draft version September 24, 2018
ABSTRACTWe analyze synthetic neutral hydrogen (H i ) absorption and emission spectral lines from a high-resolution, three-dimensional hydrodynamical simulation to quantify how well observational methodsrecover the physical properties of interstellar gas. We present a new method for uniformly decompos-ing H i spectral lines and estimating the properties of associated gas using the Autonomous GaussianDecomposition (AGD) algorithm. We find that H i spectral lines recover physical structures in thesimulation with excellent completeness at high Galactic latitude, and this completeness declines withdecreasing latitude due to strong velocity-blending of spectral lines. The temperature and columndensity inferred from our decomposition and radiative transfer method agree with the simulated valueswithin a factor of < < T s < INTRODUCTION
Neutral hydrogen (H i ) in the interstellar medium(ISM) plays a crucial role in the life cycles of galaxies.The atomic medium provides the main fuel reservoir formolecular gas and, ultimately, star formation. Further-more, the structure of interstellar H i bears importantclues to the nature of gas recycling via radiative anddynamical feedback and Galactic winds (e.g., McClure-Griffiths et al. 2015).Throughout the ISM, H i exists in a “multi-phase”state, characterized by two thermally-stable phases inpressure equilibrium (Field et al. 1969; McKee & Os-triker 1977; Wolfire et al. 2003): the cold neutral medium(CNM) and warm neutral medium (WNM). An effec-tive constraining observable for the balance between theCNM and WNM is the excitation temperature (a.k.a.,spin temperature, T s ) of the gas. However, both emis-sion and absorption by the 21 cm hyperfine transition ofH i are required to measure T s . Therefore, although theCNM ( T s ∼ −
200 K) has been extensively analyzedwith H i absorption (e.g., Crovisier et al. 1978; Dickeyet al. 2003; Heiles & Troland 2003a), excellent sensitiv-ity is required to constrain the temperature of the WNM Department of Astronomy, University of Wisconsin, Madi-son, WI 53706, USA Department of Astrophysical Sciences, Princeton University,Princeton, NJ 08544, USA Radio Astronomy Lab, UC Berkeley, 601 Campbell Hall,Berkeley CA 94720, USA University of Tasmania, School of Maths and Physics, Ho-bart, TAS 7001, Australia † [email protected] ( T s ∼ − i properties, comparisons be-tween observations and theory are necessary. Syntheticdatasets from numerical simulations provide a means to(1) assess the power of observational diagnostics to revealthe inherent state of astronomical systems, and (2) testwhether complex simulations recover all the propertiesof real systems. For example, the velocity structures ofsynthetic spectral lines provide important diagnostics ofinterstellar turbulence (e.g., Falgarone et al. 1994), andthe nature of CNM dynamics (Hennebelle et al. 2007;Saury et al. 2014). Furthermore, synthetic observationshave been used extensively to investigate molecule for-mation (Shetty et al. 2011; Smith et al. 2014; Duarte-Cabral et al. 2015; Duarte-Cabral & Dobbs 2016) andGalactic morphology (Douglas et al. 2010; Acreman et al.2012; Pettitt et al. 2014). Important observational biasescan be directly quantified using these comparisons. Forexample, considering correspondence between the truepositions and observed velocities of molecular clouds,Beaumont et al. (2013) showed that the superposition ofclouds along the line of sight introduces significant un-certainty to observational estimates of cloud mass, sizeand velocity dispersion.However, numerical simulations with suitable dynamicrange and resolution for describing the dynamics of boththe CNM and WNM have only recently been performed.Kim et al. (2014) constructed a sample of synthetic H i absorption and emission spectral lines from their three- a r X i v : . [ a s t r o - ph . GA ] J a n . . . . . . . . l og N ( H I ) [ c m − ] Fig. 1.—
Galactic H i column density ( N (H i )) map from the Lei-den Dwingeloo Survey (Hartmann & Burton 1997). The locationsof the 52 21-SPONGE H i absorption line sources (orange circles)are overlaid. dimensional hydrodynamical simulations (Kim et al.2013). Comparing conditions in the simulated datawith properties inferred from synthetic spectra, Kimet al. (2014) found excellent agreement between “true”and “observed” per-channel and line of sight (LOS)-integrated properties such as column density and spintemperature. Furthermore, they showed that columndensities computed in the optically-thin limit signifi-cantly underestimate the true column density when theH i optical depth is greater than τ ∼
1. This agrees withprevious comparisons of observed and simulated LOS col-umn density by Chengalur et al. (2013), who used MonteCarlo simulated spectra to test the role of optically-thickH i . However, Kim et al. (2014) found the discrepancyfactor to be much smaller than Chengalur et al. (2013),indicating that when proper dynamics are considered,spectral line blending due to overlapping cold clouds isnot significant.Although LOS-integrated ISM properties provide im-portant diagnostics, interpretation is more complicatedfor multi-temperature, as opposed to isothermal, condi-tions. Observations show that even high galactic latitudelines of sight contain several components of varying spintemperature (e.g., Heiles & Troland 2003a; Begum et al.2010; Roy et al. 2013; Murray et al. 2015). The tech-nique of Gaussian decomposition is one method that canidentify individual spectral components from disparateH i phases, and has been used extensively to disentan-gle complex spectral lines (e.g., Lazareff 1975; Meboldet al. 1982; Dickey & Lockman 1990; Mohan et al. 2004).When applied to both H i absorption and emission spec-tra, Gaussian decomposition can be used to estimate thespin temperatures of individual spectral features and thefraction of CNM along the line of sight (Heiles & Troland2003a; Stanimirovi´c et al. 2014; Murray et al. 2015).However, the method suffers from non-uniqueness com-plications, as Gaussian functions do not form an orthog-onal basis (e.g., Heiles & Troland 2003a) and the numberof Gaussian functions used to produce reasonable spec-tral fits can vary significantly. For example, comparingthe Gaussian decompositions of the Galactic H i absorp-tion spectrum towards 3C138, there are no fitted compo-nents which agree between the 6 found by Murray et al.(2015) and the 13 found by Roy et al. (2013). Further-more, no quantitative estimates of how well Gaussian functions recover the properties of interstellar gas exist.One of the major scientific goals for future H i ob-servations with the Australian Square Kilometer ArrayPathfinder (ASKAP) and the Square Kilometer Array(SKA) will be to understand the temperature distribu-tion of the ISM and how it relates to the life cycles ofgalaxies using thousands of spectra (e.g., Dickey et al.2013; McClure-Griffiths et al. 2015). However, the firststep in this undertaking is to understand the biases andlimitations of our observational and analysis methods inreproducing interstellar gas properties. This is best doneby analyzing synthetic H i data from numerical simula-tions, which include realistic physical processes, and alsoprovide full 3D information on simulated H i structures(e.g. density, temperature, velocity).Accordingly, we begin this paper by quantifying howwell Gaussian analysis of H i spectral lines via simple ra-diative transfer recovers “true” interstellar gas proper-ties by analyzing synthetic 21 cm spectral profiles de-rived from 3D hydrodynamical simulations from Kimet al. (2013, 2014). To analyze the synthetic spectralprofiles, we use the Autonomous Gaussian Decomposi-tion (AGD) algorithm (Lindner et al. 2015). AGD im-plements derivative-based computer vision to performGaussian decomposition of spectral lines, enabling effi-cient, reproducible and objective spectral decompositionIn the second half of the paper, we compare syn-thetic and observed H i spectra objectively using the samemethodology. For this we use data from the 21-cm Spec-tral Line Observations of Neutral Gas with the VLA(21-SPONGE) survey, one of the most sensitive Galac-tic 21 cm surveys (Murray et al. 2015), as well as theKim et al. synthetic H i spectra. We assess the waysin which the detailed statistical properties of syntheticspectra may agree or disagree with the statistics of ob-served spectra. This in turn reflects the influence of thestar formation feedback mechanisms and other physicsof the simulations. We especially focus on the impor-tance of Ly α resonant scattering for H i excitation andthe temperature distribution.In Section 2, we describe the 21-SPONGE observa-tions, and in Section 3 we describe Kim et al. simulationsand synthetic data. We present and discuss our analy-sis method in Section 4. In Section 5, we compare theproperties inferred from synthetic spectra with the sim-ulated properties of gas along the same LOS. We thencompare the synthetic spectra with 21-SPONGE obser-vations in Section 6. Finally, we present our summaryand conclusions in Section 7. OBSERVATIONS
For observations of Galactic H i , we use data from the21-SPONGE survey (Murray et al. 2015). 21-SPONGEis the most sensitive survey of Galactic H i absorption atthe Karl G. Jansky Very Large Array (VLA) to date.We target strong extragalactic radio continuum sourcesmostly at high Galactic latitudes ( | b | > ◦ ), and consis-tently reach RMS noise in H i optical depth of σ τ < − per 0 . − channels.21-SPONGE utilizes H i emission observations alongthe same LOS from the Arecibo Observatory ( ∼ . (cid:48) beam at 21 cm). The emission spectrum in the direc-tion of each source is computed by observing a patternof 16 off-source positions and interpolating across thetarget position (see, e.g., Heiles & Troland 2003a, here-after HT03). We note that the Arecibo H i emissionprofiles are not corrected for an effect known as “strayradiation”, wherein radiation enters the main telescopebeam through higher-order side lobes. Although strayradiation can be modeled and removed from H i emis-sion data (e.g., LAB and GASS surveys; Kalberla et al.2005; McClure-Griffiths et al. 2009; Kalberla et al. 2010),it is a complex process requiring stable beam shapeswhich are not achieved at Arecibo. Comparison be-tween the GALFA-H i survey at Arecibo and the strayradiation-corrected LAB survey suggested that stray ra-diation likely does not contribute more than ∼
500 mKover ∼
50 km s − to observed H i brightness temperature(Peek et al. 2011). We emphasize that the effect is onlysignificant for emission, not absorption.Armed with high-sensitivity H i absorption and emis-sion spectra along each line of sight, 21-SPONGE is sen-sitive to H i column densities and temperatures from allneutral ISM phases, including the CNM, WNM and ther-mally unstable medium.In Murray et al. (2015), we presented the surveydesign, analysis methods and initial results for 21-SPONGE. To derive physical properties of interstellargas along each line of sight, we decomposed (by hand)H i absorption and emission spectral pairs simultaneouslyinto Gaussian functions, and solved radiative transferequations to derive the column density and spin tem-perature of individual spectral components, taking intoaccount the presence of self-absorption and the order offeatures along each line of sight (as done, e.g., in HT03,Stanimirovi´c & Heiles 2005; Stanimirovi´c et al. 2014).We found excellent agreement with previous H i absorp-tion surveys. The high sensitivity of 21-SPONGE al-lowed us to extend the maximum H i spin temperaturesdetected directly in absorption and emission from 600 Kin HT03 to ∼ i column density map from theLeiden Dwingeloo Survey (LDS; Hartmann & Burton1997). The targets probe a large range in Galactic lati-tude. NUMERICAL SIMULATIONS
We analyze recent high-resolution Galactic ISM simu-lations by Kim et al. (2013, hereafter KOK13). Thesesimulations include momentum feedback from super-novae, time-varying heating, interstellar cooling appro-priate for warm/cold gas, galactic differential rotation,gaseous self-gravity and external gravity from dark mat-ter and stars. We refer the reader to KOK13 for a fulldescription of the numerical setup, and methods.Using these simulations, Kim et al. (2014, hereafterKOK14) constructed a set of synthetic spectral lines sam-pling the local ISM. Assuming an observer sits in thecenter of the simulation, they selected 10 positions ran-domly distributed in Galactic latitude ( l ) and longitude( b ) and extracted the number density ( n ), temperature(kinetic, T k , and spin, T s ) and velocity ( v ) as functions ofpath length ( s ). These lines of sight (LOS) are restrictedto | b | > . ◦ and s ≤ − and gas surface density Σ = 10 M (cid:12) pc − (KOK13).For each observed LOS, KOK14 produced syntheticH i
21 cm emission and absorption as functions of radialvelocity using analytical radiative transfer and a simpleprescription for line excitation prescriptions. The readeris referred to Section 2.3 of KOK14 for a complete de-scription of the methods used to construct the syntheticspectra.In particular, as part of their model for synthetic 21 cmlevel populations, KOK14 considered indirect radiativetransitions due to resonant scattering by Ly α photons(the Wouthuysen-Field (WF) effect; Wouthuysen 1952;Field 1959) in addition to collisions and direct radiativetransitions. They parameterized the WF effect followingField (1959) with a constant value for the Galactic Ly α photon number density, n α , inferred from Liszt (2001)to be n α = 10 − cm − . This value of n α is highly un-certain and difficult to constrain observationally or nu-merically. Given that observed LOS-integrated and per-channel properties are dominated by high-optical depthgas, wherein the 21 cm transition is already thermalizedby collisions due to high densities, the WF effect does notsignificantly affect these values. The WF effect should bemost important for the WNM, where generally T s ≤ T k due to the inefficiency of collisions at thermalizing the21 cm transition (e.g., Liszt 2001). Indeed, at high T s theWF effect is significant (c.f., Figures 9a, 10a of KOK14).We note that the KOK13 simulations do not includechemistry and the H i -to-H conversion. In addition, theirimplemented supernova feedback injects momentum andnot thermal energy, resulting in the absence of a hot ( T ∼ − K) medium. Although for the warm and coldmedium these are secondary effects, and they are beingaddressed in ongoing simulations with thermal supernovafeedback to create a hot ISM (Kim & Ostriker 2016), weneed to keep these limitations in mind when consideringproperties of synthetic spectra from KOK14.In this paper, we analyze components within the syn-thetic KOK14 H i spectral pairs to investigate how ra-diative transfer-based Gaussian fitting reproduces realphysical quantities. From their catalog of 10 spectralpairs, we selected those without saturated (defined hereas τ ≥
3) or NaN-valued absorption lines, for a final cat-alog of 9355 H i spectral pairs. To simulate observationalconditions, we added Gaussian-distributed noise with anamplitude of σ τ = 10 − to each absorption spectrum(equal to the median RMS noise in τ per channel from21-SPONGE) and σ T B = 0 . T B per channelin 21-SPONGE). GAUSSIAN DECOMPOSITION WITH AGD
To perform Gaussian fits to H i spectra (either real orsynthetic), we use the Autonomous Gaussian Decom-position algorithm (AGD; Lindner et al. 2015). AGDimplements derivative spectroscopy and machine learn-ing techniques to efficiently and objectively provide ini-tial guesses (i.e., amplitude, width, mean velocity) formultiple-Gaussian fits to spectral line data.Before implementing AGD, we trained the algorithm tomaximize the decomposition accuracy. We began by con-structing a synthetic H i dataset from the Gaussian com- TABLE 1AGD Summary
Source LOS Absorption ( N AGD ) Emission ( N AGD ) Matched b (number) (total) (per LOS) a (total) (per LOS) a (total) (per LOS) a KOK14 9355 14023 1 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . a : Mean and standard deviation over all LOS b : “Matched” statistics will be discussed in Section 6. ponents detected by the Millennium Arecibo 21 cm Ab-sorption Line Survey (HT03, Heiles & Troland 2003b).The synthetic training dataset construction and trainingare described fully in Lindner et al. (2015), and summa-rized here for clarity. We selected the number of compo-nents in each synthetic spectrum to be a uniform randominteger ranging from the mean number of components inthe survey (3) to the maximum number (8; HT03), andthen drew the component parameters from the publishedHT03 amplitude, FWHM and mean velocity distribu-tions with replacement. As done with the KOK14 syn-thetic spectra, we added Gaussian-distributed noise withan amplitude of σ τ = 10 − to each absorption spectrum(equal to the median RMS noise in τ per channel from 21-SPONGE) and σ T B = 0 . T B per channel in21-SPONGE). The synthetic training sets for absorptionand emission consist of 20 spectra each.After constructing the synthetic training dataset, weused the Python implementation of AGD, GaussPy, todecompose the synthetic training dataset for differentvalues of the “two-phase” smoothing parameters α and α . These smoothing parameters serve to identify thetypes of spectral properties present in the data. Begin-ning with initial choices for α and α and a signal-to-noise (S/N) threshold below which the algorithm will notselect components (S/N=3.0), GaussPy computes the ac-curacy of the decomposition (i.e. how closely the derivedmodel parameters are to the true model parameters), foriteratively different values of α and α until it convergeson minimal model residuals and maximum decomposi-tion accuracy. After training, we found accuracies of 80%and 70% for H i absorption and emission decompositions,respectively. The resulting values are α = 1 .
12 and α = 2 .
73 for absorption and α = 1 .
70 and α = 3 . α and α in hand, we usedGaussPy to apply the AGD algorithm identically (i.e.same values of α , α and S/N) to the observed 21-SPONGE and simulated KOK14 spectra to derive listsof Gaussian parameters for each dataset. Table 1 sum-marizes the decomposition results for the emission andabsorption spectra from 21-SPONGE and KOK14 in-cluding the WF effect and without the WF effect (“noWF”). The typical uncertainties in the fitted parametersare ∼ −
10% from the least-squares fit applied by AGD. ASSESSING THE POWER OF THE GAUSSIAN-FITTINGMETHOD WITH SYNTHETIC H i SPECTRA
From the KOK14 simulations, we have informationabout the density and spin temperature as a function ofdistance ( n ( s ), T s ( s )), as well as the optical depth andbrightness temperature as a function of velocity ( τ ( v ), T B ( v )) for each line of sight. Therefore, following AGD analysis, we can compare inferred properties from spec-tral lines to the true gas properties within the simulatedISM. This will allow us to quantify the biases and limita-tions of Gaussian analysis in reproducing realistic phys-ical properties. Defining gas structures in position and velocity
Given that τ ∝ n/T s , we define simulated gas struc-tures by selecting peaks in n/T s along each LOS. To se-lect a threshold value, we consider the parameters ofthe simulated ISM from KOK13. The gas tempera-ture and density PDFs in their Figure 8 display strongbi-modality indicative of multiple H i phases. The ra-tio n/T s is high for the CNM and low for the WNM.In identifying gas “structures” along the LOS, we wishto mark concentrations using CNM-like peaks. We se-lect n ∼ − and T ∼ K as representative val-ues between the peaks of the published bi-modal PDFs(KOK13). These values correspond to a threshold of( n/T s ) thresh = 0 .
002 cm − / K. We experimented withdifferent values of this threshold, and the subsequent re-sults do not change significantly.In Figure 2, we display ( n/T s )( s ) (left), τ ( v ) (middle)and T B ( v ) (right) for 5 example LOS from KOK14. Thepositions of peaks above ( n/T s ) thresh = 0 .
002 cm − / K(“structures”) are plotted as colored circles. Acrossall 9355 synthetic LOS, there are 7582 structures with( n/T s ) > .
002 cm − / K.To compare the properties of simulated gas structureswith synthetic spectral lines, we first determine the po-sition and velocity range (i.e. line width) of each phys-ical gas structure. We estimate the velocity of each gasstructure, v sim , by computing the average velocity ( v ( s ))of channels spanned by each peak in n/T s weighted bytheir densities ( n ( s )), specifically, v sim = (cid:82) structure n ( s ) v ( s ) ds (cid:82) structure n ( s ) ds , (1)where “structure” refers to all pixels spanned by eachpeak above ( n/T s ) thresh = 0 .
002 cm − / K. These valuesare plotted in the middle and right columns of Figure 2as circles with colors corresponding to the labels in theleft column ( n/T s ). Next, we estimate the FWHM of thestructure based on its thermal and turbulent properties.We compute the inferred thermal line width, ∆ v therm , bysolving (e.g., Eq. 9.31, Draine 2011),∆ v therm = 2 . (cid:115) T mean /
100 K
M/m H = 0 . (cid:112) T mean km s − (2) − . . . . . . n / T s [ c m − / K ] i7716 − . . . . . . . . . τ T B [ K ] . . . . n / T s [ c m − / K ] i5549 − . . . . . . . τ − T B [ K ] − . . . . . . . . n / T s [ c m − / K ] i4952 0 . . . . τ − T B [ K ] − . . . . . . . n / T s [ c m − / K ] i1354 − . . . . . . . . τ − T B [ K ] s [pc] − . . . . . . . . . n / T s [ c m − / K ] i0341 − − − − v [km s − ] − . . . . . . . τ − − − − v [km s − ] − T B [ K ] Fig. 2.—
Example LOS from KOK14. Left: density over spin temperature ( n/T s ) as a function of distance (s) along the LOS; Middle:synthetic optical depth as a function of velocity ( τ ( v )); Right: synthetic brightness temperature as a function of velocity ( T B ( v )). Gasstructures defined by peaks above ( n/T s ) thresh = 0 .
002 cm − / K (dashed line, left column) are indicated by colored circles in each panel. Ifa structure from the left column matches with an AGD absorption component in the middle column according to Equations 6 and 8, thecircles are filled and the matching AGD component is plotted in the corresponding color (unmatched components are plotted in dashedlines). If a structure matches with an AGD absorption and emission component according to Equations 9 and 10, the circle symbol has awhite star within it. where we assume M = µ m H for mean molecular weight µ = 1 .
27 (c.f., KOK13), and T mean is the harmonic meankinetic temperature of the gas spanned by each peak,given by, T mean = (cid:82) structure n ( s ) ds (cid:82) structure ( n/T )( s ) ds . (3)We then estimate the contribution from turbulent linebroadening, ∆ v turb , by computing the standard devia-tion of the velocities spanned by each structure, multi-plied by a factor of 2 .
355 to convert to a FWHM, or,∆ v turb = 2 . (cid:115) (cid:82) structure n ( s ) ( v ( s ) − v sim ) ds (cid:82) structure n ( s ) ds . (4)The final estimate of the velocity FWHM of each struc-ture, ∆ v sim , is a quadratic sum of the thermal and tur-bulent contributions (Equations 2 and 4), or ∆ v sim = (cid:112) ∆ v + ∆ v , and has values between ∼ −
10 km s − for a range of density threshold choices. Matching gas structures with H i absorption lines To match AGD-fitted Gaussian absorption lines withgas structures along each LOS, we use two matching cri-teria. First, we define δ v to be the difference in mean ve-locity of a Gaussian component fitted by AGD in absorp-tion ( v ) with the estimated velocity of the gas structure( v sim ) in terms of the measured FWHM from the AGDfit (∆ v ), or, δ v ≡ | v − v sim | ∆ v / . , (5)For a gas structure to match a Gaussian component, werequire that their positions in velocity be less than onestandard deviation away from each other, so that, δ v ≤ . (6)Second, we define R FWHM to be the ratio of the FWHMof a component in absorption (∆ v ) and the estimatedFWHM of the gas structure in velocity (∆ v sim ), or, R FWHM ≡ ∆ v sim / ∆ v . (7)For a structure to match a Gaussian component, werequire that the structure’s simulated velocity FWHM N structure N A G D < | b | < ◦ < | b | < ◦ < | b | < ◦ Fig. 3.—
The number of gas structures defined by n/T s > .
002 cm − / K along an LOS ( N structure ) compared with the num-ber of AGD-fitted features in the synthetic line profile ( N AGD ).Large symbols show N AGD matched in absorption according toEquations 6 and 8, and small symbols show N AGD matched inabsorption and emission according to Equations 6, 8, 9 and 10.Symbols and shading indicate the mean and standard deviationsof N AGD for each unique N structure . The samples are binned bylatitude according to the inset legend. (∆ v sim ), including the thermal and turbulent contribu-tions, be similar to the FWHM of the Gaussian compo-nent (within a factor of 3), or,0 . ≤ R FWHM ≤ . (8)We note that choices of cutoff values for R FWHM does notsignificantly change the results, as the criterion describedby Equation 6 dominates the matching. In addition, weemphasize that the matching criteria were designed tobe as simple as possible to minimize imposed selectionbiases.In Figure 2, the circle markers for structures whichmatch with AGD absorption components according toEquations 6 and 8, are filled, and the matching AGD ab-sorption component is plotted in the corresponding colorin the middle panel. If a structure does not have anAGD match, the circle marker is unfilled. Of the 7582total structures, there are 6097 structures with matchesto AGD absorption components.
Matching H i absorption lines with H i emission lines Both H i absorption ( τ ( v )) and emission ( T B ( v )) infor-mation are required to constrain the spin temperature( T s ) and column density ( N (H i )) of neutral gas in theISM via radiative transfer. Therefore, to compare thedensity and temperature of simulated gas structures withproperties inferred from observations, we need to deter-mine the optical depth and brightness temperature ofeach structure. To match H i absorption lines with H i emission linesfitted by AGD, we apply a similar set of match criteriaas described by Equations 6 and 8. Specifically, δ v, AGD ≡ | v − v , em | ∆ v / . ≤ , (9)1 ≤ R FWHM , AGD ≡ ∆ v , em / ∆ v ≤ , (10)where ( v , em , ∆ v , em ) are the mean velocity and FWHMof an AGD component fitted to T B ( v ). We impose therequirement that R FWHM , AGD ≥ R FWHM , AGD ≤ Quantifying Match Completeness
For most examples in Figure 2, gas structures (left) areaccounted for by the majority of the total optical depthalong the LOS (middle). This suggests that the AGDabsorption lines can be mapped to real structures. How-ever, in the presence of strong line blending, as shownby the third row of Figure 2 (e.g., case i4952), sev-eral absorption lines have nearly the same central ve-locity and the majority of the absorption feature cannotbe matched. Although the majority (90%) of LOS inKOK14 have < > N structure ) along each LOS with the number of matchedAGD-fitted components of the synthetic line profile( N AGD ). For each unique value of N structure , the mean(symbol) and standard deviation (shading) of N AGD areshown. Furthermore, we break the sample into latitudebins according to the inset legend. Large symbols indi-cate the number of AGD absorption matches accordingEquations 6 and 8, and small symbols indicate the num-ber of AGD absorption and emission matches followingthe subsequent application of Equations 9 and 10.Using Figure 3, we quantify the completeness ( C ) ofrecovery by, C = (cid:80) N AGD (cid:80) N structure . (11)For the matches between gas structures and absorptionlines only (large symbols), the completeness for the low(0 < | b | < ◦ ), mid (20 < | b | < ◦ ) and high latitude( | b | > ◦ ) bins are C absorption = (0 . , . , .
99) re-spectively. The recovery is best at the highest latitudeswhere the number of gas structures and AGD compo-nents are smallest and the LOS complexity is minimized,allowing for simple and robust AGD fits. At low lat-itudes, the blending of gas velocities and AGD compo-nents makes it difficult to associate unambiguous spectralcomponents with ( n/T s ) peaks.When matching gas structures to H i emission insteadof absorption, the recovery completeness is C emission =(0 . , . , .
93) for low, mid and high latitudes re-spectively. The completeness is worse in emission thanabsorption. As observed in Figure 3, broad compo-nents associated with high-temperature H i are promi-nent, thereby making the match to corresponding gasstructures more difficult. Although Gaussian analysishas been used extensively in the past to identify gas pop-ulations in local and external galaxies, this is among thefirst statistical quantification of the correspondence be-tween Gaussian H i emission components and individualgas structures.When matching structures to both absorption andemission (small symbols in Figure 3), the completenessfor the low, mid and high latitude bins are: C both =(0 . , . , .
83) respectively. At all latitudes, the struc-ture recovery completeness declines when the match be-tween absorption and emission is performed. The bot-tom row of Figure 2 displays an example of this decline.Whereas 4/5 structures along the LOS are recovered byAGD absorption components, only 2/4 of those absorp-tion components have matches in emission according toEquations 9 and 10. The structures selected by thematching process with both absorption and emission arebiased towards unambiguous features in all three spaces.Nevertheless, the completeness of Gaussian decomposi-tion for the multi-phase structures seen in both emissionand absorption is good at | b | > ◦ . This is certainlypromising for future large data sets – even with ∼ N AGD ≥
2, although thosecomprise the minority of cases in KOK14. In the fu-ture, improving our selection method for gas structuresalong the LOS beyond a single cutoff value in ( n/T s )will improve this completeness of structure recovery. Inaddition, developing additional criteria for structure-component matching based on their amplitudes and/ortotal column densities will enable us to better quantifythe range of gas structures that can be recovered reli-ably by fitted spectral lines. The analysis presented hererepresents a first step in our ongoing investigation. Observed vs. “True” Gas Properties T s, true [K]10 T s , A G D [ K ] Fig. 4.— “True” simulated spin temperature ( T s, true , Equa-tion 14) versus inferred spin temperature ( T s, AGD , Equation 13)for all structures which match fitted absorption and emission linesaccording to Equations 6, 8, 9 and 10. Contours indicate the1 , σ limits. Given a sample of gas structures with matches to AGDcomponents in absorption and emission, we compare thetrue temperatures in the simulation with the values in-ferred from AGD-fitting of the spectra.
Spin Temperature
With the goal of estimating spin temperature auto-matically for a large number of spectra, we take a moresimplified approach than what has been done in HT03 orMurray et al. (2015). For each AGD match between ab-sorption and emission, we start with the isothermal spintemperature as a function of velocity, T s, AGD ( v ), givenby, T s, AGD ( v ) = T B, AGD ( v )1 − e − τ AGD ( v ) , (12)where T B, AGD ( v ) and τ AGD ( v ) are the matched set ofGaussian functions fitted by AGD to T B ( v ) and τ ( v ) re-spectively. This method assumes a single temperaturegas within each velocity channel. To estimate averagespin temperature per AGD component, we compute theoptical depth-weighted spin temperature per component, T s, AGD ≡ (cid:82) τ AGD ( v ) T s, AGD ( v ) dv (cid:82) τ AGD ( v ) dv . (13)This approach produces a weighted mean temperaturefor each component, given that T s, AGD ( v ) is smaller nearthe peak of τ AGD ( v ) and larger away from it. We notethat there are several possible ways to estimate meantemperature from τ ( v ) and T B ( v ) observations, and somediscussion of the pros and cons of each method are givenin HT03 and Dickey et al. (2003). It is important tonote that Equation 13 works well if, within a multi-phasestructure, the CNM and WNM are centered at a similarradial velocity. However, if the CNM is shifted in velocityrelative to the WNM due to turbulent motions, so thatthe peaks of T B, AGD ( v ) and τ AGD ( v ) are slightly offset,Equation 13 will overestimate T s, AGD . HT03 and Murray T s, true T s , A G D −
20 0 20Velocity [km s − ]05101520250 . . . . . . . −
20 0 20Velocity [km s − ]01020304050600 . . . . . . . −
20 0 20Velocity [km s − ]01020304050607080 T B [ K ] . . . . . . . e x p ( − τ ) −
20 0 20Velocity [km s − ]01020304050 T B [ K ] −
20 0 20Velocity [km s − ]0 . . . . . . . . e x p ( − τ ) −
20 0 20Velocity [km s − ]05101520 −
20 0 20Velocity [km s − ]0 . . . . . . . . −
20 0 20Velocity [km s − ]020406080 −
20 0 20Velocity [km s − ]0 . . . . . −
20 0 20Velocity [km s − ]0102030405060 T B [ K ] −
20 0 20Velocity [km s − ]0 . . . . . e x p ( − τ ) Fig. 5.—
Example synthetic H i emission and absorption spectral pairs in which the AGD-derived spin temperature ( T s, AGD ) overestimatesthe simulated spin temperature ( T s, true ). The top-left panel is reproduces Figure 4, with 7 examples around T s, true = 400 K and T s, AGD =1000 K highlighted in red. The matched H i absorption and emission pairs corresponding to these highlighted points to are plotted andshaded in purple in the accompanying panels, along with the full AGD decomposition for each spectrum (black). et al. (2015) have allowed for the CNM motion relativeto the WNM in their temperature estimates by using amore complex fitting approach where spin temperatureis fitted simultaneously with all WNM components. Thisis, however, computationally expensive for us to imple-ment at this stage.To estimate the temperature of a simulated gas struc-ture, we compute the harmonic mean temperature, T s, true , within the pixels spanned by each peak in n/T s -space. Specifically, T s, true = (cid:82) structure n ( s ) ds (cid:82) structure ( n/T s )( s ) ds . (14)In Figure 4, we compare the simulated spin tempera-ture with the inferred spin temperature derived using ourradiative transfer approach for all structures with AGDmatches in absorption and emission according to Equa-tions 6, 8, 9 and 10. We recover nearly the full rangeof spin temperatures found in KOK14, which indicatesthat our structure selection method is likely not missinga significant gas population. Furthermore, the AGD andtrue estimates agree within the 1 σ contours of Figure 4.However, at high temperatures, where T s, true >
400 K,the AGD temperature overestimates the true spin tem-perature.To understand why AGD overestimates spin temper-ature at high temperatures, Figure 5 displays a set ofexample matched spectral lines with T s, true ∼
400 K and T s, AGD ∼ i emission and absorption com-ponents in the accompanying panels, highlighted withpurple shading. In all highlighted cases, the emissioncomponent is slightly offset in velocity from the corre- sponding absorption component.As discussed above, this offset is caused by interstellarturbulence. When Equation 13 is applied, the resultingspin temperature will be overestimated if we do not ac-count for this velocity offset. For example, if we estimatethe spin temperature using the peak brightness temper-ature and peak optical depth of highlighted components N (HI) true [cm − ]10 N ( H I ) A G D [ c m − ] . . . . . . . . l og10 ( T s , A G D ) Fig. 6.— “True”, simulated column density ( N (H i ) true , Equa-tion 17), versus “observed” column density ( N (H i ) AGD , Equa-tion 16) for all peaks which match fitted absorption and emissioncomponents according to Equations 6, 8, 9 and 10. Contours indi-cate the 1 , σ limits. l [ ◦ ] − . − . . . . . . l og10 ( A G D / T r u e ) (a) (a) T s N (HI)
10 20 30 40 50 60 70 80 | b | [ ◦ ] (b)(b) − . − . − . − . − . . . τ ) (c) Fig. 7.—
The ratio of inferred (“AGD”) to simulated (“True”) properties of gas structures matched with Gaussian spectral lines fittedto H i emission and absorption lines as a function of various LOS parameters. (a): Galactic longitude, l [ ◦ ]; (b): absolute Galacticlatitude, | b | [ ◦ ]; (c): peak optical depth of the matched absorption line, τ ; Spin temperature, T s, AGD /T s, true (red) and column density, N (H i ) AGD /N (H i ) true (blue). Dotted lines indicate factors of 2. in Figure 5, we get a value that agrees much more closelywith T s, true , since T B, peak / (1 − e − τ peak ) ∼
400 K. A morecomplex radiative transfer treatment such as the methodof HT03 and Murray et al. (2015), accounts for this ef-fect, which is strongest for those components with thehighest turbulent velocity offset between the CNM andWNM. We find that T s, AGD most strongly over-estimates T s, true when the velocity offset between absorption andemission is highest. However, most components are notaffected. We will fine-tune our radiative transfer treat-ment in future work. Column Density
The H i column density ( N (H i )) is given by, N (H i ) = C (cid:90) T s ( v ) τ ( v ) dv, (15)where C = 1 . × cm − K − (km s − ). For a pairof matched AGD absorption and emission lines, we com-pute the column density per component ( N (H i ) AGD ) as, N (H i ) AGD = C T s, AGD (cid:90) τ AGD ( v ) dv, (16)where T s, AGD is computed using Equation 13 and τ AGD ( v ) is the Gaussian function fitted by AGD to τ ( v ).The simulated column density of each gas structure isgiven by, N (H i ) true = (cid:90) structure n ( s ) ds . (17)In Figure 6, we compare the “true” and inferred col-umn density for all structures with AGD matches in ab-sorption and emission according to Equations 6, 8, 9and 10. As in Figure 4, the AGD and true estimatesagree within the 1 σ contours. However, outside the 1 σ contours, N (H i ) AGD overestimates N (H i ) true , in part be-cause the uncertainty in T s, AGD is propagated to N (H i )via Equation 16. We color the points in Figure 6 by T s, AGD to illustrate this. The most discrepant pointscorrespond to the highest values of T s, AGD , where theuncertainty in the matching process is highest (c.f., Fig-ure 4).In Figure 7, we investigate the scatter present inFigures 4 and 6 by plotting the ratios of the inferred(“AGD”) to direct (“true”) estimates of spin tempera-ture and column density as a function of various LOSparameters. These include Galactic longitude ( l ; a), ab-solute Galactic latitude ( | b | ; b), and the peak opticaldepth of the matched AGD line ( τ ; c). In each panel,the data points are colored according to T s, AGD /T s, true (red) and N (H i ) AGD /N (H i ) true (blue).In Figure 7, the ratio of AGD to true spin temperatureand column density falls within a factor of 2 for the ma-jority (68% , σ contours) of structures at all longitudes(a), latitudes (b) and peak optical depths (c) probed.This indicates that the AGD method is able to recoverthese properties reasonably well. The scatter in the ra-tio of AGD to true column density is also larger than forspin temperature, due to the fact that any uncertaintyin T s, AGD is propagated to N (H i ) AGD via Equation 16.The scatter in AGD/True appears to be constant withlongitude ( l ), yet increases at low latitudes ( | b | ) and lowpeak optical depths ( τ < . τ component to be affected by velocity blending.Examples of strong line blending are shown in Figure 4.However, the agreement between AGD and true spintemperature and column density for the majority of sim-ulated gas structures given our simple structure selec-tion and matching prescription is encouraging, and indi-cates that automatic routines for identifying and analyz-ing spectral components from H i observations – essentialfor future large observed and simulated datasets – can besuccessful in recovering true properties for a large frac-tion of interstellar gas structures. COMPARING REAL AND SYNTHETIC H i SPECTRA . . . . τ J2232 T B [ K ] . . . τ . T B [ K ] . . . τ . T B [ K ] . . τ T B [ K ] − − − −
10 0 10 20 30 40 v [km s − ]012 τ − − − −
10 0 10 20 30 40 v [km s − ]0204060 T B [ K ] Fig. 8.—
Example LOS H i absorption spectra (left) and corre-sponding H i emission spectra (right) from 21-SPONGE, with AGD-fitted Gaussian decompositions overlaid (dashed black). Compo-nents which match between absorption (left) and emission (right)according to Equations 9 and 10 are indicated by matching colorsbetween the left and right columns. After analyzing the biases of Gaussian analysis in re-covering gas structures and their properties using syn-thetically spectra from simulations, we proceed to com-pare observed (21-SPONGE) and simulated (KOK14) H i spectra via AGD-fitted Gaussian parameters.With the AGD decompositions of H i emission and ab-sorption for all 21-SPONGE and KOK14 LOS, we applythe criteria described by Equations 9 and 10 to matchas many AGD lines between absorption and emission aspossible. For KOK14, this does not take into accountmatching with gas structures along the LOS as describedin Section 5, in the interest of eliminating as many bi-ases as possible in our comparison between the match-ing statistics of KOK14 and 21-SPONGE (which doesnot have LOS density and temperature information fordefining structures).Figure 8 displays the matches between H i absorptionand H i emission for a set of 5 example observed LOS from21-SPONGE. All components fitted by AGD to each LOSare shown in dashed black, and the components whichsatisfy the matching criteria described by Equations 9and 10 are shown in colors. Number of components along LOS
In Table 1 we list the total number of AGD componentsin each H i absorption and emission dataset, as well as themean and standard deviation of the number of compo-nents ( N AGD ) per LOS. In addition, we list the totalnumber of matched components between absorption andemission, and the mean and standard deviation numberof matches per LOS.From Table 1, the mean value of N AGD is more thana factor of two greater for the observed 21-SPONGE H i absorption and emission spectra than KOK14, despite large scatter. Figure 9 displays histograms of N AGD for 21-SPONGE (top left panel) and KOK14 (bottomleft panel) absorption (black solid) and emission (orangedashed) components. In agreement with the statisticsshown in Table 1, the maximum number of componentsfitted to 21-SPONGE absorption (12) is a factor of twohigher than KOK14 (6).In the right panels of Figure 9, we account for theeffect or different viewing angles by multiplying N AGD bysin | b | for all 21-SPONGE and KOK14 components. Thisquantity is the effective number of components in thevertical direction of the simulated or observed volume.Since N AGD × sin | b | is still larger for the 21-SPONGEthan KOK14, it suggests that the discrepancy betweenis not a sin | b | effect.The larger number of components found in the 21-SPONGE spectra, after correcting for observing angle,indicates that the velocity range used to produce theKOK14 synthetic spectra is smaller than what is sam-pled by observations. The KOK13 simulations are knownto have a relatively low vertical velocity dispersion ( ∼ − − ), somewhat smaller than observed values.The lower velocity dispersion also yields a scale heightsomewhat smaller than observations. We will return tothis effect in Section 6.2.2. In more recent simulations(Kim & Ostriker 2016) with a better treatment for su-pernovae, velocity dispersions are in fact larger, and itwill be interesting to test whether this will lead to anincrease in the number of AGD features per LOS.We emphasize that the results shown in Figure 9 arederived from identical implementation of AGD to realand synthetic H i spectra, and thus the comparison isunaffected by biases introduced in spectral line analy-sis. Therefore, although the caveats described above areknown from external analysis of the KOK13 simulations,Figure 9 suggests that N AGD reflects the total velocityrange and path length.Furthermore, from Table 1, although the number ofmatches per LOS is consistent with the number of fit-ted lines per LOS ( N AGD ) in KOK14, there are com-paratively fewer matches per LOS than N AGD in 21-SPONGE. This difference likely comes from the so-calledmismatch of angular resolution. In 21-SPONGE, the an-gular resolution of the H i absorption measurements isdetermined by the size of the background source, not thetelescope beam (and is therefore < − (cid:48)(cid:48) ). However,the H i emission spectrum has an angular resolution of3 . (cid:48) . Therefore, the H i emission spectrum may not sam-ple the same structures seen in absorption, especially ifthere is significant emission structure on angular scalesbelow the resolution limit. This mismatch complicatesthe matching process and causes a larger attrition ratefor observations. Simulations, on the other hand, do notsuffer from this problem, as emission and absorption arederived using the same angular resolution. In the future,we plan to quantify this effect by smoothing simulatedspectra, and we further compare the 21-SPONGE andKOK14 emission properties in detail in Section 6.3. Properties of H i absorption components To compare 21-SPONGE and KOK14, Figure 10 dis-plays cumulative distribution functions (CDFs) of Gaus-sian parameters fitted by AGD to H i absorption spec-tra observed by 21-SPONGE (black) and simulated by1 . . . . . . . . . F r e q u e n c y N AGD . . . . . . . F r e q u e n c y KOK14 absorptionKOK14 emission . . . . . F r e q u e n c y N AGD × sin | b | . . . . . F r e q u e n c y KOK14 absorptionKOK14 emission
Fig. 9.—
Left: Histograms displaying number of AGD lines ( N AGD ) fit to H i absorption (black) and emission (orange) observations from21-SPONGE (top) and synthetic observations by KOK14 (bottom). Right: Histograms displaying number of AGD lines per unit pathlength in the “vertical” direction ( N AGD × sin | b | ) for H i absorption (black) and emission (orange) observations from 21-SPONGE (top)and synthetic observations by KOK14 (bottom). KOK14 with (blue) and without (orange) the WF ef-fect, normalized by the total number of components (seeTable 1). These parameters include amplitude ( τ ; leftpanel), FWHM (∆ v in km s − ; center panel) and meanvelocity ( v in km s − ; right panel). For each dataset, wealso plot the CDFs of 1000 bootstrapped samples (shownin lighter-shaded colors according to the legend) to illus-trate the effect of sample size and outliers on the shapeof the CDF. Comparison with previous studies
We include the results of the by-hand Gaussian decom-position of the first 31/52 21-SPONGE sources (dashedpurple; Murray et al. 2015, “DR1”) and the MillenniumArecibo 21cm Absorption Line Survey (dashed green;HT03) in Figure 10. With lower sensitivity in opti-cal depth, the HT03 distribution contains fewer τ < − components than are found in the 21-SPONGEor KOK14 AGD decompositions. However, the 21-SPONGE DR1 τ distribution agrees very well with the21-SPONGE AGD distribution, which indicates that al-though the AGD algorithm was trained using compo-nent parameters from HT03, it is successfully able to re-cover lower- τ amplitudes found in the higher-sensitivity21-SPONGE and KOK14 spectra. This agreement wasalso noted in the comparison between by-hand and AGDanalysis of a subset of the 21-SPONGE sample (Lind-ner et al. 2015). In addition, the 21-SPONGE AGD ∆ v distribution agrees very well with DR1 and HT03, indi-cating that for a wide range in optical depth sensitivity,a similar range in Gaussian spectral line widths can berecovered. Influence of local box simulations
From the righthand panel of Figure 10, the ob-served 21-SPONGE (AGD and DR1) and HT03 abso-lute mean velocities ( | v | ) agree very well. However,the KOK14 spectra are dominated by components with v <
10 km s − . This difference may be caused by thefact that the KOK14 spectra are constructed with a lim-ited path length ( s < i lines are limited bynearby gas with small variations of galactic rotation ve-locity. The limited range of v in KOK14 spectra causesmore components to have similar central velocity.To test the influence of the local box and lack of globalrotation effects in the simulation, we consider the effectof latitude on the matching statistics. For high latitudeLOS ( | b | > ◦ ) in KOK14, the number of absorptionfits, emission fits and matches per LOS are consistentwith the full KOK14 sample (i.e., all latitudes). How-ever, for 21-SPONGE, at high latitudes ( | b | > ◦ ) thereare 1 . ± . . ± . . ± . Minimum CNM temperature?
To highlight the comparison between 21-SPONGE andKOK14 we plot τ vs. ∆ v in Figure 11, includingmarginal histograms of both parameters. The median2 ⌧ . . . . . C D F v [km s ] 10 | v | [km s ] KOK14KOK14 (no WF)SPONGESPONGE (DR1)HT03
Fig. 10.—
Cumulative distribution functions (CDFs) of Gaussian parameters, including optical depth amplitude ( τ ), FWHM (∆ v )and absolute mean velocity ( | v | ), of the components fitted by AGD to observed (21-SPONGE), synthetic (KOK14) H i absorption spectra,including previous by-hand results from 21-SPONGE (purple, DR1; Murray et al. 2015) and HT03 (green) for comparison. For eachdataset, we plot the CDFs of 1000 bootstrapped samples (shown in lighter-shaded colors according to the legend) to illustrate the effect ofsample size and outliers on the CDF. − ∆ v [km s − ]10 − − − τ − − − − F r e q u e n c y KOK14KOK14 , noWFSPONGE − − − Frequency 10 − − − τ Fig. 11.—
Parameters of Gaussian components (∆ v , τ ) fittedby AGD to H i absorption spectra from 21-SPONGE (black) andKOK14 with the WF effect (blue) and without the WF effect (or-ange). Contours indicate the 1, 2, and 3 σ limits for the KOK14distributions. Marginal histograms display the same results ac-cording to the legend. σ sensitivity limit in τ is indicated by the dashed hor-izontal line ( σ τ = 10 − ), and the 21-SPONGE velocityresolution of 0 . − is indicated by the dashed verti-cal line.From the top panel of Figure 11, we observe a sharpcutoff in ∆ v ∼ − − in 21-SPONGE andKOK14. If we assume a limiting line width of ∼ − − , in the case of no turbulent broadening, thecorresponding CNM kinetic temperature is ∼ −
30 K,which is also equal to the spin temperature. The factthat 21-SPONGE and KOK14 agree in their lower limitto ∆ v , together with the fact that the AGD method is We note that the components with ∆ v < − in 21-SPONGE and KOK14 are likely spurious fits, given that the accu-racy of the AGD decomposition is known to be 80% in absorption(c.f., Section 4). a good measure of T s, true at similar temperatures (c.f.,Figure 4), suggest that the simulation and observationshave a similar lower limit for the CNM temperature of ∼ −
30 K.As shown in Figure 11, the peak optical depth spansthe whole parameter space all the way to our sensitiv-ity limit with no obvious evidence for the existence of aminimum optical depth for the CNM. In addition, onlya small fraction, < τ > Role of the WF effect
As seen most clearly in the main panel of Figure 11,the KOK14 spectra with and without the WF effect in-clude significant populations of components with ∆ v >
10 km s − and 0 . < τ < .
01. Although this regionis located well above the 21-SPONGE median sensitivityin optical depth, we find very few 21-SPONGE compo-nents there. In addition, in KOK14, these componentsare often found without narrow (CNM-like) componentssuperimposed along the same LOS. This is a type of pro-file not seen in 21-SPONGE observations. There are noobservational biases that would prevent us from seeingsimple Gaussian line profiles with a peak optical depth of ∼ .
01 and a velocity FWHM of 10 km/sec. In addition,the lack of isolated (devoid of CNM), broad, WNM-likefeatures in observations is supported by additional high-sensitivity H i absorption studies (e.g., HT03, Roy et al.2013). An example isolated, broad absorption line fromKOK14 is shown in the top row of Figure 2.We note that these broad, low- τ features appear in thesynthetic KOK14 spectra regardless of our treatment ofthe WF effect, although agreement with observations issomewhat improved when this is included (c.f., Figure11). The origin of these low- τ components is not wellunderstood and future comparisons with synthetic 21 cmprofiles from simulations will explore the effect of a morerealistic feedback treatment (e.g., Kim & Ostriker 2016).The present results already suggest that H i absorptionspectra may be able to provide discriminating tests ofthe input physics in simulations. Properties of H i emission components T B , [K]0 . . . . . C D F v , em [km s ] KOK14KOK14 , no WFSPONGESPONGE , DR1HT03 | v , em | [km s ] Fig. 12.—
Cumulative distribution functions (CDFs) of Gaussian parameters, including brightness temperature amplitude ( T B ), FWHM(∆ v ) and mean velocity ( v ), of the components fitted to observed (21-SPONGE) and synthetic (KOK14) H i emission spectra, includingprevious by-hand results from 21-SPONGE (purple, DR1; Murray et al. 2015) and HT03 (green) for comparison. For each dataset, weplot the CDFs of 1000 bootstrapped samples drawn from the full sample with replacement (shown in lighter-shaded colors according to thelegend) to illustrate the effect of sample size and outliers on the CDF. ∆ v , em [km s − ]10 − − T B , [ K ] − − − F r e q u e n c y − − − Frequency 10 − − T B , [ K ] Fig. 13.—
Parameters of Gaussian components (FWHM, T B, )fitted by AGD to H i emission spectra from 21-SPONGE (black)and KOK14 with the WF effect (blue) and without the WF effect(orange). Contours indicate the 1, 2, and 3 σ limits for the KOK14distributions. Marginal histograms display the same results ac-cording to the legend. Before comparing the properties of H i emission com-ponents in the same manner as in our comparison of ab-sorption components in Section 6.2, we emphasize againthat the 21-SPONGE and KOK14 emission profiles arederived from different angular scales. The angular reso-lution of the KOK14 emission spectra is the same as forthe absorption lines. However, the 21-SPONGE emissionspectra were observed with the Arecibo radio telescope,with a ∼ . (cid:48) beam at 21 cm derived from off-target po-sitions, and therefore have different angular resolutionthan the 21-SPONGE VLA absorption spectra. In thecase of the KOK14 spectra, which are derived from asimulation with a physical resolution of 2 pc, the pathlength would need to be longer than ∼ i emission spectra observed by 21-SPONGE(black), simulated by KOK14 with the WF effect (blue)and without the WF effect (orange), in addition to theresults of HT03 (green) and 21-SPONGE DR1 (purple).These parameters include amplitude in brightness tem-perature ( T B , in K; left), FWHM (∆ v , em in km s − ;center) and mean velocity ( v , em in km s − ; right).To illustrate the comparison between 21-SPONGE andKOK14 further, we display T B, vs. ∆ v , em for 21-SPONGE and KOK14 in Figure 13, with marginal his-tograms for both parameters. We note that the WF ef-fect does not make a difference to components fitted toH i emission spectra (i.e., orange and blue lines are in-distinguishable in Figure 12). This indicates that futuretesting of the implementation of the WF effect shoulduse absorption, rather than emission spectra. Boundaries in brightness temperature
In the left panel of Figure 12 (as well as Figure 13),the amplitudes fitted by AGD to 21-SPONGE agree wellwith 21-SPONGE DR1 and HT03. All three datasetswere obtained using the Arecibo radio telescope, andtherefore they have similar angular resolution. However,all three observed distributions are shifted to slightlylower amplitude in brightness temperature ( T B, <
10 K)relative to KOK14 in Figure 12. For the KOK14 LOSwith lower effective angular resolution than 21-SPONGE,the synthetic brightness temperature spectrum averagesany simulated emission over larger solid angles, andtherefore the KOK14 T B, should tend to be smallerthan the 21-SPONGE values derived from smaller an-gular scales. However, we observe more low- T B, compo-nents in 21-SPONGE than KOK14.The slight excess of components with high T B, andlow ∆ v in KOK14 may be caused by the lack of chem-istry and H i -to-H transition in the simulation. Further-more, as noted previously, the relative lack of high-∆ v components in KOK14 may be partly attributed to thereduced velocity dispersion in the simulations comparedto observations. Our observations may even suggest an4upper-limit on T B, of ∼ −
70 K. In addition, on thelow-end of the distribution the absence of a large filling-factor ( ∼ T ∼ − K) medium in thesimulations produces too much neutral gas per LOS, andtherefore too few LOS with T B, < a few K. The pres-ence of a hot medium occupying a large fraction of thevolume would also tend to reduce the incidence of de-tectable low- τ features, since many of the LOS withoutCNM would be primarily hot rather than primarily warmmedium. Broad emission components
In the middle panel of Figure 12, the 21-SPONGEand KOK14 ∆ v , em distributions agree very well below∆ v , em ∼
10 km s − . However, the 21-SPONGE AGD,DR1 and HT03 distributions extend to higher values of∆ v , em than KOK14. This is especially noticeable in Fig-ure 13 where 21-SPONGE components with large ∆ v , em and small T B, form a prominent tail of the distribution.As discussed previously, stray radiation the 21-SPONGEH i emission observations have not been corrected forstray radiation. which would appear in the form of weakand broad spectral components. To test how many 21-SPONGE components could be affected by stray radia-tion, we extracted H i emission spectra from the stray-radiation-corrected LAB survey (Kalberla et al. 2005) atthe positions of our sources and implemented AGD insame manner to decompose the LAB spectra into Gaus-sian components. Consequently, LAB data contain manycomponents with ∆ v , em ∼ −
70 km s − (c.f., Ap-pendix). However, components with ∆ v , em (cid:38)
70 km s − are not seen in LAB data, and therefore those compo-nents in 21-SPONGE are likely caused by stray radi-ation. There are 9 such components and they are alllocated in the large ∆ v , em and small T B, tail. Peeket al. (2011), did a similar comparison between LAB andArecibo data and concluded that the effect is unlikelyto exceed 500 mK, in agreement with our with our con-clusion that the broadest 21-SPONGE components with T B, < . v , em (cid:38) −
70 km s − ), shallow ( T B, <
10 K) components that areprominent in 21-SPONGE but absent in KOK14. Pos-sibly a higher velocity dispersion and the inclusion ofGalactic fountain in the simulation (see e.g., Kim & Os-triker 2016) may be able to reproduce such broad lines.Alternatively, this could signify the presence of the WNMat temperature > T s ∼ i emission spectra have absolute mean velocities up to ∼
50 km s − , whereas the KOK14 mean velocities appearlimited to v , em <
10 km s − (c.f., right panel Figure 10).In addition to the nature of the local box reducing | v , em | ,the KOK13 simulations did not include a Galactic foun-tain of WNM with velocities up to tens of km s − . Theinclusion of this important mechanism may produce rel-atively more structures with larger | v , em | and ∆ v , em − − − F r e q u e n c y KOK14KOK14 , no WFSPONGE T s, AGD [K]0 . . . . . . C D F KOK14KOK14 , no WFSPONGE Fig. 14.—
Top: histograms of T s, AGD (Equation 13) for allcomponents which “match” between H i emission and absorption(Equations 9 and 10) following the AGD analysis of 52 21-SPONGEH i spectral pairs (thick, solid black) and 9355 KOK14 H i spectralpairs with the WF effect (thin, solid blue) and without the WF ef-fect (thin, solid orange). Bottom: CDFs of the same results, with1000 bootstrapped samples drawn from the full sample with re-placement (shown in lighter-shaded colors according to the legend)to illustrate the effect of sample size and outliers on the CDF. and improve similarities between the 21-SPONGE andsimulation results. Observed Spin Temperature
We now compare distribution functions of the inferredspin temperature, from 21-SPONGE spectra and KOK14synthetic spectra. As we have shown in Section 5.4.1, forthe majority of cases the inferred spin temperature usingour AGD and radiative transfer approach, T s, AGD , is inagreement with the true simulated temperature, T s, true .We also discussed how at >
400 K, T s, AGD over-estimates T s, true . However, this bias will affect both observationsand simulations in the same way, as we apply the sameAGD fit and radiative transfer method to KOK14 and21-SPONGE. In addition, our main focus in this paperis on the shape of distribution functions, not the exactfractions. In future work, we will focus on the fractionsof H i in CNM, WNM and unstable phases using updatedsimulations.Figure 14 displays histograms and CDFs of T s, AGD (Equation 13) for 21-SPONGE (black) and KOK14 with(blue) and without (orange) the WF effect. The ob-served and simulated T s, AGD distributions follow eachother well until T s, AGD ∼ −
500 K, when they start todiverge. Although the 21-SPONGE observations appear5 ∆ v [km s − ]10 − − − − τ KOK14 . . . . . . . . l og10 ( T s , A G D ) [ K ] Fig. 15.—
AGD absorption properties ( τ , ∆ v ) for all compo-nents fitted to KOK14 with the WF effect (blue contours), andall components which match between absorption and emission ac-cording to Equations 9 and 10 for KOK14 (small circles) and 21-SPONGE (large circles with black outlines), colored by the AGDspin temperature ( T s, AGD ). to have a higher relative fraction of components at low T s, AGD ∼ −
30 K, these bins are determined by smallnumber of components, and the fractions agree withinuncertainties as illustrated by the bootstrapped samplesshown in the CDF panel. As discussed in Section 6.2,the observations and simulations display consistent cut-off in CNM line width, and Figure 14 indicates that thefractions of material at corresponding spin temperaturesof 20 −
30 K are consistent. HT03 detected a similarpopulation of cold CNM components with (cid:46)
20 K (17%by number, and 4% by mass), and suggested that thisis evidence for the absence of photoelectric heating bydust (Wolfire et al. 1995). We note that the simulationshave uniform heating throughout and constant metallic-ity, while in reality the photoelectric heating would bereduced in high-column regions.However, the KOK14 spectra show more H i with T s, AGD = 300 − T s, AGD .The AGD components with T s, AGD ∼ − i in absorption and has ex-cellent optical depth sensitivity (RMS noise σ τ < − per channel; Murray et al. 2015). To estimate the tem-peratures we are sensitive to in 21-SPONGE observa-tions, we assume a WNM column density of a few × (e.g., Stanimirovi´c et al. 2014, , KOK14), a FWHMof 10 −
20 km s − and a conservative RMS sensitivityin optical depth of 10 − − × − (per 0.4 km/s ve- − − − F r e q u e n c y KOK14KOK14 , no WFSPONGE T s, obs ( v ) [K]0 . . . . . . C D F KOK14KOK14 , no WFSPONGE Fig. 16.—
Top: histograms of per-channel spin temperature( T s ( v )) for all 52 21-SPONGE H i spectral pairs (thick, solid black)and 9355 KOK14 H i spectral pairs with the WF effect (thin, solidblue) and without the WF effect (thin, solid orange). Bottom:CDFs of the same results. locity channels), which results in T s ∼ − T s > ∼
400 K tendsto over-estimate the true spin temperature. Therefore,we infer that the lack of observed components with T s > T s > i absorption lines ofa subset of sources following by-hand Gaussian decom-position and found a residual H i absorption signal at 5 σ significance with an inferred excitation temperature of T s = 7200 +1800 − K, with a FWHM of 50 km s − , and anH i column density of 2 × cm − (Murray et al. 2014).This temperature is higher than analytical predictions forcollisional excitation of H i , and indicated that additionalH i excitation mechanisms (e.g., the WF effect) may bemore important for coupling the hyperfine spin states of6 − − − F r e q u e n c y KOK14KOK14 , no WFSPONGE N (HI) AGD [cm − ]0 . . . . . . C D F KOK14KOK14 , no WFSPONGE Fig. 17.—
Top: histograms of N (H i ) AGD (Equation 16) for allcomponents which “match” between H i emission and absorption(Equations 9 and 10) following the AGD analysis of 52 21-SPONGEH i spectral pairs (thick, solid black) and 9355 KOK14 H i spectralpairs with the WF effect (thin, solid blue) and without the WF ef-fect (thin, solid orange). Bottom: CDFs of the same results, with1000 bootstrapped samples drawn from the full sample with re-placement (shown in lighter-shaded colors according to the legend)to illustrate the effect of sample size and outliers on the CDF. H i to the local thermodynamic temperature than previ-ously thought.If the spin temperature of the WNM is actually T s ∼ < T s < n α , its spatial variations across the Milky Way,as well as the effect of turbulence, is essential to reconcileH i observations and theory. Per-channel T s Instead of using Gaussian-based temperature esti-mates, KOK14 considered per-channel spin temperaturein their analysis of simulated H i properties, They foundthat this quantity agrees well with the true per-channeltemperature extremely well (within a factor of 1.5) forall channels with τ (cid:46) i phases (e.g., Roy et al. 2013). To com- pare with their results, we derive per-channel spin tem-perature, T s, obs ( v ), by applying an equation similar toEquation 12 to the full τ ( v ) and T B ( v ) spectra from 21-SPONGE and KOK14, where, T s, obs ( v ) = T B ( v )1 − e − τ ( v ) . (18)For each LOS, we compute T s, obs ( v ) for only those chan-nels with optical depths greater than 3 × − , to con-servatively exclude all channels with low S/N.In Figure 16, we display histograms and CDFs of T s, obs ( v ) for 21-SPONGE (black) and KOK14 with(blue) and without (orange) the WF effect. There isa similar discrepancy between observations and simula-tions in Figure 16 as seen in Figure 14. The KOK14distributions are shifted to higher temperatures, while21-SPONGE spectra contain more channels with lowtemperature ( <
200 K). In addition, the 21-SPONGECDF has a more gradual rise, while simulated data showan abrupt jump near T s, obs ( v ) ∼ T s, obs ( v )derived from spectral channels without detectable ab-sorption. It is important to keep in mind, however,that observed and simulated spectra probe different LOSlengths, as discussed in Section 6.2, which could affectthe shape of per-channel CDFs. In addition, the absenceof a hot medium in the present simulations may lead totoo many LOS with T s ∼ Observed Column Density
As a further benefit of AGD analysis, by resolving theproperties of individual spectral components along eachLOS, we can analyze the column densities of individualgas structures in contrast with the total LOS columndensity. In Figure 17, we display histograms and CDFs of N (H i ) AGD for individual matched spectral componentsfrom the KOK14 (blue) and 21-SPONGE (thick black)H i emission and absorption spectral pairs. The columndensity distributions shown in Figure 17 agree well athigh- N (H i ) ( > cm − ), although the 21-SPONGEdistribution extends further below N (H i ) = 10 cm − .The absence of low-column lines in the KOK14 spectramay be caused by insufficient angular resolution for de-tecting small CNM features, or, as has been mentionedthroughout, the absence of a hot medium in the KOK13simulations which would serve to reduce the observedcolumn densities of the matched lines. The discrepancyaround 10 cm − is likely caused by the discrepancyin T s, AGD discussed above in Section 6.4. It is interest-ing to note that the application of the WF effect doesnot significantly affect the N (H i ) AGD distribution. TheWF effect influences the optical depth and spin temper-ature in opposite ways, i.e., increases T s and decreases τ .These quantities are the main ingredients of N (H i ) AGD (c.f., Equation 16), and therefore the two results of theWF effect may cancel each other out when computingcomponent-based column density. We conclude that thecomplexity of factors incorporated into the H i columndensity make it a less useful tool for isolating the impor-tance of the WF effect. SUMMARY AND CONCLUSIONS
Detailed comparisons between observations and simu-lations are crucial for understanding the physics behind7the observed properties of the ISM. Armed with synthetic21 cm emission and absorption profile data created fromthe 3D hydrodynamical simulations from KOK14 andhigh-sensitivity H i observations from 21-SPONGE, weaddress two main questions: (1) how well do H i spectrallines and our analysis methods recover simulated proper-ties of interstellar gas structures? (2) how do simulatedH i spectra compare with real observations? To analyze9355 synthetic and 52 real observations in an unbiasedand uniform way, we apply the Autonomous GaussianDecomposition (AGD) algorithm (Lindner et al. 2015)identically to both datasets. With these fits in hand,we compare simulated properties of gas structures alongeach LOS with observed properties of the Gaussian com-ponents.We summarize the main results:1. For gas structures defined by peaks in n/T s alongrandom LOS in the KOK13 simulations, Gaussianfits by AGD to synthetic H i absorption lines areable to recover gas structures successfully (Fig-ure 3). The recovery completeness (Equation 11)is 99% for high-latitude LOS ( | b | > ◦ ), 67% formid-latitude LOS (20 < | b | < ◦ ) and 53% forlow-latitude LOS (0 < | b | < ◦ ). The complete-ness declines with decreasing latitude because theLOS complexity is highest at the lowest latitudes.When these structures are matched to spectral linecomponents in both H i absorption and emission,the completeness is 83%, 38% and 29% for high,mid and low latitudes respectively. The declinein recovery completeness when matches betweengas structures and both H i absorption and emis-sion components are required reflects the difficultyin associating unambiguous spectral features in thepresence of line blending and turbulence.2. We use AGD fits to synthetic lines and simple ra-diative transfer to compute observational estimatesof spin temperature ( T s, AGD ) and column density( N (H i ) AGD ) for matched pairs of H i absorption andemission lines. We compare these estimates withthe simulated spin temperatures ( T s, true ) and col-umn densities ( N (H i ) true ) of corresponding struc-tures in the simulation. The observed and simu-lated spin temperatures agree within a factor of2 for the majority of structures (68%; Figure 4).At high temperatures, T s, AGD overestimates the T s, true due to velocity offsets between H i absorp-tion and emission lines caused by turbulent motions(Figure 5).The observed and simulated H i column densitiesalso agree well for the majority of structures. How-ever, the scatter is slightly larger than in the case ofspin temperature, because N (H i ) AGD incorporatesall uncertainty in T s, AGD (Figure 6). Furthermore,the agreement between inferred and true propertiesdeclines at low Galactic latitude and for low- τ com-ponents, where LOS-blending components hinderclear associations between emission and absorptionspectral lines (Figure 7).Overall, the agreement between temperature andcolumn densities inferred from synthetic spectra and computed from physical conditions in the sim-ulation is encouraging. Future comparisons withnext-generation simulations will allow us to con-struct “correction functions” for observed spin tem-perature and column density distributions.3. We find more fitted absorption and emission linesper LOS ( N AGD ) in the 21-SPONGE observationsthan the KOK14 synthetic observations (Table 1).This difference reflects the fact that the simulatedscale heights of the CNM and WNM in the KOK13simulations are lower than in observations, due tovelocity dispersions lower than seen in observations( ∼ − − ). These results are derived fromidentical implementation of AGD to 21-SPONGEand KOK14, and thus the comparison is unaffectedby biases introduced by AGD analysis. The dis-crepancy reflects the limitations of local box sim-ulations, with a simplified treatment of supernovafeedback, in producing realistic synthetic spectrallines.In addition, there are comparatively fewer matchesper LOS in 21-SPONGE than KOK14. The so-called mismatch in angular resolution between 21-SPONGE H i emission ( ∼ . (cid:48) ) and absorption (1 − (cid:48)(cid:48) ) complicates the matching process describedby Equations 9 and 10. In the future, we plan toquantify this effect and correct the observationalresults by smoothing simulated spectra.4. Using AGD, we objectively compare the propertiesof spectral lines fitted by AGD to H i absorptionfrom 21-SPONGE and KOK14. The 21-SPONGEspectra have a wider range in mean velocity (rightpanel, Figure 10), due to the limited horizontalbox size of the simulation. Furthermore, at highGalactic latitudes where the influence of the globaleffect is weakest, N AGD and matching statisticsagree between 21-SPONGE and KOK14. This in-dicates that simulated Galactic rotation plays animportant role in observed H i properties, and im-proved implementations of global effects will im-prove the completeness statistics of H i structurerecovery, and improve the completeness character-ization discussed in Summary points 1, 2 and 3.5. We find that KOK14 spectra include more low- τ and high-∆ v absorption lines than are seen in 21-SPONGE (Figures 10 and 11), despite being wellabove the 21-SPONGE sensitivity and resolutionlimits. These broad spectral lines are often foundwithout narrower, blended lines, which is a profilenot seen in observations by 21-SPONGE or previ-ous H i absorption line surveys (e.g., HT03, Roy etal. 2013). These are likely the result of the absenceof a hot, large filling-factor gas phase in the KOK13simulations (which would increase the number ofLOS without detectably-absorbing neutral gas), orpossibly some aspect of the simple treatment ofthe WF effect. In particular, we find that exclud-ing the WF effect enhances the population of thesediscrepant H i absorption components, and suggeststhat the WF effect is important for realistic spec-tral line properties. These features are not obvious8 in comparisons of integrated or per-channel prop-erties, and reflect the utility of studying velocity-resolved spectral components.6. We find that H i absorption spectra are more usefulprobes of ISM physics in comparison with simula-tions than H i emission spectra. Properties of com-ponents fitted to H i emission profiles are affectedby angular resolution mismatch and stray radia-tion, and are not sensitive to the implementationof the WF effect.7. The AGD-derived spin temperature from KOK14has more high-temperature gas (1000 < T s, AGD < T s > T s ∼ < T s < T s, AGD ∼ − i data. Upcoming large H i absorption surveys suchas GASKAP at the Australian Square Kilometer ArrayPathfinder (ASKAP) telescope (Dickey et al. 2013) willcontribute many more sources and improve the observa-tional statistics. The objective and efficient nature ofthe AGD analysis strategy presented here is well-suitedfor future large observed and simulated datasets, and willbe important for understanding the balance of CNM andWNM in the local and extragalactic ISM.This work was supported by the NSF Early Career De-velopment (CAREER) Award AST-1056780. C. E. M.acknowledges support by the National Science Founda-tion Graduate Research Fellowship and the WisconsinSpace Grant Institution. S. S. thanks the Research Cor-poration for Science Advancement for their support. Thework of E.C.O. and C.-G. Kim was supported by NSFgrant AST-1312006. 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To test for the effect of stray radiation on H i emission spectra, we analyze data from the LAB survey (Kalberla et al.2005). The LAB spectra were measured with a ∼ (cid:48) beam, whose shape was carefully modeled in order to removecontamination from stray radiation. We apply the AGD algorithm in the same manner as described in Section 6.3 toLAB spectra in the direction of the 52 21-SPONGE sources. In Figure 18, we reproduce Figure 13 including the LABdecomposition in red.From Figure 18, the LAB ∆ v , em distribution is shifted to larger values relative to KOK14 and 21-SPONGE. Asa result of the much larger angular resolution of the LAB survey, the AGD decomposition of LAB data in Figure 18feature fewer narrow velocity components (∆ v , em < − ), likely because the H i emission is smoothed over largerangular scales than in the 21-SPONGE spectra (from Arecibo Observatory, with ∼ . (cid:48) angular resolution).Furthermore, the LAB decomposition contains large-line width (∆ v , em >
30 km s − ), low-brightness temperature( T B, < v , em ∼ −
100 km s − outside the LAB distribution. Given that stray radiation has been removed from the LABspectra, this type of spectral feature may be indicative of stray radiation.Future work is needed to understand and remove the stray radiation contamination from Arecibo spectra. Kalberlaet al. (2010) found that stray radiation contributed up to 35% to some GASS H i spectra, however the effect is typically <
10% and is not significant at Galactic latitudes below ∼ ◦ (e.g., McClure-Griffiths et al. 2009). In Murray et al.(2015), we compared 21-SPONGE emission spectra with the GALFA-H i (Peek et al. 2011) and LAB surveys (Kalberlaet al. 2005) and concluded that difference are generally within 3 σ uncertainties. In a future paper, we will comparethe KOK14 synthetic H i emission spectra and data from the LAB and GALFA-HI surveys objectively using AGD inorder to statistically quantify the effect of stray radiation on observational data. ∆ v , em [km s − ]10 − − T B , [ K ] − − − F r e q u e n c y − − − Frequency 10 − − T B , [ K ] Fig. 18.—
Parameters of Gaussian components (∆ v , em , T B, ) fitted by AGD to H i emission spectra from 21-SPONGE (black) andKOK14 synthetic observations including the WF effect (blue) and without the WF effect (orange). Contours indicate the 1, 2, and 3 σ limits for the KOK14 distribution. We also include the AGD decomposition of H ii