The HI Kinematics of NGC 4013: a Steep and Radially Shallowing Extra-planar Rotational Lag
aa r X i v : . [ a s t r o - ph . GA ] J un Accepted to Astrophysical Journal on June 16, 2015
Preprint typeset using L A TEX style emulateapj v. 5/2/11
THE
H I
KINEMATICS OF NGC 4013: A STEEP AND RADIALLY SHALLOWING EXTRA-PLANARROTATIONAL LAG
Laura K. Zschaechner & Richard J. Rand Accepted to Astrophysical Journal on June 16, 2015
ABSTRACTNGC 4013 is a distinctly warped galaxy with evidence of disk-halo activity. Through deep
H i observations and modeling we confirm that the
H i disk is thin (central exponential scale height ofwith an upper limit of 4” or 280 pc), but flaring. We detect a vertical gradient in rotation velocity(lag), which shallows radially from a value of − +7 − km s − kpc − at 1.4’ (5.8 kpc), to a value of zeronear R (11.2 kpc). Over much of this radial range, the lag is relatively steep. Both the steepnessand the radial shallowing are consistent with recent determinations for a number of edge-ons, whichhave been difficult to explain. We briefly consider the lag measured in NGC 4013 in the context ofthis larger sample and theoretical models, further illuminating disk-halo flows. INTRODUCTION
To fully comprehend star formation (SF) processes,we must understand the properties and origins of thegas that fuels them. If galaxies are to avoid exhaustingtheir star forming fuel in a short time compared to theirage, then there must exist some means of gaining mate-rial, possibly through stellar mass loss as suggested byLeitner & Kravtsov (2011), or through the accretion ofprimordial neutral and ionized gas. Thus, extra-planarlayers are of great interest as they are the interface be-tween the main disk and the intergalactic medium (IGM)and all material accreted onto the midplane to fuel SFmust first pass through them.With the exception of neutral hydrogen
H i , aconnection has been made between the presence ofextra-planar components and SF both on local andglobal scales (e.g. Putman et al. 2012 and referencestherein). In particular, connections have been madebetween SF and hot gas (e.g. T¨ullmann et al. 2006,Hodges-Kluck & Bregman 2013, Li & Wang 2013), rel-ativistic particles (e.g. Irwin et al. 1999), dust (e.g.Howk & Savage 1999), and diffuse ionized gas (DIG)(e.g. Rand 1996; Rossa & Dettmar 2003). Extra-planar
H i has been shown to have some connection to disk-halo flows within some galaxies (e.g. Irwin (1994) andBoomsma et al. (2005)), but evidence for
H i accretion[e.g. HVCs in the Milky Way (Wakker & van Woerden1997), counterrotating clouds in NGC 891 seen byOosterloo et al. 2007] exists, leaving the relative contri-butions of each process ambiguous until further study.A powerful method to determine the origins of extra-planar layers is to study their kinematics. In particular,decreases in rotation speed with height above the mid-plane (lags) have come to the forefront in recent years(e.g. Oosterloo et al. 2007, Heald et al. 2007) leading tovarious models attempting to explain them in terms ofdisk-halo flows or an interaction between disk-halo cy-cling gas and accreting gas. Purely ballistic models of Max Planck Institute f¨ur Astronomie - K¨onigstuhl 17, 69117Heidelberg - Germany; [email protected] Department of Physics and Astronomy, University of NewMexico, 1919 Lomas Blvd NE, Albuquerque, New Mexico 87131- USA; [email protected] disk halo flow (e.g. Collins et al. 2002, Barnab`e et al.2006, Fraternali & Binney 2006) consider only gravita-tional effects: Material is ejected from the midplane indisk-halo flows and then moves outward radially due toits being farther from the gravitational potential. Toconserve angular momentum, the rotational velocity ofthe material decreases, creating a lag with height abovethe midplane. A clear shortcoming of such models is thatthey greatly under-predict lag magnitudes by up to anorder of magnitude (Heald et al. 2007). Thus, additionalprocesses may be at work such as pressure gradients ormagnetic tension (Benjamin 2002, 2012). However, sim-ulations involving these mechanisms have yet to be de-veloped in earnest, with future progress made possiblethrough observational constraints, in particular those in-volving lags.Pressure gradients and magnetic tension aside, sometheoretical simulations have gone beyond purely ballis-tic effects and disk-halo flows considered in isolation.Those presented by Marinacci et al. (2011) involve mo-mentum and heat exchanges occurring in disk-halo flows.When cool clouds are launched from the disk into ahot, initially static corona built up by accretion, gas isstripped from the clouds, which streams in the cloud’swake, mixing with the hot coronal gas. The mixed gasforms additional
H i clouds before falling to the mid-plane. This mixing results in substantially more efficientcooling of the hot coronal gas than is described in theearlier, related model of Fraternali & Binney (2008), inwhich hot coronal gas experienced cooling times ∼ − − kpc − to the deceleration of clouds, amount-ing to 50% of the total −
15 km s − kpc − observedin NGC 891. If one also considers ballistic effects, thismomentum exchange would nearly eliminate discrepan-cies between purely ballistic models and observed lagsin NGC 891 and the Milky Way. However, lags havebeen shown to differ substantially among galaxies (e.g.Heald et al. 2007, Sancisi et al. 2008, Zschaechner et al.2012, Kamphuis et al. 2013). Thus, additional galaxies Zschaechner & Randmust be observed and modeled to comparable sensitiv-ity and detail. The Westerbork Hydrogen Accretion inLOcal GAlaxieS (HALOGAS) survey (Heald et al. 2011)has sought to remedy this situation through extensive H i observations of a carefully selected sample of nearby spi-ral galaxies.Not only are lag magnitudes of interest, but variationsin lag magnitude with radius are expected based on con-servation of angular momentum. From ballistic consider-ations only, in a disk-halo flow, material at smaller radiiexperiences larger relative changes in the gravitationalpotential when ejected from the midplane to a givenheight (and thus a greater lag) than material at largeradii. Oosterloo et al. (2007), Zschaechner et al. (2011,2012), and Kamphuis et al. (2011) measure a shallowingof the lags in several nearby galaxies, with a current sum-mary given in Zschaechner et al. (2015). A general trendnoted in that paper is that the lags begin to shallow neara radius of 0.5R , and reach their shallowest point nearR itself. Considering the mismatch between observedlag magnitudes and ballistic models, it is likely that theorigin of the shallowing is more complicated as well.This work builds upon the HALOGAS sample andmodels, as well as supplemental galaxies presented inZschaechner et al. (2015). To this existing collection ofmodeled galaxies, we add NGC 4013, a nearby, well-studied, edge-on spiral galaxy with multiple extendedextra-planar components. NGC 4013
NGC 4013 is a member of the NGC 4151 groupcontaining 16 galaxies, but is itself rather isolated(Giuricin et al. 2000). Arguably, its most noteworthyfeature is its substantial warp originally seen in
H i byBottema et al. (1987). Here we present detailed
H i ob-servations and tilted-ring models of this galaxy. A briefsummary of NGC 4013’s observational properties is pro-vided in Table 1. Before presenting our work, we willreview some of the key findings throughout the litera-ture concerning NGC 4013.Rand (1996) presents an analysis of the DIG in ninenearby, edge-on spiral galaxies, including NGC 4013. Inthat work, four extra-planar filaments are found to ex-tend 2.5 kpc from the disk, defining a nearly “H” or“X” shape. These filaments could be related to out-flows fueled by SN activity near the center. The extra-planar DIG is not nearly as prominent as in NGC 891or NGC 5775, which have higher SFRs. The pres-ence of extended extra-planar DIG is confirmed byMiller & Veilleux (2003), who also see indications of gra-dients in rotation speed with height above the midplane.Rueff et al. (2013) observe the DIG, but also find extra-planar dust up to approximately 2 kpc above the disk.The dust is highly structured and filamentary - far moreso than the DIG. They conclude that direct evidence fora connection between the extra-planar DIG and extra-planar dust is lacking, suggesting the gas associated withthe dust is either atomic or molecular.Verstappen et al. (2013) observe the dust in sevennearby, edge-on spiral galaxies using Herschel. Accord-ing to their analysis, NGC 4013 is the only galaxy withintheir sample to display substantial extra-planar dust.All of these results suggest disk-halo cycling at a levelabove those of galaxies such as NGC 4244 and NGC 4565, but lower than that of NGC 891.Garc´ıa-Burillo et al. (1999) present CO observationsof NGC 4013, emphasizing its box-shaped bulge that islikely due to a bar. They note extra-planar extensionsshowing some connection to the extra-planar DIG de-scribed by Rand (1996). Additionally, they find a 130km s − spread in velocities near the center, which theynote has no corresponding H i feature.Mart´ınez-Delgado et al. (2009) discovered a tidal stel-lar stream extending 3 kpc above the plane of the diskin NGC 4013, possibly indicating a minor merger hastaken place approximately 3 Gyr ago. Such a scenariowas previously unexpected, as NGC 4013 appears to berelatively isolated and undisturbed.Rueff et al. (2014) observe an extra-planar
H ii regionin NGC 4013 that is 850 pc above the midplane andhas a substantially lower metallicity than the main disk.They interpret this as a mixing of material in the gasthat formed the associated stars.Perhaps most relevant to the work presented here,
H i observations and modeling were presented in Bottema(1995). Those observations were done using the WSRT,achieving a spatial resolution of 19” × − , and a single channel rms noise of0.3 mJy beam − . As will be shown in the next section,our data improve upon these numbers.(Bottema 1996) presented tilted-ring models of NGC4013 using the aforementioned WSRT data. Those mod-els include a central inclination of 90 ◦ , and a central po-sition angle of 66 ◦ . A substantial warp is also modeled.The rotation curve peaks at 195 km s − and decreasesto 165 km s − at large radii. Naturally, the models pre-sented in this paper will also constrain these parame-ters, which we derive independently but compare in § OBSERVATIONS & DATA REDUCTION
Observations of NGC 4013 were completed with theVLA in B and C configurations. Data reduction wasperformed in AIPS using standard spectral line meth-ods. Self-calibration was performed after combining alltracks, but before continuum subtraction, and led to sub-stantial improvement. Although the velocity resolutionof these data is 6.7 km s − , due to incomplete (not all ofthe awarded time was observed) observations, we aver-age three channels to obtain our desired rms noise levelfor the final cube. A variety of weighting schemes wereused to make cubes, with the final, full resolution cleanedcube using Briggs weighting and a robust parameter of3 after attempting a range of robust parameters to findthe optimal balance between resolution and sensitivity.A low spatial resolution cube (used only for the lowestcontour in Figure 1) uses a robust parameter of 2 withan outer uv taper of 7 kilo-lambda. Observation datesand cube parameters are given in Table 2. The primarybeam correction was performed in AIPS. THE DATA
Immediately clear from the zeroth-moment map (Fig-ure 1) is the prominent, nearly symmetric warp in NGC4013. Hints of a line of sight warp component, flare, orpossibly a thickened
H i layer may also be seen extend-inematic Modeling of NGC 4013 3
TABLE 1NGC 4013 Parameters
Parameter Value ReferenceDistance (Mpc) 14.6 a Nasa Extragalactic DatabaseSystemic velocity (km s − ) 835 Bottema (1995)Inclination 90 ◦ This workMorphological Type SBc Tully et al. (1988)SFR ( M ⊙ yr − ) 0.48 Weigert et al. (2015, submitted to AJ )Kinematic Center α (J2000.0) 11h 58m 31.6s This workKinematic Center δ (J2000.0) 43d 56m 52.2s This workD (kpc) 22.3 de Vaucouleurs et al. (1991)Total Atomic Gas Mass (10 M ⊙ ) 2.1 c This work a Distance is the median value of distances found on the NED database, excluding those obtainedusing the Tully-Fisher relation to be consistent with HALOGAS. b Includes neutral He via a multiplying factor of 1.36. Value obtained using single dish data at ouradopted distance is 2.4 × M ⊙ (Springob et al. 2005). WSRT value obtained by Bottema (1995)using our adopted distance is 2.3 × M ⊙ . TABLE 2Observational and Instrumental Parameters
Parameter ValueObservation Dates − C Config. 2010 Nov 162010 Nov 202010 Nov 28B Configuration 2011 Mar 21-232011 Apr 022011 Apr 222011 May 022011 May 042011 May 09Total Time on source - C Config. (hours) 9 a Total Time on source - B Config. (hours) 9.5 b Pointing Center 11h 58m 31.30s43d 56m 48.00sNumber of channels 256Velocity Resolution 20.1 km s − Primary Cube Beam Size 12.7 × ×
700 pcPA 4.92 ◦ RMS Noise − σ − Secondary Cube Beam Size 23.4 × × ◦ RMS Noise − σ − Although nine total hours were observed, approximately three ofthese hours had substantial RFI and were flagged heavily. b Approximately two hours of these data were omitted due to poorquality, resulting in 7.5 hours used in the final cubes. c Averaged from 6.7 km s − to 20.1 km s − for modeling in order toimprove SNR. ing to an apparent height of ∼ §
4) is required to determine which of thesecomponents are actually present in NGC 4013. The finalconstraints on these components will be given in § Fig. 1.—
A zeroth-moment map of NGC 4013. The full resolu-tion cube is represented in black, while an outline of the smoothedcube is in gray. Contour levels for the full resolution cube begin at2.6 × cm − and increase by factors of two. The gray contouris at 9.5 × cm − . Black lines represent the slice locations of bvdiagrams in Figure 5. The beams are shown in the lower left-handcorner (white is full resolution, black is the smoothed cube). Note theprominent warp. Also note the apparent thickness of the warped re-gions, possibly due to a flare, thick disk, or a warp component alongthe line of sight. responding to 736 km s − and 756 km s − . It is clearthat the disk is very thin near the center. Note that morethan half of the total radial extent is clearly affected bythe warp. The component of the warp across the lineof sight extends a substantial 3’ (12.8 kpc) above themidplane at its highest point.The total mass we derive from our observations (Ta-ble 1) is close to that derived from single-dish data(Springob et al. 2005), which implies that there is nosubstantial flux missing. A first assessment going by ap-parent extent only indicates that 32% of the total H i mass is at a height greater than 1 kpc (14”). If onlymaterial within a radius of 7.9 kpc (2.2’), thus avoidingthe clearly warped regions, this percentage amounts to18%. Again, we emphasize that this is only the
H i withan apparent extent above 1 kpc. Naturally if a line of Zschaechner & Randsight warp component is present, much of this gas willbe closer to the midplane.We now consider tilted-ring models created to fit thesedata. THE MODELS
For tilted-ring modeling, in which we create modelgalaxies comprised of a series of concentric rings forwhich parameters such as the rotational velocity, incli-nation, position angle, etc. may be specified, we use theTilted Ring Fitting Code (TiRiFiC, J´ozsa et al. 2007).Through tilted-ring modeling, features such as warps(via changes in inclination and position angle) and flares(vial radial increases in scale height) may be constrained.In addition to capabilities included in previous tilted-ring modeling software, TiRiFiC allows for the fitting ofasymmetries and localized features, as well as lags, whichwe utilize in our models of NGC 4013.The approaching and receding halves of NGC 4013are modeled separately as it is clear upon initial inspec-tion that their morphologies differ greatly, and fittingthem together could lead to unnecessary complexity inthe models. Additional asymmetries almost certainly ex-ist, but due to projection effects, these are not obviousupon inspection. Thus, we only allow for asymmetry be-tween the two halves. The initial estimate for the incli-nation was 90 ◦ and the position angle (PA) for the non-warped disk is 65 ◦ , both determined by eye (with onlyvery slight modifications to the central PA during laterstages of the modeling and no modification to the cen-tral inclination). The initial rotation curve was estimatedbased on matching the velocities on the terminal sides ofthe data and model lv (major-axis position-velocity) di-agrams. Subsequent modifications to the rotation curvewere made in the same manner until optimal fits werereached. The initial surface brightness distribution wasset by examining the flux distribution along the midplaneby eye. The surface brightness distribution was alteredonly slightly during the modeling, with the final distribu-tion optimized for our final preferred model, and imposedon all other models. The reasoning behind imposing thesurface brightness distribution from the final model ontothe other models is 1) any differences between modelswere minute and could be attributed to degeneracies be-tween the rotation curve and surface brightness, and 2)to maintain a clear and consistent picture of the changesin the models due to each additional component. Allmodels presented here share the same surface brightnessdistribution, velocity dispersion (15 km s − ), systemicvelocity (835 km s − ), central position (11h 58m 31.6s,43 ◦
56’ 52.2”), central inclination (90 ◦ ), radial positionangle dependence, and rotation curve. The only excep-tion is that the models including a lag have a rotationcurve that is 10 km s − higher in the approaching halfthan those that do not. The parameters not given hereare shown in Figure 3 for final models. Individual Models
We now address individual models and their definingcharacteristics, starting with a simple one-componentmodel with a warp component across the line of sight(1C in figures).As stated previously, it is clear that there exists a warpcomponent in NGC 4013 that lies across the line of sight. Thus, we model it with a single disk having an exponen-tial scale height optimally found to be 7” ± ± ±
60” (4.3 kpc) fromthe center in order to avoid the warped outer regions asmuch as possible (Figure 4)]. The width of the verti-cal profile is overestimated, yet the model cannot repro-duce the spread near the systemic side of bv (minor-axisposition-velocity) diagrams (Figure 5). Thus, the modelis too wide on the terminal side, but too narrow on thesystemic. Similarly, as may be seen in channel mapsshown in Figure 6, the disk is too thick at small radii,but too narrow at large radii. The splitting of most ofthe data bv diagrams near the systemic side is indicativeof a warp component along the line of sight, which wenow address.We add a line of sight warp component (“W” modelin figures, inclination parameters given in Figure 3) thatbegins near the radial onset of the other warp component( ∼ ◦ from the line ofsight (of course, there is a degeneracy due to projectioneffects in terms of the near and far sides). The scaleheight of the layer is reduced to 5” (350 pc) to accom-modate for the effects of the warp in this model. One cansee clear improvements to the bv diagrams and channelmaps, but can also see that the emission near the sys-temic side is not entirely matched in the former, nor arethe outer edges of the latter. The warp also improvesthe fit to the vertical profile, although a closer match isstill desirable. In the channel maps, the data show anincreasing minor axis thickness with radius, especially inthe unwarped component, which is not present in thismodel. This suggests a flare.Adding a substantial flare to the model (“WF” in fig-ures) by increasing the scale height with radius allowsfor a better fit to the vertical profile (parameters shownin Figure 3). Additionally, the scale height of the ap-proaching half decreases slightly for radii larger than 3.1’(13.2 kpc). The emission on the systemic sides of bv di-agrams is now spread out more, and the aforementionedthickening in the channel maps is well reproduced. Theimprovements due to the flare are best seen on the sys-temic sides (going from systemic to ∼
75 km s − abovesystemic) of bv diagrams corresponding to radii greaterthan 1.0’ (4.4 kpc) as seen in Figure 5. Note the in-creased spread between the contours, which is in betteragreement with the data. Also note the increase in thespread along the minor axis. One issue to note is that tofit this model, the central scale height is reduced to 3”(210 pc), pushing the limits of our ability to constrainthis parameter given our resolution. However, we do seean extremely slight distinction between a scale height of3” and 4”, mainly near the peak, shown in Figure 4. Inthe end, the significance of this difference is debatable,but we use the 3” (210 pc) scale height for subsequentmodels.For the sake of completeness, a second, thicker disk isbriefly considered, but as one may have already assumed,this yields no improvement to the models and is quicklydismissed. For instance, the thickening in the minor axisinematic Modeling of NGC 4013 5 Fig. 2.—
Channel maps of NGC 4013 show the substantial warp. Velocities in km s − are given in each panel. Contours begin at 2 σ (0.36mJy beam − , or 6.4 × cm − ) and increase by factors of two. The cube is rotatied 66 ◦ . Fig. 3.—
The parameters used in the final models of NGC 4013for the approaching and receding halves. Parameters presented hereare consistent for all models except for the scale height and inclina-tion distribution in the case of the 1C model, and the scale heightdistribution for the W model, for which these parameters are heldconstant by definition. Note the relatively symmetric distributionsof each quantity. The bold line in the bottom-right panel representsthe inclination direction evident in the channel maps as discussed aboveis naturally well matched by a flare. A thick disk wouldnot reproduce such behavior and add unnecessary com-plexity to the models. THE ADDITION OF A LAG
Improvements are seen via the addition of a lag of − +7 − km s − kpc − starting at a radius of 1.4’ (5.8kpc). The lag causes the observed velocity of the gas tobe shifted in the direction of the systemic velocity, withthis shift increasing with height. This causes the ter- Fig. 4.—
Vertical profiles of the data, one-component (1C), lineof sight warp (W), and warp with a flare (WF), summed over arange of ±
60” (4.3 kpc) from the center in order to avoid as muchof the warped regions as possible. Additionally, we present the WFmodel with a central exponential scale height of 4” as opposed to3” in order to illustrate the subtle difference between the two sincesuch differences are pushing the limits of the resolution of the data.Note how the 4” model slightly underestimates the peak. Given thisvery slight preference, we use the 3” scale height in our final models,although the central scale height could indeed be closer to 4”. A colorfigure is available in the online version. minal side of the contours in bv diagrams to shift froma relatively shallow U-shape to a steeper V-shape. Al-though we attempt to alter as few parameters as pos-sible, a slight adjustment to the rotation curve in the Zschaechner & Rand
Fig. 5.— bv diagrams showing all of the models presented in this paper. Note the clearly poor fit of the 1C model, especially in the narrowsystemic and wide terminal sides of each panel, as well as near the center. The W model is an improvement, but still lacks the properdistribution near the systemic sides that is achieved through the addition of a flare. Contours are as in Figure 2. Note the improved match inthe curvature on the terminal sides of panels corresponding to ± ± inematic Modeling of NGC 4013 7 Zschaechner & Rand Fig. 6.—
Representative channel maps showing the approaching(A) and receding (B) halves of NGC 4013. Note the splitting inthe panel corresponding to 736 km s − and how this is achieved byadding a warp component along the line of sight. The 1C, W, andWF models are shown here as these are the models exhibiting themost substantial changes from one another. The WFL and WFLvmodels are omitted from this figure as they are very similar to theWF model in channel maps since the lag is only in the flat part ofthe disk, which is visually overwhelmed by the outer, warped regionhere. Contours are as in Figure 2 approaching half (190 km s − to 200 km s − ) yielded abetter fit for the lag models. The subtle improvementsfrom the addition of such a lag are best seen in panelscorresponding to the terminal sides of ± H i present at high z atlarge radii, making it easier to detect a lag.) As for thecentral regions within ∼ ± ± ± THE UNCERTAINTIES OF THE MODELS
The rotation curves on each half nearly match eachother, and the average uncertainty in individual rings is
Fig. 7.—
Select bv diagrams of the data, warped model with aflare (WF), warped model with a flare and constant lag (WFL) , andwarped model with a flare and radially shallowing lag (WFLv) forthe approaching (A) and receding (B) halves. The orange lines tracethe approximate shape of the terminal sides of the bv diagrams inthe data. In Figure A, note the overall rounded appearance of theWF model compared to the data. Note also how the terminal sideof the WFL model is too pointed in the panel corresponding to 2.3’,while the WFLv is somewhat rounded, in closer agreement with thedata. In Figure B, the WF model is again too rounded comparedto the data. Regarding the radially shallowing lag, note the roundedappearance of the second and third contours on the terminal side ofthe data at − − − − − . Thus, to fit the data at smaller radii,the lag would need to become steeper for the WFL model, resultingin too much of a V-shape in the − approximately 5 km s − . The runs of the position an-gle and inclination are both extremely sensitive to smallchanges of a degree or two in a single ring. The narrowscale height near the center should be considered some-what skeptically as it approaches the limits of our abilityto constrain this parameter given the resolution of thedata and should be considered an upper limit. Still, aslight change can be seen in the vertical profile when thescale height near the center is changed from 3” to 4”(Figure 4). It is also clear that there is a flaring layer inNGC 4013.We consider uncertainties in the lag by creating min-imum and maximum value models which are judged toinematic Modeling of NGC 4013 9 Fig. 8.—
The distributions of lag magnitudes for the minimum,optimal, and maximum lag models shown in Figure 10. Constant lagvalues are kept to as large of radii as possible before it is obvious thatradially shallowing is necessary. Thus, the lag may shallow over itsentire range. Note the increased uncertainty in the upper limits dueto the spatial resolution along the minor axis of the data leading tosmaller changes introduced by steeper lags.
Fig. 9.— lv diagrams showing the data, one component (1C), andfinal (WFLv) models. The most substantial changes are near thesystemic side and at large minor axis offsets. Contours are as inFigure 2
Fig. 10.—
Select bv diagrams of the data, minimum, optimal,and maximum lag models for the approaching (A) and receding (B)halves. The various lags are parametrized in Figure 8. Note in par-ticular the subtle differences in the curvature along the terminal sidesin each model. The minimum lag model is too rounded, while themaximum model has a distinct “V” shape not present in the data.Contours are as in Figure 2 be still consistent with the data in addition to the op-timal lag (Figure 10). The various lag distributions areparametrized in Figure 8. Note the high maximum lag al-lowed, which is due to the thinness of the disk in relationto the resolution of the data. This difficulty is because ofthe thinning along the minor axis of the terminal side ofthe bv diagrams when steeper lags are introduced. Thethickness on this side becomes insensitive to the lag forrelatively steep lags. For a similar reason, an upper limitfor the magnitude of the lag is increasingly difficult todetermine with decreasing radius. Within ∼ can only beachieved through the addition of a lag . The presence ofthis V-shape is independent of the rotation curve at themidplane. Furthermore, the rounding of this V-shapeat large radii, also demonstrated in the aforementionedgalaxies, is achieved through shallowing of the lag. It ispossible that the rounding which we attribute to shallow-ing here, could be indicative of a constant lag startinghigher above the midplane with increasing radius (notshown). However, such a scenario, in which the round-ing of the terminal sides of bv diagrams is attributedsolely to starting height of the lag, is unlikely, althoughit, or a combination of shallowing and variable startingheight cannot be ruled out based on these data alone. DISCUSSION & CONCLUSIONS
NGC 4013
The central
H i layer in NGC 4013 is rather thin withan upper limit of 4” (280 pc) for the central scale height.The layer then flares up to 15” (1 kpc) at radii greaterthan 7 kpc. The rotation curve peaks at 195 km s − andis approximately flat, indicating a large dynamical mass(calculated as 1.1 × M ⊙ using a rotational velocity of190 km s − at a radius of 13.3 kpc). We see no evidencefor substantial amounts of extra-planar H i , but see evi-dence for a radially shallowing lag with a peak magnitudeof − +7 − km s − kpc − where the H i layer flares. Thelag then shallows with radius, going to zero near R .The lag is modeled starting from the midplane.Our models agree remarkably well with those presentedin Bottema (1996), the most notable difference being a20-30 km s − decrease in rotational velocity in the outerwarped radii in the Bottema (1996) model, while ourrotation curve remains flat. This difference is likely dueto degeneracies in modeling substantially warped regions.Recall that the lag is considered only at radii interior tothe warp, thus this discrepancy has no bearing on thefinal conclusions involving the lag.The lack of extra-planar H i is intriguing given the
Fig. 11.—
Measured radial variations of lags in several nearbyedge-on spiral galaxies including NGC 891 (Oosterloo et al. 2007),NGC 4244 (Zschaechner et al. 2011), NGC 4565 (Zschaechner et al.2012), NGC 5023 (Kamphuis et al. 2013), NGC 4302 and NGC 3044(Zschaechner et al. 2015). Note the general trend of lags beginningto shallow near 0.5R , and reaching their shallowest values nearR itself. A color figure is available in the online version. extra-planar DIG and dust. However, it is possible thatthere is a relatively large escape of ionizing radiationinto the gaseous halo, resulting in prominent extra-planarDIG but not H i .As a side note, we see no evidence for the potential
H i clouds noted by Bottema (1995) that are slightly beyondand near the highest point in the warp on the approach-ing half and conclude (as Bottema (1995) suggested) thatthey are likely artifacts in those data.As is also noted by Bottema (1995), there are no in-dications of a bar in
H i , although one appears to existin CO and at optical wavelengths (Garc´ıa-Burillo et al.1999). The location of the bar they describe coincideswith the central region of lower column densities beforethe initial peak in
H i (Figure 3).
The Lag in NGC 4013 in the Context of OtherGalaxies
The lag in NGC 4013 is the steepest measured for
H i to date. It also shows radial shallowing similar to severalexisting measured lags (Figure 11). For a more exten-sive presentation involving lag properties among a largersample of galaxies see Zschaechner et al. (2015), in par-ticular Table 5. The addition of NGC 4013 does notchange any conclusions concerning lags and SF proper-ties already presented in that work.
Implications for Theoretical Scenarios
As has been previously noted, observed lag magni-tudes are substantially steeper than those predicted bypurely ballistic models (e.g. Collins et al. 2002). Thelag in NGC 4013 is no exception. Recent models by(Marinacci et al. 2011) consider a momentum exchangebetween galactic fountain clouds and an initially static,but subsequently slowly-rotating hot halo that is builtup via accretion. As noted by (Zschaechner et al. 2015),the radial shallowing observed in these galaxies indicatesthat if lags are indeed due to such a scenario, then thismomentum exchange must be occurring predominantlywithin R .The shallowing itself is not yet understood, and mustbe considered in future simulations. Benjamin (2002,2012) suggests that extra-planar pressure gradients couldimpact lags and their radial variations. There havebeen few observational constraints to date for such pres-sure gradients, a situation soon to be remedied by deepcontinuum observations of edge-on galaxies (Irwin et al.inematic Modeling of NGC 4013 112012). Their sample includes seven galaxies with mea-sured lags, including five (as well as the receding side ofNGC 4302) with radially shallowing lags. ACKNOWLEDGMENTS