Tremaine-Weinberg integrals for gas flows in double bars
aa r X i v : . [ a s t r o - ph ] J a n **FULL TITLE**ASP Conference Series, Vol. **VOLUME**, **YEAR OF PUBLICATION****NAMES OF EDITORS** Tremaine-Weinberg integrals for gas flows in double bars
Witold Maciejewski
Astrophysics Research Institute, Liverpool John Moores University,Twelve Quays House, Egerton Wharf, Birkenhead CH42 1LD
Hannah Singh
Astrophysics, University of Oxford, Keble Road, Oxford OX1 3RH
Abstract.
We report on our attempts to achieve a nearly steady-state gasflow in hydrodynamical simulations of doubly barred galaxies. After exploringthe parameter space, we construct two models, for which we evaluate the photo-metric and the kinematic integrals, present in the Tremaine-Weinberg method,in search of observational signatures of two rotating patterns. We show that suchsignatures are often present, but a direct fit to data points is likely to returnincorrect pattern speeds. However, for a particular distribution of the tracer,presented here, the values of the pattern speeds can be retrieved reliably evenwith the direct fit.
1. Introduction
Bars within bars are common in disc galaxies – up to 30% of early-type barredgalaxies contain them (Erwin & Sparke 2002). It is generally expected that theyplay a role in the feeding of active galactic nuclei, though the direct theoreticalconfirmation is difficult, as the interaction between the bars considerably narrowsthe range of possible systems (Maciejewski & Sparke 2000, hereafter MS00; Ma-ciejewski & Athanassoula 2008). Corsini, Debattista, & Aguerri (2003) demon-strated that in NGC 2903 the two bars rotate with two different pattern speeds.Maciejewski (2006) and Merrifield, Rand, & Meidt (2006) proposed extensionsto the Tremaine-Weinberg method (Tremaine & Weinberg 1984) in order to de-rive multiple pattern speeds. There were also attempts to derive them fromdirect fitting of straight lines to the Tremaine-Weinberg (TW) integrals. Herewe analyse in detail the behaviour of the TW integrals in hydrodynamical mod-els of gas flow in double bars. Since only one dynamically plausible model ofdouble bars has been constructed so far (MS00), we approach the problem fromanother direction here: by changing the parameters of the two bars, we searchfor steady-state flows, which may indicate preferred systems.
2. Code and models
We used the grid-based Eulerian hydrodynamical code CMHOG written byJames M. Stone and adopted to the polar grid in two dimensions by Piner,Stone, & Teuben (1995). The code implements the piecewise parabolic method1
Maciejewski and Singh . . . . Figure 1. Snapshots of gas density in two models of gas flow in double bars:Setup 1 at 955 Myr (left panel) and at 980 Myr (central panel); Setup 2 at1445 Myr (right panel). Darker shading indicates higher density. Both barsrotate counterclockwise and the outer bar is vertical on the plots. Units onthe axes are in kpc. (PPM), it uses the isothermal equation of state and it does not account forself-gravity in gas. From the set of models by Regan & Teuben (2003, hereafterRT03) of gas flow in a single bar, we picked the ones in which the flow is likelyto become a steady state after the secondary bar is introduced. This is likely ifin the single bar the gas settles on an oval around that bar, with no features inthe centre, where the secondary bar is going to be placed. The models in RT03,which consider cold gas (5 km s − speed of sound), are particularly suitable,since for warmer gas a nuclear spiral develops in the centre (Maciejewski 2004).The model from the bottom-right panel in fig.3 of RT03 (axial ratio of thebar a/b = 2 .
0, bar quadrupole moment Q M = 12 . × M ⊙ kpc − , centralmass concentration ρ c = 1 . × M ⊙ kpc − ) is the best case of the oval flowaround the bar within which the secondary bar can be placed (Setup 1). We alsosearched for models with the nuclear ring sufficiently large that the flow patternsinduced by the secondary bar could fit within it (Setup 2). As the starting pointfor that setup, we used the models from the top row of fig.3 in RT03 ( a/b = 2 . ρ c = 3 . r L (measured at theL1 point) from 6 kpc in RT03 to 11 kpc. In total we constructed 21 modelsby varying a/b between 1.8 and 2.5, Q M between 4.5 and 12.5, and ρ c between1.0 and 4.8, in the units above. Five out of these 21 models reproduce wellthe two desired setups described above: gas flows for Setup 1 appear to reachsteady-state within the oval flow, but not outside it, while gas flows for Setup 2are steady-state throughout, although strong shocks persist along the bar.For the introduction of the secondary bar we used the parameters of the fivemodels above. We constructed 5 models of gas flow in double bars, out of whichtwo closest to the steady-state have the following parameters for Setup 1 (2): a/b = 1 . . Q M = 12 . . ρ c = 1 . . r L = 6(11), in the units above,with the parameters of the inner bar close to those from Model 2 in MS00: theratio of the semi-major axes 0.2, the mass ratio 0.10 (0.15), the small bar axialratio 2, and its pattern speed 110 km s − kpc − . Snapshots of gas density inthese two models are showed in Fig.1. The flow in the region of the inner bar inSetup 1 is far from a steady state. A gaseous oval following that bar is flattened as flows in double bars . Figure 2. The photometric TW integral P , on the horizontal axis, plottedagainst the kinematic integral K , on the vertical axis, both measured alongthe slits parallel to the line of nodes. Stars mark the integrals for the slitsoffset by less than 2 kpc from the line of nodes (horizontal in Fig.1), while filledcircles correspond to the slits further out. The slope of the solid (dotted) lineis equal to the imposed pattern speed of the outer (inner) bar. The diagramsare constructed for Setup 1 at 835 Myr (left panel), and Setup 2 at 755 Myr(central panel) and at 1005 Myr (right panel). when the bars are aligned, and nearly circular at bars perpendicular. This isinconsistent with the orbital structure found in Model 2 of MS00. However, thelow central mass concentration in Setup 1 may imply that the x orbital familythat gives rise to the orbits supporting the inner bar in Model 2 from MS00 isabsent there. Throughout both bars, the gas flow in Setup 1 is likely to followorbits that originate from the x orbits in a single bar. In Setup 2, most ofthe gas accumulates in the region dominated by the inner bar and the flow ismuch closer to the steady-state than in Setup 1 or in Model 2 of MS00 (seeMaciejewski et al. 2002).
3. Tremaine-Weinberg integrals for the models with two bars
In the two models described above, we followed the gas flows for 3 Gyr, recordingthe hydrodynamical variables (density, velocity) on the grid every 5 Myr. Twopattern speeds are imposed in our models, and the flow is often far from a steadystate, hence the assumptions of the Tremaine-Weinberg method are violatedhere. However, even with the assumptions of the method violated, the TWintegrals remain well defined, and the hydrodynamical variables that we recordedare sufficient to determine them.For each setup, we calculated the TW integrals at 600 snapshots in time,and searched for a signal of rotating patterns. As our models evolve, we firstintroduce the outer bar, and then the inner bar. In the early evolutionary stages,with only the outer bar present, the relation between the TW integrals is linear,having the slope consistent with the imposed pattern speed, despite the flow notbeing completely settled. Once the secondary bar is introduced in Setup 1, theflow becomes very unsettled, and for the great majority of snapshots no relationbetween the TW integrals is observed. On rare occasions when there is a linear
Maciejewski and Singh relation, its slope is inconsistent with the imposed pattern speeds, and there isno indication of two distinct rotating patterns – see Fig.2, left panel.In Setup 2, the modifications of the flow in the outer bar caused by theintroduction of the inner bar are much smaller than in Setup 1, and two steadyflows, generated by the two bars, are being established. In the majority ofsnapshots, the points in the diagram for the TW integrals gather on two distinctlines (Fig.2, central and right panels). The points for the slits far from the lineof nodes, where the outer bar dominates the flow, gather on a line whose slopeis consistent with the imposed pattern speed of the outer bar. However, in theearly evolution of the model, points for the slits close to the line of nodes gatheron a line whose slope can vary with time, and is clearly inconsistent with theimposed constant pattern speed of the inner bar (Fig.2, central panel). Thischanges later in the run, when both sets of points indicate two correct patternspeeds at most of the snapshots (Fig.2, right panel).
4. Discussion and conclusions
Correct pattern speeds in Setup 2 are recovered only at the late stages of evolu-tion, because by then the majority of gas is accumulated in the region dominatedby the inner bar. Earlier in the run, the slits close to the line of nodes sampleboth flows dominated by the inner and the outer bar. At the end of the run, thecontribution from the outer bar is reduced, so that in the slits close to the line ofnodes most of the tracer follows the inner bar.
This is the only gas distributionfor which two pattern speeds can be derived with good confidence from a directfit of straight lines to the TW integrals. Otherwise, the points indicating theTW integrals may gather on two lines, but the slope of the second line mayshow no relation to the pattern speed of the inner bar. Moreover, for manysystems with two pattern speeds, points representing the TW integrals may notgather on any line, as in Setup 1. We conclude that if there is more than onerotating pattern in a galaxy, the direct fits to the TW integrals may providereliable pattern speeds only for a very specific distribution of the tracer. Fullycomprehensive work on this subject will be possible once we know the range ofsystems with two pattern speeds that are dynamically possible.
Acknowledgments.
This work was supported by the Polish Committee forScientific Research as a research project 1 P03D 007 26 in the years 2004–2007.