Comparing Gaseous and Stellar Orbits in a Spiral Potential

1.6 MB·English

by  Gilberto C. Gómez

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed21January2013 (MNLATEXstylefilev2.2)
Comparing Gaseous and Stellar Orbits in a Spiral Potential
Gilberto C. Go´mez1 ⋆, B´arbara Pichardo2 † and Marco A. Martos2 ‡
1Centro de Radioastronom´ıa y Astrof´ısica, Universidad Nacional Auto´noma de M´exico, Apdo. Postal 3-72, Morelia Mich. 58089, M´exico
3 2Instituto de Astronom´ıa, Universidad Nacional Auto´noma de M´exico, Apdo. Postal 70-264, Ciudad UniversitariaD.F. 04510, M´exico
1
0
2 Accepted forpublicationinMNRAS

n
a
J ABSTRACT

8
1 It is generally assumed that gas in a galactic disk follows closely non self-
intersecting periodic stellar orbits. In order to test this common assumption, we have

] performed MHD simulations of a galactic-like disk under the influence of a spiral
A
galactic potential. We also have calculated the actual orbit of a gas parcel and com-
G pared it to stable periodic stellar orbits in the same galactic potential and position.
. We found that the gaseous orbits approach periodic stellar orbits far from the major
h
orbitalresonancesonly.Gasorbitsinitializedatagivengalactocentricdistancebutat
p
different azimuths can be different, and scattering is conspicuous at certain galacto-
-
o centric radii. Also, in contrast to the stellar behaviour, near the 4:1 (or higher order)
r resonance the gas follows nearly circular orbits, with much shorter radial excursions
t
s than the stars. Also, since the gas does not settle into a steady state, the gaseous
a orbits do not necessarily close on themselves.
[

Key words: Galaxy:disk – Galaxy:kinematics anddynamics – Galaxy:structure –
1
galaxies: spiral structure – MHD
v
4
8
2
4
1 INTRODUCTION From resonances emerge strong orbital families like
.
1 the x1 type that manage to survive in models up to ap-
Differencesbetweenstellarorbitaldynamicsandgasdynam-
0 proximately the 4:1 resonance. After that periodic orbits
ics in disk galaxies play a fundamental role in the interpre-
3 from many families are found. This means that at reso-
1 tation (or misinterpretation) of kinematics observations in nancesnewfamilies ofperiodicorbitsstart existing.Atres-
: ordertodeducegeneralphysicalpropertiesofgalaxies. One
v onanceswemayencountermorethanonefamilyofperiodic
example(G´omez2006)isthattheinferred quantityof dark
i orbits (Contopoulos & Grosbøl 1986; Heller & Shlosman
X matteringalaxiescanbeaffectedbyinconsistenciesbetween
1996; Kaufmann & Patsis 2005). This is in part intrinsic to
thewell-studiedgasrotationcurveandtherelativelypoorly
r the problem, due to the non-linear character of the pertur-
a studied stellar disc rotation curve.
bations.Innormalnon-barredspiralgalaxieswithamoder-
Several observational and theoretical studies have
ate non-axisymmetric component, such as spiral arms, the
been devoted to the investigation of the relationship
stellarorbitsandgaseousflowcanbehighlynon-linear(this
between the gas streamlines and the corresponding stel-
means,theresponseofadynamicalsystemtoaperturbation
lar orbits. In particular, it is customary to identify the
dependsnon-linearlyontheamplitudeoftheperturbation),
non self-intersecting stable periodic orbits of a given
evenfarfromresonances.Also,phenomenalikegasshocksor
astronomical system, with plausible regions for gas
stellarchaosarefoundeveninthecaseoflow-amplitudespi-
streams to settle (e.g. Kalnajs 1973; Simonson & Mader
ralperturbations(Pichardo et al.2003;P´erez-Villegas et al.
1973; Lindblad 1974; Wielen 1974; van der Kruit
2012).
1976; Sanders& Huntley 1976; Vandervoort & Keene
Large scale galactic gaseous and stellar flows present
1978; vanAlbada & Sanders 1982; Sanderset al. 1983;
various important differences in their kinematics. Some in-
Contopoulos et al.1989;Athanassoula1992a,b;Patsis et al.
teresting known differences between the response of gas
1994; Piner et al. 1995; Englmaier & Gerhard 1997;
and stars to an imposed galactic potential model are due
Patsis et al. 1997; England et al. 2000; Vega-Beltr´an et al.
to the fact that gas is very responsive and suffers violent
2001;Regan & Teuben 2003; Patsis et al. 2009).
shocks, unlike stars. In resonant regions, it has been sug-
gested that shocks occur close to the intersections of stable
⋆ E-mail:[email protected] periodic orbits(Schwarz1981; van Albada& Sanders1982;
† E-mail:[email protected] Athanassoula 1992b). Gas is attracted to the spiral arms
‡ E-mail:[email protected] by their gravitational field and, unlike stars, it responds to
(cid:13)c 0000RAS
2 G´omez, Pichardo & Martos
pressuregradients. Innon-barredgalaxies withlarge bulges
or thick disks (such as early spirals), stars are supported
notonlybyrotationbutbystrongradialvelocitiesorveloc-
ity dispersion, while gas settles down in muchthinnerdisks
supported by rotation. The later the morphological type of
the galaxy, the more similar are gas and stellar kinematics
(Beckman, Zurita & Beltr´an 2004). In this work we study
the interplay between gas flow and stellar orbital dynami-
calpropertiesofspiral galaxies. Wehavecalculated aset of
MHD and particle simulations in a static background grav-
itational potential based on a full 3D spiral galactic model
(Pichardo et al. 2003). This includes spiral arms different
in nature to the classic tight winding approximation, in or-
der to contribute to the understanding of the relationship
between stellar orbits and gas orbits in non-barred spiral
galaxies, and the relation with periodic orbits inside and
outside resonance regions.
This paper is organized as follows. In §2, the galactic
potential used to compute the MHD and stellar orbits and
the initial simulation set up is described. In §3, the stellar
and MHDorbits obtained with thespiral potentialare pre-
sented and compared with each other. In §4, we present a
brief summary and discussion of our results.
2 NUMERICAL MODEL
ForthepurposeofthisworkwehaveproducedMHDsimula-
tionsundertheeffectofadetailedmodelofthebackground
galacticpotential.Wedescribenextthepotentialmodeland
theperformed simulations.
2.1 The Galactic Background Model
For the orbital study we have employed a detailed galac-
ticsemi-analyticmodel,resemblingaMilky-Way-likepoten-
tial. Rather than using a simple ad hoc model for a three-
dimensional spiral perturbation (suchas acosine function),
weconstructedathree-dimensionalmassdistributionforthe
spiralarmsandderivetheirgravitationalpotentialandforce
fieldsfrompreviouslyknownresultsinpotentialtheory.The
spiral arms distribution consists of inhomogeneous oblate
spheroids superposed along a given spiral locus. It is phys-
ically simple, with continuous derivatives and density laws.
In principle, the dimensions and mass density of the oblate
spheroids will depend on the type of spiral arms that are
modeled,gaseousorstellar.Themodel,calledPERLAS,isde-
scribed in detail in Pichardo et al. (2003). Comparisons of
the model with other models have been already published,
in particular for the most relevant parts of our study, the
spiral arms perturbation (Antoja et al. 2009; Martos et al.
2004;Pichardo et al.2003).Thecorrespondingobservation-
ally motivated parameters of the model used for this work
are presented in Table 1. The spiral perturbation is a 3D
steady two-armed model that traces the locus reported by
Drimmel & Spergel(2001),usingK-bandobservations.The
solar radius, at 8.5kpc, is close to the spiral arms 4:1 res- Figure 1. Density distribution after 1Gyr (top), 2.5Gyr (mid-
onance. The mean force ratio between the arms and the dle), and 4Gyr (bottom). Although the computational grid ex-
axisymmetricbackgroundisaround 10%,which isin agree- tends to r =22kpc, only the inner region is shown. The dashed
ment with the estimations by Patsis et al. (1991) for Milky circles show the position of the inner Lindblad, 4:1, corotation,
Way type galaxies. At the solar circle, the radial compo- -4:1andouterLindbladresonances.Gasrotates intheclockwise
nentoftheforceduetothespiral armsis4.4% ofthemean direction(counterclockwise outsidethecorotationresonance).
(cid:13)c 0000RAS,MNRAS000,000–000
Gaseous and Stellar Orbits 3
Table 1.Non-axisymmetricgalacticmodel(Pichardoetal.2003)
Parameter Value Reference
Spiralarmslocus Bi-symmetric(Logthm) Churchwelletal.(2009)
Spiralarmspitchangle 15.5◦ Drimmel(2000)
Spiralarmsexternallimit 12kpc Drimmel(2000)
Spiralarms:exp.withscale-length 2.5kpc Diskbased
Spiralarmsforcecontrast ∼10% Patsisetal.(1991)
Spiralarmspatternspeed(ΩP) 20kms−1kpc Martosetal.(2004)
axisymmetricbackground.Theself-consistencyofthespiral
arms has been tested through the reinforcement of the spi-
ralpotentialbythestableperiodicorbits(Patsis et al.1991;
Pichardo et al. 2003).
2.2 The MHD Setup
The initial setup of the MHD simulations consists of
a gaseous disk with a density profile given by n(r) =
n0exp[−(r − r0)/rd], where n0 = 1.1cm−3, r0 = 8kpc,
and rd = 15kpc. The gas behaves isothermally with a
temperature T = 8000◦K and is permeated by a mag-
netic field, initially in the azimuthal direction, with inten-
sity B(r) = B0exp[−(r −r0)/rB] where B0 = 5µG and
rB = 25kpc. The disk is initially in rotational equilibrium
between the centrifugal force, the thermal and magnetic
pressures, magnetic tension and a background axisymmet-
ric galactic potential (Allen & Santill´an 1991) consisting of
a bulge, stellar disk and halo.
ThisequilibriumisperturbedbythePERLASspiralarm
potential, which rotates with a pattern speed of ΩP = Figure2. Comparisonofthestellar(contours)andgaseousden-
20km s−1kpc−1. This pattern speed and background po- sities (grayscale) in the simulation. The stellar density is that
tential place the inner Lindblad, 4:1, corotation, -4:1, and correspondingtothebackgroundaxisymmetricgalacticdiskplus
outer Lindblad resonances at 2.81, 6.98, 10.92, 14.37 and the(imposed) spiralarmspotential, whilethegas density corre-
17.72kpc, respectively. spondstotothatofasnapshottaken1Gyrafterthestartofthe
We solved the MHD equations with the ZEUS code simulation. Since the model corresponds to a trailingspiral, the
gasflowsdownfromthetopoftheplot.Thedifferencesinnumber
(Stone& Norman1992a,b),whichisafinitedifference,time
andpositionofthestellarandgaseous armsareapparent.
explicit, operator split, Eulerian code for ideal magnetohy-
drodynamics. We used a 2D grid in cylindrical geometry,
with R∈[1.5kpc,22kpc] and a full circle in the azimuthal It is noticeable the presence of an MHD instability at
coordinate, φ,using750×1500 grid points.Both boundary corotation radius(furtherdescribedinMartos2013),which
conditions in the radial direction were outflowing. All the begins to develop 2.8Gyr after the start of the simulation.
calculations are performed in the spiral pattern reference This instability is present in a variety of simulations with
frame. Noself-gravity of the gas was considered. differentmagneticfieldintensities,andabsentinpurelyhy-
The simulation starts with the gas in circular orbits in drodynamical ones. With exception of this instability, ab-
equilibriumwiththebackgroundgalacticpotential,thermal sence of a magnetic field does not affect our conclusions.
andmagneticpressuregradientsandmagnetictension.After Figure2compares theimposed stellar densitywith the
the perturbation is activated, the gas very rapidly settles response gas density at a time 1Gyr after the start of the
into a spiral pattern, with two arms (plus an oval-shaped simulation. The stellar density is obtained by taking the
ring1) inside the inner Lindblad resonance (ILR) and four Laplacianofthepotentialscorrespondingtotheaxisymmet-
armsoutside,uptothe4:1resonance(seefig.1).Theinner ricbackgrounddiskandspiralarmdistribution,asgivenby
pair of arms follows a ∼9◦ pitch angle, while thetwo pairs the PERLAS model. In this figure, the differences in position
outside the ILR follow a ∼ 9◦ and ∼ 13◦ pitch angles; in and pitch of stellar and gaseous arms are apparent.
◦
contrast,theperturbingarmpotentialfollowsa15.5 pitch.
Theexactpositionandpitchangleofthegaseousspiralarms
oscillate slightly around the abovequoted values.
3 ORBITS IN THE SPIRAL POTENTIAL
1 This ring at r < 2kpc falls in a region where the dynamics Figure 3 shows a comparison of stable periodic stellar
should be dominated by a galactic bar, which is not included in and gaseous orbits in the background axisymmetric+spiral
thiswork.Therefore,itisdisregarded. galacticpotential.Thatfigurealsoshowsthepositionsofthe
(cid:13)c 0000RAS,MNRAS000,000–000
4 G´omez, Pichardo & Martos
give rise to the second pair of spiral arms (apparent in the
rightmostcolumnoffig.4).Asopposedtothestars,thegas
may change direction rapidly by developingoblique shocks,
which generate pressure gradients that further change the
path of the orbit. Between the ILR and 4:1 resonance, the
stellarandgaseousorbitsaresimilarintheradialrangethey
span,buttheshape differswhen thesecond pair of gaseous
arms becomes stronger, since the shocks at the arm posi-
tions significantly change the path of the gas parcel. Just
outside the 4:1 resonance, at the same Jacobi constant, we
have more than one main family of periodic orbits coexist-
ing, some of them stable, but the most of them unstable
(see fig. 7 in Contopoulos & Grosbøl 1986). Most of stars
approach an orbit with a four-fold symmetry; the gas, on
theotherhand,selectsatwofold symmetricorbitrelated to
the2:1coexistingperiodicorbitsfamily,associatedwiththe
only pair of arms found in thegas in thisregion.
Neartheresonances, thegas andstars follow quitedif-
ferentorbits(fig.5).Whilethestarsfollowthefamiliargeo-
Figure3. Comparisonofsomestellar(red)andgaseous(green) metricalshapes, thegas isunabletofollow thelarge depar-
orbitsnearmajorresonances.ThelocationoftheinnerLindblad tures from circularity of the stellar orbits. While the stars
(ILR), 4:1 and corotation are indicated by dashed circles. The follow oval-shaped periodic orbits neartheILR,thegas ac-
gaseousorbitneartheILRstartssimilartothestellarorbit,but
tually tries to avoid that region. Integrations starting near
it rapidly decays to a ring. Around other resonances, the gas
orinsidetheILRdecaytothering/spurstructurejustinside
followsanearcircularorbit,evenasthestellarorbitsexperience
2kpc.Thisstructureisnotexpectedtoappearinrealgalax-
large radial excursions. As reference, the gas density (grayscale)
ies,sinceusuallyitisaffectedbythestartingpositionofthe
andthelocusofthestellararms(black line)arealsoshown.
spiral arms and should be strongly influenced by a galactic
bar not included in the present model). In fact, in order to
finda gaseous orbit that resembles those found in thestars
majorresonancesandthelocusofthespiralarmsasdefined near the ILR, we must look for an orbit that remains out-
inthePERLASmodel.SinceZEUSisanEuleriancode,itdoes sidetheILRregionduringitstimeevolution(thetoprowin
not providetheactual path a gas parcel follows. Therefore, fig.4,forexample).Also,thesegaseous orbitspresentsome
in order to reconstruct a gaseous orbit, we integrated the dispersion, since they are not quite periodic and actually
velocity field interpolating in space and time between data cross themselves. This is possible since the simulation does
files for an array of initial positions, following the trajecto- not really reach a steady state, but instead settles on a pe-
ries of individual gas parcels in the resulting velocity field. riodic cycle, with the arms oscillating slightly around given
The integration was performed over 4Gyr, starting 1Gyr positions.
into the simulation in order to avoid the initial settling of
the gas into the spiral pattern. If during that time the gas
Nearthe4:1resonance,theperiodicstellarorbitfollows
parcelreacheseitherofthenumericalboundaries(atR=1.5
thefamiliar rounded-square,while thegas actually remains
and22kpc),westoptheintegration.Thestellarorbitswere
close to a circular orbit, with deviations from circularity of
obtained bydirect integration in thepotential, locating the
less than 400pc (although these deviations are systematic
periodic ones using a Newton-Raphsonalgorithm.
and allow for one pair of the gaseous spiral arms to extend
When comparing the stellar and gaseous orbits pre-
beyond this resonance, as opposed to the stellar pattern).
sented in Figure 3, it is apparent that some of them are
The strong evolution of the gaseous orbits around the ILR
similar, but also some may be quite different, even when
is absent near the 4:1 resonance.
both gas and stars are subjected to the same gravitational
potential.
Starsandgasfolloworbitsthataresimilarintheregions Furtherout,theexpectedbanana-shapedorbitsappear
between the resonances. In Figure 4, we show some of the at some positions, although the gaseous orbits are greatly
orbitscalculated in similar regions; wealso show thestellar distorted dueto the turbulencerelated to the instability at
massandgasdensitiesforcomparisonwiththeorbit.Inthe corotation(fig.6).Thestrongvelocityperturbationsrelated
case of the stellar orbits, it is noticeable how the familiar to this instability appear to generate points that act as at-
ovalshape(associated withtheILR)gradually morphsinto tractors. Future work will try to determine whether these
aroundedsquareastheorbitapproachesthe4:1resonance. positions are suitable for star formation. Outside corota-
Theselastorbitsarelesssupportiveofthespiralpattern,in tion,thegasorbitsarenearlycircularagain,withdeviations
accordance with analytical theory. lessthan0.5kpcfrom theinitial radius.Thisisnotsurpris-
Although the gas orbits are complex and quite sensi- ingsincetheunderlyingspiralperturbationisalreadyweak,
tivetotheinitialposition,theyfollowasimilarevolutionto representingonly1%oftheradialaxisymmetricforceatthe
the stars when regions away from the ILR are considered. -4:1 resonance (the arm potential tapers-off at a radius of
Itisnoticeablethatthegaseousorbitsdevelopfeaturesthat 12kpc, see Pichardo et al. 2003).
(cid:13)c 0000RAS,MNRAS000,000–000
Gaseous and Stellar Orbits 5
Figure 4. Comparison of stellar and gaseous orbits between resonances. The first two columns show stellar orbits in cartesian (first
column) andpolar coordinates (second column) plotted over the stellarmass density (grayscale) corresponding tothe PERLAS potential
model. The last two columns show gaseous orbits, also in cartesian and polar coordinates, plotted over the gas density (grayscale)
averaged over the 4Gyr period of the orbitintegration. In all cases, the initial position of the integration is marked by the smallopen
circle.Sincethemodelcorrespondstoatrailingspiral,thesenseofthegaseousorbitsisclockwiseinthecartesiancoordinateplots,and
downfromthetopinthepolarcoordinateones;onlystellarorbitswiththesamesenseofrotationwereconsidered.Thepositionofthe
innerLindblad,4:1andcorotationresonances isshown(dashed circles).
4 DISCUSSION AND CONCLUSIONS follow orbits close to the periodic stellar orbits that do not
cross themselves. We found that the gas does not always
WeperformedMHDsimulationsofagaseousdisksubjected behavethat way.
to the PERLAS spiral potential (disregarding the effect of a
galactic bar,which will beexplored in thefuture).Wethen The most obvious difference in the stellar and gas be-
studied the actual path thegas follows as it rotates around haviour is that the gas responds to the two-arm poten-
the galaxy in order to compare it with the stable periodic tial with four spiral arms (Shu,Milione & Roberts 1973;
stellarorbitsexistinginthatsamegalacticpotentialinorder Martos et al. 2004), organized in two pairs, each with a
totestthefrequentlystatedassumptionthatthegasshould tighter pitch angle than the underlying potential. The four
(cid:13)c 0000RAS,MNRAS000,000–000
6 G´omez, Pichardo & Martos
Figure 5. Similartofig.4,fororbitsneartheresonances.
Figure 6. Gaseous orbitsnearthecorotationresonance.
gaseous arms are well defined between the ILR and the Patsis et al. 1997 and Vorobyov 2006) and will be explored
4:1 resonance, more or less straddling the stellar arms in in future work.
similar fashion to the secondary compressions described by
Shu,Milione & Roberts (1973) (see fig. 2). Outside the 4:1 As opposed to stars, the gas may develop shocks at
resonance, the gaseous arms follow the stellar arms more thearmpositions,whichallowitsorbits(understoodasthe
closely. The gas response to stellar arms depend on the de- pathagasparcelfollows)tochangedirectionabruptly.This
tails of the imposed potential (compare, for example, with behaviourgeneratescuspsinthegaseousorbitsthatarenot
present in thestellar orbits.
(cid:13)c 0000RAS,MNRAS000,000–000
Gaseous and Stellar Orbits 7
Setting aside the extra cusps in thegaseous orbits, the ACKNOWLEDGMENTS
stellar and gaseous orbits are most similar in between the
The authors wish to thank P. Patsis for useful discussions
resonances, when the general shape and radial range are
on the subject, W. Henney for reviewing the manuscript,
considered. These are theregions where theperiodic stellar
andananonymousrefereeforcommentsthatledtoamuch
orbitsarerounderand theradial excursionsaresmaller, al-
improved paper. This work has received financial support
lowing thegas tomoreclosely follow thestellar orbitssince
from UNAM-DGAPA PAPIIT grants IN106511 to G.C.G.
it will be less influenced by forces other than gravity. Near
and IN110711 to B.P.
the resonances, however, the stars suffer large radial excur-
sions that imply regions with small curvature radii so that
theorbitmaycloseonitself(seefig.5).Agasparcelcannot
follow such an orbit2 and either looses angular momentum
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