The Contribution of Spiral Arms to the Thick Disk along the Hubble Sequence

1.1 MB·English

by  L. A. Martinez-Medina

February2,2015
PreprinttypesetusingLATEXstyleemulateapj
THE CONTRIBUTION OF SPIRAL ARMS TO THE THICK DISK ALONG THE HUBBLE SEQUENCE
L. A. Martinez-Medina 1, B. Pichardo2, A. P´erez-Villegas3 & E. Moreno2
1 DepartamentodeF´ısica,CentrodeInvestigacio´nydeEstudiosAvanzados delIPN,A.P.14-740,07000M´exicoD.F.,M´exico;
lmedina@fis.cinvestav.mx
2Instituto deAstronom´ıa,UniversidadNacionalAut´onomadeM´exico,A.P.70–264,04510, M´exicoD.F.,M´exico;
[email protected]
3 CentrodeRadioastronom´ıayAstrof´ısica,UniversidadNacionalAut´onomadeM´exico,ApartadoPostal3–72, 58090Morelia,
5 Michoac´an,M´exico;[email protected]
1
0 ABSTRACT
2
The firstmechanisminvokedto explainthe existence ofthe thickdisk inthe MilkyWay Galaxy,were

n the spiral arms. Up-to-date work summon several other possibilities that together seem to better
a explain this component of our Galaxy. All these processes must affect differently in distinct types
J of galaxies, but the contribution of each one has not been straightforward to quantify. In this work,

0 we present a first comprehensive study of the effect of the spiral arms in the formation of thick
3 disks, as going from early to late type disk galaxies, in an attempt to characterize and quantify this
specific mechanism in galactic potentials. To this purpose, we perform numerical simulations of test

] particles in a three-dimensional spiral galaxy potential of normal spiral galaxies (from early to late
A
types). By varying the parameters of the spiralarms we found that the verticalheating of the stellar
G disk becomes very important in some cases, and strongly depends on the galaxy morphology, pitch
angle, arms mass and its pattern speed. The later the galaxy type, the larger is the effect on the
.
h disk heating. This study shows that the physical mechanism causing the vertical heating is different
p from simple resonantexcitation. The spiral pattern induce chaotic behavior not linked necessarily to
-
resonancesbut to direct scattering of disk stars, which leads to an increaseof the velocity dispersion.
o
r We appliedthis study tothe specific example ofthe Milky WayGalaxy,forwhichwehavealsoadded
t an experiment that includes the Galactic bar. From this study we deduce that the effect of spiral
s
a arms of a Milky-Way-like potential, on the dynamical vertical heating of the disk is negligible, unlike
[ later galactic potentials for disks.

Keywords: galaxies: evolution — galaxies: kinematics and dynamics — galaxies: spiral — galaxies:
1
structure
v
9
4
1. INTRODUCTION mechanism responsible for the vertical heating of the
6
disk from observations, specially since more likely it
7 Simulations of galaxy formation are coming to point
might be rather a combination of several possibil-
0 where detailed processes of galaxies have never been
ities. Among the mechanisms proposed there are
. explored before in detail, such as random and rota-
1 some external to the disk such as hits by satel-
tional velocities can be better studied and understood
0 lite galaxies or minor mergers (Huang & Carlberg
(Scannapieco et al.2011). Details ondisk potentials can
5 1997; Velazquez & White 1999; Benson et al. 2004;
be probed and compared with observations, and we are
1 Font et al. 2001; Quinn et al. 1993; Villalobos & Helmi
nowabletoshedsomelightonevolutionofgalaxiesstart-
: 2008; Di Matteo et al. 2011); scattering by dark
v ing now on small scale stellar motions.
i DynamicalheatingoftheMilkyWaydiskhasnowbeen halo objects or globular clusters (Vande Putte et al.
X 2009; Ha¨nninen & Flynn 2002). And some of in-
known for over 60 years,mainly throughobservations in
r thesolarvicinity. Fromthoseobservationswelearntthat ternal origin, such as, dynamical heating by di-
a rect encounters with giant molecular clouds (Carlberg
stellar random motions correlate nicely with their ages
1987; Villumsen 1983; Lacey & Ostriker 1985; Lacey
known as the age-σ relation (Wielen 1977; Binney et al.
1984; Spitzer & Schwarzrchild 1951; Inoue & Saitoh
2000). In particular, in the case of the Milky Way disk,
2014); heating by encounters with the potential
it is knownthat the radialvelocitydispersionis twice as
produced by long-lasting spiral arms (Faure et al.
much as the verticaldispersion and that the radial scale
2014) or irregular and transient spiral structure
length of the thick disk is much shorter than that of the
(Minchev & Quillen2006;Fuchs2001;Jenkins & Binney
thin disk (Bensby et al. 2011).
1990;Barbanis & Woltjer1967);perturbationsfromstel-
Recent studies also show that many, if not all, edge-
lar bars (Saha, Tseng & Taam 2010); dissolution of
on spiral galaxies appear to host dual disk systems
young stellar clusters (Kroupa 2002); or during an in-
(Gerssen & Shapiro Griffin2012;van der Kruit & Searle
tense star formation phase in a period of intense accre-
1981), a younger, dynamically colder and thinner com-
tionveryearlyinthehistoryoftheGalaxy(Snaith et al.
ponent: the thin disk and at least one older compo-
2014).
nent, (mainly) stellar, dynamically hotter and thicker
Because of the nature of these theories, the effects are
component: thethickdisk(Yoachim & Dalcanton 2006;
dependentongalaxymorphology,particularlytheintrin-
Yoachim & Dalcanton 2008; Comero´n et al. 2011).
sic mechanisms such as bars and spiral arms. Therefore,
It is still not straightforward to dilucidate the
2 Martinez-Medina et al. 2015
to deeply understand secular evolution of disk galaxies, to isolate the effect for example, of the arms or es-
itiscriticaltostudydynamicalheatinginagoodsample tablish in detail the role that each one of the pa-
of different disk galaxy types. Finally, from the point of rameters that characterize the spiral pattern play over
view of observations, the radial heating agent seems to the vertical heating of the disk. On the other hand,
varyexactlyasexpectediftheagentwerethespiralarms, when treated as steady spirals exerted as a perturba-
which provides a good chance that the spiral structure tion to the axisymmetric potential, according to the
has at least an important role as a heating mechanism hypothesis of Lin & Shu (1969), the heating is mini-
in the plane of galactic disks (Gerssen & Shapiro Griffin mum and linked only to the resonant regions of the
2012). The importance and influence of spiral arms, spirals (Lynden-Bell & Kalnajs 1972; De Simone et al.
and even their very same nature is still under debate, 2004), being more efficient in the radial direction
there is no straightforward observational prove yet of (Sellwood & Carlberg 1984; Minchev & Quillen 2006).
their effect on stars, however it is nowadays consid- Hereweusespiralarmsthatalthoughsteady,arevery
ered to have a key role on large-scale galactic dynam- different in nature to the typically and widely employed
ics (Sellwood 2013), for a review, (Roskar et al. 2012; in literature. The gravitationalpotential due to the spi-
Minchev et al. 2012; L´epine et al. 2011; Quillen et al. ral pattern is not a simple perturbation but is rather
2011; Antoja et al. 2009). One plausible observational based on a mass density distribution. With this model
example are the stellar features seen in the velocity of spiral arms our studies on the vertical heating of the
space, known as “moving groups” (Proctor 1869; Eggen disk contrast in general with the density wave approach
1959, 1977, 1990, 1996a,b; Wilson et al. 1923; Roman (except of course,for really small spiral arm masses and
1949; Soderblom & Mayor 1993; Majewski 1994; 1996 pitch angles).
); these structures might become the first clear, undi- For the orbital study we employed then three-
rect though, evidence of the effect of the spiral arms dimensional galactic potentials to model normal spiral
(Chereul et al. 1998, 1999; Dehnen 1998; Famaey et al. galaxies (Sa, Sb and Sc). The motion equations are
2005, 2008; Antoja et al. 2012; Pomp´eia et al. 2011). Of solved in the non-inertial reference system of the spiral
′ ′ ′
course,the spiral arms are not the only nor the prefered armsandinCartesiancoordinates(x,y ,z ). The orbits
mechanismto explainmoving groups,the galactic bar is are integrated for 5 Gyr with a Bulirsh-Stoer algorithm
other possibility. (Press et al. 1992), with a conservation of the Jacobi
In this work we focus on the very first proposal to constantapproximatelyupto10−12. Thediskheatingis
explain the vertical heating (Wielen 1977), disregarded computedthroughthemeasureofthevelocitydispersion
at some point in the history because of their negligi- at different times.
ble effect on the Milky Way disk: the dynamical heat-
2.1. Models for Normal Spiral Galaxies
ing by effect of the spiral arms. We attempt to isolate
and quantify the contribution of the spiral arms to the The models include an axisymmetric component
disk heating of galaxies. We performed numerical sim- (bulge, massive halo and disk), as the background po-
ulations of test particles in a three-dimensional galactic tential, formed by a Miyamoto & Nagai (1975) disk
potentialthatmodelsspiralarms(Pichardo et al.2003), and bulge, and a massive halo (Allen & Santilla´n 1991).
adjusted to simulate spiral galaxies, from early to late The parameters used to model normal spiral galaxies
types (P´erez-Villegaset al. 2012, 2013). We produced a (Sa, Sb and Sc) are presented in Table 1 (compiled by
set-up with relaxed initial conditions for a stellar disk. P´erez-Villegaset al. (2013)).
Finallywecalculatedtheeffectontheverticalheatingof Superposed to the axisymmetric components, for the
the stellar disk produced by the nonaxisymmetric large spiral arms potential, we employed a bisymmetric self-
scale structures. We have included a preliminary study gravitatingthree-dimensional potential, based on a den-
onaself-gravitatingpotential,knownasPERLASmodel sity distribution, called PERLASmodel (Pichardo et al.
(Pichardo et al. 2003, 2004) for the Milky Way Galaxy, 2003). This potential consists of individual inhomo-
that includes the spiral arms and the galactic bar. geneous oblate spheroids superposed along a logarith-
This paper is organized as follows. The galactic mod- micspirallocus(Roberts, Huntley & van Albada 1979).
els, initial conditions and methodology are described in Each spheroid has a similar mass distribution, i.e., sur-
Section 2. The role of each one of the parameters of faces of equal density are concentric spheroids of con-
the model is studied with detail in Section 3, where we stant semiaxis ratio. The model considers a linear fall
present calculations of dispersion velocity, the velocity in density within each spheroid. The minor and major
ellipsoid, time evolution for spiral galaxies from early to semiaxesofeachoblatespheroidare0.5and1.0 kpc, re-
late types, andanapplicationto the Milky Way Galaxy. spectively (this givesa width of the spiralarmsof 2 kpc
Finally, in Section 4, we present our conclusions. andheightof0.5 kpc fromthe diskplane)andthe sepa-
ration among the spheroid centers along the spiral locus
2. METHODOLOGYANDNUMERICALIMPLEMENTATION is 0.5 kpc. The superposition of the spheroids begins
The effect of the spiral arms over the stellar disk and ends, in the ILR and CR, respectively. The density
has been studied profusely in either N-body simula- fallsexponentiallyalongthe spiralarm,wherethe radial
tions(Sellwood & Binney2002;Roskar et al.2012,2013; scale length of the galactic disk is used (depending on
Kawata et al. 2014) or with spiral patterns treated as morphologicaltype, see Table 1).
perturbationstotheaxisymmetricbackgroundandmod- The mass assigned to build the spiral pattern is sub-
elling it as a density wave (De Simone et al. 2004; tracted from the disk mass to keep the given model in-
Minchev & Quillen 2006; Faure et al. 2014). variable in mass. PERLAS is a more realistic potential
In N-body simulations, although self-consistent, it since it is based on a density distribution and considers
is not plausible to adjust a given specific galaxy, or the force exerted by the whole spiral structure, obtain-
Contribution of Spiral Arms to the Thick Disk 3
Table 1
ParametersoftheGalacticModels
Parameter Value Reference
Sa Sb Sc
Axisymmetric Components
MB /MD 0.9 0.4 0.2 1,2
MD /MH 1 0.07 0.09 0.1 2,3
Rot. Velocity(kms−1) 320 250 170 4
M(VDD/(V1R01o1t)M2R⊙h) 10..2581 10..2615 00..5710 3
MB (1011M⊙) 1.16 0.44 0.10 MB/MD based
MH (1011M⊙) 16.4 12.5 4.8 MD/MH based
Diskscale-length(kpc) 7 5 3 1,3
b1 2 (kpc) 2.5 1.7 1.0
a2 2 (kpc) 7.0 5.0 5.3178
b2 2 (kpc) 1.5 1.0 0.25
a3 2 (kpc) 18.0 16.0 12.0
Spiral Arms
locus Logarithmic 5,9,10
armsnumber 2 6
pitchangle(◦) 8-40 9-45 10-60 4,7
Msp/MD 1-5% 9
scale-length(kpc) 7 5 3 diskbased
Ωsp (kms−1kpc−1) -30 -25 -20 5,8
ILRposition(kpc) 3.0 2.29 2.03
CRposition(kpc) 10.6 11.14 8.63
innerlimit(kpc) 3.0 2.29 2.03 ∼ILRpositionbased
outerlimit(kpc) 10.6 11.14 8.63 ∼CRpositionbased
References. — (1) Weinzirl et al. 2009; (2) Block et al. 2002;
(3) Pizagno et al. 2005; (4) Brosche 1971; Ma et al. 2000; Sofue &
Rubin2001; (5)Grosbøl &Patsis1998; (6)Drimmeletal. 2000; Grosbøl
et al. 2002; Elmegreen & Elmegreen 2014; (7) Kennicutt1981; (8) Patsis
et al. 1991; Grosbøl &Dottori2009; Egusaetal. 2009; Fathietal. 2009;
Gerhard2011; (9)Pichardoetal. 2003; (10)Seigar&James1998; Seigar
et al. 2006;
1 Up to 100 kpchalo radius.
2 b1, a2, b2, and a3 are scale lengths.
ing a more detailed shape for the gravitational poten-
tial,unlikeatwo-dimensionallocalarmsuchasthetight
winding approximation (TWA) represented for a simple
b2M a R2+(a +3 z2+b2)(a + z2+b2)2
cosine function. ρ = 2 d 2 2 2 2 2 ,
Spiralarmsnatureisa matterofdiscussionnowadays, MN 4π (R2+(a2+ pz2+b22)2)5/2(zp2+b22)3/2
particularly their long-lasting or transient nature. We p (1)
have performed experiments with constant, transient, whereM isthe massofthe galaxydisk, a andb are
d 2 2
gradual and sudden presence of the spiral arms. Al- the radial and vertical scale-length, respectively. This
though the growth rate is an unknown parameter in three parameters span a range of values in our simula-
galaxies, we have considered different cases to test. On tions in order to capture different galaxy morphologies
one hand we produce a set of experiments where the to- and kind of spiral arms.
tal mass of the spiral arms is introduced at once (t = 0 TodistributetheparticlesaccordingtotheMiyamoto-
Gyr). The second set of experiments, inserts the spiral Nagai density law we solved equation (1) with a root
arms mass linearly in a timelapse of 1 Gyr. And a third finder method. This is done by expressing the den-
set of experiments for which the spiral arms are simu- sity ρ(R,z) in terms of the ratios ρ(R,0)/ρ(0,0) and
lated as transient, they vanish and grow with a given ρ(R,z)/ρ(R,0),whichprovidesusanequationforRand
periodicity. zintermsofthedensity. Thevalueoftheseratiosranges
from0to1,thereforewecanexploreallthe possibleval-
ues of the density with a random function and solve it
for R and z.
Toassignvelocitiestotheparticleswefollowthestrat-
2.2. Initial Conditions and Equilibrium of a Stellar Disk
egy proposed by Hernquist (1993) where velocities are
The initial conditions set-up follows the Miyamoto- distributed by an approximation using moments of the
Nagai density profile we are imposing. This to avoid collisionless Boltzmann equation plus the epicycle ap-
transient effects induced by differences between the ini- proach.
tial particle distributionand the imposed disk potential. Thus we proceeded as follow: once the density pro-
In this manner, the initial condition for the stellar disk file has been established, it is necessary to obtain the
is given by rotation velocity. This can be derived from Φ, the grav-
4 Martinez-Medina et al. 2015
itational potential of the model 2.3. Dispersion analysis
Thediskheatingisoftenreferredastheincreaseinthe
1 ∂Φ 1/2 velocity dispersion over the lifetime of a star. Any disk
Ωc(R)= (2) thickening is then related to an increase in the vertical
(cid:18)R∂R(cid:19)
velocity dispersion of the disk stars. In this paper we
analyze the spiral arms effects on the stellar disk, based
and v =RΩ (R), so the circular velocity is given by
c c
on the study of the vertical velocity dispersion and its
dependence with the parameters that characterize the
1/2
∂Φ spiral pattern.
v (R)= R . (3)
c (cid:18) ∂R(cid:19) The vertical velocity dispersion σz, is then calculated
in the simulations by dividing the plane z =0 into 1kpc
Once known Ω at any radii we obtain binsandcomputing,asusual,thesquaredrootoftheav-
c
eragedsquaredvertical velocity for all the particles that
dΩ2 1/2 fall into a given bin. This provide us the vertical veloc-
κ= 4Ω2+R c , (4) ity dispersion as a function of R. In order to establish
(cid:18) c dR (cid:19) the contribution of the spiral arms to the disk thickness
we also compute σ as a function of time by measuring
z
known as the epicyclic frequency, necessary to calcu-
the velocitydispersionata fixedradiusR for everytime
late the velocity dispersion at R and to correct for the
codeunit(inthiscase,every100Myracross5Gyrinthe
asymmetric drift.
simulation).
To achieve the requirement for the stellar disk to be
in equilibrium, it is necessary to introduce a given dis- 2.4. Control Simulations: Testing the Initial Conditions
persion in the velocity as a function of R. The velocity Equilibrium
dispersions in the three polar coordinates are
As described in section 2.2, the first goal is to build
an initial stellar disk in equilibrium to be sure that any
Σ(R)Q
σ =3.358 (5) change seen in radial and vertical velocity dispersion is
R
κ strictlyduetotheinteractionofthespiralarmswiththe
stellardisk andnotoriginatedby aspuriousnon-relaxed
1σ κ initial condition set-up.
R
σφ = (6) For this test a control simulation was produced with
2 Ω
c only the axisymmetric components for the potential
model. Stars are run by 5Gyr, for all galaxytypes. Fig-
σz = πGΣb2, (7) ure 1 shows σz as a function of R at different stages in
p the temporal evolution for an Sa, Sb, and Sc galaxy.
whereκisthe epicyclicfrequency,Σ(R)the surfaceden- From the figure it is clear that the vertical velocity
sity, b is the vertical scale-lenght of the disk, and Q dispersion do not evolve or deviate from the initial dis-
2
is the known Toomre parameter. According to Toomre persion, as expected for a disk in equilibrium with the
(1964), localstability requiresQ>1, we choose Q=1.1 axisymmetric potential.
and found this value to be sufficient for the three galaxy
types. Inthiswaythevelocitydispersiondependsonthe 3. RESULTS
mass of the components that form each galaxy. We present in this section a set of controlled experi-
The asymmetric drift correction is defined as ments to study the dynamical heating of disks on spiral
galaxies considering the spiral arms as the driver. The
v 2 =v2 σ2 σ2 1 2R∂Σ , (8) generalpurpose is to shed some light on the relative im-
h φi c − φ− R(cid:18)− − Σ∂R(cid:19) portance of these large-scale structures to other sources
of dynamical heating in different morphological types.
isacorrectionthathastobeimplementedinthesetup The experiments include studies of different structural
forthe initialconditionsgiventhefactthatstellarorbits and dynamical parameters of the spiral arms such as,
arenotingeneralincircularorbits,instead,orbitsfollow different pitch angles, total spiral arms masses, angular
epicycles around a guiding point at the position of the speeds, transient, and one final case modeling the Milky
circularorbitandtheseepicyclesarecharacterizedbythe WayGalaxy(withpreliminaryresultsthatwillbebetter
epicyclic frequency κ. developed in a future work).
Finally the particles are distributed in the velocity
space as 3.1. Dependence of the disk heating induced by spiral
arms with galaxy morphology
Spiralgalaxiespresentawidevarietyonmorphological
v = v xσ
φ h φi± φ types, from massive bulge-dominated galaxies to practi-
vR = vR xσR (9) cally bulgeless disks,spanning a wide range ofvalues for
h i±
v = v xσ the parameters that characterize different galaxy types.
z z z
h i± Figure 2 shows the velocity dispersion, σ , as a func-
z
where x is a random number between 0 and 1, v is tion of the galactocentric radius, R, at different times in
φ
h i
given by Eq. (8) and the average radial and vertical the simulation for our galactic models: Sa, Sb and Sc
velocities are taken as v = v =0. (introduced in Section 2). For these three simulations
R z
h i h i
Contribution of Spiral Arms to the Thick Disk 5
tttiiimmmeee(((GGGyyyrrr))) tttiiimmmeee(((GGGyyyrrr)))
000 111 222 333 444 555 000 111 222 333 444 555
140 140
120 120
Sa Sa
100 100
80 80
60 60
40 40
20 20
110
100 Sb 100 Sb
80 80
) )
1 1
-s -s
m m
k 60 k 60
( (
z z
σ σ
40 40
20 20
50
Sc
40 Sc
40
30
30
20
20
10
10
0 0
0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20
R(kpc) R(kpc)
Figure 1. Test for the initial conditions equilibrium: velocity Figure 2. As in Figure 1, but including now the spiral arms
dispersionofthestellardiskintheaxisymmetricpotentialonly,as contribution to the potential. The vertical velocity dispersion is
afunctionofR,along5Gyrtimeevolutionforeachgalaxytype. plottedasafunctionofRfor5Gyrtimeevolution.
the mass of the spiral arms, M , is 5% of the total
arms
diskmasswithaPitchangleof40◦ fortheSagalaxy,45◦ The dependence of the effect of the spiral arms with
for the Sb galaxy, and 40◦ for the Sc galaxy. For these the morphology is such that for an Sc galaxy the effect
experiments we have employed the largest spiral arms is evident in the spatial distribution of the stellar disk
massesandpitchangles,forplausible(non-fullychaotic) particles. Figure 3 shows the x z projection of the
−
galactic models, to identify clearly spiral arms effects, if stellar distribution plotted at t = 0, t = 2.5Gyr, and
any. The three plots show a distinct increase in the ver- t=5Gyr.
tical velocity dispersion caused by the spiral arms. Also Also, a thickening of the disk is discernible during the
fromfigure2itisclearthatthechangeinσ withrespect orbital evolution when comparing with the initial distri-
z
to the initial dispersion is smaller for the Sa galaxy and bution.
grows with the morphological type, being much larger Thethicknesscanbequantifiedbycomputingtheroot
for the Sc galaxy model. mean square of the coordinate z, i.e. z = √<z2 >.
rms
6 Martinez-Medina et al. 2015
3
2
t = 0
1
0
-1
-2
2
t = 2.5 Gyr
) 1
c
p 0
k
z(-1
-2
2
t = 5 Gyr
1
0
-1
-2
-3
-20 -15 -10 -5 0 5 10 15 20
x(kpc)
Figure 3. Edgeondisksshowtheevolutionofstellarorbitsatthreedifferenttimes(0,2.5and5Gyr). ThemodelcorrespondstoanSc
galaxy. Thethickeningofthediskduetothespiralarmspresenceisclear.
0.7 the intermediate galactic type (Sb), the isolated effect
t=0
of spiral arms corresponds to a maximum of increment
t=2.5Gyr
0.6 t=5Gyr 20%percent of the initial velocity dispersion, againcon-
sidering the most massive and the largest pitch angles
possible to produce plausible galactic models. Finally,
0.5
for the latest type (Sc), the isolatedeffect of spiral arms
c) 0.4 corresponds to a maximum of 62% percent of the ini-
p tial velocity dispersion for this example, where we have
k
z(rms 0.3 nraoltaursmeds.thWehmeanxtimheummapxliamusuimbleppitacrhamanegtelersafnodr tmhaessspiis-
employed, the percentage goes up to almost 90% of the
0.2
initial velocity dispersion.
Since a visible effect in the vertical dispersion when
0.1 spiral arms are included is considerably larger for the
latest galactic types, compared with early and interme-
0 diatetypes,inthenextsectionsweconcentrateonamore
0 2 4 6 8 10
R(kpc) detailedstudyofthe diskthickeningfocusingonlyinthe
late type galaxies.
Figure 4. DiskthicknesszrmsasafunctionofRfort=0,2.5Gyr
and5Gyr. 3.1.1. The pitch angle effect
Thepitchangleisoneofthemostinfluentialstructural
Figure 4 shows the thickness as a function of R for the parametersthatcharacterizespiralpatterns. Inthis sec-
stellar disks in Figure 3. tion a range of values is explored in order to quantify
From these first set of experiments, we conclude that the dependence of the increment in the vertical velocity
the sharpest effect on the velocity dispersion, is present dispersion with pitch angle in the most affected galactic
on the latest morphological types. It is worth mention- models, that are the latest types.
ing here that, although we are separating the models in Figure 5 shows three plots of σ vs R, where the mass
z
Sa, Sb and Sc galaxies, with strong gaps in between the ofthespiralarmsissettoaconstantforeachplotandthe
differentmodels in the initial scale height, the resultson pitchangle varyaccordingto Table 1. Eachplot has the
thediskheatingdrivenbyspiralarmspresentedhere,are initial dispersion curve σ (R,t = 0) and the dispersion
z
moregeneral,i.e. theobservedheatingresultssignificant after a 5Gyr evolution σ (R,t = 5Gyr) for each pitch
z
in thinner disks, which, as a consequence, has implica- angle value.
tions on galaxy types, in this case, particularly on later From Figure 5, it is clear that regardless the mass of
types. the spiralarms,the verticalvelocity dispersionincreases
For the earliest type (Sa), the isolated effect of spi- notablywiththe pitchangle. Thelessmassivethe spiral
ral arms corresponds to a maximum increment of 7% arms the smaller is their effect in general, as expected.
percent of the initial velocity dispersion, this consider- Indeed, for spiral arms masses smaller that 1% of the
∼
ing the most massive and the largest pitch angles pos- disk, the contribution of spiral arms to the dynamical
sible to produce plausible galactic models. Likewise, for heating becomes negligible.
Contribution of Spiral Arms to the Thick Disk 7
PPPiiitttccchhh AAAnnngggllleee velocity dispersion. We find that this occurs at approxi-
000ooo 111000ooo 222000ooo 333000ooo 444000ooo 555000ooo 666000ooo
mately R = 1.5kpc for this model. We then keep R
max
constant at that value and plot ∆σ there as a function
50 z
of the pitch angle. This will show us the tendency seen
inthe previoussection. Now repeating this for the three
40 Marms = 1% (Mdisk) spiral arms masses employed will show how ∆σz scales
with this second parameter.
Figure6shows∆σ asafunctionofthepitchanglefor
z
30 three different masses of the spiralarms. Here it is clear
that both parameters, the pitch angle and the mass of
the spiral arms, affect considerably the disk thickening
20 effect driven by the spiral structure.
35
10 M = 1% (M )
arms disk
M = 3% (M )
arms disk
30 M = 5% (M )
arms disk
25
50
)
1
Marms = 3% (Mdisk) -m s 20
40 k
(z 15
σ
) ∆
1
-m s 30 10
k
(
σz 5
20
0
10 20 30 40 50 60
10 Pitch Angle(o)
Figure 6. Difference in the vertical velocity dispersion ∆σz (at
t=0and t=5Gyr) vsthe pitch angle, plotted forthe three masses
60
ofthespiralarms,asascalingfactor.
50 Marms = 5% (Mdisk) We produce with these results an empirical functional
relation between ∆σ and the Pitch Angle. The fit of
z
40 data plotted in figure 6 is made by noting that ∆σ ,
z
increases slowly at small angles, then the slope of the
30 curve grows with the angle and flattens for the largest
values of the pitch angle. This behavior could be inter-
20 preted as a saturationeffect of ∆σz after a certaintime,
this time being shorter for higher angles and masses.
The saturation effect is also seen in a σ - time re-
10
lation (Seabroke & Gilmore 2007; Soubiran et al. 2008;
Calberg et al. 1985), where the dispersion remains con-
0 0 2 4 6 8 10 12 14 16 18 20 stant after 5Gyr. Based on this observation, a nice
R(kpc) ∼
fit to the results would be a Boltzmann sigmoidal func-
tion which is characterized by displaying a progression
Figure 5. Final velocity dispersion after a 5Gyr evolution as a from small beginnings that accelerates and approaches
functionofradiusforspiralarmmassesof1%,3%,and5%ofthe a climax over the independent variable. The Boltzmann
totaldiskmass.
sigmoidalfunction, for this particular case, is defined by
3.1.2. The spiral arm mass effect (A A )
1 2
∆σ =A + − (10)
As it is showninthe previous section,the effect of the z 2 1+exp((x x )/d)
0
pitch angle can be significant to the disk thickening. It −
alsoresultsintuitivelyclearthatthisalsoscaleswiththe where:
mass of the spiral arms. To address this point, we pro- x = Pitch Angle (o)
duced several experiments condensed in Figure 6, that x0 = center (o)
show the disk thickening dependence on the spiral arms d = width (o)
mass and the pitch angle together. For this purpose, A1 = initial ∆σz value (kms−1)
first we identified the radius R at which occurs the A = final ∆σ value (kms−1)
max 2 z
maximumdifferencebetweenthefinalandinitialvertical
8 Martinez-Medina et al. 2015
The center x is the pitch angle at which ∆σ is
0 z
halfwaybetweenA andA . Thewidthdisrelatedwith
the steepness of th1e curve,2with a larger value denoting Pitch Angle = 40o MMarms ifnixcereda s i n g
arms
a sFhiaglulorew7cusrhvoew.s the fit of the sigmoidal function (Eq. 50 Marms/Mdisk = 0.03 MarmsM inacrmresa fsiixnegd
10) to the data. We see that this function reproduces
well the behavior of ∆σ with the pitch angle for the
three masses of spiral armzs used in our simulations. -1m s) 40 R = 1.5kpc
k
(
35 M = 1% (M ) σz 30
arms disk
M = 3% (M )
30 Marms = 5% (Mdisk) R = 3.5kpc
arms disk
20
25
)
1
-s 20
m Marms increasing
k Marms fixed
∆σ(z 15 60 PMitacrmh sA/Mngdilsek == 400.0o5 MarmsM inacrmresa fsiixnegd
10
50
)
5 -1m s 40 R = 1.5kpc
k
0 (
10 20 30 4o0 50 60 σz
Pitch Angle( )
30
R = 3.5kpc
Figure 7. Fit of the ∆σz - pitch angle relation with the Boltz- 20
mannsigmoidalfunctionforthethreemassesofthespiralarms.
Taking Eq. (10) as the functional form for the ∆σz - Marms increasing
pitch angle relation, in Table 2, we summarize the pa- Marms fixed
rameters that describe the fits plotted in figure 7 for the 60 Pitch Angle = 50o MarmMs incr efaixsiendg
three spiral arms masses. Marms/Mdisk = 0.03 arms
50
)
1
Gradually Increasing the Mass of the Spiral Arms -s
As explained before, the spiralstructure in our galaxy km 40 R = 1.5kpc
(
potential models is imposed and fully introduced since z
σ
thestartingpointofthesimulation. However,onemight 30
wonder if spurious effects (such as a drastic dispersion
increase in the early stages of the simulation) are intro- R = 3.5kpc
20
duced with this method; also, in real galaxies the birth
and death of spiral arms is most probable not a sudden
process.
M increasing
In order to explore these two scenarios, we prepared arms
M fixed
arms
a set of representative simulations modifying the model 70 Marms increasing
in such a way that the arms are allowedto have a grow- Pitch Angle = 50o Marms fixed
ing period until the total assigned mass is reached. For 60 Marms/Mdisk = 0.05
this set of simulations we started with an Sc galaxy
wmiatshsdainffdertehnetpciotmchbiannagtlieo.nsWoefcvhaoluseestwfoorrtahdeiaspliproasliatiromnss -1m s) 50 R = 1.5kpc
and measured in there the evolution of σ with time as k
showedinFigure8. Inthisfigurethetempzoralevolution σ(z 40
of σ for the sudden full mass arms is compared with a
z
30
modelwithlinear(1Gyrperiod)growingmass. Itisclear R = 3.5kpc
thatthereisashiftinthe dispersionachievedatthe end
of the simulation, but most important, for both kinds of 20
imposed spiral arms the velocity dispersion increases at
roughly the same rate. 0 10 20 8 30 40 50
With this simple experiment we are not pretending to time(10 yr)
capture the great variety of processes that lead to the
formation of a real spiral pattern, nor their timelapse of Figure 8. σ-trelationthatshows,regardlesstheradialposition,
a continuous heating for both kind of imposed spiral arms: those
thatgrowwithtime(bluelines)andthosetotallyformedsincethe
beginning(blacklines).
Contribution of Spiral Arms to the Thick Disk 9
Table 2
∆σz -PitchAngleFitDetails
Marms (%Mdisk) x0 (o) d(o) A1 (kms−1) A2 (kms−1)
1% 64.73 24.05 -0.33 5.32
3% 54.73 14.95 -0.057 24.84
5% 35.68 6.82 1.69 28.44
Note. — Parameters that combined with equation 10, describe
the∆σz -pitchanglerelation. Centerx0,widthd,initial∆σz value
A1, and final ∆σz value A2. For the three spiral arms masses (in
percentageofthediskmassMdisk).
full action, but this is still likely a better approximation Ω(km s-1 kpc-1)
15 20 25 30 35
than a sudden action of spiral arms. With this excer-
cise we notice however that, for any radial position, the 120
verticaldispersionis onlyshifted whenusingspiralarms
that have a growingperiod. This do not suppose impor-
tant differences with our previous simulations or results 100
derived from them.
Sb
80
3.1.3. Varying the Angular Velocity of the Spiral Pattern 1)
-s
Theverticalheatingofthestellardiskproducedbythe m 60
k
spiralarmsdependsontheparametersofthenonaxisym- σ(z
metric structure, particularly for the late galaxy types.
40
By varying the pitch angle and the mass of the arms we
have found a correlation between this parameters and
the thickness of the stellar disk.
20
In this section, we study now the pattern speed effect
on the disk vertical heating. For this purpose, we ran
a set of simulations with different values of the spiral 0
0 2 4 6 8 10 12 14 16 18 20
pattern for the latest types of galaxies: Sb and Sc, that R(kpc)
are more clearly affected for the structural parameters
of the arms. In Figure 9 we show first the evolution of Figure 10. Final velocity dispersion after a 5Gyr evolution for
the vertical velocity dispersion σz over time up to 5Gyr differentspiralpatternangularspeeds inanSbgalaxy.
for different values of the pattern speed, Ω, for an Sc
galaxy. There seems to be a clear relation between the
angularvelocityandthedynamicalheating. Thevelocity
Ω(km s-1 kpc-1)
dispersion increases for slower rotating patterns. 10 15 20 25 30
Figures 10 and 11 summarize the results of the set of
simulationswhenvaryingthepatternspeedforSbandSc 60
galaxies. The final stage of σ is shown acrossthe entire
z
diskforeachoneofthepatternspeedsused. Asexpected,
50
the final vertical velocity dispersion and disk thickness
are largerfor slower rotating arms, independently of the Sc
galaxy type. Smaller values of Ω allow the spiral arms
40
to heat more efficiently the stellar disk, likely due to the
1)
minorrelativeangularvelocitybetweenthearmsandthe -s
stars, this allows the stars to interact with the potential km 30
of the arms for longer periods of time. σ(z
Other studies link the heating of the stellar disk
20
and the role of the pattern speed only to the reso-
nantregionsofthespirals(Lynden-Bell & Kalnajs1972;
De Simone et al.2004),incontrast,inthisstudywefind 10
that the heating, radial and vertical, occurs along the
entire length of the spiral arms.
On the other hand N-body simulations had shown to 0 0 2 4 6 8 10 12 14 16 18 20
R(kpc)
develop transient spiral structure that spans a range of
pattern speeds (Sellwood & Binney 2002; Roskar et al.
2012),butasthearmsaretransientandisnotpossibleto Figure 11. Final velocity dispersion after a 5Gyr evolution for
isolateitseffectinN-bodysimulations,itresultsdifficult differentspiralpatternangularspeeds inanScgalaxy.
to establish a dependence of the heating on the pattern
speed. The experiments and the model shown here allows us
10 Martinez-Medina et al. 2015
60 555
50 Ω = 10 km s-1 kpc-1 Ω = 20km s-1 kpc-1 Ω = 30 km s-1 kpc-1
444
40
-1σ(km s)z 30 232323time(Gyr)time(Gyr)time(Gyr)
20
111
10
0 2 4ILR 6 8 10 12 14 16CR 18 2ILR 4 6 8 CR 10 O12LR 14 16 18 2 4 C6R 8 OLR 10 12 14 16 18 000
R(kpc)
Figure 9. Time evolution of the vertical velocity dispersion for three different spiral pattern angular speeds in an Sc galaxy. This
comparisonshowsthatslowerrotatingspiralarmsheatmoreefficientlythestellardisk.
toestablisharelationbetweentheverticalheatingofthe log(σ ) dependency. The most of the points fall on a
z
stellar disk and the pattern speed. straightline,this revealsabehaviorofthe formσ tα.
z
∝
Figure 13 shows the evolution of σ with time at
z
3.1.4. σz - Time Relation R = Rmax, Marms/Mdisk = 0.05 and at different pitch
It is already known that the age and velocity dis- angles: 20o,30o,40o and50o. Theblacklineineachplot
persion of stars are correlated. This has been estab- is the best fit of the data with a power law of the form
lished from observations in the solar neighbourhood σz tα. We made the same analysis for a spiral arm
∝
as well as from numerical simulations (Holmberg et al. mass of Marms/Mdisk = 0.03, an interesting outcome is
2009; Roskar et al. 2013; 2014 ). The σ - t rela- thatthevalueofαisindependentofthespiralarmmass.
tion shows a smooth, general increase of the veloc- Different masses for the spiral arms will just change the
ity dispersion with time and is best parametrized by proportionality constant in the relation σz tα. Conse-
∝
a power law with exponents ranging between 0.2 - 0.5 quently, α depends only on the pitch angle, and for the
(Gerssen & Shapiro Griffin 2012). angles used in our simulation α varies within the range
We explorethe σ - t relationinoursimulationsto find 0.27 0.56 for Sc galaxies.
−
outifthevelocitydispersioninthestellardiskduetothe Although we have presented the time evolution of σz
spiralarmsfitswithapowerlawtα andmoreimportant, at R = Rmax, it is possible to made the measure at
to establish a range of values for α. any value of R. Figure 14 shows the same analysis at
Tomeasurethetimeevolutionofσ inoursimulations a different radial position, and this time corresponds to
z
firstwelocatetheradiusR atwhichoccursthemax- the half of the arm length, R = 4.2kpc. We notice that
max
imum increase in the vertical velocity dispersion. This measuring the time evolution of σz at different radius
radius is R =1.5kpc andis the same for allthe sim- give us the same power law behavior.
max
ulations with the Sc galaxy, independently of the pitch With a similar analysisfor Sbgalaxies,whereRmax =
angle or the mass of the spiral arms. 2.5kpc, we find again that a power law provides a nice
fit to the data. Figure 15 shows the σ temporal evolu-
z
tionfor pitchanglesof 36o and45o, andthe bestfits are
4.1 reached with values for α of 0.32 and 0.37, respectively.
Smaller angles than those give us plots with more scat-
teredpoints wherea powerlaw fit is notstraightforward
4 Sc
Pitch angle = 50o toobtain,i.e.,wearenotableto establishthe valueofα
for angles smaller than 36o. For Sb galaxies we can only
3.9 Marms/MDisk = 0.05
provide an upper bound for α of 0.37.
∼
) 3.8
z
σ
(
g
o 3.7
l
3.6
3.5
3.4
0 0.5 1 1.5 2 2.5 3 3.5 4
log(Time)
Figure 12. log-logplotofthetimeevolutionofσz.
Since a log-log plot is useful to recognize a possible
power law relationship, Figure 12 shows the log(t) -