Download - Polygonal structure of spiral galaxies

ISSN 1063-7729, Astronomy Reports, 2012, Vol. 56, No. 1, pp. 9–15. c© Pleiades Publishing, Ltd., 2012.Original Russian Text c© E.A. Filistov, 2012, published in Astronomicheskii Zhurnal, 2012, Vol. 89, No. 1, pp. 12–18.

Polygonal Structure of Spiral Galaxies

E. A. Filistov*

Moscow, RussiaReceived July 29, 2011; in final form, August 17, 2011

Abstract—We have carried out numerical simulations of hydrodynamical processes occurring in the disksof spiral galaxies. The initial state of the disk is an equilibrium stellar–gaseous configuration. The sphericalcomponent is described by a standard analytical model for the gravitational potential. The behavior of themodeled disk in the presence of an external perturbation is analyzed. The results of numerical simulationsof stellar–gaseous galactic disks aimed at studying the formation of polygonal structures in spiral galaxiesare presented. The possible influence of spur-like formations on the appearance of polygonal structure isstudied.

DOI: 10.1134/S1063772912010027

1. INTRODUCTION

The presence of linear segments in the arms ofspiral galaxies is not rare (see, for example, [1, 2]),at least for well-ordered late-type spirals (Grand De-sign Sbc–Sc galaxies). The main geometrical andphysical characteristics of multi-angle structures wasstudied in some detail in [3–6], based on a num-ber of well observed galaxies. The development ofhydrodynamical instability in the spiral density wavein the stellar disk, leading to a restructuring of thespiral shock front to a new transient configurationconsisting of linear segments, has been proposed as apossible physical mechanism to explain the formationof linear segments in galactic spiral arms [5]. Anothercharacteristic manifestation of this type of hydrody-namical instability is the development of small-scalespurs, which are clearly visible in spiral structuresin galaxies of various morphological types (see, forexample, [7–9]).

Observations of disk galaxies show that spurs,which are among the most prominent secondarylong-lived features, emerge from a primary spiral armand then, as a rule, are drawn out in the directionopposite to the rotation of the spirals. Spur-likeformations are not unusual features in galaxies witheven spiral patterns [10–12]; their presence is animportant aspect of the spiral structure, which mustbe taken into account in theoretical modeling of diskgalaxies.

Here, we use numerical simulations to demon-strate the formation of polygonal structures of spiralgalaxies, and discuss a gas-dynamical approach tostudying the physical nature of this phenomenon: the

*E-mail: [email protected]

appearance of spur-like formations and their influenceon the development of polygonal structure in galaxies.

2. DYNAMICS OF A GASEOUS GALACTICDISK

In the computations described below, we usedthe radius and mass of the galaxy RG and MG andthe gravitational constant G to obtain dimensionlessvariables for the problem. The corresponding char-acteristic particle velocity V0, density ρ0, pressure P0,and potential Φ0 can be written

V0 =

(GMG

RG

)1/2

, ρ0 =GMG

R3G

,

P0 = V 20 ρ0, Φ0 = V 2

0 .

Further, all equations and quantities are given indimensionless variables.

In order to exclude the influence of gas-dynamicalprocesses associated with non-equilibrium aspects ofthe initial state of the disk, we adopted initial condi-tions corresponding to a stationary rotating gaseousconfiguration using the method proposed in [13]. Inparticular, the class of equilibrium gaseous configu-rations with boundaries of the form

Z(r) = α

√1π

exp(−βr2), α, β > 0 (1)

contains distributions of the density and the velocityof the isothermal gas-dynamical flow

ρ(r, z) = ρ0gas exp

(1r

Φ(r) − Ψ(r)c2T

), (2)

9

10 FILISTOV

v2ϕ(r, z) =

1r

(1 + rΨ′(r) − Ψ(r)

), vr = 0, (3)

Φ(r) = 1 − r√r2 + Z(r)2

,

Ψ(r) = 1 − r√r2 + z2

, |z| < Z(r),

where r and z are the coordinates of the particles inthe cylindrically symmetrical gas configuration, cT isthe isothermal sound speed, and ρ0gas is the constantof integration. The profile of the absolute temperatureT along the disk radius rd is described by the law [14]

T = T0

(1 +

r2

r2d

)−q/2

, (4)

where q = 1/2 and T0 are constants satisfying thecharacteristic parameters for stability of the stellar–gaseous disk [15].

When studying internal motions in the gaseousgalactic disk, we neglected light pressure comparedwith the gas pressure. Therefore, the Clapeyronequation has the form of an isothermal equation ofstate:

P = c2T ρ. (5)

The distributions of the density (2), velocity (3),temperature (4), and pressure (5) were adopted as theinitial data inside the subregion r ∈ [0, rd].

Further, for a some of gaseous disks modelled inthis work, we used the model density equation in orderto exclude strong density gradients in the galacticplane and at the galactic center:

ρ(r, z) =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

0.52ρ0gassech( r

a

)sech

( |z|b

),

if r < a, |z| < b,

ρ0gasexp(−r

a

)exp

(−|z|

b

),

if r ≥ a, |z| ≥ b,

(6)

where a and b are parameters of the model. In thiscase, we modify the function (1) to have the form

Z1(r) = α1

√14π

exp(−2πr2). (7)

The computational results presented here neglectthe effects of viscosity, the diffusion transport of mat-ter, radiation, and evolution of the electromagneticfield. The gaseous component is fully immersedin the stellar component of the disk, and its self-gravitation is included in the gas-dynamical equationas a macroscopic external force. The gaseous disk ismodeled in an inertial coordinate system. Thus, thereare several factors acting on the gaseous medium,

namely the pressure and centrifugal force tending tomove the gas toward the periphery, and the gravita-tion of the stellar disk (as well as the halo and bulge),which balance this motion.

We solved the gas-dynamical equations using themonotonic, first-order TVD scheme of Roe [16] withlimits on the anti-diffusion flows in the form given byOsher [17], which enhances the approximation order(to third order in the spatial coordinates) with minimalnumerical dissipation and preservation of the mono-tonicity of the scheme. Moreover, this initial schemewas modified using the method of Einfeldt [18] toenhance its stability. To exclude the influence of theboundary conditons on the behavior of inner regionsof the disk, the configuration was chosen such thatthe density is appreciably lower at the boundary thannear the center. In the absence of any perturbations,the gaseous disk retains its equilibrium state over afairly extended time interval.

Perturbations of the density and velocity of theinitial equilibrium state of the gaseous disk can berepresented

ρ = ρ(1 + ε0dψ), (8)

vr = −εvdψ, (9)

where ψ = U1sinΘ, U1 = r2Z1(r)n ln Z1(r), εv ≤ ε0

(we used εv = 0.5ε0), n = const (we used n = 0.3),d ≡ ∂/∂r + ∂/r∂ϕ is the sum of the partial deriva-tives with respect to the polar coordinates, and themeaning of the remaining parameters is given below.

3. DYNAMICS OF THE STELLARGALACTIC DISK

The initial state of the stellar disk was chosensuch that the initial rotating configuration remainedin equilibrium over a time corresponding to a quarterrotation of the disk. In the numerical computations,the spherical component (halo) was described usinga standard analytical model for the potential [19]. Thedynamics of the exponential, self-gravitating stel-lar disk (with surface density σ∗ = σ0∗ exp(−r/L∗),where r is the radial coordinate in the disk, L∗ is thescale factor, and σ0∗ = const) is determined not onlyby its own self-gravitation and the gravitation of thespherical component, but also by the presence of anadditional gravitational (spiral) potential. Thus, thepotential of the stellar disk was represented as thealgebraic sum of its axially symmetric Ψ0(r, z) (thehalo, stellar disk) and non-axially symmetric Ψd(r, z)parts:

Ψ(r, ϕ, z) = Ψ0(r, z) + Ψd(r, z),

where

Ψd(r, z) = ε0εtr{Z(r) + 2πrρ∗ (10)

ASTRONOMY REPORTS Vol. 56 No. 1 2012

POLYGONAL STRUCTURE OF SPIRAL GALAXIES 11

× ln[cosh(2πz/L∗)

]}cos Θ/L2

∗;

here, the function Z(r) is determined by (1) and

Θ = N

(ϕ − ϕp(rd) + Ωpt −

π ln(r/rd)tan i

),

where N is the number of spirals, i the pitch angle ofthe spirals, Ωp is the angular velocity of the rotationof the spiral pattern, ϕp(rd) is the initial coordinate,and ρ∗ is the volume density of the stellar disk. Thecoefficients ε0 and εt can be used to regulate themagnitude of the additional spiral potential relative tothe main (external) potentials of the model.

4. PARAMETERS OF THE MODEL

The specification of the initial configuration of thegaseous–stellar disk can appreciably influence thephysical character of the results. In the results de-scribed below, we chose the values MG = 1010M�and RG = 1022 cm as scale factors; the characteristicvelocity was

V0 =

(GMG

RG

)1/2

∼=(

1014 × M11R

RG

)1/2

cm/s.

Here, M11 = MG/1011M� and R ∼= 0.8 × 1023 cm isa dimension of length composed of a combination ofuniversal physical constants [20].

The size and mass of an object in virial equi-librium is determined by its characteristic tempera-ture. Adopting the temperature in the center to be104 K, we obtain the isothermal sound speed 0.13 ×107 cm/s.

The parameters in (1) defining the initial equilib-rium configuration of the gaseous disk and the formof the additional spiral potential (10) were specified tobe α = 0.1, β = 4.0, and L∗ = 0.25. The rotationalangular velocity of the spiral pattern was Ωp ≤ 1.0;the corotation radius was located at the periphery ofthe disk. The mass of the halo was 40% of MG,and the mass of ideal gas in the equilibrium con-

figuration was Mgas = 0.1MG; ρ0gas =0.1R3

G

2πa2b(a =

7 kpc, b = 0.4 kpc). The “input” data for the model ofthe stellar disk were taken from the studies [21, 22],which develop one of the most detailed models of theGalaxy (see also [23–28]). In these studies, basedon a system of galactic constants, the parameters ofsubsystems are determined and various descriptivefunctions are computed: the distributions of the den-sity (σ0∗, ρ∗), stellar velocity dispersion, gravitationalforce, etc. Parameters of the three-dimensional struc-ture of the subsystems based on photometric studiesof a sample of spiral galaxies are presented in [29, 30].

Fig. 1. Initial stage of formation of the spiral pattern.

We carried out two versions of the computationsin an annular region in the polar coordinates Ω ={(r, ϕ), r ∈ [r1, r2], ϕ ∈ [0, 2π]}, r1 = 0.1, with r2 =1 in the first version and r2 = 2 in the second. Theradius of the disk was rd = 1 in both cases.

We specified free boundary conditions at theboundaries of the region:

∂ρ

∂n= 0,

∂u

∂n= 0,

∂v

∂n= 0, (11)

where �u = (u, v) is represented by its components inthe polar coordinate system.

The external additional spiral potential was mod-eled using the coefficient ε0 = 0.1. The coefficientεt increased from zero to unity with time, making itpossible to gradually increase the amplitude of thepotential Ψd. This excluded abrupt relaxation pro-cesses in the initial equilibrium configuration due tothe instantaneous switching on of external forces.

5. FORMATION OF THE POLYGONALPATTERN

The general structure of the solution obtainedfor the ideal-gas flow in the equatorial plane of thegaseous–stellar system considered is presented inFigs. 1–7. These depict the density distributionin a rectangular region in Cartesian coordinates,Ω = {(x, y), x ∈ [−1, 1], y ∈ [−1, 1]}. Analysis of thecomputational results enables us to identify the mainproperties of the flow, and to draw some conclusionsabout the character of the development of perturba-tions that restructure the laminar flow. Let us nowconsider the flow patterns obtained in more detail.

Figures 1–4 show the flow pattern of the gasunder the influence of the self-gravitating stellar disk


12 FILISTOV

Fig. 2. Further development in the stage of formation ofthe spiral pattern.

Fig. 3. Growing spurs.

immersed in the gravitational field of the sphericalcomponent (halo), taking into account the additionalstellar spiral potential with ε0 = 0.1. With time,two large-scale, spiral formations develop, represent-ing the elements of the global spiral pattern (GrandDesign). These spiral formations are spiral strongshocks, which exist over the entire computation timewith a fairly high degree of order. The flow in a sub-stantial region of the gaseous disk undergoes an ap-preciable restructuring, especially regions with highdensity. The part of the disk that is not subject toperturbations preserves its laminar flow.

The spiral structure actively develops in regions ofthe gaseous disk outside the central region. Due tothe drop in pressure and density toward the periphery,

Fig. 4. Polygonal geometry of the spiral pattern.

Fig. 5. Polygonal geometry of the inner region of thegaseous disk.

the shocks become more extended with time as theyapproach the boundary of the disk, essentially be-coming replaced by rarefaction waves that disappearin the peripheral regions of the disk (Fig. 1). Withapproach toward the boundary, the ends of the spiralarms thicken, the density (and pressure) in the regionjust outside the arms gradually increases, and theinhomogeneity on the polar-angle distribution grows(Fig. 2). The inter-arm region has an appreciablylower pressure, which serves as an additional reasonfor the development of hydrodynamical instability inthe spiral density wave in the stellar disk. The pres-sure gradient at the ends of the arms leads to theirsplitting into spur-like formations (Fig. 3), whichgrow in both the polar and the radial directions. These



Fig. 6. The appearance of spurs at places where segmentsmeet preceeds the formation of the polygonal structure.

formations represent a new channel for the outflowof matter from the spiral arms, and provide a sortof buffer for the free outflow of gas along the arms.As a consequence, the flow speed along the bulgingpart of the arm becomes lower and the gas pressurein this region grows (in accordance with Bernoulli’stheorem in hydrodynamics), with time coming todominate the gas pressure on the other side of thearm. Since the inner boundary of the arm retainsits smoothness, the flow intensity is not decreased,and the pressure remains substantially lowered. Anegative radial component appears in the velocity ofpropagation of the outer layers of gas adjacent tothe arms, leading to a decrease in their mean cur-vature. The arms are subject to stress and twisting,and become appreciably stretched, so that they arerestructured into linear segments. The twisting andstress of the segments are transferred along the armsin the direction of winding of the spirals. As a result,the polygonal structure that is characteristic for manyobserved galaxies encompasses virtually the entirespiral pattern (Fig. 4).

It is possible to explain the hydrodynamical mech-anism for the formation of spur-like features in galac-tic spiral arms as follows. Since the trajectory of theparticles in the gas cross the arm at some (oblique)angle, the cross product of the particle velocity andthe unit normal vector to the arm at the crossing pointwill be non-zero. This means that the motion of thegas crossing the arm acquires a vortex character, withthe rotor of the velocity being perpendicular to theplane of the galactic disk. These vortical motionsarising in the outer parts of the arm are subsonic, andtheir velocities grow with increasing gas density in

Fig. 7. The polygonal structure is formed, the intensity ofthe spurs is reduced.

this region, so that the perturbation is amplified andfairly rapidly acquires the characteristics of a spur-like formation.

A polygonal geometry of the spiral pattern verysimilar to that in the spiral galaxy M51 in CanumVenaticorum is clearly traced in Fig. 5. The pres-ence of non-linear filamentary features at the endsof abutting linear arm segments is clearly visible.These points are regions of instability of the spiralshocks, where Kelvin–Helmholtz instability devel-ops due to the action of the external perturbations,stimulating the growth and development of spur-likeformations (simple hydrodynamical models includingthe Kelvin–Helmholtz instability as a mechanism forthe growth of spurs are analyzed in [31, 32], andthree-dimensional, self-gravitating models, includinga detailed analysis of the effects of Kelvin–Helmholtzinstability on the vertical structure of the galaxy, areconsidered in [33]). Growth and development of spur-like features are shown in the innermost regions of thegaseous disk, near relative maxima of the gas density(and of the gas pressure) in the spiral arms. Theseareas are characterized by the strongest and mostclearly defined rectilinear segments, indicating theirrelationship to spur-like, filamentary features of thespiral structure. In turn, numerical modeling showsthat dense, spur-like, inter-arm, gas-dynamical fea-tures of the spiral shocks most actively grow anddevelop in models with fairly strong external spiralpotentials (compare with [34], where the appearanceand development of spurs is associated with the wig-gle instability).


14 FILISTOV

6. INFLUENCE OF BOUNDARYCONDITIONS

Imposing the free boundary conditions (11) at theouter boundary of the region can lead to an uncon-trolled flow of gas mass, which, in turn, can substan-tially influence the flow inside the region. We thereforecarried out computations in which the boundariesof the computational region were located a distancefrom the disk center comparable to its diameter, in or-der to further study the adequacy of our reproductionof physical processes in the gaseous–stellar disk.

Figures 6 and 7 show the polygonal structure inthe spiral density waves. Spur-like formations at theends of the linear sections are clearly visible in Fig. 6,with intense spur formation preceeding the appear-ance and development of the polygonal structure inthe outer regions of the spiral arms. By the timethe polygonal structure has fully formed in the innerregions, the spur-like formations in these regions aremanifest less intensely.

The pattern shown in Fig. 7 differs in the directionof rotation of the spiral pattern, which is opposite tothe flow of matter in the disk. The polygonal shape ofthe segments becomes less well defined in this case(compare, for example, with Fig. 6).

Thus, our computations show that the observedflow pattern does not depend significantly on ourchoice of boundary conditions: the manifestation ofthe polygonal structure associated with the appear-ance and development of spur-like formations in thespiral structure of the gaseous disk is due to gas-dynamical processes that develop under the action ofexternal forces associated with the presence of thespiral potential of the stellar disk and the rotation ofthe system as a whole.

7. CONCLUSION

We have analyzed the formation of polygonalstructures in spiral density waves in galactic disksas a result of the development of hydrodynamicalinstability leading to the appearance of spur-likeformations. We summarize here the main results wehave obtained.

• Our numerical simulations of a gaseous–stellar,differentially rotating, inhomogeneous galactic diskindicate that the formation of a two-armed globalspiral structure can, under certain conditions, lead tothe appearance and development of hydrodynamicalinstability in the spiral density wave.

• A characteristic manifestation of this type of hy-drodynamical instability is the development of small-scale spurs, which are clearly observed in the spiralstructures of galaxies of various morphological types.

• The splitting of the arms into spur-like for-mations leads to the development and formation ofpolygonal structures similar to those observed.

• Strong spirals can form stronger spur-like for-mations, and therefore more well defined polygonalstructure.

ACKNOWLEDGMENTS

The author thanks A.D. Chernin for numerousfruitful discussions and useful recommendations inthe course of this work, and A.V. Zasov for his readingof the manuscript and valuable comments and sug-gestions for improvement.

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Translated by D. Gabuzda


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