# Polygonal structure of spiral galaxies

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<ul><li><p>ISSN 1063-7729, Astronomy Reports, 2012, Vol. 56, No. 1, pp. 915. c Pleiades Publishing, Ltd., 2012.Original Russian Text c E.A. Filistov, 2012, published in Astronomicheskii Zhurnal, 2012, Vol. 89, No. 1, pp. 1218.</p><p>Polygonal Structure of Spiral Galaxies</p><p>E. A. Filistov*</p><p>Moscow, RussiaReceived July 29, 2011; in nal form, August 17, 2011</p><p>AbstractWe have carried out numerical simulations of hydrodynamical processes occurring in the disksof spiral galaxies. The initial state of the disk is an equilibrium stellargaseous conguration. 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 stellargaseous galactic disks aimed at studying the formation of polygonal structures in spiral galaxiesare presented. The possible inuence of spur-like formations on the appearance of polygonal structure isstudied.</p><p>DOI: 10.1134/S1063772912010027</p><p>1. INTRODUCTION</p><p>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 SbcSc galaxies). The main geometrical andphysical characteristics of multi-angle structures wasstudied in some detail in [36], 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 congurationconsisting 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, [79]).</p><p>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 [1012]; their presence is animportant aspect of the spiral structure, which mustbe taken into account in theoretical modeling of diskgalaxies.</p><p>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</p><p>*E-mail: filistov.ru@mail.ru</p><p>appearance of spur-like formations and their inuenceon the development of polygonal structure in galaxies.</p><p>2. DYNAMICS OF A GASEOUS GALACTICDISK</p><p>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 0, pressure P0,and potential 0 can be written</p><p>V0 =</p><p>(GMGRG</p><p>)1/2, 0 =</p><p>GMGR3G</p><p>,</p><p>P0 = V 20 0, 0 = V20 .</p><p>Further, all equations and quantities are given indimensionless variables.</p><p>In order to exclude the inuence of gas-dynamicalprocesses associated with non-equilibrium aspects ofthe initial state of the disk, we adopted initial condi-tions corresponding to a stationary rotating gaseousconguration using the method proposed in [13]. Inparticular, the class of equilibrium gaseous congu-rations with boundaries of the form</p><p>Z(r) = </p><p>1</p><p>exp(r2), , > 0 (1)</p><p>contains distributions of the density and the velocityof the isothermal gas-dynamical ow</p><p>(r, z) = 0gas exp</p><p>(1r</p><p>(r)(r)c2T</p><p>), (2)</p><p>9</p></li><li><p>10 FILISTOV</p><p>v2(r, z) =1r</p><p>(1 + r(r)(r)</p><p>), vr = 0, (3)</p><p>(r) = 1 rr2 + Z(r)2</p><p>,</p><p>(r) = 1 rr2 + z2</p><p>, |z| < Z(r),</p><p>where r and z are the coordinates of the particles inthe cylindrically symmetrical gas conguration, cT isthe isothermal sound speed, and 0gas is the constantof integration. The prole of the absolute temperatureT along the disk radius rd is described by the law [14]</p><p>T = T0</p><p>(1 +</p><p>r2</p><p>r2d</p><p>)q/2, (4)</p><p>where q = 1/2 and T0 are constants satisfying thecharacteristic parameters for stability of the stellargaseous disk [15].</p><p>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><p>P = c2T . (5)</p><p>The distributions of the density (2), velocity (3),temperature (4), and pressure (5) were adopted as theinitial data inside the subregion r [0, rd].</p><p>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:</p><p>(r, z) =</p><p>0.520gassech( ra</p><p>)sech</p><p>( |z|b</p><p>),</p><p>if r < a, |z| < b,0gasexp</p><p>(r</p><p>a</p><p>)exp</p><p>(|z|</p><p>b</p><p>),</p><p>if r a, |z| b,</p><p>(6)</p><p>where a and b are parameters of the model. In thiscase, we modify the function (1) to have the form</p><p>Z1(r) = 1</p><p>14</p><p>exp(2r2). (7)</p><p>The computational results presented here neglectthe eects of viscosity, the diusion transport of mat-ter, radiation, and evolution of the electromagneticeld. 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,</p><p>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.</p><p>We solved the gas-dynamical equations using themonotonic, rst-order TVD scheme of Roe [16] withlimits on the anti-diusion ows 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 modied using the method of Einfeldt [18] toenhance its stability. To exclude the inuence of theboundary conditons on the behavior of inner regionsof the disk, the conguration 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.</p><p>Perturbations of the density and velocity of theinitial equilibrium state of the gaseous disk can berepresented</p><p> = (1 + 0d), (8)</p><p>vr = vd, (9)where = U1sin, U1 = r2Z1(r)n lnZ1(r), v 0(we used v = 0.50), n = const (we used n = 0.3),d /r + /r is the sum of the partial deriva-tives with respect to the polar coordinates, and themeaning of the remaining parameters is given below.</p><p>3. DYNAMICS OF THE STELLARGALACTIC DISK</p><p>The initial state of the stellar disk was chosensuch that the initial rotating conguration 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 = 0 exp(r/L),where r is the radial coordinate in the disk, L is thescale factor, and 0 = 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 0(r, z) (thehalo, stellar disk) and non-axially symmetric d(r, z)parts:</p><p>(r, , z) = 0(r, z) + d(r, z),</p><p>where</p><p>d(r, z) = 0tr{Z(r) + 2r (10)</p><p>ASTRONOMY REPORTS Vol. 56 No. 1 2012</p></li><li><p>POLYGONAL STRUCTURE OF SPIRAL GALAXIES 11</p><p> ln[cosh(2z/L)]} cos/L2;here, the function Z(r) is determined by (1) and</p><p> = N</p><p>( p(rd) + pt ln(r/rd)tan i</p><p>),</p><p>where N is the number of spirals, i the pitch angle ofthe spirals, p is the angular velocity of the rotationof the spiral pattern, p(rd) is the initial coordinate,and is the volume density of the stellar disk. Thecoecients 0 and t can be used to regulate themagnitude of the additional spiral potential relative tothe main (external) potentials of the model.</p><p>4. PARAMETERS OF THE MODEL</p><p>The specication of the initial conguration of thegaseousstellar disk can appreciably inuence thephysical character of the results. In the results de-scribed below, we chose the values MG = 1010Mand RG = 1022 cm as scale factors; the characteristicvelocity was</p><p>V0 =</p><p>(GMGRG</p><p>)1/2=</p><p>(1014 M11 R</p><p>RG</p><p>)1/2cm/s.</p><p>Here, M11 = MG/1011M and R = 0.8 1023 cm isa dimension of length composed of a combination ofuniversal physical constants [20].</p><p>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.</p><p>The parameters in (1) dening the initial equilib-rium conguration of the gaseous disk and the formof the additional spiral potential (10) were specied tobe = 0.1, = 4.0, and L = 0.25. The rotationalangular velocity of the spiral pattern was p 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-</p><p>guration was Mgas = 0.1MG; 0gas =0.1R3G2a2b</p><p>(a =</p><p>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 [2328]). 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 (0, ), 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].</p><p>Fig. 1. Initial stage of formation of the spiral pattern.</p><p>We carried out two versions of the computationsin an annular region in the polar coordinates ={(r, ), r [r1, r2], [0, 2]}, r1 = 0.1, with r2 =1 in the rst version and r2 = 2 in the second. Theradius of the disk was rd = 1 in both cases.</p><p>We specied free boundary conditions at theboundaries of the region:</p><p>n= 0,</p><p>u</p><p>n= 0,</p><p>v</p><p>n= 0, (11)</p><p>where u = (u, v) is represented by its components inthe polar coordinate system.</p><p>The external additional spiral potential was mod-eled using the coecient 0 = 0.1. The coecientt increased from zero to unity with time, making itpossible to gradually increase the amplitude of thepotential d. This excluded abrupt relaxation pro-cesses in the initial equilibrium conguration due tothe instantaneous switching on of external forces.</p><p>5. FORMATION OF THE POLYGONALPATTERN</p><p>The general structure of the solution obtainedfor the ideal-gas ow in the equatorial plane of thegaseousstellar system considered is presented inFigs. 17. These depict the density distributionin a rectangular region in Cartesian coordinates, = {(x, y), x [1, 1], y [1, 1]}. Analysis of thecomputational results enables us to identify the mainproperties of the ow, and to draw some conclusionsabout the character of the development of perturba-tions that restructure the laminar ow. Let us nowconsider the ow patterns obtained in more detail.</p><p>Figures 14 show the ow pattern of the gasunder the inuence of the self-gravitating stellar disk</p><p>ASTRONOMY REPORTS Vol. 56 No. 1 2012</p></li><li><p>12 FILISTOV</p><p>Fig. 2. Further development in the stage of formation ofthe spiral pattern.</p><p>Fig. 3. Growing spurs.</p><p>immersed in the gravitational eld of the sphericalcomponent (halo), taking into account the additionalstellar spiral potential with 0 = 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 ow 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 ow.</p><p>The spiral structure actively develops in regions ofthe gaseous disk outside the central region. Due tothe drop in pressure and density toward the periphery,</p><p>Fig. 4. Polygonal geometry of the spiral pattern.</p><p>Fig. 5. Polygonal geometry of the inner region of thegaseous disk.</p><p>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</p><p>ASTRONOMY REPORTS Vol. 56 No. 1 2012</p></li><li><p>POLYGONAL STRUCTURE OF SPIRAL GALAXIES 13</p><p>Fig. 6. The appearance of spurs at places where segmentsmeet preceeds the formation of the polygonal structure.</p><p>formations represent a new channel for the outowof matter from the spiral arms, and provide a sortof buer for the free outow of gas along the arms.As a consequence, the ow speed along the bulgingpart of the arm becomes lower and the gas pressurein this region grows (in accordance with Bernoullistheorem 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 ow 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).</p><p>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 di...</p></li></ul>

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