Cellular Automaton experiments on local galactic structure. II. Numerical simulations

Astronomy and Astrophysics Supplement Series, Jul 2018

This paper is a step towards demonstrating that the multi–parameter Cellular Automaton framework designed for the simulation of local galactic structure, developed in a companion paper (Perdang & Lejeune 1996, Paper I), is capable of duplicating the local irregularities observed in the structure of flocculent spiral galaxies. The numerical simulations exhibit the development of fractured galactic arms and the formation of fractal geometries associated with the matter distribution (fractal structure of the arms, of bulk dimension ≈ 1.7 and border dimension ≈ 1.3; distribution of different stellar components on fractal supports, of dimension ≈ 1.6, for reasonable estimates of the free model parameters). The prediction of fractional values for the different dimensions specifying the simulated structures can be exploited as a qualitative test of adequacy of the proposed model. The precise quantitative values of the observed dimensions, in conjunction with the observable global mass fractions of the different galactic components, play the parts of constraints for the free model parameters. We show that the currently theoretically inaccessible values of the free parameters of the formulation can be recovered from observation.

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Cellular Automaton experiments on local galactic structure. II. Numerical simulations

Astron. Astrophys. Suppl. Ser. Cellular Automaton experiments on local galactic structure. II. Numerical simulations A. Lejeune 1 J. Perdang 0 0 Institute of Astronomy , Madingley Road, Cambridge CB3 OHA , UK , and Institut d'Astrophysique , 5, Avenue de Cointe, B 1 Institut de Physique , Sart 2 4000 Liege , Belgium | This paper is a step towards demonstrating that the multi{parameter Cellular Automaton framework designed for the simulation of local galactic structure, developed in a companion paper (Perdang & Lejeune 1996, Paper I), is capable of duplicating the local irregularities observed in the structure of flocculent spiral galaxies. The numerical simulations exhibit the development of fractured galactic arms and the formation of fractal geometries associated with the matter distribution (fractal structure of the arms, of bulk dimension 1.7 and border dimension 1.3; distribution of di erent stellar components on fractal supports, of dimension 1.6, for reasonable estimates of the free model parameters). The prediction of fractional values for the di erent dimensions specifying the simulated structures can be exploited as a qualitative test of adequacy of the proposed model. The precise quantitative values of the observed dimensions, in conjunction with the observable global mass fractions of the di erent galactic components, play the parts of constraints for the free model parameters. We show that the currently theoretically inaccessible values of the free parameters of the formulation can be recovered from observation. galaxies; structure | star formation | methods; numerical | galaxies; evolution 1. Introduction In this Paper we report on an extended series of numerical experiments carried out in the framework of a 2{dimensional Cellular Automaton (CA) model for local galactic structure described in the companion paper (Paper I). The model takes account of stellar evolutionary e ects not dealt with in current CA models, as well as of quasi-stationary motions more complex than usual di erential rotation. Our results are indicative that the floccular character of galaxies can be simulated in the CA framework. For technical details of the implementation of the galactic physics we refer to Paper I. Our results are given in a relative reference frame F in uniform rotation with respect to the galaxy to be simulated. 2. Initial conditions. di usion Remarks on infall and In all of our experiments we start out with an initial di use non{uniform gas covering the part of the galaxy we are interested in (region Z over which the CA computations Send o print requests to: A. Lejeune ?Permanent address are carried out). Any cell is either in the empty state E, or in the gas state D; with the notations of Paper I F (0; i; j) = 0 or 1; for any cell (i; j): (2:1) The choice between an E or a D state is made on the basis of a method reminiscent of a `mole's labyrinth' (Herrmann 1983) , already adopted in previous CA simulations (Lejeune & Perdang 1991) : The cells (in the auxiliary lattice L, cf. Paper I) are all provisionally populated with gas; a number of b = 1, 2, ... empty `bubbles' are then generated by T {step random walks starting at b randomly chosen initial cells; for a selected number of di erent walks b, the number of steps T per walk is chosen such as to generate a preassigned fraction f (E) of empty space in the lattice area; the remaining fraction of the area which is lled with gas is denoted f (D) = 1 − f (E). Through the coordinate transformation Eq. (5.1a) of Paper I we obtain the initial state in the `physical' zone Z (the lattice L0). Figure 1 provides a realisation of an initial con guration (corresponding to the velocity eld of Fig. 3 of Paper I).1 1The purpose of this operation is to create an irregular cloudy shape, a locally non{uniform initial matter distribution de A special simulation should be included in our program to handle the internal dynamics of molecular clouds reponsible for the formation of the complexity of the actual cloud shapes. The random walk procedure as adopted here should be viewed as a cheap substitute for simulating the observed fragmented gas cloud border. Incidentally, the numerical estimate of the fractal dimension ( 1.3) of the border of our simulated cloud in Fig. 1 compares favourably with the reported estimates of the fractal dimensions of cloud contours. We note also that with our algorithm the gas area is either connected or it is disconnected, depending on whether or not the `empty bubble' extends from the left to the right border of our zone Z; provided that infall and diffusion are weak enough, connectivity is preserved at any later time. Under the rst alternative a signal in one cell of the gas zone can propagate over the whole zone, i.e. the state of any cell (i; j) at a given timestep can influence the state of any other cell (i0; j0) at some later time 0; under the second alternative disconnected areas evolve independently. Infall, as handled in the present simulation (Reaction (6.12) of Paper I) may generate a large number of D cells within the empty bubbles during the initial stages (Tinf 100 timesteps) over which we let it operate. If Nc ( ny2, ny number of cells in the y direction) denotes the total number of cells of our lattice ( 106), a rough estimate of the number of these new gas cells N (D) created by the implementation of infall is N (D) Nc f (E) P (E; D) Tinf : (2:2) In a T {step random walk the distance r(T ) a particle traverses on average, as estimated by the expectation E of r(T )2 is E(r(T )2) pdi T (2:3) in units of the cellsize (pdi , di usion probability factor as implemented in our program). Hence, even for the strongest possible di usion, (pdi = 1), over a timescale T ( 103, for a typical evolutionary run), the average distance (2.3), in units of the total size of the CA network is [E(r(T )2)]1=2=ny (T =Nc)1=2; (2:3a) 3 10−2 1 for our experiments. Di usion, therefore, although not completely negligible, cannot play a dominating part in our model. In the hydrodynamic representations (Fig. 4a) the e ect of di usion is obliterated for an averaging radius obeying p . This is indeed consistent with our experiments.3 Table 1 summarises the evolutionary reactions simulating the stellar evolutionary processes analysed in Paper I. The table also lists the corresponding independent transition probabilities per CA timestep (denoted here P1; P2; :::; P15) as adopted in our numerical experiments. Column (a) gives these parameter values for the galactic evolution from which we have constructed synthetic photographs. Column (b) corresponds to the reference `average' model around which our calibration models for the coe cients in representations (3.3) and (4.4) are generated. Column (c) refers to one representative of the latter collection of models; it provides an indication of the range of model parameters explored in our experiments. With the numerical values f (E) 0:25, P (E; D) = 1 10−5, we have N (D)=Nc 2:5 10−4 1, indicating that this mechanism should not play a signi cant evolutionary role.2 The action of di usion in the framework of our simulation is more complex. On the computational level of the 3The di usion algorithm adopted in our program models normal di usion, conforming to relation (2.3). For the purposes individual cellstates F ( ; i; j) (exhibited in Fig. 1a for an of simulating motions of real individual galactic components initial state, and in Fig. 3 for a later evolutionary phase), (proper motions of stars, for instance) normal di usion apthe di usive mechanism detaches border gas cells, thereby pears to constrain too strongly the allowed velocities to consticreating new disconnected patches of matter. However, on tute an acceptable model for the actual individual motions. We a macroscopic, observationally meaningful `hydrodynamic believe that a model of anomalous di usion (or overdi usion) scale', simulated in our experiments by spatial smoothing could capture more adequately the physical situation. Anoma(Fig. 1b; compare also Fig. 4), the averaged patches re- lous di usion could be implemented in principle (probabilistic main connected to the main clouds. displacement of a cell by a distance r obeying a Levy distribution; for the latter the asymptotic probability density for large void of any particular symmetry (cf. also the procedure of displacements is of the form 1=r +1, 0 < < 2; cf. Feller 1971) . Heisenberg & von Weizs¨acker 1948). The random walk method We have not attempted to incorporate this mechanism in our was chosen in our experiments since it is particularly easy to galactic model. We point out that for anomalous di usion the implement. A more evolved CA approach capable of generating variance of the displacement does not exist, as a consequence a variety of growth patterns, including realistic cloud shapes, of the contribution of the large displacements per timestep. has been developed recently by one of us (Perdang 1996) . For this very reason anomalous di usion secures a more e 2In order to lead to the formation of a single connected gas cient transport: the exponent 1=2 in the RHS of (2.3a) is to zone, and hence, in order to play a dominant part in the overall be replaced by 1= . It follows that provided that is taken evolution, the characteristic time over which the infall should small enough, anomalous di usion can play a non{negligible be operative, Tinf , should obey Tinf f(E)P1(E;D) : part over timescales as dealt with in our experiments. 3. The global evolution of the galactic species In Figs. 2 we exhibit several evolutionary sequences of the global mass fraction of our 6 galactic species. For the parameter ranges adopted, 3 phases can be distinguished: (i) An initial short transient phase, characterised by a rapid building up of the protostar population and a corresponding decline of the amount of gas. This phase subdivides into a trigger{phase of star formation, of duration determined by the probability of the cloud collapse (P1), followed in turn by a more violent star formation process of timescale xed by the catalytic reaction probability (P3). In Frame (a), corresponding to a large probability P3, we observe a prominent pulse of star formation of maximum at step 20; in Frame (b) the probability P3 is reduced by a factor 2 and the pulse of star formation is virtually suppressed; in Frame (c), where P3 is further reduced by a factor 2, the transient phase consists in a gentle rise of the protostar population over about 150 steps. (ii) A subsequent quasi{stationary phase in the protostar, main sequence, and red giant population, during which the formation rates of these species are essentially balanced by their decay rates. The leakage of active galactic matter towards inert matter (and low mass stars, which play the part of inert matter as well within our numerical approximations) ultimately implies that the `source' of protostars is progressively depleted; on average the protostar population is therefore bound to decrease. Frames (a) and (b) indicate that quasi{stationarity is practically realised beyond timestep 100. In all cases the slow decrease due to loss of active matter can be approximated by a linear time behaviour. (iii) Finally, our model conceals an asymptotic equilibrium state, for ! 1, in which all galactic matter is in the form of inert or low mass stars (if the transition probability P (L; I) is set equal to zero). Practically this asymptotic state would be reached for evolutionary times as such that as P (D; L) > 1, or as P (A; B) > 1, where P (A; B) is the smallest transition probability in the global transformation sequence D ! I via massive stars; in our numerical experiments this stage is never attained.4 We mention at this stage that depending on the precise parameter values, the quasi{stationary phase (ii) may exhibit di erent time patterns. Besides the uniform decrease, an oscillatory time-behaviour has been observed if the intermediate steps of the global reaction D ! I are arti cially set approximately equal. The low amplitude fluctuations observable in the D and P components of Fig. 2a may be a scar of such a periodic behaviour. Moreover, when instead of examining the space averaged global behaviour of the di erent species we examine the detailed spatial behaviour of the species, spatial oscilla4A semi{analytic discussion of the general time behaviour based on an examination of the structure of the transition matrix is given in Perdang (1993 Sect. 5) for a more schematic version of the present model. tions and intermittency phenomena are observed during the globally oscillatory phase. We have not followed up these types of complex behaviour in this paper. In a mean eld approximation, in which the space dependence is discarded, the global evolution equations reduce to simple iterative schemes (di erence equations in time). Approximating the di erences by di erentials, we obtain ODEs for the evolution of the global population of the species, thereby reducing our original CA formulation to an elementary model of class (1) (Paper I). The resulting ODEs can be interpreted as evolution equations of chemical kinetics in a homogeneous medium; the kinetic equations are read o from the stoichiometric equations of Table 1. As already pointed out, the existence of oscillating solutions in these kinetic equations, for some range of the rate parameters, is well known: Standard chemical kinetics can give rise to stable oscillatory behaviour, provided that the reaction schemes involve a nonlinear feedback, in the form of catalytic processes.5 In the context of the CA model of GSS oscillatory be haviour is also encountered (cf. in particular Seiden & Schulman 1990 Fig. 16) ; the GSS oscillations appear in some range of a refractory time, or period of immunity of the gas against star formation; the latter plays the part of a formal substitute for the details of the `chemical kinetics' we are adopting in our model. In principle, the relative populations of the species (mass fractions), N (Y ; ; G)=PY N (Y ; ; G) = a(Y ; ; G), Y = D; P; M; R; I; L, over some large enough xed area G of a galaxy, can be determined observationally at the present age of a galaxy gal. Assuming that the galaxy is in a stable quasi{stationary phase (ii), we nd from our numerical experiments that the relative populations are accurately represented in the form 5A triangular linear reaction scheme A ! B; B ! C; C ! A shows an oscillatory behaviour if the 3 rates are identical (or su ciently close); however, such an oscillation rapidly dies out and the system settles in a stable stationary state. The occurrence of a sustained oscillation around a stationary state requires this state to be unstable. Instability demands that the stationary state be on a `non{thermodynamic branch', which is realised, in the chemical context, by catalytic and auto{ catalytic steps (cf. Glansdor & Prigogine 1971) . The various catalytic steps in Table 1 were incorporated in our simulation precisely to investigate the possibility of `chemical instabilities' within the reaction network of the galactic species. In our scheme, the quasi{stationary phase (ii) plays the part of a non{thermodynamic branch parametrised by the `secular time'. Asymptotically this branch merges with the thermodynamic equilibrium state, namely state (iii). Small deviations from the thermodynamic state inherit the stability of the latter, so that the oscillations can occur su ciently far away from the asymptotic state (iii) only; this thermodynamic property implies that the oscillatory behaviour can survive over a nite `secular' time only. R ( a) where the subscript o refers to the onset of phase (ii), or more generally to any time su ciently exceeding the lifetime of phase (i) of the galaxy. In our numerical experiments we have set o = 100; except for a few experiments involving small induced probabilitites this value of o corresponds to a reference time within the stationary phase. The parameter dependence of the two expansion coefcients ao(Y ; G) and a0(Y ; G) is determined from a series of test experiments. We represent this dependence by a linear ansatz written in the form ao(Y ; G) = aor(Y ; G) + a0(Y ; G) = a0r(Y ; G) + X(Pk −Pkr) k(Y ; G); (3:2a) k X(Pk − Pkr) 0k(Y ; G); (3:2b) k where Pk is the transition probability of the kth independent reaction step (notations of Table 1). A subscript r to a model quantity aor, a0r indicates the value taken by this quantity in the reference model de ned by the transition probabilities (Pk = Pkr) given in Col. (b) of Table 1 (initial amount of dust 72%). The coe cients of the expansions (3.2), aor(Y ; G), a0r(Y ; G), and k(Y ; G), 0k(Y ; G), listed in Table 2, are calibrated by a least squares t to a grid of 24 experiments in which each parameter ranges from 1/3 to 3 times its reference value. To test for the goodness of the t we compared the actual values ao and a0 of the 24 calibration models with their linear estimates (Eqs. 3.2a,b). As a rule, provided that ao is not too small (> 0:02), the relative agreement is better than 20%. The linear ansatz (3.2a) is thus satisfactory. Inaccuracies occur in runs in which the quasi{ stationary phase is not yet properly established at our reference timestep o = 100 (which occurs if P1 or P2 are too small). We observe also that for species L the parameter ao is almost always well approximated (precision better than 10%). On the other hand, the rate parameters a0 are accurately given by the linear estimate only for the inactive species I and L. For the remaining species the estimate may di er by as much as a factor of 2 or 3; within the accuracy of the linear ansatz (Eqs. 3.2a,b) it is therefore legitimate to disregard the Pk{dependence in the rates of the relative populations of the active species A = P; M; R: a0(A; G) a0r(A; G). The coe cient k(A; G) indicates how a variation around the reference model of the kth reaction probability, Pk, a ects the quasi{stationary population of the active species A (= P; M; R); the induced variation of the relative population obeys a(A; G) = k(A; G) Pk; hence, if k(A; G) is positive [negative], then species A is created [destroyed] | either directly or indirectly | in process k. We note that the magnitude of the sensitivity itself, j k(A; G) j, is not only large in a direct creation or destruction process; it may remain so also in indirect processes. As an example, the direct sensitivity ther remains a red giant, or it becomes an inert star, or it of the protostar population to the probability of the di- transforms into gas; 2 transition probabilities are indepenrect star induced protostar formation process, 3(P ; G), is dent. Since our model involves 21 transitions, or reactions 0.27; the R population, although only indirectly a ected (cf. Table 1) among 6 species, 15 transition probabilities by the same reaction, has a larger sensitivity, 3(R; G) are independent (P1; P2; :::; P15 in Table 1). = 0.71. Note that the sensitivities to spontaneous (gas Observationally we can determine 5 relative populaand dust induced) star formation, 1(P ; G) and 1(R; G), tions of species, a(Y ; gal; G)obs, Y = P; M; R; I; L. Subare much larger, 2.7 and 3.4 respectively; however, for stitution of Eqs. (3.2a,b) into Eq. (3.1) then yields the all our runs the spontaneous transition probability P1 it- following linear system in the model parameters Pk self is small ( 0.0015) in comparison with the star induced transition probability P3 ( 0.033); accordingly the overall contribution of the spontaneous formation process X[ k(Y ; G) + gal 0k(Y ; G)] Pk = a(Y ; gal; G)obs − (P1 −P1r) 1(P ; G) remains always negligible with respect k to the star induced formation (P3 − P3r) 3(P ; G). As already mentioned, an examination of the orders of [aor(Y ; G)+ gala0r(Y ; G)]+X[ k(Y ; G)+ gal 0k(Y ; G)]Pkr magnitude of the fragments (Pk − Pkr) k(Y ; G) entering k the de nition of the quasi{stationary relative population (3:3) ao(Y ; G) shows that some of them are negligible when the (5 equations in 15 unknowns). Further observational inlatter coe cient is recomputed for the calibration models. formation is available from the spatial distribution of the It is then reasonable to discard those fragments also in the di erent species. Our purpose is to show that in principle general representation (3.2a). The P population, ao(P ; G), the latter can be quanti ed in a way as to supply addiis dominated by the contributions of star induced star for- tional constraining equations in the unknowns Pk. mation, of direct and star induced decay processes of protostars into gas; star induced transformation of protostars 4. The space structure into the M species also contribute; spontaneous P formation is negligible, and so are most of the indirect processes. For the M and R populations, ao(M ; G) and ao(R; G), all steps yield essentially comparable orders of contribution. Negligible contributions (Pk − Pkr) k(Y ; G) are listed in Table 2 in the form k(Y ; G) = v 0, where v is the actual numerical value of the sensitivity as obtained in our calibration. At any timestep , the evolution of each one of the 6 formal species E; D; P; M; R; I; L is represented by a probabilistic transition (Table 1). For instance, for species R our model accounts for 3 transitions (regarded as mutually exclusive) translating the physical fact that a red giant eiThe simulation of a mere global evolution of the galactic species, summarised in a small number of time series, would not require a detailed CA model: The global evolution is satisfactorily reproduced by models of class (1), namely by the low order ODEs generated from a mean eld approximation; the only extra information supplied by the CA model is a superposition of statistical fluctuations which mimic to some extent the genuine observable fluctuations. The real interest of a CA approach is that it produces the detailed time behaviour of a complex space dependence of the physics of the galaxy (encoded in CA eld variable F ( ; i; j)). Due to the intrinsically highly involved spatial patterns, there seems to be no fully satisfactory way of exhibiting the information carried by this eld variable in any data{compressed form. A proper appreciation of the full model results requires an analysis of the sequence of frames de ned by the eld matrices F (0; i; j), F (1; i; j), F (2; i; j), ... by an animation technique; only this form of visualisation can properly reveal the local details of the dynamics. In Fig. 3 we supply a rst frame (a) showing the microscopic CA state F ( ; i; j) of the galactic region at timestep = 200 in the auxiliary lattice L together with a space{averaged macroscopically meaningful picture of the same zone (Gaussian averaging over 2 pixels), and a second frame (b), at step = 500 in the physical lattice L0 together with a space{averaged picture (radius 2 and 4 pixels) (model parameters: column a of Table 1; initial conditions shown in Fig. 1). The intricacy of the averaged computed structure at any timestep leaves little hope Reference state o = 100; G: whole lattice reaction for a duplication of these results by a low order di erential model, or even by a tractable PDE model. Figure 4a is an attempt at generating a synthetic black Figure 4b is a real HST photograph of a part of M100 of and white photograph from the computed CA state at roughly same size as the computed galactic zone of Fig. 3b. timestep 500; the grey level at each CA cell is de ned Although both pictures are not comparable in detail | we through a weighted average of the contribution of the lu- should keep in mind that the synthetic gure represents a minous matter (cf. caption); the nal picture is a spatial single realisation of a stochastic model | they do show one Gaussian average over the `microscopic' grey levels (radius small{scale characteristic structural similarity: individual = 2 pixels). The pattern of white patches is seen to match lsihgahpteps.atches in both pictures have statistically equivalent almost exactly the lightblue areas in the averaged picture of Fig. 3b. reaction 7: P + mM + rR ! In our model these patches correspond to regions of This conclusion cannot be extended to arbitrary galachigh concentrations of protostars; the existence of con- tic dynamics. In the presence of a chaotic flow eld, we nected patches is related to the SPSF mechanism (and should expect the latter to a ect, and increase the fracthe Goldreich & Lynden{Bell (1965) chaotic mechanism). tal dimension of the arm structure. On the other hand, Figure 4c gives the structure of the levels of constant even in the case of a smooth velocity eld, and hence a brightness of the synthetic photograph. smooth transformation between the physical space and From Figs. 3 and 4 we can infer several general conclu- the auxiliary space, there is no guarantee that the acsions on the space structure as generated in our models. tual technique of estimating the dimension does yield the If we accept the view that the most luminous objects (the same value in both cases. In fact, the ln l against ln r population of protostars) are the tracers of the galactic curve shows in practice a linear part over some r range, arms (collection of bright patches in the synthetic photo- and it is the slope of the latter which estimates F ; at graph Fig. 4a), then our model generates arms exhibiting small scales, r < ro (= actual resolution of the gan irregular fragmented local geometry (on scales of the ure: 5 to 10 cells), the ln l { ln r curve flattens out; at order of less than 200 pc); the large scale geometry (> large scales, r > rL (order of macroscale of structure 400 pc), on the other hand, appears to be channelled by in the gure < size of gure), the local slope of ln l { the smooth dynamics of the galaxy (gravitation induced ln r can take on arbitrary values. Under a nonlinear but shearing e ect of the stationary velocity eld). The sta- smooth transformation of the gure, if the transformation bilised arm pattern tends to establish itself in the quasi{ itself exhibits structure over the range ( ro; rL), then stationary regime (ii) of the global evolution of the species. the slope of the curve ln l0 ln r0 of the transformed gure will be a ected over the transformed range of ( ro; rL). Although theoretically the fractal dimension as de ned for r ! 0, remains the same in the original and trans4.1. Small scale geometry formed variables, practical estimates are bound to operate at nite resolutions > ro. For instance, if we have a As a measure of the local fragmentation we have deter- set of hierarchically distributed points on a straight line mined fractal dimensions associated with the arm struc- (so that 0 < F < 1), experimentally approximated by ture. This can be done in several ways: The fractal di- a nite number of dots distributed according to a nite mension F of the boundary of the area occupied by the number of hierarchical levels, then a numerical estimate luminous objects (Fig. 3) in (a) the physical lattice L0; of the fractal dimension F on the experimental realisation and in (a') the auxiliary lattice L; one could likewise di- in ( ro; rL) yields an acceptable theoretical estimate of rectly measure the fractal dimension of synthetic isophotes F . However, suppose we perform a stretching of the ex(Fig. 4c). The natural method for estimating the fractal perimental points, x0 = Sx (x, initial coordinate of a point dimension F of a topologically linear set (border of the on the line; x0 new coordinate;S, stretching factor); if S area occupied by a population A, or xed isophote re- is chosen such that the intrinsic resolution in the new cospectively) embedded in a plane, consists in measuring ordinates, ro0 = S ro rL, and if we estimate again the length l( r) of this set at di erent resolutions r; the dimension F' from the slope of the ln l0 { ln r0 curve, the dimension F 2 [1; 2] is then related to the slope of the at small r0 (< S ro), then we manifestly nd F 0 < F ; ln l( r) { ln r relation, which is equal to 1 − F at a ne note that F 0 ! 0 for S large enough. This example should enough resolution (provided that this relation is linear for make it clear that an a priori knowledge of the actual inr ! 0). It is practically more convenient to estimate F trinsic resolution of our experimental data is required to via the correlation dimension (which is equal to the frac- make a reliable numerical estimate of a fractal dimension. tal dimension if the latter exists; cf. for instance Perdang 1990, 1991) ; this latter method has been adopted in our The discrepancies in fractal dimension estimates of atnumerical work. Theoretically, since L0 is a smooth de- tractors in phase spaces referred to di erent variables (cf. formation of L, the estimate of the fractal dimension is Cannizzo et al. 1990 for stellar pulsation attractors) are independent of the grid L or L0. related to this e ect. The fractal dimension F shown in Table 3 is a dimension of the area occupied by the arms rather than of the border of the arm region (or length of an isophote). The fact that this dimension turns out to be non{integer reflects here the fragmented and hierarchical topology of the arms; the number of arm fragments increases as we increase the resolution; this behaviour is in agreement with the observational situation (cf. the role of resolution in Fig. 1 of Paper I). The evolution of the dimension F exhibited corresponds to the run chosen for constructing the synthetic photographs (parameter values of Col. (a) in Ta- gas distribution (empty bubbles in our simulation); the ble 1). The e ect of the comparatively low spontaneous resulting smooth geometry is then irregularly modulated star formation probability P1 is to delay the onset of the by the randomly acting physics of the star formation and catalytic star formation process. As a consequence, the evolution processes. establishment of the quasi{stationary state is delayed as In our experiments we start with large size empty well ( o 300 instead of 100 for the larger P1 values). It fragmented bubbles floating in an originally uniform gas transpires from Table 3 that the bulk fractal dimension is con guration. Since, as indicated, the additional dynamic reasonably stationary (F 1.8) beyond timestep 300. mechanism included in our treatment, namely di usion, is The dimension given in Table 3 is the correlation di- ine cient over distances of the order of a fraction of the mension of the area occupied by the arm (cf. Perdang size of our lattice, any large scale deformation of a bubble 1991) ; Fig. 5 also illustrates the behaviour of the frac- is a consequence of a stretching by the inhomogeneous vetal dimension of the border (estimated again through a locity eld only. Through the stretching the bubble transcorrelation dimension) which, as expected, is lower than forms into a single empty lane tending to become parallel the previous dimension. Other fractal dimensions attached to the shear direction; it transforms into a double lane with the arms (mass dimension of arm area, mass dimen- once the stretched bubble has acquired a length exceedsion of border) have also been estimated; the latter may ing the length of the ring it belongs to, etc. The action of di er by 10% from the corresponding correlation dimen- the shear on several bubbles leads to a system of parallel sions; beyond step 300 all dimensions are essentially sta- empty lanes. Since no activity can occur in the empty retionary (cf. Fig. 5). gions, the bubbles de ne the interarm zone. Support of the It seems natural to expect that the ne structure of observable arm, the zone lled with gas likewise acquires an arm (understood as the locus of the luminous objects) a shape of parallel arcs covering our lattice area. is directly related to the kinetics of stellar evolution, and This large scale scenario, in substance already demore speci cally to the star{induced star formation (tran- scribed by Weizsa¨cker (1951) as a mechanism responsible sition probability P3). Accordingly the fractal dimension for the formation of spiral structure in galaxies, is indeof the arm structure should help constrain the range of pendent of any reaction kinetics.6 the model parameters. If we summarise again the results We convinced ourselves of the e ciency of this mechaof a collection of numerical experiments in an interpolation nism in the CA framework by running our simulation proformula of type (3.2a) (disregarding here the time depen- gram with the evolutionary kinetics and di usion turned dence), we obtain a constraining relation in the form (cf. o : Voids and matter are then observed to arrange in arms. Eq. (3.3) with the dashed parameters left out) Test runs further make it clear that the global shape of the arms is determined by the analytic form of the stationXk kPk = F obs − Fr + Xk kPkr (4:1) tahrye vfrealoctcaitlydi melden.sWioenmoefntthieonmiantptears{sivnogidthinattetrhfaecoereitsichaelrlye time independent ( 1.30 in our experiments); it is just (F obs, observational determination of the arm dimen- xed by the construction algorithm of the bubble; physision for a real galaxy). However, a few test experiments cally this dimension should then reflect the formation prohave suggested that the dimensions of the arms are only cesses of the molecular clouds. In the actual simulation, marginally sensitive to the individual stellar processes. It however, the stretching transforms the initially irregular would seem that once the stationary arm pattern is estab- lines of constant density into almost straight lines along lished, the arm structure acquires fairly universal fractal the shear direction. This is a consequence of a loss of spadimensions which are independent of the precise physics tial resolution accompanying the stretching transformaprevailing in the formation process of the arm. Such a pic- tion which, in principle, should reveal ner and ner deture is not inconsistent with the percolation phase transi- tails (cf. our previous remark). Hence, since the CA model tion describing the arm formation as proposed in GSS (cf. is working with a nite resolution, a numerical estimate further comments in the next subsection). yields an apparent fractal dimension decreasing with time In any event, our experiments are indicative that towards 1. the knowledge of these dimensions does not provide (ii) Carriers of the collection of luminous objects, the any signi cant constraints on the transition probabilities lanes of matter then naturally trace out an underlying parametrising stellar evolution. large scale structure of progressively smoother geometry on which the observable brightness patterns of the galaxy 4.2. Large scale structure We can further infer from our simulations that the large scale pattern traced out by the arms has a twofold origin: (i) It is due to the action of the galactic dynamics (the shear of the velocity eld) on an initial inhomogeneous 6In a paper with Heisenberg, the validity of this idea was substantiated by a simulation in which an elongated irregular 2D cloud of dots is rotated about an inner point of the cloud, according to Kepler's third law. The cloud then develops into a regular two{armed spiral (Heisenberg & von Weizs¨acker 1948) . are superimposed (Fig. 4a). As already recalled, GSS have interpreted the onset of the formation of a bright arm as a percolation phase transition leading to the occurrence of a `connected in nite cluster' of active stars. Their model makes use of a single control parameter, p (probability of induced star formation, i.e. counterpart of our parameter P3); the phase transition occurs for p > pc, c a critical value. The analogue of the GSS percolation transition is observed in our more elaborate model, the cluster being identi ed by the protostar population. We mention that at xed reaction probabilities the initial fraction of gas, f (D), can likewise be chosen as a relevant control parameter for the phase transition: An `in nite' cluster develops as f (D) exceeds a critical value fcrit(D) ( 0.66 for the choice of reaction parameters in Lejeune 1993) . The value of the critical control parameter (p, P3 or f (D)) eventually depends on an arbitrary convention of what we understand by `connection' of 2 neighbouring cells. To illustrate this point, de ne two cells as being c{ connected if they are less than c cells apart; a `cluster' is then a collection of c{connected cells. It is manifest that the onset of the transition and the critical value of the control parameter depend on the speci c arbitrary choice of c. This remark is indicative that a practically relevant concept of `percolation' for the galactic context must be predicated on the observational resolution (of which the parameter c is a caricature). The formal percolation concept as introduced in GSS has no directly accessible quantitative observational counterpart. The resolution dependence of the `connected in nite cluster' can be demonstrated by image analysis of synthetic photographs (Figs. 4a,b) and real photographs (cf. Fig.1 in Paper I): As we increase the smoothing e ect and generate lower resolution images, we observe more extended connected bright areas. The GSS interpretation has the virtue, in our opinion, of supplying a qualitative explanation of the formation process of the bright arm structure in the galaxy by relating it to the statistical mechanics of percolation. The luminous arm structure (i.e. the extended connected patches covered by protostars and luminous objects in general) appear or disappear, as some xed combination of the parameters regulating the evolution of galactic components goes through a critical value. On the account of the arbitrariness involved in the de nition of the `percolation cluster', it is hard to see, however, how the GSS percolation concept could help extract quantitative information on the stellar processes from galactic observations. In fact, if the transition is a genuine percolation phase transition, then the relevant information it does supply is that it carries no quantitative information on the speci c physical mechanisms of the underlying model: The numerical coefcients characterising a real percolation phase transition (critical exponents and γ; cf. Eqs. (1.3) of Paper I), are universal, i.e. determined by global topological and geometrical factors (dimensionality and lattice symmetry); they are independent of the precise physics responsible for the formation of the cluster (exact evolutionary steps and corresponding transition probabilities). In view of this remark global parameters quantifying the arm structure are not used as constraints for the physics of our model. The fractal dimensions F already mentioned seemingly belong into this category as well. Since our model involves several species of luminous objects (P , M and R), a phase transition is possible in principle for each of these species. We have traced such a situation under the conditions of an oscillatory regime. Extended areas of the galaxy then undergo cyclical changes: P ! M ! R ! P ! :::. If realisable in genuine galaxies | the regime requires ranges of the transition probabilities probably not consistent with realistic galactic conditions | the physical nature of the luminous arm would cyclically change, from a predominantly protostar composition, to a main sequence and red giant composition. Whether the transition isolated by GSS is to be viewed as a genuine percolation phase transition remains to be substantiated, even though the power{law behaviour of the cluster size characteristic for percolation has been veri ed under various conditions (cf. Lejeune 1993) . In fact, a percolation phase transition, in its standard acceptance, describes a transition between static states under a quasi{ static change of a control parameter (transition from absence of a percolation cluster to presence of a cluster); the notion of percolation involves no dynamics (cf. Stau er 1985) . In the present physical problem, we rather observe a propagation, i.e. a dynamic phenomenon, with the in nite cluster developing in time. Accordingly, the transition appears to be more closely related to what has been referred to as self{organised criticality (Bak et al. 1987-1989) , a transition towards a dynamical state in which very large numbers of degrees of freedom are excited. Near the onset of self{organised criticality power{law dependences hold for various magnitudes attached with the clusters of excited degrees of freedom (distribution of cluster sizes, distribution of lifetimes of clusters, etc). To the best of our knowledge, and in contrast with the power{laws related with percolation, there has been no proof so far that the exponents of these laws are universal. Accordingly, if the identi cation of the in nite cluster formation with a transition to self{organised criticality applies in our case, then the exponents could carry additional information on the physics. 4.3. Space distribution of the luminous objects An inspection of the picture of the cellstates (Fig. 3) indicates that the overall space distribution of the luminous objects, P , M and R is not homogeneous over the area investigated. The luminous objects appear to be concentrated over lower{dimensional subsets, and this type of inhomogeneity is inherited by the inert objects I. As a consequence of their formation mechanism (cf. Table 1), the low mass stars L, on the other hand, are regularly scattered over the area originally covered by gas; but since the latter itself is not uniform, the L distribution inherits the lack of uniformity of the gas. Physically the spatial distribution of a given species is determined by a combination of three e ects: (a) the initial con guration; (b) the mechanisms of formation and destruction of this species; (c) the galactic kinematics (overall velocity eld, and di usion). On the assumption that the initial conditions are essentially the same for the class of galaxies under consideration, and under the proviso that we concentrate on small enough zones of these galaxies (over which the kinematic stretching e ects are ine cient), we are entitled to expect that information on the transition probabilities Pk can be extracted from conveniently chosen parameters describing the spatial distribution of the species. We have attempted to quantify the following single aspect of the space pattern traced out by the species P , M , R and I. It is a well known fact that in CA models individual states are often uniformly distributed over self{similar, or statistically self{similar sets (cf. Perdang 1993 Sect. 4 for an analytic discussion of several examples) . Therefore, we have addressed the somewhat broader question of whether a mass dimension of the set covered by species Z (= P; M; R; I) does exist; if so, this fractal dimension can serve the purpose of a constraint for the transition probabilities Pk. To this end we test whether a scaling relation of the form N (Z; i; j; G(r)) = C rf(Z); for r small; (4:2) is applicable to our experiments; here N (Z; i; j; G(r)) is the number of cells in state Z in a disc G(r) centred at an arbitrary cell (i; j) of the auxiliary lattice L, or of the physical lattice L0; the mass dimension f (Z) ( 2 [0; 2]) exists provided that the interpolation coe cients C and f (Z) of the experimental data are independent of the reference cell (i; j). In the standard numerical method of estimating f (Z) in our CA experiments di erences in lattice L and L0 may arise as a consequence of the stretching and accompanying loss of space resolution in L0 (cf. our previous remarks). As already argued, under the loss of accuracy due to a large enough stretching in one direction a numerical estimate of f (Z) in L0 will be arti cially reduced as compared to the estimate in L.7 7Test experiments have con rmed this point. For the model corresponding to our synthetic photographs we have in L0: f (P ) =1.60; f (M ) = 0.87; f (R) = 0.36 and f (I) = 0.10 at step 200 (compare with Table 4); the large di erences in the case of the M and R species are due to an additional inaccuray related to the small population of these objects. We have computed Table 4 exhibits the conclusion of our test. It shows that numerical values of f do indeed exist (plots of ln N against ln r indicate a broad linear interval, and the slope is independent of the central cell). The f values supplied in the Table are all computed in the auxiliary lattice L in order to avoid the loss of local accuracy accompanying the stretching in L0. We believe that this procedure provides the closest theoretical approximation to an observational mass dimension as given in the frame of the observer, provided only that the observational estimate is made over a small part of a high resolution photograph (with G(r) larger than the actual resolution). The mass dimension for the di erent species does show a signi cant dependence on the model parameters. In the quasi{stationary phase (ii) this dimension is practically time independent. Again, the apparently exceptional behaviour of the model on which our synthetic photographs are based, which exhibits a noticeable increase in f between timesteps 200 and 500, is due to the fact that the onset of quasi{stationarity is delayed. We approximate the mass dimensions over phase (ii) by the linear ansatz f (Z; ) ' fo(Z) = for(Z) + (4:3) (Z = P; M; R; I). The reference values for(Z) correspond to the model whose parameters are listed in Col. b of Table 1; the collection of expansion coe cients k(Z) supplied in Table 5 have been obtained from 15 calibration models whose parameters Pk scatter around the reference model. The counterpart of Eq. (3.3) becomes here X(Pk − Pkr) k(Z); k X k k(Z) Pk = f (Z)obs − for(Z) + X k(Z)Pkr; k (4:4) also the correlation dimension (cf. Perdang 1991) of the distribution of the di erent species; we nd that as a rule, the latter is smaller than the mass dimension, obeying approximately the relation f (Z)−0.1. Since in the correlation dimension we sum over all pairs of identical states, this measure includes here the e ect of the input inhomogeneity in the matter distribution (presence of empty bubbles); the mass dimension is estimated at positions where matter is actually present. In our context both measures refer to di erent aspects of the distributions. (Z = P; M; R; I). Hence, given the observational estimates of the current fractal dimensions of the distribution of the di erent species P; M; R; I in a given galaxy (assumed to be in the stable phase ii), f (Z)obs, relations (4.4) provide 4 additional linear constraining equations for the model parameters Pk.8 5. Estimate of the free parameters Our analysis has isolated 9 observational constraints in the form of algebraic equations (Eqs. 3.3 and 4.4) to be satis ed by our 15 a priori free model parameters Pk. If 6 additional constraints were available, the proposed CA formulation would have the status of a strictly phenomenological or empirical model, analogous to a Fourier representation for the simulation, the interpolation and the extrapolation of a time series. On the theoretical side, since a subset of the model parameters are recoverable from stellar evolution theory, our CA model could serve the purpose of testing current stellar evolution theory. With 6 constraints being missing, we regard the latter subset, namely the transition probabilities of isolated objects, (P [P ; M ] = P4; P [M ; R] = P8; P [R; I] = P13; P [L; I] = P15), as being given. The timescales of these processes flow from evolution theory and are converted into transition probabilities provided only that the stellar mass function be known. If we consider timescales less than a few Gy, then the decay probabilities of the low mass stars become negligible, and we may set further P [L; D] = P14 0 (together with P [L; I] = P15 0). Moreover, as already pointed out, the spontaneous star formation rate, P [D; P ] = P1, has only a minor influence on the abundance of the protostar species (except for the delay e ect mentioned). In all cases the star{induced star formation rate is the dominant mechanism. We believe therefore that we can reasonably x the speci c parameter P1. The numerical value 8We have pointed out above that the critical exponents referring to the percolation phase transitions are universal and therefore they do not carry any information on the physics. Since critical exponents are fractal dimensions in disguise, the question arises whether conversely the fractal dimensions attached with the geometric sets of the di erent species are not intrinsically universal themselves. In fact, as observed elsewhere (Perdang 1993) , in analytically solvable CA models over square lattices only speci c dimensionalities turn up for the carriers of individual states. It is not impossible then, that the parameter and time dependences we observe in the numerically determined fractal dimensions of the galactic simulation merely reflect the fact that a true fractal structure is not yet achieved. However, even if it were not a genuine fractal dimension, f (Z) then would remain a quantitative index characterising the space distribution of species Z. A valid model for a given galaxy must duplicate the observational value of this index. We believe, therefore, that even under those circumstances the observational estimates of the exponents f (Z) continue to carry information about the free parameters of our model. D + dD ! P + dD D + dD ! L + dD D + pP + mM + rR ! P + pP + mM + rR P ! R L not not ! parametrised ! parametrised I I we adopt is the value of the reference model (Col. (b) of Table 1). Within this framework of approximations we are then left with the following list of 9 parameters (P2; P3; P5; P6; P7; P9; P10; P11; P12): (5:1) The constraints we have discussed in the previous sections enable us then to derive these parameters directly from observation, namely (i) from relative mass fractions of the galactic components, and (ii) from fractal dimensions of the space distributions of the components. for(P ) for(R) 1(P ) 1(M) 1(R) 1(I) 3(P ) 3(M) 3(R) 3(I) 5(P ) 5(M) 5(R) 5(I) 7(P ) 7(M) 7(R) 7(I) 9(P ) 9(M) 9(R) 9(I) 11(P ) 11(M) 11(R) 11(I) 6. Conclusion and outlook We believe that our experiments provide arguments that the galactic arm structure results from a combination of two physical ingredients which have already been isolated separately in models of class (2) (cf. the Heisenberg{ Weizs¨acker simulation, 1948) , and in the simplest models of class (3) (3{state CA experiments of MA and GSS). The two ingredients, rather than being mechanisms associated with what seemed to be regarded as rival theories, appear here not only as complementary, but as simultaneously necessary for the development of a realistic structure of a luminous arm pattern: On the one hand, the overall large scale pattern of the matter distribution | the Grand Design | is determined by the gravitation induced general flow eld (the di erential rotation in a lowest order approximation). This structure is derivable from classical N {body experiments. On the other hand, the brightness distribution, as directly accessible by photographs, reflecting the distribution of the bright components, is the immediate outcome of star formation and evolution processes. Since the latter processes occur in regions of high matter density, the high density regions are the carriers of the brightness distribution. The visible arm pattern, while then globally shaped by the matter distribution, possesses a structure of its own conditioned by the kinetics of star formation and evolution, which is superposed on the overall density distribution. The small scale structure of the brightness distribution is chiefly determined by this `chemical kinetics'. We believe that this is the main conclusion of our analysis. It is borne out by a comparison of our synthetic photographs (Fig. 4a) with the pictures of real galaxies (HST photograph of NGC 4321, Fig. 4c, or enhanced AG photograph of NGC 5457, Fig. 1 of Paper I). We are well aware that our treatment of the galactic dynamics in terms of a stationary velocity eld in a rotating reference frame is not fully satisfying. It ignores not only a feedback of stellar evolution on the dynamics, but above all it discards the alternative of a chaotic flow in the region under investigation. It is indeed obvious that chaos in the dynamics must a ect both qualitatively and quantitatively the large scale and the small scale pattern of the arms. Dynamical chaos is expected to increase the fragmented character of the visible arm structure, and presumably could become measurable via fractal dimensions of the arms. The ideal model of galactic structure should therefore not divorce the dynamics from star formation and evolution. In principle, to the extent that gravitation can be treated as an external force (as it is essentially done in standard self{consistent dynamic models; cf. for instance the model by Patsis et al. 1991) , the gravitational e ects could be incorporated in the CA model in a formally straighforward way. Unfortunately, such a procedure is highly time consuming. The second quantitatively relevant aspect of our model is that it demonstrates that global observational galactic data contain easily accessible information on stellar evolution rates (including mass loss). We nd that the 9 model parameters (5.1) are determined by a system of 9 equations which can be approximated by linear equations (Eqs. 3.3, 4.4) whose coe cients are observable. Four among the latter are fractal dimensions associated with the spatial distribution of the galactic species; while not yet available, there should be no major di culty in deriving them from surveys of nearby galaxies. We hope to address this problem in the future. The remaining parameters are mass fractions of the di erent galactic components which can be found in the literature. Tables 2 and 3 provide a list of tentative input coe cients to carry out the inversion problem. It would be useful to recalibrate these coe cients, using a more extended array of test models covering also wider ranges of transition probabilities. Finally, we should keep in mind that the arti cial construction of the initial state as incorporated in our program should be replaced by a simulation of a physically more realistic process of molecular cloud formation (aggregation model, for instance). Detailed physics of star formation may be implemented as well, such as a simulation of a hierarchical fragmentation process of the cloud complex. Standard hydrodynamic simulations indicate in fact that the isothermal collapse of rotating dense clouds may lead to a succession of levels of condensations in condensations (Boss 1991) . Such a mechanism could enhance local irregularities in the arm structure, thereby leading to a higher dimension F of the arm pattern. We plan to address these questions in a later work. Acknowledgements. The authors are indebted to G. 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A. Lejeune, J. Perdang. Cellular Automaton experiments on local galactic structure. II. Numerical simulations, Astronomy and Astrophysics Supplement Series, 249-263, DOI: 10.1051/aas:1996242