THE DRIPPING HANDRAIL: AN ASTROPHYSICAL ACCRETION MODEL

Transcription

1 June 4, 26 Preprint typeset using L A TEX style emulateapj v. 2/4/5 THE DRIPPING HANDRAIL: AN ASTROPHYSICAL ACCRETION MODEL A. Dey, M. Low, E. Rensi, E. Tan, J. Thorsen, M. Vartanian, and W. Wu San José State University June 4, 26 ABSTRACT The Dripping Handrail is an abstract mathematical model that describes several features of the chaotically fluctuating x-rays emitted by neutron stars in astronomy. In order to better understand certain observations that have more recently become available, we have extended the model by adding to it the effect of physical radiation forces. Comparison of simulated and actual data indicates that this approach may have promise. In addition, we have carried out a theoretical study of the extended model and have developed a variety of visualization and analysis tools in a search for a Dripping Handrail signature in astronomical objects of a more general nature. Subject headings: accretion disk dripping handrail chaos return map. INTRODUCTION Neutron stars orbiting stellar companions often emit intense, chaotically fluctuating x-rays. Although the x- rays are assumed to originate from the hot inner edge of an accretion disk, because standard hydrodynamic models break down under the extreme physical conditions present, the exact nature of the x-ray emitting regions remains largely unknown. In face of these difficulties, Scargle et al. (993) and Young and Scargle (996) (hereafter, YS) proposed an abstract mathematical model, called the Dripping Handrail (DHR), as a phenomenological description of the accreting system. The DHR pictures the chaotic x-rays as being produced by drops of matter falling onto the neutron star under its intense gravitational field. These authors found that the DHR is able to account for both the chaotic behavior as well as certain low frequency quasi-periodic oscillations (LFQPO) seen in the power spectra obtained by telescopes of that era. In their conclusion, YS stated: [B]y adding more physics to the DHR, we hope to develop a model that is less artificial and able to be compared in more detail to existing observations and those that will be obtained by the new generation of x-ray telescopes but that retains its straightforward phenomenological interpretation. Since that time, more sensitive astronomical observations have become available. These have revealed new and unexpected structure in the power spectra of these objects; in particular, high-frequency quasi-periodic oscillations (HFQPO) are now known to be characteristic (van der Klis 997, 25). Although the HFQPO provide a wealth of information about the x-ray emitting regions, understanding their physical origin is a challenging problem. The present study is a first step in the direction indicated by YS. As a source of more physics, we have followed the work by Miller and Lamb entitled Effect of Radiation Forces on Disk Accretion By Weakly Magnetic Neutron Stars (Miller and Lamb 993) (hereafter, ML). Since these authors concluded that such effects are likely to be very important, we decided to investigate the possibility of extending the DHR by incorporating these effects and to see whether this might shed light on the HFQPO phenomenon. The DHR is an example of a discrete dynamical system. Dynamical systems are systems that evolve with time, which can be regarded as discrete, as in our case, or continuous, as in the case of continuous dynamical systems or flows. More specifically, a discrete dynamical system is a map f : M M, where M is the phase space, describing all possible states of the system. Usually M is a metric space or a smooth manifold. An orbit of a point x M is the set of iterates {f k (x) : k =,, 2,...} of x. The orbit of x represents all future states of the system whose initial state is x. Given a dynamical system f, the goal is to describe the global orbit structure of f. That is, what happens in the long run for most points x? In this paper, our goal is two-fold: first, starting with the DHR and based on existing astronomical observations, to find a more suitable dynamical system model for the astrophysical phenomenon of interest, and second, to understand the asymptotic behavior of orbits of that system. The outline of the paper is as follows: Section 2 presents the DHR model and the physical basis for extending it. Section 3 contains a theoretical analysis of the extended model. Section 4 compares simulated power spectra of the DHR model with recent astronomical observations. Section 5 presents other visualization methods for extracting information from the model. Section 6 presents a return map study. In Section 7, we collect our conclusions and discuss directions for further research. 2. PHYSICAL BASIS FOR THE EXTENDED MODEL The DHR postulates a standing circular structure, the so-called handrail, at the inner edge of the accretion disk. See Figure (a). This structure is supported by a balance between inward gravitational and outward radiation forces. Accretion of matter onto the rail from greater radii occurs at a constant rate at all points, and diffusion of matter along the rail tends to even out lateral density gradients. When the density at some point along the rail reaches a critical value, an unspecified in- Only a brief description of the DHR model is presented here; the reader is referred to YS for a complete account.

2 2 stability causes the material to drip onto the central star. The instantaneous x-ray intensity is assumed proportional to the total mass on the rail or, alternatively, to the total mass in drops. YS showed that starting from a random density profile along the rail, these factors in combination and in certain parameter regimes are able to produce a chaotically fluctuating x-ray intensity. They further speculated that although the chaotic behavior generated in this manner was only transient, long term, asymptotic chaos would occur if the accretion rate were assumed to slowly vary. 2.. Effect of Radiation Forces on Accretion ML present the following sketch of the inner edge of the accretion disk [here, ω and ω E denote the accretion rate and the maximum ( Eddington ) accretion rate, respectively]: For accretion rates ω.ω E, the flow near the star is likely to be optically thick. In this case radiation forces act primarily on accreting gas within one mean free path of the inner edge and surfaces of the disk. However, the increase in radial velocity of the gas in these layers caused by rapid removal of angular momentum by radiation drag reduces the density, allowing radiation from the star and boundary layer to penetrate further into the disk. This is likely to produce a more widespread change in the velocity structure of the accretion flow... Although a quantitative determination of the radiation field and the accretion flow requires a self-consistent computation of the radiation and flow... the arguments just given show that radiation forces are likely to be very important. We view the DHR as a kind of abstract scaffolding upon which we can hang physical effects without having to consider, in the first instance at least, questions of consistency. Since the effects of radiation forces occur within one mean free path of the optically thick inner edge of the accretion disk, we assume the DHR is describing processes in this region. Thus, if we assume the standard disk model and that the inner edge of the disk is 2 km from the neutron star center, the DHR is, to rough order of magnitude, about one km thick in its radial dimension (Shapiro and Teukolsky 983). This is consistent with the thin rail approximation adopted by YS Outward Force Opposing Accretion We assume that the radiation originates from near the surface of the neutron star. The dominant component of the flux acting on the rail is therefore in the outward radial direction (see Figure (b)). According to ML, there are two radial forces: a static force acting on particles near rest, and a drag force acting on moving particles. Since the accretion velocity is much less than the speed of light, the latter can be ignored. Furthermore, although the static force results from Thomson scattering of radiation by free electrons bound electrostatically to the ions (Frank et al. 99), we do not attempt to incorporate such detailed physics into the DHR. Instead, we simply try to account for any gross effects that these forces might exert on the DHR dynamics. In particular, it is assumed that the outward radiation force acts to modulate the accretion onto the outer edge of the rail. Since the DHR is the site where radiation penetrates more or less depending the density, the force opposing accretion will therefore decrease, exponentially, as the density increases. This approximates the physical picture presented by ML: [reduced] density... allows radiation to penetrate further into the disk, [and] produces a more widespread change in the velocity of the accretion flow. From this, it is clear that the accretion rate should increase with increasing density. In the first approximation, therefore, accretion is taken to be an increasing linear function of density. As a result, the rate ω, originally constant in both space and time, is now dependent on both variables through its dependence on the density; we term this dynamical accretion. Labeling time by n and spatial cell by i, we therefore have ω i n = αρ i n + ω where α, ω is the (constant) accretion rate at zero density and ρ i n is the density in cell i at time n (see Section 3). Setting α = reduces to the original model. The above expression for ω = ω i n may then be substituted into equation 6 of YS, the map lattice implementation of the dynamical system. A theoretical analysis of this extended system (hereafter, edhr ) is given Section 3 below Drag Force and Asymmetrical Diffusion According to ML, the orbital (i.e., azimuthal) velocity of the inner edge of the accretion disk is a significant fraction (.2.5) of the speed of light. On this basis, they concluded that radiation drag opposing the orbital motion has a very important effect on the accretion dynamics. We have therefore considered how to include an azimuthal force in the DHR. The original model assumes that the diffusion of material occurs at equal rates (proportional to density differences) in the clockwise (CW) and in the counterclockwise (CCW) directions. As a rough approximation, we have therefore modeled the effect of the drag force by assuming the diffusion rates differ in these two directions. Although there may be other ways to include this effect, this seems the simplest Drag Force and the Dripping Phenomenon ML were primarily interested in the fact that the drag force, by removing angular momentum from the orbiting matter, allows it to spiral inwards towards the central star. It is therefore natural to consider whether such a force could somehow produce the dripping instability assumed in the DHR picture. It is easily seen that an instability might indeed result from the fact that the accretion rate in the extended model increases with density. However, without a solution of the coupled equations governing the radiation field and the density, it is not clear how the drag force might be involved in this. For this reason, the original modulo mechanism for the dripping phenomenon has been retained for the present (see YS).

3 3 (a) (b) Fig.. (a) Diagram for the original model. (b) Diagram for the extended model. 3. THEORY As remarked above, given a dynamical system f : M M, the main goal is to understand the long-term behavior of its orbits. Since this behavior may in general be very complicated, one first focuses on the simplest types of orbits, namely fixed and periodic ones. A point p M is called a fixed point if f(p) = p. It is called periodic if there exists an integer k > such that f k (p) = p. (In this report, we will only be concerned with fixed points leaving the study of periodic points for future work.) If p is a fixed point of f, it is natural to ask: what can be said about the asymptotic behavior of nearby points? If f is a differentiable dynamical system and its derivative A = Df(p) has no eigenvalues of absolute value one (i.e., p is a hyperbolic fixed point), then the Hartman- Grobman theorem (Palis and de Melo 982) states that for all points in some neighborhood U of p, the behavior of f is qualitatively the same as the behavior of the linear dynamical system A (more precisely, f and A are topologically conjugate; see Palis and de Melo 982). So understanding the dynamics of A helps us understand the dynamics of f near p. Let E s denote the sum of the eigenspaces of A corresponding to eigenvalues λ with λ < and let E u denote the sum of the eigenspaces corresponding to eigenvalues with λ >. It is not hard to see that if v E s, then A k v, as k, whereas if w E u, then (assuming A is invertible) A k w, as k. A natural question is: are there directions in the phase space M along which f exhibits similar behavior? If p is a hyperbolic fixed point, then the Stable Manifold Theorem (Palis and de Melo 982) guarantees the existence of the stable manifold W s (p) and unstable manifold W u (p) such that: (a) for every x W s (p), f k (x) p, as k ; (b) for every x W u (p), f k (x) p, as k (assuming f is invertible); (c) W s (p) and W u (p) are smooth manifolds tangent at p to the linear spaces E s, E u, respectively. In this section we extend the original DHR model of Young and Scargle based on the physical observations from Section 2 and study its dynamical behavior along the lines described above. 3.. Theoretical Introduction to the Extended Dripping Handrail As in the original Dripping Handrail model proposed by Young and Scargle, the extended system treats time and space as discrete variables. For example, we consider time steps (n =, 2, 3...) and divide the space of the inner edge of the accretion disk into discrete cells. The only continuous variable is the density of each cell. Keeping with the notation established in the Young and Scargle paper, we represent the index of the cells on the DHR by i ranging from to N, the number of cells in the handrail. Time is quantized and denoted by the index n. The continuous density variable in cell i at time n is given by ρ i n. Whereas the original model opted for a constant accretion rate, the extended model adds a parameter for time-varying accretion. Instead in the extended DHR (edhr) model, the initial accretion rate is denoted ω and the local accretion rate at a given time n and at cell i is represented as ωn i = αρ i n + ω where α is the constant of dynamic accretion, the new addition to the DHR. In addition to matter accreting into cells from the star, matter also diffuses from one cell to its two neighboring cells and this is represented by Γ, the diffusion parameter. To further condense the notation, we encapsulate the density values at each cell at time n into an N column vector: ρ n ρ 2 ṇ X n =.. ρ N n ρ N n

4 4 Then we can define the discrete dynamical system in terms of a map f : H N H N on the unit hypercube H N = {(x,..., x N ) R N : x i < } such that X n+ = f(x n ) where the map f is defined thus f(x) = AX + b (mod ), δ Γ Γ Γ δ Γ Γ δ A =......,.. δ Γ Γ Γ δ b = ω.., δ = 2Γ + α. The matrix A of size N N is a slight modification of the matrix in Appendix B of the Young and Scargle paper. The addition of α is to account for the time-varying accretion rate. The modulo operation in the map represents the dripping of matter off the rail when a threshold is reached. Therefore, as in the original model, the threshold dictates that ρ i n <. Hence the map always returns to the unit hypercube, H N. Note that the map X AX + b does not necessarily preserve the integer lattice Z N (or any other lattice) in R N, so f cannot be considered as a map on the N- dimensional torus R N /Z N Fixed Points It is not hard to see that the original DHR map (with α = ) does not have any fixed points. Recall that X is a fixed point of f if f(x) = X, i.e., if there exists a vector l Z N with integer coordinates such that AX + b = X + l. Since A I is invertible (see subsection 3.3), we obtain X = (A I) (l b). Since X must belong to the unit hypercube H N, we conclude that every l Z N such that (A I) (l b) H N gives a fixed point of f. We have the following particular solution to this equation. Proposition. If there exists an integer m such that < (m ω )/α <, then X = m ω α.. is a fixed point of f. Proof. Suppose that < (m ω )/α <. Then X lies in the unit hypercube H N. We have to show that AX + b = X + V, where V is a vector with integer coordinates. Observe that v = [,..., ] T is an eigenvector of A corresponding to the eigenvalue + α. Since X is a scalar multiple of v, we have AX = ( + α)x. Therefore, AX + b = ( + α) m ω v + ω v [ α = ( + α) m ω ] + ω v α = m ω v + mv α = X + mv. Since mv is an integer vector, it follows that f(x ) = X Eigenvalue Analysis of the edhr To understand the dynamics of the edhr near its fixed point X, we compute the eigenvalues of its linearization A = Df(X ). (Note that Df(X) = A for every point X in the interior of H N.) In the original model, the eigenvalues of the matrix J (which lacked the α term for time-varying accretion) were computed to be ( ( )) 2πi µ i = 2Γ cos, N i =,,..., N 2 Using this eigenvalue formula for the original matrix, we arrive at the following result for the eigenvalues of the edhr matrix. Theorem (Eigenvalues and Eigenvectors of A). If µ is an eigenvalue of J, then λ = µ + α is an eigenvalue of A. They have the same associated eigenspaces.. Proof. We can write the matrix A in terms of J as such A = J + αi, where I is the N N identity matrix. Consider an arbitrary eigenvalue µ and corresponding eigenvector v of J. By definition of eigenvalues, we have Therefore, Jv = µv. Av = (J + αi)v = Jv + αiv = µv + αv = (µ + α)v. Thus v is also an eigenvector of A and has corresponding eigenvalue λ = µ + α. Since the edhr is a discrete dynamical system, we are further interested in classifying the eigenvalues of A as having magnitude greater than, less than or equal to. Using the above theorem, this can be easily done. An interesting consequence of introducing non-constant accretion rate is the guaranteed existence of a specific eigenvalue, namely λ = + α with corresponding eigenvector v, since Au = ( + α)u

5 5 regardless of the number of cells in the dripping handrail. In the original model where α =, this eigenvalue is simply, whereas in the extended model with α >, this eigenvalue is strictly greater than. Therefore, if α is positive but small, the eigenvalues of A split into two groups: one eigenvalue ( + α) strictly greater than one with eigenspace E u spanned by v, and all the remaining eigenvalues in the interval (, ). By the Stable Manifold Theorem mentioned above, there exists a -dimensional unstable curve W u (X ) and an (N )- dimensional stable surface W s (X ). It is not hard to verify that W u (X ) is not only tangent to E u at X, but is in fact an arc in E u. We do not have such a description for W s (X ). Although we do not have enough rigorous evidence to conclude that the edhr system is chaotic (in the standard mathematical sense of the word), the presence of expansion due to an eigenvalue of A greater than one (in other words, a positive Lyapunov exponent log( + α) of f) is an indicator that f may well exhibit very complicated long-term behavior. One way of rigorously proving that f is indeed chaotic would be to find a hyperbolic periodic point p for f whose stable and unstable manifold intersect transversely; cf., Palis and de Melo (982). We leave these investigations for future work. 4. COMPARISON WITH OBSERVATIONS We have developed Matlab c code that simulates the Poincaré map lattice implementation of the extended model (see YS). 2 When the dynamical accretion parameter α is set to zero and the CW and CCW diffusion constants are set equal, the code reproduces the results shown in YS, Figure. As a baseline, we adopted the parameter value θ = given by YS as a typical chaotic setting. Starting with random initial densities assigned to 37 spatial cells arranged around the circular rail, the code generates a sequence of 6,384 time steps whose values are the total mass on the rail at each step. For purposes of interpretation, these values are then assumed proportional to the x-ray intensity. The power spectrum of the sequence was then calculated in the standard way in order to compare with published power spectra Comparison Using Original Model We first compared simulations of the original model with observations of Scorpius X-, a typical neutron star source, made with a more sensitive x-ray telescope than that available to Scargle and others when they first studied this object (Scargle et al. 993). 4 For this preliminary study, we simply made side-by-side comparisons between the simulated and the published power spectra (see Figure 2). 2 See file extended dhr.m and others, which are included in the accompanying software. 3 The power spectrum is the modulus squared of the fourier transform of the time sequence; intuitively, the spectrum is obtained by analysing the sequence into its various sinusoidal components and then plotting the amplitude squared of each component versus its frequency. See for example Press et al. (986). 4 The data shown in Figure 2(d) was obtained with NASA s RXTE satellite (van der Klis 997); shown is the power spectra of Scorpius X- obtained at 7 different times over a 4 day period. The plots have been displaced vertically for clarity; their absolute normalizations are approximately equal. First, it can immediately be seen that the DHR does indeed display what appears to be chaotic behavior; this is evident from the fact that all frequency components in the power spectrum have significant amplitudes. Secondly, there is a marked correspondence between the spectra in the broad peak appearing at low frequencies (in particular, compare the second plot from the top in Figure 2(d)). This is the low frequency quasi-periodic oscillation (LFQPO) that Scargle and others were able to reproduce (Scargle et al. 993). 5 Thirdly, the simulated spectrum falls off as a power law (i.e., linear on a log-log plot) at high frequencies, whereas the published spectra remain flat. For this reason, we have drawn in a flat (white noise) component on the simulated spectrum to provide a better fit as shown in Figure 2(a). Presumably, the origin of this noise is extrinsic to the DHR. Lastly, certain significant peaks in the observed data at high frequencies are seen that do not appear in the original model. These are the high frequency quasi-periodic oscillations (HFQPO) which are currently of great interest to astronomers Comparison Using Extended Model We repeated the comparison using the edhr model described in Section 2. Starting from the baseline model θ = , we first increased the dynamic accretion parameter α to non-zero values while the CW and CCW diffusion constants were kept equal. 7 Figure 2(b) shows a typical simulated spectrum corresponding to α =.4, with white noise added. Two main effects occur as α is increased. First, the relative amplitude of the LFQPO peak decreases, and, secondly, a series of equally-spaced peaks on the high frequency side of the peak appears. Except for the fact that the HFQPO in the actual data are not equally-spaced, this trend is in rough qualitative agreement with the trend seen in Figure 2(d) as we pass from the second to the seventh plot down from the top. Next, we increased the splitting between CW and CCW diffusion rates from zero, and adjusted the α parameter to obtain a rough best-fit with the data. A typical simulated spectrum with % splitting and α =. is shown in Figure 2(c). The general effect of asymmetric diffusion, even when α =, is again to produce a series of high frequency peaks. These are generally sharper than the peaks seen without splitting and are not as equallyspaced. There is a rough qualitative similarity between the simulated spectrum and the observed spectra of Figure 2(d) Transient Chaos Study 5 It is worth noting that the LFQPO correspondence here is much more striking than that appearing in Scargle et al. (993); this is because the spectrum shown in Scargle et al. (993) is similar to the fourth spectrum from the top in Figure 2(d), where the LFQPO peak happens to be much less pronounced. 6 Although small in amplitude, the HFQPO are in fact detected with very high statistical precision; furthermore, they always appear as twin peaks whose frequencies vary with the overall spectrum. The HFQPO were first discovered in observations using the RXTE satellite after the original DHR was developed. 7 In this process, the ω constant was fixed at the value of the baseline model.

6 Log Power Density (arb.units) (a) Log Power Density (arb.units) (b) Log Power Density (arb.units) (d) (c) Fig. 2. (a) Power spectrum of the original model (see text). A white noise component has been drawn in as a dashed line. (b) Power spectrum of the extended model with α =.7,diffusion splitting =, and white noise added. (c) Power spectrum of the extended model with α =., diffusion splitting =., and white noise added. (d) Published power spectrum of Scorpius X- (van der Klis 997).

7 7 The theoretical analysis in Section 3 showed that when the α parameter is greater than zero, f acquires a positive Lyapunov exponent. This indicated that one effect of including radiation forces may be to produce permanent, asymptotic chaos without the need to assume long-term drifts in the accretion rate. In order to verify this prediction, we therefore undertook to study the way in which the power spectrum evolves over long periods of time. In Figure 3(a) is shown the spectrum of the original model formed using the first 6,384 time steps; in Figure 3(b) is shown the spectrum formed using the last 6,384 steps after having let the simulation run for a total time of 5 6, 384 steps. Confirming the results of YS, one sees that the originally chaotic spectrum (i.e., one in which all frequencies have significant amplitudes) has become periodic and non-chaotic (i.e., almost all of the power is in sharp, equally-spaced harmonics) after 5 time blocks. In Figures 3(c) and (d) are shown corresponding spectra for the extended model with α =.4 (with symmetrical diffusion). In this case, no significant change in the chaotic spectrum is observed after 5 time blocks, in agreement with the theoretical predictions VISUALIZATION WITH COMPUTER SIMULATIONS Two animated visualizations of the DHR were developed in order to gain intuitive understanding of the model s real-time behavior. The driving motivation behind creating such visualizations is to present the model as a directly observable entity, thus making it possible to see features of its behavior in action. Observable patterns in the behavior of the model over a relatively short duration sometimes suggest the emergent behavior we seek. A snapshot of the ring visualization is included as figure 4(a). In the ring visual, colors represent each cell s state. They range from black (ρ = ) to red, orange, yellow, then white (ρ = ). One of our goals was to possibly reproduce quasi-periodic oscillations, in particular the twin peak oscillation from van der Klis (997). One way to accomplish that would be to coax our model into producing two roaming hot spots, rotating at different frequencies. In addition, the process would have to occur unpredictably, and completely periodic behavior would be undesirable. Another possible source of QPOs could be a pulsing of several cells in semi-unison at two different frequencies. Both behaviors were reproduced visually, but no quantitative analysis of the model at those settings was done. A second method of visualization was provided as a bar graph. Here, the density of each cell is represented as the height (or depth) of a bar. When animated, the model has the appearance of a flowing, dripping fluid, especially when inverted as in figure 4(b). A feature of this visual is that we can see the real-time dripping action of the model. Setting the diffusion parameter at a high value relative to accretion produced the most visually fluid-like behavior, suggesting that those settings are more realistic if we are to assume that we are modeling a dripping fluid. Unfortunately, since visual experiments were done independently of the quantitative analysis presented in this 8 Note: in comparing the various spectra, it is necessary to distinguish between the quasi-periodic peaks seen in Figures 3(c) and (d) and the purely periodic peaks seen in Figure 3(b). document, the data we acquired from them was mostly superficial. If further research is to be done with this model, we suggest a more integrated approach of the two methods; perhaps producing desired behavior visually first, then doing more in-depth analysis to produce data which can be compared with that from an astrophysical source. 6. RETURN MAP In the search for ways to characterize the behavior of the DHR model, the return map was brought to our attention. We would like to emphasize that the phrase return map is something of a misnomer. Nothing is returned in any way we could reasonably interpret, nor do any of the figures in this section depict a map (i.e., a well-defined function). But we suspect that from the figures, one can extract useful information about the DHR model (e.g., what happens to the mass in a cell in the future as a function of what we know about it now). A return map compares the mass in a particular cell at a future time to the mass in the same cell at the present time. More specifically, the return map is a plot of x n+k versus x n, where x n+k and x n are the masses in the cell at times n + k and n respectively. Our goal here is to study the return map for various values of n and k and then try to draw conclusions about the DHR model based on the graphs generated. Currently we cannot analyze the map well enough to draw any definite conclusions, so most of what we present here will be conjectures about the behavior of the model. We interpreted the description of the return map in two different ways, giving rise to two precise definitions and therefore two investigations into the behavior of the DHR model. 6.. First Approach In the first approach, both n and k are fixed. Many simulations of the accretion disk are run with random initial conditions, and from each simulation we extract a single point (x n, x n+k ). One advantage to this approach is that the order in which the points are generated is irrelevant; all that matters are the areas on which points tend to fall on (or avoid) the graph. On the other hand, one disadvantage would be the amount of computation involved. To generate sufficient number of points, say,, one would have to simulate the accretion disk, times, which requires a nontrivial amount of time. Figure 5 shows several graphs generated using rm.m, a Matlab function which implements the return map (the source code for rm.m is included in the accompanying software). Below each graph is the function call used to generate it. For each run of the return map in the first approach, we used a symmetrical diffusion scheme. Unless otherwise specified, the parameter values are those in Young and Scargle (996). More specifically, n = 37 (the number of cells), θ = (the angle), Γ =.5 sin θ (the diffusion parameter), and ω =.5 cos θ and α = (the accretion parameters). Figure 5(a) is not surprising. Little time passes when k =, and so we do not expect the mass in any particular cell to change by very much. Consequently, the dots should fall close to the line y = x (also shown in the figure).

8 8 Log Power (arb.units) 5 Log Power (arb.units) (a) 2 3 (b) Log Power (arb.units) Log Power (arb.units) (c) 2 3 (d) Fig. 3. Transient chaos figures. See text for descriptions. (a) (b) Fig. 4. Snapshots of the animated visualizations of the edhr model.

9 9 xn+k versus xn for n = and k = xn+k versus xn for n = and k = x3 x x x6 x x.6.8 (d) rm(,9,5) xn+k versus xn for n = and k = 39 xn+k versus xn for n = and k = x x4.4 x (c) rm(,5,2).4.2 xn+k versus xn for n = and k = (b) rm(,2,) xn+k versus xn for n = and k = 5.6 x (a) rm(,,) x (e) rm(,39,7) x (f) rm(,,8) and ω =.55 Fig. 5. Plots of the return map for various values of n and k. Each has ω =.5 cos unless otherwise specified. (a), (b), (c) As k increases, the line becomes wider and tilts slightly. (d) Lines form in the lower right. (e) Lines become more pronounced and span the entire width of the graph. (f) If k is increased further and ω increased to.55, then the banding pattern repeats itself.

10 Figures 5(b) and (c) show graphs for when k increases slightly to 2 and 5 respectively. The line noticeably widens and tilts. Each time step thereafter, the line continues to widen and tilt until about k = 9, when a banding pattern begins to appear in the lower-right area of the graph. See Figure 5(d). These faint lines naturally suggest that the mass in a cell becomes discretized after enough time has passed (i.e., when k is large enough). Figure 5(e) shows a graph when k = 39. Here the banding pattern becomes much more pronounced and the lines now span the entire width of the graph. It is also interesting to note that the dots tend to avoid a small area on the left side of the graph. If we increase the accretion parameter ω to.55 and look further into the future, e.g., k =, then Figure 5(f) is the result. Figure 5(f) appears to contain Figure 5(e) repeated twice; another interesting phenomenon which suggests periodic behavior. Our experiments with the return map, and in particular the graphs generated in Figure 5, lead us to the following conjectures: Where the dots are more concentrated on the graph is where the mass of the cell is more likely to be located. For example, that the dots are more concentrated along the lines x =.5 and x = in Figure 5(f) would mean that the mass in a particular cell is most likely to be near.5 or at time. After enough time has passed, the mass in a cell becomes discretized, i.e., the mass in a cell can take on only one of finitely many values. One should note that our conjectures are purely of the mathematical model and not of the actual star itself. It would be interesting to examine raw astronomical data in order to determine if the neutron star actually exhibits the behavior predicted in the two preceding bullet points Second Approach In the second approach, n is allowed to vary while k remains fixed. Only one simulation of the accretion disk is performed, and from this simulation, many points are collected. All points which appear on the graph are of the form (x n, x n+k ), where k is a positive integer chosen before the experiment is run. For example, if k =, then the points on the graph have the form (x, x ), (x 5, x 5 ), and so on. For each run of the return map in the second approach, we used a symmetrical diffusion scheme (as opposed to an asymmetrical diffusion scheme). Our parameter values are the same as in the first approach except n = 32 (the number of cells) and we now vary the value of θ in order to see how the accretion/diffusion ratio affects the return map. The purpose of the return map, as mentioned above, is to figure out how the phase changes as time passes. With the same initial condition, plots on the left side of Figure 6 show how the relationship changes between the mass density of one single cell at time n and at 5 time steps later with regard to time; plots on the right side of Figure 6 show how the relationship changes between the mass density of total cells at time n and at time step later with regard to time. From these plots, we can draw some conclusions: For the mass density of a single cell, the mass appears to repeat itself but with a slight shift. For total mass density of all cells on the disk, the graph somewhat resembles a fractal. These conclusions are based on preliminary observations. Further experiments are required. 7. CONCLUSIONS AND FUTURE DIRECTIONS The comparisons in Section 4 show that the extended DHR may provide a unified explanation for most features seen in the power spectra of certain types of accreting neutron star. Most notably, high frequency oscillations(hfqpo) appear to arise naturally when the effects of radiation forces are added. Although detailed agreement cannot be expected in view of the very simplified nature of the present model, our results nevertheless support the conjecture expressed in Scargle et al. (993); Young and Scargle (996) that the observed x-rays derive from a chaotic, dripping-like phenomenon at the inner edge of an accretion disk. Indeed, it seems entirely plausible that this is an accurate physical description of the interplay between two extreme counterbalancing forces, radiation and gravity. Even in terms of the present model, however, we have yet to explore the effect of varying the baseline parameters from θ = , or of assuming x-ray intensity proportional to total mass in drops. Beyond this, a more careful treatment of the effect of radiation forces, in particular the drag force, is clearly desirable; and there are other possibilities too numerous to mention. Moreover, the visualization studies of Section 5 have made clear, among other things, that summing density over all cells at each time step when forming power spectra could severely reduce the sensitivity to QPO phenomena; in addition, the return map studies in Section 6 have shown the need to obtain the raw astronomical time sequence data for any future comparisons. A more general conclusion is that an abstract mathematical model may sometimes serve as a very useful means of bootstrapping one s way to a more realistic physical picture when no (stable) solution of the detailed equations of motion is expected to exist. Indeed, adding physics to a model which can only be changed in certain ways serves to reduce a complicated problem to a much simpler one. In addition, there is often a synergy between theory and model-building, exemplified in this case by the theoretical suggestion that long-term chaos may result if accretion is allowed to vary dynamically within the system. For the future, we plan to further investigate the abstract model, find its periodic points and a possible transverse homoclinic point; this would be a firm indication of the presence of asymptotic chaos in the system. In addition, development of the theory in the fourier domain might provide more direct insight into the QPO phenomenon. 8. ACKNOWLEDGMENTS We thank Jeff Scargle (NASA Ames) and Karl Young (UCSF) for originally suggesting this study and for guidance during its progress; Slobodan Simić (SJSU) for

11 (a) θ =.53999; density of one cell (b) θ =.53999; total density of all cells (c) θ = ; density of one cell (d) θ = ; total density of all cells (e) θ = ; density of one cell (f) θ = ; total density of all cells (g) θ = ; density of one cell (h) θ = ; total density of all cells Fig. 6. Plots of the return map for various values of θ. As time elapses, the color of the data point changes from red at time =, to blue at the end of time. (a), (c), (e), (g) are the collection of data points of (x n, x n+5 ). (b), (d), (f), (h) are the collection of data points of ( P x n, P x n+ ).

12 2 his expertise on dynamical systems; and Timothy Hsu (SJSU) for making this all possible. This research was supported in part by a grant from the Woodward Fund of the San José State University Foundation. J. D. Scargle, T. Steiman-Cameron, and K. Young, Astrophysical Journal 4, L9 (993). K. Young and J. D. Scargle, Astrophysical Journal 468, 67 (996). M. van der Klis, Astrophysical Journal 48, L97 (997). M. van der Klis, Astronomische Nachrichten 326, 798 (25). M. C. Miller and F. K. Lamb, Astrophysical Journal 43, L43 (993). S. L. Shapiro and S. A. Teukolsky, Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects (Wiley, New York, 983). REFERENCES J. Frank, A. R. King, and D. J. Raine, Accretion Power in Astrophysics (Cambridge University Press, 99), 2nd ed. J. Palis and W. de Melo, Geometric theory of dynamical systems (Springer-Verlag, 982). W. H. Press, B. P. Flamery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipies: The Art of Scientific Computing (Cambridge University Press, Cambridge, 986).