June 14, 2006Preprint typeset using LATEX style emulateapj v. 12/14/05

THE DRIPPING HANDRAIL: AN ASTROPHYSICAL ACCRETION MODELA. Dey, M. Low, E. Rensi, E. Tan, J. Thorsen, M. Vartanian, and W. Wu

San Jose State UniversityJune 14, 2006

ABSTRACTThe 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 understandcertain observations that have more recently become available, we have extended the model by addingto it the effect of physical radiation forces. Comparison of simulated and actual data indicates thatthis approach may have promise. In addition, we have carried out a theoretical study of the extendedmodel and have developed a variety of visualization and analysis tools in a search for a DrippingHandrail signature in astronomical objects of a more general nature.Subject headings: accretion disk — dripping handrail — chaos — return map

1. INTRODUCTION

Neutron stars orbiting stellar companions often emitintense, chaotically fluctuating x-rays. Although the x-rays are assumed to originate from the hot inner edge ofan accretion disk, because standard hydrodynamic mod-els break down under the extreme physical conditionspresent, the exact nature of the x-ray emitting regionsremains largely unknown.

In face of these difficulties, Scargle et al. (1993)and Young and Scargle (1996) (hereafter, YS) proposedan abstract mathematical model, called the DrippingHandrail (DHR), as a phenomenological description ofthe accreting system. The DHR pictures the chaoticx-rays as being produced by “drops” of matter fallingonto the neutron star under its intense gravitational field.These authors found that the DHR is able to accountfor both the chaotic behavior as well as certain low fre-quency quasi-periodic oscillations (LFQPO) seen in thepower spectra obtained by telescopes of that era.

In their conclusion, YS stated: “[B]y adding morephysics to the DHR, we hope to develop a model thatis less artificial and able to be compared in more detailto existing observations and those that will be obtainedby the new generation of x-ray telescopes but that retainsits straightforward phenomenological interpretation.”

Since that time, more sensitive astronomical observa-tions have become available. These have revealed newand unexpected structure in the power spectra of theseobjects; in particular, high-frequency quasi-periodic os-cillations (HFQPO) are now known to be characteris-tic (van der Klis 1997, 2005). Although the HFQPOprovide a wealth of information about the x-ray emittingregions, understanding their physical origin is a challeng-ing problem.

The present study is a first step in the direction in-dicated by YS. As a source of “more physics,” we havefollowed the work by Miller and Lamb entitled “Effect ofRadiation Forces on Disk Accretion By Weakly MagneticNeutron Stars” (Miller and Lamb 1993) (hereafter, ML).Since these authors concluded that such effects are “likelyto be very important”, we decided to investigate the pos-sibility of extending the DHR by incorporating these ef-fects and to see whether this might shed light on theHFQPO phenomenon.

The DHR is an example of a discrete dynamical sys-tem. Dynamical systems are systems that evolve withtime, which can be regarded as discrete, as in our case,or continuous, as in the case of continuous dynamicalsystems or flows. More specifically, a discrete dynamicalsystem is a map f : M → M , where M is the phase space,describing all possible states of the system. Usually M isa metric space or a smooth manifold. An orbit of a pointx ∈ M is the set of iterates {fk(x) : k = 0, 1, 2, . . .}of x. The orbit of x represents all future states of thesystem whose initial state is x. Given a dynamical sys-tem f , the goal is to describe the global orbit structureof f . That is, what happens in the long run for mostpoints x? In this paper, our goal is two-fold: first, start-ing with the DHR and based on existing astronomicalobservations, to find a more suitable dynamical systemmodel for the astrophysical phenomenon of interest, andsecond, to understand the asymptotic behavior of orbitsof 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 ex-

tended model.Section 4 compares simulated power spectra of the

DHR model with recent astronomical observations.Section 5 presents other visualization methods for ex-

tracting 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, theso-called handrail, at the inner edge of the accretiondisk.1 See Figure 1(a). This structure is supported bya balance between inward gravitational and outward ra-diation forces. Accretion of matter onto the rail fromgreater radii occurs at a constant rate at all points, anddiffusion of matter along the rail tends to even out lat-eral density gradients. When the density at some pointalong the rail reaches a critical value, an unspecified in-

1 Only a brief description of the DHR model is presented here;the reader is referred to YS for a complete account.

2

stability causes the material to “drip” onto the centralstar. The instantaneous x-ray intensity is assumed pro-portional to the total mass on the rail or, alternatively,to the total mass in drops. YS showed that startingfrom a random density profile along the rail, these fac-tors in combination and in certain parameter regimes areable to produce a chaotically fluctuating x-ray intensity.They further speculated that although the chaotic be-havior generated in this manner was only transient, longterm, “asymptotic” chaos would occur if the accretionrate were assumed to slowly vary.

2.1. Effect of Radiation Forces on AccretionML present the following sketch of the inner edge of

the accretion disk [here, ω and ωE denote the accretionrate and the maximum (“Eddington”) accretion rate, re-spectively]:

For accretion rates ω À 0.01ωE , the flow nearthe star is likely to be optically thick. In thiscase radiation forces act primarily on accret-ing gas within one mean free path of the in-ner edge and surfaces of the disk. However,the increase in radial velocity of the gas inthese layers caused by rapid removal of an-gular momentum by radiation drag reducesthe density, allowing radiation from the starand boundary layer to penetrate further intothe disk. This is likely to produce a morewidespread change in the velocity structure ofthe accretion flow. . . Although a quantitativedetermination of the radiation field and theaccretion flow requires a self-consistent com-putation of the radiation and flow. . . the ar-guments just given show that radiation forcesare likely to be very important.

We view the DHR as a kind of abstract “scaffolding”upon which we can “hang” physical effects without hav-ing to consider, in the first instance at least, questionsof consistency. Since the effects of radiation forces occurwithin one mean free path of the optically thick inneredge of the accretion disk, we assume the DHR is de-scribing processes in this region. Thus, if we assume thestandard disk model and that the inner edge of the diskis 20 km from the neutron star center, the DHR is, torough order of magnitude, about one km thick in its ra-dial dimension (Shapiro and Teukolsky 1983). This isconsistent with the “thin rail” approximation adoptedby YS.

2.1.1. Outward Force Opposing AccretionWe assume that the radiation originates from near the

surface of the neutron star. The dominant component ofthe flux acting on the rail is therefore in the outward ra-dial direction (see Figure 1(b)). According to ML, thereare two radial forces: a static force acting on particlesnear rest, and a drag force acting on moving particles.Since the accretion velocity is much less than the speed oflight, the latter can be ignored. Furthermore, althoughthe static force results from Thomson scattering of ra-diation by free electrons bound electrostatically to theions (Frank et al. 1991), we do not attempt to incor-porate such detailed physics into the DHR. Instead, we

simply try to account for any gross effects that theseforces might exert on the DHR dynamics.

In particular, it is assumed that the outward radiationforce acts to modulate the accretion onto the outer edgeof the rail. Since the DHR is the site where radiationpenetrates more or less depending the density, the forceopposing accretion will therefore decrease, exponentially,as the density increases. This approximates the physicalpicture presented by ML: “[reduced] density. . . allows ra-diation to penetrate further into the disk, [and] producesa more widespread change in the velocity of the accretionflow.”

From this, it is clear that the accretion rate shouldincrease with increasing density. In the first approxi-mation, therefore, accretion is taken to be an increasinglinear function of density. As a result, the rate ω, origi-nally constant in both space and time, is now dependenton both variables through its dependence on the density;we term this “dynamical accretion.” Labeling time by nand spatial cell by i, we therefore have

ωin = αρi

n + ω0

where α ≥ 0, ω0 is the (constant) accretion rate at zerodensity and ρi

n is the density in cell i at time n (see Sec-tion 3). Setting α = 0 reduces to the original model. Theabove expression for ω = ωi

n may then be substitutedinto equation 6 of YS, the map lattice implementationof the dynamical system. A theoretical analysis of thisextended system (hereafter, “eDHR”) is given Section 3below.

2.1.2. Drag Force and Asymmetrical DiffusionAccording to ML, the orbital (i.e., azimuthal) velocity

of the inner edge of the accretion disk is a significantfraction (0.2–0.5) of the speed of light. On this basis,they concluded that radiation drag opposing the orbitalmotion has a “very important” effect on the accretiondynamics.

We have therefore considered how to include an az-imuthal force in the DHR. The original model assumesthat the diffusion of material occurs at equal rates (pro-portional to density differences) in the clockwise (CW)and in the counterclockwise (CCW) directions. As arough approximation, we have therefore modeled the ef-fect of the drag force by assuming the diffusion ratesdiffer in these two directions. Although there may beother ways to include this effect, this seems the simplest.

2.1.3. Drag Force and the Dripping PhenomenonML were primarily interested in the fact that the drag

force, by removing angular momentum from the orbitingmatter, allows it to spiral inwards towards the centralstar. It is therefore natural to consider whether sucha force could somehow produce the dripping instabilityassumed in the DHR picture.

It is easily seen that an instability might indeed re-sult from the fact that the accretion rate in the extendedmodel increases with density. However, without a so-lution of the coupled equations governing the radiationfield and the density, it is not clear how the drag forcemight be involved in this. For this reason, the original“modulo 1” mechanism for the dripping phenomenon hasbeen retained for the present (see YS).

3

(a) (b)

Fig. 1.— (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 behav-ior of its orbits. Since this behavior may in general bevery complicated, one first focuses on the simplest typesof orbits, namely fixed and periodic ones. A point p ∈ Mis called a fixed point if f(p) = p. It is called periodic ifthere exists an integer k > 0 such that fk(p) = p. (Inthis report, we will only be concerned with fixed pointsleaving the study of periodic points for future work.) Ifp is a fixed point of f , it is natural to ask: what canbe said about the asymptotic behavior of nearby points?If f is a differentiable dynamical system and its deriva-tive 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 1982) states thatfor all points in some neighborhood U of p, the behav-ior of f is qualitatively the same as the behavior of thelinear dynamical system A (more precisely, f and A aretopologically conjugate; see Palis and de Melo 1982). Sounderstanding the dynamics of A helps us understandthe dynamics of f near p.

Let Es denote the sum of the eigenspaces of A corre-sponding to eigenvalues λ with |λ| < 1 and let Eu denotethe sum of the eigenspaces corresponding to eigenvalueswith |λ| > 1. It is not hard to see that if v ∈ Es, thenAkv → 0, as k →∞, whereas if w ∈ Eu, then (assumingA is invertible) A−kw → 0, as k → ∞. A natural ques-tion is: are there directions in the phase space M alongwhich f exhibits similar behavior? If p is a hyperbolicfixed point, then the Stable Manifold Theorem (Palis andde Melo 1982) guarantees the existence of the stable man-ifold W s(p) and unstable manifold Wu(p) such that: (a)for every x ∈ W s(p), fk(x) → p, as k → ∞; (b) forevery x ∈ Wu(p), f−k(x) → p, as k →∞ (assuming f isinvertible); (c) W s(p) and Wu(p) are smooth manifoldstangent at p to the linear spaces Es, Eu, respectively.

In this section we extend the original DHR model of

Young and Scargle based on the physical observationsfrom Section 2 and study its dynamical behavior alongthe lines described above.

3.1. Theoretical Introduction to the Extended DrippingHandrail

As in the original Dripping Handrail model proposedby Young and Scargle, the extended system treats timeand space as discrete variables. For example, we considertime steps (n = 1, 2, 3 . . .) and divide the space of theinner edge of the accretion disk into discrete cells. Theonly continuous variable is the density of each cell.

Keeping with the notation established in the Youngand Scargle paper, we represent the index of the cells onthe DHR by i ranging from 1 to N , the number of cellsin the handrail. Time is quantized and denoted by theindex n. The continuous density variable in cell i at timen is given by ρi

n.Whereas the original model opted for a constant ac-

cretion rate, the extended model adds a parameter fortime-varying accretion. Instead in the extended DHR(eDHR) model, the initial accretion rate is denoted ω0

and the local accretion rate at a given time n and at celli is represented as

ωin = αρi

n + ω0

where α is the constant of dynamic accretion, the newaddition to the DHR. In addition to matter accretinginto cells from the star, matter also diffuses from one cellto its two neighboring cells and this is represented by Γ,the diffusion parameter.

To further condense the notation, we encapsulate thedensity values at each cell at time n into an N×1 columnvector:

Xn =

ρ1n

ρ2n...

ρN−1n

ρNn

.

4

Then we can define the discrete dynamical system interms of a map f : HN → HN on the unit hypercubeHN = {(x1, . . . , xN ) ∈ RN : 0 ≤ xi < 1} such that

Xn+1 = f(Xn)

where the map f is defined thus

f(X) = AX + b (mod 1),

A =

δ Γ 0 · · · 0 ΓΓ δ Γ · · · 0 00 Γ δ · · · 0 0...

......

. . ....

...0 0 0 · · · δ ΓΓ 0 0 · · · Γ δ

,

b = ω0

11...1

,

δ = 1− 2Γ + α.

The matrix A of size N × N is a slight modification ofthe matrix in Appendix B of the Young and Scargle pa-per. The addition of α is to account for the time-varyingaccretion rate. The modulo 1 operation in the map rep-resents the dripping of matter off the rail when a thresh-old is reached. Therefore, as in the original model, thethreshold dictates that

0 ≤ ρin < 1.

Hence the map always returns to the unit hypercube,HN . Note that the map X 7→ AX + b does not necessar-ily preserve the integer lattice ZN (or any other lattice)in RN , so f cannot be considered as a map on the N -dimensional torus RN/ZN .

3.2. Fixed PointsIt is not hard to see that the original DHR map (with

α = 0) does not have any fixed points. Recall that X is afixed point of f if f(X) = X, i.e., if there exists a vector` ∈ ZN with integer coordinates such that AX + b =X + `. Since A− I is invertible (see subsection 3.3), weobtain

X = (A− I)−1(`− b).

Since X must belong to the unit hypercube HN , we con-clude that every ` ∈ ZN such that (A−I)−1(`−b) ∈ HN

gives a fixed point of f . We have the following particularsolution to this equation.

Proposition. If there exists an integer m such that 0 <(m− ω0)/α < 1, then

X∗ =m− ω0

α

11...1

is a fixed point of f .

Proof. Suppose that 0 < (m − ω0)/α < 1. Then X∗lies in the unit hypercube HN . We have to show thatAX∗ + b = X∗ + V , where V is a vector with integer

coordinates. Observe that v0 = [1, . . . , 1]T is an eigen-vector of A corresponding to the eigenvalue 1 + α. SinceX∗ is a scalar multiple of v0, we have AX∗ = (1 + α)X∗.Therefore,

AX∗ + b = (1 + α)m− ω0

αv0 + ω0v0

=[(1 + α)

m− ω0

α+ ω0

]v0

=m− ω0

αv0 + mv0

= X∗ + mv0.

Since mv0 is an integer vector, it follows that f(X∗) =X∗.

3.3. Eigenvalue Analysis of the eDHRTo understand the dynamics of the eDHR near its fixed

point X∗, we compute the eigenvalues of its linearizationA = Df(X∗). (Note that Df(X) = A for every pointX in the interior of HN .) In the original model, theeigenvalues of the matrix J (which lacked the α term fortime-varying accretion) were computed to be

µi = 1− 2Γ(

1− cos(

2πi

N

)),

i = 0, 1, . . . ,

⌊N

2

⌋.

Using this eigenvalue formula for the original matrix, wearrive at the following result for the eigenvalues of theeDHR matrix.

Theorem (Eigenvalues and Eigenvectors of A). If µ isan 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 arbi-trary eigenvalue µ and corresponding eigenvector v of J .By definition of eigenvalues, we have

Jv =µv.

Therefore,

Av =(J + αI)v=Jv + αIv

=µv + αv

=(µ + α)v.

Thus v is also an eigenvector of A and has correspondingeigenvalue λ = µ + α. 2

Since the eDHR is a discrete dynamical system, weare further interested in classifying the eigenvalues of Aas having magnitude greater than, less than or equal to1. Using the above theorem, this can be easily done.An interesting consequence of introducing non-constantaccretion rate is the guaranteed existence of a specificeigenvalue, namely λ = 1 + α with corresponding eigen-vector v0, since

Au=(1 + α)u

5

regardless of the number of cells in the dripping handrail.In the original model where α = 0, this eigenvalue issimply 1, whereas in the extended model with α > 0,this eigenvalue is strictly greater than 1.

Therefore, if α is positive but small, the eigenvalues ofA split into two groups: one eigenvalue (1 + α) strictlygreater than one with eigenspace Eu spanned by v0, andall the remaining eigenvalues in the interval (0, 1). By theStable Manifold Theorem mentioned above, there existsa 1-dimensional unstable curve Wu(X∗) and an (N −1)-dimensional stable surface W s(X∗). It is not hard toverify that Wu(X∗) is not only tangent to Eu at X∗, butis in fact an arc in Eu. We do not have such a descriptionfor W s(X∗).

Although we do not have enough rigorous evidence toconclude that the eDHR system is chaotic (in the stan-dard mathematical sense of the word), the presence ofexpansion due to an eigenvalue of A greater than one (inother words, a positive Lyapunov exponent log(1 + α) off) is an indicator that f may well exhibit very compli-cated long-term behavior. One way of rigorously provingthat f is indeed chaotic would be to find a hyperbolic pe-riodic point p for f whose stable and unstable manifoldintersect transversely; cf., Palis and de Melo (1982). Weleave these investigations for future work.

4. COMPARISON WITH OBSERVATIONS

We have developed Matlab c© code that simulates thePoincare map lattice implementation of the extendedmodel (see YS).2 When the dynamical accretion param-eter α is set to zero and the CW and CCW diffusionconstants are set equal, the code reproduces the resultsshown in YS, Figure 10.

As a baseline, we adopted the parameter value θ =1.262797 given by YS as a typical “chaotic” setting.Starting with random initial densities assigned to 37 spa-tial cells arranged around the circular rail, the code gen-erates a sequence of 16,384 time steps whose values arethe total mass on the rail at each step. For purposesof interpretation, these values are then assumed propor-tional to the x-ray intensity. The power spectrum of thesequence was then calculated in the standard way in or-der to compare with published power spectra.3

4.1. Comparison Using Original ModelWe first compared simulations of the original model

with observations of Scorpius X-1, a typical neutron starsource, made with a more sensitive x-ray telescope thanthat available to Scargle and others when they first stud-ied this object (Scargle et al. 1993).4 For this preliminarystudy, we simply made side-by-side comparisons betweenthe simulated and the published power spectra (see Fig-ure 2).

2 See file extended dhr.m and others, which are included in theaccompanying software.

3 The power spectrum is the modulus squared of the fouriertransform of the time sequence; intuitively, the spectrum is ob-tained by analysing the sequence into its various sinusoidal compo-nents and then plotting the amplitude squared of each componentversus its frequency. See for example Press et al. (1986).

4 The data shown in Figure 2(d) was obtained with NASA’sRXTE satellite (van der Klis 1997); shown is the power spectraof Scorpius X-1 obtained at 7 different times over a 4 day period.The plots have been displaced vertically for clarity; their absolutenormalizations are approximately equal.

First, it can immediately be seen that the DHR doesindeed display what appears to be chaotic behavior; thisis evident from the fact that all frequency components inthe power spectrum have significant amplitudes.

Secondly, there is a marked correspondence betweenthe spectra in the broad peak appearing at low frequen-cies (in particular, compare the second plot from the topin Figure 2(d)). This is the low frequency quasi-periodicoscillation (LFQPO) that Scargle and others were ableto reproduce (Scargle et al. 1993).5

Thirdly, the simulated spectrum falls off as a powerlaw (i.e., linear on a log-log plot) at high frequencies,whereas the published spectra remain flat. For this rea-son, we have drawn in a flat (white noise) component onthe simulated spectrum to provide a better fit as shownin Figure 2(a). Presumably, the origin of this noise isextrinsic to the DHR.

Lastly, certain significant peaks in the observed data athigh frequencies are seen that do not appear in the orig-inal model. These are the high frequency quasi-periodicoscillations (HFQPO) which are currently of great inter-est to astronomers.6

4.2. Comparison Using Extended ModelWe repeated the comparison using the eDHR model

described in Section 2. Starting from the baseline modelθ = 1.262797, we first increased the dynamic accretionparameter α to non-zero values while the CW and CCWdiffusion constants were kept equal.7

Figure 2(b) shows a typical simulated spectrum cor-responding to α = 0.04, with white noise added. Twomain effects occur as α is increased. First, the relativeamplitude of the LFQPO peak decreases, and, secondly,a series of equally-spaced peaks on the high frequencyside of the peak appears. Except for the fact that theHFQPO in the actual data are not equally-spaced, thistrend is in rough qualitative agreement with the trendseen in Figure 2(d) as we pass from the second to theseventh plot down from the top.

Next, we increased the “splitting” between CW andCCW diffusion rates from zero, and adjusted the α pa-rameter to obtain a rough best-fit with the data. A typ-ical simulated spectrum with 10% splitting and α = 0.01is shown in Figure 2(c). The general effect of asymmetricdiffusion, even when α = 0, is again to produce a series ofhigh frequency peaks. These are generally sharper thanthe peaks seen without splitting and are not as equally-spaced. There is a rough qualitative similarity betweenthe simulated spectrum and the observed spectra of Fig-ure 2(d).

4.3. Transient Chaos Study

5 It is worth noting that the LFQPO correspondence here ismuch more striking than that appearing in Scargle et al. (1993);this is because the spectrum shown in Scargle et al. (1993) is similarto the fourth spectrum from the top in Figure 2(d), where theLFQPO peak happens to be much less pronounced.

6 Although small in amplitude, the HFQPO are in fact detectedwith very high statistical precision; furthermore, they always ap-pear as “twin peaks” whose frequencies vary with the overall spec-trum. The HFQPO were first discovered in observations using theRXTE satellite after the original DHR was developed.

7 In this process, the ω0 constant was fixed at the value of thebaseline model.

6

102

103

−10

−9

−8

−7

−6

−5

−4

−3

−2

Frequency (arb.units)

Log

Pow

er D

ensi

ty (

arb.

units

)

(a)

(d)

102

103

−10

−9

−8

−7

−6

−5

−4

−3

−2

Frequency (arb.units)

Log

Pow

er D

ensi

ty (

arb.

units

)

(b)

102

103

−10

−9

−8

−7

−6

−5

−4

−3

−2

Frequency (arb.units)

Log

Pow

er D

ensi

ty (

arb.

units

)

(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) Powerspectrum of the extended model with α = 0.017,diffusion splitting = 0, and white noise added. (c) Power spectrum of the extended modelwith α = 0.010, diffusion splitting = 0.10, and white noise added. (d) Published power spectrum of Scorpius X-1 (van der Klis 1997).

7

The theoretical analysis in Section 3 showed that whenthe α parameter is greater than zero, f acquires a posi-tive Lyapunov exponent. This indicated that one effect ofincluding radiation forces may be to produce permanent,asymptotic chaos without the need to assume long-termdrifts in the accretion rate. In order to verify this predic-tion, we therefore undertook to study the way in whichthe power spectrum evolves over long periods of time.

In Figure 3(a) is shown the spectrum of the originalmodel formed using the first 16,384 time steps; in Fig-ure 3(b) is shown the spectrum formed using the last16,384 steps after having let the simulation run for a to-tal time of 50 × 16, 384 steps. Confirming the results ofYS, 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 ofthe power is in sharp, equally-spaced harmonics) after 50time blocks.

In Figures 3(c) and (d) are shown corresponding spec-tra for the extended model with α = 0.04 (with sym-metrical diffusion). In this case, no significant change inthe chaotic spectrum is observed after 50 time blocks, inagreement with the theoretical predictions.8

5. VISUALIZATION WITH COMPUTER SIMULATIONS

Two animated visualizations of the DHR were devel-oped in order to gain intuitive understanding of themodel’s real-time behavior. The driving motivation be-hind creating such visualizations is to present the modelas a directly observable entity, thus making it possibleto see features of its behavior in action. Observable pat-terns in the behavior of the model over a relatively shortduration sometimes suggest the emergent behavior weseek. A snapshot of the ring visualization is included asfigure 4(a).

In the ring visual, colors represent each cell’s state.They range from black (ρ = 0) to red, orange, yellow,then white (ρ = 1). One of our goals was to possibly re-produce quasi-periodic oscillations, in particular the twinpeak oscillation from van der Klis (1997). One way toaccomplish that would be to coax our model into pro-ducing two roaming hot spots, rotating at different fre-quencies. In addition, the process would have to occurunpredictably, and completely periodic behavior wouldbe undesirable. Another possible source of QPOs couldbe a pulsing of several cells in semi-unison at two differ-ent frequencies. Both behaviors were reproduced visu-ally, but no quantitative analysis of the model at thosesettings was done. A second method of visualization wasprovided as a bar graph. Here, the density of each cellis represented as the height (or depth) of a bar. Whenanimated, 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-timedripping action of the model. Setting the diffusion pa-rameter at a high value relative to accretion produced themost visually fluid-like behavior, suggesting that thosesettings are more realistic if we are to assume that weare modeling a dripping fluid.

Unfortunately, since visual experiments were done in-dependently of the quantitative analysis presented in this

8 Note: in comparing the various spectra, it is necessary todistinguish 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 mostlysuperficial. If further research is to be done with thismodel, we suggest a more integrated approach of thetwo methods; perhaps producing desired behavior visu-ally first, then doing more in-depth analysis to producedata which can be compared with that from an astro-physical source.

6. RETURN MAP

In the search for ways to characterize the behavior ofthe DHR model, the return map was brought to our at-tention. 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, nordo any of the figures in this section depict a “map” (i.e., awell-defined function). But we suspect that from the fig-ures, one can extract useful information about the DHRmodel (e.g., what happens to the mass in a cell in thefuture as a function of what we know about it now).

A return map compares the mass in a particular cellat a future time to the mass in the same cell at thepresent time. More specifically, the return map is a plotof xn+k versus xn, where xn+k and xn are the masses inthe cell at times n + k and n respectively.

Our goal here is to study the return map for variousvalues of n and k and then try to draw conclusions aboutthe DHR model based on the graphs generated. Cur-rently we cannot analyze the map well enough to drawany definite conclusions, so most of what we present herewill be conjectures about the behavior of the model.

We interpreted the description of the return map intwo different ways, giving rise to two precise definitionsand therefore two investigations into the behavior of theDHR model.

6.1. First ApproachIn the first approach, both n and k are fixed. Many

simulations of the accretion disk are run with randominitial conditions, and from each simulation we extract asingle point (xn, xn+k). One advantage to this approachis that the order in which the points are generated isirrelevant; all that matters are the areas on which pointstend to fall on (or avoid) the graph. On the other hand,one disadvantage would be the amount of computationinvolved. To generate sufficient number of points, say10,000, one would have to simulate the accretion disk10,000 times, which requires a nontrivial amount of time.

Figure 5 shows several graphs generated using rm.m, aMatlab function which implements the return map (thesource code for rm.m is included in the accompanyingsoftware). Below each graph is the function call used togenerate it.

For each run of the return map in the first approach, weused a symmetrical diffusion scheme. Unless otherwisespecified, the parameter values are those in Young andScargle (1996). More specifically, n = 37 (the numberof cells), θ = 1.262797 (the angle), Γ = 0.05 sin θ (thediffusion parameter), and ω0 = 0.05 cos θ and α = 0 (theaccretion parameters).

Figure 5(a) is not surprising. Little time passeswhen k = 1, and so we do not expect the mass in anyparticular cell to change by very much. Consequently,the dots should fall close to the line y = x (also shownin the figure).

8

102

103

−10

−5

0

Frequency (arb.units)

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Pow

er (

arb.

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)

(a)

102

103

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−5

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Frequency (arb.units)

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(b)

102

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(c)

102

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(d)

Fig. 3.— Transient chaos figures. See text for descriptions.

(a) (b)

Fig. 4.— Snapshots of the animated visualizations of the eDHR model.

9

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

xn+k

versus xn for n = 1 and k = 1

x1

x 2

(a) rm(1,1,1000)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

xn+k

versus xn for n = 1 and k = 2

x1

x 3

(b) rm(1,2,1000)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

xn+k

versus xn for n = 1 and k = 5

x1

x 6

(c) rm(1,5,2000)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

xn+k

versus xn for n = 1 and k = 19

x1

x 20

(d) rm(1,19,5000)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

xn+k

versus xn for n = 1 and k = 39

x1

x 40

(e) rm(1,39,7000)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

xn+k

versus xn for n = 1 and k = 100

x1

x 101

(f) rm(1,100,8000) and ω0 = 0.515

Fig. 5.— Plots of the return map for various values of n and k. Each has ω0 = 0.05 cos 1.262797 unless otherwise specified. (a), (b), (c) As kincreases, the line becomes wider and tilts slightly. (d) Lines form in the lower right. (e) Lines become more pronounced and span theentire width of the graph. (f) If k is increased further and ω0 increased to 0.515, then the banding pattern repeats itself.

10

Figures 5(b) and (c) show graphs for when k increasesslightly to 2 and 5 respectively. The line noticeablywidens and tilts.

Each time step thereafter, the line continues to widenand tilt until about k = 19, when a banding patternbegins to appear in the lower-right area of the graph. SeeFigure 5(d). These faint lines naturally suggest that themass in a cell becomes “discretized” after enough timehas passed (i.e., when k is large enough). Figure 5(e)shows a graph when k = 39. Here the banding patternbecomes much more pronounced and the lines now spanthe entire width of the graph. It is also interesting tonote that the dots tend to avoid a small area on the leftside of the graph.

If we increase the accretion parameter ω0 to 0.515 andlook further into the future, e.g., k = 100, then Fig-ure 5(f) is the result. Figure 5(f) appears to contain Fig-ure 5(e) repeated twice; another interesting phenomenonwhich suggests periodic behavior.

Our experiments with the return map, and in partic-ular the graphs generated in Figure 5, lead us to thefollowing conjectures:

• Where the dots are more concentrated on the graphis where the mass of the cell is more likely to be“located”. For example, that the dots are moreconcentrated along the lines x101 = 0.5 and x101 =1 in Figure 5(f) would mean that the mass in aparticular cell is most likely to be near 0.5 or 1 attime 101.

• After enough time has passed, the mass in a cellbecomes “discretized,” i.e., the mass in a cell cantake on only one of finitely many values.

One should note that our conjectures are purely of themathematical model and not of the actual star itself. Itwould be interesting to examine raw astronomical datain order to determine if the neutron star actually exhibitsthe behavior predicted in the two preceding bullet points.

6.2. Second ApproachIn the second approach, n is allowed to vary while k

remains fixed. Only one simulation of the accretion diskis performed, and from this simulation, many points arecollected. All points which appear on the graph are ofthe form (xn, xn+k), where k is a positive integer chosenbefore the experiment is run. For example, if k = 10,then the points on the graph have the form (x1, x11),(x5, x15), and so on.

For each run of the return map in the second approach,we used a symmetrical diffusion scheme (as opposed to anasymmetrical diffusion scheme). Our parameter valuesare the same as in the first approach except n = 32 (thenumber of cells) and we now vary the value of θ in orderto see how the accretion/diffusion ratio affects the returnmap.

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 ofFigure 6 show how the relationship changes between themass density of one single cell at time n and at 5 timesteps later with regard to time; plots on the right side ofFigure 6 show how the relationship changes between the

mass density of total cells at time n and at 1 time steplater with regard to time.

From these plots, we can draw some conclusions:

• For the mass density of a single cell, the mass ap-pears to repeat itself but with a slight shift.

• For total mass density of all cells on the disk, thegraph somewhat resembles a fractal.

These conclusions are based on preliminary observa-tions. Further experiments are required.

7. CONCLUSIONS AND FUTURE DIRECTIONS

The comparisons in Section 4 show that the extendedDHR may provide a unified explanation for most fea-tures seen in the power spectra of certain types of ac-creting neutron star. Most notably, high frequency os-cillations(HFQPO) appear to arise naturally when theeffects of radiation forces are added. Although detailedagreement cannot be expected in view of the very simpli-fied nature of the present model, our results neverthelesssupport the conjecture expressed in Scargle et al. (1993);Young and Scargle (1996) that the observed x-rays derivefrom a chaotic, dripping-like phenomenon at the inneredge of an accretion disk. Indeed, it seems entirely plau-sible that this is an accurate physical description of theinterplay between two extreme counterbalancing forces,radiation and gravity.

Even in terms of the present model, however, we haveyet to explore the effect of varying the baseline param-eters from θ = 1.262797, or of assuming x-ray inten-sity proportional to total mass in drops. Beyond this, amore careful treatment of the effect of radiation forces, inparticular the drag force, is clearly desirable; and thereare other possibilities too numerous to mention. More-over, the visualization studies of Section 5 have madeclear, among other things, that summing density over allcells at each time step when forming power spectra couldseverely reduce the sensitivity to QPO phenomena; in ad-dition, the return map studies in Section 6 have shownthe need to obtain the raw astronomical time sequencedata for any future comparisons.

A more general conclusion is that an abstract math-ematical model may sometimes serve as a very usefulmeans of “bootstrapping” one’s way to a more realisticphysical picture when no (stable) solution of the detailedequations of motion is expected to exist. Indeed, addingphysics to a model which can only be changed in cer-tain ways serves to reduce a complicated problem to amuch simpler one. In addition, there is often a synergybetween theory and model-building, exemplified in thiscase by the theoretical suggestion that long-term chaosmay result if accretion is allowed to vary dynamicallywithin the system. For the future, we plan to furtherinvestigate the abstract model, find its periodic pointsand a possible transverse homoclinic point; this wouldbe a firm indication of the presence of asymptotic chaosin the system. In addition, development of the theoryin the fourier domain might provide more direct insightinto the QPO phenomenon.

8. ACKNOWLEDGMENTS

We thank Jeff Scargle (NASA Ames) and Karl Young(UCSF) for originally suggesting this study and for guid-ance during its progress; Slobodan Simic (SJSU) for

11

(a) θ = 0.153999; density of one cell (b) θ = 0.153999; total density of all cells

(c) θ = 0.769982; density of one cell (d) θ = 0.769982; total density of all cells

(e) θ = 0.985597; density of one cell (f) θ = 0.985597; total density of all cells

(g) θ = 1.262797; density of one cell (h) θ = 1.262797; 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 = 0, toblue at the end of time. (a), (c), (e), (g) are the collection of data points of (xn, xn+5). (b), (d), (f), (h) are the collection of data pointsof (P

xn,P

xn+1).

12

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 Fundof the San Jose State University Foundation.

REFERENCES

J. D. Scargle, T. Steiman-Cameron, and K. Young, AstrophysicalJournal 411, L91 (1993).

K. Young and J. D. Scargle, Astrophysical Journal 468, 617 (1996).M. van der Klis, Astrophysical Journal 481, L97 (1997).M. van der Klis, Astronomische Nachrichten 326, 798 (2005).M. C. Miller and F. K. Lamb, Astrophysical Journal 413, L43

(1993).S. L. Shapiro and S. A. Teukolsky, Black Holes, White Dwarfs,

and Neutron Stars: The Physics of Compact Objects (Wiley,New York, 1983).

J. Frank, A. R. King, and D. J. Raine, Accretion Power inAstrophysics (Cambridge University Press, 1991), 2nd ed.

J. Palis and W. de Melo, Geometric theory of dynamical systems(Springer-Verlag, 1982).

W. H. Press, B. P. Flamery, S. A. Teukolsky, and W. T.Vetterling, Numerical Recipies: The Art of Scientific Computing(Cambridge University Press, Cambridge, 1986).