Mechanical model of a boundary layer for the parallel tracks of kilohertz quasi-periodic oscillations in accreting neutron stars
aa r X i v : . [ a s t r o - ph . H E ] J a n Astronomy & Astrophysicsmanuscript no. blslab_print © ESO 2021January 6, 2021
Mechanical model of a boundary layer for the parallel tracks ofkilohertz quasi-periodic oscillations in accreting neutron stars
Pavel Abolmasov , and Juri Poutanen , , Department of Physics and Astronomy, FI-20014 University of Turku, Finland Sternberg Astronomical Institute, Moscow State University, Universitetsky pr. 13, 119234 Moscow, Russia Space Research Institute, Russian Academy of Sciences, Profsoyuznaya 84 /
32, 117997 Moscow, Russia Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, SE-10691 Stockholm, Sweden
ABSTRACT
Kilohertz-scale quasi-periodic oscillations (kHz QPOs) are a distinct feature of the variability of neutron star low-mass X-ray binaries.Among all the variability modes, they are especially interesting as a probe for the innermost parts of the accretion flow, including theaccretion boundary layer (BL) on the surface of the neutron star. All the existing models of kHz QPOs explain only part of their richphenomenology. Here, we show that some of their properties may be explained by a very simple model of the BL that is spun up byaccreting rapidly rotating matter from the disk and spun down by the interaction with the neutron star. In particular, if the characteristictime scales for the mass and the angular momentum transfer from the BL to the star are of the same order of magnitude, our modelnaturally reproduces the so-called parallel tracks e ff ect, when the QPO frequency is correlated with luminosity at time scales of hoursbut becomes uncorrelated at time scales of days. The closeness of the two time scales responsible for mass and angular momentumexchange between the BL and the star is an expected outcome of the radial structure of the BL. Key words. accretion, accretion disks – stars: neutron – stars: oscillations – X-rays: binaries
1. Introduction
Stitching an accretion disk rotating at about Keplerian rate withthe central object rotating much slower leads to the concept ofaccretion boundary layer (BL). The reason for talking about BLas an entity separate from the accretion disk is the inevitablebreakdown of the basic assumptions of the standard disk theoryin a very narrow region just above the surface of the accretor(Lynden-Bell & Pringle 1974; Papaloizou & Stanley 1986).In neutron star (NS) low-mass X-ray binaries (LMXBs),the BL is thought to be an important source of radiation,when the magnetic field of an accreting NS is too weakto support a magnetosphere. Shining at a luminosity com-parable to that of the accretion disk (Lynden-Bell & Pringle1974; Sibgatullin & Sunyaev 2000), but being much more com-pact, BL is expected to have a harder spectrum and shortervariability time scales. Such a component has indeed beenidentified in LMXBs spectrally (Suleimanov & Poutanen 2006;Revnivtsev et al. 2013) as well as via its timing properties, inparticular, as a source of kilohertz quasi-periodic oscillations(kHz QPOs) (Gilfanov et al. 2003). The position of the BL at thesurface of the NS makes it a valuable probe for the fundamentalproperties of the star: its size, radius, and the physical conditionson its surface.The kHz QPOs have been observed in many NS LMXBs(van der Klis 2000). Their frequencies span the range betweenabout 200 Hz and the Keplerian frequency near the surface(about 1.3 kHz; see Méndez et al. 1999; Belloni et al. 2005). Ei-ther one or two peaks, with the frequency di ff erence of about300 Hz (Méndez & Belloni 2007), are observed. In individualsources, kHz QPO frequencies may vary by a factor of 1.5–2.The frequencies are correlated with flux on time scales of hours(Méndez et al. 1999, 2001), while the correlation disappears on time scales of days. This phenomenon, known as QPO paral-lel tracks, was explained in a purely phenomenological way byvan der Klis (2001). He suggested that the instantaneous X-rayluminosity of the source is a linear combination of the mass ac-cretion rate ˙ M and its running average h ˙ M i , while the oscillationfrequency is a function of ˙ M / h ˙ M i only. In this paper, we pro-pose a mathematically similar but a more physically motivatedsolution to the parallel tracks problem.Here, we develop a simple mechanical model of the BL,which is treated as a thin massive belt supplied by the mass andangular momentum from the accretion disk and at the same timelosing mass and angular momentum to the NS. We will considerthe rotation frequency of the BL as the characteristic frequencyresponsible for kHz QPOs, though the real situation is probablymuch more complicated (e.g. Abolmasov et al. 2020). In Sect. 2,we introduce the main equations based on conservation laws. InSect. 3, we consider the properties of the model by solving theequations numerically. We discuss the results in Sect. 4.
2. Model setup
We consider the BL as an infinitely thin equatorial belt on thesurface of a NS of radius R and mass M NS rotating at an angularfrequency Ω NS . Rotation of the layer is aligned both with therotation of the star and the disk. Dynamics of the layer may bereduced to two equations, one for the mass and the other for theangular momentum conservation. The conservation law for theBL mass M may be written asd M d t = ˙ M − Mt depl , (1)where ˙ M is the mass supply rate from the disk. The second termdescribes mass precipitation from the BL onto the NS surface Article number, page 1 of 8 & Aproofs: manuscript no. blslab_print with the depletion time scale t depl that exceeds the characteristicdynamical (Keplerian) time scale t dyn = / Ω K = p R / GM NS ,where Ω K is the Keplerian frequency.Conservation of the angular momentum also involvessources and sinks related to the interaction with the surface of thestar. Hydrodynamic numerical simulations (e.g. Belyaev et al.2013) suggest that the interaction between the BL and the surfaceof the star mediated by Reynolds stress is relatively weak. Therelevant tangential stress W r ϕ ∼ − P , where P is the pressureat the bottom of the layer. The impact of magnetic fields on theinternal dynamics of the layer is probably important (Armitage2002), but it is unclear if they can provide an e ffi cient angularmomentum transfer between the BL and the star. We will as-sume that the stress at the bottom of the BL is proportional tothe pressure with a small proportionality coe ffi cient α ≪ W r ϕ = α P = αg e ff Σ , (2)where Σ is the BL surface density and g e ff = GM NS R − Ω R (3)is the e ff ective surface gravity, where Ω is the rotation frequencyof the layer. This allows to express the braking torque acting onthe layer as T − = ARW r ϕ = αg e ff MR , (4)where A is the surface area of the BL (projected onto the surfaceof the star) and the BL mass is M = A Σ .The angular momentum conservation law including mass de-pletion and friction takes the formd J d t = ˙ M j d − Jt depl − αg e ff MR , (5)where J = Ω MR is the total angular momentum of the layer, j d = √ GM NS R is the specific angular momentum of the mat-ter entering from the disk. We ignore viscous interaction be-tween the disk and the BL. This corresponds to the “accretiongap” scenario (Kluzniak & Wagoner 1985) when the last stableorbit is located above the surface of the NS, and thus the diskis causally disconnected from the BL. Recent constraints forthe NS radius (Nättilä et al. 2017; Miller et al. 2019; Riley et al.2019; Capano et al. 2020) suggest that this should be the case, atleast below the Eddington limit.Two equations (1) and (5) are su ffi cient to describe the evo-lution of the physical parameters of the BL with time, given ˙ M ( t )and initial conditions. In our framework, the energy released dur-ing accretion and dissipation does not a ff ect the dynamics of thelayer. However, luminosity is an important observable. Some ofthe kinetic energy of the flow contributes to the spin-up of thestar and the rest is converted to heat and contributes to the lumi-nosity. The dissipated luminosity may be found as the change inthe kinetic energy (see e.g. Appendix B of Popham & Narayan1995). Our model splits this spin-down of the gas being accretedinto two episodes: some dissipation occurs when the matter fromthe disk enters the BL at the rate ˙ M , and some during the matterdepletion from the BL (at the rate of M / t depl ). In addition to thesetwo components, there is viscous dissipation unrelated to massexchange, equal to one half of the stress W r ϕ times the strain R d Ω / d R (see Landau & Lifshitz 1987, section 16). Together, the luminosity associated with the BL may be written as the sum ofthree terms L =
12 ˙ MR (cid:16) Ω − Ω (cid:17) + αg e ff MR ( Ω − Ω NS ) + Mt depl R (cid:16) Ω − Ω (cid:17) . (6)The first term on the right-hand side is the kinetic energy lostby the matter that enters the BL from the disk with the angularfrequency Ω d = j d / R . The second term is the viscous dissipa-tion associated with the Reynolds stress (2). The last term corre-sponds to the kinetic energy of the BL material that precipitatesonto the NS and acquires its rotation velocity.Below, we will assume that the BL is fed by a variable sourceof mass. We will assume stochastic variability of the mass accre-tion rate, modeled as a white noise source convolved with a ker-nel corresponding to a power-law power-density spectrum (PDS)with a random Fourier image phase (that corresponds to a ran-dom moment in time and unsynchronized variability at di ff erentfrequencies). Integrating white noise leads (as it involves sum-mation of a large number of independent random numbers) to anormally distributed quantity. To reproduce the log-normal fluxdistribution reported in many observational works (Uttley et al.2005), we then exponentiate the result of the convolution andre-normalize it to match the mean value of ˙ M .
3. Results
For a fixed BL mass and mass accretion rate, rotation of the BLmay be described in terms of approach to a single equilibriumstate. Using Eqs. (1) and (5), we can derive an evolutionary equa-tion for Ω :d Ω d t = dd t (cid:18) JMR (cid:19) = JMR ˙ M j d J − ˙ MM − αg e ff MRJ ! = ˙ MM ( Ω d − Ω ) − α (cid:16) Ω − Ω (cid:17) . (7)The right-hand side of this equation is quadratic in Ω , that allowsto re-write it in the formd Ω d t = α ( Ω − − Ω ) ( Ω + − Ω ) , (8)where Ω ± = ˙ M α M ± s ˙ M α M − Ω K ! + ˙ M α M ( Ω K − Ω d ) . (9)For Ω d = Ω K , one of the frequencies becomes Ω + = Ω K , andthe other Ω − = ˙ M / ( α M ) − Ω K . The lower of the two roots, thatis always Ω − for the parameter values we consider (see Sect. 3.3for more details), is stable.Our approximation is valid only if Ω < Ω K , otherwise ef-fective gravity becomes negative and the flow is unbound. Un-less Ω − becomes smaller than Ω NS , BL will evolve towards thisequilibrium state. Otherwise, the layer stalls at Ω = Ω NS , and W r ϕ works as static friction.Mass equilibrium is reached when M = M eq = ˙ Mt depl . (10) Article number, page 2 of 8avel Abolmasov and Juri Poutanen: QPOs from a neutron-star boundary layer
When, at a fixed mass accretion rate, the system reaches bothequilibrium mass and rotation frequency, the position of the sta-ble stationary point depends, apart from Ω d / Ω K that we fix to1, on a single parameter α M eq / ˙ M . It is easy to check that thisquantity, multiplied by Keplerian frequency, is equal to the ratioof the characteristic depletion and friction time scales, q = t depl t fric = α Ω K t depl . (11)For Ω d = Ω K , the equilibrium rotation frequency is Ω eq = q − ! Ω K . (12)When the friction becomes more e ffi cient than depletion, thelayer brakes down to Ω = Ω NS , that leads to trivial rotationalevolution. Hence, in the simulations with variable mass accre-tion rate, we will keep α . / ( Ω K t depl ). If the mass accretion inflow to the layer is variable, the BL worksas a filter for the variability of ˙ M . The system of equations weconsider is practically linear, though there is non-linearity intro-duced by g e ff in the friction term in Eq. (5). The characteristicdepletion and friction time scales are presumably much longerthan the dynamical time, and probably also exceed the viscoustime scales in the inner disk. The outer disk, however, evolveseven slower. In the relevant frequency range, the shapes of thePDSs of LMXBs are generally close to a power law PDS ∝ f − p with the slope of p ≃ . M = L Edd / c . The exact value does not a ff ect the qualitative pictureof accretion but sets the accretion time scale and equilibriummass of the layer. As it was mentioned in Sect. 2, the variationsof the mass accretion rate logarithm were considered as an in-tegral of a white noise process. This allows to introduce oneextra parameter, the dispersion of ln ˙ M . In our simulations, weset the root-mean-square deviation of mass accretion rate loga-rithm D = r(cid:28)(cid:16) ∆ ln ˙ M (cid:17) (cid:29) to 0 .
5. This value allows to reproducethe relative variations of the characteristic frequencies withoutstrong inconsistency with flux variation amplitudes in LMXBs(Hasinger & van der Klis 1989; Méndez et al. 1999).In our model, the BL does not have any variability of its own,hence the variations of its luminosity are essentially smaller thanthat of the mass accretion rate, especially at high frequencies. Inreality, of course, there is an additional variability componentoriginating in the layer. The BL light curve is smoother and lagsthe mass accretion rate, as one would expect from the propertiesof the model where the BL emission depends on the history ofmass accretion rate.We computed the cross-spectra of BL luminosity L (seeEq. 6) and its rotation frequency Ω which are the proxies forthe flux and QPO frequency, respectively. The argument of thecross-spectrum gives the phase lags, which we show in Fig. 1as a function of Fourier frequency. We also computed the coher-ence (Vaughan & Nowak 1997; Nowak et al. 1999) shown in thelower panel of the figure. Both are averaged over a series of 10 light curves.Quite expectedly, the quantities are correlated at lower fre-quencies but uncorrelated at f ≫ / t depl , fric . Maximal coher-ence, however, occurs at intermediate frequencies f ∼ (0 . − − − − π − π π π ∆ ϕ − − − − − f , Hz . . . . . . . c oh e r e n ce Fig. 1.
Phase lags (upper panel) and coherence (lower panel) betweeninstantaneous luminosity of the BL L and its rotation frequency Ω . Pos-itive phase lag means that Ω lags L . The green vertical lines show thefrequencies corresponding to the depletion t depl (dot-dashed) and thefriction 1 /α Ω K (dashed) time scales. Additional error bars (vertical dot-ted) show variability of the quantities within the bin. The parameters are α = − , t depl ≃
740 s (corresponding to q ≃ . / t depl , fric . At higher frequencies, luminosity becomes sensitiveto rapid variations in ˙ M , uncorrelated with Ω . Phase lags at lowfrequencies are negative, as the variations of L lag the varia-tions of ˙ M , while Ω follows the variations of Ω − ( ˙ M , M ) (seeSect. 3.3). The phase lags increase with frequency and becomepositive at the time scales somewhat longer than the time scalesof the BL ( ∼ t depl and t fric ). At high frequencies, they approach ∆ ϕ = π /
2. Such a flat phase lag spectrum is a natural outcomeof the mathematical properties of the initial system of equations.The luminosity given by Eq. (6) contains one term proportionalto ˙ M (first term, related to the variable mass inflow to the BL).The other two terms depend only on M = R ˙ Mdt + const andon Ω . The spectral slope of ˙ M is always shallower than that of M . At a given frequency f , the friction and depletion terms havecontributions ∼ / ( f t fric , depl ) with respect to the first term. Thusat high frequencies, variability of L is dominated by variationsof the mass accretion rate. Rotation frequency at high f (when M ≃ const) is a result of integration of Ω − (see Eq. 9) that is afunction of ˙ M and M . Taking Fourier transform of Eq. (8) in thehigh-frequency limit yields2 π i f ˜ Ω ≃ ( Ω K − Ω ) ˜˙ MM , (13)where all the higher-order terms in f are neglected, Ω + replacedwith Ω K , and the Fourier transform of Ω − replaced by ˜˙ M /α M . Article number, page 3 of 8 & Aproofs: manuscript no. blslab_print . . . . . . . q . . . . . . . . . Ω / Ω K .
05 0 .
10 0 . L / L Edd . . . . . .
05 0 .
10 0 . L / L Edd . . . . . . Ω / Ω K .
05 0 .
10 0 . L / L Edd . . . . . . . t , s Fig. 2.
Upper panel : mean BL rotation frequency (black dots; error bars show the root mean square variations of Ω ) as a function of q comparedto the equilibrium value Ω − (red solid line). Dashed red line shows the rotation frequency of the NS (3ms). Lower panels : BL rotation frequencydependence on instantaneous luminosity for sample light curves with three di ff erent values of depletion time corresponding to q = . , .
6, and0 .
8. In all the simulations, α = − . Time in the lower panels is color-coded, see the colorbar on the right. The crosses in the lower panels are theaverage values calculated for 64 s-long time bins, the error bars show standard deviation. Hence, in this limit, the Fourier image of rotation frequency is˜ Ω ≃ π i f ˜˙ MM ( Ω K − Ω ) . (14)As L is mainly a ff ected by the first term, the cross-spectrum be-comes C ( L , Ω ) ≃ R (cid:16) Ω − Ω (cid:17) ˜˙ M ˜ Ω ∗ ≃ i4 π f R (cid:16) Ω − Ω (cid:17) ( Ω K − Ω ) (cid:12)(cid:12)(cid:12)(cid:12) ˜˙ M ∗ (cid:12)(cid:12)(cid:12)(cid:12) . (15)The argument of this expression is π / Behavior of the BL, including its rotation frequency, dependsstrongly on the balance between mass and angular momentumloss, that may be described by the dimensionless quantity q (see Eq. 11). In Fig. 2 we show the mean rotation frequency and itsvariations for di ff erent values of q . Apparently, the mean valueis well predicted by Ω − given by Eq. (9). When the depletiontime scale is much shorter ( q . . Ω NS .Strong variations in Ω are present only when the two time scales(friction and depletion) are comparable ( q ≃ . − . t depl , fric ,variability of the luminosity is dominated by the first term inEq. (6), uncorrelated with Ω . However, if the luminosity is av-eraged in time bins several times smaller than the time scalesof the BL, it becomes correlated with Ω . On these time scales,variations of Ω − in Eq. (8) dominate over variations of Ω (seeFig. 3), hence rotation frequency derivatived Ω d t ≃ ( Ω K − Ω ) ˙ MM . (16) Article number, page 4 of 8avel Abolmasov and Juri Poutanen: QPOs from a neutron-star boundary layer t , s . . . . . Ω / Ω K t , s . . . . . . . . L / L E dd Fig. 3.
Upper panel : portion of the light curve of the simulation with α = − and q = .
6. Solid black curve shows the total 64 s-averagedluminosity (Eq. 6). We also show three contributions to the luminos-ity separately: first, second, and third terms from Eq. (6) are plottedwith green dashed, blue dotted, and red dot-dashed lines. Black dots areinstantaneous luminosity values (every 2 s).
Lower panel : rotation fre-quency Ω (black solid) and Ω ± (blue dotted) for the same model. Greendashed horizontal line corresponds to the spin of the NS. Neglecting mass depletion, this yields M ∝ Ω K − Ω , (17)where the proportionality coe ffi cient is a slowly variable func-tion of time. This is a scaling well reproduced in the evolutionof the BL on the time scales several times smaller than frictionand depletion scales (Fig. 4). Luminosity variations also followa similar trend L ∝ ( Ω K − Ω ) − . In spite of its simplicity, the model has several parameters, thevalues of which are not derived from the basic principles. The in-fluence of the rotation frequency of the star Ω NS does not changethe overall behavior. For the solutions with q .
1, it only limitsthe possible values of Ω and slightly modulates the spin-downterm. The mean mass accretion rate in the framework of ourmodel also plays a secondary role, a ff ecting only the luminos-ity of the BL.The variability spectrum of the mass accretion rate is en-coded by two parameters, the root-mean-square variation ofmass accretion rate logarithm D and the slope of the power-lawspectrum p . Their influence on the parallel tracks e ff ect is shownin Figs. 5 and 6. Redder variability spectrum allows the systemto accrete longer at a steady rate di ff erent from the mean value, and thus increases the variations of mass and angular momen-tum. Thus, the parallel tracks e ff ect is much more prominent forthe case of red noise (right panel in Fig. 5). Harder variabilityspectrum ( p .
1) makes the parallel tracks closer. However, the L ∝ ( Ω K − Ω ) − scaling still holds well.Di ff erent values of D (see Fig. 6) also a ff ect the prominenceof the parallel tracks e ff ect. As the amplitude of mass accretionrate variations increases by several times, the spacing betweenthe short-term tracks increases from about 30% to nearly twoorders of magnitude.
4. Discussion
As mentioned in the Introduction, the observed kHz QPO fre-quencies vary by a factor 1.5–2 in individual sources. While ourmodel reproduces the parallel tracks e ff ect in a broad range ofparameters, strong variations in the rotation frequency of the BLappear only when the characteristic friction and mass depletiontime scales are comparable. If friction is more e ffi cient ( q & . q . . ff ectively, the second independent parameternecessary to reproduce the parallel tracks behavior exists onlyin a narrow range of q , meaning that there should be a physicalreason for the depletion and friction time scales to be close toeach other.Such a similarity in the time scales may be explained if theBL is resolved in radial direction. The radial flux of angularmomentum consists of two parts, viscous w r ϕ R and advective ω R ρv , where v is vertical velocity, h ≪ R is the height abovethe NS surface, w r ϕ = w r ϕ ( h ) is the viscous stress component,and ω = ω ( h ) is the rotation frequency, decreasing from Ω some-where inside the BL to Ω NS at the NS surface. Because the vis-cous angular momentum transfer is directed outwards in the diskand inwards at the bottom of the BL, at some altitude it shouldbe zero. Let us assume that w r ϕ = ω = Ω , and write down angular momentum transfer along theradial coordinate ∂∂ t (cid:16) ω R (cid:17) + v ∂∂ h (cid:16) ω R (cid:17) = − R ρ ∂∂ h (cid:16) w r ϕ R (cid:17) . (18)In a steady-state case, ρv = const, and the time derivative inEq. (18) is zero. Integration yields ρvω R + w r ϕ R = ρv Ω R . (19)At the surface of the NS, ω ( h ) = Ω NS and w r ϕ = W r ϕ , that implies ρv ( Ω − Ω NS ) R = RW r ϕ . (20)Multiplying this equation by A and taking into account Eq. (4)yields Mt depl ( Ω − Ω NS ) R = αg e ff MR . (21)Note that the mass flux ρv is related to the mass motion fromthe BL onto the surface of the star, hence we replaced ρv A with M / t depl . Substituting g e ff from Eq. (3), we can express the q pa-rameter using Eq. (11) as q = Ω K ( Ω − Ω NS ) Ω − Ω . (22) Article number, page 5 of 8 & Aproofs: manuscript no. blslab_print . . . M , 10 g . . . . . Ω / Ω K .
05 0 .
10 0 . L / L Edd . . . . . Ω / Ω K t , s Fig. 4.
Parallel tracks on the M − Ω and L − Ω planes for a simulation with α = − , q = . D = .
5, and p = .
3. Solid black lines are thelines of ( Ω K − Ω ) M = const and ( Ω K − Ω ) L = const. Time is color-coded (see the color bar on the right). To demonstrate the parallel tracks e ff ectduring multiple observation runs, we show only the data points in 10 s intervals separated by 10 s gaps. .
05 0 .
10 0 . L / L Edd . . . . . . . . Ω / Ω K t , s .
05 0 .
10 0 . L / L Edd . . . . . . Ω / Ω K t , s p =2 p =1 Fig. 5.
Same as right panel of Fig. 4 but for p = left panel ) and p = right panel ). .
05 0 .
10 0 . L / L Edd . . . . . . Ω / Ω K t , s .
05 0 .
10 0 . L / L Edd . . . . . . . . Ω / Ω K t , s D =0.25 D =1 Fig. 6.
Same as right panel of Fig. 4 but for D = .
25 ( left panel ) and D = right panel ). These estimates suggest that, instead of being an independentparameter, q should depend on the rotation frequency of the BL.It is unclear if q should change with the variations of Ω . If q de-pends on the mean or instantaneous value of Ω , Eq. (12) predictsan attractor for Ω / Ω K and q . Combining Eqs. (12) and (22), we get q = + Ω NS / Ω K , (23) Article number, page 6 of 8avel Abolmasov and Juri Poutanen: QPOs from a neutron-star boundary layer .
05 0 .
10 0 . L / L Edd . . . . . . . . Ω / Ω K t , s Fig. 7.
Same as the right panel of Fig. 4 for a model with α = − and q given by Eq. (22). The horizontal red line corresponds to Ω eq given byEq. (24) and for the equilibrium rotation frequency Ω eq = Ω K + Ω NS . (24)In Fig. 7, we show how our dynamical model behaves ifthe depletion time depends on rotation frequency as t depl = ( Ω − Ω NS ) /α (cid:16) Ω − Ω (cid:17) for a fixed value of α , that implies q following Eq. (22). The parallel tracks e ff ect is still reproducedin this version of the model. Here, we considered the rotation frequency of the BL as a char-acteristic QPO frequency. Though it is probably not the case, thereal dynamical processes behind kHz QPOs are likely sensitiveto Ω . If the real oscillation frequencies are functions of Ω and L or M , the parallel tracks e ff ect is equally well reproduced, thoughthe parameters of the correlation with radiation flux become dif-ferent.In particular, for the Rossby-wave model considered inAbolmasov et al. (2020), the characteristic oscillation frequen-cies are the epicyclic frequency Ω e ≃ Ω cos θ, (25)where θ is the co-latitude of the region where the oscillationsare excited, and its aliases with rotation frequency, Ω e + n Ω ,where n is a whole number. The oscillations are likely excitedin the region of strongest latitudinal velocity shear, that is un-stable to supersonic shear instability. This naturally explains themultiplicity of kHz QPO frequencies and the di ff erence betweenthe frequencies that tends to be close to Ω NS (though not nec-essarily, see Méndez et al. 2001). Such a model also explainsthe characteristic values of the QPO frequencies and their cor-relation with the flux (cos θ is likely a growing function of L ,see Inogamov & Sunyaev 1999; Suleimanov & Poutanen 2006),and the di ff erent quality factors of the two QPO peaks (qualityfactors of the axisymmetric mode n = ff er, as visibility e ff ects enhance the periodic component in anon-axisymmetric case). It is unclear, however, how to explain the existence of only two QPO peaks (probably, n = − ff erent. If, instead of rotation frequency,we plot Ω e ( L ), the qualitative picture remains the same: tightcorrelation on the time scales about the time scales of the BL,that becomes worse on longer scales. The crucial point is the ex-istence of the second variable, BL mass, slowly changing withtime.In beat-frequency models of kHz QPO (Miller 2001), thehigher peak corresponds to rotation frequency somewhere in thedisk, and the lower – to the beat between the higher frequencyand stellar rotation. Both frequencies in such models changewith a single variable parameter, the radius in the disk where theoscillations are excited. This radius apparently should changeon the viscous timescale of the inner disk, and on longer times,the flux from the disk and the characteristic frequency shouldtightly correlate. A way to reproduce a parallel-track picture inthe framework of such a model is to add a contribution from theBL to the flux. The QPO frequency depends on the disk ratherthan total flux, and the dependence Ω ( L ) retains its slope butnot the constant. Apparently, this is not the case, as the slope ofthe short-time relation between flux and frequency also changesconsiderably (Méndez et al. 1999), suggesting that the frequencyitself is sensitive to the parameters of the BL rather than the disk.
5. Conclusions
We show that a very simple, zero-dimensional model of a BLaccumulating mass and angular momentum from the disk allowsto explain some of the properties of kHz QPOs. In particular, themodel naturally reproduces the parallel tracks e ff ect: the rotationfrequency of the BL correlates with its luminosity at small timescales, but becomes uncorrelated at longer time scales.Such a ‘integrator’ BL should have a distinct phase-lag sig-nature: at high frequencies, its mass and rotation frequencyshould lag the variations of the mass accretion rate by ∆ ϕ ≃ π / Acknowledgements.
This research was supported by the grant 14.W03.31.0021of the Ministry of Science and Higher Education of the Russian Federation andthe Academy of Finland grants 322779 and 333112. PA acknowledges supportfrom the Program of Development of M.V. Lomonosov Moscow State University(Leading Scientific School ‘Physics of stars, relativistic objects and galaxies’).We thank the anonymous referee for the valuable comments.
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