Oscillations of the Eddington capture sphere
^{1} Copernicus Astronomical Center, ul. Bartycka 18, 00716 Warszawa, Poland
email: wlodek@camk.edu.pl
^{2} Institute of Micromechanics and Photonics, ul. św A. Boboli 8, 02525 Warszawa, Poland
email: maciek.wielgus@gmail.com
^{3} Physics Department, Gothenburg University, 41296 Göteborg, Sweden
email: gusstaad@student.gu.se; marek.abramowicz@physics.gu.se
^{4} Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava, Bezručovo nám. 13, 74601 Opava, Czech Republic
Received: 14 August 2012
Accepted: 28 August 2012
We present a toy model of mildly superEddington, optically thin accretion onto a compact star in the Schwarzschild metric, which predicts periodic variations of luminosity when matter is supplied to the system at a constant accretion rate. These are related to the periodic appearance and disappearance of the Eddington capture sphere. In the model the frequency is found to vary inversely with the luminosity. If the input accretion rate varies (strictly) periodically, the luminosity variation is quasiperiodic, and the quality factor is inversely proportional to the relative amplitude of mass accretion fluctuations, with its largest value Q ≈ 1/(10 δṀ/Ṁ) attained in oscillations at about 1 to 2 kHz frequencies for a 2 M_{⊙} star.
Key words: accretion, accretion disks / gravitation / relativistic processes / stars: neutron / Xrays: binaries
© ESO, 2012
1. Introduction
Abramowicz et al. (1990, hereafter AEL) argued that the luminosity of a relativistic star that is accreting at a superEddington rate, should periodically change. They have shown that in the combined gravitational and radiation fields of a spherical, compact star, radially moving test particles are captured by a sphere on which the gravitational and radiative forces balance. This is because in Einstein’s general relativity the radiative force diminishes more strongly with the distance than the gravitational force, and radiation may be superEddington close to the star, but subEddington further away, reaching the Eddington value at the Eddington capture sphere (ECS), whose radius is given by a simple expression in the Schwarzschild coordinates (Phinney 1987), (1)Here R is the radius of the star, R_{G} = GM/c^{2} its gravitational radius, and L, which is assumed to satisfy Eq. (2), is the stellar luminosity at its surface. Bini et al. (2009); Oh et al. (2011), and Stahl et al. (2012) have shown that the ECS captures particles from a wide class of nonradial orbits as well^{1}, so the following discussion is not restricted to the case of radial accretion. However, it is important to keep in mind that the balance of forces necessary for the existence of the ECS requires a large, radial radiative flux. In nonspherical accretion this can only be realized in the optically thin regime.
In this paper we present a simple model in which the idea of periodic luminosity changes is realized. AEL assumed that the radiation power was provided by the kinetic energy of the accreted particles, released when they hit the surface of the star. As stressed by Stahl et al. (2012), particles settle on the ECS rather gently, so the ECS itself is not particularly luminous. If all particles are captured at the Eddington sphere, they do not reach the surface of the star, and the stellar accretion luminosity goes to zero. When it does, and actually already when L/L_{Edd} < (1−2R_{G}/R)^{−1/2}, the ECS disappears, accretion is resumed by the star and eventually the luminosity may build up to its former value, so the ECS will reappear and accretion will stop again. Thus, the accretion process may be quasiperiodic, alternating between states of high luminosity and no luminosity.
2. The model
The toy model considered here assumes a steady^{2} supply of optically thin fluid at some distance above the stellar surface. As shown by Stahl et al. (2012), unless its velocity is extraordinarily large the fluid will settle on the ECS, regardless of its point of origin, if only (2)In our model the stellar luminosity will be assumed to be either zero, or to have a certain definite value, L = L_{0}, satisfying Eq. (2). In this sense the model states are binary (onoff). This property implies that the ECS is located at a specific radius, r_{ECS} = r_{0} > R, whenever present (Eq. (1)). The model is described by three equations: where L_{ECS} is the stellar luminosity at the ECS, L_{ ∗ } is the luminosity at the stellar surface, Ṁ_{ ∗ } is the accretion rate at the stellar surface, μ is a redshift factor between R and r_{0}, η is the conversion factor between accretion rate and stellar luminosity.
The simplest possibility, which we will ignore, is that ηṀ_{0} < L_{Edd}(1−2R_{G}/R)^{−1/2}, so the critical luminosity for establishing the ECS is never reached, and the equations describe a steady state solution. We will assume instead that L_{0} = ηṀ_{0}, i.e., the stellar luminosity in the “on” state has the necessary value to establish an ECS at the radius r_{0}. This assumption allows nontrivial time behaviour of the model system.
There are three timescales that determine the behaviour of the model, the reaction time δt_{r}, which is the timescale for converting accreting matter into radiation, the light travel time δt_{l}, which is the travel time of light from the stellar surface to the ECS, and the infall time δt_{i}, which is the infall time of matter from the ECS to the stellar surface in the absence of radiation. Now, δt_{l} and δt_{i} can be determined from R_{G},R and r_{0}. So, for a given star these two timescales are a function of the stellar luminosity alone, i.e., of ηṀ_{0}. The reaction time introduces a phase shift between accretion and radiation at the stellar surface. In numerical examples we will adopt two extreme values δt_{r} = 0, or δt_{r} = 0.1 ms, the latter being the estimated cooling time of a clump of fluid that falls on a neutron star surface (Kluźniak et al. 1990).
In reality, the time delays would not be as sharp as we have assumed them to be. For instance, even at time δt_{l} after the luminosity is turned off, particles outside the ECS suffer from radiation drag for an additional interval of time (r − r_{0})/c, where r is the position of the particle at time δt_{l} + (r − r_{0})/c after the radiation is turned off. Conversely, after the luminosity is turned on again, all particles between the ECS and the stellar surface will feel the full impact of radiation pressure after a delay of only (r − R)/c < δt_{l}, where again, r is the position of the particle when the radiation front passes it, (r − R)/c after the radiation is turned on. In the toy model, we neglect such effects entirely.
Combining Eqs. (3)–(5) we find that Ṁ_{∗}(t) is determined by Ṁ_{∗}(t − T), with T = δt_{i} + δt_{r} + δt_{l}: where Θ is the Heaviside step function. Thus, and clearly, the model system shows periodic behaviour with the period 2T: (6)
Fig. 1 Stellar accretion rate and luminosities at the stellar surface and at the ECS, according to the model. 

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Fig. 2 Stellar accretion rate and the luminosity at the ECS, for another choice of initial conditions. 

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3. Results of the model
A typical behaviour of the model is shown in Fig. 1 which shows a solution of Eqs. (3) − (5) with δt_{l}:δt_{i}:δt_{r} = 8:17:10. The black solid line traces Ṁ_{∗}. In the figure, we identify the time b − a = e − d = δt_{r} as the reaction time, c − b = h − e = δt_{l} as the light travel time, and d − c = j − h = δt_{i} as the infall time. The luminosities at the stellar surface and at the ECS are shifted relative to the accretion rate by δt_{r} and δt_{r} + δt_{l}, respectively. We note also that d − a = j − d = h − c = k − e = e − b = δt_{r} + δt_{l} + δt_{i} = T, and the pulses are as long as the periods without activity. Thus the period of the oscillation is P = 2T, as expected from Eq. (6).
In constructing Fig. 1 it was assumed that matter first arrives at the stellar surface at t = 0 and that there is a continual inflow of matter to the system at the rate Ṁ_{0} for all t ≥ 0. Different initial conditions can lead to a more complicated pulse shape of Ṁ_{∗}(t) for t in the intervals (2nT,2nT + T), with its complement Ṁ_{∗}(t) = Ṁ_{0} − Ṁ_{∗}(t − T) in the intervals (2nT + T,2nT + 2T). Here, and elsewhere, n = 0,1,2,3 .... The state of accretion (Ṁ_{0} or 0) at time t has no influence on the future value of Ṁ_{∗} until the time t + T, so one can assume that at any instant in the initial interval t ∈ [0,T) the accretion rate has any of the two values, 0 or Ṁ_{0}, i.e., in this interval Ṁ_{∗}(t) = Ṁ_{0}g(t), where g(t) is an arbitrary binary function, mapping the initial time interval into onoff states, g: [0,T) → { 0,1 } . Figure 2 provides an example of such behaviour. Thus, in principle, the harmonic content of the signal may be quite rich, although the fundamental is still at f = 1/P = 1/(2T).
As can be seen from Eqs. (3)–(5) the model takes no account of accumulation of matter on the ECS, that is to say, here we ignore the arrival at t = 2(n + 1)T (and at other moments as well in the case of Fig. 2), of the additional matter that had accumulated on the ECS. In reality, this matter would produce a brief flash of radiation of energy η_{ECS}Ṁ_{0}ΔT, where η_{ECS} is the conversion efficiency to radiation of the kinetic energy of matter falling from the ECS, and ΔT is the accumulation time (ΔT = T in Fig. 1). If in the steady accretion phase matter is falling in from r ≫ r_{0}, one would expect η_{ECS} ≪ η.
Fig. 3 The semiperiod T of the oscillation in geometrical units as a function of peak luminosity at infinity in Eddington units, when kinetic energy is instantaneously converted to luminosity (δt_{r} = 0). The corresponding frequency f = 1/(2T) can be read off in kHz from the right vertical axis. The curves are labeled with the stellar radius (3, ...,10) in units of R_{G}. Filled squares indicate a value of the slope dlog T/dlog L_{∞} = 50, filled circles a value of 100, and the crosses the minimum value of the logarithmic derivative for each curve. 

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In a physically realistic situation the infall time is the dominant time scale, δt_{i} ≫ δt_{l}. The radius of the ECS, and hence also the infall time, increases rapidly with L_{∞}, and without bounds as L_{∞} → L_{Edd}: (7)Here, L_{∞} is the redshifted luminosity at infinity. Hence, except for the lowest values of the luminosity parameter L_{0}, the infall time δt_{i} is the dominant time scale, and the frequency of oscillations f = 1/(2T) varies inversely with the luminosity. Neglecting δt_{r}, we can compute the semiperiod T, and the frequency of oscillation as a function of the stellar radius R and of the luminosity L_{∞} aloneT and f will, respectively, scale directly or inversely with M. These are shown in Fig. 3 for various stellar radii. Figure 4 shows the frequency as a function of luminosity, for a star with M = 2 M_{⊙}, when δt_{r} = 0.1 ms.
Fig. 4 The frequency of the oscillations as a function of peak luminosity at infinity in Eddington units, when there is a delay in converting kinetic energy to luminosity δt_{r} = 0.1 ms (see Kluźniak et al. 1990). The stellar mass is assumed to be M = 2 M_{⊙}. The curves are labeled with the stellar radius (3, ...,10) in units of R_{G}. Filled squares indicate a value of the slope dlog T/dlog L_{∞} = 30, filled circles a value of 50, and the crosses the minimum value of the slope for each curve (boxed values). 

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4. Conclusions and discussion
We have shown that supplying mass to the vicinity of a compact star, continuously and at a constant accretion rate, can lead to periodic tophat variations of luminosity, if only the mass accretion rate corresponds to a mildly superEddington luminosity at the stellar surface. This oscillatory behaviour of luminosity is related to the phenomenon of the ECS, which is a consequence of the interplay of radiation drag and general relativity.
If such an oscillation occurs in the real world, for example in the Z sources, where rapid variations of the inferred inner radius of the accretion disk have been reported (Lin et al. 2009), the actual mechanism is likely to be more complex than the strictly periodic oscillation in the toy model considered here. E.g., the accretion rate is not likely to be constant (contrary to our assumption of simple onoff behaviour at the stellar surface).
As a minor extension of the model, consider an accretion rate that is varying on a timescale comparable to the period of the ECS oscillator. Even strictly periodic variation of Ṁ_{0} will lead to a decoherence of the ECS oscillation, because the position r_{0} of the ECS and (hence) the infall time from the ECS, and (hence) the oscillator frequency, are strongly varying functions of the luminosity, L_{0} = ηM_{0}. Indeed, (8)Thus, the Q factor of the oscillation is (9)We have indicated some values of the logarithmic derivative dlnT/dlnL_{∞} in Figs. 3, 4. Comparing the figures, we see that delayed emission (reaction time δt_{r} > 0) at the stellar surface acts as a lowpass filter, cutting out the highest frequencies. At the same time it improves the quality factor of the oscillator.
Accreting neutrons stars in low mass Xray binaries are expected to have a radius in the range of 4 to 7GM/c^{2} (Arnett & Bowers 1977; Kluźniak & Wagoner 1985), and a mass of about two solar masses. It is seen from Fig. 4 that the minimum values of the logarithmic derivative attained for radii of 4, 5, 6, 7GM/c^{2} are between 5 and 15. Hence, the maximum expected value of the oscillator quality factor would be Q ≈ 1/(10  δṀ_{0}/Ṁ_{0}  ), occurring for frequencies between about 1 and 2 kHz, as can be seen from Fig. 4.
As shown in detail by Oh et al. (2011), azimuthal radiation drag efficiently removes particle’s angular momentum, typically making motions in the combined gravitational and radiation fields asymptotically radial.
The assumption of a constant Ṁ will be relaxed in Sect. 4.
Acknowledgments
Research supported in part by Polish NCN grants UMO2011/01/B/ST9/05439 and N N203 511238.
References
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All Figures
Fig. 1 Stellar accretion rate and luminosities at the stellar surface and at the ECS, according to the model. 

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In the text 
Fig. 2 Stellar accretion rate and the luminosity at the ECS, for another choice of initial conditions. 

Open with DEXTER  
In the text 
Fig. 3 The semiperiod T of the oscillation in geometrical units as a function of peak luminosity at infinity in Eddington units, when kinetic energy is instantaneously converted to luminosity (δt_{r} = 0). The corresponding frequency f = 1/(2T) can be read off in kHz from the right vertical axis. The curves are labeled with the stellar radius (3, ...,10) in units of R_{G}. Filled squares indicate a value of the slope dlog T/dlog L_{∞} = 50, filled circles a value of 100, and the crosses the minimum value of the logarithmic derivative for each curve. 

Open with DEXTER  
In the text 
Fig. 4 The frequency of the oscillations as a function of peak luminosity at infinity in Eddington units, when there is a delay in converting kinetic energy to luminosity δt_{r} = 0.1 ms (see Kluźniak et al. 1990). The stellar mass is assumed to be M = 2 M_{⊙}. The curves are labeled with the stellar radius (3, ...,10) in units of R_{G}. Filled squares indicate a value of the slope dlog T/dlog L_{∞} = 30, filled circles a value of 50, and the crosses the minimum value of the slope for each curve (boxed values). 

Open with DEXTER  
In the text 