Oscillations of the Eddington Capture Sphere

We present a toy model of mildly super-Eddington, 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 quasi-periodic, and the quality factor is inversely proportional to the relative amplitude of mass accretion fluctuations, with its largest value approximately Q= 1/(10 |delta Mdot/Mdot|) attained in oscillations at about 1 to 2 kHz frequencies for a 2 solar mass star

Radiative corrections to the neutron star mass inferred from QPO frequencies

The frequencies of kHz QPOs are widely interpreted as being indicative of the values of characteristic frequencies related to orbital motion around neutron stars, e.g., the radial epicyclic frequency. In regions directly exposed to the radiation from the luminous neutron star these frequencies change with the luminosity. Including radiative corrections will change the neutron star mass value inferred from the QPO frequencies. Radiative forces may also be behind the puzzling phenomenon of parallel tracks.Comment: 6 pages including 1 figur

Escape, capture, and levitation of matter in Eddington outbursts

Context: An impulsive increase in luminosity by one half or more of the Eddington value will lead to ejection of all optically thin plasma from Keplerian orbits around the radiating star, if gravity is Newtonian and the Poynting-Robertson drag is neglected. Radiation drag may bring some particles down to the stellar surface. On the other hand, general relativistic calculations show that gravity may be balanced by a sufficiently intense radiation field at a certain distance from the star. Aims: We investigate the motion of test particles around highly luminous stars to determine conditions under which plasma may be ejected from the system. Results: In Einstein's gravity, if the outburst is close to the Eddington luminosity, all test particles orbiting outside an "escape sphere" will be ejected from the system, while all others will be captured from their orbits onto the surface of another sphere, which is well above the stellar surface, and may even be outside the escape sphere, depending on the value of luminosity. Radiation drag will bring all the captured particles to rest on this "Eddington capture sphere," where they will remain suspended in an equilibrium state as long as the local flux of radiation does not change and remains at the effective Eddington value.Comment: 6 pages, 6 figures. To be published in Astronomy and Astrophysic

Mass of a Black Hole Firewall

Quantum entanglement of Hawking radiation has been supposed to give rise to a Planck density "firewall" near the event horizon of old black holes. We show that Planck density firewalls are excluded by Einstein's equations for black holes of mass exceeding the Planck mass. We find an upper limit of $1/(8\pi M)$ to the surface density of a firewall in a Schwarzschild black hole of mass $M$, translating for astrophysical black holes into a firewall density smaller than Planck density by more than 30 orders of magnitude. A strict upper limit on the firewall density is given by the Planck density times the ratio $M_{\rm Pl}/(8\pi M)$.Comment: 6 pages, version published in Phys. Rev. Let

Epicyclic orbital oscillations in Newton's and Einstein's dynamics

We apply Feynman's principle, ``The same equations have the same solutions'', to Kepler's problem and show that Newton's dynamics in a properly curved 3-D space is identical with that described by Einstein's theory in the 3-D optical geometry of Schwarzschild's spacetime. For this reason, rather unexpectedly, Newton's formulae for Kepler's problem, in the case of nearly circular motion in a static, spherically spherical gravitational potential accurately describe strong field general relativistic effects, in particular vanishing of the radial epicyclic frequency at the marginally stable orbit.Comment: 8 page

The upper kHz QPO: a gravitationally lensed vertical oscillation

We show that a luminous torus in the Schwarzschild metric oscillating along its own axis gives rise to a periodically varying flux of radiation, even though the source of radiation is steady and perfectly axisymmetric. This implies that the simplest oscillation mode in an accretion flow, axisymmetric up-and-down motion at the meridional epicyclic frequency, may be directly observable when it occurs in the inner parts of accretion flow around neutron stars and black holes. The high-frequency modulations of the X-ray flux observed in low-mass X-ray binaries at two frequencies (twin kHz QPOs) could then be a signature of strong gravity both because radial and meridional oscillations have different frequencies in non-Newtonian gravity, and because strong gravitational deflection of light rays causes the flux of radiation to be modulated at the higher frequency.Comment: 8 p., 4 fig

A precise determination of angular momentum in the black hole candidate GRO J1655-40

We note that the recently discovered 450 Hz frequency in the X-ray flux of the black hole candidate GRO J1655-40 is in a 3:2 ratio to the previously known 300 Hz frequency of quasi-periodic oscillations (QPO) in the same source. If the origin of high frequency QPOs in black hole systems is a resonance between orbital and epicyclic motion of accreting matter, as suggested previously, the angular momentum of the black hole can be accurately determined, given its mass. We find that the dimensionless angular momentum is in the range $0.2<j<0.65$ if the mass is in the (corresponding) range of 5.5 to 7.9 solar masses

The centrifugal force reversal and X-ray bursts

Heyl (2000) made an interesting suggestion that the observed shifts in QPO frequency in type I X-ray bursts could be influenced by the same geometrical effect of strong gravity as the one that causes centrifugal force reversal discovered by Abramowicz and Lasota (1974). However, his main result contains a sign error. Here we derive the correct formula and conclude that constraints on the M(R) relation for neutron stars deduced from the rotational-modulation model of QPO frequency shifts are of no practical interest because the correct formula implies a weak condition R* > 1.3 Rs, where Rs is the Schwarzschild radius. We also argue against the relevance of the rotational-modulation model to the observed frequency modulations.Comment: 3 pages, Minor revisions, A&A Letters, in pres

Epicyclic frequencies derived from the effective potential: simple and practical formulae

We present and discuss a short and simple derivation of orbital epicyclic frequencies for circular geodesic orbits in stationary and axially symmetric spacetimes. Such spacetimes include as special cases analytically known black hole Kerr and Schwarzschild spacetimes, as well as the analytic Hartle-Thorne spacetime and all numerically constructed spacetimes relevant for rotating neutron stars. Our derivation follows directly from energy and angular momentum conservation and it uses the concept of the effective potential. It has never been published, except for a few special cases, but it has already become a part of the common knowledge in the field.Comment: Invited lecture at the conference "From X-ray Binaries to Quasars: Black Hole Accretion on All Mass Scales", 13-15 July, 2004, Amsterda

Eddington Capture Sphere around luminous stars

Test particles infalling from infinity onto a compact spherical star with a mildly super-Eddington luminosity at its surface are typically trapped on the "Eddington Capture Sphere" and do not reach the surface of the star. The presence of a sphere on which radiation pressure balances gravity for static particles was first discovered some twenty five years ago. Subsequently, it was shown to be a capture sphere for particles in radial motion, and more recently also for particles in non-radial motion, in which the Poynting-Robertson radiation drag efficiently removes the orbital angular momentum of the particles, reducing it to zero. Here we develop this idea further, showing that "levitation" on the Eddington sphere (above the stellar surface) is a state of stable equilibrium, and discuss its implications for Hoyle-Lyttleton accretion onto a luminous star. When the Eddington sphere is present, the cross-section of a compact star for actual accretion is typically less than the geometrical cross-section (pi Rsquared), direct infall onto the stellar surface only being possible for relativistic particles, with the required minimum particle velocity at infinity typically ~1/2 the speed of light. We further show that particles on typical trajectories in the vicinity of the stellar surface will also be trapped on the Eddington Capture Sphere.Comment: 6 pages, 13 panels in 8 figure