Relativistic Thin Disks Research Articles

Using X-ray continuum-fitting to estimate the spin of MAXI J1305–704

ABSTRACT MAXI J1305–704 is a transient X-ray binary with a black hole primary. It was discovered on 2012 April 9, during its only known outburst. MAXI J1305–704 is also a high inclination low-mass X-ray binary with prominent dip features in its light curves, so we check the full catalogue of 92 Swift/XRT continuous observations of MAXI J1305–704, focusing only on the stable spectra. We select 13 ‘gold’ spectra for which the rms <0.075 and the coronal scattered fraction $f_{\mathrm{sc}} \lesssim 25$ per cent. These ‘gold’ data are optimal thermal-state observations for continuum-fitting modelling, in which the disc extends to the innermost stable circular orbit and is geometrically thin. The black hole spin was unknown for this object before. By utilizing the X-ray continuum fitting method with the relativistic thin disc model kerrbb2 and supplying the known dynamical binary system parameters, we find MAXI J1305–704 has a moderate spin ($a_{*}=0.87_{-0.13}^{+0.07}$) at a 68.3 per cent confidence level. This is the first determination of MAXI J1305–704’s spin.

Asymptotic Green’s function solutions of the general relativistic thin disc equations

ABSTRACT The leading order Green’s function solutions of the general relativistic thin disc equations are computed, using a pseudo-Newtonian potential and asymptotic Laplace mode matching techniques. This solution, valid for a vanishing innermost stable circular orbit (ISCO) stress, is constructed by ensuring that it reproduces the leading order asymptotic behaviour of the near-ISCO, Newtonian, and global Wentzel–Kramers–Brillouin limits. Despite the simplifications used in constructing this solution, it is typically accurate, for all values of the Kerr spin parameter a and at all radii, to less than a per cent of the full numerically calculated solutions of the general relativistic disc equations. These solutions will be of use in studying time-dependent accretion discs surrounding Kerr black holes.

A full relativistic thin disc – the physics of the plunging region and the value of the stress at the ISCO

ABSTRACT The widely used Novikov–Thorne relativistic thin disc equations are only valid down to the radius of the innermost stable circular orbit (ISCO). This leads to an undetermined boundary condition at the ISCO, known as the inner stress of the disc, which sets the luminosity of the disc at the ISCO and introduces considerable ambiguity in accurately determining the mass, spin, and accretion rate of black holes from observed spectra. We resolve this ambiguity by self-consistently extending the relativistic disc solution through the ISCO to the black hole horizon by calculating the inspiral of an average disc particle subject to turbulent disc forces, using a new particle-in-disc technique. Traditionally it has been assumed that the stress at the ISCO is zero, with material plunging approximately radially into the black hole at close to the speed of light. We demonstrate that in fact the inspiral is less severe, with several (∼4–17) orbits completed before the horizon. This leads to a small non-zero stress and luminosity at and inside the ISCO, with a local surface temperature at the ISCO between ∼0.15 and 0.3 times the maximum surface temperature of the disc, in the case where no dynamically important net magnetic field is present. For a range of disc parameters we calculate the value of the inner stress/surface temperature, which is required when fitting relativistic thin disc models to observations. We resolve a problem in relativistic slim disc models in which turbulent heating becomes inaccurate and falls to zero inside the plunging region.

The spectral evolution of disc dominated tidal disruption events

ABSTRACT We perform a detailed numerical and analytical study of the properties of observed light curves from relativistic thin discs, focussing on observational bands most appropriate for comparison with tidal disruption events (TDEs). We make use of asymptotic expansion techniques applied to the spectral emission integral, using time-dependent disc temperature profiles appropriate for solutions of the relativistic thin disc equation. Rather than a power law associated with bolometric disc emission L ∼ t−n, the observed X-ray flux from disc-dominated TDEs will typically have the form of a power law multiplied by an exponential (see equation 91). While precise details are somewhat dependent on the nature of the ISCO stress and disc-observer orientational angle, the general form of the time-dependent flux is robust and insensitive to the exact disc temperature profile. We present numerical fits to the UV and X-ray light curves of ASASSN-14li, a particularly well observed TDE. This modelling incorporates strong gravity optics. The full 900 d of ASASSN-14li X-ray observations are very well fit by a simple relativistic disc model, significantly improving upon previous work. The same underlying model also fits the final 1000 d of ASASSN-14li observations in three different UV bandpasses. Finally, we demonstrate that the analytic formulae reproduce the properties of full numerical modelling at both UV and X-ray wavelengths with great fidelity.

Evolution of relativistic thin discs with a finite ISCO stress – I. Stalled accretion

ABSTRACT We present solutions to the relativistic thin disc evolutionary equation using an α-model for the turbulent stress tensor. Solutions with a finite stress at the innermost stable circular orbit (ISCO) give rise to bolometric light curves with a shallow power-law time dependence, in good agreement with those observed in tidal disruption events. A self-similar model based on electron scattering opacity, for example, yields a power-law index of −11/14, as opposed to −19/16 for the case of zero ISCO stress. These solutions correspond to an extended period of relaxation of the evolving disc which, like the light curves they produce, is not sustainable indefinitely. Cumulative departures from the approximation of exact circular orbits cause the power-law index to evolve slowly with time, leading eventually to the steeper fall-off associated with traditional zero ISCO stress models. These modified solutions are discussed in detail in a companion paper.

Evolution of relativistic thin discs with a finite ISCO stress – II. Late time behaviour

ABSTRACT We present solutions to the relativistic thin disc evolutionary equation using a modified description of the mean fluid flow within the disc. The model takes into account the effects of sub-circular velocities in the innermost disc regions, and resolves otherwise unsustainable behaviour present in simple finite innermost stable circular orbit (ISCO) stress disc models. We show that the behaviour of a relativistic thin disc evolving with a finite ISCO stress is comprised of three distinct stages which join the ordinarily distinct finite and vanishing ISCO stress solutions into a fully continuous model parametrization. The most important prediction of our model is the existence of an intermediate stage of ‘stalled accretion’, controlled by a single dimensionless parameter. The hallmarks of this evolutionary phase appear to have been seen in General Relativistic MHD simulations as well as in the late time X-ray observations of tidal disruption events, but dedicated simulations and extended observations are needed for a deeper understanding.

HST unveils a compact mildly relativistic broad-line region in the candidate true type 2 NGC 3147

ABSTRACT NGC 3147 has been considered the best case of a true type 2 AGN: an unobscured AGN, based on the unabsorbed compact X-ray continuum, which lacks a broad-line region (BLR). However, the very low luminosity of NGC 3147 implies a compact BLR, which produces very broad lines, hard to detect against the dominant background host galaxy. Narrow (0.1 arcsec × 0.1 arcsec) slit HST spectroscopy allowed us to exclude most of the host galaxy light, and revealed an H α line with an extremely broad base (FWZI${\sim }27\, 000$ km s−1). The line profile shows a steep cut-off blue wing and an extended red wing, which match the signature of a mildly relativistic thin accretion disc line profile. It is indeed well fit with a nearly face on thin disc, at i ∼ 23°, with an inner radius at 77 ± 15 rg, which matches the prediction of $62^{+18}_{-14}$rg from the RBLR–L1/2 relation. This result questions the very existence of true type 2 AGN. Moreover, the detection of a thin disc, which extends below 100 rg in an L/LEdd ∼ 10−4 system, contradicts the current view of the accretion flow configuration at extremely low accretion rates.

Broad-band reflection spectroscopy of MAXI J1535–571 using AstroSat: estimation of black hole mass and spin

We report the results from \textit{AstroSat} observations of the transient Galactic black hole X-ray binary MAXI J1535-571 during its hard-intermediate state of the 2017 outburst. We systematically study the individual and joint spectra from two simultaneously observing \textit{AstroSat} X-ray instruments, and probe and measure a number of parameter values of accretion disc, corona and reflection from the disc in the system using models with generally increasing complexities. Using our broadband ($1.3-70$ keV) X-ray spectrum, we clearly show that a soft X-ray instrument, which works below $\sim 10-12$ keV, alone cannot correctly characterize the Comptonizing component from the corona, thus highlighting the importance of broadband spectral analysis. By fitting the reflection spectrum with the latest version of the \textsc{relxill} family of relativistic reflection models, we constrain the black hole's dimensionless spin parameter to be $0.67^{+0.16}_{-0.04}$. We also jointly use the reflection spectral component (\textsc{relxill}) and a general relativistic thin disc component (\texttt{Kerrbb}), and estimate the black hole's mass and distance to be $10.39_{-0.62}^{+0.61} M_{\odot}$ and $5.4_{-1.1}^{+1.8}$ kpc respectively.

Relativistic disks with two charged perfect fluids components

A method to describe exact solutions of the Einstein-Maxwell field equations in terms of relativistic thin disks constituted by two perfect charged fluids is presented. Describing the surface of the disk as a single charged fluid we find explicit expressions for the rest energies, the pressures and the electric charge densities of the two fluids. An explicit example is given. The particular case of the thin disks composed by two charged perfect fluids with barotropic equation of state is also presented.

Self-trapping of g-mode oscillations in relativistic thin discs, revisited – II. Revision of boundary condition

In the previous paper, we have examined how the self-trapping of g-mode oscillations in geometrically thin relativistic discs is affected by the presence of uniform vertical magnetic fields. Discs considered are isothermal in the vertical direction and are truncated at a certain height by presence of hot corona. After the correction of simple analytical error, we showed that the self-trapping of axisymmetric g-mode oscillations in non-magnetized discs is destroyed by weak magnetic fields as Fu and Lai showed. In this paper, however, we re-examine the same problem by imposing a different, more relevant boundary condition on the disc-corona surface and find that the self-trapping of axisymmetric g-mode oscillations seems to still exist like in the case of non-magnetized discs.

The general relativistic thin disc evolution equation

In the classical theory of thin disc accretion discs, the constraints of mass and angular momentum conservation lead to a diffusion-like equation for the turbulent evolution of the surface density. Here, we revisit this problem, extending the Newtonian analysis to the regime of Kerr geometry relevant to black holes. A diffusion-like equation once again emerges, but now with a singularity at the radius at which the effective angular momentum gradient passes through zero. The equation may be analysed using a combination of WKB, local techniques, and matched asymptotic expansions. It is shown that imposing the boundary condition of a vanishing stress tensor (more precisely the radial-azimuthal component thereof) allows smooth stable modes to exist external to the angular momentum singularity, the innermost stable circular orbit, while smoothly vanishing inside this location. The extension of the disc diffusion equation to the domain of general relativity introduces a new tool for numerical and phenomenolgical studies of accretion discs, and may prove to be a useful technique for understanding black hole X-ray transients.

Stability aspects of relativistic thin magnetized disks

We adapt the well known "displace, cut and reflect" method to construct exact solutions of the Einstein-Maxwell equations corresponding to infinitesimally thin disks of matter endowed with dipole magnetic fields, which are entirely supported by surface polar currents on the disk. Our starting point is the Gutsunaev-Manko axisymmetric solution describing massive magnetic dipoles in General Relativity, from which we obtain a continuous three-parameter family of asymptotically flat static magnetized disks with finite mass and energy. For strong magnetic fields, the disk surface density profile resembles some well known self-gravitating ring-like structures. We show that many of these solutions can be indeed stable and, hence, they could be in principle useful for the study of the abundant astrophysical situations involving disks of matter and magnetic fields.

On the Influence of the Mass Definition in the Stability of Axisymmetric Relativistic Thin Disks

Uno de los criterios más importantes para la caracterización de modelos galácticos o discos de acreción astrofísıcamente relevantes, es el análisis de la estabilidad de dichos modelos. En este trabajo, se realiza un análisis de la estabilidad de discos delgados estáticos relativistas con simetría axial, mediante la introduccón de una perturbación de primer orden en el tensor de energía-impulso del fluido. El formalismo se aplica a tres modelos construidos con el método de desplazamiento-corte-reflexión (DCR), previamente considerados en la literatura (Ujevicand Letelier, 2004), pero modificando el criterio de masa, es decir, usando la masa de Komar en lugar de la masa total superficial. Bajo estas condiciones, se encuentra que los valores de masa total son independientes de los parámetros del método DCR, lo que permite elegir la condición de frontera para el radio de corte que tome el valor máximo y a la vez permita una perturbación apreciable y bien comportada en el disco. Como resultado general, se encuentra que para la mayoría de modos de oscilación, la masa de Komar es más apropiada para definir el radio de corte.

Erratum: Self-trapping of g-mode oscillations in relativistic thin discs, revisited

Erratum: Self-trapping of g-mode oscillations in relativistic thin discs, revisited

Variational thermodynamics of relativistic thin disks

We present a relativistic model describing a thin disk system composed of two fluids. The system is surrounded by a halo in the presence of a non-trivial electromagnetic field. We show that the model is compatible with the variational multi-fluid thermodynamics formalism, allowing us to determine all the thermodynamic variables associated with the matter content of the disk. The asymptotic behaviour of these quantities indicates that the single fluid interpretation should be abandoned in favour of a two-fluid model.

Axially symmetric relativistic thin disks immersed in spheroidal matter haloes

Two infinite families of axially symmetric relativistic thin disks of dust immersed in spheroidal matter haloes are presented. The disks are obtained from solutions of the Einstein equations for an axially symmetric conformastaticspacetimein which the metric tensor is characterised only by one metric function. By introducing a finite discontinuity on the first derivatives of the metric tensor, solutions with a singularity of the delta function type are obtained, so describing thin disks. The nonzero components of the energy-momentum tensor, both for the disk and the halo, are obtained from the Einstein equations. In this way, the energy densities and pressures of the sources are determined. By imposing the fulfilment of all the energy conditions we obtain a constraint over the solutions, in such a way that the metric function can be properly expressed in terms of a solution of the Laplace equation. By using the solution to Laplace equation in cylindrical coordinates we find infinite disks and by using the solution to Laplace equation in oblate spheroidal coordinates we find finite disks. In both cases we obtain particular solutions with energy densities and pressures well behaved everywhere. We also show that the masses of de disks and the haloes are finite. Finally we solve the geodesic equation for circular orbits in the plane of the disk.

Stationary axially symmetric relativistic thin discs with nonzero radial pressure

A detailed analysis of the surface energy–momentum (SEMT) tensor of stationary axially symmetric relativistic thin discs with nonzero radial pressure is presented. The physical content of the SEMT is analysed and expressions for the velocity vector, energy density, principal stresses and heat flow are obtained. We also present the counter-rotating model interpretation for these discs by considering the SEMT as the superposition of two counter-rotating perfect fluids. We analyse the possibility of counter-rotation along geodesics as well as counter-rotation with equal and opposite tangential velocities, and explicit expressions for the velocities are obtained in both the cases. By assuming a given choice for the counter-rotating velocities, explicit expressions for the energy densities and pressures of the counter-rotating fluids are then obtained. Some simple thin disc models obtained from the Kerr solution are also presented.

A fully relativistic twisted disc around a slowly rotating Kerr black hole: derivation of dynamical equations and the shape of stationary configurations

(abbreviated) In this paper we derive equations describing dynamics and stationary configurations of a twisted fully relativistic thin accretion disc around a slowly rotating black hole. We find that the disc dynamics and stationary shapes are determined by a pair of equations for two complex variables describing orientation of the disc rings and velocity perturbations in the disc. We analyse shapes of stationary twisted configurations. It is shown that the stationary configurations depend on two parameters - the $\alpha $parameter and $\tilde \delta = \delta_{*}/\sqrt a$, where $\delta_{*}\sim h/r$ is the disc opening angle (h is the disc halfthickness) and $a$ is the black hole rotational parameter. When $a > 0$ and $\tilde \delta \ll 1$ the shapes depend drastically on value of $\alpha$. When $\alpha $ is small the disc inclination angle oscillates with radius with amplitude and radial frequency of the oscillations dramatically increasing towards the last stable orbit. For moderate values of $\alpha $ the oscillations do not take place but the disc does not align with the equatorial plane at small radii. Its inclination angle is either increasing towards $R_{ms}$ or exhibits a non-monotonic dependence on the radial coordinate. Finally, when $\alpha $ is sufficiently large the disc aligns with the equatorial plane at small radii. When $a < 0$ the disc aligns with the equatorial plane for all values of $\alpha $. The results reported here may have implications for determining structure and variability of accretion discs close to $R_{ms}$ as well as for modelling of emission spectra coming from different sources, which are supposed to contain black holes.

General magnetized Weyl solutions: disks and motion of charged particles

We construct three families of general magnetostatic axisymmetric exact solutions of Einstein-Maxwell equations in spherical coordinates, prolate, and oblates. The solutions obtained are then presented in the system of generalized spheroidal coordinates which is a generalization of the previous systems. The method used to build such solutions is the well-known complex potential formalism proposed by Ernst, using as seed solutions vacuum solutions of the Einstein field equations. We show explicitly some particular solutions among them a magnetized Erez-Rosen solution and a magnetized Morgan-Morgan solution, which we interpret as the exterior gravitational field of a finite dislike source immersed in a magnetic field. From them we also construct using the well known "displace, cut and reflect" method exact solutions representing relativistic thin disks of infinite extension. We then analyze the motion of electrically charged test particles around these fields for equatorial circular orbits and we discuss their stability against radial perturbations. For magnetized Morgan-Morgan fields we find that inside of disk the presence of magnetic field provides the possibility of to find relativist charged particles moving in both prograde and retrograde direction.

Lindblad resonance torques in relativistic discs - I. Basic equations

Lindblad resonances have been suggested as an important mechanism for angular momentum transport and heating in discs in binary black hole systems. We present the basic equations for the torque and heating rate for relativistic thin discs subjected to a perturbation. The Lindblad resonance torque is written explicitly in terms of metric perturbations for an equatorial disc in a general axisymmetric, time-stationary spacetime with a plane of symmetry. We show that the resulting torque formula is gauge-invariant. Computations for the Schwarzschild and Kerr spacetimes are presented in the companion paper.