Relativistic reflection within an extended hot plasma geometry

TL;DR

Relativistic reflection near black holes depends sensitively on the corona geometry. The authors develop relxill_ring, a fast general-relativistic extension of relxill that models a radially extended, ring-like corona and its two-dimensional positioning, enabling constraints on corona size from X-ray spectra. Applying the method to ESO 033-G002 with XMM-Newton and NuSTAR yields $x<2.1\,r_g$ and $h<2.4\,r_g$, indicating a compact corona near the ISCO, with the angular position largely unconstrained; the data remain well described by both ring-like and lamp-post geometries. This ring model thus lifts the previous limitation of assuming point-like coronæ, offers a direct spectroscopic size constraint, and is publicly available for broader use in accretion physics studies.

Abstract

The reflection of X-rays at the inner accretion disk around black holes imprints relativistically broadened features in the observed spectrum. Aside from the black hole properties and the ionization and density of the accretion disk, these features also depend on the location and geometry of the primary source of X-rays, often referred to as the corona. We present a fast general relativistic model for spectral fitting of a radially extended, ring-like corona above the accretion disk. A common approach used to explain observed X-ray reflection spectra is the lamp post geometry, which assumes a point-like source on the rotational axis of the black hole. While it is typically able to explain the observations, this geometric model does not allow for any constraint to be placed on the radial size of the corona. We therefore extended the publicly available relativistic reflection model relxill by implementing a radially extended, ring-like primary source. With the new RELXILL model allowing us to vary the position of the primary source in two dimensions, we present simulated line profiles and spectra and discuss the implications of carrying out a data fitting, in comparison to the lamp post model. We applied this extended RELXILL model to XMM-Newton and NuSTAR data of the radio-quiet Seyfert-2 active galactic nucleus (AGN) ESO 033-G002. The new model describes the data well and we are able to constrain the distance of the source to the black hole to be less than three gravitational radii, while the angular position of the source is poorly constrained. We show that a compact, radially extended corona close to the innermost stable circular orbit is able to explain the observed relativistic reflection as well as the lamp post corona does. This model has been made freely available to the community.

Figures (10)

• Figure 1: Individually selected isotropically distributed photon trajectories emitted by a point source at $h = 5r_\mathrm{g}\xspace$, $x\xspace = 2r_\mathrm{g}\xspace$. The BH has a spin of $a=0.998$ and is located at $x^\prime = 0, y^\prime = 0, z^\prime = 0$. The event horizon is illustrated in black. The accretion disk plane is $z^\prime = 0$. The dark red ring is the primary source after axial averaging. Red trajectories plunge into the BH, green trajectories escape the system, and blue trajectories hit the accretion disk, where they will be reflected. The point source is rotating in the direction denoted by the dark red arrow with a velocity $v_{\varphi}$ as given by Eq. \ref{['eq:velocity']}. We also display the corresponding spherical coordinates of the source, $(r\xspace, \theta)$, related to $(h, x\xspace)$ by Eq. \ref{['eq:metric_to_hx']}.
• Figure 2: Energy shift, $g$, of photons propagating from off-axis sources, located at $h = 3r_\mathrm{g}\xspace$, at two ring radii $x\xspace = 2r_\mathrm{g}\xspace$ (blue lines) and $x\xspace = 5r_\mathrm{g}\xspace$ (red lines), and BH spin of $a = 0.998$. Photon trajectories are selected from an isotropic distribution of the initial photon directions in the rest frame of the sources. Lighter lines show energy shifts of photons emitted by the off-axis sources located on the same ring at different azimuthal angles. Thick lines represent the resulting averaged energy shifts over the disk annuli. The dashed black line corresponds to the energy shift of the photons from a lamp post source with $h = 3r_\mathrm{g}\xspace$.
• Figure 3: Disk irradiation flux profiles for varying emitting ring radii, $x$, for (a) a ring at $h = 3r_\mathrm{g}\xspace$ and (b) a ring of varying height with radius fixed at $x\xspace = 3r_\mathrm{g}\xspace$, a BH spin of $a = 0.998$, and an irradiating source with $\Gamma = 2$. The solid black line is for the lamp post case, $x\xspace = 0$, for $h=3r_\mathrm{g}\xspace$ in (a) and $h=6r_\mathrm{g}\xspace$ in (b). The dashed line shows a power-law flux $\propto r_\mathrm{d}\xspace^{-3}$. Markers of the same color on each curve show the disk radius equal to the primary source radius, $r_\mathrm{d}\xspace \equiv x\xspace$. The normalization of the fluxes is set to the non-relativistic limit at large radii (see Appendix \ref{['app:implementation']}).
• Figure 4: Line profiles for a ring geometry simulated with relxill. We display the geometric parameters of the ring in terms of spherical radii, $r$, and polar angles, $\theta$. The parameters are uniquely related to height and ring radius through Eq. \ref{['eq:metric_to_hx']}. For ease of comparison, we also convert these values to heights and radii in Table \ref{['table:rtheta2hx']}. (a) Varying polar angle of the primary source, $\theta$, for two spherical radii, $r\xspace = 3r_\mathrm{g}\xspace$ (dashed lines) and $r\xspace = 9r_\mathrm{g}\xspace$ (solid lines), with $a = 0.998$. (b) Varying the spherical radius of the primary source, $r$, with two polar angles of the primary source, $\theta = 30^{\circ}$ (solid lines) and $\theta = 0^{\circ}$ (dashed lines, the lamp post case), with $a = 0.998$. (c) Varying the BH spin, $a$, for a source spherical radius fixed at ISCO value, $r\xspace = r_\mathrm{ISCO}\xspace$, and $\theta = 60^{\circ}$. ISCO radii are $1.24r_\mathrm{g}\xspace$, $6r_\mathrm{g}\xspace$, $9r_\mathrm{g}\xspace$ for $a = 0.998$, $0.0$, $-0.998$, respectively. To convert the quantities $(r\xspace, \theta)$ to $(h, x\xspace)$, we provide the Table \ref{['table:rtheta2hx']}. All lines are normalized to have the same integrated photon flux. The photon index is $\Gamma = 2$ and the inclination to the observer is $i = 30^{\circ}$.
• Figure 5: Relativistic reflection spectra for varying ring radius, $x\xspace$. (a) Solid lines show simulated reflected spectra for BH spin of $a = 0.998$, height of the primary source, $h = 3r_\mathrm{g}\xspace$, and varying ring radius. Dashed lines show the primary spectra. Different colored lines correspond to different ring radii. The black line corresponds to the lamp post case, $x\xspace = 0$. The spectra are normalized setting $\textsc{boost}\xspace = 1$ in relxill, such that the reflection fraction is predicted from the geometry of the isotropically emitting primary source. Returning radiation is turned off. All other parameters of relxill are set to their default values, i.e., the iron abundance is $A_{\mathrm{Fe}}\xspace = 1$, the inclination is $i = 30^{\circ}$, the disk inner edge is at $r_\mathrm{ISCO}\xspace$, the disk outer edge is at $400r_\mathrm{g}\xspace$, the photon index is $\Gamma = 2$, the disk ionization is $\log\xi\xspace = 3.1$, the disk density $\log n\xspace = 10^{15}\,\mathrm{cm}^{-3}$, and the electron temperature $k T_{\mathrm{e}}\xspace=60$ keV. (b) Ratio between the given reflected spectrum of the ring and the lamp post spectrum, re-normalized to the same flux.