Relativistic radiative transfer problems require the calculation of photon trajectories in curved spacetime. We present a novel technique for rapid and accurate calculation of null geodesics in the Kerr metric. The equations of motion from the Hamilton-Jacobi equation are reduced directly to Carlson's elliptic integrals, simplifying algebraic manipulations and allowing all coordinates to be computed semi-analytically for the first time. We discuss the method, its implementation in a freely available FORTRAN code, and its application to toy problems from the literature.
Deep Dive into A Fast New Public Code for Computing Photon Orbits in a Kerr Spacetime.
Relativistic radiative transfer problems require the calculation of photon trajectories in curved spacetime. We present a novel technique for rapid and accurate calculation of null geodesics in the Kerr metric. The equations of motion from the Hamilton-Jacobi equation are reduced directly to Carlson’s elliptic integrals, simplifying algebraic manipulations and allowing all coordinates to be computed semi-analytically for the first time. We discuss the method, its implementation in a freely available FORTRAN code, and its application to toy problems from the literature.
1. INTRODUCTION Efficient and accurate computation of null geodesics in the vicinity of spinning black holes is important for studies of active galaxies, X-ray binaries, and other accreting black hole systems. The radiated flux from accretion disks mostly originates in the innermost radii, where relativistic effects are important for understanding observations. Proper calculation of the bending of light requires integration along rays (Broderick 2006). In general, propagation through the plasma will influence the photon trajectories, leading to non-geodesic paths (Broderick & Blandford 2003, 2004). However, these effects are mostly important at low frequencies, comparable to the expected plasma and cyclotron frequency. When plasma effects can be neglected, the rays correspond to null geodesics, and these circumstances are assumed throughout this paper.
The first applications of general relativistic radiative transfer to accreting systems were of two main types. Cunningham (1975) packaged all radiative effects for optically thick, geometrically thin disks as a transfer function to go from local emissivity to that observed at infinity. Luminet (1979) used the simple relationships between impact parameters at infinity and constants of the motion to shoot rays backwards in time from an observer’s photographic plate to the object under study. More recently, Viergutz (1993) and Beckwith & Done (2005) considered the so-called emitter-observer problem. That is, given locations of the emitter and the observer, determine the constants of the motion for null geodesics connecting the two. This approach is much more efficient when the source is highly localized, such as an orbiting star or hotspot. Here, backwards ray shooting is impractical since most of the rays miss the target.
Such techniques have been applied to the study of emission lines and spectra from active galactic nuclei (AGN) accretion disks and tori (Cadez et al. 1998;Wu & Wang 2007) as well as their quasi-periodic oscillations (QPOs) (Schnittman et al. 2006). Li et al. (2005) used a ray tracing approach to study the spectra of X-ray binaries.
Electronic address: jdexter@u.washington.edu Noble et al. (2007) created images of galactic center black hole candidate Sagittarius A* (Sgr A*) using axisymmetric general relativistic MHD (GRMHD) simulations, and Bromley et al. (2001) studied its polarization from a simplified accretion model. Broderick & Loeb (2006) modeled the frequency dependence of its centroid position, and Reid et al. (2008) used ray tracing to compare hot spot accretion models with the observed astrometric motion of its mean position as a function of wavelength. Finally, although the spacetime surrounding neutron stars only asymptotically approaches the Kerr metric, using its null geodesics for ray tracing has still found application in modeling spectra of neutron stars (Braje et al. 2000).
Despite all of this work, numerical integration of Kerr null geodesics is computationally expensive in certain applications. Rauch & Blandford (1994) (hereafter RB94) described a method for calculating null geodesics in the Kerr metric semi-analytically using the Hamilton-Jacobi formulation of the equations of motion and used it to study the primary caustic. Bozza (2008) used a similar method to investigate caustics of all orders, building on earlier analytic work (Bozza 2002). Fanton et al. (1997) used a fast analytic version for creating line profiles and accretion disk images, and Agol (1997) applied this method to the case of polarization from thin disk accretion. Falcke et al. (2000) went on to use this code along with a simple model for the Galactic center black hole to create images of its accretion flow.
All of this work used Legendre’s formulation of elliptic integrals (e.g., Abramowitz & Stegun 1965), and treated the φ and t coordinates numerically, if at all. The tables given in Carlson (1988Carlson ( , 1989Carlson ( , 1991Carlson ( , 1992) ) greatly simplify the reductions of the equations of motion to elliptic integrals. The primary aim of this paper is to use Carlson’s integrals to calculate all geodesic coordinates semi-analytically for the first time.
Section 2 gives the geodesic equations in Kerr spacetime. Sections 3 and 4 present the reductions to elliptic integrals and the specifics of our implementation. Section 5 outlines a variety of checks performed to ensure its validity and accuracy, and discusses the speed improve-ment that should be expected from using an analytic code. Section 6 provides an overview of our code for readers who are not interested in all of its detail, and the code is applied to toy problems and test cases in Section 7. Finally, Section 8 discusses future work both in extending the code and in applying it to more realistic astrophysical situations.
In Boyer-Lindquist coordinates (t,r,θ,φ), the Kerr line element can be written,
with the definitions
where a is the angular momentum of the black hole and we
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