A new approximation of photon geodesics in Schwarzschild spacetime

TL;DR

This paper presents a new approximation method for photon paths in Schwarzschild spacetime, accurately modeling highly bent trajectories near the black hole with less than 1% deviation from numerical solutions.

Contribution

It introduces a novel approximation technique for photon geodesics that is especially accurate for trajectories behind the central mass, improving modeling near the ISCO.

Findings

Deviation from numerical results below 1% up to ISCO

Effective for highly bent photon trajectories

Enhances computational efficiency in modeling photon paths

Abstract

In this research note we introduce a new approximation of photon geodesics in Schwarzschild spacetime which is especially useful to describe highly bent trajectories, for which the angle between the initial emission position and the line of sight to the observer approaches $\pi$: this corresponds to the points behind the central mass of the Schwarzschild metric with respect to the observer. The approximation maintains very good accuracy overall, with deviations from the exact numerical results below $1\%$ up to the innermost stable circular orbit (ISCO) located at $6~GM/c^2$.

Equations

$$1-\cos \alpha \approx (1-\cos \psi )(1-2r_g/r),$$

$$1-\cos \alpha \approx (1-\cos \psi )(1-\frac{2r_g}{r})[1+k_1\frac{2r_g}{r}(1-\cos (\psi -k_2){)}^{k_3}];$$

Full text

A new approximation of photon geodesics in Schwarzschild spacetime

Riccardo La Placa

INAF - Osservatorio Astronomico di Roma, Monte Porzio Catone, Italy

Research Centre for Computational Physics and Data Processing, Silesian University in Opava, Czech Republic

Pavel Bakala

Research Centre for Computational Physics and Data Processing, Silesian University in Opava, Czech Republic

M. R. Štefánik Observatory and Planetarium, Hlohovec, Slovak Republic

INAF - Osservatorio Astronomico di Roma, Monte Porzio Catone, Italy

Luigi Stella

INAF - Osservatorio Astronomico di Roma, Monte Porzio Catone, Italy

Maurizio Falanga

International Space Science Institute (ISSI), Bern, Switzerland

International Space Science Institute Beijing, P.R. China

1

Photon geodesics in the Schwarzschild metric lie on the plane defined by the central mass in the origin, the emitting point and the observer. The elliptic integral which describes them has often been approximated analytically in terms of the impact parameter $b$, a conserved quantity which in practical (numerical) applications often requires a priori knowledge of the trajectory’s characteristics (see e.g. Semerák, 2015). Beloborodov (2002) proposed the following approximation (later derived in De Falco et al. (2016) by Taylor-expanding the geodesics equation)

[TABLE]

where $r_{g}=GM/c^{2}$ is the gravitational radius. Knowing the emission radius $r$ and the angle $\psi$ between the line of sight and the radial direction $\hat{r}$, Eq. 1 gives the emission angle $\alpha$ between $\hat{r}$ and the initial direction needed for the photon to reach the observer at infinity parallel to the line of sight with $b=r(1-2r_{g}/r)^{-\frac{1}{2}}\sin\alpha$ (see Fig. 1 in Beloborodov, 2002). This is an especially convenient set of variables for applications in which fast calculations of photon geodesics from the emitting point to the observer are required (see e.g. Chang et al., 2006; Poutanen & Beloborodov, 2006). Eq. 1 yields a small deviation from the exact trajectories ($\delta\alpha/\alpha<1$%) for direct photons ($\alpha\leqslant\pi/2$); its accuracy, however, degrades from $\delta\alpha/\alpha\sim 1$% up to $10$% for a wide range of geodesics with a turning point ($\alpha>\pi/2$).

We present the following approximate equation which holds for $\psi$ between 0 and $\pi$ (and thus up to the maximum $\alpha$ we consider, $\alpha(r,\psi=\pi))$ and retains $\delta\alpha/\alpha\lesssim 1$% throughout:

[TABLE]

here $k_{1}=0.1416,~{}k_{2}=1.196,~{}k_{3}=2.726$. The form of Eq. 2 and the values of the $k$-parameters were determined by fitting the relationship between $\alpha$ and $\psi$ to the geodesics computed numerically for various radii between 6 and 100 $r_{g}$, and by tuning the parameters for $r=10~{}r_{g}$. Fig. 1 shows the accuracy of Eq. 2: for radii larger than $r_{ISCO}=6~{}r_{g}$ (the radius of the innermost stable circular orbit for massive particles) our approximation yields $\delta\alpha/\alpha\lesssim 0.7\%$ over all angles. For lower radii $\delta\alpha/\alpha$ crosses the $1\%$ threshold in a small region with $r\lesssim 4.7~{}r_{g}$, reaching a maximum of $1.35\%$ for a radius of $4~{}r_{g}$, below which the approximation breaks down. Also note that by using Eq. 2 while calculating $\partial b/\partial r$ in the expression of the solid angle element (see e.g. Bao, 1992), a higher accuracy is attained than applying the approximation derived in De Falco et al. (2016). Eq. 2 is especially useful in geometries for which $\psi$ approaches $\pi$, where Eq. 1 would produce overly bent trajectories (see e.g. the application to the “far side” of a relativistic accretion disk in La Placa et al., 2019).