Constraining Black Hole Parameters from Shadow and Inner-Shadow Morphology Considering Effects from Thick Disk Accretion Flows

TL;DR

The paper addresses the challenge of inferring black hole spacetime parameters from shadow and inner-shadow images when near-horizon emission geometry is uncertain. It develops a ray-tracing framework in static, spherically symmetric spacetimes (Reissner–Nordström) to compute the shadow radius $r_c$ and inner-shadow metrics $\overline{r_h}$ and $\alpha_h$ as functions of $M/D$, $q$, $i$, and $s$ for M87$^*$- and Sgr A$^*$-like systems. It finds that independent measurements of $r_c$ and $\overline{r_h}$ can constrain $M/D$ and $q$ only if the inclination $i$ and the near-horizon emission geometry $s$ are known; mis-specifying $s$ biases or defeats the inference. The results, relevant for ngEHT and BHEX observations, quantify how emission geometry affects parameter inference and guide observational strategies for robust spacetime constraints.

Abstract

We study the effects of emission geometry on the capability to constrain black hole parameters from measurements of the shadow and inner-shadow of a Reissner-Nordström black hole. We investigate the capability to constrain mass, charge, observer inclination, and emission co-latitude from images of black hole accretion flows that would arise from thick and thin accretion disks. We confirm previous studies that have shown that independent radii measurements of the shadow and inner-shadow can constrain black hole parameters if the viewing inclination is known, but find that it is only possible if the true emission geometry is also assumed. We study the constraining capabilities of the shadow and inner-shadow observations of M87* and Sgr A* like systems within the context of the BHEX and NgEHT future observatories.

Figures (6)

• Figure 1: Critical curve (red, dashed lines) and inner-shadow (orange lines) of accretion disks with varying horizon intersection co-latitude around a Schwarzschild black hole as seen by observers at varying inclinations with respect to the symmetry axis of the accretion disk. We show differences in images seen by an observer viewing a thin disk accretion system at inclinations of $20\degree$ and $80\degree$ with respect to the symmetry axis of the disk. We also show differences in the size of the inner-shadow (orange curve) at emission co-latitudes of $0\degree$ and $20\degree$. The un-lenzed image of the black hole horizon, a radius of $r_h=2GM/c^2$, in the absence of gravitational lensing is shown for scale (solid black disk).
• Figure 2: (Top left panel) normalized 2-D density plot of $\overline{r_h}$ and $r_c$ value over $M/D \in [3,5]$, $q\in[0,1]$, $s\in [-30\degree,30\degree]$, and $i\in[0\degree,85\degree]$. The top-right, bottom-left and bottom-right panels are the same as the top-left, but after restricting $s=0$, $i=0$ and $q=0$ respectively.
• Figure 3: $\alpha_h$ vs. $i$, for a Reisner-Nordstrom black hole over a range of charges $q\in[0,1]$ and emission co-latitudes $s\in[0\degree,30\degree]$. We find a best fit quadratic curve $\alpha_h(i)=-1\times 10^{-4}(i-3.1\degree)^2+1$ that is traced with purple points for values of $i\in[0\degree,50\degree]$. The orange region represents the range of inner-shadow asymmetry values that are consistent with each inclination parameter. The thinness of the orange region implies a strong relationship between $\alpha_h$ and $i$.
• Figure 4: (Left panel) Filled contour plot of all potential $\overline{r_h}$ values across different inclinations and charges. Our values are chosen for a black hole with mass-to-distance ratio $M/D=3.78 \;\mu as$ and emission co-latitude $s=0$. Isoradial contours of $\overline{r_h}$ at whole number intervals are shown in black. (Right panel) Filled contour map of all potential $\overline{r_h}$ values across different emission co-latitudes and charges of a black hole with mass-to-distance ratio $M/D=3.78\mu as$ and viewing inclination $i=0$. Although only positive co-latitudes are relevant for observations, we plot values of $\overline{r_h}$ for the range $s\in[-30\degree,30\degree]$ to help illustrate general trends. (Bottom panel) Example of constraints made on black hole viewing inclination, $i$, and charge, $q$, from measurement of the average radius of the inner-shadow, $\overline{r_h}$. The measurement shown is for a black hole with true parameters of $q=0.2$ and $i = 20\degree$. A measurement of $r_c$ could be used to establish an independent constraint on charge (horizontal black line). The gray region covers a 1% uncertainty region for the value of $\overline{r_h}$.
• Figure 5: (Left) Filled contour map of all potential $\overline{r_h}$ values across different emission co-latitudes and charges of a black hole with mass-to-distance ratio $M/D=3.78\mu as$ and viewing inclination $i=0$. Isoradial contours of $\overline{r_h}$ at whole number intervals are shown in black. (Right) Example of constraints on black hole emission co-latitude, $s$, and charge, $q$, from measurement of the average radius of the inner-shadow, $\overline{r_h}$. The measurement shown is for a black hole with true parameters of $q=0.2$ and $s = 0\degree$. A measurement of $\overline{r_c}$ could be used to establish an independent constraint on charge (horizontal black line). The dark gray band shows the 1% uncertainty region and the light gray band shows the 5% uncertainty region for our calculation of $\overline{r_h}$.