Event Horizon Telescope Pattern Speeds in the Visibility Domain

TL;DR

This work extends the measurement of apparent rotation in EHT sources from the image domain to the visibility domain by defining and extracting the visibility-domain pattern speed $Ω_{\mathrm{VA}}$ from time-resolved visibility amplitudes. Applying an autocorrelation-based, reconstruction-free method to the Illinois Sgr A* GRMHD library, the authors show $Ω_{\mathrm{VA}}$ is sub-Keplerian, increases with baseline length, and strongly depends on inclination and mass, with spin modulating the rotation curve differently for MAD and SANE models. They provide analytic fits for $Ω_{\mathrm{VA}}(ρ;i,a_{\ast},R_{\rm high})$ and discuss how space VLBI (e.g., BHEX) could extend baselines to reveal the full pseudo-rotation curve and improve spin constraints. The results establish a robust, reconstruction-independent diagnostic of black hole spin and inclination that complements image-domain analyses, with practical implications for future EHT campaigns and space-based mm-VLBI observations.

Abstract

The Event Horizon Telescope is preparing to produce time sequences of black hole images, or movies. In anticipation, we developed an autocorrelation technique to measure apparent rotational motion using the image-domain pattern speed $Ω_p$. Here, we extend this technique to the visibility domain and introduce the visibility amplitude pattern speed $Ω_{\mathrm{VA}}$. We show that in the Illinois v3 library of EHT source models, $Ω_{\mathrm{VA}}$ depends on the source inclination, black hole mass, black hole spin, accretion state (MAD or SANE), and baseline length, and then provide approximate fits for this dependence. We show that $Ω_{\mathrm{VA}}$ is particularly sensitive to baseline length for MAD (strongly magnetized) models, and that the slope of this dependence can be used to constrain black hole spin. As with $Ω_p$, models predict that $Ω_{\mathrm{VA}}$ is well below the Keplerian frequency in the emission region for all model parameters. This is consistent with the idea that $Ω_{\mathrm{VA}}$ measures an angular phase speed for waves propagating through the emission region. Finally, we identify the information that would be provided by space-based millimeter VLBI such as the proposed BHEX mission.

Figures (9)

• Figure 1: A 230 GHz snapshot image from a typical Sgr A* model (MAD, $a_{\ast} = 0.5$, $i = 30^\circ$, $R_{\rm high} = 160$) (left), and the accompanying visibility amplitudes (right). The visibility amplitude colormap is logarithmic. The image-domain ring produces a series of damping lobes and nulls, while spiral waves produce spiral features in the $(u,v)$ domain. The helicity of the spiral has flipped, as expected.
• Figure 2: A $400 \, t_g$ window of a $(u,v)$ cylinder plots (left) and the autocorrelations (right) from the same Sgr A* model (MAD, $a_{\ast} = 0.5$, $i = 30^\circ$, $R_{\rm high} = 160$), measured at $\rho = 3 \, G\lambda$ (top) and $10 \, G\lambda$ (bottom). The bounded region of the autocorrelation $\xi > \xi_{\rm crit}$ is surrounded by a black contour and the measured pattern speed $\Omega_{\mathrm{VA}}$ is marked by the dashed green line.
• Figure 3: Pattern speed for a toy model in the image domain (shown in warm tones, with color denoting mean brightness at the angular radius, or impact parameter, marked by the top x-axis) and in the $(u,v)$ domain (cool tones, with color denoting mean amplitude for a given $\rho$ marked by the bottom x-axis). The image domain values are measured from an analytic iNoisy model with a known power-law dependence. The best-fit curve to the $(u,v)$ pattern speed (blue) asymptotically approaches the brightness-weighted image domain pattern speed (dashed purple).
• Figure 4: Measured $(u,v)$ pattern speeds as a function of baseline length, shown for all MAD face-on models ($i = 10^\circ, 170^\circ$). Color corresponds to dimensionless black hole spin $a_{\ast}$. Notice that prograde ($a_{\ast} > 0$) models slope more steeply, while retrograde ($a_{\ast} < 0$) models have a shallower slope.
• Figure 5: Measured $(u,v)$ pattern speeds as a function of baseline length, shown for all SANE face-on models ($i = 10^\circ, 170^\circ$). Color corresponds to the electron temperature distribution function (left) and black hole spin (right). To make the spin dependence clearer, we have not plotted $R_{\rm high} = 160$ on the right plot. SANE models demonstrate a strong dependence on $R_{\rm high}$ and a weaker dependence on $a_{\ast}$.