Photon ring polarimetry with next-generation black hole imaging I. M87*

The work by the Event Horizon Telescope (EHT) collaboration in imaging the horizon scale emission of black holes Messier 87* (hereafter M87*) (Event Horizon Telescope Collaboration et al., 2019a, b, 2024a) and Sagittarius A* (hereafter Sgr A*) (Event Horizon Telescope Collaboration et al., 2022a) using very-long-baseline interferometry (VLBI) has opened up a new window in probing the astrophysical features of the strong field gravity regime. The observation of polarised structure in these near-horizon images (Event Horizon Telescope Collaboration et al., 2021a; Roelofs et al., 2023; Event Horizon Telescope Collaboration et al., 2023, 2024b) allows us to study the behaviour of magnetic fields in the vicinity of black holes (Event Horizon Telescope Collaboration et al., 2021b; Tsunetoe et al., 2020; Event Horizon Telescope Collaboration et al., 2024c). Indeed, the Astro2020 Decadal Survey emphasized the role of the EHT in this regime, noting the importance of probing the base of the relativistic jet of M87*, while also recognising the role of such observations to inform our understanding energy extraction from black holes (National Academies of Sciences et al., 2021). More recently, images of the base of the M87* jet at 86 GHz (Lu et al., 2023) have shown a bright ring of light around a flux depression, which taken with EHT results suggest very optically thin emission with rich prospects for accessing gravity signatures near the horizon, in polarization and otherwise.

In the context of future extensions of the EHT, polarimetry has also served as a useful tool for probing the so-called “$n=1$ photon ring” (Johnson et al., 2020; Gralla et al., 2020a; Wong et al., 2022; Palumbo et al., 2023), a strong lensing feature of photons orbiting the black hole with an appearance that is only weakly dependent on the astrophysical details of the accretion. The “order” $n$ indexes the photon ring by the number of half-orbits made by the photon (in the $\theta$ direction of the Boyer-Lindquist co-ordinate system) (Gralla et al., 2020b; Johnson et al., 2020). Here the $n=0$ is the image arising from the “direct” emission and $n=1$ is the secondary image arising from strong lensing. For an equivalent indexing scheme that is covariant but only works when performing “forward” ray tracing from the emitting source to the observer, see Appendix A.3 of Chang et al. (2024). This indexing has been used to probe photon ring signatures in general relativistic magnetohydrodynamic (GRMHD) simulations, (Ricarte et al., 2021; Palumbo & Wong, 2022; Palumbo et al., 2023), EHT observations (Broderick et al., 2022) and for analytic modelling of photon ring structures (Tiede et al., 2022; Cárdenas-Avendaño et al., 2023; Cárdenas-Avendaño & Lupsasca, 2023).

Through both analytic work and GRMHD simulations, there have been several studies of linear (LP) (Himwich et al., 2020; Jiménez-Rosales et al., 2021; Palumbo & Wong, 2022; Palumbo et al., 2023) and circular polarization (CP)(Ricarte et al., 2021; Mościbrodzka et al., 2021) features as proxies for inferring the presence of the photon ring, also leading to a rich interplay with polarimetric features in jets (Mościbrodzka et al., 2017; Davelaar et al., 2019; Kawashima et al., 2021; Tsunetoe et al., 2022; Ogihara et al., 2024).

Now, the EHT observations are performed using only Earth-based stations and therefore the nominal diffraction-limited angular resolution for a given wavelength $\lambda$ is limited by the Earth’s diameter $D$. For observations at 230 GHz, this is approximately $\lambda/D\approx 21\,\mu as$ and the maximum possible baseline length in units of wavelength, is $D/\lambda\approx 9.6$ G$\lambda$. In order to improve the limitations on dynamical range and temporal resolution arising due to EHT’s sparse $(u,v)$ coverage (Event Horizon Telescope Collaboration et al., 2019c), as well to increase the achievable baseline lengths, there are several next-generation black hole imaging missions potentially offering sensitivity to the photon ring signals. The next-generation Event Horizon Telescope (ngEHT) is a ground-based extension to the EHT that aims to install new telescope sites, expand the observing bandwidth to 16 GHz and perform simultaneous multi-frequency observations at 86-230-345 GHz, all of which can lead to high-fidelity, real-time movies of black holes and their associated jets (Doeleman et al., 2023). The Black Hole Explorer (BHEX) mission (Johnson et al., 2024), aims to perform Earth-space VLBI with a single space-based orbiter, providing long baselines $(\gtrapprox 20G\lambda)$ enabling photon ring science while also obtaining a dense coverage of the (u,v) plane on short baselines as well, with the aid of simultaneous dual-band observations by leveraging the expansions on the ground.

To study the efficacy of these missions to probe the photon ring, this paper studies its polarimetric features through both, geometric modelling and time-averaged GRMHD simulations. Indeed, while geometric modelling allows to probe lensing structures that are largely agnostic to astrophysical source profiles (Gralla et al., 2020a), simulations help identify features of the accretion flow that are strongly sensitive to the presence of the photon ring (Ricarte et al., 2021; Jiménez-Rosales et al., 2021). The geometric modelling of photon ring signatures of M87* has been studied by several authors in total intensity (Johnson et al., 2020; Gralla et al., 2020a; Lockhart & Gralla, 2022a; Paugnat et al., 2022; Cárdenas-Avendaño & Lupsasca, 2023; Jia et al., 2024), and in LP (Himwich et al., 2020). Gralla (2020) has also studied signatures on long baselines of general narrow features of arbitrary shape and intensity. The polarimetric signature of M87*’s photon ring in GRMHD simulations has also been studied by several authors (Ricarte et al., 2021; Mościbrodzka et al., 2021; Jiménez-Rosales et al., 2021; Palumbo et al., 2023).

The paper is organised as follows. Section 2 introduces the geometric model of the ring. Section 3 explores the photon ring transition signatures of the geometric model in all polarizations. Section 4 analyses transition behavior in a large library of GRMHD simulations in light of the geometric analysis. Section 5 describes detection requirements for ngEHT and BHEX in terms of permissible thermal noise and antenna diameter. Section 6 presents the conclusions and discusses the outlook for future work. Extended derivations of formulae used in the paper, as well as tabulated numerical results from GRMHD, can be found in the appendices.

Let the image plane be spanned by some two dimensional co-ordinate system $\vec{r}=(x,y)$ and the baseline between the two stations spanned by a dimensionless vector $\vec{b}=(b_{1},b_{2})$ drawn perpendicular to the line of sight. Then, the visibility response $V(\vec{b})$ of an interferometer formed by this baseline, for an image signature $I(\vec{r})$, is given by a two-dimensional Fourier transform:

Here the “tilde” ($\tilde{V}$) denotes a quantity in the Fourier/visibility domain. Note that this relation applies for all Stokes parameters $(I,Q,U,V)$ where $I$, $P=Q+iU$ and $V$ represent total intensity, LP and CP respectively.

We consider the visibility domain to be spanned by polar co-ordinates $(\rho,\theta)$. These co-ordinates are related to the Cartesian $(u,v)$ co-ordinates via the relation:

We now study the transition in $\rho$ at which the $n=1$ signature overtakes the $n=0$ signature in each polarization. Though we consider rings of equal radii for the sake of completeness, an analytic description of such transitions without using this assumption is provided in the Appendix B. The calculation for CP is given below, while for total intensity and LP is given in Appendix B.

For CP, using Equations 7 and 9, we get:

Imposing the condition of equal radii cancels out $J_{0}(...)$ terms, and taking $v_{0}F_{0}e^{-2\pi^{2}\sigma_{0}^{2}\rho^{2}}$ and $F_{0}e^{-2\pi^{2}\sigma_{0}^{2}\rho^{2}}$ common from the numerator and denominator respectively gives:

The condition for $\breve{v}_{01}$ to have a negative sign is,

Solving Equation 16 for $\rho$ and using the notation from Equation 10 gives the transition value $\rho\rightarrow(\rho_{T})_{V}$ at which the sign changes as,

A key insight that arises from Equation 16 is that the physical condition of the $n=1$ signal dominating over $n=0$, is exactly equivalent to the mathematical condition for $\breve{v}_{01}$ to change its sign. This further motivates the subsequent analysis of the GRMHD simulations to probe the photon ring signatures in these quantities.

Moreover, noting the fact that in Equation 17 the quantities inside the square root are all ratios and hence dimensionless, it can be inferred that the strongest determinant of $\rho_{T}$ is the “resolving out” of the direct image by exceeding $\rho=1/\sigma_{0}$. This is evident from rewriting Equation 17 as:

The thickness of the ring is is a crucial factor in comparing simulations with observations (Lockhart & Gralla, 2022b, a) and so Equation 18 and can be a useful tool for geometric modelling of polarimetric signatures of the photon ring. Moreover, the ensuing geometric features can be explored solely by fixing the one parameter of the $n=0$ ring, namely $\sigma_{0}$, and then defining the ratios of quantities relating the $n=0$ and $n=1$ ring.

The formulae for total intensity and LP, reproduced from Equations B3 and B8 with the notation of Equation 10, are given by:

Using Equations 17, 19 and 20, Figure 2 shows example interferometric quantities as a function of $\rho$ for combinations of model parameters that vary the relative contribution of the photon ring. We investigate the visibility domain signatures in $\tilde{I}_{01},\text{arg}(\breve{m}_{01}e^{-2i\theta})\equiv\text{arg}(\bar{m}_{01})$ and $\breve{v}_{01}$. We include the ratio $\mathcal{F}\approx e^{\pi}$ to mimic the “universal regime” prediction for isotropic emitters (Johnson et al., 2020) and to make contact with the results of Tiede et al. (2022) for M87*. The Figure demonstrates the advantages of the geometric modelling scheme since after fixing geometric parameters of the rings (i.e $\sigma_{0}$ and $\sigma_{0}/\sigma_{1}$), the onset of the photon ring transition is primarily governed by the relative polarization ratios ($\mathcal{F}$, $|\beta_{r}|$ and $|v_{r}|$) of the $n=0$ to the $n=1$ image. We find that the the photon ring signal starts to dominate at longer baselines ($\gtrapprox 10G\lambda)$, tuned by model parameters. Specifically for LP, as the $n=1$ ring becomes more linearly depolarised with respect to the $n=0$ ring (i.e as $|\beta_{r}|$ increases), the transition radius $\rho_{T}$ at which the photon ring begins to dominate occurs at longer baselines. For CP, similar trends are observed for the values of the ratio $|v_{r}|$.

We now investigate the range of values of $\rho_{T}$ for GRMHD simulations of M87*, focusing on LP and CP signatures. We consider a library of time-averaged ray-traced GRMHD $n=0$ and $n=1$ images produced in Palumbo & Wong (2022), which hold fixed the mass at $6.5\times 10^{9}M_{\odot}$, the distance at $16.8$ Mpc, and the inclination at $17^{\circ}$. The library contains five values of dimensionless spin $a_{*}$: -0.94, -0.5, 0, +0.5, and +0.94, where a minus/plus sign indicates retrograde/prograde spin with respect to the large-scale accretion disk. In all cases, the black hole spin is directed away from the observer. The fluid models themselves were produced by iharm3D (Gammie et al., 2003; Prather et al., 2021) and ray-traced using IPOLE (Mościbrodzka & Gammie, 2018) with six values of the electron heating parameter $R_{\rm h}$ (Mościbrodzka et al., 2016): 1, 10, 20, 40, 80, and 160. This parameter tunes the relative temperatures of ions and electrons as a function of the magnetization of the plasma.

In order to address whether ngEHT and BHEX can indeed observe the LP and CP photon ring signatures discussed above, we focus on the four polarimetric best-bet models and consider two main quantities of interest, namely the maximum thermal noise and the antenna size. The thermal noise is a proxy for sensitivity requirements of a baseline while greater antenna diameter implies better performance of the given station.

In this paper, we have introduced two approaches to obtain the transition radii in the Fourier domain beyond which the photon ring signal begins to dominate over the direct image. Firstly, by geometrical modelling of the $n=0$ and $n=1$ signals as symmetric, concentric Gaussian rings of equal radii, exact analytical expressions for the transition radii have been found for total intensity, LP and CP. The corresponding formulae are functions solely of tunable model parameters and have a similar mathematical structure. Secondly, a general diagnostic tool, namely CAA, is introduced to systematically obtain the transition radii from the suite of time-averaged GRMHD images. This is applied to the simulation library of M87* at 230 GHz and its polarimetric best-bet models are treated in detail. Lastly, by studying the reference sensitivity and antenna diameter specifications for the ngEHT and BHEX missions, we’ve explored the extent to which these missions can observe the LP and CP photon ring signatures for these best-bet models.

In the context of the M87* model space probed in this paper, we find that Earth-based photon ring detection may have its strongest prospects in sensitivity-limited, astrophysics-forward CP signatures rather than high-resolution imaging. These signatures are relatively insensitive to morphology but may sense the presence of the photon ring in the near future. For BHEX, the prospects for probing the CP signal are sub-optimal, but an overwhelming majority of the signatures in LP are accessible and the nominal sensitivity of BHEX is sufficiently high that Stokes $I$ and $P$ signatures will likely provide strong morphological sensitivity to the photon ring. However, in the case of morphology that disobeys the trends exploited in this work (for example, if Stokes $V$ flips sign across the direct image, as has been seen in some simulations (Tsunetoe et al., 2021)), large discrepancies from rotational symmetry should be apparent even in the Fourier domain. In such cases, there is no substitute for polarimetric imaging.

A natural extension of this work is to performing a similar analysis of M87* at 345 GHz, possibly with the inclusion of more sophisticated geometric models for the image that are currently being used by the EHT (Event Horizon Telescope Collaboration et al., 2019d). It is also timely to perform broad GRMHD library analyses for photon ring detection prospects for Sgr A* as well. Due to its smaller mass, the dynamical timescales of Sgr A* are much shorted than M87* (Event Horizon Telescope Collaboration et al., 2022b) and so it would be instructive to apply the CAA scheme developed in this paper to quantify the range of transition radii as the source evolves. While there has been recent encouraging results in studies of LP detections of the Sgr A* photon ring with the ngEHT (Shavelle & Palumbo, 2024), BHEX may face challenges in detecting Sgr A* due to the diminishment of long-baseline signals caused by diffractive scattering (Johnson & Narayan, 2016). Appropriate orbit selection and observation strategies for the Galactic Center are the object of ongoing work; however, interferometric quotients such as those studied in this paper are innately robust to convolutional corruptions to the image, but will be sensitivity-limited in applications to Sgr A*.

We thank Paul Tiede for his assistance in the application of VIDA, as well as Michael Johnson and Sheperd Doeleman for many helpful conversations. A.T. gratefully acknowledges the Center for Astrophysics $|$ Harvard and Smithsonian, and the Black Hole Initiative for several fruitful discussions and providing a welcoming environment during an extended research visit during May-July 2024. We acknowledge financial support from the National Science Foundation (AST-2307887). This project was funded in part by generous support from Mr. Michael Tuteur and Amy Tuteur, MD. This work was supported by the Black Hole Initiative, which is funded by grants from the John Templeton Foundation (Grant 62286) and the Gordon and Betty Moore Foundation (Grant GBMF-8273) - although the opinions expressed in this work are those of the author(s) and do not necessarily reflect the views of these Foundations.