Measuring Photon Rings with the ngEHT

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Citation: Tiede, P.; Johnson, M.D.;

Pesce, D.W.; Palumbo, D.; Chang,

D.O.; Galison, P. ngEHT Photon

Rings. Preprints 2022,1, 0.

https://doi.org/

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4.0/).

Article

Measuring Photon Rings with the ngEHT

Paul Tiede1,2,∗, Michael D. Johnson1,2 , Dominic W. Pesce1,2 , Daniel C. M. Palumbo1,2 , Dominic O. Chang1,2,

and Peter Galison2,3,4

1Center for Astrophysics |Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA

2Black Hole Initiative at Harvard University, 20 Garden Street, Cambridge, MA 02138, USA

3Department of Physics, Harvard University, Cambridge, MA 02138, USA

4Department of History of Science, Harvard University, Cambridge, MA 02138, USA

*Correspondence: paul.tiede@cfa.harvard.edu

Abstract:

General relativity predicts that images of optically thin accretion ﬂows around black holes

should generically have a “photon ring,” composed of a series of increasingly sharp subrings that

correspond to increasingly strongly lensed emission near the black hole. Because the effects of lensing are

determined by the spacetime curvature, the photon ring provides a pathway to precise measurements

of the black hole properties and tests of the Kerr metric. We explore the prospects for detecting and

measuring the photon ring using very long baseline interferometry (VLBI) with the Event Horizon

Telescope (EHT) and the next generation EHT (ngEHT). We present a series of tests using idealized

self-ﬁts to simple geometrical models and show that the EHT observations in 2017 and 2022 lack the

angular resolution and sensitivity to detect the photon ring, while the improved coverage and angular

resolution of ngEHT at 230 GHz and 345 GHz is sufﬁcient for these models. We then analyze detection

prospects using more realistic images from general relativistic magnetohydrodynamic simulations by

applying “hybrid imaging,” which simultaneously models two components: a ﬂexible raster image

(to capture the direct emission) and a ring component. Using the Bayesian VLBI modeling package

Comrade.jl

, we show that the results of hybrid imaging must be interpreted with extreme caution for

both photon ring detection and measurement — hybrid imaging readily produces false positives for a

photon ring, and its ring measurements do not directly correspond to the properties of the photon ring.

Keywords: Black Holes; Photon Rings; Radio Astronomy; VLBI

1. Introduction

Simulated images of optically thin accretion ﬂows around supermassive black holes

(SMBHs) generically exhibit a nested series of “photon rings” produced from strong gravita-

tional lensing of photon trajectories near the black hole [e.g.,

]. These increasingly sharp

ring-like features are exponentially demagniﬁed as they converge on an asymptotic critical

curve [

], and they can be indexed by the number

of half-orbits that light takes around

the black hole, as shown in Figure 1 [

–

]. Because the null geodesics that deﬁne the photon

ring are determined by the spacetime curvature and are negligibly affected by accreting plasma,

detection of an

0 photon ring would provide striking evidence that the supermassive

compact objects in galactic cores are Kerr black holes and would provide a pathway to precisely

measuring their properties.

To date, measurements of the horizon-scale emission structure around black holes are

only possible using millimeter-wavelength very long baseline interferometry (VLBI). The Event

Horizon Telescope (EHT) is a globe-spanning network of (sub)millimeter radio telescopes that

has carried out VLBI observations of the SMBHs M87

∗

and Sgr A

∗

on horizon scales [

–

The next-generation EHT (ngEHT) plans to build on the capabilities of the EHT by adding

multiple new telescopes to the array, increasing the frequency coverage, and improving the

sensitivity by observing with wider bandwidths [

]. Though the ngEHT will operate with

arXiv:2210.13498v1 [astro-ph.HE] 24 Oct 2022

2 of 19

an unprecedentedly ﬁne diffraction-limited angular resolution of

∼

as, the

1 photon

ring is anticipated to be ﬁner still; the expected thickness of the

1 photon ring in M87

∗

corresponds to an angular size of less than

∼

as [

]. Direct imaging of the

1 photon ring

will thus likely remain unachievable for the foreseeable future, and studies of this feature using

ground-based VLBI will require some degree of “superresolution” via judicious application of

parameterized models of the source structure.

At least two classes of modeling methodology currently show promise for extracting

superresolved photon ring signatures from VLBI measurements of black holes: models that

parameterize the three-dimensional distribution of the material in the vicinity of the black hole

[e.g.,

–

], and models that parameterize the two-dimensional distribution of the emission

morphology as seen on the sky [

]. In either case, because the additional information

supplied by the model speciﬁcation is supporting the extraction of superresolved structural

information, it is important to quantify precisely what deﬁnes a photon ring “detection.” For

instance, the most compelling detection might not require the assumption that general relativity

(GR) is true, while a somewhat weaker claim of detection might test for the presence of this

feature under the assumptions of GR. Likewise, methods could utilize models that assume the

existence of the photon ring to make measurements of black hole parameters without needing

to meet potentially more stringent criteria for an unambiguous detection of the same feature.

A parameterized modeling approach to study the photon ring was recently developed

by Broderick et al.

[24]

(hereafter B20), who employ a “hybrid imaging” technique that ﬁts

a thin geometric ring component alongside a more ﬂexible pixel-based image component,

where the pixel ﬂuxes are treated as model parameters. Broderick et al.

[25]

(hereafter B22)

applied this technique to the EHT observations of M87

∗

, ﬁnding that the diameter of the thin

ring component is well-constrained by the EHT data; the authors associate this component

with the

1 photon ring. While the value and stability of the diameter of this component

across different datasets support its identiﬁcation as an image feature that is determined by the

spacetime, other aspects – particularly the fraction of the total ﬂux density that is recovered in

the thin ring – challenge its association with the

1 ring. This ambiguity underscores the

need to quantify exactly what constitutes a photon ring detection.

In this paper, we aim to investigate the efﬁcacy of tools such as hybrid imaging to extract

photon ring signatures from EHT- and ngEHT-like data and to determine what VLBI measure-

ments are necessary and sufﬁcient to reliably detect a photon ring. In section 2, we conduct tests

using simple geometric models, deriving necessary conditions to detect the

1 photon ring.

Next, in section 3, we explore the application and limitations of the hybrid imaging approach

to detect and measure the photon ring, and we perform tests using more realistic synthetic

data from general relativistic magnetohydrodynamic (GRMHD) simulations. In section 4, we

summarize these results and discuss their implications for the EHT, ngEHT, and other future

VLBI arrays.

2. Geometric Modeling

We begin with a series of idealized tests, generating simulated data from a simple geomet-

ric on-sky model that includes a proxy for the photon ring, and then ﬁtting the same model

to these data. This so-called self-ﬁt procedure guarantees that model parameter posteriors are

directly interpretable. However, the clarity of this procedure comes with the penalty of being

artiﬁcially optimistic; it provides requirements for detecting the photon ring that are likely

necessarily but almost certainly not sufﬁcient. Hence, if these self-ﬁts to simulated data cannot

detect a photon ring with a given array, then we expect that photon ring detection with the

same array in realistic settings will be impossible.

The structure of this section is as follows. First, we describe our geometric model (sub-

section 2.1). Next, we outline our construction of simulated data and the ﬁtting procedure

3 of 19

n=0+1

n=1

n=0

Baseline distance (Gλ)

0 5 10 15 20

Visi

lity

mplitude (mJy)

10⁰

10¹

10² EHT

lim.

ngEHT

lim.

n=0+1

n=1

Figure 1.

The image of a black hole can be decomposed into subimages that are indexed by the number

of half orbits that their photons traveled around the black hole before reaching the observer. In this

scheme, the

0 emission (top right panel) is the “direct” image of the accretion ﬂow and is dominated

by astrophysical emission structure. The

1 emission (top middle panel) is the “secondary” image,

consisting of photons that have traveled a half orbit around the black hole before reaching the observer.

The actual observed image is a sum of all

subimages (top right panel). The bottom panel shows visibility

amplitudes of these (sub)images for projected baselines that are parallel (blue) and perpendicular (pink)

to the black hole spin axis. The longest EHT and ngEHT baselines, indicated with vertical black lines,

occur at baseline lengths for which the

0 and

1 contributions are comparable, raising the prospect

of distinguishing them through modeling.

(subsection 2.2). Finally, we perform self-ﬁts for a variety of EHT and ngEHT arrays to assess

the requirements for detecting the n=1 photon ring (subsection 2.3).

2.1. Specifying the Geometric Model

Our simple parametric model is motivated by the expected image structure for optically

thin emission near a black hole consisting of multiple ring-like structures. For each component,

we use the m-ring model from Johnson et al.

[19]

and Event Horizon Telescope Collaboration

et al.

[12]

. This model is an inﬁnitesimally thin ring with azimuthal brightness modulation

determined by angular Fourier coefﬁcients, which is then convolved with a Gaussian kernel

4 of 19

Table 1.

Arrays used for synthetic data. For additional details on EHT sites, see [

]; for additional details

on ngEHT sites, see [

]. Note that the SPT cannot observe M87

∗

so does not contribute to the tests shown

in this paper. New ngEHT phase 1 sites use speciﬁcations for existing facilities (HAY: 37-m, OVRO: 10.4-m)

and are 6.1-m for new locations (BAJA, CNI, LAS); new ngEHT phase 2 sites assume 8-m diameters with

the exception of the AMT, which is planned to be 15-m [33].

Array Freq. (GHz) Sites

EHT 2017 230 (8) ALMA, APEX, JCMT, LMT, IRAM, SMA, SMT, SPT

EHT 2022 230 (11) EHT 2017, KP, NOEMA, GLT

ngEHT phase 1 230, 345 (16) EHT 2022, BAJA, CNI, HAY, LAS, OVRO

ngEHT phase 2 230, 345 (22) ngEHT phase 1, GARS, AMT, CAT, BOL, BRZ, PIKE

We restrict ourselves to a ﬁrst-order Fourier expansion, giving the following intensity proﬁle

for the thin ring:

M(r,θ|di,ai,bi,Fi) = Fi

πdi

δ(r−di/2)(1+aicos(θ)−bisin(θ)), (1)

where we parameterize

using a polar representation

ai=Aicos φi

and

bi=Aisin φi

where

is the amplitude and

φi

is the phase of the ﬁrst-order Fourier coefﬁcient. Finally,

and

are the ﬂux and diameter of the ring, respectively. Note that we have included a

subscript,

, in anticipation of the nested photon rings. The location of the observed

photon

rings relative to the emitting plasma are shifted as a function of spin and inclination. Therefore,

we allow the centroid of the rings to be displaced by an amount

. To give the ring ﬁnite

width, we convolve the m-ring with a symmetric Gaussian:

G(r,θ|wi) = 4 log(2)

πw2

exp −4 log(2)r2

i!(2)

where

is the Gaussian’s full width at half maximum (FWHM). We denote the thick m-ring

model by

T(x

y) = M?G

, where

is the convolution operator. Finally, the shape of the ring

is also of interest since it encodes information about the spin and inclination of the central

black hole [see, e.g.,

–

]. To add ring ellipticity, we modify the intensity map of the thick

m-ring

T(x,y)→RξT(x,(1+τ)y) = I(x,y)(3)

where

Rξ

rotates the image by

radians counter-clockwise, and

τ>

0 parametrizes the

ring ellipticity. Formally,

is related to the eccentricity

of the elliptical (stretched) ring via

e=p1−1/(1+τ)2≈√2τ

. We denote this model by

and call it the stretched thick m-ring.

The stretched thick m-ring forms the base image for each nested photon ring. The ﬁnal

model that we use is a sum of multiple stretched thick m-ring components:

I0:N(x,y) =

∑

n=0

I(x,y|Fi,di,wi,Ai,φi,τi,ξi,xi,yi). (4)

2.2. Simulated Observations and Fitting Procedure

To create simulated data, we use Equation 4 with m-ring parameters motivated by the

observed structure and expected gravitational lensing of M87

∗

[

]. Because we are focused on

distinguishing the

0 and

1 structure, our model for the construction of the simulated

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Citation:Tiede,P.;Johnson,M.D.;Pesce,D.W.;Palumbo,D.;Chang,D.O.;Galison,P.ngEHTPhotonRings.Preprints2022,1,0.https://doi.org/Publisher'sNote:MDPIstaysneutralwithregardtojurisdictionalclaimsinpublishedmapsandinstitutionalafl-iations.Copyright:©2022bytheauthors.LicenseeMDPI,Basel,Switzerland.Thisarti...

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