Toward Determining the Number of Observable Supermassive Black Hole Shadows

Our strategy for determining N(θ_r , σ_ν ) by considering the global distribution of SMBHs as a function of mass M and redshift z,

$$\begin{eqnarray}&&{\rm{\Phi }}(M,z)\equiv \displaystyle \frac{{dN}}{{dz}\,{dM}}\end{eqnarray} \tag{ 1 }$$

to which we then sequentially apply the above three criteria to narrow down the number of potentially detectable sources. The distribution Φ(M, z) is described by the black hole mass function (BHMF), which we discuss in Section 2.2.

For a given SMBH mass M and redshift z, applying our first criterion—that the angular shadow size ϑ is larger than some resolution θ_r —amounts to requiring that the black hole mass exceed some minimum mass m₀(z). A black hole of mass M situated at an angular diameter distance D_A has an angular shadow size that is given by

$$\begin{eqnarray}&&\vartheta \approx \sqrt{27}\displaystyle \frac{{R}_{{\rm{S}}}}{{D}_{{\rm{A}}}}=\displaystyle \frac{2\sqrt{27}{GM}}{{c}^{2}{D}_{{\rm{A}}}},\end{eqnarray} \tag{ 2 }$$

where R_S is the Schwarzschild radius and the numerical prefactor $\sqrt{27}$ is determined by the shadow diameter for a Schwarzschild black hole (Hilbert 1917; Bardeen 1973). At a particular redshift z, the condition ϑ ≥ θ_r corresponds to

$$\begin{eqnarray}&&M\geqslant {m}_{0}({\theta }_{r})\equiv \displaystyle \frac{{\theta }_{r}{c}^{2}D(z)}{2\sqrt{27}G\left(1+z\right)},\end{eqnarray} \tag{ 3 }$$

where m₀(θ_r ) is the critical mass for which an SMBH at redshift z has a shadow with angular size θ_r , and where we have cast the expression in terms of the comoving distance, D(z) = (1 + z)D_A (see also Bisnovatyi-Kogan & Tsupko 2018).

Applying our second condition—that the flux density S_ν be greater than some threshold σ_ν —requires knowing the distribution P(S_ν ∣M, z) of flux densities for an SMBH of mass M at redshift z. The flux density S_ν (ν₀) observed at a frequency ν₀ is related to the emitted luminosity density L_ν by (Peacock 1999),

$$\begin{eqnarray}&&{S}_{\nu }({\nu }_{0})=\displaystyle \frac{{L}_{\nu }([1+z]{\nu }_{0})}{4\pi (1+z)D{\left(z\right)}^{2}}.\end{eqnarray} \tag{ 4 }$$

Here, L_ν ([1 + z]ν₀) denotes the luminosity density evaluated at the redshifted frequency (1 + z)ν₀, and we have assumed that the emission is isotropic. ⁸ L_ν is determined by the spectral energy distribution (SED) of the source, which we model as described in Section 2.3 (with more comprehensive details provided in Appendix A). Within our SED model, L_ν depends not only on the mass of the SMBH, but also on its mass accretion rate $\dot{M}$, which we cast in terms of the Eddington ratio λ,

$$\begin{eqnarray}&&\lambda \equiv \displaystyle \frac{\dot{M}}{{\dot{M}}_{\mathrm{Edd}}}.\end{eqnarray} \tag{ 5 }$$

Here, ${\dot{M}}_{\mathrm{Edd}}\equiv {L}_{\mathrm{Edd}}/\eta {c}^{2}$ is the Eddington mass accretion rate, and η is a nominal radiative efficiency that relates ${\dot{M}}_{\mathrm{Edd}}$ to the Eddington luminosity L_Edd; for this paper, we take the radiative efficiency to be η = 0.1 (e.g., Yuan & Narayan 2014). Determining P(S_ν ∣M, z) thus further requires knowledge of the Eddington ratio distribution function (ERDF), which we describe in Section 2.4.

Applying our third condition—that the horizon-scale emission be optically thin at the observing frequency ν₀—can also be achieved using our SED model, which provides an optical depth prediction for an SMBH with any given M, λ, and ν₀. Practically, we can absorb this condition into the definition of the flux density distribution by considering only those systems that are optically thin, that is, by determining P(S_ν ∣M, z, τ ≤ 1). The fraction f(σ_ν ) of SMBHs for which we could expect to detect the horizon-scale emission is then given by

$$\begin{eqnarray}&&f({\sigma }_{\nu })={\int }_{{\sigma }_{\nu }}^{\infty }P({S}_{\nu }| M,z,\tau \leqslant 1)\,{{dS}}_{\nu },\end{eqnarray} \tag{ 6 }$$

where σ_ν is some specified sensitivity threshold.

Combining all three criteria, we can compute the source counts expected for any choice of θ_r and σ_ν by integrating the global distribution over mass and redshift,

$$\begin{eqnarray}&&N({\theta }_{r},{\sigma }_{\nu })={\int }_{0}^{\infty }{dz}{\int }_{{m}_{0}({\theta }_{r})}^{\infty }f({\sigma }_{\nu }){\rm{\Phi }}(M,z)\,{dM}.\end{eqnarray} \tag{ 7 }$$

Many of the results presented in this paper are derived from evaluation of Equation (7). When computing this integral, we must keep in mind that m₀(θ_r ) is a function of z and that f(σ_ν ) is a function of both z and M. Figure 1 illustrates the procedure we follow to determine N(θ_r , σ_ν ) for an example set of angular resolution and flux density thresholds (in this case, θ_r = 1 μas and σ_ν = 10⁻⁵ Jy).

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Any evaluation of Equation (7) requires a choice of BHMF, which commonly takes the form

$$\begin{eqnarray}&&{\rm{\Phi }}^{\prime} =\displaystyle \frac{{dN}}{{dVdM}},\end{eqnarray} \tag{ 8 }$$

where dN is the number of SMBHs in the mass range (M, M + dM) and the comoving volume range (V, V + dV). ⁹ For our purposes, it is more useful to work with Φ(M, z), the number of black holes in the redshift range (z, z + dz; see Equation (1)), which is related to ${\rm{\Phi }}^{\prime}$ by

$$\begin{eqnarray}\begin{array}{rcl}{\rm{\Phi }}(M,z) & = & {\rm{\Phi }}^{\prime} \displaystyle \frac{{dV}}{{dz}}\\ & = & \displaystyle \frac{4\pi {{cD}}^{2}{\rm{\Phi }}^{\prime} }{{H}_{0}E(z)}.\end{array}\end{eqnarray} \tag{ 9 }$$

Here, $E(z)=H(z)/{H}_{0}=\sqrt{{{\rm{\Omega }}}_{m}{\left(1+z\right)}^{3}+{{\rm{\Omega }}}_{{\rm{\Lambda }}}}$ is the dimensionless Hubble parameter (Peebles 1993).

Estimating the BHMF from observations is difficult because astronomical surveys are inevitably incomplete in ways that impose poorly known selection functions on the SMBH count in any mass bin, and because there are currently no SMBH mass measurement techniques that are both precise and broadly applicable (Kelly & Merloni 2012). Many variants of the BHMF thus exist in the literature (e.g., Salucci et al. 1999; Aller & Richstone 2002; Marconi et al. 2004; Greene & Ho 2007; Lauer et al. 2007; Natarajan & Treister 2009; Kelly & Shen 2013). Recognizing that no single one of these BHMFs is likely to be uniquely correct, in this paper we consider two different BHMF prescriptions—which we will refer to as our "lower" and "upper" BHMFs—that aim to capture a reasonable range of possibilities.

We take as our lower BHMF the phenomenological model developed by Shankar et al. (2009) and shown in the left panel of Figure 2. This BHMF is evolved self-consistently forward in time within a continuity equation formalism (Cavaliere et al. 1971; Small & Blandford 1992) tuned to match an estimate of the bolometric active galactic nucleus (AGN) luminosity function based primarily on the X-ray observations compiled by Ueda et al. (2003). The Shankar et al. (2009) BHMF is a function of both M and z, covering SMBH masses in the range 10⁵–10^9.5 M_⊙ and redshifts in the range 0–6. To account for the known existence of SMBHs with masses exceeding 10^9.5 M_⊙ (e.g., Event Horizon Telescope Collaboration et al. 2019f), we extrapolate the BHMF using a power law with an exponential cutoff,

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{\mathrm{extrapolated}}^{{\prime} }\propto {M}^{-a}\exp \left(-\displaystyle \frac{M}{{M}_{\mathrm{cutoff}}}\right).\end{eqnarray} \tag{ 10 }$$

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As a counterpart to the model-based lower BHMF, we also consider an upper BHMF derived empirically using the UniverseMachine stellar mass function (SMF) from Behroozi et al. (2019). The UniverseMachine SMF is constructed as part of a comprehensive model for galaxy growth spanning redshifts 0 ≤ z ≤ 10 and accommodating many observational constraints, including among them a number of observational SMFs determined in various bands (Baldry et al. 2012; Ilbert et al. 2013; Moustakas et al. 2013; Muzzin et al. 2013; Tomczak et al. 2014; Song et al. 2016). From the UniverseMachine SMF, we convert from stellar mass M_* to SMBH mass M using the scaling law from Kormendy & Ho (2013), as done in Ricarte & Natarajan (2018),

$$\begin{eqnarray}&&\mathrm{log}\left(\displaystyle \frac{M}{\ {M}_{\odot }}\right)=8.69+1.16\mathrm{log}\left(\displaystyle \frac{{M}_{* }}{{10}^{11}\ {M}_{\odot }}\right).\end{eqnarray} \tag{ 11 }$$

After converting from stellar to SMBH mass, we convolve the SMBH mass distributions with a Gaussian kernel with a 0.3 dex FWHM to account for the intrinsic scatter in the scaling relations. The resulting upper BHMF is shown in the center panel of Figure 2.

Relative to the lower BHMF, the upper BHMF predicts systematically more SMBHs at low to intermediate redshifts (i.e., z ≲ 3) and at all masses, though at the highest redshifts the lower BHMF predicts more SMBHs with M ≳ 10^9.5 M_⊙ (see right panel of Figure 2). The low-redshift behavior of the lower BHMF agrees well with a BHMF derived from the UniverseMachine SMF using the McConnell & Ma (2013) scaling relation (see also Saglia et al. 2016). ¹⁰ To remain conservative in our estimates, throughout this paper we treat the lower BHMF as our fiducial case and use it for all computations and figures unless otherwise specified; we use the upper BHMF primarily to determine plausible uncertainty ranges for computed values. For this paper, we treat both BHMFs as being nonzero only in the range 0 ≤ z ≤ 6 and 10⁵ ≤ M ≤ 10¹¹ M_⊙.

For the analyses carried out in this paper, the high-mass end of the BHMF is most important. To assess the fidelity of the high-mass end of the lower and upper BHMFs, we compare their predictions against the number of known massive SMBHs in the local universe. In this regard, the MASSIVE galaxy survey provides a convenient comparison point because it is a volume-limited survey targeting massive early-type galaxies with stellar masses above 10^11.5 M_⊙ within a distance of 108 Mpc, or z ≈ 0.02 (Ma et al. 2014). To date, four ¹¹ SMBHs in this volume have dynamically measured masses at or above M87s M ≥ 6.5 × 10⁹ M_⊙: M87 (Event Horizon Telescope Collaboration et al. 2019f), NGC 1600 (Thomas et al. 2016), NGC 3842, and NGC 4889 (McConnell et al. 2011). Our lower and upper BHMFs predict that the number of SMBHs within z ≤ 0.02 and M > 6.5 × 10⁹ M_⊙ should be ∼5 and ∼29, respectively, which are consistent with the MASSIVE survey results. The specific behavior of the BHMF at low masses is less important because these black holes do not contribute significantly at the angular resolutions and flux densities of most interest for this paper.

Given the global distribution of SMBHs across mass and redshift, Equation (7) selects only the fraction f(σ_ν ) that have optically thin emission with a flux density that exceeds the sensitivity threshold. This fraction is defined in Equation (6), and it results from integrating over the distribution of flux densities P(S_ν ∣M, z) at a given M and z. The first piece of information we need to compute this integral is an SED model, which will permit us to determine the flux density S_ν (λ∣M, z) corresponding to a particular choice of Eddington ratio, black hole mass, and redshift, and also to assess when the observed emission will be optically thin.

Observational constraints on SMBH growth indicate that SMBHs spend the majority of their time accreting at well below the Eddington rate (Hopkins et al. 2006). At these low accretion rates, the material in the vicinity of the black hole is thought to follow the advection-dominated accretion flow (ADAF) solution to the hydrodynamic equations describing viscous and differentially rotating flows around black holes (Narayan & Yi 1995a; Narayan et al. 1998; Yuan & Narayan 2014). An ADAF accretion disk has a two-temperature structure in which the ion temperature is greater than the electron temperature. The electrons are able to cool via a combination of synchrotron, bremsstrahlung, and inverse Compton radiation, which together define the SED for the observed emission.

For SMBHs observed in the radio to submillimeter wavelength range, as relevant for this work, the SED is dominated by synchrotron and Compton emission. Mahadevan (1997, hereafter M97) provides a convenient formalism for computing the gross spectral properties of an ADAF system given a black hole mass M and accretion rate $\dot{M}$ (see also Narayan & Yi 1994, 1995a, 1995b). We use a modified version of the M97 formalism for the SED models in this paper, and Appendix A provides a detailed description of our updated model. We note that this SED model only considers emission from the accretion flow, and it does not incorporate a jet component.

Our SED model provides an estimate of the emitted luminosity density L_ν as a function of frequency for any input values of M and λ. Given a particular redshift z, we convert L_ν to S_ν using Equation (4). We determine whether the system is optically thin by comparing the rest-frame observing frequency, (1 + z)ν₀, to the peak synchrotron frequency in the source, ν_p (see Equation (A16)). So long as (1 + z)ν₀ ≥ ν_p , we consider the system to be optically thin.

The last piece of information we need to compute the integral in Equation (6) is an ERDF, which provides a probabilistic description of what fraction of SMBHs should be accreting at any particular Eddington rate λ. In this paper, we consider every SMBH to be active at some level, rather than considering the accretion to have only binary "on" and "off" states. We thus dispense with the notion of a "duty cycle" often adopted for AGNs (or equivalently, we take the duty cycle to be unity), and we instead work exclusively in terms of an ERDF (e.g., Merloni & Heinz 2008) to account for the differences in accretion rates.

There is emerging evidence that luminous ("Type 1"; unobscured; λ ≳ 10⁻²) and low-luminosity ("Type 2;" obscured; λ ≲ 10⁻²) AGNs follow different distributions (Kauffmann & Heckman 2009; Trump et al. 2011; Weigel et al. 2017). Though the ERDF for luminous AGNs appears to be consistent with a log-normal distribution (Lusso et al. 2012), there is no clear consensus in the literature on a specific form for the ERDF of low-luminosity AGNs (LLAGNs). Different authors have used variants that include a power law (Aird et al. 2012; Bongiorno et al. 2012), a Schechter function (Hopkins & Hernquist 2009; Cao 2010; Hickox et al. 2014), and a log-normal (Kauffmann & Heckman 2009; Conroy & White 2013). In addition, while there seems to be broad agreement on a power-law behavior toward low Eddingtion ratios in the local universe (i.e., z ≲ 1), few observational constraints currently exist for the ERDF of LLAGNs at z ≳ 1.

We proceed with a form for the ERDF adapted from the analytic prescription used by Tucci & Volonteri (2017) and updated using the more recent measurements from Aird et al. (2018). For their ERDF, Tucci & Volonteri (2017) used a Schechter function with an exponential cutoff value of λ = 1.5, but for our purposes (i.e., LLAGN with λ ≪ 1) only the power-law component of the ERDF is relevant. Furthermore, the LLAGN portion of the ERDF from Tucci & Volonteri (2017) was constructed to match the low-redshift behavior from Hopkins & Hernquist (2009), Kauffmann & Heckman (2009), and Aird et al. (2012). None of these previous papers included observational constraints for AGNs accreting below λ ≈ 10⁻⁵. To avoid the strong dependence on the low-end cutoff that comes from continuing the power law to arbitrarily small values, we posit instead that the distribution breaks (as in, e.g., Weigel et al. 2017). Specifically, we modify the power-law ERDF from Tucci & Volonteri (2017) such that it flattens out for Eddington ratios smaller than some value λ₀. That is, we have

$$\begin{eqnarray}P(\lambda )=\left\{\begin{array}{ll}A, & {\lambda }_{\min }\leqslant \lambda \leqslant {\lambda }_{0}\\ A{\left(\displaystyle \frac{\lambda }{{\lambda }_{0}}\right)}^{\alpha }, & {\lambda }_{0}\lt \lambda \lt {\lambda }_{\max },\end{array}\right.\end{eqnarray} \tag{ 12 }$$

where P(λ) is the probability density per unit logarithmic interval in λ, ${\lambda }_{\min }$ and ${\lambda }_{\max }$ are the lowest and highest permitted values, and the coefficient A is constructed such that the distribution integrates to unity:

$$\begin{eqnarray}&&A={\left[\mathrm{log}\left(\displaystyle \frac{{\lambda }_{0}}{{\lambda }_{\min }}\right)+\displaystyle \frac{1}{\alpha {\lambda }_{0}^{\alpha }\mathrm{ln}(10)}\left({\lambda }_{\max }^{\alpha }-{\lambda }_{0}^{\alpha }\right)\right]}^{-1}.\end{eqnarray} \tag{ 13 }$$

In this paper, we use values of ${\lambda }_{\min }={10}^{-10}$, λ₀ = 10⁻⁵, and ${\lambda }_{\max }={10}^{-2}$ (see Appendix A.4).

In addition to permitting the power-law index α to evolve with redshift, we also allow for additional evolution with SMBH mass,

$$\begin{eqnarray}\alpha (z,M)=a(M)\times \left\{\begin{array}{ll}-1, & z\leqslant a(M)\\ -1/\left[1+z-a(M)\right], & z\gt a(M).\end{array}\right.\end{eqnarray} \tag{ 14 }$$

Here, a(M) encodes the mass dependence of the power-law index. Though there is some prior observational evidence indicating that the ERDF is approximately independent of SMBH mass (Kauffmann & Heckman 2009; Kelly & Shen 2013; Weigel et al. 2017), recent measurements by Aird et al. (2018) found that more massive SMBHs tend to be accreting at higher rates. We thus treat a(M) as being essentially bimodal, with low-mass SMBHs having one power-law index value and high-mass SMBHs having another, and we use a logistic function to smoothly vary a(M) between these two extremes,

$$\begin{eqnarray}&&a(M)=\displaystyle \frac{{a}_{\mathrm{hi}}+{a}_{\mathrm{lo}}{\left(M/{M}_{0}\right)}^{-1/{\rm{\Delta }}}}{1+{\left(M/{M}_{0}\right)}^{-1/{\rm{\Delta }}}}.\end{eqnarray} \tag{ 15 }$$

Here, a_lo describes the power-law index at small masses, a_hi describes the power-law index at large masses, M₀ denotes the midpoint mass, and Δ is the logistic width in $\mathrm{log}(M)$ that controls how quickly the transition from the low-mass regime to the high-mass regime occurs. We determine the values of these four parameters by fitting Equation (12) to the Aird et al. (2018) measurements; our fitting procedure is described in Appendix B. We find best-fit values of a_lo = 0.55, a_hi = 0.20, $\mathrm{log}({M}_{0})=7.5$, and Δ = 0.3, and the resulting ERDF is shown in Figure 3.

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Equation (12) defines the probability P(λ) per unit $\mathrm{log}(\lambda )$ for any particular SMBH to be accreting at the rate λ. Given some specified M and z, we determine the probability P(S_ν ∣M, z) by numerically sampling from P(λ) and using our SED model (see Section 2.3) to associate each sample with a particular S_ν . Efficient sampling of P(λ) can be achieved by transforming a random variable x that is distributed according to a unit uniform distribution through the inverse cumulative distribution function (CDF) of Equation (12). This inverse CDF is given by

$$\begin{eqnarray}&&\begin{array}{l}{\mathrm{CDF}}^{-1}(x)=\\ \left\{\begin{array}{ll}{\lambda }_{\min }{10}^{x/A}, & 0\leqslant x\leqslant A\mathrm{log}\left(\displaystyle \frac{{\lambda }_{0}}{{\lambda }_{\min }}\right)\\ {\lambda }_{0}{\left(\alpha \mathrm{ln}(10)\left[\displaystyle \frac{x}{A}-\mathrm{log}\left(\displaystyle \frac{{\lambda }_{0}}{{\lambda }_{\min }}\right)\right]+1\right)}^{1/\alpha }, & A\mathrm{log}\left(\displaystyle \frac{{\lambda }_{0}}{{\lambda }_{\min }}\right)\lt x\leqslant 1,\end{array}\right.\end{array}\end{eqnarray} \tag{ 16 }$$

which we can use to generate random samples distributed according to Equation (12). The associated distribution of S_ν provides an estimate of P(S_ν ∣M, z), which we then integrate per Equation (6) for the purposes of evaluating Equation (7).

Putting it all together, Figure 4 shows the result of evaluating Equation (7) over a range of values for both the angular resolution threshold θ_r and the flux density sensitivity σ_ν at an observing frequency of ν₀ = 230 GHz. The top panel shows the source counts predicted without imposing the optical depth condition, while the bottom panel restricts the sources to those that satisfy τ ≤ 1 (see Equation (6)). Each point in both panels of Figure 4 is computed from an integral over the remaining (M, z) space. These plots thus represent an observation-independent prediction about the character of the SMBH population; namely, how many SMBHs are expected to have angular shadow sizes in excess of θ_r , horizon-scale flux densities at 230 GHz greater than σ_ν , and (in the case of the bottom panel) an optically thin accretion flow. An approximate analytic description of the resulting N(θ_r , σ_ν ) is provided in Appendix C.

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The two panels of Figure 5 show the behavior of N(θ_r , σ_ν ) in the limit as σ_ν = 0 (left panel) and θ_r = 0 (right panel); these limits correspond approximately to one-dimensional slices through the top panel of Figure 4 along the horizontal and vertical axes, respectively. The black curve in the left panel shows N(ϑ > θ_r , σ_ν = 0), while the colored curves show the contribution from SMBHs in different mass ranges. At large θ_r we see that the source counts follow the $N\propto {\theta }_{r}^{-3}$ behavior expected from simple volume scaling. The upturn around θ_r ≈ 1 μas occurs because this is the resolution threshold below which the most massive SMBHs can be seen at any redshift (because of the turnover in angular diameter distance at z ≈ 1.6), and the re-flattening at smaller θ_r is caused by the finite redshift coverage of the BHMF. The black curve in the right panel shows N(θ_r = 0, S_ν > σ_ν ), while the colored curves again split out the contribution by SMBH mass. Throughout most of the space, we see the source counts climbing volumetrically as the flux density decreases, following $N\propto {\sigma }_{\nu }^{-3/2}$. Cosmological effects become noticeable at the lowest σ_ν values, where the curve starts flattening out due to a combination of the luminosity distance increasing more rapidly and the finite redshift coverage of the BHMF.

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We base our expectations for the horizon-scale emission structure from an SMBH on the observational and theoretical understanding of the M87 system. Johnson et al. (2020) provide an approximate analytic expression for the expected flux density of the photon ring emission as a function of baseline length for optically thin emission, which we adapt to take the following form:

$$\begin{eqnarray}&&\begin{array}{l}S(u)=\eta {S}_{0}{J}_{0}(\pi \vartheta u)\displaystyle \sum _{n=0}^{\infty }{e}^{-n\pi }{e}^{-\displaystyle \frac{{\left(\pi {{uW}}_{n}\right)}^{2}}{4\mathrm{ln}(2)}},\end{array}\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&{W}_{n}\approx {W}_{0}{e}^{-n\pi }.\end{eqnarray} \tag{ 18 }$$

Here, S₀ is the total flux density (i.e., the value provided by the SED model, given the redshift of the SMBH), ϑ is the angular diameter of the photon ring (which for our purposes is given by Equation (2)), u is the length of the baseline in units of wavelengths, W₀ is the FWHM angular thickness of the lowest-order (i.e., n = 0) photon ring, and η = 1 − e^−π is a normalizing prefactor. We assume W₀ = ϑ/5 (Event Horizon Telescope Collaboration et al. 2019e, 2019f). Equation (17) is shown as the gray curve in Figure 6.

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On long baselines (i.e., u ≫ 1/ϑ), the bandwidth-averaged flux density will be given by

$$\begin{eqnarray}&&\displaystyle \bar{S}(u)\approx \displaystyle \frac{2\eta {S}_{0}}{{\pi }^{2}}\sqrt{\tfrac{2}{\vartheta u}}\sum _{n=0}^{\infty }{e}^{-n\pi }{e}^{-\tfrac{{\left(\pi {{uW}}_{n}\right)}^{2}}{4\mathrm{ln}(2)}},\end{eqnarray} \tag{ 19 }$$

which is smaller by a factor 2/π than the envelope of Equation (17) as a result of averaging over many periods; Equation (19) is shown as the dashed blue curve in Figure 6. By replacing S_ν in Equation (6) with $\bar{S}(1/{\theta }_{r})$ from Equation (19) and then recomputing the SMBH source counts via Equation (7), the form of N(θ_r , σ_ν ) becomes that shown in Figure 7. Unlike in Figure 4, the source counts no longer monotonically increase as angular resolution improves (i.e., as θ_r decreases), because Equation (19) ensures that longer baselines see lower flux densities from any given SMBH. An analytic approximation for the resulting N(θ_r , σ_ν ) is provided in Appendix C.

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Figure 8 shows the integrand from Equation (7) plotted over the domain of integration for several values of θ_r and σ_ν , providing the distribution of observable SMBHs as a function of M and z. The number of objects generally increases with increasing redshift (at fixed mass) and with decreasing mass (at fixed redshift), though the density peaks at z ≈ 2 for the smallest values of σ_ν . For certain configurations, such as θ_r = 0.1 μas and σ_ν = 10⁻⁵ Jy, the impact of Equation (19) is visually apparent as a lack of monotonicity in the source counts with increasing redshift (at fixed mass). This behavior reflects the fact that a fixed baseline becomes sensitive to emission from larger spatial scales around a particular SMBH as that SMBH is moved to larger distances; that is, $\bar{S}$ increases as ϑ decreases. On certain intervals in z, this flux increase associated with smaller ϑ is more than sufficient to compensate for the flux decrease associated with the increased distance to the SMBH.

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The expression in Equation (19) for the horizon-scale flux density contains contributions from all orders of photon rings, and in Figure 6 we can see that rings of different order are expected to dominate the observed flux density on different baseline length intervals. Depending on the value of u relative to 1/ϑ, a telescope may thus be primarily sensitive to emission from photon rings with n > 0. To determine the number of sources from which we expect to be able to detect higher-order photon rings, we can decompose the total source counts into bins corresponding to which order of photon ring dominates the emission.

We take as our resolution requirement to "see" the nth sub-ring that θ_r ≤ 2w_n−1, where ${w}_{n}={W}_{n}/\sqrt{8\mathrm{ln}(2)}$ is the Gaussian width corresponding to the FWHM W_n (Equation (18)). This angular resolution requirement can be recast as a mass threshold m_n for a given redshift, analogous to Equation (3); for n > 0, we have

$$\begin{eqnarray}&&M\geqslant {m}_{n}\equiv 5\sqrt{2\mathrm{ln}(2)}{e}^{(n-1)\pi }{m}_{0},\end{eqnarray} \tag{ 20 }$$

where m₀ is defined in Equation (3). Figure 6 marks the n > 0 and n > 1 resolution thresholds using vertical dashed cyan lines. To ensure that the emission is optically thin enough to see down to the nth sub-ring, we further impose a more stringent condition on the optical depth of

$$\begin{eqnarray}&&\tau \leqslant {\tau }_{n}=\displaystyle \frac{1}{n+1}.\end{eqnarray} \tag{ 21 }$$

By replacing the lower mass limit m₀ in Equation (7) with m_n , and by replacing the τ ≤ 1 condition in Equation (6) with τ ≤ τ_n , we can compute the source counts associated with objects for which a photon ring of order n or greater is detectable.

Figure 9 shows these source counts for the first three orders of photon ring at observing frequencies of 86, 230, 345, and 690 GHz, corresponding to standard atmospheric transmission windows (Thompson et al. 2017). At each observing frequency, we see qualitatively similar behavior: the source counts corresponding to the higher-order photon rings look approximately like scaled-down versions of the n ≥ 0 counts. For each additional order, the same source count value is achieved at an angular resolution threshold that is ∼20 times finer and a sensitivity threshold that is approximately ∼100 times fainter than was necessary at the previous order. The angular resolution increment is associated with the factor e^−π ≈ 1/23 in Equation (18) that sets the angular size ratio between consecutive photon rings. The flux density increment comes from a combination of a similar exponential suppression factor (a factor e^−π from the summand of Equation (19)) as well as the fact that the flux density profile is being observed on baselines that are typically a factor of e^π longer, thereby incurring an additional flux density factor of e^−π/2 ≈ 1/5 from the u^−1/2 proportionality in Equation (19).

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The evolution of the source counts with frequency primarily affects the required sensitivity, with higher-frequency observations achieving the same source counts at a higher value of σ_ν than lower-frequency observations. The flux density threshold required to detect a particular number of objects is approximately 1 order of magnitude smaller at 86 GHz than at 690 GHz; that is, approximately 1 order of magnitude better sensitivity—in terms of Jy—is required at 86 GHz than at 690 GHz. The angular resolution requirement does not show substantial evolution with frequency across this range.

The analysis presented in this section thus far has assumed that an interferometric baseline can observe the entire sky with the same angular resolution. However, in reality, any physical baseline between two stations will have a different projected length as seen from different locations in the sky. The resolving power of the baseline will thus be a function of source location on the sky, which means that the number of black hole shadows a baseline can detect per unit solid angle will also vary across the sky.

For a particular baseline, we can define a spherical coordinate system (θ, ϕ) such that θ is a polar angle measured from the axis defined by the baseline orientation and ϕ is measured azimuthally around this axis. θ_r (θ) is then the effective angular resolution of the baseline when projected toward a source at a sky position with polar angle θ,

$$\begin{eqnarray}&&{\theta }_{r}(\theta ,b)=\displaystyle \frac{1}{b\sin \theta }=\displaystyle \frac{{\theta }_{r,0}}{\sin \theta },\end{eqnarray} \tag{ 22 }$$

where b is the baseline length in units of the observing wavelength and θ_r,0 = 1/b is the angular resolution achieved when θ = π/2 (i.e., the finest resolution achievable by the baseline). Denoting the number density of sources per unit solid angle as $\tfrac{{d}^{2}N}{d{\rm{\Omega }}}$, we can express the total number of sources observable by this baseline as

$$\begin{eqnarray}&&N(b)={\iint }_{{{\rm{\Omega }}}_{\mathrm{vis}}}\sin \theta \displaystyle \frac{{d}^{2}N}{d{\rm{\Omega }}}\left[{\theta }_{r}(\theta ,b)\right]\,d\theta \,d\phi ,\end{eqnarray} \tag{ 23 }$$

where we have explicitly indicated that the number density is a function of the angular resolution, θ_r (θ, b), and we have assumed that sources are distributed isotropically on the sky such that there is no ϕ dependence. The integral is carried out over the solid angle Ω_vis on the sky that is visible to the baseline.

To illustrate the impact of this geometric effect on source counts, we consider the concrete example of an interferometric baseline formed between two space-based antennas, each of which can see the entire sky. In this case, the function d² N/dΩ is given simply by N(θ_r , σ_ν )/4π, and the domain of integration for Equation (23) will be all (θ, ϕ); Figure 10 shows the result of this evaluation. Relative to the source counts in Figure 7, at large values of θ_r,0 (e.g., ∼20 μas) the source counts in Figure 10 are reduced because some fraction of the sky is not observed with sufficient angular resolution to see SMBHs with shadow sizes that are close to θ_r,0. The magnitude of this reduction is modest, amounting to a factor of 3π/16 ≈ 0.59 for uniformly distributed sources in flat space (see Equation (D3) with α = 4). However, a much more pronounced impact can be seen in the region of fine angular resolution and poor sensitivity (e.g., the region around θ_r,0 ≈ 10⁻¹ μas and σ_ν ≈ 10⁻⁴ Jy), where the source counts in Figure 10 are significantly increased relative to Figure 7. This difference arises because N(θ_r , σ_ν ) increases rapidly toward larger θ_r in this region, and so the coarser angular resolutions arising from baseline projection provide access to many SMBHs that a baseline with a fixed angular resolution of θ_r,0 across the entire sky would be unable to see. In this region of the (θ_r,0, σ_ν ) space, the impact of baseline projection is to increase the accessible number of SMBH shadows by several orders of magnitude.

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While Equation (23) provides the source counts appropriate for a fixed baseline, in real-world arrays the baseline will typically be changing orientation with time. For instance, a spaceborne antenna forming a baseline with another antenna situated on the Earth would execute a complete revolution once every orbital period, as observed by a distant source. One effect of this rotation is to make a larger fraction of the sky observable with the finest resolution than would otherwise be possible with just the instantaneous configuration, up to a unit fraction if both stations are spaceborne and thus can view the entire sky. The net impact of rotating the baseline is to bring more SMBH shadows into view than would be accessible by a static baseline. Appendix D provides a more detailed exposition of the sampling behavior of such a baseline as it rotates.

As of the writing of this paper, the SMBH in M87 is the only one whose shadow size (∼40 μas) and horizon-scale flux density (∼0.5 Jy at 230 GHz) have been directly imaged ¹⁴ (Event Horizon Telescope Collaboration et al. 2019a, 2019b, 2019c, 2019d, 2019e, 2019f). M87 thus presents a natural test case against which to compare our source counts predictions from Section 2. Our model predicts that the number of SMBHs having ϑ > 40 μas and S_ν > 0.5 Jy should be between ∼0.03 and ∼0.23 for the lower and upper BHMF prescriptions, respectively. Compared against the 1 object known to adhere to the chosen criteria, our model is systematically underpredicting the prevalence of M87. This underprediction may be explained at least in part if the local density of galaxies exceeds the cosmic mean, as suggested by, for example, Dálya et al. (2018), ¹⁵ which violates our model assumption of a homogeneous distribution of SMBHs. However, any such overdensity likely does not explain a discrepancy larger than a factor of ∼2, indicating that we may simply be finding ourselves on the high end of sampling variance. We thus expect that using the existence and properties of M87 to extrapolate the number of SMBHs with smaller shadows or weaker flux densities will result in systematically overoptimistic predictions; that is, more sources will be predicted than our modeling suggests the real universe likely contains.

Similarly, the EHT is currently the only telescope to have successfully carried out shadow-resolved observations of an SMBH. The number of sources that the EHT is able to resolve and detect the shadows for thus presents a test case against which to compare our source counts predictions from Section 3. The EHT currently relies on observing with ALMA as part of the array, and during the 2017 observing campaign that led to the published M87 black hole images, ALMA itself required in-beam sources with flux densities of ≥0.5 Jy to perform the array phasing necessary for it to participate in VLBI observations (Matthews et al. 2018). For the purposes of estimating source counts, this phasing threshold effectively sets the sensitivity limit of the EHT. In this case, our model predicts that for θ_r = 20 μas and σ_ν = 0.5 Jy we should expect to resolve and detect up to ∼0.4 sources, similar to the projected number based on the above extrapolation using M87 as a benchmark.

However, the 0.5 Jy phasing threshold has since been relaxed by permitting the transfer of phase corrections to faint targets from nearby but bright out-of-beam calibrators, and even the on-source phasing threshold can potentially be lowered through refinement of the phasing algorithm. Moving forward, the EHT may thus be able to observe much fainter targets. In a best-case scenario in which the phasing threshold is reduced to mJy levels, these improvements could permit the nominal sensitivity of the EHT to be used for source count estimates. Observing at 230 GHz, the EHT achieves θ_r ≈ 20 μas and σ_ν ≈ 10⁻³ Jy, for which our model predicts the number of accessible SMBHs to be between ∼0.6 and ∼5.7 for the lower and upper BHMF prescriptions, respectively. We thus predict that the EHT could potentially gain access to approximately an order of magnitude more shadow-resolved sources by improving its effective sensitivity to mJy levels in this way.

More generally, the behavior of N(θ_r , σ_ν )—in particular, the behavior of its gradient—has implications for how an existing array can be most efficiently augmented to increase the number of accessible black hole shadows. As mentioned in Section 4.1, the EHT is currently operating with an angular resolution of θ_r ≈ 20 μas and an effective flux density sensitivity between σ_ν ≈ 10⁻³ Jy and σ_ν ≈ 0.5 Jy. This sensitivity range straddles the Pareto front for θ_r ≈ 20 μas (see Figure 11), such that with σ_ν ≈ 0.5 Jy the EHT array could most significantly increase the number of horizon-resolved black hole targets through improvements in sensitivity. However, once the sensitivity improves beyond the Pareto threshold of ∼70 mJy then the EHT will require enhanced angular resolution to increase the source counts further. For instance, at a fixed sensitivity of σ_ν = 10⁻³ Jy, an order-of-magnitude improvement in the angular resolution would yield an increase of approximately 2 orders of magnitude in the number of detectable black hole shadows; in contrast, while keeping the angular resolution fixed at 20 μas, arbitrary improvements in sensitivity beyond 10⁻³ Jy would not yield many additional sources.

In practice, an Earth-based array like the EHT is limited to a maximum physical baseline length of one Earth diameter, meaning that any significant angular resolution improvements must come from increasing the observing frequency. A near-future aspiration for the EHT (Event Horizon Telescope Collaboration et al. 2019b), and a defining capability for the next-generation EHT (ngEHT; Doeleman et al. 2019; Raymond et al. 2021) will be to observe at a frequency of 345 GHz. At a fixed long-baseline sensitivity of σ_ν = 10⁻³ Jy, we expect that the effective 50% improvement in angular resolution over the current EHT should correspond to a factor of ∼3 increase in the number of detectable black hole shadows. In contrast, at a fixed θ_r = 20 μas, the doubling of the baseline sensitivity that the ngEHT is expected to provide will only increase the source counts by ∼10%.

While angular resolution may ultimately limit the number of observable black hole shadows for ground-based interferometers like the EHT and ngEHT, sensitivity is expected to be the limiting factor for many prospective interferometers that network with space-based stations. For instance, a baseline connecting a station on Earth to one located at the Earth–Sun L2 Lagrange point—such as may be possible using the proposed Millimetron (Kardashev et al. 2014) or Origins (Wiedner et al. 2021) space telescopes—would have a finest 230 GHz angular resolution of θ_r ≈ 0.2 μas. At this resolution, we expect that a sensitivity of σ_ν ≲ 10⁻⁴ Jy would be required to detect even a single object. To achieve this sensitivity level, a 10-m dish observing at 230 GHz as part of a baseline with the phased ALMA array would require a time-bandwidth product of ∼3 × 10¹² (e.g., 3 min of on-source integration time using 16 GHz of bandwidth), which is already larger than achieved by the EHT. Improving the sensitivity to 10⁻⁵ Jy would require a time-bandwidth product that is 2 orders of magnitude larger still (e.g., 2 hr of on-source integration time using 32 GHz of bandwidth), and pushing to 10⁻⁶ Jy would require an additional 2 orders beyond that (e.g., 5 days of on-source integration time using 64 GHz of bandwidth). Achieving the ∼10⁻⁶ Jy Pareto front flux density corresponding to a ∼0.2 μas angular resolution thus imposes demanding sensitivity and stability requirements, and we expect that the number of sources accessible using long (≫1 Earth diameter) Earth-space baselines will be sensitivity limited rather than resolution limited.

We take the underlying accretion flow properties to be described by the self-similar models developed by NY95, in which the relevant parameters are the black hole mass M, the accretion rate $\dot{M}$, the radius R, the viscosity parameter α, the ratio of gas to magnetic pressure β, ¹⁸ and the fraction f of viscously dissipated energy that is advected into the black hole. Following M97, we use scaled quantities,

$$\begin{eqnarray}&&M=\left(1\ {M}_{\odot }\right)m,\end{eqnarray} \tag{ A3a }$$

$$\begin{eqnarray}\begin{array}{rcl}R & = & {{rR}}_{S}\\ & = & \left(2.953\times {10}^{5}\ \mathrm{cm}\right)m\,r,\end{array}\end{eqnarray} \tag{ A3b }$$

$$\begin{eqnarray}\begin{array}{rcl}\dot{M} & = & \dot{m}{\dot{M}}_{\mathrm{Edd}}\\ & = & \left({\dot{m}}_{0}{\dot{M}}_{\mathrm{Edd}}\right){\left(\displaystyle \frac{R}{{R}_{S}}\right)}^{s}\\ & = & \left(1.399\times {10}^{18}\ {\rm{g}}\ {{\rm{s}}}^{-1}\right)m\,{\dot{m}}_{0}\,{r}^{s},\end{array}\end{eqnarray} \tag{ A3c }$$

where R_S = 2GM/c² is the Schwarzschild radius, ${\dot{M}}_{\mathrm{Edd}}={L}_{\mathrm{Edd}}/\eta {c}^{2}$ is the Eddington accretion rate, and we have taken the radiative efficiency η to be 0.1. Here, the difference between Equation A3(c) and M97 Equation (4) comes from our adoption of the radius-dependent accretion rate from Blandford & Begelman (1999), which accounts for outflowing material via a radial dependence of the mass accretion rate with power-law index s.

The self-similar equations describing the accretion flow, M97 Equation (5), become

$$\begin{eqnarray}\begin{array}{rcl}\rho & = & \displaystyle \frac{\dot{M}}{4\pi H\alpha {c}_{1}\sqrt{{GMR}}}\\ & = & \left(6.022\times {10}^{-5}\ {\rm{g}}\ {\mathrm{cm}}^{-3}\right){\alpha }^{-1}\,{c}_{1}^{-1}\,{m}^{-1}\,{\dot{m}}_{0}\,{r}^{-(3/2)+s},\end{array}\end{eqnarray} \tag{ A4a }$$

$$\begin{eqnarray}\begin{array}{rcl}{n}_{e} & = & \displaystyle \frac{\rho }{{\mu }_{e}{m}_{p}}\\ & = & \left(3.158\times {10}^{19}\ {\mathrm{cm}}^{-3}\right){\alpha }^{-1}\,{c}_{1}^{-1}\,{m}^{-1}\,{\dot{m}}_{0}\,{r}^{-(3/2)+s},\end{array}\end{eqnarray} \tag{ A4b }$$

$$\begin{eqnarray}\begin{array}{rcl}B & = & \sqrt{\tfrac{24\pi {c}_{3}{GM}\rho }{(1+\beta )R}}\\ & = & \left(1.428\times {10}^{9}\ {\rm{G}}\right){\left(1+\beta \right)}^{-1/2}{\alpha }^{-1/2}\,{c}_{1}^{-1/2}\,{c}_{3}^{1/2}\,{m}^{-1/2}\,{\dot{m}}_{0}^{1/2}\,{r}^{-(5/4)+(s/2)},\end{array}\end{eqnarray} \tag{ A4c }$$

where ρ is the mass density, B is the magnetic field strength, n_e is the number density of electrons, α is the disk viscosity parameter (Shakura & Sunyaev 1973), H is the disk scale height (we have followed M97 in setting H = R), μ_e = 1.14 is the mean molecular weight (NY95), and c₁ ≈ 0.5 and c₃ ≈ 0.3 are constants defined in NY95 and specified in Table 3.

We adopt a power-law radial profile for the electron temperature T_e of the form

$$\begin{eqnarray}&&{T}_{e}=\displaystyle \frac{{T}_{e,0}}{{r}^{1-t}},\end{eqnarray} \tag{ A5 }$$

with t ≤ 1. From NY95 Equation (2.16), the two-temperature accretion flow must satisfy

$$\begin{eqnarray}&&{T}_{i}+1.08{T}_{e}=\left(6.66\times {10}^{12}\ {\rm{K}}\right){\left(1+\beta \right)}^{-1}\beta \,{c}_{3}\,{r}^{-1},\end{eqnarray} \tag{ A6 }$$

where T_i is the ion temperature. Setting T_i = T_e at some maximum radius $r={r}_{\max }$ yields an expression for t,

$$\begin{eqnarray}&&t=\displaystyle \frac{1}{\mathrm{ln}\left({r}_{\max }\right)}\mathrm{ln}\left(\displaystyle \frac{\left(6.66\times {10}^{12}\ {\rm{K}}\right)\beta {c}_{3}}{2.08{T}_{e,0}\left(1+\beta \right)}\right),\end{eqnarray} \tag{ A7 }$$

such that T_i > T_e for all $r\lt {r}_{\max }$.

The plasma in an ADAF is heated by viscous forces, with the total heating rate per unit volume denoted as q⁺. Some fraction δ of this energy is deposited into the electrons, while the remaining fraction (1 − δ) heats the ions. The ions can transfer thermal energy to the electrons via Coulomb collisions, with the rate of this transfer denoted by q^ie, and the electrons can radiate energy away at a rate q⁻. Taken altogether, energy balance yields advected energy rates of

$$\begin{eqnarray}&&{q}^{\mathrm{adv},{\rm{i}}}=(1-\delta ){q}^{+}-{q}^{\mathrm{ie}},\end{eqnarray} \tag{ A8a }$$

$$\begin{eqnarray}&&{q}^{\mathrm{adv},{\rm{e}}}=\delta {q}^{+}+{q}^{\mathrm{ie}}-{q}^{-},\end{eqnarray} \tag{ A8b }$$

for the ions (q^adv,i) and electrons (q^adv,e). The ion heating is driven by viscous dissipation, while the dominant electron heating source depends on the accretion rate; at high accretion rates the ion-electron heating is dominant, whereas at low accretion rates the viscous heating is more important.

NY95 give an expression for the viscous heating rate per unit volume,

$$\begin{eqnarray}\begin{array}{rcl}{q}^{+} & = & \displaystyle \frac{3\rho \alpha {c}_{1}{c}_{3}}{2{Rf}(1+\beta )}{\left(\displaystyle \frac{{GM}}{R}\right)}^{3/2}\\ & = & \left(2.914\times {10}^{21}\ \mathrm{erg}\ {\mathrm{cm}}^{-3}\ {{\rm{s}}}^{-1}\right){f}^{-1}{\left(1+\beta \right)}^{-1}{c}_{3}^{1/2}{m}^{-2}{\dot{m}}_{0}{r}^{-4+s},\end{array}\end{eqnarray} \tag{ A9 }$$

where f is the fraction of viscously dissipated energy that is advected into the black hole. The total (i.e., volume-integrated) viscous heating rate is then given by

$$\begin{eqnarray}&&\begin{array}{l}{Q}^{+}=\left(9.430\times {10}^{38}\ \mathrm{erg}\ {{\rm{s}}}^{-1}\right){f}^{-1}{\left(1+\beta \right)}^{-1}{c}_{3}^{1/2}m{\dot{m}}_{0}\\ \times \ \left\{\begin{array}{ll}{\left(1-s\right)}^{-1}\left({r}_{\min }^{-1+s}-{r}_{\max }^{-1+s}\right), & s\ne 1\\ \mathrm{ln}\left({r}_{\max }/{r}_{\min }\right), & s=1\end{array}\right.,\end{array}\end{eqnarray} \tag{ A10 }$$

where ${r}_{\min }$ and ${r}_{\max }$ are the minimum and maximum radius, respectively. ¹⁹

The heating rate per unit volume of the electrons from Coulomb interactions with protons is given by Stepney & Guilbert (1983),

$$\begin{eqnarray}\begin{array}{rcl}{q}^{\mathrm{ie}} & = & \displaystyle \frac{3{m}_{e}}{2{m}_{p}}{n}_{e}{n}_{i}{\sigma }_{{\rm{T}}}c\mathrm{ln}({\rm{\Lambda }})\left(\displaystyle \frac{k({T}_{i}-{T}_{e})}{{K}_{2}(1/{\theta }_{e}){K}_{2}(1/{\theta }_{i})}\right)\times \ \left[\displaystyle \frac{2{\left({\theta }_{e}+{\theta }_{i}\right)}^{2}+1}{{\theta }_{e}+{\theta }_{i}}{K}_{1}\left(\displaystyle \frac{{\theta }_{e}+{\theta }_{i}}{{\theta }_{e}{\theta }_{i}}\right)+2{K}_{0}\left(\displaystyle \frac{{\theta }_{e}+{\theta }_{i}}{{\theta }_{e}{\theta }_{i}}\right)\right]\\ & \approx & \left(5.624\times {10}^{-32}\ \mathrm{erg}\ {\mathrm{cm}}^{3}\ {{\rm{s}}}^{-1}\ {{\rm{K}}}^{-1}\right)\displaystyle \frac{{n}_{e}^{2}({T}_{i}-{T}_{e})}{{K}_{2}(1/{\theta }_{e})}\left(2+2{\theta }_{e}+\displaystyle \frac{1}{{\theta }_{e}}\right){e}^{-1/{\theta }_{e}},\end{array}\end{eqnarray} \tag{ A11 }$$

where θ_e = kT_e /m_e c² is the dimensionless electron temperature, θ_i = kT_i /m_p c² is the dimensionless ion temperature, $\mathrm{ln}({\rm{\Lambda }})\approx 20$ is a Coulomb logarithm, K_n represents a modified Bessel function of the nth order, and we have assumed n_e = n_i . In the second line we have adopted the approximation from M97. ²⁰ We note that in evaluating the prefactor in Equation (A11) we have followed NY95 and multiplied by an additional factor of 1.25 to account for the ions containing a mixture of roughly 75% hydrogen and 25% helium.

The volume-integrated ion-electron heating rate is given by

$$\begin{eqnarray}&&{Q}^{\mathrm{ie}}=\left(3.236\times {10}^{17}\ {\mathrm{cm}}^{3}\right){m}^{3}{\int }_{{r}_{\min }}^{{r}_{\max }}{q}^{\mathrm{ie}}{r}^{2}\,{dr},\end{eqnarray} \tag{ A12 }$$

which does not have an analytic form and so must be integrated numerically.

The observed emission in radio and (sub)millimeter bands is dominated by synchrotron radiation, but the primary electron cooling mechanisms also include bremsstrahlung and inverse Compton radiation. Each of these emission mechanisms contributes to q⁻, and each depends on the electron temperature T_e .

A.3.1. Synchrotron Emission

We use a form for the synchrotron spectrum from Mahadevan et al. (1996; see also NY95; M97), which assumes an isotropic distribution of relativistic electrons. The synchrotron spectral emissivity is given by

$$\begin{eqnarray}&&{\epsilon }_{\mathrm{synch},\nu }=\left(4.43\times {10}^{-30}\ \mathrm{erg}\ {{\rm{s}}}^{-1}\ {\mathrm{Hz}}^{-2}\right)\displaystyle \frac{4\pi {n}_{e}\nu }{{K}_{2}(1/{\theta }_{e})}M({x}_{M}),\end{eqnarray} \tag{ A13 }$$

where we assume the relativistic limit for M(x_M ),

$$\begin{eqnarray}&&M({x}_{M})=\displaystyle \frac{4.0505}{{x}_{M}^{1/6}}\left(1+\displaystyle \frac{0.40}{{x}_{M}^{1/4}}+\displaystyle \frac{0.5316}{{x}_{M}^{1/2}}\right)\exp \left(-1.8899{x}_{M}^{1/3}\right),\end{eqnarray} \tag{ A14 }$$

and x_M is a dimensionless frequency,

$$\begin{eqnarray}&&{x}_{M}=\displaystyle \frac{2\nu }{3{\nu }_{b}{\theta }_{e}^{2}},\end{eqnarray} \tag{ A15a }$$

$$\begin{eqnarray}\begin{array}{rcl}{\nu }_{b} & = & \displaystyle \frac{{eB}}{2\pi {m}_{e}c}\\ & = & \left(3.998\times {10}^{15}\ \mathrm{Hz}\right){\left(1+\beta \right)}^{-1/2}{\alpha }^{-1/2}{c}_{1}^{-1/2}{c}_{3}^{1/2}{m}^{-1/2}{\dot{m}}_{0}^{1/2}{r}^{-(5/4)+(s/2)}.\end{array}\end{eqnarray} \tag{ A15b }$$

Equation (A13) assumes optically thin emission, but below some critical frequency ν_c (which is a function of radius) we expect the synchrotron to be optically thick and thus described by a blackbody spectrum. We follow M97 and determine ν_c (r) by equating emission within a volume of radius r to the Rayleigh–Jeans blackbody emission from a spherical surface at that radius,

$$\begin{eqnarray}&&{\epsilon }_{\mathrm{synch},{\nu }_{c}}\left(\displaystyle \frac{4\pi {R}^{3}}{3}\right)=4\pi {R}^{2}\left(\displaystyle \frac{2\pi {\nu }_{c}^{2}{{kT}}_{e}}{{c}^{2}}\right),\end{eqnarray} \tag{ A16 }$$

which lends itself to a prescription for estimating the optical depth more generally of

$$\begin{eqnarray}\begin{array}{rcl}\tau & = & \displaystyle \frac{{\epsilon }_{\mathrm{synch},\nu }{{Rc}}^{2}}{6\pi {\nu }^{2}{{kT}}_{e}}\\ & = & {\left(\displaystyle \frac{\nu }{{\nu }_{c}}\right)}^{-2}\left(\displaystyle \frac{{\epsilon }_{\mathrm{synch},{\nu }_{c}}}{{\epsilon }_{\mathrm{synch},\nu }}\right).\end{array}\end{eqnarray} \tag{ A17 }$$

We numerically solve Equation (A16) for ν_c at ${R}_{\min }$ and ${R}_{\max }$, yielding a peak frequency ν_p at ${R}_{\min }$ (with luminosity L_p ) and a minimum frequency ν_m at ${R}_{\max }$ (with luminosity L_m ); the left panel of Figure 14 shows how ν_p changes with m and $\dot{m}$. We take the synchrotron spectrum to be blackbody (i.e., optically thick) at frequencies below ν_m , optically thin with an emissivity described by Equation (A13) at frequencies above ν_p , and a power law at intermediate frequencies. That is,

$$\begin{eqnarray}{L}_{\nu ,\mathrm{synch}}=\left\{\begin{array}{ll}\left(1.058\times {10}^{-24}\ \mathrm{erg}\ {{\rm{s}}}^{-1}\ {\mathrm{Hz}}^{-3}\ {{\rm{K}}}^{-1}\right){T}_{e,0}{m}^{2}{\nu }^{2}{r}_{\max }^{1+t}, & \nu \lt {\nu }_{m}\\ {L}_{m}{\left(\displaystyle \frac{\nu }{{\nu }_{m}}\right)}^{\mathrm{ln}({L}_{p}/{L}_{m})/\mathrm{ln}({\nu }_{p}/{\nu }_{m})}, & {\nu }_{m}\leqslant \nu \leqslant {\nu }_{p}\\ \left(1.896\times {10}^{8}\ \mathrm{erg}\ {{\rm{s}}}^{-1}\ {\mathrm{Hz}}^{-2}\right)\displaystyle \frac{M({x}_{M})}{{K}_{2}(1/{\theta }_{e})}{\alpha }^{-1}{c}_{1}^{-1}{m}^{2}{\dot{m}}_{0}\nu {r}_{\min }^{(3/2)+s}, & \nu \gt {\nu }_{p}.\end{array}\right.\end{eqnarray} \tag{ A18 }$$

The total emitted synchrotron power is then the integral of L_ν,synch over frequency,

$$\begin{eqnarray}&&{P}_{\mathrm{synch}}={\int }_{0}^{\infty }{L}_{\nu ,\mathrm{synch}}\,d\nu ,\end{eqnarray} \tag{ A19 }$$

which we evaluate numerically.

A.3.2. Bremsstrahlung Emission

We use an expression for the bremsstrahlung emission that follows M97 Equation (27),

$$\begin{eqnarray}\begin{array}{rcl}{q}_{\mathrm{brems}} & = & \left(1.48\times {10}^{-22}\ \mathrm{erg}\ {\mathrm{cm}}^{3}\ {{\rm{s}}}^{-1}\right){n}_{e}^{2}F({\theta }_{e})\\ & = & \left(1.476\times {10}^{17}\ \mathrm{erg}\ {\mathrm{cm}}^{-3}\ {{\rm{s}}}^{-1}\right){\alpha }^{-2}{c}_{1}^{-2}{m}^{-2}{\dot{m}}_{0}^{2}F({\theta }_{e}){r}^{-3+2s},\end{array}\end{eqnarray} \tag{ A20 }$$

where

$$\begin{eqnarray}F({\theta }_{e})=\left\{\begin{array}{ll}4{\left(\displaystyle \frac{2{\theta }_{e}}{{\pi }^{3}}\right)}^{1/2}\left(1+1.781{\theta }_{e}^{1.34}\right)+1.73{\theta }_{e}^{3/2}\left(1+1.1{\theta }_{e}+{\theta }_{e}^{2}-1.25{\theta }_{e}^{5/2}\right), & {\theta }_{e}\leqslant 1\\ \left(\displaystyle \frac{9{\theta }_{e}}{2\pi }\right)\left[\mathrm{ln}\left(1.123{\theta }_{e}+0.48\right)+1.5\right]+2.30{\theta }_{e}\left[\mathrm{ln}\left(1.123{\theta }_{e}\right)+1.28\right], & {\theta }_{e}\gt 1.\end{array}\right.\end{eqnarray} \tag{ A21 }$$

The volume-integrated power emitted in bremsstrahlung radiation will then be

$$\begin{eqnarray}\begin{array}{rcl}{P}_{\mathrm{brems}} & = & \left(3.236\times {10}^{17}\ {\mathrm{cm}}^{3}\right){m}^{3}{\displaystyle \int }_{{r}_{\min }}^{{r}_{\max }}{q}_{\mathrm{brems}}{r}^{2}\,{dr}\\ & = & \left(4.776\times {10}^{34}\ \mathrm{erg}\ {{\rm{s}}}^{-1}\right){\alpha }^{-2}{c}_{1}^{-2}m{\dot{m}}_{0}^{2}{\displaystyle \int }_{{r}_{\min }}^{{r}_{\max }}F({\theta }_{e}){r}^{-1+2s}\,{dr},\end{array}\end{eqnarray} \tag{ A22 }$$

with a spectral dependence given by

$$\begin{eqnarray}\begin{array}{rcl}{L}_{\nu ,\mathrm{brems}} & = & \left(2.292\times {10}^{24}\ \mathrm{erg}\ {{\rm{s}}}^{-1}\ {\mathrm{Hz}}^{-1}\ {\rm{K}}\right){\alpha }^{-2}{c}_{1}^{-2}m{\dot{m}}_{0}^{2}{T}_{e,0}^{-1}{\displaystyle \int }_{{r}_{\min }}^{{r}_{\max }}F({\theta }_{e})\exp \left(-\displaystyle \frac{h\nu }{{{kT}}_{e}}\right){r}^{-2+2s+t}\,{dr}.\end{array}\end{eqnarray} \tag{ A23 }$$

We integrate both of the above expressions numerically.

A.3.3. Inverse Compton Emission

We follow M97 in considering Comptonization only of synchrotron photons emitted predominantly at the peak frequency ν_p , for which the spectrum in the temperature range of interest is expected to be a power law,

$$\begin{eqnarray}&&{L}_{\nu ,\mathrm{compt}}={L}_{p}{\left(\displaystyle \frac{\nu }{{\nu }_{p}}\right)}^{-{\alpha }_{c}}.\end{eqnarray} \tag{ A24 }$$

The power-law index α_c is determined both by how frequently photons are scattered (which is determined by the optical depth of the scattering process) and by how much a photon is amplified during a typical scattering event. We use an expression for the optical depth to electron scattering τ_es adapted from NY95 Equation (2.15),

$$\begin{eqnarray}\begin{array}{rcl}{\tau }_{\mathrm{es}} & = & 2{n}_{e}{\sigma }_{T}{R}_{\min }\\ & = & 6.205{\alpha }^{-1}{c}_{1}^{-1}{\dot{m}}_{0}{r}_{\min }^{-(1/2)+s}.\end{array}\end{eqnarray} \tag{ A25 }$$

We take the mean amplification factor A from M97 Equation (32) (originally inspired from Rybicki & Lightman 1979),

$$\begin{eqnarray}&&A=1+4{\theta }_{e,0}+16{\theta }_{e,0}^{2},\end{eqnarray} \tag{ A26 }$$

which together with τ_es determines the power-law slope for the Compton emission,

$$\begin{eqnarray}&&{\alpha }_{c}=-\displaystyle \frac{\mathrm{ln}\left({\tau }_{\mathrm{es}}\right)}{\mathrm{ln}(A)}.\end{eqnarray} \tag{ A27 }$$

The total Compton power will then be given by the integral of L_ν,compt up to the maximum final frequency of a Comptonized photon (ν_f = 3kT_e,0/h),

$$\begin{eqnarray}\begin{array}{rcl}{P}_{\mathrm{compt}} & = & {\displaystyle \int }_{{\nu }_{p}}^{{\nu }_{f}}{L}_{\nu ,\mathrm{compt}}\,d\nu \\ & = & \displaystyle \frac{{\nu }_{p}{L}_{p}}{1-{\alpha }_{c}}\left[{\left(\displaystyle \frac{{\nu }_{f}}{{\nu }_{p}}\right)}^{1-{\alpha }_{c}}-1\right].\end{array}\end{eqnarray} \tag{ A28 }$$

A.3.4. Electron Advection

The M97 model assumed that T_e ≪ T_i , and it therefore ignored electron energy advection. This assumption was reasonable for the parameters considered in that paper, particularly the choice of δ = m_e /m_p . However, the modern view is that δ is much larger (≈0.3; see Yuan & Narayan 2014). Such large values of δ make electrons significantly hotter, especially at very low $\dot{m}$, and so energy advection in electrons can no longer be ignored.

When electron advection is included, M97 Equation (8) gains an additional term Q^adv,e and becomes Equation (A1). This advective cooling term is given by

$$\begin{eqnarray}&&{Q}^{\mathrm{adv},{\rm{e}}}={\int }_{{R}_{\min }}^{{R}_{\max }}4\pi {R}^{2}\left({n}_{e}{{vT}}_{e}\displaystyle \frac{{{ds}}_{e}}{{dR}}\right)\,{dR},\end{eqnarray} \tag{ A29 }$$

where s_e is the entropy per electron. Let us write

$$\begin{eqnarray}&&{T}_{e}{{ds}}_{e}={{du}}_{e}+{p}_{e}d\left(\displaystyle \frac{1}{{n}_{e}}\right)=\displaystyle \frac{{{kdT}}_{e}}{{\gamma }_{\mathrm{CV}}-1}-\displaystyle \frac{{{kT}}_{e}}{{n}_{e}}{{dn}}_{e},\end{eqnarray} \tag{ A30 }$$

where, following the approach described in Narayan & Yi (1994), we express the specific heat at constant volume C_V in terms of an effective γ_CV. Substituting in Equations A4(b) and (A5) and differentiating with respect to R yields

$$\begin{eqnarray}&&{T}_{e}\displaystyle \frac{{{ds}}_{e}}{{dR}}=\displaystyle \frac{{{kT}}_{e,0}}{{R}_{S}{r}^{2-t}}\left(\displaystyle \frac{3}{2}-s-\displaystyle \frac{1-t}{{\gamma }_{\mathrm{CV}}-1}\right).\end{eqnarray} \tag{ A31 }$$

Sa̧dowski et al. (2017) provide an accurate fitting function for γ_CV, which we write as

$$\begin{eqnarray}&&{\gamma }_{\mathrm{CV}}=\displaystyle \frac{20\left(2+8{\theta }_{e}+5{\theta }_{e}^{2}\right)}{3\left(8+40{\theta }_{e}+25{\theta }_{e}^{2}\right)}.\end{eqnarray} \tag{ A32 }$$

Noting further that n_e = ρ/μ_e m_p and $\dot{M}=-4\pi {R}^{2}v\rho$, we finally obtain

$$\begin{eqnarray}&&{Q}^{\mathrm{adv},{\rm{e}}}=\left(1.013\times {10}^{26}\ \mathrm{erg}\ {{\rm{s}}}^{-1}\ {{\rm{K}}}^{-1}\right)m{\dot{m}}_{0}{T}_{e,0}{\int }_{{r}_{\min }}^{{r}_{\max }}\left(\displaystyle \frac{1-t}{{\gamma }_{\mathrm{CV}}-1}-\displaystyle \frac{3}{2}+s\right){r}^{s+t-2}\,{dr},\end{eqnarray} \tag{ A33 }$$

which we integrate numerically. We note that there are conditions under which Equation (A33) can yield a negative value for Q^adv,e; in these cases, we impose Q^adv,e = 0.

The ADAF solution ceases to exist above some critical mass accretion rate, ${\dot{m}}_{\mathrm{crit}}$, where the accretion flow is no longer advection-dominated (NY95; M97). Within the context of our SED model, this condition manifests as a maximum accretion rate above which there is no equilibrium temperature (i.e., the heating and cooling curves never cross). We numerically determine a value of ${\dot{m}}_{\mathrm{crit}}\approx {10}^{-1.7}$, and so in this paper we only work with values of ${\dot{m}}_{0}\leqslant {10}^{-2}$.

In addition to the critical $\dot{m}$ above which no ADAF solutions exist, there is also a softer threshold accretion rate above which solutions do exist but our assumed input radiative efficiency of η = 0.1 is no longer consistent with the output of the SED model. Figure 15 shows the predicted radiative efficiency from the model as a function of ${\dot{m}}_{0}$; we take the model radiative efficiency to be the ratio of the bolometric luminosity,

$$\begin{eqnarray}&&{L}_{\mathrm{bol}}={\int }_{0}^{\infty }\left({L}_{\nu ,\mathrm{synch}}+{L}_{\nu ,\mathrm{compt}}+{L}_{\nu ,\mathrm{brems}}\right)\,d\nu ,\end{eqnarray} \tag{ A34 }$$

to the accretion rate equivalent luminosity, $\dot{M}{c}^{2}$. Regardless of the input mass, the output radiative efficiency exceeds the assumed input value for ${\dot{m}}_{0}\gtrsim {10}^{-2.5}$. Though this inconsistency reflects a physical limitation of the model, we note that given the ERDF prescription used in this paper (see Section 2.4),it impacts only a small fraction of SMBHs (<1% for most M and z, reaching a peak of ∼5% for M > 10⁹ M_⊙ and z > 5).

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For a static universe in which SMBHs are distributed uniformly, the number of SMBHs that could be spatially resolved at an angular resolution θ_r by a telescope with arbitrary sensitivity will be proportional to ${\theta }_{r}^{-3}$. Similarly, the number of SMBHs that could be detected at a sensitivity σ_ν by a telescope with arbitrary angular resolution will be proportional to ${\sigma }_{\nu }^{-3/2}$. A simple function that captures both limiting behaviors is

$$\begin{eqnarray}&&N({\theta }_{r},{\sigma }_{\nu })\approx {\left[{\left(\displaystyle \frac{{\theta }_{r}}{40\ \mu \mathrm{as}}\right)}^{3}+{\left(\displaystyle \frac{{\sigma }_{\nu }}{1\ \mathrm{Jy}}\right)}^{3/2}\right]}^{-1}.\end{eqnarray} \tag{ C1 }$$

Here, we have chosen the normalization to be such that we would expect to see ≳ 1 black hole shadows with angular sizes smaller than θ_r = 40 μas and with flux densities less than σ_ν = 1 Jy—approximately matching the values appropriate for the SMBH in M87 (Event Horizon Telescope Collaboration et al. 2019c, 2019d)—and that at this angular resolution and flux density sensitivity we would expect to see N ≈ 1 black hole shadow.

Following these expectations, we fit a simple functional form to the source counts of

$$\begin{eqnarray}&&{ \mathcal N }({\theta }_{r},{\sigma }_{\nu })={\left[{\left(\displaystyle \frac{{\theta }_{r}}{{\theta }_{r,0}}\right)}^{\gamma }+{\left(\displaystyle \frac{{\sigma }_{\nu }}{{\sigma }_{\nu ,0}}\right)}^{\kappa }\right]}^{-1}.\end{eqnarray} \tag{ C2 }$$

The model parameters θ_r,0, γ, σ_ν,0, and κ are determined by minimizing the squared logarithmic differences between ${ \mathcal N }({\theta }_{r},{\sigma }_{\nu })$ from Equation (C2) and the complete numerical evaluation of N(θ_r , σ_ν ) from Equation (7) (see Section 2.5), assuming an observing frequency of 230 GHz. Both functions are evaluated on a 200 × 200 grid of (θ_r , σ_ν ) points, logarithmically spaced between [10⁻², 10²] μas in θ_r and between [10⁻⁹, 10¹] Jy in σ_ν .

We find best-fit parameter values of θ_r,0 = 21.8 μas, γ = 2.95, σ_ν,0 = 0.080 Jy, and κ = 1.32; this best fit is shown in the left panel of Figure 17. The power-law indices, γ and κ, have best-fit values that are close the initial expectations (i.e., γ = 3 and κ = 1.5), indicating that the cosmological effects are not causing large deviations from simple volumetric scaling relations. The normalization factors, θ_r,0 and σ_ν,0, are substantially different from the values in Equation (C1), in line with the model's known underprediction of M87 (see Section 4.1). Overall, the best-fit Equation (C2) provides a description of the source counts that deviates from the numerical computation by less than an order of magnitude across most of the (θ_r , σ_ν ) space. Only for θ_r ≲ 0.1 μas and σ_ν ≲ 10⁻⁷ Jy does the analytic approximation deviate from the numerical computation by more than an order of magnitude in N.

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When observing with an interferometric baseline, the correlated flux density depends on the brightness temperature T_b of the source emission. For a source that is marginally resolved and which subtends a solid angle Ω, the brightness temperature can be expressed as

$$\begin{eqnarray}\begin{array}{rcl}{T}_{b} & \approx & \displaystyle \frac{{c}^{2}{\sigma }_{\nu }}{2{\nu }^{2}k{\rm{\Omega }}}\\ & \approx & \left(1.64\times {10}^{10}\ {\rm{K}}\right)\left(\displaystyle \frac{{\sigma }_{\nu }}{1\ \mathrm{Jy}}\right){\left(\displaystyle \frac{{\theta }_{r}}{40\ \mu \mathrm{as}}\right)}^{-2}{\left(\displaystyle \frac{\nu }{230\ \mathrm{GHz}}\right)}^{-2}.\end{array}\end{eqnarray} \tag{ C3 }$$

Here, σ_ν represents the total flux density of the source, and θ_r is its angular size on the sky. For synchrotron sources, self-absorption and energy equipartition are expected to limit T_b to some maximum value of approximately 10¹¹ K (Kellermann & Pauliny-Toth 1969; Readhead 1994, though see also Kovalev et al. 2016). More specifically, the emitted brightness temperature should never exceed the electron temperature, which for our SED model described in Appendix A does not go above ∼7 × 10¹⁰ K (see the right panel of Figure 14). Following the considerations from Appendix C.1 while also accounting for this brightness temperature limit, we can modify Equation (C1) using an exponential cutoff to smoothly suppress the the high brightness temperature emission,

$$\begin{eqnarray}&&N({\theta }_{r},{\sigma }_{\nu })\approx {e}^{-{T}_{b}/\left({10}^{10}\ {\rm{K}}\right)}{\left[{\left(\displaystyle \frac{{\theta }_{r}}{40\ \mu \mathrm{as}}\right)}^{3}+{\left(\displaystyle \frac{{\sigma }_{\nu }}{0.1\ \mathrm{Jy}}\right)}^{3/2}\right]}^{-1}.\end{eqnarray} \tag{ C4 }$$

Here, θ_r should now be understood to represent a single-baseline angular resolution, and we have adjusted the σ_ν normalization to match the 230 GHz flux density observed from M87 on long baselines (Event Horizon Telescope Collaboration et al. 2019c). We have set the brightness temperature cutoff to 10¹⁰ K because we are selecting for SMBHs that are optically thin and which therefore should not typically saturate the brightness limit. We note that magnetohydrodynamic simulations of the M87 system also exhibit brightness temperatures that peak between 10¹⁰ and 10¹¹ K (Event Horizon Telescope Collaboration et al. 2019e).

Following these expectations, we expand on the results of Appendix C.1 and fit a simple functional form to the source counts of

$$\begin{eqnarray}&&{ \mathcal N }({\theta }_{r},{\sigma }_{\nu })=\sum _{n=0}^{\infty }{e}^{-{\left({T}_{b}/{T}_{b,n}\right)}^{\mu }}{\left[{\left(\displaystyle \frac{{\theta }_{r}}{{\theta }_{r,n}}\right)}^{3}+{\left(\displaystyle \frac{{\sigma }_{\nu }}{{\sigma }_{\nu ,n}}\right)}^{3/2}\right]}^{-1}.\end{eqnarray} \tag{ C5 }$$

Here, we have fixed the exponents of the θ_r and σ_ν terms to the values expected from initial considerations and further motivated by the fitting results of Appendix C.1. We also incorporate the understanding from Section 3.2 into the scale-setting parameters θ_r,n and σ_ν,n for the nth sub-ring, which are defined to be

$$\begin{eqnarray}&&{\theta }_{r,n}={\theta }_{r,0}{e}^{-n\pi }\quad {\sigma }_{\nu ,n}={\sigma }_{\nu ,0}{e}^{-3n\pi /2}.\end{eqnarray} \tag{ C6 }$$

We do not have an a priori expectation for the scaling behavior of the brightness temperature T_b,n in each sub-ring, so we simply include it as an additional parameter,

$$\begin{eqnarray}&&{T}_{b,n}={T}_{b,0}{C}^{n}.\end{eqnarray} \tag{ C7 }$$

The model has five free parameters—θ_r,0, σ_ν,0, T_b,0, μ, and C—which we fit in the same manner described in Appendix C.1. We find best-fit parameter values of θ_r,0 = 23.2 μas, σ_ν,0 = 0.17 Jy, T_b,0 = 2.2 × 10⁸ K, μ = 0.50, and C = 2.39; this best fit is shown in the right panel of Figure 17. The fit quality is similar to that in Appendix C.1, with the analytic source counts throughout most of the (θ_r , σ_ν ) space agreeing to better than an order of magnitude with the numerical results. The deviations become worse than an order of magnitude at small values of θ_r ≲ 0.1 μas and σ_ν ≲ 10⁻⁷ Jy, as well as wherever the source counts contain substantial contributions from n > 0 photon rings.