230 GHz VLBI OBSERVATIONS OF M87: EVENT‐HORIZON‐SCALE STRUCTURE DURING AN ENHANCED VERY‐HIGH‐ENERGY $\gamma$‐RAY STATE IN 2012

We detected fringes on baselines to all three sites, consistent with the results of Doeleman et al. (2012). Furthermore, we detected closure phases on the Arizona–Hawaii–California triangle. Figure 1 shows the measured closure phase on the SPF/SPD triangles (upper; hereafter VLBI triangles) listed in Table 2 and SFD/PFD triangles (lower; hereafter trivial triangles). The average error bar on closure phases is 103 for VLBI triangles and 50 for trivial triangles. The error-weighted average of the closure phases by the square of S/N is $-0\buildrel{\circ}\over{.} 7\pm 2\buildrel{\circ}\over{.} 9$ for VLBI triangles and $-0\buildrel{\circ}\over{.} 1\pm 0\buildrel{\circ}\over{.} 6$ for trivial triangles. The closure phase is consistent with zero on trivial triangles, as would be expected if the source is point-like on arcsecond scales. All closure phases on VLBI triangles coincide with zero within the 1σ level except for one data point, which is consistent with zero within 2σ levels. We note that non-detections in VLBI triangles during 7:00–10:00 UTC are attributable to non-detections on the SP baseline (Figure 2).

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Table 2.  Closure Phase of M87

Year	DOY	UTC		Triangle	${u}_{\mathrm{XY}}$	${v}_{\mathrm{XY}}$	${u}_{\mathrm{YZ}}$	${v}_{\mathrm{YZ}}$	${u}_{\mathrm{ZX}}$	${v}_{\mathrm{ZX}}$	Closure Phase	1σ Error
		(h)	(m)	(XYZ)	(Mλ)	(Mλ)	(Mλ)	(Mλ)	(Mλ)	(Mλ)	(deg)	(deg)
2012	81	5	49	SPF	−2336.473	−426.660	1997.198	847.338	339.275	−420.677	3.60	9.98
2012	81	6	3	SPF	−2483.493	−458.173	2113.293	874.212	370.200	−416.039	−3.80	5.05
2012	81	6	26	SPF	−2704.560	−513.928	2286.622	921.497	417.939	−407.569	−6.30	8.67
2012	81	5	49	SPD	−2336.473	−426.660	1997.150	847.284	339.323	−420.624	11.40	11.96
2012	81	6	3	SPD	−2483.493	−458.173	2113.246	874.158	370.247	−415.985	3.50	7.65
2012	81	6	26	SPD	−2704.560	−513.928	2286.577	921.442	417.983	−407.514	−5.10	7.69
2012	81	6	49	SPD	−2898.263	−574.140	2436.772	972.202	461.491	−398.062	−12.80	9.12
2012	81	5	49	SPF	−2336.473	−426.660	1997.198	847.338	339.275	−420.677	2.40	8.38
2012	81	6	3	SPF	−2483.493	−458.173	2113.293	874.212	370.200	−416.039	0.70	5.91
2012	81	6	26	SPF	−2704.560	−513.928	2286.622	921.497	417.939	−407.569	−0.50	5.84
2012	81	6	49	SPF	−2898.263	−574.140	2436.814	972.258	461.449	−398.118	2.70	7.36
2012	81	10	43	SPD	−3029.578	−1284.919	2439.510	1556.894	590.067	−271.975	−0.20	9.21
2012	81	11	6	SPD	−2858.690	−1348.199	2289.791	1607.719	568.899	−259.520	−7.30	9.74
2012	81	11	33	SPD	−2621.320	−1417.355	2084.597	1662.922	536.723	−245.567	3.00	5.07
2012	81	10	43	SPF	−3029.578	−1284.919	2439.508	1556.955	590.070	−272.036	−1.20	8.04
2012	81	11	6	SPF	−2858.690	−1348.199	2289.783	1607.779	568.906	−259.581	−0.60	6.99
2012	81	11	33	SPF	−2621.320	−1417.355	2084.583	1662.983	536.737	−245.628	−0.80	4.59

Machine-readable versions of the table is available.

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The correlated flux density of M87 is shown in Figure 2 and Table 3. The arcsecond-scale core flux density of 2.2 Jy is ∼17% higher than the 1.9 Jy measured in 2009 (D12). This brightening on arcsecond scales is not accompanied by changes on VLBI scales. The visibility amplitudes are broadly consistent with 2009 results of D12, confirming the presence of the event-horizon-scale structure. This indicates that the region responsible for the higher flux density must be resolved out in these observations and therefore located somewhere down the jet.

Most of the missing flux on VLBI scales is most likely attributed to the extended jet inside the arcsecond-scale radio core including the bright and stable knots such as HST-1. In the last decade, no radio enhancement was detected in such bright knots except for the 2005 VHE flare at HST-1. Even for the exceptionally variable HST-1, the radio flux has been decreasing from the 2005 VHE flare to at least the end of the 2012 event (see Abramowski et al. 2012 and H14). The observed increment in the missing flux seems incompatible with this trend in the bright knots, favoring that the missing flux originates in the vicinity of the radio core on milliarcsecond scales rather than the bright knot features. We discuss it in a physical context related with the 2012 event in Section 5.3.

The structure of M87 is not yet uniquely constrained, since millimeter VLBI detections of M87 remain limited in terms of baseline length and orientation, similar to previous observations in D12. Even with our detections of closure phase, our small data set is consistent with a variety of geometrical models (see Section 5.1 for physically motivated models). It is still instructive to investigate single-Gaussian models, which inherently predict a zero closure phase, to estimate the flux and approximate size of VLBI-scale structure and compare with the results of the previous observations.

Circular Gaussian fits to the visibility amplitudes on VLBI baselines are shown in Table 4. The parameters of the best-fit circular Gaussian model agree with values obtained by D12. The compact flux density of 0.98 ± 0.05 Jy is precisely consistent with the D12 value, while the size of 42.9 ± 2.2 μas (corresponding to 5.9 ± 0.2 ${R}_{{\rm{s}}}$) is slightly larger but still consistent within 3σ uncertainty. We find no evidence of significant changes in event-horizon-scale structure between the 2009 and 2012 observations.

Table 4.  Geometrical Models of M87. Errors are 3σ

Model	Date (Year/DOY)	Compact Flux Density (Jy)	FWHM Size(μas)	${\chi }_{\nu }^{2}$ (dof)
2012	2012/81	0.98 ± 0.05	42.9 ± 2.2	2.2 (54)
2009a	2009/95–97	0.98 ± 0.04	40.0 ± 1.8	0.6 (102)

Note.

^aModel obtained from all three days of data in the 2009 observations.

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Physically motivated structural models have been proposed for the Schwarzschild-radius-scale structure at 230 GHz in M87 for both jet and disk models (Broderick & Loeb 2009; Dexter et al. 2012; Lu et al. 2014). Although all proposed models predict the existence of a feature at the last photon orbit illuminated by a counter jet and/or accretion disk in the close vicinity of the black hole, there are significant differences between model images. The closure phase is an ideal tool to constrain physically motivated models, since relativistic effects such as gravitational lensing, light bending, and Doppler beaming generally induce asymmetric emission structure in the vicinity of the central black hole, causing the closure phase to be nonzero.

Figure 3 shows images and visibilities of the approaching-jet-dominated models (Broderick & Loeb 2009; Lu et al. 2014), counter-jet-dominated models (J2 in Dexter et al. 2012), and the accretion-disk-dominated models (DJ1 in Dexter et al. 2012). For jet models, 230 GHz emission structure can be categorized into two types. One is the approaching-jet-dominated models, where emission from the approaching jet is predominant at 230 GHz (Broderick & Loeb 2009; Lu et al. 2014). The model images consist of bright blob-like emission from the approaching jet and a weaker crescent or ring-like feature around the last photon orbit illuminated by a counter jet. The emission from the approaching jet dominates the 230 GHz emission regardless of the loading radius of non-thermal particles where leptons are accelerated and the jet starts to be luminous, although the crescent-like feature appears more clearly at smaller particle loading radii (see Figure 3 in Lu et al. 2014). In counter-jet-dominated models, the counter-jet emission is predominant instead of the approaching jet. Such a situation could happen if the bright emission region in the jet is very close to the central black hole (within few ${R}_{{\rm{s}}}$), suppressing the approaching jet emission due to gravitational lensing. Photons from the counter jet illuminate the last photon orbit, forming a crescent-like feature. It is worth noting that Dexter et al. (2012) and Lu et al. (2014) have clear differences in their images even at the same particle loading radius of a few ${R}_{{\rm{s}}}$, most likely due to differences in magnetic field distribution and also the spatial and energy distribution of non-thermal particles in their models. The accretion disk models are well characterized by a crescent-like or ring-like feature around the last photon orbit. The 230 GHz emission arises in the inner portion of the accretion flow ($r\sim 2.5$ ${R}_{{\rm{s}}}$) near the mid-plane.

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Measured closure phases on the Hawaii–Arizona–California triangle are consistent with these three models. In Figure 4, we show the model closure phases calculated in the MIT Array Performance Simulator²⁷ for an approaching-jet-dominated model, a counter-jet-dominated model, and an accretion-disk-dominated model in Figure 3. The closure phase of the approaching-jet-dominated model is almost zero. On the other hand, the model closure phases of counter-jet-dominated and accretion-disk-dominated models are systematically smaller than the observed closure phase in the later GST range, but the models and observed closure phases are consistent within a 3σ level. We note that the results for counter-jet-dominated and accretion-disk-dominated models shown in Figure 4 disagree with Figure 9 of Dexter et al. (2012), due to a mistake in Dexter et al. (2012) in constructing the closure phase triangles.

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All three models commonly predict small closure phases on the Hawaii–Arizona–California triangle. Visibility phases on the Arizona–California baseline, which barely resolves the source, are nearly zero. The closure phases on current VLBI triangles are almost the same as differences in the visibility phase between long baselines between Hawaii and the US mainland. For the case of the approaching-jet-dominated models, the phase gradient between long baselines is expected to be moderate, particularly at large particle loading radii, since emission is blob-like and fairly symmetric on spatial scales corresponding to the current long baselines. Models with a clear crescent-like or ring-like feature generally predict a steep phase gradient around the null amplitude region (see Figure 3), which would be detectable not on the current baselines but on longer ones such as the Hawaii–Mexico baseline.

The observed closure phases cannot distinguish between models with different dominant origin of 230 GHz emission on the current VLBI triangle due to the large errors on the data points. However, future observations with a higher recording rate of 16 Gbps will be able to measure the closure phase with an accuracy within a few degrees at 1 minute integration, which can constrain physical models more precisely. In addition, models can be effectively distinguished by near future observations with additional telescopes such as the Large Millimeter Telescope (LMT) in Mexico or the Atacama Large Millimeter/submillimeter Array (ALMA) in Chile. Differences in closure phases between models become more significant on larger triangles, as shown in the middle and bottom panels of Figure 4.

While all of the physically motivated models are broadly consistent with the currently measured closure phases and amplitudes, they make dramatically different predictions for forthcoming measurements. Models in which the image is dominated by contributions close to the horizon (counter-jet-dominated and accretion-disk-dominated models) exhibit large closure phases on triangles that include the LMT in stark contrast to those dominated by emission further away (approaching-jet-dominated). This extends to the visibility amplitudes: the compact emission models predict nulls on baselines probed by ALMA and the LMT (see Figure 3).

New VLBI observations of M87 at 230 GHz in 2012 confirm the presence of the event-horizon-scale structure reported in D12. The compact flux density and effective size derived from the circular Gaussian models allow us to estimate the effective brightness temperature of this structure, which is given by (e.g., Akiyama et al. 2013)

$$\begin{eqnarray}&&{T}_{b}=\displaystyle \frac{{c}^{2}}{2{k}_{B}{\nu }^{2}}\displaystyle \frac{F}{\pi {\phi }^{2}/4\mathrm{ln}2}\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&=\;1.44\times {10}^{10}\;{\rm{K}}\times {\left(\displaystyle \frac{\nu }{230\;\mathrm{GHz}}\right)}^{-2}\left(\displaystyle \frac{F}{1\;\mathrm{Jy}}\right){\left(\displaystyle \frac{\phi }{40\;\mu \mathrm{as}}\right)}^{-2},\end{eqnarray} \tag{ 2 }$$

where F, ν, and ϕ are the total flux density, observation frequency, and the FWHM size of the circular Gaussian. The effective brightness temperature is ${1.42}_{-0.10}^{+0.11}\times {10}^{10}$ K for the 2009 model and $(1.23\pm 0.11)\times {10}^{10}$ K for the 2012 model, where errors are 3σ. These brightness temperatures of $\sim 2\times {10}^{10}$ K are below the upper cutoff in the intrinsic brightness temperature of $\sim {10}^{11}$ K on the "inverse Compton catastrophe" argument (e.g., Kellermann & Pauliny-Toth 1969; Blandford and Königl 1979; Readhead 1994). Although the 1.3 mm VLBI structure has been poorly constrained, particularly in the N–S direction, possibly inducing additional uncertainties, it is still instructive to discuss the effective brightness temperature and its physical implications for both the jet and accretion disk scenarios.

In the case of the jet scenarios, the brightness temperature would not be highly affected by the Doppler beaming, and then not significantly differ from the intrinsic (i.e., not Doppler-boosted) brightness temperature. The brightness temperature is amplified by a factor of δ for an isotropic blob-like source (e.g., Urry & Padovani 1995). The Doppler factor is 1–3 at a moderate viewing angle of 15°–25° (e.g., Hada et al. 2011) and the Lorentz factor of 1–2 in the inner 10² ${R}_{{\rm{s}}}$ region (e.g., Asada et al. 2014) inferred for the M87 jet.

Interestingly, the 230 GHz brightness temperature is broadly consistent with the peak brightness temperature of $\sim {10}^{9}-{10}^{10}$ K at the radio cores at lower frequencies from 1.6 to 86 GHz (e.g., Dodson et al. 2006; Ly et al. 2007; Asada & Nakamura 2012; Hada et al. 2012; Nakamura & Asada 2013) located within $\sim {10}^{2}$ ${R}_{{\rm{s}}}$ from the jet base (Hada et al. 2011). This would provide some implications also for the magnetic field structure of the jet. If we assume that the radio core surface corresponds to the spherical photosphere of the synchrotron self-absorption at each frequency, the magnetic field strength at the radio core can be estimated as (e.g., Marscher 1983; Hirotani 2005; Kino et al. 2014)

$$\begin{eqnarray}&&B=b(p){\nu }^{5}{\phi }^{4}\;{F}^{-2}\displaystyle \frac{\delta }{1+z}\propto \nu {T}_{b}^{-2}\displaystyle \frac{\delta }{1+z}.\end{eqnarray} \tag{ 3 }$$

The constant Doppler factor and brightness temperature give the magnetic field strength as roughly proportional to the observation frequency at the radio core (i.e., ${B}_{\mathrm{core}}\propto {\nu }_{\mathrm{obs}}$). Using the frequency dependence of the radio core position (${r}_{\mathrm{core}}\propto {\nu }_{\mathrm{obs}}^{-0.59\pm 0.09}$; Hada et al. 2011), the magnetic field strength at the radio core is inversely proportional to the distance from the jet base approximately (i.e., ${B}_{\mathrm{core}}\propto {r}_{\mathrm{core}}^{-1}$) in the inner $\sim {10}^{2}$ ${R}_{{\rm{s}}}$. This magnetic field profile can be obtained if the transverse (i.e., nearly toroidal) magnetic field dominates on this scale rather than the longitudinal (i.e., nearly poloidal) field along the conical stream with no velocity gradient under the flux frozen-in condition (Blandford and Königl 1979; also see Section 5 in Baum et al. 1997). This profile can also be obtained if the longitudinal (i.e., nearly poloidal) field dominates the transverse (i.e., nearly toroidal) magnetic field along the paraboloidal stream under the flux frozen-in condition, although recent observations favor a conical stream of the jet in the inner $\sim {10}^{2}$ ${R}_{{\rm{s}}}$ (Hada et al. 2013).

Even though the above assumptions might not work well for M87, this simple analysis suggests that the dominance of toroidal or poloidal magnetic fields starts to become a major concern on the jet formation in the inner $\sim {10}^{2}$ ${R}_{{\rm{s}}}$. Future EHT observations with additional stations and space VLBI observations (e.g., Radio Astron; Kardashev et al. 2013) will provide more detailed structure of the radio core including the profile of the stream line, enabling more precise analysis of the magnetic field structure of the relativistic jet in M87.

The measurements of the brightness temperature also give some implications for the energetics of the jet base. The equipartition brightness temperature (Readhead 1994) of the non-thermal plasma with the flux density of ∼1 Jy at 230 GHz is ${T}_{\mathrm{eq}}\leqslant {10}^{12}$ K, where the equality is given if 230 GHz emission is fully optically thick. This gives the ratio between the internal energy of non-thermal leptons and the magnetic field energy density ${U}_{B}/{U}_{e}={T}_{\mathrm{eq}}/{T}_{b}\leqslant {10}^{2}$ (Readhead 1994). This implies that, if the 230 GHz emission is dominated by optically thick non-thermal synchrotron emission, the magnetic field energy dominates the internal energy of the non-thermal particles at the jet base. We note that, recently, Kino et al. (2015) performed more careful analysis on the energetics at the jet base, stating that the magnetic field energy is dominant even in the fully optically thin case unless protons are relativistic.

The brightness temperature is broadly consistent with the electron temperature of $\sim {10}^{9-10}$ K as expected for RIAF-type accretion disks (e.g., Manmoto et al. 1997; Narayan et al. 1998; Manmoto 2000; Yuan et al. 2003). The brightness temperature is a factor of ∼2–3 smaller than that of Sgr A* with similar size and higher flux density (Doeleman et al. 2008; Fish et al. 2011). If the 230 GHz emission is dominated by thermal synchrotron emission from the accretion disk in both Sgr A* and M87, it seems broadly consistent with a theoretical prediction that a disk with higher accretion rate has a lower electron temperature due to enhanced electron cooling (see Figure 2 in Mahadevan 1997).

Our observations were performed in the middle of the VHE enhancement reported in Beilicke & VERITAS Collaboration (2012) and H14. There are several observational pieces of evidence for the existence of a radio counterpart to the VHE enhancement around our observations. First, the onset of the radio brightening at 22 and 43 GHz occurs simultaneously with the VHE enhancement, indicating that the radio and VHE emission regions are not spatially separated. Since the radio brightening starts ∼20–30 days before our observations, 230 GHz emission is also expected to be enhanced at the epoch of our observations. The radio flux measured with the SMA in H14 indeed shows a local maximum in its light curve during our observations, which is consistent with our results showing a radio flux greater than in 2009 April on arcsecond scales when M87 was in a quiescent state (Abramowski et al. 2012). Second, the radio counterpart was not resolved in the radio core in VERA observations, suggesting that the radio counterpart of the 2012 event should exist near the radio core at 43 GHz located at a few tens of Schwarzschild radii downstream from the central black hole and/or the jet base visible at 230 GHz.

The geometrical model (described in Section 4.2) suggests that there are no obvious structural changes on event-horizon scales between 2009 and 2012, despite the increase in the core flux on arcsecond scales. One plausible scenario for explaining the different behavior between event-horizon scales and arcsecond scales is that the structure of the flare component at 230 GHz has extended structure that is resolved out with the current array. The shortest VLBI baseline in our observations, SMT-CARMA, has a length of 600 Mλ. If we consider the Gaussian-like structure for the flaring region with a radio flux of a few ×100 mJy corresponding to the flux increment at the local peak in the 230 GHz light curve of H14, the flaring region should be extended enough to have a correlated flux smaller than the standard deviation on SMT-CARMA baselines of ∼90 mJy so that the increment in the radio flux is not significantly detected on those baselines. The minimum FWHM size can be estimated to be ∼140 μas ∼20 ${R}_{{\rm{s}}}$, which has an HWHM size of ∼600 Mλ in the visibility plane. This limitation is consistent with at least two aspects of VHE flares.

First, the 2 months duration of the 2012 VHE event implies that the size of the emission region is $\lt 60\delta$ light days $\sim 0.6\delta$ mas, from causality considerations. Similar constraints of $\lt 0.44$ mas $\sim 60{R}_{{\rm{s}}}$ are provided with VERA at 43 GHz in H14, since the flare component was not spatially resolved during their observations. Combining our measurement with these size limits, the size of the VHE emission region during our observations is constrained to be in the tight range of ∼0.14–0.44 mas, corresponding to ∼20–60 ${R}_{{\rm{s}}}$.

Second, when the size of the emission region is larger than ∼20 ${R}_{{\rm{s}}}$, the emitted VHE photons will not be affected by absorption due to the process of photon–photon pair creation ($\gamma \gamma$-absorption). In principle, $\gamma$-ray photons with energy E interact most effectively with target photons in the infrared (IR) and optical photon field of energy (e.g., Rieger 2011)

$$\begin{eqnarray}&&\varepsilon (E)\sim {\left(\displaystyle \frac{E}{1\;\mathrm{TeV}}\right)}^{-1}\;\mathrm{eV}.\end{eqnarray} \tag{ 4 }$$

Since the 2012 enhancement was detected at ∼0.3–5 TeV in the VHE regime (see Beilicke & VERITAS Collaboration 2012), the target photon wavelength is ∼0.4–6 μm in the near-infrared (NIR) and optical regimes. The optical depth of $\gamma$-rays of energy E for the center of an IR source with a size R and luminosity $L(\varepsilon )$ can be written as (e.g., Neronov & Aharonian 2007)

$$\begin{eqnarray}{\tau }_{\gamma \gamma }(E,R) & \eqsim & \displaystyle \frac{{\sigma }_{T}}{5}\displaystyle \frac{L(\varepsilon (E))}{4\pi {R}^{2}c\varepsilon }R\\ & \eqsim & 0.25\left(\displaystyle \frac{L\{[E/(1\;\mathrm{TeV})]\;\mathrm{eV}\}}{{10}^{40}\;\mathrm{erg}\;{{\rm{s}}}^{-1}}\right)\\ & & \times {\left(\displaystyle \frac{R}{20\;{R}_{s}}\right)}^{-1}\left(\displaystyle \frac{E}{5\;\mathrm{TeV}}\right).\end{eqnarray} \tag{ 5 }$$

The NIR and optical luminosity is $L\sim {10}^{40}$ erg s⁻¹ within a few tens of parsecs of the nucleus (e.g., Biretta et al. 1991; Boksenberg et al. 1992). Even in the extreme case that the flaring region accounts for all nucleus emission in the NIR and optical regime, the optical depth is smaller than unity at $E\lt$ a few TeV, where the enhancement was detected in Beilicke & VERITAS Collaboration (2012), for the size of ∼20 ${R}_{{\rm{s}}}$. This allows $\gamma$-ray photons up to a few TeV to escape from the vicinity of the black hole, explaining why the 2012 event was detectable without introducing a special geometry of emission regions. Note that more careful calculation increases ${\tau }_{\gamma \gamma }$ by a factor of several (Brodatzki et al. 2011; Broderick & Tchekhovskoy 2015, in preparation), but even in this case the optical depth is smaller than unity for the upper half of the size range (∼40–60 ${R}_{{\rm{s}}}$).

The scenario limiting the size to a range of ∼20–60 ${R}_{{\rm{s}}}$ during our observations in the middle of the 2012 event can naturally explain our results and other observational results. It is instructive to compare this scenario to the numerous physical models proposed for the VHE emission in M87 (see H14 and Abramowski et al. 2012 for a review). Here, we briefly discuss general implications for the existing VHE models of M87 based on our scenario.

The size of ∼20–60 ${R}_{{\rm{s}}}$ is presumably incompatible with many existing models that assume extremely compact regions of ≲1–10 ${R}_{{\rm{s}}}$, ascribing the VHE emission to particle acceleration in the black hole magnetosphere (e.g., Neronov & Aharonian 2007; Rieger & Aharonian 2008; Levinson & Rieger 2011; Vincent 2014), multiple leptonic blobs in the jet launch/formation region (e.g., Lenain et al. 2008), leptonic models involving a stratified velocity field in the transverse direction (Tavecchio & Ghisellini 2008), mini-jets within the main jet (e.g., Giannios et al. 2010; Cui et al. 2012), and interactions between a red giant star/gas cloud and the jet base (e.g., Barkov et al. 2010, 2012). These models can reasonably explain the very short variable timescale of ≲1 day in the past three flares in 2005, 2008, and 2010, but are not favored for this particular event in 2012.

Consistency with the size limitation is less clear for models assuming different emission regions or different kinds of emitting particles for radio and VHE emissions, such as hadronic models (e.g., Reimer et al. 2004; Barkov et al. 2010, 2012; Reynoso et al. 2011; Cui et al. 2012; Sahu & Palacios 2013) and some multi-zone leptonic models with a stratified velocity field in radial or transverse directions of the jet by involving the deceleration flow or the spine-layer structure, respectively (Georganopoulos et al. 2005; Tavecchio & Ghisellini 2008). Since the relation between radio and VHE emission has not been well formulated for these models, more detailed predictions particularly on the radio–TeV connection are required for further discussions.

Interestingly, a homogeneous one-zone synchrotron self-Compton (SSC) model (i.e., the standard leptonic model) predicts a comparable source size (∼0.1 mas) to our scenario for a broadband spectral energy distribution (SED) in a relatively moderate state (Abdo et al. 2009). It can also naturally explain the radio–VHE connection in H14. The simple leptonic one-zone SSC model seems more plausible than other existing models for M87 to explain some properties such as the size constraint and the radio–VHE connection, but further dedicated modeling for the 2012 event would be required to test consistency with overall observational properties such as the broadband SED, which is not discussed here. Note that leptonic models might be problematic for explaining the hard VHE spectrum, which is common in the previous three VHE flares and the 2012 events, against the Klein–Nishina and $\gamma$-ray opacity effects softening the VHE spectrum (see discussions in Tavecchio & Ghisellini 2008).

Our new observations clearly show that short-millimeter VLBI is a useful tool to constrain the size of the radio counterpart, which is a new clue to understanding the VHE activities in M87. In particular, new constraints can be obtained by combining simultaneous EHT observations with measurements of VHE spectra at ≳10 TeV highly affected by $\gamma \gamma$-absorption (see Equation (5)) with the Cherenkov Telescope Array (CTA, Actis et al. 2011). In addition, the complementary dedicated monitoring with lower frequency monitoring on mas and arcsec scales is also important to study details on radio/VHE connections as well as constrain the important physical parameters.