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An accretion-jet model for M87: interpreting the spectral energy distribution and Faraday rotation measure

M87 is arguably the best supermassive black hole (BH) to explore the jet and/or accretion physics due to its proximity and fruitful high-resolution multi-waveband observations. We model the multi-wavelength spectral energy distribution (SED) of M87 core that observed at a scale of 0.4 arcsec ($\sim 10^5R_{\rm g}$, $R_{\rm g}$ is gravitational radius) as recently presented by Prieto et al. Similar to Sgr A*, we find that the millimeter bump as observed by Atacama Large Millimeter/submillimeter Array (ALMA) can be modeled by the synchrotron emission of the thermal electrons in advection dominated accretion flow (ADAF), while the low-frequency radio emission and X-ray emission may dominantly come from the jet. The millimeter radiation from ADAF dominantly come from the region within $10R_{\rm g}$, which is roughly consistent with the recent very long baseline interferometry observations at 230\,GHz. We further calculate the Faraday rotation measure (RM) from both ADAF and jet models, and find that the RM predicted from the ADAF is roughly consistent with the measured value while the RM predicted from the jet is much higher if jet velocity close to the BH is low or moderate (e.g., $v_{\rm jet}\lesssim0.6\,c$). With the constraints from the SED modeling and RM, we find that the accretion rate close to the BH horizon is $\sim (0.2-1)\times10^{-3}{M}_{\odot} \rm yr^{-1}\ll\dot{\it M}_{\rm B}\sim 0.2\it {M}_{\odot} \rm yr^{-1}$ ($\dot{M}_{\rm B}$ is Bondi accretion rate), where the electron density profile, $n_{\rm e}\propto r^{\sim -1}$, in the accretion flow is consistent with that determined from X-ray observation inside the Bondi radius and recent numerical simulations.