High Resolution Imaging of a Black Hole Shadow with Millimetron Orbit around Lagrange Point L2

Imaging of the shadow around supermassive black hole (SMBH) horizon with a very long baseline interferometry (VLBI) is recognized recently as a powerful tool for experimental testing of Einstein's General relativity. The Event Horizon Telescope (EHT) has demonstrated that an Earth-extended VLBI with the maximum long base ($D=10,700$ km) can provide a sufficient angular resolution $\theta\sim 20~\mu$as at $\lambda=1.3$ mm ($\nu=230$ GHz) for imaging the shadow around SMBH located in the galaxy M87$^\ast$. However, the accuracy of critically important characteristics, such as the asymmetry of the crescent-shaped bright structure around the shadow and the sharpness of a transition zone between the shadow floor and the bright crescent silhouette, both of order $\Delta\theta\sim 4~\mu$as, is still to be improved. In our previous paper we have shown that Space-Earth VLBI observation within a joint Millimetron and EHT configuration at the near-Earth high elliptical orbit (HEO) can considerably improve the image quality. Even more solid grounds for firm experimental validation of General relativity can be obtained with a higher resolution available within the joint Millimetron and EHT program at the Lagrangian point L2 in the Sun-Earth system with an expected imaging resolution at 230~GHz of $\Delta\theta\sim 5~\mu$as. In this paper we argue that in spite of limitations of L2 orbit, an adequate sparse $(u,v)$ coverage can be achieved and the imaging of the shadows around Sgr A$^\ast$ and M87$^\ast$ can be performed with a reasonable quality.