A telescope to see a black hole

Adapted from IOAA 2019 (Keszthely), theory problem 3. We have restated it in our own words and kept the original data. The problem takes the radius of the shadow to be 3 Schwarzschild radii, a little more than the true 2.60, and the mass of Sagittarius A* to be 3.6 million solar masses, less than the 4.3 million measured since. We keep its values. The solution was last checked on 23 October 2026.

The Event Horizon Telescope imaged the shadow of the black hole at the centre of the galaxy M87 at a wavelength of 1.3 mm. Take the radius of the shadow to be 3 Schwarzschild radii.

(a) How large must a telescope be to resolve the shadow of a black hole of mass $M$ at distance $d$? (6 points)

(b) Give this size in Earth radii for (i) the black hole of M87, of $6.5\times {10}^9$ solar masses at $5.5\times {10}^7$ light-years, and (ii) Sagittarius A*, of $3.6\times {10}^6$ solar masses at 8.3 kpc. (1 point each)

(c) Which technique makes such a telescope possible: (A) gravitational lensing by dark matter, (B) interferometry with an array of radio telescopes, (C) slowing photons in a dense medium, (D) correcting the distortion of the incoming wavefront, or (E) focusing neutrinos with strong electromagnetic fields? (2 points)

Answer(a) $D\ge 1.22\lambda c^{2}d/(12GM)$. (b) (i) 1.12 Earth radii, (ii) 1.0 Earth radius. (c) (B), interferometry with an array of radio telescopes.