The Sun as a telescope

Adapted from IOAA 2016 (Bhubaneswar), theory problem T10. We have restated it in our own words and kept the original data. The solution was last checked on 23 October 2026.

A ray that passes a massive body at the distance $r$ from its centre is bent by ${\theta }_{\text{b}}=2R_{\text{s}}/r$, where $R_{\text{s}}$ is the Schwarzschild radius of the body (the original writes $R_{\text{sch}}$). Take the bending to happen at one point. Rays from a distant source that pass the body at the same $r$ meet on the axis at a distance $f_r$ from its centre, where an observer would see the source greatly amplified.

(a) Find the shortest distance $f_{min}$ from the centre of the Sun, in astronomical units, at which rays can be focused. (6 points)

(b) A small circular detector of radius $a$ is placed at $f_{min}$, centred on the axis and facing the Sun. Only rays that pass the Sun in a thin ring of width $h$, much less than $R_{\odot }$, reach it. The amplification is the ratio of the light falling on the detector with the Sun to the light without it. Find the amplification in terms of $R_{\odot }$ and $a$. (8 points)

(c) For a round cloud of mass, such as the dark matter of a cluster of galaxies, let a ray with impact parameter $r$ be bent only by the mass $M(r)$ inside $r$. How must $M(r)$ grow with $r$ for the cloud to act as a perfect lens, bringing all parallel rays to one focus? (6 points)

Answer(a) $f_{min}=R_{\odot }^2{c}^2/(4G{M}_{\odot })=548$ au. (b) The amplification is $R_{\odot }/a$, which for a detector of radius 1 m is $6.96\times {10}^8$, or 22.1 magnitudes. (c) $M(r)\propto r^2$.