The Einstein ring

Adapted from IOAA 2007 (Chiang Mai), theory problem 4. We have restated it in our own words and kept the original data, adding the wavelength of 500 nm that the official solution uses for part (c). The official solution prints the Einstein radius of part (b) as 0.956 × 10⁻⁹ rad (1.97 × 10⁻⁴″), and the data give 0.978 × 10⁻⁹ rad (2.02 × 10⁻⁴″). The solution was last checked on 23 October 2026.

A spherical mass $M$ bends a ray that passes it at the distance $\xi$ by the small angle $4GM/(\xi c^2)$. A source S, a lens L and an observer O are nearly in line, with the lens at the distance $D_{\text{L}}$ and the source at $D_{\text{S}}$. The source is seen at the small angle $\theta$ from the lens, where without the lens it would be at $\beta$.

(a) When the source is exactly behind the lens, $\beta =0$, show that the image is a ring of angular radius ${\theta }_{\text{E}}=\sqrt{(4GM/c^2)(D_{\text{S}}-D_{\text{L}})/(D_{\text{L}}{D}_{\text{S}})}$. (2 points)

(b) A star is about 50 kpc away, and a lens of one solar mass is 10 kpc in front of it. Find the angular radius of the Einstein ring. (1 point)

(c) What is the resolution of the Hubble Space Telescope, with its mirror of 2.4 m, at 500 nm? Could it resolve this ring? (2 points)

(d) For a source not exactly behind the lens there are two images. Find their positions ${\theta }_1$ and ${\theta }_2$ in terms of $\beta$ and ${\theta }_{\text{E}}$. (2 points)

(e) With $\eta =\beta /{\theta }_{\text{E}}$, find ${\theta }_1/\beta$ and ${\theta }_2/\beta$ in terms of $\eta$. (2 points)

(f) Find the ratio $\mathrm{\Delta }\theta /\mathrm{\Delta }\beta$ for each image, when the source moves by a small $\mathrm{\Delta }\beta$. (1 point)

Answer(a) As shown. (b) $9.78\times {10}^{-10}$ rad, or $2.02\times {10}^{-4}$ arcseconds. (c) $2.54\times {10}^{-7}$ rad, so the ring cannot be resolved. (d) ${\theta }_{1,2}=\beta /2\pm \sqrt{{\beta }^2/4+{\theta }_{\text{E}}^2}$. (e) $\frac{1}{2}(1\pm \sqrt{{\eta }^2+4}/\eta )$. (f) $\frac{1}{2}(1\pm \eta /\sqrt{{\eta }^2+4})$.