2023 DA2 | Weighing an isolated black hole with astrometric microlensing

In 2022, two groups reported an isolated black hole from the microlensing event OGLE-2011-BLG-0462. Here we analyse Hubble Space Telescope data to reproduce their result.

In gravitational microlensing, light from a distant source star is bent and magnified by an intervening lens. The angular Einstein radius depends on the lens mass $M$ and distance $D_\ell$:

where $D_s\approx8$ kpc for a source in the Galactic bulge.

(a) Calculate $\theta_E$ in mas for a $1\,M_\odot$ lens at 1 kpc. (2)

When the lens and source are separated by $\theta = u\,\theta_E$, two images form on the line through them, at angular distances $\theta_\pm = \tfrac12\left(u\pm\sqrt{u^2+4}\right)\theta_E$ from the lens, with magnifications

(b) Telescopes cannot resolve the two images, only their brightness-weighted centroid. Derive the angular separation $\theta_c$ of the centroid from the lens as a function of $u$ and $\theta_E$. (8)

(c) Derive the deflection $\Delta\theta = \theta_c - \theta$ of the centroid from the unlensed source position. What is it when $u\approx0$? (4)

The source and lens move relative to each other, so both magnification and centroid position change with time. The OGLE light curve gives an Einstein timescale $t_E = 247$ days (time to move one $\theta_E$), a peak on 21 July 2011 (HJD 2455763), and a minimum separation $u_0\approx0$. Assume rectilinear relative motion. The table gives HST positions of the source relative to background objects.

(d) Plot the East and North positions against time. (10)

(e) The observed motion is the source's rectilinear proper motion plus the microlensing deflection. Calculate the proper motion (mas/yr) and its uncertainty in East and North. (8)

(f) Subtract the proper motion, then calculate and plot the total deflection against $u$. Neglect the uncertainty of the proper motion. (20)

(g) Determine $\theta_E$ and its uncertainty. (Hint: linearise $\Delta\theta$.) (16)

(h) Including the Earth's orbital motion gives the microlensing parallax $\pi_E = (\pi_\ell - \pi_s)/\theta_E = 0.095\pm0.009$. Find the lens mass and its uncertainty. (7)