Notes on Mesolensing of a Background Galaxy by Luhman's Dwarf

Einstein radius of nearby stars lensing very distant sources

Nearby stars have two properties that make them potentially useful as gravitational lenses. The first is that their proximity makes their Einstein radius for very distant sources large enough to be resolved by modern telescopes. When the distance between the source and lens \(D_{\rm LS} \approx D_{\rm S}\), the distance to the source, we are in the mesolensing regime: \[\theta_E = \sqrt{\frac{4GM}{c^2} \frac{D_{\rm LS}}{D_{\rm L}D_{\rm S}}} \approx 90.3 \mbox{mas} \sqrt{\frac{(M/M_\odot)}{(D_{\rm L}/\mbox{pc})}}\]

The second is that their high proper motions mean they will lens very faint, extragalactic background sources more frequently than more distant stars.

The possibility of mesolensing by vB10 was discussed by (Lépine et al., 2012), and the overall statistics of detecting such events has been analyzed by (Stefano 2008, Stefano 2008a).

Finite lens sizes

One potential complication is the physical size of the star itself, which projects to an angular size of \[\theta_*=R/D_{\rm L}=4.65 \mbox{mas} \frac{R/R_\odot}{D_{\rm L}/pc}\] Using the rough rule of thumb that \(R/R_\odot \approx M/M_\odot\), the angular size of the star thus equal to its Einstein radius at a distance of \[D_{\rm L} = 5.7\times10^{-4} \mbox{pc} (M/M_\odot)\] So we see that all dwarf stars except the Sun have Einstein radii larger than their angular radii. Brown dwarfs have size \(\sim 0.1 R_\odot\), which even at 2.5 pc yields \(\sim\)0.2 mas, much smaller than its Einstein ring.

Typical values for nearby stars

The nearest stars (within 3 pc) have masses ranging from \(\approx 0.01\)\(2 M_\odot\) (e.g. Luhman, 2014), and proper motions of order 1000 mas/yr. This leads to typical Einstein ring radii of 40 mas, and typical crossing times of \(\sim\) 2 weeks. For Luhman’s Dwarf specifically, we have M \(\gtrsim 3 M_{\rm Jup}\) and \(D_L = 2.5\) pc, so the diameter of the ring is

\[d = 2\theta_E = 1.2 {\rm mas} \sqrt{M/M_{\rm Jup}} \gtrsim 7 {\rm mas}\]

Effects of Mesolensing

There are two primary observational effects of mesolensing: to distort the image of the background source, and to magnify the background light, leading the system to increase in brightness. The latter effect is most pronounced for point sources— for extended sources the magnitude of the effect is unclear, but flux conservation requires that the net effect of a lens be zero on large angular scales. This would seem to imply zero brightness increase of a uniform background extended object for unresolved lenses, and possibly for resolved lenses, as well.