# How do OGLE-III and GAIA measure the mass of free microlensing black holes?

The angle of deflection of light passing a massive object is given by:

$$\theta = \frac{4GM}{r c^2}$$

where $$r$$ is the minimum distance from the mass that the light passes.

If two black holes pass by a line-of-sight to a distant object and their speeds and distances of closest approach $$r$$ both scale linearly with their mass, they produce the identical deflection magnitude and time dependence.

Question: How then can such an observation of a free-floating black hole be used to determine its mass? What additional information is necessary? I see from the article that parallax is involved, but without knowing the distance to the black hole I don't see how this is enough to work out a mass.

• @HDE226868 If I understand correctly these are black holes not associated with other objects, so I'm assuming that the only observation are the deflection of more distant luminous objects, and even if they are close enough for a parallax measurement of their distance, I don't see how that helps. – uhoh Jun 30 at 6:40
• @RobJeffries thanks; I've adjusted the title. – uhoh Jun 30 at 7:31
• While it is true that "if their speeds and distances of closest approach r both scale linearly with their mass, they produce the identical deflection magnitude and time dependence" their is no physical reason that their speeds or distance would scale linearly with their mass (although they acknowledge the possibility that black holes having larger intrinsic velocities from "natal kicks" would confound their analysis). – antlersoft Jun 30 at 14:41
• @antlersoft no, what that means is that there is an ambiguity; if all you have is a curve of deflection of a background luminous object versus time, there is a whole family of solutions of different masses that can do it. There's no way to say "yep that was about 4 solar masses" because a 10 solar mass object could have done the same thing. – uhoh Jun 30 at 21:37