What is the "hypothesized lower mass gap" between 2.5 and 5 solar masses? eventually links to Constraining the masses of microlensing black holes and the mass gap with Gaia DR2.

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.

  • $\begingroup$ @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. $\endgroup$
    – uhoh
    Commented Jun 30, 2020 at 6:40
  • $\begingroup$ @RobJeffries thanks; I've adjusted the title. $\endgroup$
    – uhoh
    Commented Jun 30, 2020 at 7:31
  • $\begingroup$ 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). $\endgroup$
    – antlersoft
    Commented Jun 30, 2020 at 14:41
  • $\begingroup$ @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. $\endgroup$
    – uhoh
    Commented Jun 30, 2020 at 21:37
  • 1
    $\begingroup$ @uhoh. The acknowledgement that antlersoft mentions suggests that they are making assumptions about the velocities based on general knowledge of the movement of stars and other bodies in the galaxy. $\endgroup$ Commented Jul 24, 2020 at 9:34

1 Answer 1


The introduction of the Wyrzykowski & Mandel paper gives the following information about estimating the lens mass.

In order to obtain the mass of the lens (Gould 2000a), it is necessary to measure both the angular Einstein radius of the lens ($\theta_\mathrm{E}$) and the microlensing parallax ($\pi_\mathrm{E}$)

$$M = \frac{\theta_\mathrm{E}}{\kappa \pi_\mathrm{E}}$$

where $\kappa = 4G / (c^2\ \mathrm{AU}) = 8.144\ \mathrm{mas/M_\odot}$; and $\pi_\mathrm{E}$ is the length of the parallax vector $\mathbf{\pi_\mathrm{E}}$, defined as $\pi_\mathrm{rel}/\theta_\mathrm{E}$, where $\pi_\mathrm{rel}$ is relative parallax of the lens and the source. The microlensing parallax vector $\mathbf{\pi_\mathrm{E}}$ is measurable from the non-linear motion of the observer along the Earth’s orbital plane around the Sun. The effect of microlensing parallax often causes subtle deviations and asymmetries relative to the standard Paczynski light curve in microlensing events lasting a few months or more, so that the Earth’s orbital motion cannot be neglected. The parameter $\mathbf{\pi_\mathrm{E}}$ can also be obtained from simultaneous observations of the event from the ground and from a space observatory located ∼1 AU away (e.g., Spitzer or Kepler, e.g., Udalski et al. 2015b, Calchi Novati et al. 2015, Zhu et al. 2017).

In particular, the Gould 2000a paper gives a good summary of the various relationships between the quantities. The Udalski et al. 2015b notes that the distance between the Earth and Spitzer (which would also apply to Gaia) means that Spitzer would see differences in the light curve, allowing the parallax to be determined.

Note that things get more complicated if the source is a binary, in which case a "reverse parallax" effect from the source's orbital motion, usually called "xallarap" needs to be taken into account — but that's a matter for another question...

The other relevant quantity is the angular Einstein radius of the lens. In their discussion of measuring $\theta_\mathrm{E}$, Wyrzykowski & Mandel reference Rybicki et al. 2018. That paper notes that precision astrometry can help measure $\theta_\mathrm{E}$ because microlensing also changes the apparent position of the source:

The positional change of the centroid depends on the $\theta_\mathrm{E}$ and separation $u$. Contrary to the photometric case, the maximum shift occurs at $u_0 = \sqrt{2}$ and reads (Dominik & Sahu 2000)

$$\delta_\mathrm{max} = \frac{\sqrt{2}}{4} \theta_\mathrm{E} \approx 0.354 \theta_\mathrm{E}$$

Thus, for the relatively nearby lens at $D_l = 4\ \mathrm{kpc}$, source in the bulge $D_s = 8\ \mathrm{kpc}$ and lensing by a stellar BH with the mass $M = 4M_\odot$, the astrometric shift due to microlensing will be about 0.7 milliarcsecond.

The bulk of the paper goes on to determine that these shifts should be observable by Gaia.

Another way to measure the size of the lens is to measure the lens-source proper motion by searching for the lens several years after the event, this has been done for a couple of exoplanet-hosting lenses but would not be possible for a dark lens like a black hole.

  • $\begingroup$ This is beautiful, what an elegant problem! Thank you for taking the time to dig in and then compose such a thorough answer. I will go get these sources to day and pour through them. $\endgroup$
    – uhoh
    Commented Jul 29, 2020 at 3:17

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