<p><a href="https://astronomy.stackexchange.com/q/36742/7982">What is the &quot;hypothesized lower mass gap&quot; between 2.5 and 5 solar masses?</a> eventually links to <a href="https://arxiv.org/abs/1904.07789" rel="nofollow noreferrer">Constraining the masses of microlensing black holes and the mass gap with Gaia DR2</a>.</p>
<p>The angle of deflection of light passing a massive object is <a href="https://en.wikipedia.org/wiki/Gravitational_lens#Explanation_in_terms_of_spacetime_curvature" rel="nofollow noreferrer">given by</a>:</p>
<p><span class="math-container">$$\theta = \frac{4GM}{r c^2}$$</span></p>
<p>where <span class="math-container">$r$</span> is the minimum distance from the mass that the light passes.</p>
<p>If two black holes pass by a line-of-sight to a distant object and their speeds and distances of closest approach <span class="math-container">$r$</span> both scale linearly with their mass, they produce the identical deflection magnitude and time dependence.</p>
<p><strong>Question:</strong> 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.</p>

CC BY-SA 4.0
2020-06-30T06:40:30.193Z
@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
CC BY-SA 4.0
2020-06-30T07:31:15.270Z
@RobJeffries thanks; I've adjusted the title.
uhoh
CC BY-SA 4.0
2020-06-30T14:41:10.077Z
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
CC BY-SA 4.0
2020-06-30T21:37:11.917Z
@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
CC BY-SA 4.0
2020-07-24T09:34:29.697Z
@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.
Steve Linton
CC BY-SA 4.0
2020-07-24T09:59:44.720Z
@SteveLinton I think you can write an answer based on that. The title of the paper begins "*Constraining* the masses..." and so my use of "determine" in the body and "measure" in the title are the problem. If they're not doing that and they're just *constraining* them, then that's the answer!
uhoh
<observational-astronomy><black-hole><mass><gaia><mass-gap>
How do OGLE-III and GAIA measure the mass of free microlensing black holes?
uhoh
# How do OGLE-III and GAIA measure the mass of free microlensing black holes?

(Asked by: username_0 on 2020-06-30)

[What is the "hypothesized lower mass gap" between 2.5 and 5 solar masses?](https://astronomy.stackexchange.com/q/36742/7982) eventually links to [Constraining the masses of microlensing black holes and the mass gap with Gaia DR2](https://arxiv.org/abs/1904.07789).

The angle of deflection of light passing a massive object is [given by](https://en.wikipedia.org/wiki/Gravitational_lens#Explanation_in_terms_of_spacetime_curvature):

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

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.

**Tags:** observational-astronomy, black-hole, mass, gaia, mass-gap

> **Comments:**
> **username_0**: @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.  
> **username_0**: @RobJeffries thanks; I've adjusted the title.  
> **username_1**: 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).  
> **username_0**: @username_1 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.*  
> **username_2**: @username_0. The acknowledgement that username_1 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.  
> **username_0**: @username_2 I think you can write an answer based on that. The title of the paper begins "*Constraining* the masses..." and so my use of "determine" in the body and "measure" in the title are the problem. If they're not doing that and they're just *constraining* them, then that's the answer!  

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## 0 answers

