Has gain in mass from accretion onto a black hole been observed?

Question

Have we observed a black hole directly gain mass via accretion? That is, have we observed the black hole to have mass $m_1$ at time $t_1$ and then observed its mass to be $m_2 > m_1$ at a later time $t_2>t_1$?

I understand this would most likely be plausible for stellar-mass X-ray binaries or for supermassive black holes, and that the uncertainty on the mass measurement of such a black hole would need to be smaller than the change in the mass due to accretion. But I'm wondering if we have any observational evidence that a black hole's mass increases as it accretes (of course, theoretically it should, regardless of whether the gravitational singularity "really exists" or not!).

Answer 1

Unfortunately, the answer is "No", because accretion rates are far too low -- and our ability to measure black hole masses is far too uncertain -- for this to be visible in reasonable times. Given our current ability to measure black hole masses, you'd typically have to wait millions or tens of millions of years to see any accretion-related changes.

Let's look at supermassive black holes (SMBHs)[1]. The most precise mass measurement for an external galaxy's SMBH[2] is for NGC 4258 (M106): $3.98 \pm 0.04 \times 10^{7} M_{\odot}$ (Reid et al. 2019). Other SMBH measurements are at best uncertain at the 10% level, and many are uncertain by factors of several. So at a minimum, you need the SMBBH mass to grow by at least a percent to have any chance of detecting the change. How would long that take?

Daly (2021) has some nice tables with both accretion-rate estimates (in solar masses per year) and corresponding black hole masses. The highest accretion rate is about 10 solar masses per year, for the quasar 3C 268.4 (Table 4). Since this quasar has an estimated SMBH mass of $6 \times 10^{9} M_{\odot}$, you would need about 60 million years years to get a 10% increase in mass, or 6 million years to get a 1% increase. (Assuming the accretion rate held steady, which is not guaranteed!)

For NGC 4258, where we can measure the SMBH mass at about the 1% level, the estimated accretion rate (for the Seyfert nucleus) is about 0.002 solar masses per year. So we'd have to wait about 200 million years to see a measurable increase in its mass.

Table 1 of that paper has some mean values for accretion rates and SMBH masses, which show the general trend is the same as for those two specific cases: you'll have wait at least several tens of millions of years to see a measurable increase in the SMBH mass.

The same paper also has some galactic ("stellar-mass") accreting black hole measurements for X-ray binaries. Although the BH masses are much smaller (some less than $10 M_{\odot}$), so are the accretion rates. The lowest-mass BH (GX 339-4, about $6 M_{\odot}$) has an accretion rate of about $3 \times 10^{-9} M_{\odot}$ per year, so you'd need about 20 million years to see a 1% increase in mass. (I suspect the uncertainty in the BH mass is probably at least 10%, so you're more likely to need several hundred million years.)

[1] Partly because they're the kind of black holes I study, so I know more about the data.

[2] The Milky Way's own SMBH (Sgr A*) has a mass-measurement of $4.152 \pm 0.014 \times 10^{6} M_{\odot}$, which is an uncertainty of $\sim 0.3$%, but its current accretion rate (Daly 2021) is $\sim 6 \times 10^{-7} M_{\odot}$ per year, so you'd have to wait about ten billion years....