<p>Unfortunately, the answer is &quot;No&quot;, 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.</p>
<p>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): <span class="math-container">$3.98 \pm 0.04 \times 10^{7} M_{\odot}$</span> (<a href="https://ui.adsabs.harvard.edu/abs/2019ApJ...886L..27R/abstract" rel="noreferrer">Reid et al. 2019</a>). Other SMBH measurements are <em>at best</em> 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?</p>
<p><a href="https://ui.adsabs.harvard.edu/abs/2021MNRAS.500..215D/abstract" rel="noreferrer">Daly (2021)</a> has some nice tables with both accretion-rate estimates (in solar masses per year) and corresponding black hole masses. The <em>highest</em> 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 <span class="math-container">$6 \times 10^{9} M_{\odot}$</span>, 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!)</p>
<p>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.</p>
<p>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 <em>at least</em> several tens of millions of years to see a measurable increase in the SMBH mass.</p>
<p>The same paper also has some galactic (&quot;stellar-mass&quot;) accreting black hole measurements for X-ray binaries. Although the BH masses are much smaller (some less than <span class="math-container">$10 M_{\odot}$</span>), so are the accretion rates. The lowest-mass BH (GX 339-4, about <span class="math-container">$6 M_{\odot}$</span>) has an accretion rate of about <span class="math-container">$3 \times 10^{-9} M_{\odot}$</span> 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.)</p>
<p>[1] Partly because they're the kind of black holes I study, so I know more about the data.</p>
<p>[2] The Milky Way's own SMBH (Sgr A*) has a mass-measurement of <span class="math-container">$4.152 \pm 0.014 \times 10^{6} M_{\odot}$</span>, which is an uncertainty of <span class="math-container">$\sim 0.3$</span>%, but its current accretion rate (Daly 2021) is <span class="math-container">$\sim 6 \times 10^{-7} M_{\odot}$</span> per year, so you'd have to wait about <em>ten billion years</em>....</p>

CC BY-SA 4.0
2021-07-16T13:24:55.590Z
Thank you for this comprehensive answer! It's very helpful to see some key example systems with their masses and accretion rates. I suspected that it would be hopeless to measure the gain in mass directly. Especially with Sgr A*!!! The sun might die before we see its mass change appreciably ;D
Daddy Kropotkin
CC BY-SA 4.0
2021-07-16T13:25:32.743Z
So, before accepting your answer, I have one follow-up question: are there indirect methods by which to detect a change in mass? For example, in principle, can X-ray emission from accretion onto a compact body effect surrounding stars in a way that can be used to infer a change in mass, i.e., the luminosity changes by $x$, which would imply that the mass of the BH changed by $y$? OR is that also not feasible since the change in luminosity would be essentially undetectable on human timeframe?
Daddy Kropotkin
CC BY-SA 4.0
2021-07-16T13:30:01.970Z
@DaddyKropotkin I strongly suspect that wouldn't work, not just because (as you suggest) the change in luminosity would be too slow, but also because it would be dependent on so many other things (especially the accretion rate itself, instabilities in the accretion disk, etc.).
Peter Erwin
CC BY-SA 4.0
2021-07-16T13:35:01.303Z
If you look at Table 3 of the Daly (2021) paper, you can see all sorts of fluctuations in the accretion rate (which is based on the accretion luminosity) for the X-ray binaries -- e.g., variations of an order of magnitude or more. So it would be *worse* than measuring the mass dynamically, where you don't have to worry about the secondary star (or maser emission spots or whatever) jumping around on short time scales.
Peter Erwin
CC BY-SA 4.0
2021-07-17T06:58:08.783Z
Whilst I agree that precision is a problem, I think your assessments may be too pessimistic. Can you explain what governs the uncertainties in the mass measurement. If it were because the distance to the BH is uncertain for example, then this wouldn't play any role in determining how precisely you could detect a change in the BH mass.
ProfRob
CC BY-SA 4.0
2021-07-17T15:21:10.080Z
@ProfRob For the vast majority of SMBH measurements, the uncertainties are formal statistical uncertainties from the modeling (e.g., $\Delta \chi^{2}_{\nu} = 1$, cumulative marginalized likelihood distributions, etc.; see, for example, Sec.7 of [Erwin+2018](https://ui.adsabs.harvard.edu/abs/2018MNRAS.473.2251E/abstract)). Uncertainties from the distance are almost never included in this.
Peter Erwin
CC BY-SA 4.0
2021-07-17T15:50:56.267Z
ok, but the paper you refer to for NGC 4258 appears to be all about simultaneously determining the distance with the parameters of the black hole. So my point is, one *should* look at the BH parameters uncertainties with the distance (and any other "uninteresting" parameters)  fixed for the purposes of answering the question. I am sure the precisions are still too poor to have a hope of measuring a mass increase.
ProfRob
CC BY-SA 4.0
2021-07-17T16:50:35.580Z
Sure, though that's a unique case where the distance is more interesting than the BH mass (since it's a direct determination that allows an independent calibration of its Cepheid-star distance). If you look at the corner plot in Fig.1, it's hard to see how you'd get more than a factor of 2-5 improvement on the BH mass uncertainty if you held the distance fixed. So, maybe 40 million years instead of 200 million for that galaxy...
Peter Erwin
CC BY-SA 4.0
2021-07-16T12:50:11.193Z
2021-07-16T12:50:11.193Z
Peter Erwin
<p>Have we observed a black hole directly gain mass via accretion? That is, have we observed the black hole to have mass <span class="math-container">$m_1$</span> at time <span class="math-container">$t_1$</span> and then observed its mass to be <span class="math-container">$m_2 &gt; m_1$</span> at a later time <span class="math-container">$t_2&gt;t_1$</span>?</p>
<p>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 &quot;really exists&quot; or not!).</p>

CC BY-SA 4.0
2021-07-15T13:50:45.077Z
We've observed with gravitational waves that black holes gain mass as a result of absorbing other black holes or neutron stars.  I think the mass gain from accretion disks would be too slow to observe in a reasonable amount of time.
antlersoft
CC BY-SA 4.0
2021-07-15T14:26:11.387Z
Indeed, but I really mean to ask about accretion, not mergers. So if a quantitative argument could be shown that demonstrates that the amount of mass gained via accretion over human observational timescales is too small to measure? And in what regime would it be possible to observe a high enough accretion rate for sufficiently long to observe a change in mass from accretion?
Daddy Kropotkin
CC BY-SA 4.0
2021-07-15T22:13:52.973Z
`+1` for an interesting question! I hope following question(s) extend this to other objects as well. The effects of mass infall can make flares, but have subsequent measurements demonstrated the mass increase is a challenge.
uhoh
<black-hole><general-relativity><supermassive-black-hole><accretion-discs><accretion>
Has gain in mass from accretion onto a black hole been observed?
Daddy Kropotkin
# Has gain in mass from accretion onto a black hole been observed?

(Asked by: username_0 on 2021-07-15)

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!).

**Tags:** black-hole, general-relativity, supermassive-black-hole, accretion-discs, accretion
> **Comments:**
> **username_1**: We've observed with gravitational waves that black holes gain mass as a result of absorbing other black holes or neutron stars.  I think the mass gain from accretion disks would be too slow to observe in a reasonable amount of time.  
> **username_0**: Indeed, but I really mean to ask about accretion, not mergers. So if a quantitative argument could be shown that demonstrates that the amount of mass gained via accretion over human observational timescales is too small to measure? And in what regime would it be possible to observe a high enough accretion rate for sufficiently long to observe a change in mass from accretion?  
> **username_2**: `+1` for an interesting question! I hope following question(s) extend this to other objects as well. The effects of mass infall can make flares, but have subsequent measurements demonstrated the mass increase is a challenge.  

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## 1 answers

### Answer by username_3 on 2021-07-16. Score: 7 | Accepted answer

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](https://ui.adsabs.harvard.edu/abs/2019ApJ...886L..27R/abstract)). 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)](https://ui.adsabs.harvard.edu/abs/2021MNRAS.500..215D/abstract) 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*....

> **Comments:**
> **username_0**: Thank you for this comprehensive answer! It's very helpful to see some key example systems with their masses and accretion rates. I suspected that it would be hopeless to measure the gain in mass directly. Especially with Sgr A*!!! The sun might die before we see its mass change appreciably ;D  
> **username_0**: So, before accepting your answer, I have one follow-up question: are there indirect methods by which to detect a change in mass? For example, in principle, can X-ray emission from accretion onto a compact body effect surrounding stars in a way that can be used to infer a change in mass, i.e., the luminosity changes by $x$, which would imply that the mass of the BH changed by $y$? OR is that also not feasible since the change in luminosity would be essentially undetectable on human timeframe?  
> **username_3**: @DaddyKropotkin I strongly suspect that wouldn't work, not just because (as you suggest) the change in luminosity would be too slow, but also because it would be dependent on so many other things (especially the accretion rate itself, instabilities in the accretion disk, etc.).  
> **username_3**: If you look at Table 3 of the Daly (2021) paper, you can see all sorts of fluctuations in the accretion rate (which is based on the accretion luminosity) for the X-ray binaries -- e.g., variations of an order of magnitude or more. So it would be *worse* than measuring the mass dynamically, where you don't have to worry about the secondary star (or maser emission spots or whatever) jumping around on short time scales.  
> **username_4**: Whilst I agree that precision is a problem, I think your assessments may be too pessimistic. Can you explain what governs the uncertainties in the mass measurement. If it were because the distance to the BH is uncertain for example, then this wouldn't play any role in determining how precisely you could detect a change in the BH mass.  
> **username_3**: @ProfRob For the vast majority of SMBH measurements, the uncertainties are formal statistical uncertainties from the modeling (e.g., $\Delta \chi^{2}_{\nu} = 1$, cumulative marginalized likelihood distributions, etc.; see, for example, Sec.7 of [Erwin+2018](https://ui.adsabs.harvard.edu/abs/2018MNRAS.473.2251E/abstract)). Uncertainties from the distance are almost never included in this.  
> **username_4**: ok, but the paper you refer to for NGC 4258 appears to be all about simultaneously determining the distance with the parameters of the black hole. So my point is, one *should* look at the BH parameters uncertainties with the distance (and any other "uninteresting" parameters)  fixed for the purposes of answering the question. I am sure the precisions are still too poor to have a hope of measuring a mass increase.  
> **username_3**: Sure, though that's a unique case where the distance is more interesting than the BH mass (since it's a direct determination that allows an independent calibration of its Cepheid-star distance). If you look at the corner plot in Fig.1, it's hard to see how you'd get more than a factor of 2-5 improvement on the BH mass uncertainty if you held the distance fixed. So, maybe 40 million years instead of 200 million for that galaxy...  

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