Black hole mass-sigma correlation

Hans Aberg wrote:

Only approximately, as there is a GR correction showing that they will in
fact slowly drop into the center (due to emission of "gravity waves").
This is also why this development is so exiting:

As GR, unlike Newtonian physics, predicts that the galaxy mass should drop
into the center, one would think that the belief that there are black
holes at the centers of the galaxies should have come along very early,
but in fact it took a long time. Now there are even formulas for telling
the size of that black hole mass.



This effect should be completely negligible compared to other
effects taking place in a galaxy, such as the transport of angular
momentum by the spiral arms, which may let some of the material in the
galactic disc drift inwards while angular momentum is transported
outwards, and the scattering of stars against giant molecular clouds.
There may be even more dramatic effects taking place during the
formation of a galaxy, and as far as we can tell today the black holes
at the centers of the galaxies form early in the life of the galaxies.

Angular momentum loss through the emission of gravitational waves can be
important in the evolution of close binaries though, and provides a perfect
explanation for the timings of the double pulsar PSR 1916+13.

Ulf Torkelsson

[Mod. note: reformatted to 80 characters per line -- mjh]

Hans Aberg wrote:

I am not an expert at GR computations, but my intuition about it was that
sufficiently massive objects will emit those gravitational waves. Thus,
other forces may be stronger in the short term, but unless there is some
outward push by some non-gravitational forces, say solar winds from the
stars or the like, in the long term, mass should drop into the center.


For the general relativistic deviations from Newtonian gravity to become
dominating we need to have (GM/c^2)/L to be of order unity. For a galaxy
with a mass of 10^(12) solar masses GM/c^2 is of the order of 10^(12) km,
and the galaxy has a size of 10 kpc, which is 10^(16) km, so you can forget
general relativity on the scale of individual galaxies. Assuming that there
is a black hole of 10^8 solar masses at the center of the galaxy we see that
you need to get within 10^8 km of that before general relativity becomes
really important, that is about the distance between the Earth and the Sun.

The damping time scale of the orbit due to gravitational radiation is
L/c (L/(GM/c^2))^3, see for instance

Shapiro, S. L., Teukolsky, S. A., 1983,
Black holes, white dwarfs, and neutron stars,
John Wiley & Sons, New York,

which for our galaxy is 10^(17) years.

This BBC World program described a theory explaining the black hole
mass-sigma correlation in terms of a black hole formed even before the
galaxy had formed. It suggested that this was in reality the quasars,
galaxies in very early formation, emitting radiation from the black hole.
The idea was that this black hole should then stop feeding.


Right, but keep in mind that this radiation is formed in the gas
that is accreting onto the black hole, and the process is essentially
one described by classical hydrodynamics. The exception being if the
black hole is rotating, in which case there is a theoretical
possibility to tap the black hole of its rotational energy. In the
latter case you need to do relativistic (magneto)hydrodynamics in a
Kerr metric.


However, this program also presented some even later research, suggesting
that the black hole is feeding also in later stages of the galaxy life. In
addition, I have vague memory of research a few years ago suggesting that
our Milky Way black hole is feeding, even swallowing up some stars, and it
should be in its middle age.

Therefore, taking those circumstantial pieces of evidence into account, it
might perhaps be the case that the galaxy center black holes are feeding a
lot more than those facts you are mentioning suggest.


Actually the accretion rate onto the black hole in the centre of the
Milky Way is quite low for the moment, definitely much lower than in
an ordinary active galaxy or quasar, see for instance

Mezger, P. G., Duschl, W. J., Zylka, R., 1996,
The Galactic Center: a laboratory for AGN?,
Ann. Rev. Astron. & Astrophys., 7, 289

My guess is that such observations have in the past been used in order to
confirm GR.

Actually the observations of PSR 1916+13 are still the only
observations showing that gravitational waves are generated.

Ulf Torkelsson

[Mod. note: quoted text trimmed -- mjh.]

Hans Aberg wrote:

I am not an expert at GR computations, but my intuition about it was that
sufficiently massive objects will emit those gravitational waves. Thus,
other forces may be stronger in the short term, but unless there is some
outward push by some non-gravitational forces, say solar winds from the
stars or the like, in the long term, mass should drop into the center.


For the general relativistic deviations from Newtonian gravity to become
dominating we need to have (GM/c^2)/L to be of order unity. For a galaxy
with a mass of 10^(12) solar masses GM/c^2 is of the order of 10^(12) km,
and the galaxy has a size of 10 kpc, which is 10^(16) km, so you can forget
general relativity on the scale of individual galaxies. Assuming that there
is a black hole of 10^8 solar masses at the center of the galaxy we see that
you need to get within 10^8 km of that before general relativity becomes
really important, that is about the distance between the Earth and the Sun.

The damping time scale of the orbit due to gravitational radiation is
L/c (L/(GM/c^2))^3, see for instance

Shapiro, S. L., Teukolsky, S. A., 1983,
Black holes, white dwarfs, and neutron stars,
John Wiley & Sons, New York,

which for our galaxy is 10^(17) years.

This BBC World program described a theory explaining the black hole
mass-sigma correlation in terms of a black hole formed even before the
galaxy had formed. It suggested that this was in reality the quasars,
galaxies in very early formation, emitting radiation from the black hole.
The idea was that this black hole should then stop feeding.


Right, but keep in mind that this radiation is formed in the gas
that is accreting onto the black hole, and the process is essentially
one described by classical hydrodynamics. The exception being if the
black hole is rotating, in which case there is a theoretical
possibility to tap the black hole of its rotational energy. In the
latter case you need to do relativistic (magneto)hydrodynamics in a
Kerr metric.


However, this program also presented some even later research, suggesting
that the black hole is feeding also in later stages of the galaxy life. In
addition, I have vague memory of research a few years ago suggesting that
our Milky Way black hole is feeding, even swallowing up some stars, and it
should be in its middle age.

Therefore, taking those circumstantial pieces of evidence into account, it
might perhaps be the case that the galaxy center black holes are feeding a
lot more than those facts you are mentioning suggest.


Actually the accretion rate onto the black hole in the centre of the
Milky Way is quite low for the moment, definitely much lower than in
an ordinary active galaxy or quasar, see for instance

Mezger, P. G., Duschl, W. J., Zylka, R., 1996,
The Galactic Center: a laboratory for AGN?,
Ann. Rev. Astron. & Astrophys., 7, 289

My guess is that such observations have in the past been used in order to
confirm GR.

Actually the observations of PSR 1916+13 are still the only
observations showing that gravitational waves are generated.

Ulf Torkelsson

[Mod. note: quoted text trimmed -- mjh.]

Hans Aberg wrote:

However, it depends on how that 1983 formula was derived: Perhaps it makes
some approximations about heavy objects moving in orbit in order to
filtrate out GR effects.

It is just a simple order of magnitude estimate to check whether
gravitational radiation is relevant in these circumstances and it is
not.

If mass is spread over an area, then one may have
to integrate over the matter distribution to find more accurate GR
corrections. Perhaps this gains some magnitudes in the GR corrections.
Then it might the case that more significant amounts of matter drops into
the center.

Yes, much more matter drops into the center in the beginning because of a
pure Newtonian effect called free fall. The reason that not all the matter
drops into the center is due to another Newtonian effect called conservation
of angular momentum. There is no need to invoke relativistic complications
to understand this.

By the standards of this development, this paper is should be old.

I do not remember exactly what they said on the BBC Horizon program, but I
got the impression that they said that one thought the Milky Way(?) center
black hole should be stable (i.e., non-feeding), but now they had some
evidence that it was not so. I got the impression that these results are
more recent than the observational verification of the black hole
mass-sigma correlation.


No, the existence of a black hole at the center of our galaxy was
discussed already in the seventies, and in the nineties it was clear
that the accretion rate onto this black hole is actually quite small,
though not negligible, compared to that of active galaxies. The black
hole mass-sigma correlation is fresher than that, but they are in
accordance with each other.

Ulf Torkelsson

[Mod. note: quoted text trimmed, reformatted, &c -- mjh.]

Hans Aberg wrote:

However, it depends on how that 1983 formula was derived: Perhaps it makes
some approximations about heavy objects moving in orbit in order to
filtrate out GR effects.

It is just a simple order of magnitude estimate to check whether
gravitational radiation is relevant in these circumstances and it is
not.

If mass is spread over an area, then one may have
to integrate over the matter distribution to find more accurate GR
corrections. Perhaps this gains some magnitudes in the GR corrections.
Then it might the case that more significant amounts of matter drops into
the center.

Yes, much more matter drops into the center in the beginning because of a
pure Newtonian effect called free fall. The reason that not all the matter
drops into the center is due to another Newtonian effect called conservation
of angular momentum. There is no need to invoke relativistic complications
to understand this.

By the standards of this development, this paper is should be old.

I do not remember exactly what they said on the BBC Horizon program, but I
got the impression that they said that one thought the Milky Way(?) center
black hole should be stable (i.e., non-feeding), but now they had some
evidence that it was not so. I got the impression that these results are
more recent than the observational verification of the black hole
mass-sigma correlation.


No, the existence of a black hole at the center of our galaxy was
discussed already in the seventies, and in the nineties it was clear
that the accretion rate onto this black hole is actually quite small,
though not negligible, compared to that of active galaxies. The black
hole mass-sigma correlation is fresher than that, but they are in
accordance with each other.

Ulf Torkelsson

[Mod. note: quoted text trimmed, reformatted, &c -- mjh.]

In article , Ulf Torkelsson
wrote:

However, it depends on how that 1983 formula was derived: Perhaps it makes
some approximations about heavy objects moving in orbit in order to
filtrate out GR effects.

It is just a simple order of magnitude estimate to check whether
gravitational radiation is relevant in these circumstances and it is
not.

My worry is that Newtonian 1/r forces are very special mathematically,
making formulas line up very nicely, making angular momentum around
different points independent. GR has some compensations thrown in, so it
could be that one has to take the interactions of the different components
into account in order to get better GR correction.

If mass is spread over an area, then one may have
to integrate over the matter distribution to find more accurate GR
corrections. Perhaps this gains some magnitudes in the GR corrections.
Then it might the case that more significant amounts of matter drops into
the center.

Yes, much more matter drops into the center in the beginning because of a
pure Newtonian effect called free fall. The reason that not all the matter
drops into the center is due to another Newtonian effect called conservation
of angular momentum. There is no need to invoke relativistic complications
to understand this.

If some large cloud already has angular momentum, why should mass drop
into the center? It does not happen with the planets in orbit.

By the standards of this development, this paper is should be old.

I do not remember exactly what they said on the BBC Horizon program, but I
got the impression that they said that one thought the Milky Way(?) center
black hole should be stable (i.e., non-feeding), but now they had some
evidence that it was not so. I got the impression that these results are
more recent than the observational verification of the black hole
mass-sigma correlation.

No, the existence of a black hole at the center of our galaxy was
discussed already in the seventies, and in the nineties it was clear
that the accretion rate onto this black hole is actually quite small,
though not negligible, compared to that of active galaxies. The black
hole mass-sigma correlation is fresher than that, but they are in
accordance with each other.

You must have misunderstood what I said: The program said that one thought
that the black hole accretion rate was low, in seeming accordance with the
model that seems to explain the black hole mass-sigma correlation. But
some research later then the discovery of the black hole mass-sigma
correlation suggested otherwise, namely that it might in fact be higher,
at a rate not explainable by this theory. It was mentioned at the very end
of that program.

The comment did not have anything to do with the history of the discussion
of that there should be a black hole there.

Hans Aberg

In article , Ulf Torkelsson
wrote:

However, it depends on how that 1983 formula was derived: Perhaps it makes
some approximations about heavy objects moving in orbit in order to
filtrate out GR effects.

It is just a simple order of magnitude estimate to check whether
gravitational radiation is relevant in these circumstances and it is
not.

My worry is that Newtonian 1/r forces are very special mathematically,
making formulas line up very nicely, making angular momentum around
different points independent. GR has some compensations thrown in, so it
could be that one has to take the interactions of the different components
into account in order to get better GR correction.

If mass is spread over an area, then one may have
to integrate over the matter distribution to find more accurate GR
corrections. Perhaps this gains some magnitudes in the GR corrections.
Then it might the case that more significant amounts of matter drops into
the center.

Yes, much more matter drops into the center in the beginning because of a
pure Newtonian effect called free fall. The reason that not all the matter
drops into the center is due to another Newtonian effect called conservation
of angular momentum. There is no need to invoke relativistic complications
to understand this.

If some large cloud already has angular momentum, why should mass drop
into the center? It does not happen with the planets in orbit.

By the standards of this development, this paper is should be old.

I do not remember exactly what they said on the BBC Horizon program, but I
got the impression that they said that one thought the Milky Way(?) center
black hole should be stable (i.e., non-feeding), but now they had some
evidence that it was not so. I got the impression that these results are
more recent than the observational verification of the black hole
mass-sigma correlation.

No, the existence of a black hole at the center of our galaxy was
discussed already in the seventies, and in the nineties it was clear
that the accretion rate onto this black hole is actually quite small,
though not negligible, compared to that of active galaxies. The black
hole mass-sigma correlation is fresher than that, but they are in
accordance with each other.

You must have misunderstood what I said: The program said that one thought
that the black hole accretion rate was low, in seeming accordance with the
model that seems to explain the black hole mass-sigma correlation. But
some research later then the discovery of the black hole mass-sigma
correlation suggested otherwise, namely that it might in fact be higher,
at a rate not explainable by this theory. It was mentioned at the very end
of that program.

The comment did not have anything to do with the history of the discussion
of that there should be a black hole there.

Hans Aberg

Hans Aberg wrote:
My worry is that Newtonian 1/r forces are very special mathematically,
making formulas line up very nicely, making angular momentum around
different points independent. GR has some compensations thrown in, so it
could be that one has to take the interactions of the different components
into account in order to get better GR correction.

The order of magnitude estimate that I was using is based on GR. There
are no gravitational waves in Newtonian gravity theory. Apart from
that the differences between GR and Newtonian physics is small as long
as GM/(c^2 r) is small.

If some large cloud already has angular momentum, why should mass drop
into the center? It does not happen with the planets in orbit.

The difference between a gas cloud and the planets is that different
parts of the gas cloud are interacting with each other through
hydrodynamic forces. Thus the angular momentum can be re-distributed
within the gas cloud allowing some parts of the cloud to gain most of
the angular momentum, while other gas elements fall to the center of
the cloud.

Ulf Torkelsson

[Mod. note: quoted text trimmed and posting reformatted, again -- mjh]

Hans Aberg wrote:
My worry is that Newtonian 1/r forces are very special mathematically,
making formulas line up very nicely, making angular momentum around
different points independent. GR has some compensations thrown in, so it
could be that one has to take the interactions of the different components
into account in order to get better GR correction.

The order of magnitude estimate that I was using is based on GR. There
are no gravitational waves in Newtonian gravity theory. Apart from
that the differences between GR and Newtonian physics is small as long
as GM/(c^2 r) is small.

If some large cloud already has angular momentum, why should mass drop
into the center? It does not happen with the planets in orbit.

The difference between a gas cloud and the planets is that different
parts of the gas cloud are interacting with each other through
hydrodynamic forces. Thus the angular momentum can be re-distributed
within the gas cloud allowing some parts of the cloud to gain most of
the angular momentum, while other gas elements fall to the center of
the cloud.

Ulf Torkelsson

[Mod. note: quoted text trimmed and posting reformatted, again -- mjh]

In article , Ulf Torkelsson
wrote:

My worry is that Newtonian 1/r forces are very special mathematically,
making formulas line up very nicely, making angular momentum around
different points independent. GR has some compensations thrown in, so it
could be that one has to take the interactions of the different components
into account in order to get better GR correction.

The order of magnitude estimate that I was using is based on GR.

Sure, that was clear from your posts.

There
are no gravitational waves in Newtonian gravity theory.

And this is, of course the reason one wants to put in GR corrections.

Apart from
that the differences between GR and Newtonian physics is small as long
as GM/(c^2 r) is small.

What might make a difference in the magnitude of this smallness is that it
may behave differently when just considering say two point masses, or a
large distribution of mass where individual components may emit
gravitational waves by mutual GR interaction (not just against the center
mass). In picture, this difference might be like the difference between
planets and a gas, with GR influences replacing gas hydrodynamic forces.

If the paper you quoted with the rough GR estimate does not do that latter
thing, integrating over the GR corrections, then that might be one thing
to look up before writing off GR influences as wholly negligible. --
Especially, if there is someone out there doing observations that might
suggest that the Milky Way black hole feeding rate is in fact higher than
explainable by the suggested the black hole mass-sigma correlation model,
the latter which only derives from Newtonian physics.

The most important thing, though, would be to pin down the feeding rate of
the Milky Way black hole. But if it eventually turns out to be higher than
earlier research suggested, and you are positively sure that there is an
estimate that once and for all rules out all GR effects, what should cause
it? -- The alternative explanation (to that of GR waves) would be that
there is some gas that for some reason is loosing its angular momentum.

Hans Aberg