Virial Theorem and Dark Matter conclusion

In most studies arriving at the conclusion of galactic dark matter,
the virial theorem is applied to compute the mass of the structure in
question from the observed velocities. For instance, Xue et al.
( http://arxiv.org/abs/0801.1232 ) find a 'virial mass' for the Milky
Way of 10^12 solar masses from the observation of Blue Horizontal-
Branch Stars in the Halo of the Milky way. Ignoring for the moment the
fact that this particular value is substantially lower than those
obtained by other methods (but still implying dark matter), the
question is what justifies the use of the virial theorem here? The
point is that the virial theorem implies a bound system, so what
justifies the assumption that the Halo stars are in bound orbits? If
we look at a system of many interacting masses in general, then the
energy distribution function will be continuous, containing masses
with energies lower then the virial (circular orbit) energy (i.e.
bound masses) as well as particles with an energy higher than the
virial energy (unbound masses). And it is obvious that in the course
of the development of the whole structure these components will become
separated: the ensemble of masses with small energies will contract
into a smaller volume, the masses with higher energy will expand into
a larger volume. In this sense, it should be questionable whether Halo
stars are bound at all. Stars with a velocity of the order of 100 km/
sec travel a pathlength of 100 kpc in 10^9 years, so the Halo stars
might well be unbound but have not managed yet to fully escape from
the Milky Way.

Thomas

On May 30, 1:59*am, Thomas Smid wrote:
In *most studies arriving at the conclusion of galactic dark matter,
the virial theorem is applied to compute the mass of the structure in
question from the observed velocities. For instance, Xue et al.
(http://arxiv.org/abs/0801.1232) find a 'virial mass' for the Milky
Way of 10^12 solar masses from the observation of Blue Horizontal-
Branch Stars in the Halo of the Milky way. Ignoring for the moment the
fact that this particular value is substantially lower than those
obtained by other methods (but still implying dark matter),

No it isn't. ~10^12 M_sun has been a consistent answer for years now.

the
question is what justifies the use of the virial theorem here? The
point is that the virial theorem implies a bound system, so what
justifies the assumption that the Halo stars are in bound orbits?

This is one of those questions that I would argue that if you have to
ask it seriously, you don't know enough about the subject to complain.

A galaxy is a bound system. It is really as simple as that.

That is the justification for usage of the Virial theorem, as the
system is both bound and close enough to being Newtonian that the
approximation is valid.

If you want to argue that some of the stars won't be bound, nobody is
going to care because you can chuck a whole cluster out of the galaxy
and it won't even change a visible decimal in the ~10^12 M_sun
determination.

If
we look at a system of many interacting masses in general, then the
energy distribution function will be continuous, containing masses
with energies lower then the virial (circular orbit) energy (i.e.
bound masses) as well as particles with an energy higher than the
virial energy (unbound masses).

No. Go forth and read:

http://math.ucr.edu/home/baez/virial.html

And it is obvious that in the course
of the development of the whole structure these components will become
separated: the ensemble of masses with small energies will contract
into a smaller volume, the masses with higher energy will expand into
a larger volume. In this sense, it should be questionable whether Halo
stars are bound at all. Stars with a velocity of the order of 100 km/
sec travel a pathlength of 100 kpc in 10^9 years, so the Halo stars
might well be unbound but have not managed yet to fully escape from
the Milky Way.

Thomas

The standard halo size estimate in the literature is something like
200 galactic radii.

Besides, your argument is irrelevant even if it were completely true
(it isn't) as the occasional corner case star being flung into
infinity does not meaningfully or even measurably alter the mass of
the galaxy at large.

On May 31, 8:26 pm, Eric Gisse wrote:
On May 30, 1:59 am, Thomas Smid wrote:

In most studies arriving at the conclusion of galactic dark matter,
the virial theorem is applied to compute the mass of the structure in
question from the observed velocities. For instance, Xue et al.
(http://arxiv.org/abs/0801.1232) find a 'virial mass' for the Milky
Way of 10^12 solar masses from the observation of Blue Horizontal-
Branch Stars in the Halo of the Milky way. Ignoring for the moment the
fact that this particular value is substantially lower than those
obtained by other methods (but still implying dark matter),

No it isn't. ~10^12 M_sun has been a consistent answer for years now.

If you had read only the abstract of the paper, you would have read
this:

"If we assume an NFW halo profile of characteristic concentration
holds, we can use the observations to estimate the virial mass of the
Milky Way's dark matter halo, Mvir = 1.0 +0.3 -0.2 ×10^12 M_sun, which
is lower than many previous estimates. We have checked that the
particulars of the cosmological simulations are unlikely to introduce
systematics larger than the statistical uncertainties. This estimate
implies that nearly 40% of the baryons within the virial radius of the
Milky Way's dark matter halo reside in the stellar components of our
Galaxy. A value for Mvir of only 1×10^12 M_sun also (re-)opens the
question of whether all of the Milky Way's satellite galaxies are on
bound orbits."


the
question is what justifies the use of the virial theorem here? The
point is that the virial theorem implies a bound system, so what
justifies the assumption that the Halo stars are in bound orbits?

A galaxy is a bound system. It is really as simple as that.

There is no direct observational evidence for the claim that all
populations of a galaxy are bound to the latter, in particular as far
as the Halo population is concerned. It is just an assumption.

If you want to argue that some of the stars won't be bound, nobody is
going to care because you can chuck a whole cluster out of the galaxy
and it won't even change a visible decimal in the ~10^12 M_sun
determination.

And still the observed velocity of such a cluster is often used to
derive the total galaxy mass via the assumption that it is bound.

we look at a system of many interacting masses in general, then the
energy distribution function will be continuous, containing masses
with energies lower then the virial (circular orbit) energy (i.e.
bound masses) as well as particles with an energy higher than the
virial energy (unbound masses).

No. Go forth and read:

http://math.ucr.edu/home/baez/virial.html

As I said, the virial theorem only applies to a bound system of
masses. Any real system of many interacting masses will however
contain a whole range of energies including energies 0, and the
virial theorem is not applicable to the latter (in fact, it is not
applicable to masses with any specific energy other than the average
energy).
If you would take a random sample of stars then the virial theorem
should hold, but it may not if you just consider a certain sub-
population of the whole system.

Thomas

On Jun 3, 1:40*am, Thomas Smid wrote:
On May 31, 8:26 pm, Eric Gisse wrote:

On May 30, 1:59 am, Thomas Smid wrote:

In *most studies arriving at the conclusion of galactic dark matter,
the virial theorem is applied to compute the mass of the structure in
question from the observed velocities. For instance, Xue et al.
(http://arxiv.org/abs/0801.1232) find a 'virial mass' for the Milky
Way of 10^12 solar masses from the observation of Blue Horizontal-
Branch Stars in the Halo of the Milky way. Ignoring for the moment the
fact that this particular value is substantially lower than those
obtained by other methods (but still implying dark matter),

No it isn't. ~10^12 M_sun has been a consistent answer for years now.

If you had read only the abstract of the paper, you would have read
this:

"If we assume an NFW halo profile of characteristic concentration
holds, we can use the observations to estimate the virial mass of the
Milky Way's dark matter halo, Mvir = 1.0 +0.3 -0.2 ×10^12 M_sun, which
is lower than many previous estimates. We have checked that the
particulars of the cosmological simulations are unlikely to introduce
systematics larger than the statistical uncertainties. This estimate
implies that nearly 40% of the baryons within the virial radius of the
Milky Way's dark matter halo reside in the stellar components of our
Galaxy. A value for Mvir of only 1×10^12 M_sun also (re-)opens the
question of whether all of the Milky Way's satellite galaxies are on
bound orbits."

To be honest I hadn't even read the abstract as I've read enough
papers on the subject that another one isn't going to add anything to
my internal understanding that isn't already there.

What I said remains correct. The mass of the milky way continues to
park in the ~10^12 M_sun range and nothing you have said has altered
that.

Though I do note the NFW assumption is observationally false for the
milky way as the galaxy's dark matter distribution has an observed
triaxiality. Which is neither really here nor there.




the
question is what justifies the use of the virial theorem here? The
point is that the virial theorem implies a bound system, so what
justifies the assumption that the Halo stars are in bound orbits?
A galaxy is a bound system. It is really as simple as that.

There is no direct observational evidence for the claim that all
populations of a galaxy are bound to the latter, in particular as far
as the Halo population is concerned. *It is just an assumption.

Since the galaxy has remained bound against the dispersive forces of
the Hubble flow, I'd consider "it is still here" to be evidence
enough.

Besides, there isn't anything that is "marginally" attached to the
galaxy that would change the mass estimate significantly if flung off
into the void.

Why are you invoking irrelevant arguments?


If you want to argue that some of the stars won't be bound, nobody is
going to care because you can chuck a whole cluster out of the galaxy
and it won't even change a visible decimal in the ~10^12 M_sun
determination.

And still the observed velocity of such a cluster is often used to
derive the total galaxy mass via the assumption that it is bound.

Really, only one cluster is used to determine the mass of the galaxy?
That strikes me as shoddy science.

Fortunately we have more than one test particle to use in order to
determine the gravitational potential of the galaxy...


we look at a system of many interacting masses in general, then the
energy distribution function will be continuous, containing masses
with energies lower then the virial (circular orbit) energy (i.e.
bound masses) as well as particles with an energy higher than the
virial energy (unbound masses).

No. Go forth and read:

http://math.ucr.edu/home/baez/virial.html

As I said, the virial theorem only applies to a bound system of
masses. Any real system of many interacting masses will however
contain a whole range of energies including energies 0, and the
virial theorem is not applicable to the latter (in fact, it is not
applicable to masses with any specific energy other than the average
energy).
If you would take a random sample of stars then the virial theorem
should hold, but it may not if you just consider a certain sub-
population of the whole system.

Thomas

Since you are so sure the application of the theorem is obtaining a
significantly wrong answer, I believe it would be straight forward for
you to show that a dynamical mass estimate is incompatible with, say,
our relative place in the Tully-Fisher relation?

Its' not even like you have a shortage of halo clusters to use.
There's something like 20 known ones, last I checked.

At this point I'm not even clear what you are complaining about beyond
"I DONT LIKE APPROXIMATIONS"

It strikes me that if your hypothesis were correct, that the virial
theorem is not a valid tool to use in determining the presence or
absence of dark matter in at least spiral galaxies, and that
significant populations of ~halo stars were routinely becoming
gravitationally unbound over time, that the following would be true:

In our various deep sky surveys, and considering the tens of thousands
of spirals examined, we would be seeing many galaxies showing various
degrees of this 'falling apart' or stellar eva- poration, to
paraphrase you, proportional to their ages and histories within their
clusters. But we don't. Regarding their various 'anatomies' and
internal motions we would expect to see clear signatures that they
violate the virial theorem But we don't. Additionally, if
significant amounts of stars were effectively 'boiling' away from
their home galaxies, their masses would still show up, virially
speaking, in the gravity budgets of the larger galaxy groups they
still inhabit, because they would still be gravitationally bound to
the home galaxy cluster. For what it's worth.