From: metzler@pablo.physics.lsa.umich.edu (Chris Metzler)
Newsgroups: sci.physics
Subject: Re: What is Omega_o?
Date: 12 Nov 1992 03:40:01 GMT
Organization: University of Michigan Department of Physics
Distribution: world
NNTP-Posting-Host: pablo.physics.lsa.umich.edu

bweiner@ruhets.rutgers.edu (Benjamin Weiner) writes:
|> metzler@pablo.physics.lsa.umich.edu (Chris Metzler) writes:
|> >It is true estimates of Omega_o from cluster galaxy dynamics or from
|> >cluster X-ray emission modelling generally result in values smaller than one.
|> >However, it has been shown numerous times, through N-body and hydrodynamical
|> >simulations, that both methods of estimating Omega_o are based on serious
|> >assumptions that affect the analysis.  
|> 
|> Yes, but most people acknowledge these estimates as lower bounds.
|> I guess I'd phrase it differently: I figure the assumptions are
|> probably true, but couldn't set upper bounds on Omega_0, regardless.
|> They're consistent with Omega_0 = 1 if you assume some kind of
|> biasing, which you have to assume anyway.  The arguments for Omega_0 = 1
|> are of a more global variety (eg horizon problem; structure formation,
|> which you outlined nicely, thanks!)
|> 
|> > [ dynamics <-> substructure problem ]
|> > [ x-rays <-> thermalization assumed ]
|> 
|> >Both of these are very questionable.  In particular, the so-called "beta
|> >discrepancy" suggests strongly that cluster gas has not thermalized since
|> >infall.
|> 
|> The substructure problem is well recognized.  Not that that stops
|> anybody.  For X-rays, the situation is more complex, because the beta
|> problem (brief summary: extended distribution of gas implies that
|> the gas should have a higher specific energy than the galactic matter,
|> but this is not observed, in eg Perseus cluster) isn't always a problem.  
|> In my hot little hands I have a paper (Watt et al MNRAS 258, 738) which gives 
|> references saying that there is no beta problem for the Coma cluster.

The description of the "beta problem" you give is not what I meant by the
"beta discrepancy."  What I meant was this . . .

(you probably already know this, but I'm going to say it anyway for the benefit
of anyone else who is bored enough to read this) . . .beta, in the study
of clusters, comes up in two ways.  X-ray surface brightness profiles of
clusters are usually fit to a form that looks like

                                                     2  -3*beta_f + 1/2
                                      (     [ theta ]  ) 
		Sig(theta) = Sig(0) * ( 1 + [-------]  )
                                      (     [theta_x]  )

where theta_x, beta_f, and Sig(0) are fitting parameters.  The principal reason
for fitting to such an ugly form is because it gives a good fit.  But if you
assume that the gas is isothermal, you get the same sort of function for the
radial density profile (although the complicated looking exponent above is
different).  Since free-free emission is very sensitive to the density 
( L \propto n^2 ), we figure that the surface brightness profile is echoing
the gas density, and thus are able to pin down the density profile of the
gas as involving the same fitting parameter --
                                             2  -3*beta_f/2
                                  (     [ r ]  ) 
		Rho(r) = Rho(0) * ( 1 + [---]  )
                                  (     [r_c]  )

On the other hand, we can mess with the equations of hydrostatic equilibrium,
which tell us about the density in clusters.  Defining beta ( NOT beta_f ) to
be the ratio of specific kinetic energy in the galaxies to specific thermal
energy in the gas,

                                2            sigma = velocity dispersion of
                           sigma                     galaxies
		beta = ----------------      T = temperature of cluster gas
                       ( kT /         )      mu = mean molecular weight of
                       (   / mu * m_p )           gas (taken to be 0.6, the
                                                  value for a solar abundance
                                                  plasma)
                                             m_p = proton mass

By plugging the density function described above into the equations for
hydrostatic equilibrium, you can get a relation between beta and beta_f.
If you then make two more assumptions -- that the collisionless (galaxies
or dark matter) density profile of the cluster follows the isothermal
distribution, and that the cluster galaxy velocity dispersons are isotropic,
then you get that beta = beta_f.

We measure beta_f through X-ray observations of clusters; we measure beta
through X-ray determinations of cluster temperatures + measurements of
cluster galaxy redshifts.  The beta discrepancy is that traditionally,
beta_f < beta.

The principle weaknesses in all this, which are probably the source of
this recurring difference, are the assumptions we made along the way.
We assumed that both the gas and the collisionless matter are isothermal.
We assumed that the galaxies were in a relaxed virial equilibrium, and the
gas in hydrostatic equilibrium, with the cluster gravitational potential.
Relaxing these assumptions makes the beta discrepancy go away.  Simulated
clusters in 3-D N-body + hydro simulations reproduce the beta discrepancy
and pin its origin down:  the gas need not be completely thermalized yet,
so beta underestimates the energy in gas by not including a kinetic component;
cluster galaxy orbits need not be isotropic, and substructure observations
suggest they are not; and the collisionless component density profile
probably doesn't follow a perfect King isothermal profile.

The important point to remember is that although the beta discrepancy
highlights the problems in these assumptions, cluster mass estimates
nonetheless make the same assumptions all over again.  We don't have
good temperature profiles of clusters; so we either assume isothermality
straight into the hydrostatic equations, or we use the X-ray surface
brightness profile and go through the same crap we just went through.
And all of these assumptions push your mass estimate downward.


|>      
|> >N-body, and N-body + hydrodynamic simulations have shown several times that
|> >such models underestimate cluster masses.  For example, Evrard (1990) showed
|> >results for clusters grown in an Omega_o = 1 cosmology which, when virial
|> >and binding mass estimates were applied to the final cluster, showed an 
|> >apparent Omega_o = 0.2.
|> 
|> This is the key.  I just heard a talk by David Weinberg in which he said the
|> same thing (paper in newest Ap J Lett).  Basically it comes down to
|> this: when you talk about dark matter, you can never rule out that
|> there's more of it on the larger length scales.

Was Dave over at Rutgers recently?  I guess that's not that much of a haul
for him (where is Rutgers, anyway?)  He and my advisor are pretty good friends;
I think they shared an office at Berkeley.