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Outline

• Main features of galaxy clusters physics
• On the total mass and its distribution
• On the Jeans method of mass determination
• Our work
• Data preparation procedures
• Calculation results
• Conclusions

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1. Main features of galaxy clusters physics
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The distribution function gives all the
information on the system
Galaxies number density
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Moments of the B.E. can be taken to deduce
simpler equations relative to these observable
quantities
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Its normally believed that clusters are
sufficiently relaxed structures

• In a first approximation, the galaxies and the
  hot gas can be thought in equilibrium within the
  global potential well dominated by the dark
  matter
• The 3D structure should be approximately spherical

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More in general, the solution to the time
independent Boltzmann equation for a given fixed
potential and in spherical symmetry is
where

• The AK profile is much used in the literature
• To fit observed galaxies number counts profiles
• As a total mass model with a soft core

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2. On the total mass and its distribution
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• Many methods are available to determine the total
  cluster mass
• Gravitational lensing (for intermediate and large
  distance cluster)
• Methods employing the hot gas (beta models,
  deprojection tecniques)
• Dynamical methods employing the galaxies

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The GALAXIES and the GAS are both tracers of the
underlying gravitational potential
BASIC HYPOTHESIS
SPHERICAL SIMMETRY DYNAMICAL EQUILIBRIUM NO
STREAMING ROTATIONAL MOTIONS
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DYNAMICAL JEANS METHODS
Jeans equation in spherical symmetry
galactic motion can be anisotropic
Radial orbits
Isotropic orbits
Tangential orbits
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N-BODY SIMULATIONS
Universal shape for the dark halo profile (from
dwarves galaxies to rich clusters)
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SOME RESULTS IN THE LITERATURE ON THE COMA CLUSTER
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Hows the total mass distribution like ?
Peaked or soft-core profile ?
Soft-Core KING
Cusp NFW
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2. Constraints to the theories of clusters and
galaxies evolution
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3. On the Jeans method of mass determination
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JEANS METHOD OF MASS DETERMINATION
Galaxies are employed as tracers of the
underlying total gravitational potential
To perform a sound Jeans analysis one needs A
rich cluster which may be retained evoluted
enough and whose 2D projection is nearly spherical
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What can we observe about the cluster galaxies ?
Sky plane projection
sky plane
Radial position on the sky
Observables
Line of sight velocity
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OBSERVABLE PROFILES
Number of galaxies per unit area
In the form of discrete, binned profiles
Line of sight velocity dispersion
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THE PROJECTION CAUSES LOSS OF INFORMATION
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FIXED A MASS MODEL THE SOLUTIONS ARE DEGENERATE
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4. Our work
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WHAT WE DID
Jeans analysis of the Coma Cluster
THE SYSTEM IS CLOSED USING TWO ANALYTICAL MODELS
FOR THE TOTAL MASS
WE TRY TO BREAK THE DEGENERACY WITH AN ACCURATE
COMPARISON TO INDEPENDENT MASS RESULTS FROM A HOT
GAS X ANALYSIS
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WHICH TASKS WE WANT TO ACCOMPLISH

• Verify the consistency of dynamical and X
  analysis
• Investigate whether the joint tecnique is able to
  restrict the allowed mass for Coma at different
  radii (MASS CONTENT)
• Verify whether a cusp model (NFW) is prefered
  over a soft core model (AK) or not (INNER MASS
  PROFILE SHAPE)
• Obtain, at the same time, information about the
  galactic dynamics (ORBITAL ANISOTROPY)

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WHY THE COMA CLUSTER ?
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5. Data preparation procedures
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THE DATASET

Kent S.M. Gunn J.E., AJ 87(7) (1982), 945
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1. Outliers elimination
F F f

• Statistical procedure needed to eliminate the
  outliers
• Exploiting the symmetry hypothesis, we devised an
  original technique to determine the mean velocity
  BEFORE outliers rejection
• Using this mean as an estimate of Coma recession
  velocity, we perform rejection employing a
  sigma-clipping technique
• To this extent we operate upon 3 radial
  sub-groups, employing the robust statistical
  estimators of biweight to calculate centre and
  width of the velocity distribution
• We perform the clipping with 3 different cuts at
  2.5, 2.7 3 sigma, thus verifying the stability
  of the method

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Are the symmetry hypothesis and clipping
procedure well posed ?
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Some properties of the cleaned dataset
290 galaxies with
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2. Subdivision of the dataset into radial bins
We use an algorithmical procedure to bin the
dataset in a non-subjective fashion
The guiding principle is the attempt to smooth
down the fluctuations, since they dont reflect
in general the average behaviour of the
average cluster
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3. Calculation of the projected observable
profiles
a. Projected discrete 2D profiles
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4. Mass models X results
Integrating for the AK NFW profiles we
obtain the mass models
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6. Calculation results
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MASS DETERMINATION TECHNIQUE
X results guide the exploration of parameter space
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TO PERFORM NUMERICAL INTEGRATION ITS NECESSARY
TO CHOOSE A Cut Radius RC
FOR R RC WE PUT
EXTRAPOLATION ZONE
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• For a fixed mass profile the solutions may be
  influenced by
• the cut radius RC
• the fit to the dispersion profile

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I.C. Vs. Mtot(2Mpc)
I.C.
Mtot(2Mpc)
I.C. increases with increasing typical mass
of the profile employed RC has little
influence on the solutions The fit to the
dispersion profile can have an important influence
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Fixed The best fit dispersion profile RC
450 (17Mpc)
We solved the Jeans system for over 600 different
mass profiles, looking for dynamical acceptable
solutions
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RESULTS FOR THE SOFT CORE AK MODEL
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RESULTS FOR THE CUSP NFW MODELS
stripe of acceptable solutions

• There are dynamically acceptable solutions
• Some are consistent with X data at 68 c.l.

The NFW model is consistent with both the optical
X observations
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THE TOTAL MASS PROFILES
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THE GALACTIC DYNAMICS
TOTAL MASS RISES FROM RED TO BLACK
rising mass at large radii
BLU PROFILES ARE SELECTED BY DYNAMICS X
RADIAL VELOCITY DISPERSION PROFILE
Radial orbits
Radial orbits
ANISOTROPY PROFILE
Isotropy
Isotropic orbits
Tangential orbits
Circular orbits
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DYNAMICAL INDETERMINATION IN THE INTERNAL RADIAL
REGION
R
L
L
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R
R
R
L
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This central uncertainty is linked to the width
of the stripe
We showed analytically that a necessary condition
for rs and c to belong to the stripe is
Thus the central uncertainty should not be a
numerical artificial effect Instead it should
be an effect of the 2D projection, which causes
loss of information
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The consistency depends upon the global mass of
the profile
The selected (rs ,c) should be almost independent
of the particular model used
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rising Mtot(1.7 Mpc)
isomass curves for r 1.7 Mpc
The continuation of the stripe of consistent
Jeans solutions should be reproduced fairly well,
both for NFW AK models, by the black curve
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The estimate of the dispersion profile is very
important
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7. Conclusions
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Finally we remark that the most recent studies on
the subject indicate that different galactic
populations have different dynamical and
distribution properties Early type galaxies
should be a well relaxed system on nearly
isotropic orbits while late type galaxies appear
less concetrated and on more anisotropic
orbits Hence, It would be interesting to repeat
the shown analysis on a larger data compilation
including morphological type information Using
only the early type galaxies as tracers of the
total potential and determining, in a second
time, the dynamics of the late type galaxies,
would thus give a more realistic picture of the
dynamical state of the Coma Cluster
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8. Extra slides
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g
Influence of the cut radius on the solutions
RC increases from red to black
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Outliers elimination search of recession
velocity
The underlying symmetry imply that the galaxies
distribution in the (R,vP) plane must be
symmetric with respect to the cluster recession
velocity
Calibrating properly and plotting
against the window position vP we expect
to find when vP vrec
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On the input profiles cumulative approach
Cumulative number counts profile
Cumulative dispersion profile
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WE DEDUCED THE EQUATION FOR THE DISPERSION
PROFILES AS
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The solution to the previous equation is given by
the dashed line
ANYWAY, in order to employ seriously this
tecnique, one should take into account the risk
of correlations connected to the cumulative
profiels