Download Measuring the masses of clusters

Measuring the masses of clusters:
•! One way is to measure the velocities of
galaxies in cluster
•! Assuming the cluster is “virialized”,
–! 2PE + KE = 0
–! KE = ! ! mivi2 = ! M<v2> = 3/2 M"2
•! M = mass of cluster
•! " = measured radial velocity dispersion
•! <v2> = 3"2
–! PE = -3/5 GM2/<R>, for a uniform spherical
distribution
–! M = 5<R>"2 / G
Virial masses of clusters:
Measuring the masses of clusters:
•! Zwicky was the first one to do this (in 1933!) for the
Coma cluster, observing only 8 galaxies
•! Found M> 5 " 1014 M!
•! Calculated mass-to-light ratio and determined that
about 90% of the mass necessary to account for
observed ratio was missing and therefore invisible, or
"dark".
But nobody believed him …
Hot x-ray gas:
•! As we have seen, clusters are full of hot x-ray gas
•! Poor clusters
–! " ~ 500-800 km/s
–! D ~ 1-3 Mpc
–! M ~ 1014 h-1 M!
•! Rich clusters
–!
–!
–!
–!
" > 800 km/s
D ~ a few Mpc
M ~ 1015 h-1 M!
Mass to light ratio of the cluster ~ several hundred M!/L!
•! Even more dark matter! It’s everywhere ….
–! T ~ 107 – 108 K, emission is from free-free emission (as in groups,
but hotter)
–! In fact, many distant clusters are now being discovered via x-ray
surveys (such as the ROSAT survey)
–! Temperatures are not uniform, we see patches of “hot spots” which
are not obviously associated with galaxies. May have been heated
as smaller galaxies (or clumps of galaxies) fell into the cluster
–! In densest regions, gas may cool and sink toward the cluster center
as a “cooling flow”
–! Unlikely that all of it has escaped from galaxies, some must be
around from cluster formation process. It is heated via shocks as
the gas falls into the cluster potential
–! But some metals, must be from stars in galaxies
–! X-ray luminosity correlates with cluster classification, regular
clusters have high x-ray luminosity, irregular clusters have low x-ray
luminosity
Masses of clusters from x-ray gas:
•! If we assume the cluster is in hydrostatic equilibrium,
#=density of the gas, P=pressure
dP
GM(r)
dr
="
r2
#
–! From the ideal gas law:
9 x 107 K
# " &
P =%
(kT
$ µ mH '
!
where µ= mean molecular weight (~0.6 for an ionized plasma), mH =
mass of the hydrogen atom, k=Boltzman constant
dP " k %" d(
dT % " GM(r) %
=$
+ ( ' = )$
'(
'$ T
dr # µ m H &# dr
dr & # r 2 &
!
X-ray spectrum of Coma, Henriksen & Mushotsky (1986)
Masses of clusters from x-ray gas:
dP " k %" d(
dT % " GM(r) %
=$
+ ( ' = )$
'(
'$ T
#
dr # µ m H & dr
dr & # r 2 &
!
Masses of clusters from x-ray gas:
•! We have:
Factor out T and #, multiply by r
# kT" &# r d" r dT & # GM(r) &
+
(" * r
%
(%
( = )%
2
$ µm H '$ " dr T dr ' $ r '
!
Then
!
kT # d ln " d lnT &
M(r )
+
%
(= )
µ mHG $ d ln r d ln R '
r
Solving for M( r)
M(r) = "
!
!
kT $ d ln # d lnT '
+
&
)r
µm H G % d ln r d ln R (
If the gas is isothermal,
Finally,
!
d lnT
=0
d ln r
kT $ d ln # '
M(r) = "
&
)r
µm H G % d ln r (
!
M(r) = "
kT $ d ln # '
&
)r
µm H G % d ln r (
•! So if we know dln#/dlnr, we can measure the mass distribution of
cluster
•! If the
! cluster is spherically symmetric this can be derived from xray intensity and spectral observations
•! For Virgo,
–! Find M(r<1.8Mpc)~1.5 – 5.5 x 1014M!
–! But the gaseous mass (from x-ray luminosity) is only ~ 4 – 5.5 x
1013M!
–! Typical cluster masses:
•!
•!
•!
•!
Total: 5 x 1014 M! to 5 x 1015 M!
Luminous mass: ~5%
Gaseous mass: 10-30%
Dark matter: 60-85% !!!!!
HI Gas Deficiency
•! Evidence for stripping of gas in cluster spirals has
been found from HI measurements
•! Most deficient spirals are found in cluster cores
(another environmental dependency)
•! HI deficiency also correlates with X-ray luminosity
(which correlates with cluster richness)
•! It is the outer disks of the spirals that are missing
HI Deficiency in 3 clusters
Giovanelli & Haynes 1985
HI Deficiency vs X-ray luminosity
HI map of Virgo
Butcher-Oemler Effect
•! In 1978, Butcher & Oemler found that the fraction of blue
galaxies in two clusters at z=0.4 was significantly higher than in
Coma
•! This was later extended to larger samples of clusters, and to
higher redshifts
•! Star formation is decreasing rapidly in clusters as a function of
redshift (why??)
•! HST allowed us to push to higher redshifts and to study the
morphologies of these high-redshift clusters
•! Clusters are a great laboratory for studying galaxy evolution!!
Fraction of
Blue galaxies
redshift
Butcher & Oemler (1984)
CL0939
z=0.41
CL0939
z=0.41
CL0939
z=0.41
RBS797
z=0.35
CL1252
z=1.24
Blakeslee et al.
2003 w/ACS
Ram Pressure Stripping:
CL1252
z=1.24
Rosati et al.
2004
Consider a spiral galaxy moving through a cluster. As it interacts
with the intracluster gas, gas from the spiral will be swept out.
this phenomenon is called ram pressure stripping. This phenomenon
is probably the cause of the observed HI gas deficiency in spirals
in clusters and from the transformations of spirals to SO’s in cluster
environments.
What is the mass density in the intracluster gas required
for this to occur?
Let "* = surface mass density of the spiral
"gas= surface mass density of gas in the spiral
#x = density of x-ray gas
What is the mass density in the intracluster gas required
for this to occur?
Let "* = surface mass density of the spiral
"gas= surface mass density of gas in the spiral
#x = density of x-ray gas
Then the force per unit mass
required to lift a particle from the
surface of the galaxy to infinity is:
Now consider the gas in the spiral disk. The mass of gas contained
within a surface area dA is related to the surface density by
dm= "gasdA. Substituting into our previous equation, we find that
the pressure required to remove gas from the disk is:
The ram pressure of the x-ray gas is related (from fluid
mechanics) to its density and velocity by:
Thus the galaxy gas will be stripped when:
This is the Gunn-Gott criterion (first derived in 1972)
An example:
•! Consider the Milky Way in a cluster environment:
–! 2x1011 stars in a disk with R~20 kpc
–! Assuming a typical star is 1M!, the stellar surface density is "* =
2x1011/$R2 ~0.03 gm/cm2
–! The surface density of the gas is "gas ~6x1020 atoms/cm2
–! Assuming a typical speed for a galaxy moving through a cluster of
v~1300 km/s, we find an x-ray gas density required for stripping of
#x ~ 5x10-4 atoms/cm3
•! This isn’t much! Ram pressure stripping is probably common in
cluster environments, especially in the dense inner regions.
•! But detailed numerical models show that ram pressure stripping
alone isn’t enough to explain the truncation in star formation
implied by the Butcher-Oemler effect. Probably is the main
cause of the transformation of spirals to S0’s.
•! Note we ignored the dark matter halo but this is only important at
large radii (>20kpc)
Ram pressure stripping
Field
Virgo DSS image
Intracluster Light:
Selection
•! Zwicky (again!) in 1951 first noted “an extended mass of
luminous intergalactic matter of very low surface brightness" in
Coma
•! Confirmed in 1998 by Gregg & West, features are extremely low
surface brightness >27 mag per arcsec2 in R
•! Also discoveries (here at Penn State!) of intracluster red giant
stars and intracluster planetary nebulae in Virgo & Fornax,
~10-20% of the cluster light
•! Probably caused by galaxy-galaxy or galaxy-cluster potential
tidal interactions
M86
M84
–! This is called galaxy harassment
M87
Where to observe?
Diffuse intracluster
light everywhere!
Field Selection
Looking for intracluster
Williams et al (2007) @PSU
Fieldstars,
Selection
Mihos et al. (2005)
Mihos et al. (2005)
M86
200 kpc
M87
200 kpc
Field Selection
The Intracluster Stars
There are lots of them!
Galaxy Harassment:
•! First proposed by Moore et al. in 1996 as frequent high speed
galaxy encounters driving the morphological transformation of
galaxies in clusters
•! Galaxies will interact with both other galaxies in the cluster and
with the cluster potential
•! Galaxy harassment will transform Sc/Sd/dIrr galaxies into dE or
dSph galaxies over the timescale of several billion years (thus
several such encounters are expected to occur between z=0.4
and today)
•! Tidal heating of disks may form S0’s
•! Tidally stripped debris is probably the source of intracluster stars
Abell 3627
Galaxy harassment
Galaxy harassment
Galaxy harassment
Dubinski 1998
4 clusters @ z~0.4
Oemler et al. 1997
MS1054 @ z=0.83
Van Dokkum et al. 2000
Post-starburst galaxies:
•! There is a significant population of poststarburst galaxies in distant clusters (~20%),
these have k+a (or E+A) spectrum, showing
both the features of a k-star (typical E
spectrum) plus the strong Balmer absorption
lines of an A star
•! This would only be seen in a galaxy that was
forming stars in the recent past (<1.5 Gyr) but
the star formation was truncated
•! This is probably related to the conversion of S
to S0 galaxies (morphology density) and the
Butcher-Oemler effect
Morphology-Density Relation (Dressler et al. 1997)
9 clusters @ z~0.4
Fraction of
Blue galaxies
redshift
Morphology-Density Relation (Dressler et al. 1997)
Butcher & Oemler (1984)
Color magnitude diagram for CL0939, z~0.41
Postman et al 2007
Postman et al. 2005
Elliptical
S0
E/SO
fraction as a function
of lookback time
merger
Spiral
Oemler et al. 1997
Evolution of spirals in cluster environments:
Cluster spectral types
Dressler et al. 1999
•! Possible scenario for spirals transforming into S0’s
–!
–!
–!
–!
–!
–!
–!
–!
–!
Infalling spiral galaxies @ z~0.5
Triggering star formation
Starburst (emission-line galaxies)
Gas is stripped by intracluster medium
Post-starburst galaxies
Tidal interactions heat disk
Stars fade
S0’s @ z~0
Morphological segregation proceeds hierarchically, affecting richer,
denser groups earlier. S0’s are only formed after cluster
virialization.
•! But how do we get S0’s in group environments??
Gravitational lensing:
Mass bending spacetime
•! We know from general relativity that mass bends light
•! Chowlson (1924) and Einstein (1936) predicted that if
a background object is directly aligned with a point
source mass, the light rays will be deflected into an
“Einstein Ring”
•! In 1937, Zwicky (again!) predicted that one could
study the mass distribution (dark matter) in clusters by
studying background galaxies that are “lensed” by the
dark matter in the cluster
•! Turns out he was right (again!), but this was not
observationally feasible until the mid-1990’s for
clusters
•! Gravitational lensing first observed in quasars in 1979
Einstein cross
Einstein ring
Gravitational lens simulation
Gravitational lensing
Consider a photon moving past a
point mass M. From classical Newtonian
Gravity, the photon will under go an acceleration
perpendicular to the direction of it’s motion.
The amount of deflection can be calculated by
integrating along the path. Substituting dx=cdt:
This integral is analytic (exercise for the reader!)
and works out to 2/%2. Thus:
and the Newtonian deflection angle is
Gravitational lensing
Gravitational lensing
In practice, &, ', ( are very small, so we can
use the small angle approximation
In general relativity, gravity affects both the
spatial and time component of the photon’s path,
From the law of sines on the triangle OSI
This is one big geometry problem. Let’s
define some terms:
sin('-&) ~ '-& and sin(180-()=sin( ~ (
We already know that the deflection angle is:
and sin' = %/Dd ~ '
Note that &, ', (, and % are vectors, so must
be added vectorially. Not important for the point-source case,
but is important when modeling complex systems (like a cluster!)
Gravitational lensing
Gravitational lensing
When & = 0, or the source is directly behind the lens, we have
an Einstein ring at an angle:
4GM
Dds
"E =
#
c2
DsDd
We can also define a critical density,
Putting it together:
$ c 2 ' $ Ds '
" crit = &
)
)*&
% 4 # G ( &% Dd Dd s )(
!
such that lensing occurs when the surface density,
M(< " E )
> $ crit
#"2
!
The geometry of a non-point source lensing mass is much
more complex, but can be modeled to determine the mass distribution
of the cluster. In general arcs (due to the non-symmetry of the
!
cluster mass distribution) are formed near the critical curve
defined by ' = 'E , when & is small. The enclosed mass can be
measured by the radius of the arcs.
Gravitational lensing preserves surface brighntess,
but changes the solid angle subtended by a source. Thus, the total
flux is magnified. The magnification is given by the ratio of the
areas of the source and of the lensed image:
magnification = '/& d'/d&
Radial caustics (images pulled out into a radial line) occur
when &= &r when d&/d' -> 0
Stretching of images both tangentially and radially
Induces a shear, round objects become elliptical. We see
arclets and shear in images of massive clusters. These
can also be used to measure the cluster mass out to
larger radii.
Fort & Mellier
1994
Zooming in on Abell 1689
z=5.58 or 12.6 billion light years
Galaxy number
density
Light distribution
Abell 2218
Squires et al. 1996
Shear map
Probable z~7.6 galaxy or 12.8 billion light
years, Bradley et al., 2008
Mass distribution
Cluster masses from gravitational lensing:
•! Strong lensing constraints:
Abell 2218
Squires et al. 1996
–!
–!
–!
–!
A370
A2390
MS2137
A2218
M~5x101 3h-1
M~8x1013 h-1
M~3x1013 h-1
M~1.4x1014 h-1
M/L ~270h
M/L ~240h
M/L ~500h
M/L ~360h
•! Weak lensing constraints (a subset):
–!
–!
–!
–!
–!
–!
–!
MS1224
A1689
CL1455
A2218
CL0016
A851
A2163
M/L ~800h
M/L ~400h
M/L ~520h
M/L ~310h
M/L ~180h
M/L ~200h
M/L ~300h
Yep, lots o’ dark matter!