Kupy galaxií – lekce III
Pavel Jáchym
Overview

Numerical simulations
◦
◦
◦
◦
◦



N-body
tree method
sticky particles
SPH vs. grid codes
genetic algorithms
Applications: ram pressure stripping
ICM - recapitulation
Cluster mass from gravitational lensing
Environmental effects






Morphological evolution: more spirals at z=0.5 than at z=0
(Dressler 1980)
Morphology-density relation
Fraction of blue galaxies increases with z (Butcher &
Oemler effect, 1978)
HI deficiency (Davies & Lewis 1973)
Dynamical perturbations (Rubin et al. 1999)
...
Interaction of spirals with environment

Gravitational interactions
◦ galaxy – cluster
◦ galaxy – galaxy

Ram pressure
◦ galaxy ISM – ICM
 cluster galaxies are HI deficient
by a factor 2 to 5 compared to field
galaxies

Hydrodynamical interactions
◦ viscous stripping
◦ thermal evaporation
◦ ...
Galaxy mergers


In the hierarchical CDM model, present-day galaxies are built up in a
sequence of mergers from originally small objects similar to
irregulars
The outcome of a merger between two galaxies depends on the
mass-ratio between the two objects, their intrinsic and orbital
angular momenta and their gas content
◦ mass-ratio < 1:4 does not change much the structure of the more massive
galaxy
◦ mergers between two late-type spirals may create an S0 or an elliptical
◦ mergers between an elliptical and a spiral could produce an elliptical or an S0
galaxy
◦ generally: merger between two galaxies produces a remnant of an earlier type
in the Hubble diagram
◦ the orbital angular momentum of the galaxies is absorbed into the angular momentum
of the remnant’s halo
◦ the gas quickly moves to the center of the merger remnant; it may feed nuclear BHs
◦ ULIRGs show the final stages of spiral-spiral merger with heavy star-formation taking
place
Evolution of spirals

Possible scenario for spirals transforming into S0’s
◦
◦
◦
◦
◦
◦
◦
◦

infalling spiral galaxies at z=0.5
triggering star formation
starburst (emission-line galaxies)
gas stripping by intracluster medium
post-starburst galaxies
tidal interactions heat disk
stars fade
S0 galaxies at z=0
morphological segregation proceeds hierarchically, affecting
richer and denser groups earlier. S0’s are only formed after
cluster virialization.
A note to the formation of clusters

Chandra survey of the Fornax
galaxy cluster revealed a vast,
swept-back cloud of hot gas
near the center of the cluster
◦ the hot gas cloud is moving rapidly
through a larger, less dense cloud of
gas
Shock fronts and cold fronts
typical relative velocity for merging
clusters is ~2000 km/s
 a cold clump of gas is moving through a
warmer medium

A sequence …

Clusters
T ~ 108 K
M ~ 1015 M sol

Groups
T ~ 107 K
M ~ 1013 M sol

Ellipticals
T ~ 107 K
M ~ 1012 M sol
Numerical simulations


Gravitational interactions
◦
◦
◦
◦
test particles
direct summation
tree codes
…
Hydrodynamical interactions
◦ SPH
◦ finite difference codes
Test particles, direct summation, PM method

Test particles

Direct method (particle-particle method, PP)
◦ e.g. Toomre & Toomre (1972)
◦ can be used in combination with other methods

d ri 
 vi
dt
 

N
rj  ri
d vi
  Gm j   2
dt
(| rj  ri |  2 )3 / 2
j i
softening
parameter
◦ integration of all the N particle’s equations of motion
◦ high computational requirements ~ O(N2)

PM method (particle in mesh)
◦ calculating the grav. force field on a grid of regularly spaced points
◦ the acceleration of each particles is obtained by interpolation
between nearest points of the grid
◦ PPPM methods …
Tree algorithm
Lagrangian technique
Uses direct summation to compute attraction
of close particles
 Detailed internal structure of distant groups
of particles may be ignored




many similar particle – distant-particle interactions
are replaced with a single particle-group interaction
Particles are organized into a hierarchic
structure of groups and cells resembling a
tree
◦ - e.g. oct-tree scheme (see Fig.)
◦ alternative AJP method
The influence of remote particles is obtained
by evaluating the multipole expansion of the
group
 computational cost scales as O(N logN)

Tree algorithm, cont.
once the tree is completed, information about masses, center-ofmass positions, and multipole moments are appended to each
cell
 the multipole expansion of a cell is used only if "opening"
criterion is fulfilled: d > l / θ

◦ d … distance of the cell
◦ l … size of the cell
◦ θ … opening angle

the opening criterion follows from comparison of the size of the
quadrupole term with the size of the monopole term
◦ higher-order multipoles of the gravitational field decay rapidly with respect to
the dominant monopole term
◦ it is possible to approximate the group's potential only by monopole term, or
low-order corrections for the group's internal structure can be included as well

multipole expansion of the potential:
 M 1 r Q r 
 (r )  G 

5
r
2
r


Qij   mk (3xk xk , j  rk2 ij )
k
Hydrodynamical calculations

Gasdynamics:
◦ continuity equation
◦ Euler equation
◦ energy equation
◦ + eq. of state

d
   v  0
dt
dv
P

 
dt

du
P  X (u,  )
  v 
dt


P  (  1)  u
SPH - methodology

Smoothed Particle Hydrodynamics

Lagrangian technique

an arbitrary physical field A(r) is interpolated as
A(r )   A(r ' )W (r  r ' ; h)dr '
◦ smoothing kernel function W(r;h) specifies the extent of the interpolation volume, it
has a sharp peak about r=0 and satisfies two conditions:
 W (r  r ' ; h)dr '  1

h 0
in numerical implementation values of A(r) are known only at locations of a
selected finite number of particles distributed with number density
n( r )    ( r  r j )
j

lim W (r  r ' ; h)   (r  r ' )
then
N
A(rj )
j 1
j
A(r )   m j
W ( r  r j ; h)
number of neighboring particles N is fixed during the calculation
SPH – methodology, cont.

gradient of function
A(r )   m j
j

density
A(rj )
j
W ( r  r j ; h )
 i   m jW (rij ; h)
j

Euler eq.

energy eq.

EOS
 Pi Pj
 ~
dvi
  m j  2  2   ij  iWij


dt
j
 i j

 Pi

Pj
dui 1
~
  m j  2  2   ij vi  v j   iWij


dt 2 j

i
j


Pi  (  1) i ui
Πij … artificial viscosity
Finite-difference method
Eulerian technique
approximates the solutions to differential equations using finite
difference equations to approximate derivatives
 grid-based codes
 for non-homogeneous systems – adaptive mesh refinement
hydrodynamics codes
 Ram pressure stripping simulation:


Roediger & Brüggen (2007)
ICM – recapitulation


Intra-cluster medium
◦ optically thin plasma
◦ thermal bremsstrahlung = braking radiation, free-free radiation
◦ radiation by an unbound particle (e-) due to acceleration by
another charged particle (ion)
Diffuse emission from a hot ICM is the direct manifestation of the
existence of a potential-well within which the gas is in dynamical
equilibrium with the cool baryonic matter (galaxies) and the DM
X-ray luminosity is well correlated with the cluster mass and the X-ray
emissivity is proportional to the square of the gas density
 cluster emission is thus more concentrated than the optical bidimensional galaxy distribution

X-ray observations



X-rays are absorbed by the Earth’s atmosphere
HEAO-1 X-ray Observatory was the first to provide a flux–limited
sample of X–ray identified clusters
XMM-Newton & Chandra
◦ we can map the gas distribution in nearby clusters from very deep
inside the core, at the scale of a few kpc with Chandra, up to very
close to the virial radius with XMM-Newton

We can measure basic cluster properties up to high z~1.3
◦ morphology from images,
◦ gas density radial profile,
◦ global temperature and gas mass
◦ total mass and entropy can be derived assuming isothermality
 X-ray luminosities LX~1043 – 1045 erg/s
◦ clusters are identifiable at large cosmological distances
X-ray surveys

From surveys several global observables can be derived:
◦ X-ray flux (luminosity if z known)
◦ temperature

using scaling relations, these can be related to physical
parameters, like mass, …
Scaling properties





baryonic matter follows the DM grav. potential well
it is heated by adiabatic compression during the halo mass
growth and by shocks induced by supersonic accretion or
merger events
gravity dominates the process of gas heating
when assuming that the gas is in hydrostatic eq. with DM and
bremsstrahlung dominates the emissivity
=> LX  T 2 , M gas  M vir  T 3/ 2
however, from observations:
◦ the luminosity-temperature relation is steeper (α=2.5-3 or even more in
groups)
◦ also the relation between the gas mass and T is steeper (α=1.7-2)
◦ this indicates that non-gravitational processes (SN, AGN feedbacks,
radiative cooling, winds, etc.) take place during the cluster formation
and left an imprint on its X-ray properties
ICM

typical cluster spectrum
◦ continuum emission dominated by thermal
bremsst.
◦ main contribution from H and He
◦ emissivity of the continuum
 sensitive to temperature for energies > kT
 rather insensitive for energies lower
◦ iron K-line complex at 6.7 eV
◦ intensity of other lines decreases with
increasing T
shape of the spectrum determines T
 its normalisation then density
 ICM is not strictly isothermal – T from an
isothermal fit is a „mean“ value
 metallicity evolution:

Cooling of ICM

cooling timescale
◦
◦
◦
◦
cooling function Λc(T)
tcool = kT / nΛc(T) > 1010 yr (n/10-3 cm-3)-1 (T/108 K)1/2
in central cluster regions it can be shorter than the age of the Universe
in fairly relaxed clusters, the decrease of the ICM temp. in the central
regions has been recognized
◦ cooling flows
◦ supernovae or AGNs as possible feedback mechanisms providing an
adequate amount of extra energy to balance overcooling

three ways how an electron can get rid of energy
◦ collisional cooling – very efficient but not for completely ionized ICM
◦ Recombination – probability ~ 1/velocity => unimportant for ICM
◦ free-free interaction = thermal bremsstrahlung
Cooling, cont.
about 1000 Msol/yr of gas can cool out of the X-Ray halo
 this gas could form stars – some cD galaxies show filaments of gas
emission and blue colors in the central region
 others do not show the lower central temperatures that would be
expected if cooling was efficient
 presumably, cooling will lead to enhanced accretion of gas onto the black
hole in the cD galaxy

◦ it in turn may become active and provide high energy particles to heat the gas
◦ thus, a quasi equilibrium may be established that prevents the gas from ever
forming stars

still under debate …
Cooling, cont.



Mixing via turbulence could counteract
cooling towards the outside
Central AGNs can produce relativistic jets
which directly inject energy into the ICM
and may cause shocks. Jets also inflate
bubbles, which rise buoyantly, pushing
colder gas upwards out of the core.
Acoustic waves produced by AGN outbursts
can also transfer energy to the ICM if it is
viscous enough
Characteristic time-scales

Mean free path for the ions and electrons of the ICM
T
8

2
/ 10 K
fp  23 kpc
nICM / 103 cm 3

◦ is << cluster size
◦ ICM can be described by fluid dynamics …

ICM

Timescale for pressure equilibrium
◦ if a region of gas undergoes a compression, how long does it take for
the pressure wave to propagate across the cluster?
1/ 2
 P

cs  
  
1/ 2
  kT 

 
  mH 
 T 
8
tp  D / cs  6.5 10 yr  8 
 10 
1/ 2
 D 


 Mpc 
◦ this is short compared to Hubble time – we can assume that the gas is
in pressure eq.
Characteristic time-scales, cont.

Timescale for cooling
1/ 2

 T   n
tcool  2.2 1010 yr  8   -3 e -3 
 10   10 cm 
◦ due to thermal emission
◦ longer than a Hubble time
◦ => hot gas stays hot!


Crossing time
Relaxation time
◦ time-change to equil.
tcross 
D
 cl
9
~ 10 yr
t relax  6 108

D / Mpc 
cl
/1000 kms -1

T / 10 K 
yr
n/10 cm 
8
-3
3/ 2
-3

1
Cluster mass estimates


From the virial theorem
From X-ray data
◦ hydrostatic equilibrium condition
kT
M (r )   B
G mp
 d ln gas d ln T 

r 

d ln r 
 d ln r
 μ ... mean molecular weight (~0.59 for primordial composition)
◦ distribution of ICM
 gas (r ) 
0
1  r / r  
2 3 / 2
c
 β ... ratio between the kinetic energy of any tracer of the grav. potential and the
thermal energy of the gas

From gravitational lensing
Mass estimate from gravitational lensing
clusters act as grav. lenses on more distant galaxies
one of the most important methods for the mass
determination of galaxy clusters
 the only method working for non-equilibrium systems!
 There are two rather different regimes


◦ Strong lensing
 background galaxies are strongly distorted
 good for massive clusters
◦ Weak lensing
 background galaxies only slightly distorted
 For a symmetric potential, the galaxies are elongated slightly in the
axial direction
 This is a "shearing" effect and only reveals the gradient in the
potential, not its integrated depth
 By measuring thousands of faint galaxy images, the effect is identified
statistically.

shape and radial trend of the weak shear and strong lensing
effects yield the cluster mass distribution independent of the
nature of the mass and therefore allow reliable total mass
estimates including the dark matter
Mass from lensing
v
2GM


 Newtonian deflection angle:  
c
 c2
4GM
 In general relativity:  
 c2


D

 ( )  ds  ( )
thin lens approximation:
Ds
lens equation:      ( )
ˆ
Dds 4GM
Dd Ds c 2
 for β=0:
4GM Dds
◦ angular radius of Einstein ring:  
E
2
c
Dd Ds
 critical surface density:
2
Ds
c
◦ lensing occurs when Σ> Σcrit  crit 
4 G Dd Dds

 ( )   


convergence:   

crit


Jacobian matrix of lens eq.: A    1     1
   1   2 … complex shear


2
α
2


1     1 
Since κ and γ are derived from the same potential, it is possible to determine the surface mass density.
Results of lensing

Comparison between the ICM
temperatures inferred by the
fitting of isothermal profiles to
the shear data, and from X-ray
measurements.

Comparison between the
velocity dispersion found by
fitting isothermal profiles to the
shear data and those estimated
through spectroscopic
measurements.