Download Galaxy Formation Leading questions for today • How do visible

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Galaxy Formation
Leading questions for today
• How do visible galaxies form inside halos ?
• Why do galaxies/halos merge so easily ?
How do visible galaxies form inside halos ?
gas
density fluctuations and gravity produce:
dark matter halos
• halos much bigger than visible part of galaxy
• halos rotate slowly < v > /σ ≈ 0.3
dark matter
Halos entirely UNLIKE visible galaxies
So what happens ?
When new halo just formed: gas is distributed like
dark matter
• gas supported by pressure
• dark matter supported by random motions
Gas can cool by radiation, and can collapse to the
center if cooling efficient
Dark matter cannot cool ! Will not collapse to center
Gas cooling is expressed as:
cooling rate = n2 Λ(T )
cooling rate is cooling per unit volume element
n is number density of gas
Λ(T ) is cooling function
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Draw figure of nlum , the number density in luminous
material (in units of particles/cm3 ), versus temperature T .
How do tcool and tdyn depend on density and temperature ?
cooling time ∝
thermal energy/volume
coolingrate/volume
√
dynamical time ∝ 1/ nlum
∝
nlum T
n2lum Λ(T )
Hence the dynamical time and cooling time are equal
when nlum ∝ T 2 /Λ2 (T )
Cooling mechanisms: bound-bound, bound-free, freefree, electron scattering. The peaks in the cooling
function are caused by:
• T=104 = ionization/recombination hydrogen
• T=105 = ionization/recombination helium
T> 106 K : thermal bremsstrahlung and Compton
scattering
Now take a halo with gas inside. Two options:
• tcool < tdyn , then cooling is efficient, and the gas
will collect in the center
• tcool > tdyn , then cooling is inefficient, and the
gas will NOT collect in the center
Now draw the line of tcool = tdyn , and put in astronomical objects:
• galaxies
• groups and clusters of galaxies
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Homework assignment
1/2
1) prove M ∝ T 3/2 /nlum . Use σ 2 ∝ T and nlum ∝
M/R3 .
We find
• gas in galaxies cools efficiently
• gas in clusters does not !
when galaxies cool, the density will increase, while T
remains roughly constant.
When the gas becomes self gravitating, the T might
increase (the circular velocity will increase, hence
the temperature ∝ vc2 )
when the gas is self-gravitating, it can start to form
stars !
As a result, the luminous galaxy forms in the
center of the halo, and is much smaller than
the dark halo
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How to get disks in spiral galaxies?
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Hence this occurs when
0.3f =
Gas cools down and contracts.
Assume that the specific angular momentum j is conserved
j=
J
= rstart < v >= 0.3rstart σdm
m
Gas contracts by factor f :
f=
rstart
rend
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√
2
or
f =5
Hence, a cooling gas cloud will settle into a disk
when it has contracted by about a factor of 5 or
so.
Ellipticals and bulges can be made out of pre-existing
disks when galaxies merge, or during mergers
As specific angular momentum is conserved:
jend = rend < vend >= jstart = 0.3rstart σdm
Hence
< vend
rstart
σdm
>= 0.3
rend
= 0.3f σdm
The gas rotates faster and faster while the gas contracts
contraction is halted when the cooled gas is in circular orbits around the center. For such orbits
√
v = vc = 2 σdm
Homework assignment:
2) Our Milky Way galaxy has a mass which is typical
for galaxies. It likely has an isothermal dark matter
halo. Calculate the temperature of isothermal gas in
the halo of our galaxy if it has a density proportional
to r−2 , where r is the radius from the center.
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Why do halos/ galaxies merge so easily ?
BT 7.1 abbreviated
• dynamical friction slows down two galaxies meeting each other
• even galaxies with hyperbolic orbits can merge !
Dynamical friction
consider massive point mass M , moving in sea of
small point masses m
• the motion of the massive object will produce a
wake behind it
• the wake is an overdensity, and will exert a force
which slows down the moving massive object
• the wake is proportional to GM , hence the gravitational force on the object is proportional to
G2 M 2
The motion of the pointmass and the massive object
is simply given by the solution for the two-body problem.
m is the mass of small pointmass.
M is the mass of big pointmass.
v0 is velocity difference at the start. ~v0 = ~vm − ~vM
b is impact parameter.
The deviations in velocity are
−1
2mbv03
b2 v04
|∆vM,⊥ | =
1+ 2
G(M + m)2
G (M + m)2
−1
b2 v04
2mv0
1+ 2
|∆vM,|| | =
(M + m)
G (M + m)2
the parallel deviation ∆vM,|| will always be in the
same direction, whereas the perpendicular component will be in a random direction.
the friction can be estimated effectively by considering individual interactions between the massive
object, and the small point masses
Hence, as a result, when the contributions from all
particles are taken (by integration), the perpendicular component can be ignored to first order, and
the parallel component will dominate.
This will slow down the moving mass M .
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Assume that phase-space density of stars is f (~v ). M
encounters stars with velocities in volume d~vm at
impact parameters between b and b + db
rate = 2πb db × v0 × f (vm )d~vm
Integrate over impact parameters b, then the change
in ~vM is
Z bmax
d~vM
2mv0
= ~v0 f (~vm )d~vm
×
dt
M
+
m
0
−1
b2 v04
1+ 2
2πbdb
G (M + m)2
where bmax is the largest impact parameter that
needs to be considered.
Now calculate the integral, using ~v0 = ~vm − ~vM . We
get:
(~vm − ~vM )
d~vM
= 2π ln(1+Λ2 )G2 m(M +m)f (~vm )d~vm
dt
|~vm − ~vM |3
where
bmax v02
Λ≡
G(M + m)
Usually Λ very large. Globular cluster in Milky Way:
M = 106 , v0 ≈ 100 km/s. Λ = 4.6 103 , ln(1+Λ2 ) ≈
17. In the following we use ln(1 + Λ2 ) ≈ 2 ln Λ. We
ignore the variations of Λ with the velocities vm of
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the background stars. We assume that the galaxy is
isotropic.
The resulting integral is valid for encounters with
stars in volume element d~vm . We have to integrate
over all ~vm to get the total force on M . Notice that
the force depends on ~vm only through the last 3
terms at the right hand side of the equation. Hence
we have to integrate these 3 terms over ~vm :
Z
(~vm − ~vM )
f (~vm )
d~vm
|~vm − ~vM |3
The equation simplifies if the distribution function is
isotropic. Notice that the equation is equivalent to
the equation for the gravitational force at a location
r = vM for a density denstribution ρ = f . Newtons
theorem states that this is equivalent to the force
caused by a pointmass M with mass equivalent to
the mass inside r.
Hence
Z
(~vm − ~vM )
d~vm =
f (~vm )
|~vm − ~vM |3
R vM
0
2
f (vm )4πvm
dvm
~vM
3
vM
Use ln(1 + Λ2 ) ≈ 2 ln Λ to obtain
d~vM
= −16π 2 ln(Λ)G2 m(M +m)
dt
Equation
R vM
0
2
f (vm )vm
dvm
~vM
3
vM
(1)
Only stars moving slower than vM contribute to the
force. The drag always opposes the motion.
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This is the CHANDRASEKHAR DYNAMICAL FRICTION FORMULA (BT 7.1)
for very large vM , one derives approximately
dvM
4π ln(Λ)G2 (M + m)ρm
=−
2
dt
vM
Two interesting points:
• the drag depends on ρm , but not m itself
• the larger M , the stonger the deceleration
massive objects fall in faster than low-mass objects
Dynamical friction for object with mass M in
isothermal halo
For mass M in orbit in isothermal halo one derives
the following. The density in the halo is given by ρ =
σ 2 /(2πGr2 ) = vc2 /(4πGr2 ).
Fill this in the equation 1 above to derive:
GM
dvM
= −0.428 ln Λ 2
dt
r
Application to Magellanic Clouds (BT 7.1
1(b)
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The force causes the mass M to lose angular momentum per unit mass L
dL
Fr
GM
=
= −0.428
ln Λ
dt
M
r
At all times L = rvc , and vc is constant. Hence
r
dr
GM
= −0.428
ln Λ
dt
vc
Solution
r(t)2 = −2 ∗ 0.428
GM
ln Λt + constant
vc
Hence, if r(0) = ri , the mass reaches r = 0 at
tf ric = 1.17ri2 vc /(GM ln Λ)
For the clouds this gives:
tf ric
1 × 1010
=
ln Λ
r
60kpc
2 vc
220km/s
2 × 1010
Mtot
approximately: ln Λ = 3
Hence they will spiral inwards quickly, in a couple of
Gyr !
This mechanism makes galaxies merge well !!
The LMC and SMC have distances of 50 and 63 kpc,
and masses of 2e10 and 2e9 solar masses.
If our galaxy has a massive halo extending to the
clouds, then the friction decay time can be estimated
in the following way:
Simulations indicate that the orbits may be elliptical. Since the Clouds are close to peri-galacticon,
the average friction time may be higher than the
previous estimate.
yr.
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Homework assignment
The Magellanic Stream (BM 8.4.1 page 530) is gas
that escaped from the clouds during their orbit
around the galaxy.
The stream can be found over 60 degrees in the
Southern Sky
The gas distribution of the magellanic stream and the magellanic clouds
magellanic stream projected on the sky
3) Calculate the dynamical friction timescale for a
globular cluster with mass 1e6 M⊙ at a radius of
10kpc from the center of the galaxy.
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Mergers of galaxies
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(BT 7.4 1(a,b))
Theories of galaxy formation predict that halos grow
by the mergers of lower mass halos. What happens
to the galaxies inside these halos ? Simulations can
give us the answer. Below is an example.
Compare to observations
Merging galaxies quickly form 1 large galaxy.
Structure of mergers of non-rotating galaxies
• generally slowly rotating, oblate
• density ρ ∝ r
−3
, or r
1/4
Radial structure
profiles - like ellipticals
• radial gradients not erased:
material that was in center in progenitors is
likely in center of merger remnant
A “standard” example is NGC 7252, which has low
luminosity tails, and a luminous, central system.
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• Conclusion: after the merger is over, it will look
very much like a normal elliptical !
How to recognize mergers ?
BT 7.4 3
Arp’s “Atlas of Peculiar Galaxies” has many merger
remnants. Examples
the radial surface brightness can be measured by averaging over circles
• The photometry follows an r1/4 law very accurately.
These were first thought to be exploding.
Now we know they are mergers.
Very “typical” mergers “the Antennae”
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BT 7.4 3(a) Stellar populations in mergers
typical is:
• two fuzzy centers
• from each “center”, a long, curved, tail
Toomre and Toomre (1972) showed that this can
exactly be produced in a merger
We can predict the velocities along the tail, and compare to observations. Usually good agreement.
In a plot of U-B against B-V, mergers generally lie
away from the relation for normal galaxies. This is
caused by bursts of starformation which happened
recently.
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How are these tails made in mergers ?
Mergers of rotating galaxies
(BT 7.4 2)
result depends on sense of rotation of disk:
• prograde: disk angular momentum pointing in
same direction as that of orbit.
• retrograde: disk angular momentum pointing in
opposite direction as that of orbit.
Tails are formed by tidal forces. Each disk can produce 1 tail.
Why is this ? Model the merger as the encounter of
two pointmasses, one with a set of rings around it
• retrograde: the pointmasses in the ring move opposite to that of the “merger” galaxy. They do
not react very strongly, because sense of rotation
opposite.
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see more detail in a “slow-motion” picture below:
The particles in the ring move with comparable angular speed as the “intruder” galaxy. The system is a
forced oscillator, with the frequency of the force term
comparable to the natural frequency of the oscillator.
Hence a resonance occurs, and a very strong reaction
follows.
The resulting merger remnants have the following
features:
• prograde: the pointmasses in the ring move in
the same sense as that of the “merger” galaxy.
A very strong reaction follows
• they look like featureless ellipticals , with de Vaucouleurs profiles
• merger remnants are slowly rotating if initial total angular momentum low
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Frequency of merging
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BT 7.4 3(b)
How efficient is merging ?
assume two equal mass galaxies, with orbits of certain energy E and angular momentum L
calculate numerically whether they merge or not, and
plot in diagram
Mergers are very efficient, even galaxies on hyperbolic
orbits can merge !
• about 11 out of 4000 NGC galaxies are typical
“mergers”
• the timescale for this merger phase is typically
5 × 108 h−1 yr.
• if the merger rate is constant, then for every observed merger now, 1010 /(5 × 108 )=20 happened
in the past. past.
• in total, 11*20 galaxies in NGC catalogue may
have undergone merging, or ≈ 200.
• there are 440 ellipticals in the NGC catalogue comparable to the expected number of mergers
• all ellipticals may have formed through mergers if
merging rate increases towards the past !
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Ripples
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BT 7.4 3(c)
Some ellipticals show ripples in the outer parts (“MalinCarter shells”).
They can also form from “major mergers”, but a cold
component is needed in both cases
• either cold disk, or cold spherical galaxy
Homework assignment
4) We can distinguish between elliptical galaxies,
which have no disk, and spiral galaxies, which have
a disk. If we see a merger remnant with 1 tail, what
kind of galaxies were involved in the merger (ellipticalelliptical, elliptical-spiral, or spiral-spiral). Justify your
answer. What kind of merger produces two tails ?
5) Try to find the equal mass merger with 2 nuclei
which is closest to us.
These can form from a small galaxy falling into a
much bigger one: