Download 3. Galactic Dynamics handout 3 Aim: understand equilibrium of

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3. Galactic Dynamics
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handout 3
Solution: assume ensemble of point masses, moving
under influence of their own gravity
Aim: understand equilibrium of galaxies
Point masses can be: stars, mini BHs, planets, very
light elementary particles ...
1.
2.
3.
4.
What are the dominant forces ?
Can we define some kind of equilibrium?
What are the relevant timescales?
Do galaxies evolve along a sequence of equilibria ?
[like stars] ?
equilibrium of stars as comparison:
• gas pressure versus gravity
isotropic pressure at a given point
• hydrostatic equilibrium, thermal equilibrium
• timescales corresponding to the above, plus nuclear burning
• structure completely determined by mass plus
age (and metallicity)
very little influence from “initial conditions”
galaxies:
Much harder: what are galaxies are made of ?
Questions addressed in this and the next handout:
• What are equations of motion ?
• Do stars collide ?
• Use virial theorem to estimate masses, binding energy, etc.
• How many equilibrium solutions for each mass ?
• What are the relevant time scales ?
dynamical timescale, particle interaction timescale
Basic reference:
Binney & Tremaine (1987): Galactic Dynamics (BT)
chapters 1 to 1.1; 2 selected parts; 3 selected parts, 4
selected parts
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3.1 Equations of motion
assume a collection of masses mi at location xi , and
assume gravitational interaction. Hence the force F~i
on particle i is given by
X ~xj − ~xi
d2
~
Gmi mj
Fi = mi 2 ~xi =
dt
|~xj − ~xi |3
j6=i
• Only analytic solution for 2 point masses
• “Easy” to solve numerically (brute force) - but
slow for 1011 particles [see http://astrogrape.org/]
• Analytic approximations necessary for a better understanding of solution
In the following we investigate properties of gravitational systems without explicitly solving the equations
of motion.
3.2 Do stars collide ?
(BT 1 to 1.1)
Is it safe to ignore non-gravitational interactions ?
Calculate number of collisions between t1 and t1 + dt
of star coming in from the left, in galaxy with homogeneous number density n.
• incoming star has velocity v
• suppose all stars have radius r∗ .
• all stars in volume V1 will cause collision −>
V1 = π(2r∗ )2 × v × dt
• number of stars in V1 : N1 = nV1
• number of collisions per unit time = 4πr∗2 nv
Typical values:
r⊙ = 7 × 1010 cm
n = 1010 /[3kpc]3 = 1.3 × 10−56 cm−3
v = 200 km/sec= 2 × 107 cm/sec
which gives a collision rate of 1.6×10−26 sec−1 =
5×10−19 yr−1 −> very rare indeed
Hence we can ignore these collisions without too much
trouble.
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3.3 Virial Theorem:
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(not in BT)
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Since
= F~i we have
X
1 d2 I
F~i ~xi + 2K.
=
2 dt2
i
relation for global properties: Kinetic Energy and Potential energy.
Again consider our system of point masses mi with
positions ~xi .
P
Construct i p~i ~xi and differentiate w.r.t. time:
d~
pi
dt
Now assume the galaxy is ’quasi-static’, i.e. its prop2
erties change only slowly so that ddt2I = 0. Then the
equation above implies
K=−
d X d~xi
d X1 d
d X
p~i ~xi =
mi
~xi =
(mi x2i )
dt i
dt i
dt
dt i 2 dt
where I =
P
i
1 d2 I
=
2 dt2
.
X d~
X d~xi
d X
pi
p~i ~xi =
~xi +
p~i
dt i
dt
dt
i
i
Then
X
i
X
d~xi
=
mi~vi2 = 2K
p~i
dt
i
with K the kinetic energy.
1X~
Fi ~xi
2 i
Then assume that the force F~i can be written as a
summation over ’pairwise forces’ F~ij
F~i =
mi x2i , which is the moment of inertia.
However, we can also write:
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X
F~ij
j,j6=i
Now realize that
X
i
F~i ~xi =
XX
i
F~ij ~xi
j6=i
This summation can be rewritten. We sum the terms
over the full area 0 < i ≤ N , 0 < j ≤ N , i 6= j.
However, we can also limit the summation over just
half this area: 0 < i ≤ N , i < j ≤ N , and add
the (j, i) term explicitly to the (i, j) term within the
summation.
Hence instead of summing over F~ij ~xi , we sum over
F~ij ~xi + F~ji ~xj
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Hence
XX
i
F~ij ~xi =
XX
i
j6=i
i
(F~ij ~xi + F~ji ~xj )
j>i
j>i
Gmi mj ~
xi −~
xj
For gravitational force F~ij = − |~xi −~
xj |2 |~
xi −~
xj |
X
F~i ~xi = −
i
X X Gmi mj
(~xi − ~xj )(~xi − ~xj )
3
|~
x
−
~
x
|
i
j
i j>i
which equals
=−
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which is the virial theorem for quasi-static systems.
The more general expression for non-static systems is:
This is simply a change in how the summation is done,
it does not use any special property of the force field.
Because F~ij = −F~ji (forces are equal and opposite for
pairwise forces) the last term can be rewritten
XX
X
F~ij (~xi − ~xj )
F~i ~xi =
i
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X X Gmi mj
1 X X Gmi mj
=−
=W
|~
x
−
~
x
|
2
|~
x
−
~
x
|
i
j
i
j
i j>i
i
j6=i
with W the total potential energy. Therefore for a
galaxy in quasi-static equilibrium:
1
K = − W,
2
1 d2 I
= W + 2K.
2 dt2
Homework assignments:
1) derive the relation for the potential energy given
above. Suppose each particle has mass f mi , 0 ≤
f ≤ 1. Slowly add an extra mass mi df to each particle from infinity and calculate how much energy is
released. Integrate f from 0 to 1.
2) How empty are galaxies ? Calculate the distance
between neighbor stars, and get the radius of a sun
from literature. Take the ratio of the two. Compare
this to the same ratio of galaxies: take the average
distance between neighbor galaxies, and the radius of
galaxies.
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3.4 Applications
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(BT pages 213, 214)
Consider a system with total mass M
Kinetic energy K = 12 M hv 2 i with
hv 2 i = mean square speed of stars (assumption: speed
of star not correlated with mass of star)
Define gravitational radius rg
GM 2
W =−
rg
Spitzer found for many systems that rg = 2.5rh , where
rh is the radius which contains half the mass
Virial theorom implies:
GM 2
M hv i =
rg
2
M = hv 2 irg G−1
Hence, we can estimate the mass of galaxies if we
know the typical velocities in the galaxy, and its size!
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Homework Assignments:
3) calculate the mass of our galaxy for two assumed
radii:
a.) 10 kpc (roughly the distance to the sun)
b.) 350 kpc (halfway out to Andromeda)
Take as a typical velocity the solar value of 200 km/s.
4) Give the correct expression for the total kinetic energy if the velocities of stars ARE correlated with their
mass.
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3.5 Binding Energy and Formation of Galaxies
3.6. Scaling Relations
The total energy E of a galaxy is
Consider a steady state galaxy with particles mi at location xi (t).
Can the galaxy be rescaled to other physical galaxies?
E = K + W = −K = 1/2W
• Bound galaxies have negative energy - cannot fall
apart and dissolve into a very large homogeneous
distribution
• A galaxy cannot just form from an unbound, extended smooth distribution −> Etotal = Estart ≈
0, Egal = −K, so energy must be lost or the
structure keeps oscillating:
(not in BT)
• scaled particle mass m̂i =am mi
• scaled particle location x̂i = ax xi (at t)
• am , ax , at are scaling parameters
In the ’rescaled’ galaxy we have
ˆ
d2
F~i =m̂i (d2 x̂i /dt2 )=am mi dt
x(at t) = am ax a2t F~i,orig
2 ax ~
The gravitational force is equal to
P
ˆ
F~G = j6=i
ax ~
xj −ax ~
xi
|ax ~
xj −ax ~
xi |3 Gam mi am mj
=
a2m ~
a2x FG,orig
Equilibrium is satisfied if the two terms above are equal
Possible energy losses through
• Ejection of stars
• Radiation (before stars would form)
ˆ
ˆ
F~i = F~G
Since we have equilibrium when all scaling parameters
are equal to 1, we obtain
am ax a2t
or
a2m
= 2,
ax
am = a3x a2t .
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Now how do velocities scale ?
d
d
v̂= dt
x̂ = dt
ax x(at t) = ax at vorig
Hence, the scaling of the velocities satisfies: av =
ax at . We can write the new scaling relation as
am = ax a2v
Hence ANY galaxy can be scaled up like this !
Notice that it is trivial to derive this from the virial
theorem - the expression for the mass has exactly the
same form.
As a consequence, if we have a model for a galaxy with
a certain mass and size, we can make many more models, with arbitrary mass, and arbitrary size.
Homework Assignments:
5) Show explicitly that the virial theorem produces the
same result for the scaling relations.
6) How does this compare to stars ? What happens
to a star when we scale it in the same way ? Which
equilibrium is maintained, which equilibrium is lost ?