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
F~i ~xi = −
i
X X Gmi mj
(~xi − ~xj )(~xi − ~xj )
3
|~
x
−
~
x
|
i
j
i j>i
which equals
=−
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
W =−
1 X X Gmi mj
2 i
|~xi − ~xj |
j6=i
Gmi mj ~
xi −~
xj
For gravitational force F~ij = − |~xi −~
xj |2 |~
xi −~
xj |
X
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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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Proceed by assuming that 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. Explain each step in
this calculation.
2) Derive how empty galaxies are. First calculate the
distance between neighbor stars, by assuming they are
located on a 3-dimensional grid. Next get the radius
of the sun from literature. Take the ratio of the two.
Compare this to the same ratio of galaxies: calculate
the average distance between neighboring galaxies in
the same way, and get 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) The kinetic energy is given by
1/2M hv 2 i
if the velocities of the stars are NOT correlated with
ther mass. Give the correct expression for the total kinetic energy if the velocities of stars ARE correlated
with their mass. This is a simple, single expression
without additions of various terms.
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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 steady state galaxies?
• 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
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)
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, the scaled system is in equilibrium if the
scaling parameters satisfy:
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 up in size by a factor of 2,
and reduce the temparature by a factor of 2 ? Which
equilibrium is maintained, which equilibrium is lost ?