Transcript Document 7840872

Gravitational Dynamics
Gravitational Dynamics can be applied to:
•
•
•
•
•
•
Two body systems:binary stars
Planetary Systems
Stellar Clusters:open & globular
Galactic Structure:nuclei/bulge/disk/halo
Clusters of Galaxies
The universe:large scale structure
Syllabus
• Phase Space Fluid f(x,v)
– Eqn of motion
– Poisson’s equation
• Stellar Orbits
– Integrals of motion (E,J)
– Jeans Theorem
• Spherical Equilibrium
– Virial Theorem
– Jeans Equation
• Interacting Systems
– TidesSatellitesStreams
– Relaxationcollisions
How to model motions of 1010stars
in a galaxy?
• Direct N-body approach (as in simulations)
– At time t particles have (mi,xi,yi,zi,vxi,vyi,vzi),
i=1,2,...,N (feasible for N<<106).
• Statistical or fluid approach (N very large)
– At time t particles have a spatial density
distribution n(x,y,z)*m, e.g., uniform,
– at each point have a velocity distribution
G(vx,vy,vz), e.g., a 3D Gaussian.
N-body Potential and Force
• In N-body system with mass m1…mN,
the gravitational acceleration g(r) and
potential φ(r) at position r is given by:
r12
N



G  m  mi  rˆ12
F  mg (r )     2  m r
i 1
r  Ri
G  m  mi


m

(
r
)


mi

 
r  Ri
i 1
N
r
Ri
Eq. of Motion in N-body
• Newton’s law: a point mass m at position r
moving with a velocity dr/dt with Potential
energy Φ(r) =mφ(r) experiences a Force
F=mg , accelerates with following Eq. of
Motion:



d  dr (t )  F   r (r )
 


dt  dt  m
m
Orbits defined by EoM & Gravity
• Solve for a complete prescription of history
of a particle r(t)
• E.g., if G=0  F=0, Φ(r)=cst,  dxi/dt =
vxi=ci  xi(t) =ci t +x0, likewise for yi,zi(t)
– E.g., relativistic neutrinos in universe go
straight lines
• Repeat for all N particles.
•  N-body system fully described
Example: 5-body rectangle problem
• Four point masses m=3,4,5 at rest of three
vertices of a P-triangle, integrate with time
step=1 and ½ find the positions at time t=1.
Star clusters differ from air:
• Size doesn’t matter:
– size of stars<<distance between them
– stars collide far less frequently than
molecules in air.
• Inhomogeneous
• In a Gravitational Potential φ(r)
• Spectacularly rich in structure because φ(r)
is non-linear function of r
Why Potential φ(r) ?
• More convenient to work with force,
potential per unit mass. e.g. KE½v2
• Potential φ(r) is scaler, function of r only,
– Easier to work with than force (vector, 3
components)
– Simply relates to orbital energy E= φ(r) +½v2
nd
2
Lec
Example: energy per unit mass
• The orbital energy of a star is given by:
1 2

E  v   (r , t )
2


dE  dv dr

v
  
dt
dt dt
t
0

dv
 
since
 dt
0 for static potential.
and dr  v
dt
So orbital Energy is Conserved in a
static potential.
Example: Force field of two-body
system in Cartesian coordinates
2

G  mi

 (r )     , where Ri  (0,0,i )  a, mi  i  m
i 1 r  Ri
Sketch the configurat ion, sketch equal potential contours
 ( x, y , z )  ?
 
  

g (r )  ( g x , g y , g z )   (r )  ( , , )
x y z
 
g (r )  ( g x2  g y2  g z2 )  ?
sketch field lines. at what positions is force  0?
Example: Energy is conserved
• The orbital energy of a star is given by:
1 2

E  v   (r , t )
2


dE  dv dr

v
  
dt
dt dt
t
0

dv
 
since
 dt
0 for static potential.
and dr  v
dt
So orbital Energy is Conserved in a
static potential.
rd
3
• Animation of GC
formation
Lec
A fluid element: Potential & Gravity
• For large N or a continuous fluid, the gravity dg and
potential dφ due to a small mass element dM is calculated
by replacing mi with dM:
r12
r
R
dM
d3R

G  dM  rˆ12
dg     2
r  Ri
G  dM
d    
r R
Potential in a galaxy
• Replace a summation over all N-body particles with the
integration:
N
 dM  m
i
i 1
RRi
• Remember dM=ρ(R)d3R for average density ρ(R) in small
volume d3R
• So the equation for the gravitational force becomes:
3


G ( R)dR
F / m  g (r )   r , with  (r )     
r R
Poisson’s Equation
• Relates potential with density
   4G (r )
2
• Proof hints:
 Gm
 
    4Gm (r  R)
r R
 
4G (r )   4G (r  R)  ( R)dR 3
2
Poisson’s Equation
• Poissons equation relates the potential to the
density of matter generating the potential.
• It is given by:
 
      g  4G (r )
Gauss’s Theorem
• Gauss’s theorem is obtained by integrating
poisson’s equation:

2
   (r )dV   4G (r )dV  4GM
V
V


   (r )dV    (r ).ds
2
V
S

   (r ).ds  4GM
S
• i.e. the integral ,over any closed surface, of the
normal component of the gradient of the potential
is equal to 4G times the Mass enclosed within
that surface.
Laplacian in various coordinates
Cartesians :
2
2
2



2  2  2  2
x
y
z
Cylindrica l :
2
2
1


1




2 
R
 2

 2
2
R R  R  R 
z
Spherical :
2
1


1


1





2
2  2
r

 2
 sin 
 2 2
r r  r  r sin   
  r sin   2
th
4
• Potential,density,orbits
Lec
From Gravitational Force to Potential
r
 (r )   g dr

d
g    
dr
From Potential to Density
Use Poisson’s Equation
The integrated form of
Poisson’s equation is
given by:
1

 2
4G

Gd r 

 (r )     
r  r
3
More on Spherical Systems
• Newton proved 2 results which enable us to
calculate the potential of any spherical system
very easily.
• NEWTONS 1st THEOREM:A body that is inside a
spherical shell of matter experiences no net
gravitational force from that shell
• NEWTONS 2nd THEOREM:The gravitational
force on a body that lies outside a closed spherical
shell of matter is the same as it would be if all the
matter were concentrated at its centre.
From Spherical Density to Mass
M(R  dr)  M(R)  dM
4 3
2
dM   (r)d  r   4r  (r) dr
3

dM
dM
 (r ) 

2
4
4r dr

3
d  r 
3

4 3
M ( R )   d  r 
3

M(r+dr)
M(r)
Poisson’s eq. in Spherical systems
• Poisson’s eq. in a spherical potential with no θ or Φ
dependence is:
1   2  
 2
r
  4G (r )
r r  r 
2
Proof of Poissons Equation
• Consider a spherical distribution of mass of
density ρ(r).
g
g
GM (r )
r2

   g (r )dr
r
since   0 at  and is  0 at r
r

GM (r )
 
dr
2
r
r

Mass Enclosed   4r 2  (r )dr
r
• Take d/dr and multiply r2 

d
2
2
r
  gr  GM (r )  G  4r  (r )dr
dr
2

• Take d/dr and divide r2
1   2   1 
1 
2
GM   4G (r )
r g  2
r
 2
2
r r  r  r r
r r

2
    .g  4G


Sun escapes if Galactic potential
well is made shallower
Solar system accelerates weakly
in MW
• 200km/s circulation
g(R0 =8kpc)~0.8a0,
a0=1.2 10-8 cm2 s-1
Merely gn ~0.5 a0 from
all stars/gas
• Obs. g(R=20 R0)
~20 gn
~0.02 a0
• g-gn ~ (0-1)a0
• “GM” ~ R if weak!
Motivates
– M(R) dark particles
– G(R) (MOND)
Circular Velocity
• CIRCULAR VELOCITY= the speed of a test
particle in a circular orbit at radius r.
2
cir
v
GM (r )
g 
  
2
r
r
2
vcir
r
 M (r ) 
G
For a point mass:
GM
vc (r ) 
r
For a homogeneous sphere
4G
4 3
vc (r ) 
r since M(r)  r 
3
3
Escape Velocity
• ESCAPE VELOCITY= velocity required in order
for an object to escape from a gravitational
potential well and arrive at  with zero KE.
1 2
1 2
 (r )   ()  vesc   vesc
2
2
 vesc (r )   2 (r )
-ve
• It is the velocity for which the kinetic energy
balances potential.
Plummer Model
• PLUMMER MODEL=the special case of the
gravitational potential of a galaxy. This is a
spherically symmetric potential of the form:
 
GM
r a
2
2
• Corresponding to a density:
3M  r 2 
1  2 

3 
4a  a 

5
2
which can be proved using poisson’s equation.
• The potential of the plummer model looks like
this:

r

GM o
a
g    GM o r (r 2  a 2 )

3
2
 0 for r  0
 r  0 is minimum of potential

GM o 1 2
2GM o
 vesc  vesc 
a
2
a
• Since, the potential is spherically symmetric g is
also given by: g   GM
r2


GM
 2  GM o r r 2  a 2
r

 M  M or 3 r 2  a


3
2

3

2 2
dM (r )
• The density can then be obtained from:   dV
• dM is found from the equation for M above and
dV=4r2dr.
5

2
2


3
M
r
• This gives
(as before from



1

4a 3  a 2 
Poisson’s)
Tutorial Question 1:
Singular Isothermal Sphere
GM 0
• Has Potential Beyond ro:  (r )   r
r
2
 (r )  v0 ln  o
• And Inside r<r0
ro
• Prove that the potential AND gravity is continuous
at r=ro if
0  GM 0 / r0  v02
• Prove density drops sharply
to 0 beyond r0, and
2
V0
inside r0
 (r ) 
4Gr 2
• Integrate density to prove total mass=M0
• What is circular and escape velocities at r=r0?
• Draw Log-log diagrams of M(r), Vesc(r), Vcir(r),
Phi(r), rho(r), g(r) for V0=200km/s, r0=100kpc.
Tutorial Question 2: Isochrone Potential
• Prove G is approximately 4 x 10-3 (km/s)2pc/Msun.
• Given an ISOCHRONE POTENTIAL
GM
 (r )  
b  (b  r )
2
2
• For M=105 Msun, b=1pc, show the central escape velocity
= (GM/b)1/2 ~ 20km/s.
• Argue why M must be the total mass. What fraction of the
total mass is inside radius r=b=1pc? Calculate the local
Vcir(b) and Vesc(b) and acceleration g(b). What is your
unit of g? Draw log-log diagram of Vcir(r).
• What is the central density in Msun pc-3? Compare with
average density inside r=1pc. (Answer in BT, p38)