Download Orbits in Spherical Potentials

th
5
• orbits
Lec
Stellar Orbits
• Once we have solved for the gravitational
potential (Poisson’s eq.) of a system we
want to know: How do stars move in
gravitational potentials?
• Neglect stellar encounters
• use smoothed potential due to system or
galaxy as a whole
Motions in spherical potential
Equation of motion
dx
v
dt
dv

 g  
dt
If no gravity
x(t )  v 0t  x 0
v (t )  v 0
If spherical

gr  
r
Conserved if spherical
1
E  v 2   (r )
2
L  J  x  v  rvt  nˆ
In static spherical potentials: star
moves in a plane (r,q)
• central force field
• angular momentum
g  grˆ
r  r  L
dL d
 r  r   r  r  r  r  r  g  0
dt
dt
• equations of motion are
– radial acceleration:
– tangential acceleration:
r  rq 2  g (r )
2rq  rq2  0
r 2q  constant  L
Orbits in Spherical Potentials
• The motion of a star in a centrally directed field of
force is greatly simplified by the familiar law of
conservation (WHY?) of angular momentum.
  dr
Lr
 const
dt
area swept
2 dq
r
2
dt
unit time
Keplers 3rd law
pericentre
apocentre
Energy Conservation
(WHY?)

1  dr  1  dq 
2
E   (r )      r

2  dt  2  dt 
 2
L
1  dr 
  (r )  2   
2r
2  dt 
2
eff
dr
  2 E  2eff (r )
dt
2
Radial Oscillation in an Effective
potential
• Argue: The total velocity of the star is slowest at
apocentre due to the conservation of energy
• Argue: The azimuthal velocity is slowest at
apocentre due to conservation of angular
momentum.
th
6
• Phase Space
Lec
dr
 0 at the PERICENTRE and APOCENTRE
•
dt
L2
• There are two roots for 2 E  2 (r )  2
r
• One of them is the pericentre and the other is the
apocentre.
• The RADIAL PERIOD Tr is the time required for
the star to travel from apocentre to pericentre and
back.
• To determine Tr we use:
dr
L2
  2E     2
dt
r
• The two possible signs arise because the star
moves alternately in and out.
ra
dr
 Tr  2 
2
L
rp
2 E  2  2
r
• In travelling from apocentre to pericentre and
back, the azimuthal angle q increases by an
amount:
r
r dq
r L
2
dq
dt
r
q  2 
dr  2 
dr  2 
dr
a
rp
a
dr
rp
a
dr
dt
rp
dr
dt
2
Tr
• The AZIMUTHAL PERIOD is Tq 
q
2
• In general q will not be a rational number.
Hence the orbit will not be closed.
• A typical orbit resembles a rosette and eventually
passes through every point in the annulus between
the circle of radius rp and ra.
2
• Orbits will only be closed if
is an integer.
q
Examples: homogeneous sphere
• potential of the form   12  2 r 2  constant
• using x=r cosq and y = r sinq
• equations of motion are then:
x   2 x;
y   2 y
x  A cos(t   x );
– spherical harmonic oscillator
y  B cos(t   y )
• Periods in x and y are the same
so every
orbit is closed ellipses centred on the centre
of attraction.
homogeneous sphere cont.
A
B
A
t=0
B
• orbit is ellipse
• define t=0 with x=A, y=0
  x  0,  y   / 2
t  Pr / 2  x  0 if
Pq  2Pr
 
Pr
Radial orbit in homogeneous sphere
d 2r
GM ( r )
4G


r
2
2
dt
r
3
– equation for a harmonic oscillator
angular frequency 2/P
P
3
t dyn 
4

16G
Altenative equations in spherical
potential
• Let
u
1
r
2E
 dr  1 2 (r )

const

 2  2  2
L2
L
 r dθ  r
2 E  du 
2 (1 / u )
2
 2 
 u 
L
L2
 dq 
 d 
Apply 

2
du


2
0
d  du 
GM (1 / u )

 u 
dq  dq 
L2
Kepler potential
g (r )  GM / r 2
• Equation of motion becomes:
d 2u
GM

u

dq 2
L2
– solution: u linear function of cos(theta):
a 1  e 2 
r q  
1  e cos q
with Pq  Pr and thus q  2
• Galaxies are more centrally condensed than
a uniform sphere, and more extended than a
point mass, so   q  2
Pr  Pq
Tutorial Question 3: Show in
Isochrone potential
 (r )   GM
b  (b
2
r )
2
1
2


L
Pr 
, and q  1 
3

 2 E  2
L2  4GMb
– radial period depends on E, not L
2GM
• Argue
  q  2

,
butq  2
for
– this occurs for large r, almost Kepler


1
2


L2  4GMb
Helpful Math/Approximations
(To be shown at AS4021 exam)
• Convenient Units
• Gravitational Constant
• Laplacian operator in
various coordinates
• Phase Space Density
f(x,v) relation with the
mass in a small position
cube and velocity cube
1km/s 
1kpc
1pc

1Myr 1Gyr
G  4  10  3 pc (km/s) 2 M - 1
sun
G  4  10  6 kpc (km/s) 2 M - 1
sun
     2   2   2 (rectangul ar)
z
y
x
 R - 1 ( R )   2  R - 2 2 (cylindric al)

z
R
R
2
 (r 2 )  (sin q )

q 
r  q
(spherical )
 r
2
2
2
2
r sin q
r
r sin q
dM  f ( x, v)dx 3dv3