Orbits in Spherical Potentials
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Transcript 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
If spherical
Equation of motion
dx
v
dt
dv
g
dt
If no gravity
x(t ) v 0t x 0
v (t ) v 0
gr
r
Conserved if spherical
1 2
v (r )
2
L J x v rvt nˆ
E
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 )
2rq rq2 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
Lr
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 2eff (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
2E 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 ); y B cos(t y )
– spherical harmonic oscillator
• 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 )
4G
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
dq
GM (1 / u )
du
u
L2
dq
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 e2
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
2GM
• Argue
q 2
,
butq 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