gravitational dynamics formulae. link phase space quantities r j(r,v) k(v) (r) vtvt e(r,v) dθ/dt...
TRANSCRIPT
Motions in spherical potential
If spherical
0
rgr
g
gdtddtd
motion ofEquation
v
vx
0
00
)(
)(
gravity no If
vv
xvx
t
tt 2
Conserved if spherical static
1E ( )
2ˆt
v r
L J rv n
x v
PHASE SPACE DENSITY:Number of stars per unit volume per unit velocity volume f(x,v).
dN=f(x,v)d3xd3v
TOTAL # OF PARTICLES PER UNIT VOLUME:
MASS DISTRIBUTION FUNCTION:
TOTAL MASS:
TOTAL MOMENTUM:
MEAN VELOCITY:
<v>=<vxvy>=0 (isotropic) & <vx2>=<vy
2>=σ2(x)
xdvdvxmfxdxM total
333 ),()(
vdxdvvxmfmdNvPtotal
33),(
dN
dNvv
NOTE: d3v=4πv2dv (if isotropic)
d3x=4πr2dr (if spherical)
GAMMA FUNCTIONS:
2
1
)1()1()(
)( 1
0
nnn
dxxen nx
GRAVITATIONAL POTENTIAL DUE TO A MASS dM:
RELATION BETWEEN GRAVITATIONAL FORCE AND POTENTIAL:
FOR AN N BODY CASE:
Rr
RGdM
)(
Rr
mRGdMmF r
)(
N
ii
iN
i i
ir
Rr
rmGm
Rr
mGmmg
12
12
1
LIOUVILLES THEOREM:
(volume in phase space occupied by a swarm of particles is a constant for collisionless systems)
IN A STATIC POTENTIAL ENERGY IS CONSERVED:
0d
dV
)(2
1 2 rmmvmE
0dt
dENote:E=energy per unit mass
2
2
2222
22
2
2
2
2
22
2
2
2
2
2
22
sin
1sin
sin
11
:
11
:
:
rrrr
rr
Spherical
zRRR
RR
lCylindrica
zyx
Cartesians
RELATING PRESSURE GRADIENT TO GRAVITATIONAL FORCE:
GOING FROM DENSITY TO MASS:
dr
dr
dr
d )(
)( 2
drrrdRM 23 4
3
4)(
SINGULAR ISOTHERMAL SPHERE MOD
r
GMrSphereOutside
r
GMrvrvrSphereInside
Gr
v
rd
dMr
r
v
r
GMr
dt
dv
rG
rvrM
vconstrvlocityCircularVe
o
o
oocc
c
rc
r
c
oc
)(:_
lnln)(:_
434
)(
)(1
)(
)(
)(_
22
2
2
3
22
2
2
ISOTROPIC SELF GRAVITATING EQUILIBRIUM SYSTEMS
0
3
0
3
2
2
3
4)()(
3
4)()(
0)(:
)(2
)(
)(2
E
r
r
re
t
vdEfr
rdrrM
assume
drrv
r
rvJ
rv
E
JEANS EQUATION (steady state axisymmetric system in which σ2 is isotropic and the only streaming motion is in the azimuthal direction)
RR
v
R
zz
rot
22
2
)(1
)(1
VELOCITY DISPERSIONS (steady state axisymmetric and isotropic σ)
OBTAINING σ USING JEANS EQUATION:
22
222222
)(
),(
rot
rotzr
vv
vvvvzR
z
rot
z
dzzR
R
RRv
dzz
zR
2
2 1),(
ORBITS IN AXISYMMETRIC POTENTIALS
ER
JzRzR
constzRzRR
dt
Rd
zz
RRR
dt
rd
z
r
2
222
2222
2
..
2..
2
2
2),()(
2
1
),(2
1
0)(
Φeff
CONDITION FOR A PARTICLE TO BE BOUND TO THE SATELLITE RATHER THAN THE HOST SYSTEM:
TIDAL RADIUS:
keplerianforkR
GM
r
R
r
GMk s 331
22
33
3
1
34
)(
34
)(
)(
r
rM
R
rm
rkM
mrR
s
sT
LAGRANGE POINTS: Gravitational pull of the two large masses precisely cancels the centripetal acceleration required to rotate with them.
EFFECTIVE FORCE OF GRAVITY:
JAKOBI’S ENERGY:
00
yxeffeff
)(2 rvggeff
02
1 2 dt
dErE J
effJ
DYNAMICAL FRICTION:
MM
v
mmm
m
o
Mm
Mmmm
m
Vv
dvvvf
mMmGdt
vd
formulafrictiondynamicalharChandrasek
velocitiesstellarofondistributiisotropicanfor
mMG
vbarithmcoulomb
vv
vvvdvfmMmG
dt
vd
M
30
2
22
2max
3322
)(
)(ln16
:
)(lnlogln
)()()(ln16
Cont:
Only stars with vvM contribute to dynamical friction.
For small vM:
For sufficiently large vM:
MM vmMmfG
dt
vd
)()0(ln3
16 22
2 MM v
dt
dv
FOR A MAXWELLIAN VELOCITY DISTRIBUTION:
MM
om
om
verfv
mnmMG
dt
vd
vnvf
)exp(2
)()(ln4
2exp
)2(
)(
23
2
2
2
2
32
ORBITS IN SPHERICAL POTENTIALS
)()(22
)(2
1
2
1
2
2
2
22
2
apocentreandpericentreatrootsr
LrE
dt
rd
rdt
dr
dt
rdE
timeunit
sweptareaconst
dt
dr
dt
rdrL
eff
RADIAL PERIOD:Time required for the star to travel from apocentre to pericentre and back.
AZIMUTHAL PERIOD:
Where:
In general θ will not be a rational numberorbits will not be closed.
a
p
r
r
r
rL
rE
drT
2
2
)(2
2
rTT
2
dr
dtdr
rL
drdr
ddr
dr
ddr
dtdr
dtd a
p
a
p
a
p
a
p
r
r
r
r
r
r
r
r 22222
STELLAR INTERACTIONS
Nb
R
Rv
GmNv
N
R
v
Gmb
b
db
Rv
GmNv
bv
Gmv
b
vt
b
GmF
min
22
2min
2
2
32
2
2
ln8
8
2
1
FOR THE SYSTEM TO NO LONGER BE COLLISIONLESS:
RELAXATION TIME:
CONTINUITY EQUATION:
ln82
2
22
N
v
vn
vv
relax
crosscrossrelaxrelax tN
Ntnt
ln8
0).(
vt
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
3dv3dx),(dM
)(spherical 2sin2r
2
sin2r
)(sin
2r
)2(
al)(cylindric 22-R2)(1-R
ar)(rectangul 222
1-sun
M2(km/s) kpc6104
1-sun
M2(km/s) pc3104
Gyr1
kpc1
1Myr
1pc 1km/s
vxf
rr
r
zRR
R
zyx
G
G