gravitational dynamics. gravitational dynamics can be applied to: two body systems:binary stars...
TRANSCRIPT
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
– TidesSatellitesStreams– Relaxationcollisions
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
Ri
r mi
N
i i
i
N
ir
i
i
Rr
mmGrm
mRr
rmmGrgmF
1
12
12
)(
ˆ)(
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:
m
r
m
F
dt
trd
dt
d r )()(
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: Force field of two-body system in Cartesian coordinates
0?force is positionsat what lines. fieldsketch
?)()(
),,()(),,()(
?),,(
contours potential equalsketch ion,configurat Sketch the
,),0,0( where,)(
222
2
1
zyx
zyx
iii i
i
gggrg
zyxrgggrg
zyx
mmaiRRr
mGr
Example: 4-body problem
• Four point masses Gm=1 at rest (x,y,z)=(0,1,0),(0,-1,0),(-1,0,0),(1,0,0). What is the initial total energy?
• Integrate EoM by brutal force with time step=1 to find the positions/velocities at time t=1. i.e., use straight-orbit V=V0+gt, R=R0+V0t+gt2/2. What is the new total energy?
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. KE½v2
• 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
2nd Lec
Example: energy per unit mass
• The orbital energy of a star is given by:
),(2
1 2 trvE
tdt
rd
dt
vdv
dt
dE
0 since
and
dt
vd
vdt
rd
0 for static potential.
So orbital Energy is Conserved in a static potential.
Example: Energy is conserved
• The orbital energy of a star is given by:
),(2
1 2 trvE
tdt
rd
dt
vdv
dt
dE
0 since
and
dt
vd
vdt
rd
0 for static potential.
So orbital Energy is Conserved in a static potential.
3rd Lec
• Animation of GC formation
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
Rr
dM
d3RRr
dMGd
212ˆ
iRr
rdMGgd
Potential in a galaxy
• Replace a summation over all N-body particles with the integration:
• Remember dM=ρ(R)d3R for average density ρ(R) in small volume d3R
• So the equation for the gravitational force becomes:
Rr
dRRGrrgmF r
3)()( with ,)(/
i
N
i
mdM
1
RRi
Poisson’s Equation
• Relates potential with density
• Proof hints: )(42 rG
3
2
)()(4)(4
)(4
dRRRrGrG
RrGmRr
Gm
Poisson’s Equation
• Poissons equation relates the potential to the density of matter generating the potential.
• It is given by:
)(4 rGg
Gauss’s Theorem
• Gauss’s theorem is obtained by integrating poisson’s equation:
• i.e. the integral ,over any closed surface, of the normal component of the gradient of the potential is equal to 4G times the Mass enclosed within that surface.
GMdsr
dsrdVr
GMdVrGdVr
S
SV
VV
4).(
).()(
4)(4)(
2
2
Laplacian in various coordinates
2
2
2222
22
2
2
2
2
22
2
2
2
2
2
22
sin
1sin
sin
11
:Spherical
11
:lCylindrica
:Cartesians
rrrr
rr
zRRR
RR
zyx
4th Lec
• Potential,density,orbits
dr
dg
drgrr
)(
From Potential to Density
Use Poisson’s Equation
The integrated form of Poisson’s equation is given by:
rr
rdGr
3
)(
From Gravitational Force to Potential
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)
M(r+dr)
3
23
23
3
4)(
434
)(
(r)43
4(r)ddM
dMM(R)dr)M(R
rdRM
drr
dM
rd
dMr
drrr
Poisson’s eq. in Spherical systems
• Poisson’s eq. in a spherical potential with no θ or Φ dependence is:
)(41 2
22 rG
rr
rr
Proof of Poissons Equation
• Consider a spherical distribution of mass of density ρ(r).
r
g
drrr
drr
rGM
drrg
r
rGMg
r
r
r
)(4Enclosed Mass
)(
rat 0 is andat0 since)(
)(
2
2
2
• Take d/dr and multiply r2
• Take d/dr and divide r2
drrrGrGMgrdr
dr )(4)( 222
Gg
rGGMrr
grrrr
rrr
4.
)(4111
2
22
22
2
Sun escapes if Galactic potential well is made shallower
Solar system accelerates weakly in MW
• 200km/s circulationg(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.
G
rvrM
r
rGM
r
vg
cir
cir
2
2
2
)(
)(
For a point mass: For a homogeneous sphere
r
GMrvc )( 3
3
4M(r)since
3
4)( rr
Grvc
Escape Velocity• ESCAPE VELOCITY= velocity required in order
for an object to escape from a gravitational potential well and arrive at with zero KE.
• It is the velocity for which the kinetic energy balances potential.
)(2)(
2
1
2
1)()( 22
rrv
vvr
esc
escesc
-ve
Tutorial Question 1: Singular Isothermal Sphere
• Has Potential Beyond ro:
• And Inside r<r0
• Prove that the potential AND gravity is continuous at r=ro if
• Prove density drops sharply to 0 beyond r0, and inside r0• 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.
r
GMr 0)(
oor
rvr ln)( 2
0
20000 / vrGM
2
20
4)(
Gr
Vr
Tutorial Question 2: Isochrone Potential
• Prove G is approximately 4 x 10-3 (km/s)2pc/Msun. • Given an ISOCHRONE POTENTIAL
• 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)
)()(
22 rbb
GMr
Example:Single Isothermal Sphere Model
• For a SINGLE ISOTHERMAL SPHERE (SIS) the line of sight velocity dispersion is constant. This also results in the circular velocity being constant (proof later).
• The potential and density are given by:
2
22
4)(ln)(
Gr
VrrVr c
c
Proof: Density
23
22
2
434
)(
)(1
)(
)(
constlocity_CircularVe
2
Gr
v
rd
dMr
r
v
r
GMr
dt
dv
rG
rvrM
v(r)v
c
rc
r
c
oc
r-2
n=-2
Log()
Log(r)
Proof: Potential
We redefine the zero of potential
If the SIS extends to a radius ro then the mass and density distribution look like this:
rvdr
r
vdrr c
rc
r
r ln)( 22
constant)ln(2 rvc
M
r
rroro
• Beyond ro:
• We choose the constant so that the potential is continuous at r=ro.
r
rGM o )(
o
oocc
oo
c
r
GMrvrvr
r
rvr
lnln)(
ln)(
22
2
r
logarithmic
r-1
So:
r
GMr
r
GMrvrvr
o
o
oocc
)(:hereOutside_Sp
lnln)(:ereInside_Sph 22
Plummer Model
• PLUMMER MODEL=the special case of the gravitational potential of a galaxy. This is a spherically symmetric potential of the form:
• Corresponding to a density:
2
5
2
2
31
4
3
a
r
a
M
which can be proved using poisson’s equation.
• The potential of the plummer model looks like this:
r
a
GM o
potential of minimum is 0r
0r for 0)( 2
322
arrGMg o
a
GMvv
a
GM oescesc
o 2
2
1 2
• Since, the potential is spherically symmetric g is also given by:
• The density can then be obtained from:• dM is found from the equation for M above and
dV=4r2dr.• This gives
2r
GMg
2
3223
2
322
2
arrMM
arrGMr
GM
o
o
dV
rdM )(
2
5
2
2
31
4
3
a
r
a
M
(as before from
Poisson’s)
Isochrone Potential
• We might expect that a spherical galaxy has roughly constant near its centre and it falls to 0 at sufficiently large radii.
• i.e.
• A potential of this form is the ISOCHRONE POTENTIAL.
r)largefor(r
r)small(forconstant1-
2
r
)()(
22 rbb
GMr
5th Lec
• orbits
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
rgr
spherical If
gdtddtd
motion ofEquation
v
vx
0
00
)(
)(
gravity no If
vv
xvx
t
tt
nrvJL
rv
t ˆ
)(2
1E
spherical if Conserved
2
vx
Proof: Angular Momentum is Conserved
•
vrL
dt
vdrvv
dt
vdrv
dt
rd
dt
vrd
dt
Ld
Since 0
then the force is in the r direction.
both cross products on the RHS = 0.
So Angular Momentum L is Conserved in Spherical Isotropic Self Gravitating Equilibrium Systems.
Alternatively: =r×F & F only has components in the r direction=0 so
0t
L
In static spherical potentials: star moves in a plane (r,)
• central force field
• angular momentum
• equations of motion are– radial acceleration:– tangential acceleration:
Lr
rr
rgrr
grrrrrrrdt
d
dt
Ld
Lrr
constant
02
)(
0
2
2
2
rgg ˆ
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.
unit time
sweptarea2r
const
2
dt
ddt
rdrL
Keplers 3rd lawpericentre
apocentre
Energy Conservation (WHY?)
2
2
2
22
2
1
2)(
2
1
2
1)(
dt
rd
r
Lr
dt
dr
dt
rdrE
eff
)(22 rEdt
dreff
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.
6th Lec
• Phase Space
• at the PERICENTRE and APOCENTRE
• There are two roots for• 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:
0dt
dr
2
2
)(22r
LrE
2
2
2r
LE
dt
dr
• The two possible signs arise because the star moves alternately in and out.
• In travelling from apocentre to pericentre and back, the azimuthal angle increases by an amount:
a
p
r
r
r
rL
E
drT
2
2
22
2
dr
dtdrr
Ldr
dtdr
dtd
drdr
d a
p
a
p
a
p
r
r
r
r
r
r
2
222
• The AZIMUTHAL PERIOD is
• In general 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.
• Orbits will only be closed if is an integer.
rTT
2
2
2
Examples: homogeneous sphere
• potential of the form
• using x=r cos and y = r sin• equations of motion are then:
– spherical harmonic oscillator
• Periods in x and y are the same so every orbit is closed ellipses centred on the centre of attraction.
)cos( );cos(
;
constant
22
2221
yx tBytAx
yyxx
r
homogeneous sphere cont.
• orbit is ellipse
• define t=0 with x=A, y=0
• One complete radial oscillation: A to -A
• azimuth angle only increased by š
•
AA
B
Bt=0
if 0 2/
2/,0
rr
yx
PxPt
P 2Pr
Radial orbit in homogeneous sphere
– equation for a harmonic oscillator
angular frequency 2/P
G
Pt
rG
r
rGM
dt
rd
16
3
4
3
4)(
dyn
22
2
Altenative equations in spherical potential
• Let
2
22
2
2
2222
)/1(0
2
)/1(22
)(212
1
L
uGMu
d
du
d
d
du
dApply
L
uu
d
du
L
E
L
r
rdθr
drconst
L
Er
u
Kepler potential
• Equation of motion becomes:
– solution: u linear function of cos(theta):
with and thus
• Galaxies are more centrally condensed than a uniform sphere, and more extended than a point mass, so
cos1
1
/)(
2
22
2
2
e
ear
L
GMu
d
ud
rGMrg
P Pr 2
2 Pr P
Tutorial Question 3: Show in Isochrone potential
– radial period depends on E, not L
• Argue , but for– this occurs for large r, almost Kepler
21
223
22
41 and ,
2
2
)()(
21
GMbL
L
E
GMP
rbbGMr
r
2 2 L2 4GMb
1st Tutorial
g
(r) (r)
2
(E)vesc
M
7th Lec
Tidal Stripping
• TIDAL RADIUS:Radius within which a particle is bound to the satellite rather than the host system.
• Consider a satellite of mass Ms inside radius R is moving in a spherical potential (r) made from a point mass M.
rR
M(r)
• The condition for a particle to be bound to the satellite rather than the host system is:
222)( R
GM
r
GM
Rr
GM s
Differential (tidal) force on the particle due to the host galaxy
Force on particle due to satellite
22 R
GMU
r
GM s
.........2
1r
R-1r then UR If
2
r
R
2,22
k
R
GM
r
R
r
GMk s
• Generally, fudge factor k=14, bigger for radial orbits, bigger for point-like mass.
• Therefore Tidal Radius is (ambiguously defined as):
• The tidal radius is smallest at pericentre where r is smallest. Often tidal radius is only defined when r(t)=pericentre.
• As a satellite losses mass, its tidal radius shrinks.
3
1
)()()(
rkM
mtrtR s
T
The meaning of tidal radius
• The inequality can be written in terms of the mean densities.
• The less dense part of the satellite is torn out of the system, into tidal tails.
33
34
)(
34
)(
r
rM
R
rms
Size and Density of a BH
• A black hole has a finite (schwarzschild) radius Rbh=2 G Mbh/c2 ~ 2au (Mbh/108Msun)
– verify this! What is the mass of 1cm BH?
• A BH has a density (3/4Pi) Mbh/Rbh3, hence
smallest holes are densest. – Compare density of 108Msun BH with Sun (or
water) and a giant star (10Rsun).
Growth of a BH by capturing objects in its Loss Cone
• A small BH on orbit with pericentre rp<Rbh is lost (as a whole) in the bigger BH.– The final process is at relativistic speed. Newtonian
theory is not adequate
• (Nearly radial) orbits with angular momentum J<Jlc =2*c*Rbh =4GMbh/c2 enters `loss cone` (lc)
• When two BHs merger, the new BH has a mass somewhat less than the sum, due to gravitational radiation.
Tidal disruption near giant BH
• A giant star has low density than the giant BH, is tidally disrupted first.
• The disruption happens at radius rdis > Rbh , Mbh/rdis
3 ~ M*/R*3
• Giant star is shreded.
• Part of the tidal tail feeds into the BH, part goes out.
Adiabatic Compression due to growing BH
• A star circulating a BH at radius r has
• a velocity v=(GMbh/r)1/2,
• an angular momentum J = r v =(GMbh r)1/2,
• As BH grows, Potential and Orbital Energy E changes (t)
• But J conserved (no torque!), still circular!
• So Ji = (GMi ri)1/2 =Jf =(GMf rf )1/2
• Shrink rf/ri = Mi/Mf < 1, orbit compressed!
Adiabatic Invariance
• Suppose we have a sequence of potentials p(|r|) that depends continuously on the parameter P(t).
• P(t) varies slowly with time.• For each fixed P we would assume that the orbits
supported by p(r) are regular and thus phase space is filled by arrays of nested tori on which phase points of individual stars move.
• Suppose 0)(
ttP
• The orbit energy of a test particle will change.• Suppose
• The angular momentum J is still conserved because rF=0.
)()( rr PfPi Initial Final
ffii vrvrJ
• In general, two stellar phase points that started out on the same torus will move onto two different tori.
• However, if potential is changed very slowly compared to all characteristic times associated with the motion on each torus, all phase points that are initially on a given torus will be equally affected by the variation in Potential
• Any two stars that are on a common orbit will still be on a common orbit after the variation in Pot is complete. This is ADIABATIC INVARIANCE.
8th Lec
• Phase Space
Stellar interactions• When are interactions important?
• Consider a system of N stars of mass m
• evaluate deflection of star as it crosses system
• consider en encounter with star of mass m at a distance b:
br
X=vtv
Fperp
2
32
22221cos
23
b
vt
b
Gm
bx
bGm
r
Gmg
Stellar interactions cont.
• the change in the velocity vperp is then
– using s = vt / b
• Or using impulse approximation: – where gperp is the force at closest approach and
– the duration of the interaction can be estimated as :
t = 2 b / v
v
b
b
Gmtgv
bv
Gmdss
bv
Gmdtgv
2
21
2
23
2
Number of encounters with impact parameter b - b + b
• let system diameter be: 2R
• Star surface density ~ N/R2/Pi
• the number encountering
– each encounter has effect vperp but each one randomly oriented
– sum is zero:
b
b+b
N b b 2 b2 N
R2
2bbN
R2
2N
R2bb
v 0
change in kinetic energy
• but suming over squares (vperp2) is > 0
• hence
• now consider encounters over all b– then
– but @ b=0, 1/b is infinte!
– need to replace lower limit with some bmin
– expected distance of closest R/N R
GNmv
N
R
v
Gmb
b
db
Rv
GmNv
bbR
N
bv
Gmv
b
b
22min
2max
min
2
2
22
used have we where,
8
22
Relaxation time
• hence v2 changes by v2 each time it crosses the system where v2 is:
Orbit deflected when v2 ~ v2
– after nrelax times across the system
and thus the relaxation time is:
crosscrosscrossrelaxrelax
2
2
relax
min
22
ln8ln8
ln8
where,ln8
tN
Nt
Ntnt
N
v
vn
Nb
R
Rv
GmNv
Relaxation time cont.
• collisionless approx. only for t < trelax !
• mass segregation occurs on relaxation timescale– also referred to as equipartition– where kinetic energy is mass independent– Hence the massive stars, with lower specific
energy sink to the centre of the gravitational potential.
12221 constant mvmvKE
• globular cluster, N=105, R=10 pc– tcross ~ 2 R / v ~ 105 years
– trelax ~ 108 years << age of cluster: relaxed• galaxy, N=1011, R=15 kpc
– tcross ~ 108 years
– trelax ~ 1015 years >> age of galaxy: collisionless
• cluster of galaxie: trelax ~ age
Dynamical Friction
• DYNAMICAL FRICTION slows a satellite on its orbit causing it to spiral towards the centre of the parent galaxy.
• As the satellite moves through a sea of stars I.e. the individual stars in the parent galaxy the satellites gravity alters the trajectory of the stars, building up a slight density enhancement of stars behind the satellite
• The gravity from the wake pulls backwards on the satellites motion, slowing it down a little
• The satellite loses angular momentum and slowly spirals inwards.
• This effect is referred to as “dynamical friction” because it acts like a frictional or viscous force, but it’s pure gravity.
• More massive satellites feel a greater friction since they can alter trajectories more and build up a more massive wake behind them.
• Dynamical friction is stronger in higher density regions since there are more stars to contribute to the wake so the wake is more massive.
• For low v the dynamical friction increases as v increases since the build up of a wake depends on the speed of the satellite being large enough so that it can scatter stars preferentially behind it (if it’s not moving, it scatters as many stars in front as it does behind).
• However, at high speeds the frictional force v-2, since the ability to scatter drops as the velocity increases.
• Note: both stars and dark matter contribute to dynamical friction
• The dynamical friction acting on a satellite of mass M moving at vs kms-1 in a sea of particles of density mXn(r) with gaussian velocity distribution
• Only stars moving slower than M contribute to the force. This is usually called the Chandrasekhar Dynamical Friction Formula.
mrnrv
rvrf s )()(2
1
2exp)(),( 32
2
• For an isotropic distribution of stellar velocities this is:
• For a sufficiently large vM, the integral converges to a definite limit and the frictional force therefore falls like vM
-2.
• For sufficiently small vM we may replace f(vM) by f(0) , define friction timescale by:
MM
v
mmm
M V
dvvvf
mMGdt
dv
M
30
2
22
)(
)(ln16
fric
MM
M
t
vvmMfG
dt
dv ))(0(ln
3
16 22
Friction & tide: effects on satellite orbit
• When there is dynamical friction there is a drag force which dissipates angular momentum. The decay is faster at pericentre resulting in the staircase-like decline of J(t).
• As the satellite moves inward the tidal force becomes greater so the tidal radius decreases and the mass will decay.
9th Lec
• Phase Space
Collisionless Systems
• stars move under influence of a smooth gravitational potential– determined by overall structure of system
• Statistical treatment of motions– collisionless Boltzman equation– Jeans equations
• provide link between theoretical models (potentials) and observable quantities.
• instead of following individual orbits
• study motions as a function of position in system
• Use CBE, Jeans eqs. to determine mass distributions and total masses
Collisionless Systems
• We showed collisions or deflections are rare• Collisionless: stellar motions under influence of
mean gravitational potential!• Rational:• Gravity is a long-distance force, decreases as r-2
– as opposed to the statistical mechanics of molecules in a box
Fluid approach:Phase Space Density
PHASE SPACE DENSITY:Number of stars per unit volume per unit velocity volume f(x,v) (all called Distribution Function DF).
The total number of particles per unit volume is given by:
313 )(kmspc
mN
volumevelocityvolumespace
mstarsofnumberv)f(x,
• E.g., air particles with Gaussian velocity (rms velocity = σ in x,y,z directions):
• The distribution function is defined by:
mdN=f(x,v)d3xd3v
where dN is the number of particles per unit volume with a given range of velocities.
• The mass distribution function is given by f(x,v).
3
2
222
o
)2(
2expnm
v)f(x,
xyx vvv
• The total mass is then given by the integral of the mass distribution function over space and velocity volume:
• Note:in spherical coordinates d3x=4πr2dr
• The total momentum is given by:
xdvdvxfxdxM total
333 ),()(
vdxdvvxfmdNvPtotal
33),(
xdxnvdNv 3)(
• The mean velocity is given by:
dN
dNv
vdvxf
vdvxfv
xmn
vdvxfvv
3
3
3
),(
),(
)(
),(
• Example:molecules in a room:
These are gamma functions
3
2
222
o
)2(
2expnm
v)f(x,
xyx vvv
• Gamma Functions:
2
1
)1()1()(
)( 1
0
nnn
dxxen nx
DF and its moments
mρ
n
dM
AdMAA
xEvvvvV
dvπvdvdΩvdvdvdvd
drπrdrdΩrdxdydzd
zyxxrddd
zyx
zyx
,
average, weighted-mass
)]([2
spherical) (if 4
spherical) (if 4
,22sin
222
223
223
222
0
2
0
v
x
vxVxVy,vx
vxx
vxxx
,)(21
,Vx ,V ),A( :
),(
),()(
222x
333
333
xVzVyVxyxVFor
dvxfddMd
dvxAfdAdMAd
kly
Additive equations
21
21
fff
BABA
1st Tutorial
g
(r) (r)
2
(E)vesc
M
10th Lec
• orbits
Liouvilles Theorem
We previously introduced the concept of phase space density. The concept of phase space density is useful because it has the nice property that it is incompessible for collisionless systems.
A COLLISIONLESS SYSTEM is one where there are no collisions. All the constituent particles move under the influence of the mean potential generated by all the other particles.
INCOMPRESSIBLE means that the phase-space density doesn’t change with time.
Consider Nstar identical particles moving in a small bundle through spacetime on neighbouring paths. If you measure the bundles volume in phase space (Vol=Δx Δ p) as a function of a parameter λ (e.g., time t) along the central path of the bundle. It can be shown that:
It can be seen that the region of phase space occupied by the particle deforms but maintains its area. The same is true for y-py and z-pz. This is equivalent to saying that the phase space density f=Nstars/Vol is constant. df/dt=0!df/dt=0!
THEOREM'LIOUVILLES',0,0 d
dNstar
d
dVol
px
x
px
x
motions in phase-space
• Flow of points in phase space corresponding to stars moving along their orbits.
• phase space coords:
• and the velocity of the flow is then:– where wdot is the 6-D vector related to w as
the 3-D velocity vector v relates to x
),(),(
),...,,(),( 621
vvxw
wwwwvx
• stars are conserved in this flow, with no encounters, stars do not jump from one point to another in phase space.
• they drift slowly through phase space
fluid analogy
• regard stars as making up a fluid in phase space with a phase space density
•
• assume that f is a smooth function, continuous and differentiable– good for N 105
f (x,v,t) f (w,t)
• as in a fluid, we have a continuity equation• fluid in box of volume V, density, and
velocity v, the change in mass is then:
– Used the divergence theorem
0
3
23
23
V
SV
SV
xdvt
SdFxdF
SdvxdtdtdM
continuity equation• must hold for any volume V, hence:
• in same manner, density of stars in phase space obeys a continuity equation:
If we integrate over a volume of phase space V, then 1st term is the rate of change of the stars in V, while 2nd term is the rate of outflow/inflow of stars from/into V.
0
0
0
3
1
3
1
6
1
6
1
i iii i
i
i
i
xvv
v
x
v
w
w
w
wf
t
f
vt
0
Collisionless Boltzmann Equation
• Hence, we can simplify the continuity equation to the CBE:
• Vector form 0
0
0
3
1
6
1
v
ffv
t
f
v
f
xx
fv
t
f
w
fw
t
f
i iiii
• in the event of stellar encounters, no longer collisionless
• require additional terms to rhs of equation
CBE cont.
• can define a Lagrangian derivative• Lagrangian flows are where the coordinates travel along with the motions
(flow)– hence x= x0 = constant for a given star
• then we have:
• and
• rate of change of phase space density seen by observer travelling with star• the flow of stellar phase points through phase space is incompressible• f around the phase point of a given star remains the same0
6
1
6
w
fw
t
f
dt
df
wtdt
d
incompressible flow
• example of incompressible flow• idealised marathon race: each runner runs at
constant speed• At start: the number density of runners is
large, but they travel at wide variety of speeds• At finish: the number density is low, but at
any given time the runners going past have nearly the same speed
11th Lec
• orbits
DF & Integrals of motion
• If some quantity I(x,v) is conserved i.e.
• We know that the phase space density is conserved i.e
• Therefore it is likely that f(x,v) depends on (x,v) through the function I(x,v), so f=f(I(x,v)).
0),(
dt
vxdI
0dt
df
Jeans theorem
• For most stellar systems the DF depends on (x,v) through generally three integrals of motion (conserved quantities), Ii(x,v),i=1..3 f(x,v) = f(I1(x,v), I2(x,v), I3(x,v))
• E.g., in Spherical Equilibrium, f is a function of energy E(x,v) and ang. mom. vector L(x,v).’s amplitude and z-component
)ˆ||,||,(),( zLLEfvxf
Analogy
• DF(x,v) Analogous to density(x,y,z), • DF(E,L,Lz) analogous to density(r,theta,phi), • E(x,v) analogous to r(x,y,z)• Integrals analogous to spherical coordinates• Isotropic F(E) analogous to spherical density(r)• Normalization dM=f(E)*dx^3*dv^3=
density(r)*dv^3
Spherical Equilibrium System
• Described by potential φ(r)• SPHERICAL: density ρ(r) depends on modulus of r.
• EQUILIBRIUM:Properties do not evolve with time.
00
),0,0()0,,0(0,0,
,
yxx
rr
000
ttt
f
Anisotropic DF f(E,L,Lz).
• Energy is conserved as:
• Angular Momentum Vector is conserved as:
• DF depends on Velocity Direction through L=r X v
0t
0
e.g., F(E,L) is an incompressible fluid
• The total energy of an orbit is given by:
),(2
1 2 trvE
00),(),(),(
dt
dL
L
LEF
dt
dE
E
LEF
dt
LEdF
0 for static potential, 0 for spherical potential
So F(E,L) constant along orbit or flow
Stress Tensor • describes a pressure which is anisotropic
– not the same in all directions
• and we can refer to a “pressure supported” system
• the tensor is symmetric.
• can chose a set of orthogonal axes such that the tensor is diagonal
• Velocity ellipsoid with semi-major axes given by
Pnx iji
2
ij2 ii
2 ij
11 , 22 , 33
n ij2
drdrdrd trr //)2(/)( 222
• To prove the above Jeans eq.
)/()(),(//
/)(),(
/)()(
///
/22
23232
223
2223
32
222
drdvdELdLEffdvrrLdrd
rvdELdLEffdv
rvdELddvvddv
rLdrddrdvv
rLEv
r
r
rrt
rr
r
222
422
2
222
00
222
4222
223
/2/)(
//),(
/)/(),(
/)/(*),(/)(
),(
)/(*),(
)/(),(
2
trr
r
r
rr
r
vL
r
rt
r
drrddrrd
vrLdELdLEf
vrdrddELdLEf
rdrdvdELdLEfrdrrd
vdELEfLdr
vrLdELdLEf
rvdELdLEffdv
r
Velocity dispersions and masses in spherical systems
• For a spherically symmetric system we have
• a non-rotating galaxy has – and the velocity ellipsoids are spheroids with
their symmetry axes pointing towards the galactic centre
• Define anisotropy
22
2222
/1
2
r
rr
vvr
dr
dnvvv
r
nvn
dr
d
v2 v
2
12th Lec
• Phase Space
Spherical mass profile from velocity dispersions.
• Get M(r) or Vcir from:
• rhs observations of dispersion and as a function of radius r for a stellar population.
2ln
ln
ln
ln
21
222
circ
2
22
rd
vd
rd
ndv
r
rGM
dr
drv
r
rGM
dr
d
r
vvn
dr
d
n
rr
rr
Total Mass of spherical systems
• E.g. Motions of globular clusters and satellite galaxies around 100kpc of MW– Need n(r), vr
2, to find M(r), including dark halo
• Several attempts all suffer from problem of small numbers N ~ 15
• For the isotropic case, Little and Tremaine TOTAL mass of 2.4 (+1.3, -0,7) 1011 Msol
• 3 times the disc need DM
• Isotropic orbits: • Radial orbits• If we assume a power law for the density
distribution
– E.g. Flat rotation a=1, Self-grav gamma=2, Radial anis.
– E.g., Point mass a=0, Tracer gam=3.5, Isotro
)(, rrMrn
0, v2 vr
2
1, v2 0
Grvvr /)(5.4M 22
Mass of the Milky Way
We find
Drop first term, solutions
cst
r
rGM
r
rvr 2
)12(
rr
GMvvr
1
5.4 22
-
2
21
2
21
22
cstr
rGM
r
rv
r
rv
dr
d
r
rGM
r
v
r
v
dr
d
rr
rr
Scalar Virial Theorem
• The Scalar Virial Theorem states that the kinetic energy of a system with mass M is just
where <v2> is the mean-squared speed of the system’s stars.
• Hence the virial theorem states that
2
2
1vMK
.2
2
rv
r
GM
M
Wv
g
Virial02 WK
• Equation of motion: dt
vd
TT
rdtTdt
vdrdt
T 00
.11
dt
rdv
dt
vrd
)(
TTT
rdtT
dtvvTT
rv
000
.11)(
rvv
This is Tensor Virial Theorem
• E.g.
• So the time averaged value of v2 is equal to the time averaged value of the circular velocity squared.
22
2
2222
)(
.
etc
cir
cir
zyx
yyx
xxx
vv
sphericalvdr
dr
r
vvvv
xvv
xvv
• In a spherical potential
rrxy
dr
rd
r
yx
r
r
y
rx
y
rx
)(
)()(
r
y
dy
dr
ydyrdr
ydrd
zyxr
)()( 22
2222
So <xy>=0 since the average value of xy will be zero.
<vxvy>=0
13th Lec
• orbits
Spherical Isotropic Self Gravitating Equilibrium Systems
• ISOTROPIC:The distribution function only depends on the modulus of the velocity rather than the direction.
02
1 2tangential
2222
yx
rzyx
vv
vf
Note:the tangential direction has and components
Isotropic DF f(E)
• In a static potential the energy of an particle is conserved.
• So,if we write f as a function of E then it will agree with the statement:
)(2
1 2 rmmvmE 0dt
dE Note:E=energy per unit mass
0),(
dt
vxdfFor incompressible fluids
• So:
• E=cst since• For a bound equilibrium system f(E) is positive
everywhere (can be zero) and is monotonically decreasing.
)(2
1,)),((),( 2 xvEvxEfvxf
0t
• SELF GRAVITATING:The masses are kept together by their mutual gravity. In non self gravitating systems the density that creates the potential is not equal to the density of stars. e.g a black hole with stars orbiting about it is not self gravitating.
Eddington Formulae
• EDDINGTON FORMULAE:These can be used to get the density as a function of r from the energy density distribution function f(E).
E
d
d
d
dE
dEf
E
dEEf
dEEEfr
0
2
0
0
8
1)()3(
)(
)(8)2(
)(2)(4)()1(
Proof of the 1st Eddington Formula
vdEfr 3)()(
0
0
)(
2
1
0
)(
2
1
)(2)(4)(
))((24
))((2
3
3
28
dEEEfr
dEEEf
dEEEf
r
r
So, from a given distribution function we can compute the spherical density.
Relation Between Pressure Gradient and Gravitational Force
• Pressure is given by:
dEEvEf
EdvEf
vdvEf
vdvEfv
P
2
122
3
2
32
32
322
2
2
3)(2
3
4
3
1
23
4)(
3
1
3
4)(
3
1
)(3
1
3
dEEEfr
rr
rr
rEEdEf
dr
d
dEEEf
Ev
dEvEEf
r
0
2
12
0
)(
2
32
2
22
1
2)(4)(since
)()(
2
3)(
3
224
)(223
4
)(2but
))((23
4
• So, this gives:
• This relates the pressure gradient to the gravitational force. This is the JEANS EQUATION.
dr
dr
dr
d )(
)( 2
Note: 2=P
drdr
d
r
2
Note:-ve sign has gone since we reversed the limits.
g
(r) (r)
2
(E)vesc
M
So, gravity, potential, density and Mass are all related and can be calculated from each other by several different methods.
2nd Tutorial
g
(r) (r)
2
(E)vesc
M
Proof: Situation where Vc2=const is a Singular
Isothermal Sphere• From Previously:
• Conservation of momentum gives:
dr
dr
dr
d
dr
dP )(
)( 2
Pg
gPt
u
1
24
24
0,rrAt
1
4
1
2
2
2
2
222
2o
2
2
222
2
2
cc
cc
cr
cr
c
oc
r
r
v
Gr
v
r
v
G
v
P
drr
v
rG
vdr
r
v
rrr
v
dr
d
• Since the circular velocity is independent of radius then so is the velocity dispersionIsothermal.
2
2
22
2
22
c
c
c
v
v
v
Finding the normalising constant for the phase space density
• If we assume the phase space density is given by:
constantconstantE
-exp
constantln2E
-exp
constant1E
-exp
constant2
exp),(
2k
2
2k
22k
2
222
r
r
nvvv
vxf zyx
constantexp)(2
E
Ef
• We can then find the normalizing constant so that (r) is reproduced.
• Note: you want to integrate f(E) over all energies that the star can have I.e. only energies above the potential
• We are integrating over stars of different velocities ranging from 0 to .
dEEEfrGr
Vr c
0
2
2
2)(4)(4
)(
)(
)(rE
dEEf
vdEfv
0
3)(
• One way is to stick the velocity into the distribution function:
• Using the substitution gives:
0
22
2
421
)(exp)( dvv
vxfmx o
2
2
2v
x
22
32 )(
exp2)( x
fx o
• Now can also be found from poissons equation. Substituting in from before gives:
• Equating the r terms gives:
2
2
2
322)(
cv
orfr
2
2
2
1 cv
rr
2cv as before.
14th Lec
• orbits
Flattened Disks
• Here the potential is of the form (R,z).• No longer spherically symmetric.• Now it is Axisymmetric
zg
Rg
zRR
RR
GzRzR
zr
2
2
4
1),(),(
Orbits in Axisymmetric Potentials (disk galaxies)
• cylindrical (R,,z) symmetry z-axis• stars in equatorial plane: same motions as in
spherically symmetric potential– non-closed rosette orbits
• stars moving out of plane– can be reduced to 2-D problem in (R,z)
– conservation of z-angular momentum, Lz
z
R
yx
R2=x2+y2
Orbits in Axisymmetric Potentials• We employ a cylindrical coordinate system (R,,z)
centred on the galactic nucleus and align the z axis with the galaxies axis of symmetry.
• Stars confined to the equatorial plane have no way of perceiving that the potential is not spherically symmetric.
Their orbits will be identical to those in spherical potentials.
• R of a star on such an orbit oscillates around some mean value as the star revolves around the centre forming a rosette figure.
Reducing the Study of Orbits to a 2D Problem
• This is done by exploiting the conservation of the z component of angular momentum.
• Let the potential which we assume to be symmetric about the plane z=0, be (R,z).
• The general equation of motion of the star is then:
• The acceleration in cylindrical coordinates is:
),(2
2
zRdt
rd
Motion in the meridional plane
• The component of angular momentum about the z-axis is conserved.
• If has no dependence on then the azimuthal angular momentum is conserved (rF=0).
• Eliminating in the energy equation using conservation of angular momentum gives:
0)( 2222 Rdt
dRLZ
constzRzRR ),(2
1 2222
ER
JzRzR z
2
222
2),()(
2
1
Specific energy density in 3D
eff
Motions in Meridional Plane• EoM in (R,z)
– :
– in cylindrical coords:
– and Lz conserved 0 02
;
),(
2
2
ˆˆ
ˆˆ
2
2
Rdt
dRR
zz
RRR
zR
zRr
zRdt
rd
zR
zR
• Thus, the 3D motion of a star in an axisymmetric potential (R,z) can be reduced to the motion of a star in a plane.
• This (non uniformly) rotating plane with cartesian coordinates (R,z) is often called the MERIDIONAL PLANE.
eff(R,z) is called the EFFECTIVE POTENTIAL.
effective potential eff(R,z)
• coupled equations for oscillations in R,z directions
• use Lz to replace by
• reduced to motion in meridional plane (R,z)
2
2
eff
effeff
2),(
;
RLzR
zz
RR
z
Lz / R2
• So the minimum in eff occurs at the radius at which a circular orbit has angular momentum Lz.
• The value of eff at the minimum is the energy of this circular orbit.
R
eff
E
2
2
2R
J z
Rcir
• The orbits are bound between two radii (where the effective potential equals the total energy) and oscillates in the z direction.
Example: Logarithmic potential• oblate galaxy with Vcirc ~ V0 =100km/s
• Draw contours of the corresponding Self-gravitating Density to show it is unphysical.
• Plot effective potential contours for Lz=100kpckm/s.
• orbits with E=Φ(1kpc,0), what is maximum z-height?
• What is Rg of a circular orbit with
E= Φ(1kpc,0)?
cstzRvzR 22202
1 2ln),(
15th Lec
• orbits
Total Angular momentum almost conserved
• These orbits can be thought of as being planar with more or less fixed eccentricity.
• The approximate orbital planes have a fixed inclination to the z axis but they process about this axis.
• star picks up angular momentum as it goes towards the plane and returns it as it leaves.
Orbital energy• Energy of orbit is (per unit mass)
• effective potential is the gravitational potential energy plus the specific kinetic energy associated with motion in direction
• orbit bound within
2
eff
eff22
21
2
222
21
22221
E
zR
R
LzR
zRRE
z
• The angular momentum barrier for an orbit of energy E is given by
• The effective potential cannot be greater than the energy of the orbit.
• The equations of motion in the 2D meridional plane then become: .
EzReff ),(
0
),(2222
zREzR eff
z
eff
eff
JR
zz
RR
2
• The effective potential is the sum of the gravitational potential energy of the orbiting star and the kinetic energy associated with its motion in the direction (rotation).
• Any difference between eff and E is simply kinetic energy of the motion in the (R,z) plane.
• Since the kinetic energy is non negative, the orbit is restricted to the area of the meridional plane satisfying .
• The curve bounding this area is called the ZERO VELOCITY CURVE since the orbit can only reach this curve if its velocity is instantaneously zero.
0E
Minimum in eff
• The minimum in eff has a simple physical significance. It occurs where:
• (2) is satisfied anywhere in the equatorial plane z=0.
• (1) is satisfied at radius Rg where
• This is the condition for a circular orbit of angular speed
)2(0
)1(03
2
z
R
L
RR
eff
zeff
23
2
0,
g
g
z
R
RR
L
Rr
conditions for a circular orbit at Rg
• minimum in effective potential at R,z = Rg,0
with angular speed
• circular orbit with angular momentum Lz
0 23
2
0,
eff
g
g
z
R
RR
L
RRg
• If the energy of the orbit is reduced the two points between which the orbit is bound eventually become one.
• You then get no radial oscillation.• You have circular orbits in the plane of the galaxy.• This is one of the closed orbits in an axisymmetric
potential and has the property that.
0
RcircularR
eff
r
(Minimum in effective potential.)
02
1),(
3
22
R
J
RRR
JzR zeffz
eff
Centrifugal Force
Gravitational force in radial direction
Nearly circular orbits: epicycles• In disk galaxies, many stars (disk stars) are
on nearly-circular orbits
• EoM:
• x=R-Rg
– expand in Taylor series about (x,z)=(0,0)
– then2/ 2/
0,at 0
;
2222
2
)0,(
2eff
2
212
)0,(
2eff
2
21
eff
effeff
effeff
zx
zz
xR
zRRzR
zz
RR
gg RR
g
• When the star is close to z=0 the effective potential can be expanded to give
22
2
2
1)0,(),( z
zz
zRzR eff
effeff
Zero, changes sign above/below z=0 equatorial plane. 2
.......2
1)0,(),( 22 zRzR effeff
zz 2
So, the orbit is oscillating in the z direction.
epicyclic approximation
• ignore all higher / cross terms:
• EoM: harmonic oscillators– epiclyclic frequency :– vertical frequency :
– with ;
3
2),(
and ,
4
2
)0,(
2
22
2
2
eff
22
g
z
R
z
R
L
R
RLzR
zzxx
g
epicycles cont.
• using the circular frequency , given by
– so that
disk galaxy: ~ constant near centre– so ~ 2
~ declines with R, » slower than Keplerian R-3/2
» lower limit is ~
in general < 2
gR
z
z
R
RR
RRR
L
RR
R
L
RRR
22
224
22
4
2
)0,(
2
4
33
1)(
R
Vrot
Example:Oort’s constants near Sun
– where R0 is the galacto-centric distance
• then 2 = -4A(A-B) + 4(A-B)2 = -4B(A-B) = -4B0
• Obs. A = 14.5 km/s /kpc and B=-12 km/s /kpc
00
21
21 ;
RR RRB
RRA
2.03.12 0
0
BA
B
the sun makes 1.3 oscillations in the radial direction per azimuthal (2) orbit– epicyclic approximation not valid for z-motions
when |z|>300 pc
16th Lec
• orbits
JEANS EQUATION for oblate rotator
: a steady-state axisymmetrical system in which 2 is isotropic and the only streaming motion is azimuthal rotation:
RR
v
R
zz
rot
22
2
)(1
)(1
• The velocity dispersions in this case are given by:
• If we know the forms of (R,z) and (R,z) then at any radius R we may integrate the Jeans equation in the z direction to obtain 2.
222
rotrot
2222
222222
isotropic sit' vfromapart since vvbut
2)(
since
),(
rot
rotrotrot
rotzr
vv
vvvvvv
vvvvzR
Obtaining 2
Inserting this into the jeans equation in the R
direction gives:
dzz
zRz
1
),(2
z
rot dzzR
R
RRv
2
1st Tutorial
g
(r) (r)
2
(E)vesc
M
• Example:
).,(density star and
stars of mass totalcalculate ./1000V assume
term),(2nd stars and (1st term) halodark todue
,)1/)(1(
)2ln(5.0),( potentialIn 2/12222
0
2220
zR
skm
kpczRv
zRvzR
s
)0,1( unphysical haverotator isotropic Show
),1()0,1(stellar Calculate
(1kpc,0)?z)(R,equator on density halodark theisWhat
2
02
kpcv
dzz
zkpckpc
rot
ss
17th Lec
• orbits
Orbits in Planar Non-Axisymmetrical Potentials
• Here the angular momentum is not exactly conserved.
• There are two main types of orbit– BOX ORBITS– LOOP ORBITS
LOOP ORBITS
• Star rotates in a fixed sense about the centre of the potential while oscillating in radius
• Star orbits between allowed radii given by its energy.
• There are two periods associated with the orbit:
– Period of the radial oscillation
– Period of the star going around 2• The energy is generally conserved and determines the
outer radius of the orbit.
• The inner radius is determined by the angular momentum.
pericentreapocentre
Box Orbits• Have no particular sense of circulation about the
centre.• They are the sum of independent harmonic
motions parallel to the x and y axes.
• In a logarithmic potential of the form
box orbits will occur when R<<Rc and loop orbits will occur when R>>Rc.
2
2222 ln
2
1),(
q
yxRvyx col
Orbits in 2D elliptical potentialsbars in nulcear regions of disc galaxies: SBs
• E.g., non-rotating logarithmic potential
– equipotential ellipses constant axial ratios, q
– for small R << Rc, expand:
– potential of a 2-D harmonic oscillator (same as for an homogenous ellipsoid)
2
22
2
2022
021
2
2222
021
2ln
1 ,ln,
qyx
R
vRv
yxRvyx
c
cL
cL
non-rotating potentials cont.• Box Orbits
– Within core, harmonic oscillators:
– unless frequencies are commensurable:– independent oscillations, stars eventually pass
close to every point inside a rectangle:
cy
cx
c
L
c
L
qRv
Rv
yqR
yv
yx
R
xv
x
00
22
20
2
20
and ;
- and ,
x / y n /m
• For large R, R >> Rc,
– numerical integrations are required
• if launched with a tangential velocity
• LOOP Orbits
• orbits rotate in fixed sense about centre of potential– oscillate in radius between Rmin and Rmax
– never approach centre!
– fills in annulus, of width determined by Lz,
Loop and Box orbits• motions in plane y=0 and with vY > 0
– with given energy E
– loop orbits are in annuli around “closed loop orbit”
» equivalent to circular orbit in axisymmetric potential
– box orbits• outermost curve denotes y=0 and vY=0; “closed box
orbit”• motion simply parallel to x-axis
x
vx
loop orbits, anticlockwise
box orbits
loop orbits, clockwise
0
0
• relative proportion of loop versus box orbits
• More box orbits when energy is lower– all orbits with E < Phi(Rc) are box orbits
• More box orbits when increasingly non-axisymmetric
• all orbits are loop orbits in axisymmetric potentials– always rotate in fixed sense (conserves L)
– loop orbits become more elongated– less epicyclic motion necessary to fill in central
hole• becomes a box orbit
Orbits in 3-D Triaxial Potentials– E.g., Elliptical Galaxies
• For given Energy have
• Within core of potential: (box orbits)– 3-D harmonic oscillator– adopt long-axis orbit as parent to family– all axial orbits stable
• Outside core : potentially 3 axial orbits and 3 loop orbits– one about each axis, stable?– short and middle-axis (box) orbits are unstable– middle-axis tube (loop) axis unstable
• have one box orbit (long axis) and two loop orbits (long-axis and short-axis) closed orbits and hence parents of non-closed families
• disc stars in spirals are short-axis tube orbits
18th Lec
• orbits
2-D rotating potentials• non-axisymmetric potentials generally
rotate!
• frame of reference (x,y,z) in which L is static rotates at angular velocity b = b ez
• EoM in this rotating cordinate system are:
centrifugal coriolis
• Effective potential:
• as used in the binary stars section
2
2
eff rr
rrr
b
bbb
22
21
eff Rb
Five Lagrange points• as in Roche-lobe potential
– L3, potential minimum at
central stationary point
– L1 and L2, saddle
points : unstable
– L4 and L5, potential
maxima: stable for
certain – Annulus bounded by L1,2,4,5 : stars appear to be
stationary, called Co-Rotation Radius
eff
x0, and
eff
y0
L5
L4
L2
L1
L3
Co-Rotation RadiusCo-Rotation: b
– where angular velocity of a star is the angular velocity of the rotating potential
• stars stay in same place relative to potential
• For R>> RCo-Rotation, all closed orbits nearly circular
• Barred potential spins much faster than stars– so asymmetry is averaged out
• For R < RCo-Rotation, most heavily populated orbits parented by long-axis (box) orbits– aligned with potential
Lindblad Resonances• barred potential• in polar coordinates (R,) in frame rotating
with potential where =0 is long axis• loop orbits as circular with small epicyclic
oscillations
• for a barred potential:– m=2 ensures a barred pot.
• small motions R1(t) are harmonic oscillator, frequency 0, driven at frequency m(0-b)
mR
RRR
tttRRtR
b cos),(
1 with
),()(),( :put then
)()( and ),()(
1
0
1
10
1010
Lindblad Resonances cont.• At Co-Rotation, no oscillations
• At Lindblad resonances, m(0-b)= ± 0, and star encounters successive crests of the potential at a frequency that coincides with natural frequency of radial oscillations.
• + sign: star overtakes potential --- Inner LR
• - sign: potential overtakes star --- Outer LR
• e.g. MW:outer LR
inner LR
bar
Lagrange Points
• There is a point between two bodies where a particle can belong to either one of them.
• This point is known as the LAGRANGE POINT.
• A small body orbiting at this point would remain in the orbital plane of the two massive bodies.
• The Lagrange points mark positions where the gravitational pull of the two large masses precisely cancels the centripetal acceleration required to rotate with them.
• At the lagrange points:
00
yxeffeff
Effective Force of Gravity
• A particle will experience gravity due to the galaxy and the satellite along with a centrifugal force and a coriolis term.
• The effective force of gravity is given by:
• The acceleration of the particle is given by:
)(2 rvggeff
Coriolis termCentrifugal term
)(2 rrr eff
Jakobi’s Energy
• The JAKOBI’S ENERGY is given by:
• Jakobi’s energy is conserved because
2
2
1rE effJ
))(()2.(
))(()2.(
.
2
1 2
rrrr
rrrrr
rrr
rdt
d
dt
d
dt
dE
effeff
eff
effJ
• For an orbit in the plane (r perpendicular to )
0
0))r(.(rr to is )r( so
0).( to is
dt
dE
rr
rrrr
J
s
sg
sg
eff
rr
GM
r
GM
rrr
rrr
)()()(2
1)()( 22
• EJ is also known as the Jakobi Integral.
• Since v2 is always positive a star whose Jakobi integral takes the value EJ will never tresspass into a region where eff(x)>EJ.
• Consequently the surface eff(x)>EJ, which we call the zero velocity surface for stars of Jakobi Integral EJ, forms an impenetrable wall for such stars.
• By taking a Taylor Expansion r=rs, you end up with:
• Where we call the radius rJ the JAKOBI LIMIT of the mass m.
• This provides a crude estimate of the tidal radius rt.
DM
m
Mm
M
mDrJ
3
13
1
33
19th Lec
• orbits
The Jeans Equations
• The DF (phase space density f) is a function of 7 variables and hence generally difficult to solve
• Can gain insights by taking moments of the equation.
• where integrate over all possible velocities– where the summation over subscripts is implicit
0
0
333
3
1
vdv
f
xvd
x
fvvd
t
f
v
f
xx
fv
t
f
iiii
i iiii
Jeans equations cont.
– first term: velocity doesn’t depend on time, hence we can take partial w.r.t. time outside
– second term, vi does not depend on xi, so we can take partial w.r.t xi outside
– third term: apply divergence theorem so that
• but at very large velocities, f 0, hence last term is zero• must be true for all bound systems
Fd 3 xV F
S d 2S
x i
fvi
d 3 v xi
fS d 2 S
ft d3 v vi
fx i
d 3 v xi
fvi
d 3 v 0
First Jeans equation
• define spatial density of stars n(x)
• and the mean stellar velocity v(x)
• then our first (zeroth) moment equation becomes
• this is the first Jeans equation• analogous to the continuity equation for a fluid.
n fd 3v
v i 1
nfvi d 3v
nt
nv i xi
0
2nd Jeans equation
• multiply the CBE by vj and then integrate over all velocities
• We get
0333
vdv
fv
xvd
x
fvvvdvf
t ij
iijij
1
0n
3vfdvvn
vv
xnvvn
xv
t
jiji
jji
ij
3rd Jeans Equation
similar to the Euler equation for a fluid flow:– last term of RHS represents some sort of pressure
force
Pvvt
v
x
n
xn
x
vvn
t
vn
i
ij
ii
j
ij
1
2
Jeans Equation
• Compact form, s=x, y, z, R, r, …
• e.g., oblate spheroid, s=[R,phi,z],– Isotropic rotator, a=[-Vrot^2/R, 0, 0],
sigma=sigma_s– Tangential anisotropic (b<0),
a=[b*sigma^2)/R, 0, 0], sigma=sigma_R=sigma_z=sigma_phi/(1-b),
ssn
na sss
2
• e.g., non-rotating sphere, s=[r,th,phi], a=[-2*b*sigma_r^2/r, 0, 0],
sigma_th=sigma_phi=(1-b)*sigma_r
20th Lec
• orbits
Applications of the Jeans Equations
• I. The mass density in the solar neighbourhood
• Using velocity and density distribution perpendicular to the Galactic disc– cylindrical coordinates.– Ignore R dependence
Vertical Jeans equation
• Small z/R in the solar neighbourhood, R~8.5 kpc, |z|< 1kpc, R-dependence neglected.
• Hence, reduces to vertical hydrostatic eq.:
z
nvnz
z
2
mass density in solar neighbourhood
• Drop R, theta in Poisson’s equation in cylindrical coordinates:
GzRR
RRR
411
2
2
2
22
Gz
42
2
local mass density = 0
Finally
• all quantities on the LHS are, in principle, determinable from observations. RHS Known as the Oort limit.
• Uncertain due to double differentiation!
Gvnznz z 4/
1- 2
local mass density
• Don’t need to calculate for all stars– just a well defined population (ie G stars, BDs etc)
– test particles (don’t need all the mass to test potential)
• Procedure– determine the number density n, and the mean square vertical velocity, v z
2, the variance of the square of the velocity dispersion in the solar neighbourhood.
– need a reliable “tracer population” of stars• whose motions do not reflect formation• hence old population that has orbited Galaxy many times• ages > several x 109 years
• N.B. problems of double differentiation of the number density n derived from observations
• need a large sample of stars to obtain vz as f(z)
– ensure that vz is constant in time
– ie stars have forgotten initial motion
local mass density
• > 1000 stars required
• Oort : 0 = 0.15 Msol pc-3
• K dwarf stars (Kuijken and Gilmore 1989)– MNRAS 239, 651
• Dynamical mass density of 0 = 0.11 Msol pc-3
• also done with F stars (Knude 1994)
• Observed mass density of stars plus interstellar gas within a 20 pc radius is 0 = 0.10 Msol pc-3
• can get better estimate of surface density
• out to 700 pc ~ 90 Msol pc-2
• from rotation curve rot ~ 200 Msol pc-2
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
21th Lec: MOND
• orbits