as4021 gravitational dynamics 1 gravitational dynamics: an introduction hongsheng zhao
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AS4021 Gravitational Dynamics 1
Gravitational Dynamics: An Introduction
HongSheng Zhao
AS4021 Gravitational Dynamics 2
C1.1.1 Our Galaxy and Neighbours
• How structure in universe form/evolve?
• Galaxy Dynamics Link together early universe & future.
AS4021 Gravitational Dynamics 3
Our Neighbours
• M31 (now at 500 kpc) separated from MW a Hubble time ago
• Large Magellanic Cloud has circulated our Galaxy for about 5 times at 50 kpc– argue both neighbours move with a typical
100-200km/s velocity relative to us.
AS4021 Gravitational Dynamics 4
Outer Satellites on weak-g orbits around Milky Way
~ 50 globulars on weak-g (R<150 kpc)~100 globulars on strong-g (R< 10 kpc)
R>10kpc: Magellanic/Sgr/Canis streamsR>50kpc: Draco/Ursa/Sextans/Fornax…
AS4021 Gravitational Dynamics 5
C1.1.2 Milky Way as Gravity Lab
• Sun has circulated the galaxy for 30 times– velocity vector changes direction +/- 200km/s
twice each circle ( R = 8 kpc )– ArgueArgue that the MW is a nano-earth-gravity Lab– Argue that the gravity due to 1010 stars only
within 8 kpc is barely enough. Might need to add Dark Matter.
AS4021 Gravitational Dynamics 6
Sun escapes unless our Galaxy has Dark Matter
AS4021 Gravitational Dynamics 7
C1.1.3 Dynamics as a tool
• Infer additional/dark matter– E.g., Weakly Interacting Massive Particles
• proton mass, but much less interactive
• Suggested by Super-Symmetry, but undetected
– A $billion$ industry to find them. • What if they don’t exist?
AS4021 Gravitational Dynamics 8
…
• Test the law of gravity: – valid in nano-gravity regime?– Uncertain outside solar system:
• GM/r2 or cst/r ?
AS4021 Gravitational Dynamics 9
Outer solar system
• The Pioneer experiences an anomalous non-Keplerian acceleration of 10-8 cm s-2
• What is the expected acceleration at 10 AU?
• What could cause the anomaly?
AS4021 Gravitational Dynamics 10
Gravitational Dynamics can be applied to:
• Two body systems:binary stars
• Planetary Systems, Solar system
• Stellar Clusters:open & globular
• Galactic Structure:nuclei/bulge/disk/halo
• Clusters of Galaxies
• The universe:large scale structure
AS4021 Gravitational Dynamics 11
Topics• 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
• MOND
AS4021 Gravitational Dynamics 12
C2.1 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.
AS4021 Gravitational Dynamics 13
C2.2 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 ∑
∑
=
=
−⋅⋅
−==Φ
∇−=−
⋅⋅⋅−==
N
i i
i
N
ir
i
i
Rr
mmGrm
mRr
rmmGrgmF
1
12
12
)(
ˆ)(
vv
rvv
rr
φ
φ
AS4021 Gravitational Dynamics 14
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
mmaiRRrmG
r
vr
vvr
vvv
vo
φφφφ
φ
φ
AS4021 Gravitational Dynamics 15
C2.3 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 vv−
⋅−=φ
212ˆ
iRr
rdMGgd vvr
−
⋅⋅−=
AS4021 Gravitational Dynamics 16
Lec 2 (Friday, 10 Feb): Why Potential φ(r) ?
• Potential per unit mass φ(r) is scalar,– function of r only,– Related to but easier to work with than force
(vector, 3 components) – Simply relates to orbital energy E= φ(r) + ½ v2
AS4021 Gravitational Dynamics 17
C2.4 Poisson’s Equation
• PE relates the potential to the density of matter generating the potential by:
• [BT2.1]
)(4 rGg ρπφ =⋅∇−=∇⋅∇rr
AS4021 Gravitational Dynamics 18
C2.5 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 )()( Φ∇−==⎥⎦
⎤⎢⎣
⎡ rrvv
AS4021 Gravitational Dynamics 19
Example 1: trajectories when G=0
• Solve Poisson’s Eq. with G=0 – F=0, Φ(r)=cst,
• Solve EoM for particle i initially at (Xi,0, V0,i)– dVi/dt = Fi/mi = 0 Vi = cst = V0,i – dXi/dt = Vi = Vi,0 Xi(t) = Vi,0 t + Xi,0, – where X, V are vectors,
straight line trajectories • E.g., photons in universe go straight
– occasionally deflected by electrons,– Or bent by gravitational lenses
AS4021 Gravitational Dynamics 20
What have we learned?
• Implications on gravity law and DM.
• Poisson’s eq. and how to calculate gravity
• Equation of motion
AS4021 Gravitational Dynamics 21
How N-body system evolves
• Start with initial positions and velocities of all N particles.
• Calculate the mutual gravity on each particle– Update velocity of each particle for a small time step dt
with EoM
– Update position of each particle for a small time step dt
• Repeat previous for next time step. N-body system fully described
AS4021 Gravitational Dynamics 22
C2.6 Phase Space of Galactic Skiers • Nskiers identical particles moving in a small bundle
in phase space (Vol =Δx Δ v),
• phase space deforms but maintains its area.
• Gap widens between faster & slower skiers– but the phase volume & No. of skiers are constants.
+x front
vx
x
+Vx Fast
AS4021 Gravitational Dynamics 23
“Liouvilles Theorem on the piste”
• Phase space density of a group of skiers is const. f = m Nskiers / Δx Δvx = const
Where m is mass of each skier,
[ BT4.1]
AS4021 Gravitational Dynamics 24
C2.7 density of phase space fluid: Analogy with air molecules
• air with uniform density n=1023 cm-3
Gaussian velocity rms velocity σ =0.3km/s in x,y,z directions:
• Estimate f(0,0,0,0,0,0) in pc-3 (km/s)-3
3
2
222
o
)2(
2expnm
v)f(x,σπ
σ ⎟⎟⎠
⎞⎜⎜⎝
⎛ ++−×
=
xyx vvv
AS4021 Gravitational Dynamics 25
Lec 3 (Valentine Tuesday)C2.8 Phase Space Distribution Function (DF)
PHASE SPACE DENSITY: No. of sun-like stars per unit volume per velocity volume f(x,v)
sun sun3 3
sun3 1 3
m number of suns mf(x,v)
space volume velocity volume
1 m
pc (100kms )
dN
dx dv
−
× ×= =
××
×:
AS4021 Gravitational Dynamics 26
C2.9 add up stars: integrate over phase space
• star mass density: integrate velocity volume
• The total mass : integrate over phase space
xdvdvxfxdxM total
vvvv 333 ),()( ∫∫ == ρ
AS4021 Gravitational Dynamics 27
• define spatial density of stars n(x)
• and the mean stellar velocity v(x)
• E.g., Conservation of flux (without proof)( ) ( ) ( )
3
3
2 31
1 2 3
flux in i-direction
0
i i
n fd v
nv fv d v
nv nv nvn
t x x x
∂ ∂ ∂∂
∂ ∂ ∂ ∂
≡
≡ =
+ + + =
∫
∫
AS4021 Gravitational Dynamics 28
C3.0 Star clusters differ from air:
• Stars collide far less frequently– size of stars<<distance between them– Velocity distribution not isotropic
• Inhomogeneous density ρ(r) in a Grav. Potential φ(r)
AS4021 Gravitational Dynamics 29
Example 2: A 4-body problem
• Four point masses with G m = 1 at rest (x,y,z)=(0,1,0),(0,-1,0),(-1,0,0),(1,0,0). Show the initial total energy
Einit = 4 * ( ½ + 2-1/2 + 2-1/2) /2 = 3.8
• Integrate EoM by brutal force for one time step =1 to find the positions/velocities at time t=1.– Use V=V0 + g t = g = (u, u, 0) ; u = 21/2/4 + 21/2/4 + ¼ = 0.95
– Use x= x0 + V0 t = x0 = (0, 1, 0).
• How much does the new total energy differ from initial? E - Einit = ½ (u2 +u2) * 4 = 2 u2 = 1.8
AS4021 Gravitational Dynamics 30
Often-made Mistakes
• Specific energy or specific force confused with the usual energy or force
• Double-counting potential energy between any pair of mass elements, kinetic energy with v2
• Velocity vector V confused with speed, • 1/|r| confused with 1/|x|+1/|y|+1/|z|
AS4021 Gravitational Dynamics 31
What have we learned?
Potential to Gravity
Potential to density
Density to potential
Motion to gravity
g φ=−∇
∫ ′−
′−=
rr
rdGr vv
vv 3
)(ρ
φ
/g dv dt=
AS4021 Gravitational Dynamics 32
Concepts
• Phase space density– incompressible– Dimension Mass/[ Length3 Velocity3 ]
– Show a pair of non-relativistic Fermionic particle occupy minimal phase space (x*v)3 > (h/m)3 , hence has a maximum phase density =2m (h/m)-3
AS4021 Gravitational Dynamics 33
Where are we heading to?Lec 4, Friday 17 Feb
• potential and eqs. of motion – in general geometry– Axisymmetric– spherical
AS4021 Gravitational Dynamics 34
Link phase space quantities
r
J(r,v)
Ek(v)
(r)
Vt
E(r,v)dθ/dt
vr
AS4021 Gravitational Dynamics 35
C 3.1: 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
φθθθ
θθ
φ
∂∂
+⎟⎠
⎞⎜⎝
⎛∂∂
∂∂
+⎟⎠
⎞⎜⎝
⎛∂∂
∂∂
=∇
∂∂
+∂∂
+⎟⎠
⎞⎜⎝
⎛∂∂
∂∂
=∇
∂∂
+∂∂
+∂∂
=∇
rrrr
rr
zRRR
RR
zyx
AS4021 Gravitational Dynamics 36
Example 3: Energy is conserved in STATIC potential
• The orbital energy of a star is given by:
),(2
1 2 trvEvφ+=
0dE dv dr
vdt dt dt t t
φ φφ ∂ ∂= + ∇ + = +
∂ ∂
v vv
0 since
and
φ−∇=dt
vdv
vdt
rd vv=
0 for static potential.
So orbital Energy is Conserved dE/dt=0 only in “time-independent” potential.
AS4021 Gravitational Dynamics 37
Example 4: Static Axisymmetric density Static Axisymmetric potential
• We employ a cylindrical coordinate system (R,,z) e.g., centred on the galaxy and align the z axis with the galaxy axis of symmetry.
• Here the potential is of the form (R,z).• Density and Potential are Static and Axisymmetric
– independent of time and azimuthal angle
zg
Rg
zRR
RR
GzRzR
zr ∂∂
−=∂∂
−=
⎥⎦
⎤⎢⎣
⎡
∂∂
+⎟⎠
⎞⎜⎝
⎛∂∂
∂∂
=⇒
φφ
φφπ
ρφ 2
2
41
),(),(
AS4021 Gravitational Dynamics 38
C3.2: Orbits in an axisymmetric potential
• 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
• Eqs. of motion in cylindrical coordinates
),(2
2
zRdt
rd φ−∇=v
Eq. of Motion
AS4021 Gravitational Dynamics 39
Conservation of angular momentum z-component Jz if axisymmetric
• The component of angular momentum about the z-axis is conserved.
• If (R,z) has no dependence on θ then the azimuthal angular momentum is conserved– or because z-component of the torque rF=0. (Show it)
2 2( ) 0Z
d dJ R Jz R
dt dtθ θ= ⇒ = =& &
AS4021 Gravitational Dynamics 40
C4.1: Spherical Static System
• Density, potential function of radius |r| only
• Conservation of – energy E, – angular momentum J (all 3-components)– Argue that a star moves orbit which confined to
a plane perpendicular to J vector.
AS4021 Gravitational Dynamics 41
C 4.1.0: Spherical Cow Theorem
• Most astronomical objects can be approximated as spherical.
• Anyway non-spherical systems are too difficult to model, almost all models are spherical.
AS4021 Gravitational Dynamics 42
Globular: A nearly spherical static system
AS4021 Gravitational Dynamics 43
C4.2: 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
πρ
ππ
ρ
ρππρ
AS4021 Gravitational Dynamics 44
C4.3: Theorems on Spherical Systems
• 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. [BT 2.1]
AS4021 Gravitational Dynamics 45
C4.4: Poisson’s eq. in Spherical systems
• Poisson’s eq. in a spherical potential with no θ or Φ dependences is:
• BT2.1.2
)(41 2
22 rG
rr
rrρπ =⎟
⎠
⎞⎜⎝
⎛∂∂
∂∂
=∇
AS4021 Gravitational Dynamics 46
Example 5: Interpretation 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
ρπ
φφ
∫
∫
∫
∞
∞
∞
=
−=
<∞==
−=
AS4021 Gravitational Dynamics 47
• Take d/dr and multiply r2
• Take d/dr and divide r2
( )∫==−= drrrGrGMgrdr
dr )(4)( 222 ρπ
φ
( ) ( )
ρπφ
ρπ
Gg
rGGMrr
grrrr
rrr
4.
)(4111
2
22
22
2
=∇−=∇→
=∂
∂=−
∂
∂=⎟
⎠
⎞⎜⎝
⎛∂
∂
∂
∂
v
AS4021 Gravitational Dynamics 48
C4.5: Escape Velocity• ESCAPE VELOCITY= velocity required in
order for an object to escape from a gravitational potential well and arrive at with zero KE.
21( ) ( )
2
( ) 2 ( ) 2 ( )
esc
esc
r v
v r r
φ φ
φ φ
= ∞ −
→ = ∞ −
=0 often
AS4021 Gravitational Dynamics 49
Example 6: Plummer Model for star cluster
• A spherically symmetric potential of the form:
• Show corresponding to a density (use Poisson’s eq): 2
5
2
2
31
4
3−
⎟⎟⎠
⎞⎜⎜⎝
⎛+=
ar
aMπ
ρ
e.g., for a globular cluster a=1pc, M=105 Sun Mass show Vesc(0)=30km/s
AS4021 Gravitational Dynamics 50
What have we learned?
• Conditions for conservation of orbital energy, angular momentum of a test particle
• Meaning of escape velocity
• How Poisson’s equation simplifies in cylindrical and spherical symmetries
AS4021 Gravitational Dynamics 51
Lec 5, (Tue 21 Feb)Links of quantities in spheres
g(r)
(r) ρ(r)
M(r)
AS4021 Gravitational Dynamics 52
A worked-out example 7: Hernquist Potential for stars in a galaxy
• E.g., a=1000pc, M0=1010 solar, show central escape velocity Vesc(0)=300km/s,
• Show M0 has the meaning of total mass
– Potential at large r is like that of a point mass M0
– Integrate the density from r=0 to inifnity also gives M0
0*
1 2
0* 3
( ) , use Poisson eq. show
( ) 12
GMr
a r
M r rr
a a a
φ
ρπ
− −
=−+
⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
AS4021 Gravitational Dynamics 53
Potential of globular clusters and galaxies looks like this:
r
a
GM o−
-1
(0) (finite well at centre)
( ) r (Kepler for large r)
is the minimum of potential with escape velocity
2(0) o
esc
const
r
Centre
GMv
a
φφ
=−
∝→
=
AS4021 Gravitational Dynamics 54
C4.6: 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
)(
)(
=⇒
=∇== φ
For a point mass M: Show in a uniform density sphere
r
GMrvc =)( ρπρπ 3
3
4M(r)since
3
4)( rr
Grvc ==
AS4021 Gravitational Dynamics 55
What have we learned?
• How to apply Poisson’s eq.
• How to relate – Vesc with potential and – Vcir with gravity
• The meanings of – the potential at very large radius,– The enclosed mass
AS4021 Gravitational Dynamics 56
Lec 6, (Fri, 24 Feb)Links of quantities in spheres
g(r)
(r) ρ(r)
vesc
M(r)
Vcir
AS4021 Gravitational Dynamics 57
C4.7: Motions in spherical potential [BT3.1]
If spherical
0
rg r
gθ
φ
φθ
∂=−
∂∂
=− =∂
φ−∇==
=
gvdtddtd
motion ofEquation
v
vx
0
00
)(
)(
gravity no If
vv
xvx
=+=
ttt 2
Conserved if spherical static
1E ( )
2ˆt
v r
L J rv n
φ= +
= = ⊗ = ⋅x v
AS4021 Gravitational Dynamics 58
Another Proof: Angular Momentum is Conserved if spherical
•
vrLvvv
×=( )
0d r vdL dr dv
v r r gdt dt dt dt
×= = × + × = + ×
v v v v vrv v v
Since 0=∂∂θφ
then the spherical force g is in the r direction, no torque
both cross products on the RHS = 0.
So Angular Momentum L is Conserved
0dL
dt=
v
AS4021 Gravitational Dynamics 59
C4.8: Star moves in a plane (r,θ) perpendicular to L
• Direction of angular momentum
• equations of motion are– radial acceleration:– tangential acceleration:
2
2
2
( )
2 0
constant
r r L
r r g r
r r
r L
θθ θθ
× =
− =
+ =
= =
&
&&&& &&&&
AS4021 Gravitational Dynamics 60
C4.8.1: 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 ==
=×=
dtddtrd
rL
θ
vvv
Keplers 3rd law
AS4021 Gravitational Dynamics 61
C 4.9: Radial part of motion• Energy Conservation (WHY?)
2
2
2
22
2
1
2)(
2
1
2
1)(
⎟⎠
⎞⎜⎝
⎛++=
⎟⎠
⎞⎜⎝
⎛+⎟⎠
⎞⎜⎝
⎛+=
dtrd
rL
r
dtd
rdtrd
rE
v
v
φ
θφ
eff
)(22 rEdt
dreffφ−±=
AS4021 Gravitational Dynamics 62
C5.0: Orbit in the z=0 plane of a disk potential (R,z).
• Energy/angular momentum of star (per unit mass)
• orbit bound within
( )2212
221
2 2
21eff2
eff
( ,0)
[ ( ,0)] 2
(R,0)
( ,0)
z
E R R R
JR R
R
R
E R
θ⎡ ⎤= + + Φ⎢ ⎥⎣ ⎦
⎡ ⎤= + + Φ⎣ ⎦
⎡ ⎤= + Φ⎣ ⎦
⇒ ≥ Φ
&&
&
&
AS4021 Gravitational Dynamics 63
C5.1: Radial Oscillation• An orbit is bound between two radii: a loop• Lower energy E means thinner loop (nearly
circular closed) orbit
R
eff
E
2
2
2R
J z
Rcir
AS4021 Gravitational Dynamics 64
C5.2: Eq of Motion for planar orbits
• EoM:
eff
eff
; z=0
If circular orbit R=cst, 0 => 0 at c
RR
z R RR
∂∂
∂∂
Φ=−
Φ= = =
&&
AS4021 Gravitational Dynamics 65
Apocenter
Pericenter
AS4021 Gravitational Dynamics 66
Apocenter
Peri
Apo
AS4021 Gravitational Dynamics 67
C5.3: Apocenter and pericenter
• No radial motion at these turn-around radii– dr/dt =Vr =0 at apo and peri
• Hence• Jz = R Vt
= RaVa = RpVp
• E = ½ (Vr2 + Vt
2 ) + Φ (R,0) = ½ Vr2 + Φeff (R,0)
= ½ Va2 + Φ (Ra,0)
= ½ Vp2 + Φ (Rp,0)
AS4021 Gravitational Dynamics 68
Lec7: Orbit in axisymmetric disk potential (R,z).
• Energy/angular momentum of star (per unit mass)
• orbit bound within
( )22 212
22 21
2 2
2 21eff2
eff
2
(R,z)
( , )
z
E R R z
JR z
R
R z
E R z
θ⎡ ⎤= + + + Φ⎢ ⎥⎣ ⎦
⎡ ⎤= + + + Φ⎣ ⎦
⎡ ⎤= + + Φ⎣ ⎦
⇒ ≥ Φ
&& &
& &
& &
AS4021 Gravitational Dynamics 69
C5.4 EoM for nearly circular orbits
• EoM:
• Taylor expand
– x=R-Rg
eff eff
eff eff
2 22 2eff eff1 1
eff 2 22 2
( ,0) ( ,0)
2 2 2 2
;
0 at , 0
/ 2 / 2 ...
g g
g
R R
R zR z
R R zR z
x zR z
x z
∂ ∂∂ ∂
∂ ∂∂ ∂
∂ ∂∂ ∂
κ ν
Φ Φ=− =−
Φ Φ= = = =
⎛ ⎞ ⎛ ⎞Φ ΦΦ = + +⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
= + +
&& &&
K
AS4021 Gravitational Dynamics 70
Sun’s Vertical and radial epicycles
• harmonic oscillator +/-10pc every 108 yr – epicyclic frequency :– vertical frequency :
2 2 R , and
R z zκ ν=− =−&& &&
κ
ν
AS4021 Gravitational Dynamics 71
Lec 8, C7.0: Stars are not enough: add Dark Matter in galaxies [BT10.4]
NGC 3198 (Begeman 1987)
AS4021 Gravitational Dynamics 72
Bekenstein & Milgrom (1984) Bekenstein (2004), Zhao & Famaey (2006)
• Modify gravity g, – Analogy to E-field in medium of varying Dielectric
• Gradient of Conservative potential
*
0 0
0
0 0
g( )
4 G
G(g/a ) = (1+ a /g) G
~ G if g= | |
~ G a / > G if g <a
r
a
g
ρπ
φ
⎛ ⎞−∇• =⎜ ⎟⎝ ⎠
∇ >
Not Standard Belief!
AS4021 Gravitational Dynamics 73
Researches of Read & Moore (2005)
MOND and DM similar in potential, rotation curves and orbits... Personal view!
AS4021 Gravitational Dynamics 74
Explained: Fall/Rise/wiggles in Ellip/Spiral/Dwarf galaxies
Views of Milgrom & Sanders, Sanders & McGaugh
AS4021 Gravitational Dynamics 75
What have we learned?
• Orbits in a spherical potential or in the mid-plane of a disk potential
• How to relate Pericentre, Apocentre through energy and angular momentum conservation.
• Rotation curves of galaxies– Need for Dark Matter or a boosted gravity
AS4021 Gravitational Dynamics 76
Tutorial: 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 diagrams of M(r), Vesc(r), Vcir(r), |Phi(r)|, rho(r), |g(r)| vs. r (assume V0=200km/s, r0=100kpc).
r
GMr 0)( −=φ
oor
rvr φφ += ln)( 20
20000 / vrGM −=−=φ
2
20
4)(
Gr
Vr
πρ =
AS4021 Gravitational Dynamics 77
Another Singular Isothermal Sphere• Consider a potential Φ(r)=V0
2ln(r). • Use Jeans eq. to show the velocity dispersion σ (assume isotropic) is
constant V02/n for a spherical tracer population of density A*r-n ; Show
we required constants A = V02/(4*Pi*G). and n=2 in order for the tracer
to become a self-gravitating population. Justify why this model is called Singular Isothermal Sphere.
• Show stars with a phase space density f(E)= exp(-E/σ2) inside this potential well will have no net motion <V>=0, and a constant rms velocity σ in all directions.
• Consider a black hole of mass m on a rosette orbit bound between pericenter r0 and apocenter 2r0 . Suppose the black hole decays its orbit due to dynamical friction to a circular orbit r0/2 after time t0. How much orbital energy and angular momentum have been dissipated? By what percentage has the tidal radius of the BH reduced? How long would the orbital decay take for a smaller black hole of mass m/2 in a small galaxy of potential Φ(r)=0.25V0
2ln(r). ? Argue it would take less time to decay from r0 to r0 /2 then from r0/2 to 0.
AS4021 Gravitational Dynamics 78
Lec 9: C8.0: Incompressible df/dt=0• Nstar identical particles moving in a small
bundle in phase space (Vol=Δx Δ p), • phase space deforms but maintains its area.
– Likewise for y-py and z-pz.
Phase space density f=Nstars/Δx Δ p ~ const
THEOREM'LIOUVILLES',0,0 ==λλ d
dNstard
dVol
px
x
px
x
AS4021 Gravitational Dynamics 79
C8.1: Stars flow in phase-space[BT4.1]
• Flow of points in phase space ~
stars moving along their orbits.
• phase space coords: ),(),(
),...,,(),( 621
Φ−∇==≡≡
vvxwwwwwvx
&&&
AS4021 Gravitational Dynamics 80
C8.2 Collisionless Boltzmann Equation
• Collisionless df/dt=0:
• Vector form
6
1
3
1
d ( , , ) ( , ) 0
dt
0
0
ii i i i
f x v t w f w tt w
f f fv
t x x v
f fv f
t v
αα α
∂ ∂∂ ∂
∂ ∂ ∂ ∂∂ ∂ ∂ ∂
∂ ∂∂ ∂
=
=
⎛ ⎞= + =⎜ ⎟
⎝ ⎠
⎡ ⎤Φ+ − =⎢ ⎥
⎣ ⎦
+ ⋅∇ −∇Φ⋅ =
∑
∑
&
AS4021 Gravitational Dynamics 81
C8.3 DF & its 0th ,1st , 2nd moments
3 3
3
3 3 3
x
2 2 2 2x y z
( , )( )
( , )
where A( , ) 1, V , ...
e.g., verify V V V V
d Af x v dA
d
d dM d f x v d
ρ
ρ
=
= =
=
= + +
∫∫∫
∫∫∫x y
x vx
x
x x v
x v ,V V
AS4021 Gravitational Dynamics 82
• E.g: rms speed of air particles in a box dx3 :2 2 2
o 2
3
m n exp2
f(x,v)( 2 )
x y zv v v
σ
πσ
⎛ ⎞+ +× −⎜ ⎟⎜ ⎟
⎝ ⎠=
AS4021 Gravitational Dynamics 83
C8.4: CBE Moment/Jeans Equations [BT4.2]
• Phase space incompressible
df(w,t)/dt=0, where w=[x,v]: CBE
• taking moments U=1, vj, vjvk by integrating over all possible velocities
6 3
1 1
3 3 3
0
0
ii i i i
ii i i
f f f f fw v
t w t x x v
f f fUd v v Ud v Ud vt x x v
αα α
∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂
∂ ∂ ∂ ∂∂ ∂ ∂ ∂
= =
⎡ ⎤Φ+ = + − =⎢ ⎥
⎣ ⎦
Φ+ − =
∑ ∑
∫ ∫ ∫
&
AS4021 Gravitational Dynamics 84
C8.5: 0th moment (continuity) eq.
• define spatial density of stars n(x)
• and the mean stellar velocity v(x)
• then the zeroth moment equation becomes( )
3
3
1,2,3
1
0
i i
i
i i
n fd v
v fv d vn
nvn
t x
∂∂
∂ ∂=
≡
≡
+ =
∫
∫
∑
AS4021 Gravitational Dynamics 85
C8.6: 2nd moment Equation
similar to the Euler equation for a fluid flow:– last term of RHS represents pressure force
( ) ( )
( )
2
1,2,3 1,2,3
2
( )( )
1
j ijji
i ii j i
i jij i i j j i j
v nvv
t x x n x
v v v v VV V V
vv v P
t
∂ ∂ σ∂ ∂
∂ ∂ ∂ ∂
σ
∂
∂ ρ
= =
Φ→ + = − −
= − − = −
+ ⋅∇ = −∇Φ − ∇
∑ ∑
( ) Pnx iji
−∇⇔− 2 σ∂
∂
AS4021 Gravitational Dynamics 86
Lec 10, C8.7: Meaning of pressure in star system
• What prevents a non-rotating star cluster collapse into a BH? – No systematic motion as in Milky Way disk.– But random orbital angular momentum of stars!
AS4021 Gravitational Dynamics 87
C8.8: Anisotropic Stress Tensor • describes a pressure which is
– perhaps stronger in some directions than other
• the tensor is symmetric, can be diagonalized– velocity ellipsoid with semi-major axes given
by
σ ij2 = σ ii
2δ ij
σ11 ,σ 22 ,σ 33
2ij ijP nσ=
AS4021 Gravitational Dynamics 88
C8.9: Prove Tensor Virial Theorem (BT4.3)
• Many forms of Viral theorem, E.g.
2 2 2 2
etc
.
Scaler Virial Theorem
2 0
x x x
x y y
x y z
v v x
v v x
v v v v
r
drdr
K W
φφ
φφ
< >=< ∂ >< >=< ∂ >
< >=< > + < > + < >
=< ∇ >
=< >
⇒ + =
v
2 ij ij
vv r
K W
φ⇒< >=< ∇ >=−
vv v
AS4021 Gravitational Dynamics 89
C9.0: Jeans theorem [BT4.4]
• For most stellar systems the DF depends on (x,v) through generally 1,2 or 3 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 ⋅=vv
AS4021 Gravitational Dynamics 90
C9.1: An anisotropic incompressible spherical fluid, e.g, f(E,L) =exp(-E/σ0
2)L2β [BT4.4.4]
• Verify <Vr2> = σ0
2, <Vt2>=2(1-β) σ0
2
• Verify <Vr> = 0
• Verify CBE is satisfied along the orbit or flow:
( , ) ( , ) ( , )0 0
df E L f E L dE f E L dL
dt E dt L dt
∂ ∂= + = +
∂ ∂
0 for static potential, 0 for spherical potential
So f(E,L) indeed constant along orbit or flow
AS4021 Gravitational Dynamics 91
C9.2: Apply JE & PE to measure Dark Matter [BT4.2.1d]
• A bright sub-component of observed density n*(r) and anisotropic velocity dispersions
<Vt2>= 2(1-β)<Vr
2>
• in spherical potential φ(r) from total (+dark) matter density ρ(r)
( )2 2
2
2
02
21JE: - *
*
( )4PE:
t rr
r
v vd dn v
n dr r dr
G r r dr d
r dr
ρ π
− Φ+ =
Φ=
∫
[
AS4021 Gravitational Dynamics 92
C9.3: Measure total Matter density ρr)
Assume anistropic parameter beta, substitute JE for stars of density n* into PE, get
• all quantities on the LHS are, in principle, determinable from observations.
( )222
2
*2 - ( ) ( )
4 *
rr
d n vvd rr
r dr G r n dr
βρ
⎡ ⎤⎢ ⎥+ =⎢ ⎥π⎢ ⎥⎣ ⎦
AS4021 Gravitational Dynamics 93
C9.4: Spherical Isotropic f(E) Equilibriums [BT4.4.3]
• ISOTROPIC β=0:The distribution function f(E) only depends on |V| the modulus of the velocity, same in all velocity directions.
Note:the tangential direction has θ and components
AS4021 Gravitational Dynamics 94
C9.5: subcomponents
• Non-SELF-GRAVITATING: There are additional gravitating matter
• The matter density that creates the potential is NOT equal to the density of stars. – e.g., stars orbiting a black hole is non-self-
gravitating.
AS4021 Gravitational Dynamics 95
C9.6: subcomponents add upto the total gravitational mass
1 2
1 2
Phase density of stars plus dark matter
density of stars plus dark matter
Total density shared total potential
A B A B
f f f
ρ ρ ρ
ρ
+ = +
= +
= +
==>Φ
AS4021 Gravitational Dynamics 96
What have we learned?
• Meaning of anistrpic pressure and dispertion.
• Usage of Jeans theorem [phase space]
• Usage of Jeans eq. (dark matter)
• Link among quantities in sphere.
AS4021 Gravitational Dynamics 97
C10.0: Galactic disk mass density from vertical equilibrium
• Use JE and PE in cylindrical coordinates.
• drop terms 1/R or d/dR ( d/dz terms dominates).– |z|< 1kpc < R~8kpc near Sun
• Combine vertical hydrostatic eq. and gravity eq. :
• RHS are observables
• => JE/PE useful in weighing the galaxy disk.
( )2 ** zn v
n z z
∂ ∂
∂ ∂
Φ≈ −
2
24 G
z
∂ρ∂Φ
π ≈
( )z=1kpc
2
1z=-1kpc
total density in a column 1kpc high
2= dz *
4 * zz kpc
n vGn z
∂ρ
∂ =
⇒
⎡ ⎤= ⎢ ⎥π⎣ ⎦∫
AS4021 Gravitational Dynamics 98
A toy galaxy
2 2 2 2 2 2 2 1/ 20 0 ( , ) 0.5 ln( 2 ) (1 ( ) /1 ) ,
0 100 / . Argue 1st & 2nd terms of above
galaxy potential resemble dark halo and stars respectively.
R z v R z v R z kpc
v km s
φ −= + − + +=
Calculate the circular velocity and dark halo density
on equator (R,z) (1kpc,0)
Estimate the total mass of stars and dark matter inside 10kpc.
Estimate the star column density inside |z|<0.1kpc, R=1kpc.
=
If you like challenge…
Learn to analyse
AS4021 Gravitational Dynamics 99
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
=
∂+
∂∂+
∂∂=
∂+∂+∂∂=
∂+∂+∂=∇•∇
−×≈
−×≈
≈≈
θ
φ
θθθθ
φ
AS4021 Gravitational Dynamics 100
Thinking of a globular cluster …
AS4021 Gravitational Dynamics 101
C10.1: Link phase space quantities
r
J(r,v)
K(v)
(r)
Vt
E(r,v)dθ/dt
vr
AS4021 Gravitational Dynamics 102
C10.2: Link quantities in spheres
g(r)
(r) ρ(r)vesc
2(r)
M(r)Vcir2 (r)
σr2(r)
σt2(r)
f(E,L)