AS 4021: Gravitational Dynamics

AS 4021 Gravitational Dynamics

AS 4021: Gravitational Dynamics

HongSheng Zhao

[email protected]

Texts:

Binney and Tremaine: Galactic Dynamics

(chpt 1-4)

notes: star-www.st-and.ac.uk/~hz4/gravdyn/gravdyn.html


Gravitational Dynamics

Can be applied to:

• Two-body systems: Binary Stars

• Planetary Systems

• Stellar clusters– open and globular

• Galactic structure– galactic nuclei/bulge/disk/halo

– black holes

• Clusters of Galaxies

• The Universe: Large Scale Structure


Example: Collisions in Globular Clusters

• Do stars hit each other?

• 105 to 106 stars in 10 pc(coll)

1

nv

• Collisional timescale

• (2 Ro)2

• n 104 pc-3

• v 10 km / s

coll 1015 years

• >> age of cluster


Importance of Gravitational DynamicsObservations:

MagnitudesSpectral lines

Proper MotionsDistribution in ( l, b)

Gravitational DynamicsVelocitiesDensities

3-D mass distribution, e.g.,Clues of Dark Matter/BHs, How galaxies form (merge)

+


Syllabus• Phase Space Fluid f(x,v)

– Eq. of motion

– Poisson’s eq. -> G

• Spherical equilibrium M(r)– Virial theorem

– Jeans eq.

• Stellar Orbits x(t),v(t)– Integrals of motion (E, J)

– Jeans theorem

• Interacting systems– Tides->Satellites->Streams

– Relaxation = Collisions

• Crisis: Fin du MOND

G

(r) x(t),v(t)


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.


Example: 5-body rectangle problem

• Four point masses m=3,4,5 at rest of three vertices of a P-triangle, integrate with time step=1 and ½ find the positions at time t=1.


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


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) ?

• Potential φ(r) is scaler, function of r only,– Easier to work with than force (vector, 3 components)

– Simply relates to specific orbital energy E= φ(r) +½v2


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: 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.


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


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


Poisson’s Equation

• Poissons equation relates the potential to the density of matter generating the potential.

• It is given by:

)(4 rGg


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


Phase Space Density f(t,x,v)

• Twinkle, twinkle, so many stars … – statistical approach

• Life is too short …– Snapshot of a galaxy/cluster

• Can you do sums? – a smooth and linear galaxy


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

AS 4021: Gravitational Dynamics

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