Gravitational Dynamics Formulae

Link phase space quantities

r

J(r,v)

K(v)

(r)

Vt

E(r,v)dθ/dt

vr

Link quantities in spheres

g(r)

(r) (r)vesc

2(r)

M(r)Vcir2 (r)

σr2(r)

σt2(r)

f(E,L)

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

POISSON’S EQUATION:

INTEGRATED FORM:

)(42 rG

rr

rr

rr

rdGrg

rr

rdGr

2

3

3

)(

)(

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

EDDINGTON FORMULAE:

)(8

1)(

)(

)(8

2)(4)(

0

2

0

0

E

d

d

d

dE

dEf

E

dEEf

dEEEfr

E

RELATING PRESSURE GRADIENT TO GRAVITATIONAL FORCE:

GOING FROM DENSITY TO MASS:

dr

dr

dr

d )(

)( 2

drrrdRM 23 4

3

4)(

GOING FROM GRAVITATIONAL FORCE TO POTENTIAL:

)()(2

2

1)()(

)(

2

rvr

vr

g

drgr

esc

esc

r

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

Conservation of momentum:

Pg

gPt

u

1

PLUMMER MODEL:

GAUSS’ THEOREM:

2

5

2

2

3

22

14

3

a

r

a

M

ar

GM

GMsdr 4).(

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

Cont:

drrr

r

v

r

rGMr

dt

vd

r

r

rcr

)()(

)()(

2

22

2

CIRCULAR SPEED:

ESCAPE SPEED:

ISOCHRONE POTENTIAL:

r

rGMvc

)(

)(2)( rrve

)()(

22 rbb

GMr

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

EQUATIONS OF MOTION IN THE MERIDIONAL PLANE:

z

eff

eff

JR

zz

RR

2

..

..

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