Laws of Gravity II

Kepler's three laws applicable if m<<M (one moving body, 1-body)

What if m & M are comparable?

Two moving bodies (2-body)Use 'reduced mass'

applications: binary stars, galaxies, (dwarf) planets (pluto-charon),detecting extra-solar planets, ...

Three-body...

N-body...

General problem

Nx3 coupled second-order diferential equations!

1-body:

Solving this yields:

d r i

d t=v i ,

d v i

d t=ai=∑ j≠i

F ji

mi

Hence, d 2 r i

d t2=∑ j≠i

G m j

∣r j−r i∣2

r j−r i

∣r j−r i∣

For N=2 with m1≫m2 , we approximate r 1≃0

So that d2 r

d t2=−

GM

r2

rr

with r=r 2 , M=m1 , and m=m2

L=m√GMa(1−e2)

E=−GMm2a

d Ad t

=L2m

=const

P2=4π

2

GMa3 (K III)

Two-body problem:reduce it to an equivalent one-body problem using the concept of 'reduced mass':

m2

m1

r2

r1

M

m

rm

An imaginary particle of mass μ,distance r away from CM (center of mass)'the reduced mass particle'

CM

Let r m=r1−r 2 , M=m1+m2

Let r m , r 1 , r2 be measured from CM

with m1 r 1+m2 r 2=0

We have r 1=m2

Mrm , r 2=−

m1

Mrm , and

∣F 12∣=∣F 21∣=G m1m2

∣r1−r2∣2=

Gm M

rm

2

where m=m1 m2

m1+m2

2-body: Energy, Angular Momentum & Kepler's Laws

M

m

Angular momentum L=m1 r1×v 1+m2 r 2×v 2=m rm×vm

(where v m=d rm

d t=v 1−v 2 )

Energy E =12

m1 v12+

12

m2v 22−

G m1 m2

∣r 1−r2∣=

12

mvm2−

GmMrm

Hence, as for 1-body case (m≪M), we have

L=m√GM am(1−e2) ,E=−

Gm M2am

together with Kepler's three Laws, e.g., P2=

4π2

GMam

3

General problem

Nx3 coupled second-order diferential equations!

1-body:

2-body:

d r i

d t=v i ,

d v i

d t=ai=∑ j≠i

F ji

mi

Hence, d2 r i

d t2=∑ j≠i

G m j

∣r j−r i∣2

r j−r i

∣r j−r i∣

For N=2 with m1≫m2 ,we approximate r1≃0

So that d 2 r

d t 2 =−GM

r2

rr

with r≃r 2 , M≃m1 , and m≃m2

For N=2 with m1∼m2 ,we use m1 r 1+m2 r 2≡0

So that d 2 r

d t 2 =−GM

r2

rr

with r≡r 2−r1 , M≡m1+m2 , and m≡m1 m2

m1+m2

L=m √GMa(1−e2)

E=−GMm

2a

d Adt

=L2m

=const

P2=4π

2

GMa3 (K III)

Kepler in the astronomer's toolkit

GM=a3 ( 2π

P )2

=a3Ω

2

v tot,circ=√GMa

a1

a2

=v 1

v 2

=m2

m1

,a1

a=

v 1

v=

m2

M

Determine masses in binary systems Mpluto

+ Mcharon

~ 1/400 ME

MCharon

/MPluto

~ 1/7

Three-body gravitational interaction:

not reducible to one-bodyIn general, motion no longer periodic (quasi-periodic or chaotic)

our closest neighbour α-Centauri (a triple system)

triple star encounter simulation(Hut)

Sun (M⊙)

Earth

Jupiter (10-3 M⊙)

3-body (practical applications):

Stability of the Solar systemvery rich phenomenology & analytical theory -- Planetary Dynamics

Climate changes on Earth

Orbital evolution of near-earthasteroids and comets.

Voyagers getting swings from outer planets -- gravity-assist

Space travel

N-body

intractable analyticallyrelies on numerical integration

globular cluster ~ 106 stars

cluster of galaxies ~ 103 galaxies

Universe ~ 1011 galaxies

Globular Cluster M4(~ 2 kpc)

Quiz: cannibalism in close binary stars

star m1 bloats up, part (d m) of its envelope becomes dominated by

the gravity of m2 and is transferred from m

1 to m

2

-- does the binary unbind or spiral-in?

How to estimate?● Use energy conservation?● Use angular momentum conservation?

Cataclysmic Variable

m1

m2

Dana Berry