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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