Laws of Gravity II General problem d r d v F d 2 r G m r −r Kepler's three laws applicable if m<<M (one moving body, 1-body) i =v , i =a = ji Hence, i = j j i i i ∑ j≠i 2 ∑j≠i 2 d t d t mi d t ∣r j−r i∣ ∣r j−r i∣ What if m & M are comparable? Nx3 coupled second-order diferential equations! Two moving bodies (2-body) L=m GM a(1−e2) Use 'reduced mass' 1-body: For N=2 with m1≫m2 , we approximate r 1≃0 √ d2 r GM r G M m applications: binary stars, galaxies, (dwarf) planets (pluto-charon), So that =− E=− 2 2 2a detecting extra-solar planets, ... d t r r d A L with r=r 2 , M=m1 , and m=m2 = =const Three-body... d t 2m N-body... Solving this yields: 4π2 P2= a3 (K III) G M 2-body: Energy, Angular Momentum & Kepler's Laws Two-body problem: reduce it to an equivalent one-body problem Angular momentum L=m1 r1×v 1+m2 r 2×v 2=m rm×vm using the concept of 'reduced mass': d r (where v = m =v −v ) m d t 1 2 1 1 G m m 1 Gm M Let r =r −r , M=m +m Energy E = m v 2+ m v 2− 1 2 = m v 2− An imaginary particle of mass μ, m 1 2 1 2 2 1 1 2 2 2 ∣r −r ∣ 2 m r distance r away from CM (center of mass) 1 2 m Let r , r , r be measured from CM 'the reduced mass particle' m 1 2 Hence, as for 1-body case (m≪M), we have with m1 r 1+m2 r 2=0 r 2 Gm M 1 L=m √GM am(1−e ),E=− m 2am 1 m2 m1 We have r 1= rm , r 2=− rm , and 2 r M M 2 4 π 3 2 together with Kepler's three Laws, e.g., P = a GM m G m1 m2 Gm M ∣F 12∣=∣F 21∣= 2 = 2 ∣r 1−r 2∣ rm m2 m m m m where m= 1 2 rm m1+m2 CM M M General problem Kepler in the astronomer's toolkit d r d v F d2 r G m r −r i =v , i =a = ji Hence, i = j j i i i ∑ j≠i 2 ∑j≠i 2 d t d t mi d t ∣r j−r i∣ ∣r j−r i∣ Nx3 coupled second-order diferential equations! 2 π 2 GM=a3 =a3 Ω2 1-body: ( P ) 2 For N=2 with m ≫m ,we approximate r ≃0 L=m G M a(1−e ) 1 2 1 √ GM v tot,circ= d 2 r GM r G Mm a So that =− E=− √ d t 2 r 2 r 2a a1 v 1 m2 a1 v 1 m2 with r≃r , M≃m , and m≃m d A L = = , = = 2-body: 2 1 2 = =const d t 2m a2 v 2 m1 a v M 2 2 4 π 3 For N=2 with m1∼m2 ,we use m1 r 1+m2 r 2≡0 P = a (K III) GM d 2 r GM r So that =− d t 2 r 2 r m1 m2 with r≡r 2−r1 , M≡m1+m2 , and m≡ m1 +m2 Determine masses in binary systems Mpluto + Mcharon ~ 1/400 ME Three-body gravitational interaction: MCharon/MPluto ~ 1/7 not reducible to one-body In general, motion no longer periodic (quasi-periodic or chaotic) our closest neighbour α-Centauri (a triple system) triple star encounter simulation (Hut) 3-body (practical applications): Orbital evolution of near-earth asteroids and comets. Stability of the Solar system very rich phenomenology & analytical theory -- Planetary Dynamics Climate changes on Earth Space travel Sun (M⊙) Earth Voyagers getting swings from outer planets -- gravity-assist -3 Jupiter (10 M⊙) Quiz: cannibalism in close binary stars N-body intractable analytically star m bloats up, part (d m) of its envelope becomes dominated by relies on numerical integration 1 the gravity of m2 and is transferred from m1 to m2 6 globular cluster ~ 10 stars -- does the binary unbind or spiral-in? cluster of galaxies ~ 103 galaxies How to estimate? 11 ● Universe ~ 10 galaxies Use energy conservation? Cataclysmic Variable Globular Cluster M4 ● Use angular momentum conservation? m1 (~ 2 kpc) m2 Dana Berry.