Resonances in orbital dynamics

Transcription

1 Resonances in orbital dynamics Maarten van de Meent University of Southampton Capra 16, July 2013

2 Outline 1 Resonant Orbits 2 Self-force 3 Resonant evolution 4 Sustained resonances

3 Outline 1 Resonant Orbits 2 Self-force 3 Resonant evolution 4 Sustained resonances

4 Outline 1 Resonant Orbits 2 Self-force 3 Resonant evolution 4 Sustained resonances

5 Outline 1 Resonant Orbits 2 Self-force 3 Resonant evolution 4 Sustained resonances

6 Kerr as an integrable system Geodesic motion in Kerr has 3 constants of motion E, L, and Q. The specic energy E. The specic angular momentum L. Carter's constant Q. (and technically the invariant mass m.)

7 Kerr as an integrable system Geodesic motion in Kerr has 3 constants of motion E, L, and Q. The specic energy E. The specic angular momentum L. Carter's constant Q. (and technically the invariant mass m.)

8 Kerr as an integrable system Geodesic motion in Kerr has 3 constants of motion E, L, and Q. The specic energy E. The specic angular momentum L. Carter's constant Q. (and technically the invariant mass m.)

9 Kerr as an integrable system Geodesic motion in Kerr has 3 constants of motion E, L, and Q. The specic energy E. The specic angular momentum L. Carter's constant Q. (and technically the invariant mass m.)

10 Kerr as an integrable system Geodesic motion in Kerr has 3 constants of motion E, L, and Q. The specic energy E. The specic angular momentum L. Carter's constant Q. (and technically the invariant mass m.)

11 Frequencies as constants of motion Simpler coordinates on phase space: ẇ r = Υ r Υ r = 0, ẇ θ = Υ θ Υ θ = 0, ẇ φ = Υ φ Υφ = 0 For Mino time the relation between (Υ r, Υ θ, Υ φ ) and (E, L, Q) is 1-1.

12 Frequencies as constants of motion Simpler coordinates on phase space: ẇ r = Υ r Υ r = 0, ẇ θ = Υ θ Υ θ = 0, ẇ φ = Υ φ Υφ = 0 For Mino time the relation between (Υ r, Υ θ, Υ φ ) and (E, L, Q) is 1-1.

13 Frequency space

14 Generic orbits Generic orbits ergodicly ll the phase plane (invariant torus). Orbit is uniquely determined by constants of motion. (E, L, Q) or (Υ r, Υ θ, Υ φ )

15 Generic orbits Generic orbits ergodicly ll the phase plane (invariant torus). Orbit is uniquely determined by constants of motion. (E, L, Q) or (Υ r, Υ θ, Υ φ )

16 Generic orbits Generic orbits ergodicly ll the phase plane (invariant torus). Orbit is uniquely determined by constants of motion. (E, L, Q) or (Υ r, Υ θ, Υ φ )

17 Generic orbits Generic orbits ergodicly ll the phase plane (invariant torus). Orbit is uniquely determined by constants of motion. (E, L, Q) or (Υ r, Υ θ, Υ φ )

18 Resonant orbits Resonant orbits close Invariant torus is foliated by resonant orbits Need phase dierence δ in addition to constants of motion to determine orbit.

19 Resonant orbits Resonant orbits close Invariant torus is foliated by resonant orbits Need phase dierence δ in addition to constants of motion to determine orbit.

20 Resonant orbits Resonant orbits close Invariant torus is foliated by resonant orbits Need phase dierence δ in addition to constants of motion to determine orbit.

21 Resonant orbits Resonant orbits close Invariant torus is foliated by resonant orbits Need phase dierence δ in addition to constants of motion to determine orbit.

22 Resonance locations

23 Include self-force In the extreme mass ratio limit (η µ/m << 1) corrections due to the nite mass of the object can be added order by order: w = Υ + η g( Υ, w) + O(η 2 ) Υ = η G (1) ( Υ, w) + η 2 G (2) ( Υ, w) + O(η 3 ), where w = (w r, w θ, w φ ), Υ = (Υ r, Υ θ, Υ φ ), etc.

24 Fourier expand Focussing on η G (1) ( Υ, w) we can Fourier expand the dependence on the phases w r and w θ. w = Υ Υ = η v( Υ) + η n,m k( Υ) cos(nwr + mw θ ) + κ( Υ) sin(nw r + mw θ )

25 Generic (non-resonant) orbits Inspiral time scale (O(η 1 )) is much longer than orbital time scale (O(1)). Generic (non-resonant) orbits ergodicly sample the invariant torus. Consequently, the oscillatory terms average to zero.

26 Generic (non-resonant) orbits Inspiral time scale (O(η 1 )) is much longer than orbital time scale (O(1)). Generic (non-resonant) orbits ergodicly sample the invariant torus. Consequently, the oscillatory terms average to zero.

27 Generic (non-resonant) orbits Inspiral time scale (O(η 1 )) is much longer than orbital time scale (O(1)). Generic (non-resonant) orbits ergodicly sample the invariant torus. Consequently, the oscillatory terms average to zero.

28 Resonant orbits Suppose system evolves through a resonant orbit with nυ r + mυ θ = 0. (Happens generically!) Adiabatic approximation fails. The nw r + mw θ = 0 harmonics remain relevant near the harmonic surface.

29 Resonant orbits Suppose system evolves through a resonant orbit with nυ r + mυ θ = 0. (Happens generically!) Adiabatic approximation fails. The nw r + mw θ = 0 harmonics remain relevant near the harmonic surface.

30 Resonant orbits Suppose system evolves through a resonant orbit with nυ r + mυ θ = 0. (Happens generically!) Adiabatic approximation fails. The nw r + mw θ = 0 harmonics remain relevant near the harmonic surface.

31 Resonant evolution Suppose there is just one resonant harmonic. Then the equations of motion become: [Gair et al. '12] ẅ r = Υ r = v r ( Υ) + k r ( Υ) cos(nw r + mw r ) ẅ θ = Υ θ = v θ ( Υ) + k θ ( Υ) cos(nw r + mw r ) Introduce convenient coordinates (and drop dependence on Υ): ẅ = v + k cos(w ) ẅ = v + k cos(w ), with X nx r + mx θ and X nx r mx θ.

32 Resonant evolution Suppose there is just one resonant harmonic. Then the equations of motion become: [Gair et al. '12] ẅ r = Υ r = v r ( Υ) + k r ( Υ) cos(nw r + mw r ) ẅ θ = Υ θ = v θ ( Υ) + k θ ( Υ) cos(nw r + mw r ) Introduce convenient coordinates (and drop dependence on Υ): ẅ = v + k cos(w ) ẅ = v + k cos(w ), with X nx r + mx θ and X nx r mx θ.

33 k << v Solution for late times: Υ = ẇ = tv + Υ = ẇ = tv + πk 2 v cos(w (0) ± π/4) + O(t 1, k 2 v 2 ) πk 2 v cos(w (0) ± π/4) + O(t 1, k 2 v 2 ) Constants of motion make a jump of order η across a resonance. Over the entire inspiral the phases accumulate a correction of order 1/ η.

34 k << v Solution for late times: Υ = ẇ = tv + Υ = ẇ = tv + πk 2 v cos(w (0) ± π/4) + O(t 1, k 2 v 2 ) πk 2 v cos(w (0) ± π/4) + O(t 1, k 2 v 2 ) Constants of motion make a jump of order η across a resonance. Over the entire inspiral the phases accumulate a correction of order 1/ η.

35 k << v Solution for late times: Υ = ẇ = tv + Υ = ẇ = tv + πk 2 v cos(w (0) ± π/4) + O(t 1, k 2 v 2 ) πk 2 v cos(w (0) ± π/4) + O(t 1, k 2 v 2 ) Constants of motion make a jump of order η across a resonance. Over the entire inspiral the phases accumulate a correction of order 1/ η.

36 Higher harmonics Easy to include other harmonic terms Υ = tv + i πk,i 2i v cos(iw (0) ± π/4)+ Υ = tv + i i i πκ,i 2i v sin(iw (0) ± π/4) + O(t 1, k 2 πk,i 2i v cos(iw (0) ± π/4)+ v 2 ) πκ,i 2i v sin(iw (0) ± π/4) + O(t 1, k 2 v 2 ) Dissipative and conservative terms appear on same footing.

37 Higher harmonics Easy to include other harmonic terms Υ = tv + i πk,i 2i v cos(iw (0) ± π/4)+ Υ = tv + i i i πκ,i 2i v sin(iw (0) ± π/4) + O(t 1, k 2 πk,i 2i v cos(iw (0) ± π/4)+ v 2 ) πκ,i 2i v sin(iw (0) ± π/4) + O(t 1, k 2 v 2 ) Dissipative and conservative terms appear on same footing.

38 Large k The equation of motion Allows a rst integral: ẅ = v + k cos(w ) 1 (ẇ 2 ) 2 = v w + k sin w + ẇ (0)

39 Potential If k < v the potential is monotonic. If k > v the potential has local minima. Sustained resonances.

40 Potential If k < v the potential is monotonic. If k > v the potential has local minima. Sustained resonances.

41 Phase portrait

42 Do sustained resonances occur? Not generically. k is typically much smaller than v. e.g. [Flanagan, Hughes Ruangrsi, '12] nd variations no larger than a few tenth of percent. Most likely to occur for low order resonance (e.g. 2:3).

43 Do sustained resonances occur? Not generically. k is typically much smaller than v. e.g. [Flanagan, Hughes Ruangrsi, '12] nd variations no larger than a few tenth of percent. Most likely to occur for low order resonance (e.g. 2:3).

44 Do sustained resonances occur? Not generically. k is typically much smaller than v. e.g. [Flanagan, Hughes Ruangrsi, '12] nd variations no larger than a few tenth of percent. Most likely to occur for low order resonance (e.g. 2:3).

45 Thank You... and that all I have to say about that.