Post-Keplerian effects in binary systems
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1 Post-Keplerian effects in binary systems Laboratoire Univers et Théories Observatoire de Paris / CNRS
2 The problem of binary pulsar timing (Credit: N. Wex)
3 Some classical tests of General Relativity Gravitational redshift of light τ t = 1 G c 2 M R R M τ t Perihelion advance of Mercury Φ = G c 2 6πM a (1 e 2 ) M Precession of Earth-Moon spin Ω prec = G ( ) c 2 v 3M 2r prec S M r
4 (Credit: N. Wex) Different regimes of gravity
5 (Credit: N. Wex) Different regimes of gravity
6 (Credit: N. Wex) Different regimes of gravity
7 (Credit: N. Wex) Different regimes of gravity quasi-stationary, weak-field regime quasi-stationary, strong-field regime highly dynamical, strong-field regime radiative regime
8 (Credit: N. Wex) Different regimes of gravity
9 Effacement of the internal structure
10 Effacement of the internal structure
11 Effacement of the internal structure Induced quadrupole moment: Q ij R 5 i j U R 5 (Gm/D 3 )
12 Effacement of the internal structure Induced quadrupole moment: Q ij R 5 i j U R 5 (Gm/D 3 ) Induced quadrupolar force: F i quad m i( Q r 3 ) R 5 (Gm 2 /D 7 )
13 Effacement of the internal structure Induced quadrupole moment: Q ij R 5 i j U R 5 (Gm/D 3 ) Induced quadrupolar force: F i quad m i( Q r 3 ) R 5 (Gm 2 /D 7 ) For a compact body with R Gm/c 2, F quad (G 6 /c 10 )(m/d) 7 F Newt Gm 2 /D 2 ( ) Gm 5 ( v ) 10 c 2 1 D c
14 Outline 1 The Newtonian two-body problem 2 Relativistic celestial mechanics Periastron advance Geodetic spin precession Gravitational radiation reaction 3 Light propagation in curved spacetime Einstein delay Shapiro delay 4 Testing relativistic gravitation
15 Outline 1 The Newtonian two-body problem 2 Relativistic celestial mechanics Periastron advance Geodetic spin precession Gravitational radiation reaction 3 Light propagation in curved spacetime Einstein delay Shapiro delay 4 Testing relativistic gravitation
16 Real two-body problem m 2 H(x a, p a ) = p2 1 2m 1 + p2 2 2m 2 Gm 1m 2 r 12 m 1
17 Real two-body problem m 2 H(x a, p a ) = p2 1 2m 1 + p2 2 2m 2 Gm 1m 2 r 12 m 1 p 1 + p 2 = 0
18 Real two-body problem m 2 H(x a, p a ) = p2 1 2m 1 + p2 2 2m 2 Gm 1m 2 r 12 m 1 p 1 + p 2 = 0 Effective one-body problem H(r, p) = p2 2µ Gmµ r m =m 1 +m 2 m = m 1 m 2 /m
19 Keplerian solution for bound orbits
20 Keplerian solution for bound orbits Keplerian elements eccentricity e semi-major axis a
21 Keplerian solution for bound orbits Keplerian elements eccentricity e semi-major axis a Constants of motion energy E = Gmµ/(2a) ang. mom. L = µ Gma(1 e 2 )
22 Keplerian solution for bound orbits Keplerian elements eccentricity e semi-major axis a Constants of motion energy E = Gmµ/(2a) ang. mom. L = µ Gma(1 e 2 ) Parametric solution r(u) = a (1 e cos u) ( 1 + e φ(u) = 2 arctan 1 e tan u ) 2 ν(u) = u e sin u = n (t t 0 )
23 Keplerian solution for bound orbits Keplerian elements eccentricity e semi-major axis a Constants of motion energy E = Gmµ/(2a) ang. mom. L = µ Gma(1 e 2 ) Parametric solution r(u) = a (1 e cos u) ( 1 + e φ(u) = 2 arctan 1 e tan u ) 2 ν(u) = u e sin u = n (t t 0 )
24 Keplerian solution for bound orbits Keplerian elements eccentricity e semi-major axis a Constants of motion energy E = Gmµ/(2a) ang. mom. L = µ Gma(1 e 2 ) Parametric solution r(u) = a (1 e cos u) ( 1 + e φ(u) = 2 arctan 1 e tan u ) 2 ν(u) = u e sin u = n (t t 0 )
25 Keplerian solution for bound orbits Keplerian elements eccentricity e semi-major axis a Constants of motion energy E = Gmµ/(2a) ang. mom. L = µ Gma(1 e 2 ) Parametric solution r(u) = a (1 e cos u) ( 1 + e φ(u) = 2 arctan 1 e tan u ) 2 ν(u) = u e sin u = n (t t 0 )
26 Keplerian solution for bound orbits Keplerian elements eccentricity e semi-major axis a Constants of motion energy E = Gmµ/(2a) ang. mom. L = µ Gma(1 e 2 ) Parametric solution r(u) = a (1 e cos u) ( 1 + e φ(u) = 2 arctan 1 e tan u ) 2 ν(u) = u e sin u = n (t t 0 ) Kepler s third law: n 2 a 3 = Gm
27 Outline 1 The Newtonian two-body problem 2 Relativistic celestial mechanics Periastron advance Geodetic spin precession Gravitational radiation reaction 3 Light propagation in curved spacetime Einstein delay Shapiro delay 4 Testing relativistic gravitation
28 Outline 1 The Newtonian two-body problem 2 Relativistic celestial mechanics Periastron advance Geodetic spin precession Gravitational radiation reaction 3 Light propagation in curved spacetime Einstein delay Shapiro delay 4 Testing relativistic gravitation
29 Relativistic perihelion advance of Mercury Leading-order relativistic angular advance per orbit: Φ = 6πGM c 2 a (1 e 2 ) Mercury Origin Amplitude ( /cent.) Other planets ± 0.69 Sun oblateness ± General relativity ± 0.04 Total ± 0.69 Observed ± 0.65 M
30 Two-body Hamiltonian at 1PN order H = H N + 1 c 2 H 1PN + O(c 4 ) H 1PN (x a, p a ) = 1 (p 2 1 )2 8 m G 2 m 1 m 2 m 4 r Gm 1 m 2 8 r 12 [ 12 p2 1 m p 1 p (n 12 p 1 )(n 12 p 2 ) m 1 m 2 m 1 m 2 ]
31 Periastron advance in binary pulsars Generic eccentric orbit parametrized by the two frequencies n = 2π P, 2π + Φ ϕ = P Leading-order periastron advance: ω Φ 2π = 3G(m 1 + m 2 ) c 2 a (1 e 2 ) PSR Amplitude ( /year) B ± J ±
32 Outline 1 The Newtonian two-body problem 2 Relativistic celestial mechanics Periastron advance Geodetic spin precession Gravitational radiation reaction 3 Light propagation in curved spacetime Einstein delay Shapiro delay 4 Testing relativistic gravitation
33 Geodetic spin precession [de Sitter, MNRAS 1916] GR predicts that a test spin S with velocity v in a gravitational potential Φ GM/(c 2 r) will precess according to ds dt = Ω prec S where Ω prec 3 2 v Φ Precession of Earth-Moon spin axis of 1.9 /cent. confirmed using Lunar Laser Ranging v prec S r [Shapiro et al., PRL 1988]
34 Geodetic effect in Gravity Probe B [Everitt et al., PRL 2011] Measured Geodetic precession (mas/yr) 6602 ± Frame-dragging (mas/yr) 37.2 ± Predicted
35 Spin-orbit coupling at leading order [Barker & O Connell, PRD 1975] ds a dt = Ω a S a spin-orbit coupling { }}{ H(x a, p a, S a ) = H orb (x a, p a ) + Ω b (x a, p a ) S b Ω 1 (x a, p a ) = G c 2 r12 2 ( 3m2 2m 1 n 12 p 1 2n 12 p 2 b ) L
36 The double pulsar PSR J [Burgay et al., Nature 2003] (Credit: M. Kramer)
37 Spin precession of pulsar B z [Breton et al., Science 2008] θ µ Ω B projected orbital motion α pulsar A z 0 R mag pulsar B φ y x (to Earth)
38 Spin precession of pulsar B [Breton et al., Science 2008] Ω B = 4.77 ± 0.66 /yr z θ µ Ω B projected orbital motion α pulsar A z 0 R mag pulsar B φ y x (to Earth)
39 Spin precession of pulsar B [Breton et al., Science 2008]
40 Outline 1 The Newtonian two-body problem 2 Relativistic celestial mechanics Periastron advance Geodetic spin precession Gravitational radiation reaction 3 Light propagation in curved spacetime Einstein delay Shapiro delay 4 Testing relativistic gravitation
41 Einstein quadrupole formula
42 Einstein quadrupole formula Gravitat. wave energy flux F(t) = G c 5 Q ij Q ij
43 Einstein quadrupole formula Gravitat. wave energy flux F(t) = G c 5 Q ij Q ij Radiation reaction force F i (t, x) = 2G 5c 5 Q(5) ij x j
44 Application to a binary pulsar [Peters, Phys. Rev. 1964] F = 32G 4 m1 2m2 2 (m ( 1 + m 2 ) 5c 5 a 5 (1 e 2 ) 7/ e G = 32G 7/2 5c 5 m 2 1 m2 2 (m 1 + m 2 ) 1/2 a 7/2 (1 e 2 ) 2 ( e2 ) 96 e4 )
45 Application to a binary pulsar [Peters, Phys. Rev. 1964] F = 32G 4 m1 2m2 2 (m ( 1 + m 2 ) 5c 5 a 5 (1 e 2 ) 7/ e G = 32G 7/2 5c 5 m 2 1 m2 2 (m 1 + m 2 ) 1/2 a 7/2 (1 e 2 ) 2 ) de 96 e4 = dt ( ) dl e2 = dt
46 Application to a binary pulsar [Peters, Phys. Rev. 1964] da = 64G 3 m 1 m 2 (m 1 + m 2 ) dt 5c 5 a 3 (1 e 2 ) 7/2 de = 304G 3 m 1 m 2 (m 1 + m 2 ) dt 15c 5 a 4 (1 e 2 ) 5/2 ( e ( e e3 96 e4 ) )
47 Application to a binary pulsar [Peters, Phys. Rev. 1964] ( ) dp 192πG 5/3 m 1 m 2 2π 5/ = e e4 dt 5c 5 m 1/3 P (1 e 2 ) 7/2
48 Binary pulsar PSR B [Hulse & Taylor, ApJ 1975]
49 Orbital decay of PSR B [Weisberg & Huang, ApJ 2016]
50 Orbital decay of PSR B [Weisberg & Huang, ApJ 2016] Ṗ = (2.398 ± 0.004) 10 12
51 Binary pulsar tests of orbital decay PSR Ṗ int /Ṗ GR Reference B ± Weisberg & Huang (2016) J ± Kramer et al. (2006) B C 1.00 ± 0.03 Jacoby et al. (2006) J ± 0.03 Ferdman et al. (2014) J ± 0.05 van Leeuwen et al. (2015) J ± 0.06 Bhat et al. (2008) B ± 0.06 Stairs et al. (2001) J ± 0.13 Freire et al. (2012) J ± 0.18 Antoniadis et al. (2013)
52 Outline 1 The Newtonian two-body problem 2 Relativistic celestial mechanics Periastron advance Geodetic spin precession Gravitational radiation reaction 3 Light propagation in curved spacetime Einstein delay Shapiro delay 4 Testing relativistic gravitation
53
54 Outline 1 The Newtonian two-body problem 2 Relativistic celestial mechanics Periastron advance Geodetic spin precession Gravitational radiation reaction 3 Light propagation in curved spacetime Einstein delay Shapiro delay 4 Testing relativistic gravitation
55 Gravitational redshift of light Light frequency ω em ω rec = 1 + G c 2 M R
56 Gravitational redshift of light Light frequency ω em ω rec = 1 + G c 2 M R Proper time τ t = 1 G c 2 M R
57 Gravitational redshift of light Light frequency ω em ω rec = 1 + G c 2 M R Proper time τ t = 1 G c 2 M R Confirmed by Gravity Probe A at 0.007% [Vessot et al., PRL 1980]
58 Einstein delay dτ p dt = 1 gravitat. redshift {}}{ U(x p) c 2 2 nd order Doppler {}}{ 1 2 v 2 p c 2 + O(c 4 )
59 Einstein delay dτ p dt = 1 gravitat. redshift {}}{ U(x p) c 2 = 1 Gm c c 2 r 2 nd order Doppler {}}{ v 2 p 1 2 c 2 + O(c 4 ) m c Gm c m c 2 + cst. r
60 Einstein delay dτ p dt = 1 gravitat. redshift {}}{ U(x p) c 2 = 1 Gm c c 2 r 2 nd order Doppler {}}{ v 2 p 1 2 c 2 + O(c 4 ) m c Gm c m c 2 + cst. r r = a (1 e cos u) t = (u e sin u)/n
61 Einstein delay dτ p dt = 1 gravitat. redshift {}}{ U(x p) c 2 = 1 Gm c c 2 r 2 nd order Doppler {}}{ v 2 p 1 2 c 2 + O(c 4 ) m c Gm c m c 2 + cst. r r = a (1 e cos u) t = (u e sin u)/n E (u) τ p t = γ sin u
62 Einstein delay dτ p dt = 1 gravitat. redshift {}}{ U(x p) c 2 = 1 Gm c c 2 r 2 nd order Doppler {}}{ v 2 p 1 2 c 2 + O(c 4 ) m c Gm c m c 2 + cst. r r = a (1 e cos u) t = (u e sin u)/n E (u) τ p t = γ sin u γ = Gm c(m p + 2m c ) c 2 a (m p + m c )n e
63 Einstein delay dτ p dt = 1 gravitat. redshift {}}{ U(x p) c 2 = 1 Gm c c 2 r 2 nd order Doppler {}}{ v 2 p 1 2 c 2 + O(c 4 ) m c Gm c m c 2 + cst. r r = a (1 e cos u) t = (u e sin u)/n E (u) τ p t = γ sin u γ = Gm c(m p + 2m c ) c 2 a (m p + m c )n e PSR Amplitude (ms) B ± J ±
64 Outline 1 The Newtonian two-body problem 2 Relativistic celestial mechanics Periastron advance Geodetic spin precession Gravitational radiation reaction 3 Light propagation in curved spacetime Einstein delay Shapiro delay 4 Testing relativistic gravitation
65 Gravitational time delay [Shapiro, PRL 1964]
66 Gravitational time delay [Shapiro, PRL 1964] t = 2GM c 3 [ ( ) ] 4rE r M ln b 2 + 1
67 Gravitational time delay [Shapiro, PRL 1964] t = 2GM c µs [ ( ) ] 4rE r M ln b 2 + 1
68 Gravitational time delay [Shapiro et al., PRL 1968]
69 Shapiro delay in binary pulsars S (θ) = 2 Gm c c 3 }{{} range r ln (1 }{{} sin i cos θ) shape s
70 Double pulsar PSR J [Kramer et al., Science 2006]
71 Binary pulsar PSR J [Demorest et al., Nature 2010] T iming residual (µs) IPTA O rbital P hase (turns)
72 Constraining the NS equation of state [Demorest et al., Nature 2010] Mass (solar) GR P < GS1 Double NS Systems causality AP3 ENG AP4 J SQM3 J FSU SQM1 PAL6 GM3 J MPA1 MS1 PAL1 MS2 MS rotation 0.0 Nucleons Nucleons+ Exotic Strange Quark Matter Radius (km )
73 Outline 1 The Newtonian two-body problem 2 Relativistic celestial mechanics Periastron advance Geodetic spin precession Gravitational radiation reaction 3 Light propagation in curved spacetime Einstein delay Shapiro delay 4 Testing relativistic gravitation
74 Three post-keplerian parameters Periastron advance ω = 3G 2/3 c 2 ( ) 2π 5/3 ( 1 e 2 ) 1 (m1 + m 2 ) 2/3 P
75 Three post-keplerian parameters
76 Three post-keplerian parameters Periastron advance ω = 3G 2/3 c 2 ( ) 2π 5/3 ( 1 e 2 ) 1 (m1 + m 2 ) 2/3 P Einstein delay γ = G 3/2 c 2 ( ) P 1/3 e m 2(m 1 + 2m 2 ) 2π (m 1 + m 2 ) 4/3
77 Three post-keplerian parameters
78 Three post-keplerian parameters m 1 = ± M m 2 = ± M
79 Three post-keplerian parameters Periastron advance ω = 3G 2/3 c 2 ( ) 2π 5/3 ( 1 e 2 ) 1 (m1 + m 2 ) 2/3 P Einstein delay γ = G 3/2 c 2 ( ) P 1/3 e m 2(m 1 + 2m 2 ) 2π (m 1 + m 2 ) 4/3 Orbital decay 192πG 5/3 Ṗ = 5c 5 ( ) 2π 5/3 f (e) P m 1 m 2 (m 1 + m 2 ) 1/3
80 Three post-keplerian parameters m 1 = ± M m 2 = ± M
81 Tests of relativistic gravity [Esposito-Farèse, Proc. MG10] 1 test 3 tests
82 Tests of relativistic gravity [Esposito-Farèse, Proc. MG10] 2 tests 5 tests
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