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

The double pulsar PSR J0737-3039 [Burgay

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

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

83 Mass measurements for neutron stars [Antoniadis et al., ApJ 2016] millisecond pulsars J J J J J J J J J J J J J B J A J B J J Mass (Solar Mass) double neutron stars J B B (c) B C B C(c) B (c) J A B J J J (c) J (c) J B J (c) Mass (Solar Mass)

Probing Relativistic Gravity with the Double Pulsar

Probing Relativistic Gravity with the Double Pulsar Marta Burgay INAF Osservatorio Astronomico di Cagliari The spin period of the original millisecond pulsar PSR B1937+21: P = 0.0015578064924327 ± 0.0000000000000004