Post-Keplerian effects in binary systems

Post-Keplerian effects in binary systems Laboratoire Univers et Théories Observatoire de Paris / CNRS

The problem of binary pulsar timing (Credit: N. Wex)

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

(Credit: N. Wex) Different regimes of gravity

(Credit: N. Wex) Different regimes of gravity quasi-stationary, weak-field regime quasi-stationary, strong-field regime highly dynamical, strong-field regime radiative regime

(Credit: N. Wex) Different regimes of gravity

Effacement of the internal structure

Effacement of the internal structure Induced quadrupole moment: Q ij R 5 i j U R 5 (Gm/D 3 )

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 )

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

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

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

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

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

Keplerian solution for bound orbits

Keplerian solution for bound orbits Keplerian elements eccentricity e semi-major axis a

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 )

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 )

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 531.63 ± 0.69 Sun oblateness 0.028 ± 0.001 General relativity 42.98 ± 0.04 Total 574.64 ± 0.69 Observed 574.10 ± 0.65 M

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 m1 3 + 1 G 2 m 1 m 2 m 4 r12 2 + 1 Gm 1 m 2 8 r 12 [ 12 p2 1 m1 2 + 14 p 1 p 2 + 2 (n 12 p 1 )(n 12 p 2 ) m 1 m 2 m 1 m 2 ]

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) B1913+16 4.226598 ± 0.000005 J0737-3039 16.89947 ± 0.00068

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]

Geodetic effect in Gravity Probe B [Everitt et al., PRL 2011] Measured Geodetic precession (mas/yr) 6602 ± 18 6606 Frame-dragging (mas/yr) 37.2 ± 7.2 39.2 Predicted

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 et al., Nature 2003] (Credit: M. Kramer)

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)

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)

Spin precession of pulsar B [Breton et al., Science 2008]

Einstein quadrupole formula

Einstein quadrupole formula Gravitat. wave energy flux F(t) = G...... 5c 5 Q ij Q ij

Einstein quadrupole formula Gravitat. wave energy flux F(t) = G...... 5c 5 Q ij Q ij Radiation reaction force F i (t, x) = 2G 5c 5 Q(5) ij x j

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 ) 7/2 1 + 73 24 e2 + 37 G = 32G 7/2 5c 5 m 2 1 m2 2 (m 1 + m 2 ) 1/2 a 7/2 (1 e 2 ) 2 (1 + 78 e2 ) 96 e4 )

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 ) 7/2 1 + 73 24 e2 + 37 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 (1 + 78 ) dl e2 = dt

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 ( 1 + 73 24 e2 + 37 ( e + 121 304 e3 96 e4 ) )

Application to a binary pulsar [Peters, Phys. Rev. 1964] ( ) dp 192πG 5/3 m 1 m 2 2π 5/3 1 + 73 24 = e2 + 37 96 e4 dt 5c 5 m 1/3 P (1 e 2 ) 7/2

Binary pulsar PSR B1913+16 [Hulse & Taylor, ApJ 1975]

Orbital decay of PSR B1913+16 [Weisberg & Huang, ApJ 2016]

Orbital decay of PSR B1913+16 [Weisberg & Huang, ApJ 2016] Ṗ = (2.398 ± 0.004) 10 12

Binary pulsar tests of orbital decay PSR Ṗ int /Ṗ GR Reference B1913+16 0.9983 ± 0.0016 Weisberg & Huang (2016) J0737-3039 1.003 ± 0.014 Kramer et al. (2006) B2127+11C 1.00 ± 0.03 Jacoby et al. (2006) J1756-2251 1.08 ± 0.03 Ferdman et al. (2014) J1906+0746 1.01 ± 0.05 van Leeuwen et al. (2015) J1141-6545 1.04 ± 0.06 Bhat et al. (2008) B1534+12 0.91 ± 0.06 Stairs et al. (2001) J1738+0333 0.94 ± 0.13 Freire et al. (2012) J0348+0432 1.05 ± 0.18 Antoniadis et al. (2013)

Gravitational redshift of light Light frequency ω em ω rec = 1 + G c 2 M R

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

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]

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 )

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

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

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

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

Gravitational time delay [Shapiro, PRL 1964]

Gravitational time delay [Shapiro, PRL 1964] t = 2GM c 3 [ ( ) ] 4rE r M ln b 2 + 1

Gravitational time delay [Shapiro, PRL 1964] t = 2GM c 3 120 µs [ ( ) ] 4rE r M ln b 2 + 1

Gravitational time delay [Shapiro et al., PRL 1968]

Shapiro delay in binary pulsars S (θ) = 2 Gm c c 3 }{{} range r ln (1 }{{} sin i cos θ) shape s

Double pulsar PSR J0737-3039 [Kramer et al., Science 2006]

Binary pulsar PSR J1614-2230 [Demorest et al., Nature 2010] 30 20 10 0-10 -20-30 -40 30 20 10 T iming residual (µs) 0-10 -20-30 -40 30 20 10 0-10 -20-30 IPTA 2017-40 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 O rbital P hase (turns)

Constraining the NS equation of state [Demorest et al., Nature 2010] Mass (solar) 2.5 2.0 1.5 GR P < GS1 Double NS Systems causality AP3 ENG AP4 J1 6 1 4-2 2 3 0 SQM3 J1903+ 0327 FSU SQM1 PAL6 GM3 J1909-3744 MPA1 MS1 PAL1 MS2 MS0 1.0 0.5 rotation 0.0 Nucleons Nucleons+ Exotic Strange Quark Matter 7 8 9 10 11 12 13 14 15 Radius (km )

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

Three post-keplerian parameters

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

Three post-keplerian parameters

Three post-keplerian parameters m 1 = 1.4398 ± 0.0002M m 2 = 1.3886 ± 0.0002M

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

Three post-keplerian parameters m 1 = 1.4398 ± 0.0002M m 2 = 1.3886 ± 0.0002M

Tests of relativistic gravity [Esposito-Farèse, Proc. MG10] 1 test 3 tests

Tests of relativistic gravity [Esposito-Farèse, Proc. MG10] 2 tests 5 tests

Mass measurements for neutron stars [Antoniadis et al., ApJ 2016] millisecond pulsars J0348+0432 J1614-2230 J1946+3417 J1012+5307 J1023+0038 J1903+0327 J0751+1807 J1909-3744 J1738+0333 J0437-4715 J0337+1715 J2234+0611 J1807-2500B J1910-5959A J1713+0747 B1855+09 J1802-2124 J1918-0642 0.0 0.5 1.0 1.5 2.0 Mass (Solar Mass) double neutron stars J0453+1559 B1913+16 B1913+16(c) B2127+11C B2127+11C(c) B1534+12(c) J0737-3039A B1534+12 J1906+0746 J1756-2251 J1906+0746(c) J1756-2251(c) J0737-3039B J0453+1559(c) 0.0 0.5 1.0 1.5 2.0 Mass (Solar Mass)