Kuo Liu Laboratoire de Physique et Chimie de l Environnement, LPCE UMR 6115 CNRS, F-45071 Orleans Cedex 02 Station de radioastronomie de Nancay, Observatoire de Paris, CNRS/INSU, F- 18330 Nancay, France Collaborators: N. Wex, R. Eatough, M. Kramer, J. M. Cordes, J. Lazio
Outline Background: a). Black holes in General Relativity (GR) b). Formation of pulsar-black hole binaries (PSR-BH) c). Pulsar timing Gravity tests with pulsar-stellar mass black hole binaries (PSR-SBH) Gravity tests with pulsar in orbit with the Sgr A* (PSR-Sgr A*)
Black hole in GR Uniqueness theorem for black holes [ Israel, Carter, Hawking, Robinson ] All stationary, asymptotically flat vacuum black hole space-times (with non-degenerate horizon) are parametrized by the Kerr metric: No-hair theorem [ Geroch 1970; Hansen 1974 ] All the higher multipole moments of the gravitational field of a Kerr black hole are uniquely related to M and S=aM: M l + is l = M(ia)l, l = 2,3,4, Cosmic censorship conjecture [ Penrose 1979] Quadrupole moment Q=M 2 :
PSR-BH formation Three paths that lead to a birth of a pulsar-stellar mass black hole system: a). Standard binary evolution scenario: the pulsar is born after the black hole companion, evolving no accretion procedure (Bhattacharya & van den Heuvel, 1991; Voss & Tauris, 2003). b). Reversal mechanism : the pulsar is formed before the black hole, possibly have an accretion phase (Tauris & Sennels, 2000; Sipior et al., 2004). c). Stellar capture: the pulsar and black hole are born independently and then form a binary system via a multiple body interaction ( Kulkarni et al., 1993; Faucher-Giguere & Loeb, 2011). Pulsars in orbit with Sgr A*: ~100 pulsars can be expected with orbital period less than ~10 years (Pfahl & Loeb, 2004). Yungelson & Portegies Zwart (1998)
Probing Sgr A* by astrometry Mass measurement: [ Gillesen et al. 2008 ] = 0.2 0.99 [ Genzel et al. 2003; Aschenbach et al. 2004; Belanger et al. 2006; Aschenbach 2010 ]
Pulsar timing Experiment procedure: Lorimer & Kramer (2005) Timing model: a). Spinning parameters: period, period derivative, glitches, b). Position parameters: Ra, Dec, proper motion, parallax, c). Keplerian parameters: orbital period (P b ), projected semi-major axis (x), longitude of periastron (ω), eccentricity (e), epoch of periastron passage (T 0 ) d). Post-Keplerian parameters: γ, x, Pb, ω, e, sin i, M 2, x, ω,
Timing residual (μs) Pulsar timing Slow pulsars (RMS of 0.1~100 ms): Millisecond pulsars (RMS of 0.01~10 μs): Lyne et al., 2010; Hobbs et al., 2010 Verbiest et al., 2009
PSR-SBH: mass measurement Masses measurement via pulsar timing come from the determination of the post-keplerian parameters. Liu et al., in prep.
PSR-SBH: spin measurement The black hole spin will induce a precession of the pulsar orbit (the Frame-Dragging effect, e.g., Barker & O Connell, 1975): Black hole spin measurement comes from the first and second derivative of x and ω (Wex & Kopeikin, 1999).
PSR-SBH: spin measurement Liu et al., in prep.
PSR-SBH: Q measurement? The quadrupole contribution will cause a periodic perturbation of the pulsar orbit (Garfinkel 1958, 1959), which leads to a periodic variation in Roemer delay (Liu, 2011): Liu et al., in prep.
PSR-SBH: T-S constrain Modified Einstein equation: Liu et al., in prep.
Conclusion I Tests with PSR-SBH systems: a). Mass measurements come from two post-keplerian parameters, and can be achieved with high precision, e.g., 10 3. b). Spin measurement is achieved by measuring the pulsar Lense-Thirring precession. Precision of 10 2 ~ 10 3 can be expected. c). Quadrupole moment may be measurable ONLY for MSP-SBH system of high eccentricity or heavy SBH. d). Constrain on Tensor-Scalar theory of gravity can be improved by more than one order of magnitude. Pulsar in orbit with super-massive black hole (i.e., Sgr A*) is highly desirable however
PSR-Sgr A*
PSR-Sgr A*
PSR-Sgr A*: mass measurement Since M PSR << M BH, ONLY ONE post-keplerian parameter is needed for mass measurement of Sgr A*. Liu et al., 2012 Not affected by uncertainty of Galactic Centre distance!
PSR-Sgr A*: spin measurement Pulsar orbit P b = 0.3 yr e = 0.5 0 = 45º 0 = 45º = 60º = 60º Weekly TOAs: 100 s Sgr A* M = 4 10 6 M = 1 Liu et al., 2012
PSR-Sgr A*: spin measurement Pulsar orbit P b = 0.3 yr e = 0.5 0 = 45º 0 = 45º = 60º = 60º Weekly TOAs: 100 s Sgr A* M = 4 10 6 M = 1.2 Liu et al., 2012
PSR-Sgr A*: Q measurement Unique feature by quadrupole moment in pulsar timing residuals, of order 1~10 ms, for orbit of P b =0.3 yr, e=0.5. Simulations: 5 years of timing, one 100 μs TOA per week No-hair theorem test with ~ 1% precision! Liu et al., 2012
Conclusion II Pulsar Sgr A* system: a). With weekly observations of 100 μs TOA precision, the mass, spin and quadurpole moment can be measured with high precision for P b < 0.3 yr. b). The property measurements will lead to a test of the no-hair theorem with ~ 1% precision.