What do I do? George Pappas Caltech, TAPIR. November 09, 2015

Transcription

1 What do I do? George Pappas Caltech, TAPIR November 09, 015

2 Brief Timeline 1 /17 PhD in 01 on neutron star (NS) spacetimes month DAAD scholarship to work with K. Kokkotas Postdoc with T. Sotiriou that started at SISSA (013) and continued at the U. Nottingham (01-15) This summer moved to Oxford, MS

3 Work Topics /17 Neutron star spacetimes. Analytic spacetimes with appropriate matching parameters. Comparison between numerical spacetimes and their analytic counterparts. Neutron star multipole moments. Calculation from numerical models, range for realistic neutron stars, and properties and relations between them (universal 3-hair relations). Properties of Neutron star spacetimes and astrophysical applications (neutron star spacetimes are not Kerr). QPOs and relativistic precession model. Properties of accretion discs in neutron star spacetimes (spectra, selftrapping c-modes). Neutron stars in alternative theories of gravity. Post-TOV formalism, a theory independent approach to neutron star properties and structure. Characterising spacetimes in alternative theories of gravity. geodesics of spacetimes in scalar-tensor theories of gravity. Multipole moments and

4 Neutron stars and neutron star spacetimes 3 /17 Rotating neutron Stars The line element for a stationary and axially symmetric spacetime (the spacetime admits a timelike, ξ a, and a spacelike, η a, killing field, i.e., it has rotational symmetry and symmetry in translations in time) is 1, ds = e ν dt + r sin θb e ν (dφ ωdt) + e (ζ ν) (dr + r dθ ). Field equations in the frame of the ZAMOs: D (BDν) = 1 [ ] (ɛ + p)(1 + u r sin θb 3 e ν Dω Dω + πbe ζ ν ) + p 1 u D (r sin θb 3 e ν Dω) = 1πr sin θb ζ ν (ɛ + p)u e 1 u, D (r sin θdb) = 1πr sin θbe ζ ν p,, Komatsu, Eriguchi, and Hechisu proposed a scheme for integrating the field equations using Green s functions. This scheme is implemented by the RNS numerical code to calculate rotating neutron stars 3. Asymptotic expansion of the metric functions: ν = ν l (r)p l (µ), ω = l=0 ω l (r)p l+1,µ (µ), B = l=0 where P l are the Legendre polynomials, µ = cos θ, and T 1/ l 1 E. M. Butterworth and J. R. Ipser, ApJ 0, 00 (197). H. Komatsu, Y. Eriguchi, and I. Hechisu, MNRAS 37, 355 (199). 3 N. Stergioulas, J.L. Friedman, ApJ,, 30 (1995). l=0 B l (r)t 1/ l (µ), are the Gegenbauer polynomials.

5 Neutron stars and neutron star spacetimes /17 One can use RNS to calculate models of rotating neutron stars for a given equation of state. For example we show here some models for the APR EOS:.5.5 M Mo 1.5 M Mo Ρ c x R e The models with the fastest rotation have a spin parameter, = J/M, around 0.7 and a ratio of the polar radius over the equatorial radius, r p /r e, around 0.5. The code, except from the various physical characteristics of the neutron stars, provides the metric functions in a grid on the coordinates x and µ in the whole space (for values from 0 to 1 for both variables), where µ = cosθ, r = xr e 1 x and r e is a length scale. But, numerical spacetimes in tabulated form are not very practical. An analytic spacetime that captures the neutron star spacetime properties would be very useful. How should we relate the neutron star spacetime to an analytic spacetime? Relativistic multipole moments characterise stationary axisymmetric spacetimes.

6 Properties of neutron star moments in GR 5 /17 Multipole moments of numerical spacetimes The RNS code can calculates the first non-zero multipole moments, i.e., M, S 1 J, M, S 3 and M. M 0 = M, S 1 = M, M = 1 + b 0 + 3q M 3, 3 S 3 = 3( + b 0 5w ) M, M = b q + 33b b 0q 5q 19b M 5 5 where J/M, and the various parameters are given by the integrals, Q l = M l+1 q l = rl+1 e W l = M l w l = rl e l B l = M l+ b l = 1 πre l+ l ds s l 1 (1 s ) l+ 0 dµ P l (µ ) S ρ (s, µ ), ds s l 1 dµ sin θ P 1 (1 s ) l+ l 1 (µ ) Sˆω (s, µ ), 1/ 0 0 ds s l+3 1 (1 s ) l+5 0 dµ sin θ p(s, µ )Be (ζ ν) T 1/ l (µ ), where in the second integral l 1, in contrast to the other two integrals where l 0. G.P. and T. A. Apostolatos, Phys. Rev. Lett. 31 (01), K. Yagi, K. Kyutoku, G. P., N. Yunes, and T.A. Apostolatos, Phys.Rev. D (01).

7 5 W.G. Laarakkers and E. Poisson, Astrophys. J. 51 (1999). G.P. and T. A. Apostolatos, Phys. Rev. Lett. 31 (01). K. Yagi, K. Kyutoku, G. P., N. Yunes, and T.A. Apostolatos, Phys.Rev. D (01). Properties of neutron star moments in GR /17 Neutron star multipole moments properties in GR Black Hole-like behaviour of the moments 5 : Kerr moments M 0 = M, J 1 = J = M, M = M 3, J 3 = 3 M, M = M 5,. M n = ( 1) n n M n+1, J n+1 = ( 1) n n+1 M n+ Neutron star moments M 0 = M, J 1 = M, M = a(eos, M) M 3, J 3 = β(eos, M) 3 M, M = γ(eos, M) M 5,. M n =?, J n+1 =?

8 provided that stars spin moderately fast (spin period less slowly-rotating stars, we extract multipole moments by than 0.1s) and the magnetic fields are not too large (less solving the Einstein equations in a slow-rotation expansionstar to quartic moments orderinin GR spin, extending previously-found 7 /17 than 1 G). Such properties are precisely Properties those one of expects millisecond pulsars to have. quadratic [37, 3] and cubic [39] solutions. Validity of the neutron Several studies have relaxed the slow-rotation approximation [3 5], leading to a small controversy. Initially, allows us to estimate the importance of quartic-order-in- quadratic solution is discussed in [0]. Such an extension Neutron Doneva star et al. [3] multipole constructed NSmoments and QS sequencesproperties by spin terms in inx-ray GRobservations of millisecond pulsars, varying the dimensional spin frequency and found that EoS independent behaviour of the moments which were neglected in [1, 3]. 1 the EoS-universality of the relation between the moment: of inertia APRand the quadrupole moment was lost. Shortly APR AU stars spinning at di erent frequencies. Reference [] already found that there exists a universal M M SLy relation after, Pappas L and Apostolatos [] and Chakrabarti et AU L SLy al. [5] UUconstructed NS sequences by varying dimensionless poly. combinations n=0.5 of the spin angular moment and 3 Shen UU Shen to leading-order in a weak-field, Newtonian expansion, poly. n=0.5 SQM1 so let us investigate SQM1 this relation first. The top panel found that SQM the relation remained EoS-universal. More SQM of Fig. 1 shows the M SQM3 M relation for various realistic recently, Stein et al. [] proved analytically that universality is preserved to leading (Newtonian) order in a fit NS and QS SQM3 fit EoSs Newton, n=0.5 and various spins, computed with the Newton, n=0.5 LORENE and RNS codes, as well as in the slow-rotation weak-field expansion, supporting the numerical calculations of [, 5]. approximation. 1 The bottom panel shows the fractional di erence between the data and the fit of Eq. (90) with 0 Recent studies have also considered whether approximately EoS independent relations exist between higher- y =( M ) 0 1/ and x = M and the coe cients given in Table I. Observe -1 that the EoS-universality is slightly weaker -1 ` multipole moments. Reference [] in fact found than the S 3 M relation, but still, it holds up to roughly one such relation between the current octupole and the 0%. This larger variation is not an artifact of numerical error, - since our calculations are valid to O(1%). This mass - quadrupole moments of NSs. This relation was not only approximately EoS-universal but also approximately spin-insensitive. The Newtonian results of [] indicates that the universality becomes worse as one considers multipole moment relations for higher ` modes, as analytically -3 confirmed this result. The latter, in fact, -3 1 first predicted 1 in []. proved that higher-` multipole M moments in the nonrelativistic Newtonian limit can be expressed in terms APR M M n = FIG. of7. ust M (Color the n mass /( online) monopole, n M n+1 (Top) S 3 spin M ), relation current J n+1 with dipole = various and J realistic quadrupole NS and QS moments EoSs and through spins, relations together that with are the approxi- fit in hexadecapole ( M ) quadrupole ( M ) moments relation with mass n+1 FIG. /( n+1 1. (ColorM n+ online) (Top) ) The (reduced dimensionless) Eq. mately (90) and EoS-universal the Newtoniandrelation spin-independent. for the n =0.5 Thispoly- tropes sality, in []. however, The meaning was found of the todotted-dashed deteriorate with vertical increasing line by Eq. (90) around and the Newtonian neutronrelation starsfound as in well []. as Observe the univer- various NS and QS EoSs and spins, together with the fit given All these are properties that characterize the spacetime is the ` multipole same as innumber. Fig. 1. Observe that the QS relation is almost the Thesame existence as the NS of approximately one. (Bottom) Fractional universaldi erence M the relation approaches the Newtonian one as one increases gravitational aspects of the stars themselves. relations is. The Newtonian relation for an n =0.5 polytrope agrees - between not only the data of academic and the fit. with the relativistic fit for various realistic EoSs within % interest, but it also has practical accuracy above the critical M denoted by the dotted-dashed, applications. For example, if one could measure any two vertical line. (Bottom) Fractional di erence between the data G.P. ical and quantities values T. and A. in Apostolatos, the a given fits. relation Observe Phys.Rev.Lett. independently, that the slow-rotation 11 one could 111 and (01). the fit. Observe the relation is universal to roughly 0%. relation perform to O( an EoS-independent ) is valid to O(1%) test for of GR < 0.3. in the strong- This means that the hexadecapole moment can be approxi- K. Yagi, K. Kyutoku, G. P., N. Yunes, and T.A. Apostolatos, Phys.Rev. 0.3<χ<0. D (01). S 3 S 3 -S 3 fit /S3 fit M RNS -M slow /M slow M M -M fit /M fit

9 Spacetime around neutron stars: An analytic spacetime /17 An analytic neutron star spacetime would be very useful to do astrophysics. The vacuum region of a stationary and axially symmetric space-time can be described by the Papapetrou line element 7, ds = f (dt wdφ) + f 1 [ e γ ( dρ + dz ) + ρ dφ ], where f, w, and γ are functions of the Weyl-Papapetrou coordinates (ρ, z). By introducing the complex potential E(ρ, z) = f(ρ, z)+iψ(ρ, z), the Einstein field equations take the form, (Re(E)) E = E E, where, f = ξ a ξ a and ψ is defined by, a ψ = ε abcd ξ b c ξ d. An algorithm for generating solutions of the Ernst equation was developed by Sibgatullin and Manko 9. A solution is constructed from a choice of the Ernst potential along the axis of symmetry in the form of a rational function E(ρ = 0, z) = e(z) = P (z) R(z), where P (z), R(z) are polynomials of z of order n with complex coefficients in general. 7 A. Papapetrou, Ann. Phys., 1, 309 (1953). F.J. Ernst, Phys. Rev., 17, 1175 (19); Phys. Rev., 1, 115 (19). 9 V.S. Manko, N.R. Sibgatullin, Class. Quantum Grav.,, 133 (1993).

10 Spacetime around neutron stars: An analytic spacetime 9 /17 Two-Soliton spacetime: This is a -parameter analytic spacetime which can be produced if one chooses the Ernst potential on the axis to have the form: (z M ia)(z + ib) k e(z) = (z + M ia)(z + ib) k The parameters a, b, k of the spacetime can be related to the first non-zero multipole moments through the equations, where M is the mass. J = am, M = (a k)m, J 3 = [ a 3 (a b)k ] M, One can use the multipole moments M, J, M, and J 3 of a numerically calculated neutron star and produce an analytic two-soliton spacetime that reproduces very accurately the numerically calculated spacetime. Instead of using a specific set of values for the moments, one could reproduce any neutron star spacetime using the universal relations ( ) ν1 ( ) ν 3 J 3 = A + B 1 M + B M, Therefore the first higher moments of a general neutron star spacetime can be expressed in terms of only three parameters, the mass M, the angular momentum J, and the quadrupole M, 11 having thus a universal analytic spacetime. G. P., and T. A. Apostolatos, MNRAS, 9, 3007 (013); Other analytic spacetimes have been proposed in the past, see for eg. E. Berti, and N. Stergioulas, MNRAS, 350, 11 (00) 11 G.P. and T. A. Apostolatos, Phys.Rev.Lett (01): 3 J 3 = ( M ) 0.5

11 Scaled Frequencies for the Kerr spacetime: Scaled Orbital frequency: MΩ = /r 3 1+(1/r) 3/ Kerr frequencies /17 Scaled Radial frequency: Mκ ρ = MΩ ( 1 (1/r) + (1/r) 3/ 3 (1/r) ) 1/ Scaled Vertical frequency: Mκ z = MΩ ( 1 (1/r) 3/ + 3 (1/r) ) 1/ Orbital, Mν φ, and precession, Mν a, scaled frequencies for Kerr black holes for various ( ). Kerr black hole Kerr black hole M v km Hz M vr, M vz km Hz Rcirc M Plots of the orbital, periastron and nodal precession scaled frequencies M v φ km Hz Plots of the periastron and the nodal precession scaled frequencies against the orbital scaled frequency. The general effect of rotation is to increase the observed frequencies (and reduce the ISCO radius; for 1 the horizon and ISCO radii go to 1M).

12 Neutron star frequencies 11 /17 Frequencies for the spacetime around neutron stars: The effect of rotation Orbital frequency, periastron precession frequency and nodal precession frequency: Orbital, Mν φ, and precession, Mν a, scaled frequencies for neutron star models constructed with the APR EOS for various up to the Kepler limit(0-0.7) and the same central density. Ρ c.31 1, M 0.9Mo,1.13Mo, 0.,0. Ρc 7.3 1, M 1.Mo,1.5Mo, 0.,0. M v km Hz M v km Hz Rcirc M Ρc. 1, M 1.7Mo,.Mo, 0.,0. M v km Hz Rcirc M Rcirc M Plots of the orbital, periastron and nodal precession frequencies for different rotations for models with ρ c =.3 1 g/cm 3. Rotation in the range, f kHz. Same plots for ρ c = g/cm 3 (upper) and ρ c = 15 g/cm 3 (lower).

13 Neutron star frequencies 1 /17 Application: Corrugation modes in accretion disks around neutron stars. Upper: Diskoseismic propagation diagram for a Kerr black hole with spin Upper: Same plot but for a neutron star spacetime with spin parameter parameter = 0. for one-armed waves with m = 1, n = 1. Waves can = 0., quadrupole rotational deformability α =, and spin-octupole propagate in the white regions exterior to risco, and are evanescent in deformability β ' 1.. Waves with frequency f = ω/π can propagate the shaded regions between the vertical resonances (VR) and Lindblad in the white regions exterior to the NS radius rns (or wherever the disk is resonances (LR). Inertial modes (g-modes), with m = 1, n = 1, can truncated). Wave regions are qualitatively similar to the Kerr black hole, become self-trapping due to the turnover of the outer Lindblad resonance, except for the low-frequency c-mode region, where ω < Ω Ω. Lower: while lower frequency g-modes are quickly damped by corotation (CR). At low frequencies c-modes can be self-trapped due to the turnover of Corrugation waves can propagate at high frequencies exterior to the outer the Lense-Thirring frequency, Ω Ω, at radius rpeak, and frequency fpeak, VR, and at low-frequencies interior to the inner VR. Lower: Enlargement as a result of the spacetime quadrupole contribution. of the propagation diagram at low frequencies.

14 Astrophysical observables: frequencies 13 /17 Orbital and precession frequencies: G.P. MNRAS 5, 0 (015) R isco /M M ν ISCO = M Ω ISCO /(π) R ISCO R eq M ν max K M ν max K & f NS M (Ω z /π) ISCO M (Ω z /π) inn

15 Accretion disc properties: Astrophysical observables: accretion disc 1 / η = 1 Ẽ in Some basic scalings: T max eff (Ṁ 0 = 1 M year 1 ) ρ = M ρ, Ω = M 1 Ω, Ẽ = M 0 Ẽ, and L = M L. Some more involved scalings: νl ν /Ṁ 0 F (ρ)/ṁ 0 = M ( F ( ρ)/ṁ 0 ), T = M 1/ (Ṁ 0 ) 1/ T, ε hν = M 1/ (Ṁ 0 ) 1/ ε hν, and finally, L ν = M 1/ (Ṁ 0 ) 3/ L ν, and νl ν = (Ṁ 0 )νl ν.

16 Combining the different properties: Measuring the moments / Contour plots of the orbital frequency at the orbit closest to the stellar surface (dashed black lines), the nodal precession frequency at the same orbit (solid black lines), and the rotation frequency of the star itself (dotted red lines) Contour plots of the maximum integrated luminosity (solid black lines), the nodal precession frequency at the orbit closest to the stellar surface (dashed black lines), and the rotation frequency of the star itself (dotted red lines).

17 ...and constraining the EOS 1 /17 Identifying the EoS: Determining the parameters α and and the independent knowledge of the mass of the neutron star (assuming for example that it is known from the binary system observations), one can evaluate the first three multipole moments. Such a measurement 1 of the first 3 moments (M, J, M ) could select an EOS 13 out of the realistic EOS candidates M km G.P. MNRAS 5 0 (015), and for an alternative proposal see, G.P., 01 MNRAS,, G.P. and T. A. Apostolatos, Phys.Rev.Lett (01)

18 Thank You. 17 /17