General Relativistic Eects on Pulsar Radiation. Dong-Hoon Kim Ewha Womans University

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1 Geneal Relativistic Eects on Pulsa Radiation Dong-Hoon Kim Ewha Womans Univesity The 2nd LeCosPA Intenational Symposium NTU, Taiwan, Dec. 14,

2 Outline 1. Electomagnetic adiation in cuved spacetime 1) Petubed EM adiation due to gavitation 2) Point-paticle & semi-elativistic appoximations 3) Synchoton adiation aound a pulsa 4) Cuvatue adiation aound a pulsa 2. Polaization of cuvatue adiation in pogess) 3. Conclusions and discussion 2

3 1. Electomagnetic adiation in cuved spacetime 1) Petubed EM adiation due to gavitation In cuved spacetime: 2 A µ R ν µ A ν = 4π c j µ ; j µ = e In at spacetime: Γ ḡ µµ zτ), zτ )) u µ τ )δ 4) zτ) zτ )) dτ. A µ = 4π c j µ ; j µ = eu µ. Consideing the gavitation eect, we may assume Then we have A µ pet) = A µ flat) + A µ. F µν pet) = µ A ν pet) ν A µ pet) = µ A ν flat) ν A µ flat) + µ A ν ν A µ 0 E x E y E z 0 E x E y E z = E x 0 B z B y E y B z 0 B x + E x 0 B z B y E y B z 0 B x E z B y B x 0 E z B y B x 0 flat) E µ pet) = Eµ flat) + E µ and B µ pet) = Bµ flat) + Bµ. 3

4 2) Point-paticle & semi-elativistic appoximations The system of a sola-mass black hole [neuton sta] and a chage can be modeled as black hole [neuton sta] + point chage. Then EM adiation to be measued by an obseve who is fa away fom this system can be obtained by solving the Maxwell's equations: A µ = 4π c j µ. The solutions ae the Lienad-Wiechet potential, Q µ t et ) A µ t, x) = c [ 1 n Ẋ t et) /c ] x X t et ) whee the chage-dipole moment is tet =t x X c ; n x X t et) x X t et ), Q µ = qx µ. In a semi-elativistic appoximation we identify the chage's tajectoy X µ with a geodesic in the backgound geomety of a BH [NS]: petubed EM adiation esults fom the chage's acceleated obital motion due to the BH [NS]. Q µ pet) = qxµ flat) + qx µ A µ pet) = Aµ flat) + Aµ. 4

5 3) Synchoton adiation aound a pulsa Fo a slowly otating pulsa, we may conside the synchoton adiation as if it was poduced fom electons gyating in a magnetic eld B ove a static neuton sta: the geomety is indistinguishable fom a Schwazschild black hole, ds 2 = 1 2GM ) c 2 dt 2 + c 2 1 2GM ) 1 d dθ 2 + sin 2 θdϕ 2). c 2 5

6 A dipole-type magnetic eld B in the Schwazschild geomety can be eectively modeled by solving the Maxwell's equations with a cuent loop I along the sta's equato [Petteson, Phys. Rev. D, 10, )]: with [solution] j µ = δ µ ϕ 2 A µ = 4π c j µ, 1 R ) 1/2 Schw Iδ R NS ) δ cos θ) 2 [ ln 1 R ) Schw ; R Schw 2GM NS c 2. ] + R2 Schw, 2 2 A ϕ = 3m2 sin 2 θ RSchw 3 + R Schw with the magnetic dipole moment m πrns 2 I [1 R Schw/R NS )] 1/2, and thus Bˆ = g ϕϕ g θθ) 1/2 Aϕ, θ Bˆθ = g ϕϕ g ) 1/2 Aϕ, 6m cos θ = = R 3 Schw 6m sin θ R 3 Schw [ ln [ ln 1 R Schw 1 R Schw ) 1 R Schw ) 1/2 ) + R Schw + R Schw R Schw ] + R2 Schw, 2 2 RSchw 2 ]. 2 R Schw ) 6

7 [Example] Magnetic eld B o at the pola cap A caeful geneal elativistic teatment of a magnetized otating body can give an estimate fo the magnetic eld B o at the pola cap of a neuton sta of R NS = 10 6 cm and M NS = 1.4 M PSR ). Fo a slowly otating body, we nd E ot = 1 2 IΩ2 = 2π2 I P, 2 whee the geneal elativistic moment of inetia can be well-appoximated by an empiical fom [Lattime and Schutz, ApJ, 629, )] [ I J Ω ± 0.008) MR M 10 5 cm M M R ) 4 ] cm. M R Now, we assume de ot /dt = EM adiation powe as measued by an obseve in an inetial efeence fame, whee de ot = d ) 1 dt dt 2 I NSΩ 2 = 4π 2 I NS P 3 P, EM adiation powe = 2 3c 3 = 8π4 27 d 2 m inet) sin 2 θ i c 3 P 4 dt 2 ) 2 = 2 3c 3 sin θi Ω 2 m inet) ) 2 1 R Schw R NS ) [ln B 2ˆ = R NS, θ = 0)RSchw 6 ) 1 R Schw R NS + R Schw R NS ] 2. + R2 Schw 2RNS 2 7

8 Then we nd B o Bˆ = R NS, θ = 0) to be ) 1/2 ) 3 1 R Schw R [ln 1 R Schw NS R NS B o = πrschw 3 sin θ i + R Schw R NS ] + R2 Schw 2RNS c3 I NS P P ) 1/2. Fo PSR with obseved values P = s and P = , we obtain B o 9.5 ± G/ sin θ i. Howeve, in the at-spacetime computation, we take I = 2 5 MR2 Newtonian) and R Schw 0, then obtain [Caol and Ostlie, An Intoduction to Moden Astophysics 2nd ed 2007)] B o G/ sin θ i. Theefoe, we nd ω cuv) ω flat) = B ocuv) B oflat) = 9.5 ± G/ sin θ i G/ sin θ i 10.5 ± 3.7 % incease! 8

9 4) Cuvatue adiation aound a pulsa The magnetic eld lines aound a pulsa fo cuvatue adiation) can be obtained fom the dieential equation [Konno and Kojima, Pog. Theo. Phys., 104, )]: 3m2 sin 2 θ R 3 Schw d dθ = B [ ln 1 R Schw ) B θ A ϕ = const., + R Schw + R2 Schw 2 2 ] = k const.) : coheent eld lines. Cuvatue adiation is chaacteized by the adius of cuvatue: ρ, θ) = 3m sin 2 θ = ) 1 dθ d = dl + dθ [ ) ln 1 R Schw 3 R 3 Schw k ) 2 and in the at spacetime limit + 2 R 2 Schw ρ, θ) ] + 2R Schw ) 1 + cot 2 ln 1 R Schw θ ) ln 1 R Schw 3m sin θ cos2 θ. k + R Schw + R Schw R Schw + R2 Schw 2 2 R2 Schw 2 R Schw ) 2, 9

10 Magnetic eld lines aound a pulsa: plotted in cuved and at spacetimes. Note ρ cuv < ρ flat. Theefoe, ω cuv = βc > ω flat = βc. c.f. ω CR Hz vs. ρ cuv ρ flat ω cyc Hz) 10

11 2. Polaization of cuvatue adiation in pogess) Geometic conguation of a chage in motion along a cuve fo cuvatue adiation) Cedit: [Gil and Snakowski, Aston. Astophys. 234, )]) While a chage moves along the cuved magnetic eld lines, cuvatue adiation will be emitted, which is stongly polaized in the plane of cuvatue. We have E ω) = E o ω δ 2 + φ 2) [ ω K 2/3 δ 2 + φ 2) ] 3/2 3ω o E ω) = ie o ωφ δ 2 + φ 2) 1/2 K1/3 [ ω 3ω o δ 2 + φ 2) 3/2 c.f. K 2/3, K 1/3 : modied Bessel functions) exp i ωr c ] exp ), i ωr c ) ; E o = 2eω o 3cR, ω o = βc ρ. 11

12 Then the Stokes paametes ae given by I = E E + E E = E 2 o ω 2 { δ 2 + φ 2) 2 K 2 2/3 [ ω +φ 2 δ 2 + φ 2) K 2 1/3 [ ω 3ω o δ 2 + φ 2) 3/2 ]}, Q = E E E E = E 2 o ω 2 { δ 2 + φ 2) 2 K 2 2/3 [ ω φ 2 δ 2 + φ 2) K 2 1/3 U = E E + E E = 0, [ ω 3ω o δ 2 + φ 2) 3/2 ]}, δ 2 + φ 2) ] 3/2 3ω o δ 2 + φ 2) ] 3/2 3ω o V = i E E E E ) = 2Eo 2 ω 2 φ δ 2 + φ 2) { 3/2 δ 2 + φ 2) [ ] 2 K 2 ω 2/3 3ω [ o ω K 2/9 δ 2 + φ 2) ] [ 9/2 ω K 1/3 δ 2 + φ 2) ] 3/2, 3ω o 3ω o Modulation of ρ in cuved spacetime leads to modulation of the Stokes paametes. The atio between linea and cicula polaization should change in cuved spacetime: atio = V L = V Q2 + U = V 2 Q. 12

13 Specta of I ω) solid), L ω) = Q ω) dashed), and V ω) dotted) coesponding to φ = 0.25 o. Cedit: [Gil and Snakowski, Aston. Astophys. 234, )]) In cuved spacetime these specta will change and shift to the ight due to incease of the eld intensity and chaacteistic fequency: E o ω o ρ 1, ω c ω o ρ 1. 13

14 3. Conclusions and discussion Point-paticle and semi-elativistic appoximations can be applied to the study of electomagnetic adiation nea compact objects - black holes o neuton stas; solving Maxwell's equations in cuved spacetime. Fequency modulation of electomagnetic adiation - synchoton/cuvatue adiation in cuved spacetime aound a pulsa can be estimated using the above computational methods. Futhe elated studies such as special elativistic beaming eects in conjunction with this should follow in the futue. Modulation of the Stokes paametes in cuved spacetime will change between linea and cicula polaization of cuvatue adiation. Futhe investigation of this should follow in the futue. Induction of electic eld in the magnetosphee of a pulsa can be appoached by the Goldeich-Julian model. Investigation of this poblem in cuved spacetime will be highly inteesting; extending the model to a non-aligned otating conducting sphee in Schwazschild spacetime. 14