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Spin-up by accretion with magnetic field and formation of millisecond pulsars M. Bejger1 on behalf of M. Fortin1,2 , P. Haensel1 and J. L. Zdunik1 1 2 N. Copernicus Astronomical Center, PAS Laboratoire Univers et Théories, Observatoire de Paris & Université Paris Diderot MODE2010 / Bordeaux / 15.11.2010 Outline ? Spin-up by accretion with and without the magnetic field, ? Description of the magnetic torque, ? Results for recently-measured massive neutron stars (PSR J1903+0327 and PSR J1614-2230), ? Remarks about the interior of an accreting star. Thin-disk accretion without the magnetic field Accretion proceeds from the Innermost Stable Circular Orbit (ISCO). ? The equation for the evolution of total stellar angular momentum J reads dJ = xl lISCO , dMB xl ∈ (0, 1)a where the total baryon mass changes in time as Z t Ṁb (t 0 )dt 0 . ∆Mb (t) = tin a xl ≈ 1 - Beckwith, Hawley & Krolik (2008), Shafee et al. (2008) In Schwarzschild metric, rISCO = 3rs = 6GM rc 2 Thin-disk accretion with the magnetic field To determine the radius r0 from which the accretion effectively takes placea , one needs: ? the magnetospheric radius rm ≡ (GM)−1/7 Ṁ −2/7 µ4/7 , ? the corotation radius rc ≡ GM ωs2 1/3 , ? disk angular-momentum transport equation, it reduces to: ! 7/2 s 3 rm rc 1 − 1 = r0 2 r03 → The stellar angular momentum J evolution equation: s ! µ2 dJ rc3 = l(r0 )− 3−2 3 dMB r03 9r0 ṀB a Kluźniak & Rapapport (2007), Wang (1995), Aarons (1993), Ghosh & Lamb (1979) Also needed: a "recipe" for the evolution of the magnetic fielda : B = B0 /(1 + ∆M/mB ) with mB = 10−5 − 10−4 M . Additionally: ? Toroidal component of the magnetic field: Bφ = Bz (1 − Ω/ωs ), ? the solutions scale with q = B 2 /Ṁ. a van den Heuvel & Bitzaraki (1995), Taam & van den Heuvel (1986) Approximate formulæ for orbital parameters in a thin-disk model Simple expression for specific angular momentuma - substitute the true orbital velocity v= r (Ω−N φ ) N with ' Schw. r (2πforb −2GJ/r 3 c 2 ) c √ 1−rs /r to obtain e l= √ rv 1−v 2 , Rotating neutron star space-time metric is ds2 = −N 2 dt 2 + A2 (dr 02 + r 02 dθ 2 ) + B 2 r 02 sin2 θ(dφ − N φ dt)2 a Bejger, Zdunik & Haensel (2010) Douchin & Haensel (2001) EOS, non-rotating mass: 1.4 M (AAKT: Abramowicz et al. 2003) PSR J1903+0327 Measured parameters: ? M = 1.67 ± 0.01 M , Ω = 465 Hz, Ṗ = 1.88 × 10−20 s/s, B ' 2 × 108 G. Obtained by measuring the periastron advance ω̇ and Shapiro parameter s (Champion et al. 2008, Freire et al. 2009) Initial braking by strong magnetic-field. Characteristic radii of the problem. PSR J1903+0327 Measured parameters: ? M = 1.67 ± 0.01 M , Ω = 465 Hz, Ṗ = 1.88 × 10−20 s/s, B ' 2 × 108 G. Obtained by measuring the periastron advance ω̇ and Shapiro parameter s (Champion et al. 2008, Freire et al. 2009) EOS used: ? APR: Akmal, Pandharipande & Ravenhall (1998), ? BGN2H1: Balberg & Gal (1997), ? DH: Douchin & Haensel (2001). Spin frequency f = 465 Hz, Ṁ = 10−10 M /yr PSR J1903+0327 Measured parameters: ? M = 1.67 ± 0.01 M , Ω = 465 Hz, Ṗ = 1.88 × 10−20 s/s, B ' 2 × 108 G. Obtained by measuring the periastron advance ω̇ and Shapiro parameter s (Champion et al. 2008, Freire et al. 2009) Spin frequency f = 465 Hz, Ṁ = 10−10 M /yr Accretion rate vs. time to form PSR J1903+0327 (initial mass Mi = 1.4 M ) PSR J1614-2230 Measured parameters: ? M = 1.97 ± 0.04 M , Ω = 317 Hz, Ṗ = 9.62 × 10−21 s/s, B ' 1.8 × 108 G. Obtained by measuring the Shapiro parameters s and r (Demorest et al. 2010) Initial magnetic field vs. initial mass (Ṁ = 10−10 M /yr ) Accretion rate vs. time to form PSR J1614-2230 Accretion and spin-up: influence on the interior Evolution of the central density nc : ? spin-up likely with xl ≈ 1 dJ = xl l dMB ? interesting interplay between gravity and centrifugal forces - most of the time the density decreases with M! ? ...seems that accreting objects are not ideal sources for creation of exotic phases _ ¨ Douchin & Haensel (2001) EOS Accretion and spin-up: influence on the interior Evolution of the central density nc : ? spin-up likely with xl ≈ 1 dJ = xl l dMB ? interesting interplay between gravity and centrifugal forces - most of the time the density decreases with M! ? ...seems that accreting objects are not ideal sources for creation of exotic phases _ ¨ ? but... the magnetic field changes a lot ^ ¨ (in this example Bini = 1012 G, Ṁ = 10−10 M /yr ) Douchin & Haensel (2001) EOS (work in progress!)