Download Spin-up by accretion with magnetic field and formation of

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!)