Revised Pulsar Spindown Page: 3 of 8
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FIG. 1. P - P evolutionary diagram that shows the effect
of the misalignment angle 0. We show spindown trajectories for
pulsars with initial period P = 10 msec, magnetic field B, =
1012 G, and Vgap = 1013 Volt. Initial braking index n = 3 is
assumed (a = 0). Trajectories from bottom to top correspond to
pulsars with increasing magnetic inclination angle from 0' to 90'
with 150 increments. Rectangular dots indicate when pulsars reach
Death and turn off. The dashed line represents the theoretical
death line as given in eq. 15 which does not take into account
the misalignment angle dependence introduced in eq. 12. Oblique
pulsars evolve faster through the diagram.
More importantly, though, eq. 8 brings a new element
to the discussion: we directly introduce pulsar death and
magnetic-rotation axes misalignment in the electromag-
netic pulsar spindown expression. Pulsar death expresses
the inability of the magnetosphere to generate the par-
ticles required in the poloidal electric currents which
generate spindown torques in the aligned rotator case.
They do not affect the orthogonal spindown component
though. Beyond the death line, the misaligned neutron
star will continue to spin down without emitting pulsar
3. PERIOD EVOLUTION AND THE P - P DIAGRAM
In order to study how pulsars spin down, we equate
the values of L in eqs. 1 & 8 and thus obtain
P - 3.3 x 10-
x (sin20+ (-
where from (5)
-1 B)2 (Po
d1012 G sec
Death = 8.12-a sec
2--a P( s
Po is the initial period at pulsar birth. Note that we
have introduced an angular dependence in the definition
of the death line P(Pdeath). As we will see below, this has
interesting observational consequences. Note also that,
in the limit 0 ~ 90' and a = 0, we obtain the following
_death 5 x 10- Pdeath 3 ( oap 2
Po 1012 Volt
which can also be written as
log Pdeath 3 log Pdeath - 17.3 + 2(log Vgap
(compare with Zhang, Harding & Muslimov 2000).
When P < Pdeath,
and pulsars evolve in the logarithmic P - P diagram
along lines of constant magnetic field P c p2a-1 As P
approaches Pdeath, however, the evolution curves down
away from the above straight lines and ends at a point
P - Pdeath and P = P(Pdeath) (see figure 1). This evo-
lution is similar to what one would expect if we had as-
sumed magnetic field decay in the standard dipole spin-
down model (e.g. Gonthier et al. 2002 and 2004, here-
after G02 and G04 respectively). In figure 1 we show
only a = 0, or evolution with initial braking index n = 3.
For n < 3 the straight portion of the lines would rotate
counterclockwise, and would become horizontal for n = 2
(a 0.5). Such pulsars would also evolve faster.
As a test of the validity of our results, we tried to re-
produce the observed distribution of pulsars in the P- P
diagram with the minimum number of assumptions pos-
sible, through a simple Monte Carlo numerical experi-
ment that follows G02 and G04. We integrated eq. 12
for random times t within a 109 year time interval (this
mimics random pulsar birth in the galaxy). The result of
the integration is recorded only if the star remains active
as a pulsar, i.e. only if P(t) < Pdeath. To make our re-
sults more realistic, we assumed a multipeaked lognormal
distribution of polar magnetic fields around B* 1012G
(eq. 1 in G02): pB E=1A~e-(logB-1-ogB)2/a . We used
two terms with A1 20.0, A2 = 2.0, log B1 12.65,
log B2 = 13.0, o-1 = 0.6, and 0-2 = 0.4.
We assumed a uniform random distribution of initial
pulse periods from 10 msec to 0.2 sec. We did account for
the observational selection effect due to finite instrument
detectability by implementing the radio pulsar luminos-
ity model of Narayan & Ostriker 1990 (eqs. 23, 24 in
G02), and a lower detectable pulsar luminosity limit of
12.5 mJy-kpc2. Finally, we assumed that there is no mag-
netic field or misalignment angle evolution with time. In
an effort to simplify the approach we did not consider
the multitude of selection effects and individual survey
detectability limits. We also did not calculate the dis-
tance to individual pulsars after tracking them in the
galactic potential as in G02 and G04, relying on the lumi-
nosity distribution function. We believe, however, that
the gross effects of the new spindown law (12) can be
understood even with our procedure.
Our results are shown in figs. 2,3 & 5. In figure 2
we show the result of our simulation for the standard
B* oa P 2P 2 --a
n-= 3 -2a
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Contopoulos, Ioannis; Academy, /Athens; Spitkovsky, Anatoly & /KIPAC, Menlo Park. Revised Pulsar Spindown, article, December 14, 2005; [Menlo Park, California]. (digital.library.unt.edu/ark:/67531/metadc873604/m1/3/: accessed September 21, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.