# Revised Pulsar Spindown Page: 3 of 8

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10-12-

V

00.01

0.10

Period, sec

,'

. . . . z's . . . . .. a . . . . .1.00

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

radiation.

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 obtainP - 3.3 x 10-

x (sin20+ (-

where from (5)-1 B)2 (Po

d1012 G sec

Pds2at)Death = 8.12-a sec

B*

X 1012GVgap

1012 Volt)2--a P( s

Po is the initial period at pulsar birth. Note that we

have introduced an angular dependence in the definitionof 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

simple expression_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(14)

12) (15)(compare with Zhang, Harding & Muslimov 2000).

When P < Pdeath,10.00

(16)

(17)

(18)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 standardB* oa P 2P 2 --a

n-= 3 -2a

m=n(n -1),10-"-

10-1n r

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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 May 24, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.