I. Contopoulos
Research Center for Astronomy, Academy of Athens, 11527 Athens, Greece
Received 18 June 2007 / Accepted 10 September 2007
Abstract
Positive and negative pulsar breaking indices
suggest that some fraction of the pulsar
spindown torque undergoes a cyclic evolution.
The observed strong correlation of
"anomalous'' breaking indices with pulsar age
implies that the characteristic periodicity
timescale is in the range
100 to 10 000 years depending on the
fraction of the spindown torque that undergoes
cyclic evolution, 1 to 100% respectively. We argue that the longest variability timescale is consistent with a neutron star magnetic cycle similar to the solar cycle.
Key words: stars: pulsars: general - stars: magnetic fields
A pulsar spins down due to the torque
on the neutron star crust generated by the electric current
flowing in its magnetosphere. In the simplified picture of a steady-state
axisymmetric force-free ideal MHD magnetosphere, Contopoulos et al. (1999, hereafter CKF) first showed
that the distribution of the magnetospheric electric current I can be determined as an eigenvalue of the problem, if one makes the natural assumption that the
magnetosphere is smooth and continuous on the light cylinder
(defined as the distance
from the rotation
axis, where
is the pulsar angular velocity).
The unique electric current distribution thus obtained yields a
unique pulsar spindown torque, and thus a unique pulsar
spindown rate
.
This is of the same order as the value obtained
for simple electromagnetic vacuum dipole radiation, namely
Strictly speaking, however, the results of CKF and subsequent
related work are only valid in steady-state. Thus, as the neutron star
spins down and the light cylinder moves to larger and larger
distances, one needs to take into account the
evolution of the pulsar magnetosphere. The first thing one
may assume is that the magnetosphere evolves through a
sequence of steady-state equilibria of the CKF type, i.e. that
it manages to readjust itself so that at all times, the region of closed lines extends all the way to the light cylinder, and the last open magnetic field line
extends to infinite distances without reconnecting accross the
equator. In addition, one may assume that B_{*} does not evolve with pulsar age. Unfortunately, the situation is more complicated than that, since Eq. (1) yields a braking index value
Figure 1: Braking index as a function of characteristic spindown time. We plot here ( according to whether n>0 or n<0) vs. , where in years. Note that |n|>1 everywhere. At large , the diagram may be fit by the simple linear relation . | |
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In Fig. 1 we plot
(with
according to whether n>0 or n<0respectively) as a function of the characteristic spindown time
in years. One may argue
(Alice Harding, personal communication) that in young pulsars (
), braking index measurements may be "corrupted'' by neutron star glitches. On the other hand, in older pulsars (
)
where glitches are not as important,
one finds the correlation
Figure 2: diagram for the 400 pulsars with measured braking index values shown in Figure 1. Here, P, are the pulsar period and period derivative respectively. The distribution is that of a standard sparse diagram without any obvious observational bias. | |
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There are several ways to reconcile Eqs. (1) and (3):
(A) assume that the approximation of a sequence of CKF-type steady-state magnetospheric equilibria holds, and that the neutron star magnetic field undergoes a cyclic evolution;
(B) relax the assumptions of the CKF analysis and assume a variable magnetospheric structure that would yield a cyclic evolution of the factor f; (C) relax the assumption of constant neutron star moment of inertia. Only (A) and (B) refer to the spindown torque itself. In any case,
Figure 3: Fit of the distribution shown in Fig. 1 assuming a 100% cyclic evolution of the pulsar spindown torque (Eqs. (4) and (5)). (in years). The fit is acceptable, even for young pulsars ( ) where some of the dispersion in the measurements of n is due to neutron star glitches. | |
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Several physical models that address the issue of cyclic variation of the pulsar spindown have been proposed in the literature, ranging from neutron star interior "wobbling'' on a timescale of a few years (e.g. Kundt 1988), to magnetospheric variability (e.g. Contopoulos 2005). In the present work, we would like to focus on our simplest (one-parameter) fit of the anomalous braking index data, namely the one with and .
F(t) becoming zero periodically is not compatible with a cyclic
evolution of the neutron star moment of inertia
(case C above). On the other hand such a scenario is compatible with a cyclic evolution of the neutron star magnetic field similar to the solar cycle (case A above).
Interestingly enough, the ten thousand year timescale that we obtain
is comparable to the neutron star cooling timescale (e.g. Blandford et al. 1983).
It is conceivable that some sort of dynamo mechanism in the neutron star interior, may support a cyclic evolution with
Figure 4: On the left, we plot the magnetic field structure in the case of no magnetospheric reconnection (CKF). Distances are normalized to the light cylinder distance . On the right, we plot the corresponding electric current distribution along the field lines that cross the light cylinder. The magnetic flux is normalized to the canonical value . The electric current is normalized to the canonical value . For comparison, we plot also (dashed line) the electric current distribution of a relativistic magnetic split monopole with the same amount of magnetic flux crossing the light cylinder (Michel 1991). | |
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Figure 5: Same as Fig. 4 with some amount of equatorial magnetospheric reconnection that corresponds to (Eq. (10)). | |
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We also tried to seek variants of the CKF solution (case B above)
that would yield values of f very different from unity.
In fact, what we need is a physical
mechanism that will periodically turn off
the neutron star magnetospheric spindown.
In a series of papers (Contopoulos 2007b,c), we relaxed
the assumption of ideal MHD in the equatorial
region of the pulsar magnetosphere beyond the light cylinder.
This is the region where the magnetospheric return current
flows, and several authors before us suggested that this
may be the region of electromagnetic energy dissipation
that would result in particle acceleration (e.g.
Coroniti 1990; Michel 1994; Lyubarsky & Kirk 2001; Kirk & Skjæraasen 2003; Romanova, Chulsky & Lovelace 2005). As we argued in Contopoulos (2007c), one cannot study equatorial
reconnection without taking into account the global topology
of the poloidal magnetic field.
The details of equatorial reconnection remain (yet) unknown.
However, it is easy to realize that, when equatorial reconnection
is present, magnetic field lines that cross the light cylinder and would
have extended to infinity in CKF, now continuously reconnect
across the equator. As a result, the equatorial condition for the magnetic flux
function
(defined as the magnetic flux crossing a circle of cylindrical radius r at height z around the axis of rotation) differs from that in CKF. In particular,
is not constant but decreases with distance. We assume for simplicity that
Figure 6: Same as Fig. 4 with maximum equatorial magnetospheric reconnection that corresponds to . | |
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Figure 7: The spindown torque parameter f (Eq. (1)) as a function of the dissipation parameter . In the absence of reconnection ( ), f=0.6 (CKF). | |
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We conclude that a cyclic component in the evolution of the magnetospheric spindown torque may account for the measured large positive and negative anomalous braking index values. If we are willing to consider a 100% cyclic evolution, this can only be due to a neutron star magnetic cycle similar to the solar cycle. In that case, the evolution timescale would be on the order of 10 000 years.
Acknowledgements
We would like to thank Drs. Alice Harding and Demos Kazanas for their hospitality at the NASA Goddard Space Flight Center in January and June 2007 where some of the ideas in the present work originated. We would also like to thank Pr. Wolfgang Kundt for an honest exchange of ideas.