Astron. Astrophys. 319, 515-524 (1997)

4. Structure function analysis

In this section we study the short term fluctuations of angular velocites by employing structure function analysis (Lindsey & Chie 1976; Cordes & Downs 1985). This technique is more sensitive to sharp fluctuations. Therefore it is more likely to show the effects of a possible decoupling between the crust and the superfluid core. A first order structure function for the angular velocity time series can be defined as

For a random walk in frequency one can express the time series as

where is the step size of events is the step function and is the events occur at random times with a rate R. The structure function of the above equation can be written as (Cordes 1985),

where t is the lag between two measurments of , S is the random walk noise strength.

If the neutron star has significiant superfluid in the core (Lamb et al., 1978a, b; Sauls 1988; Lamb 1991; Datta & Alpar 1993) then the moment of inertia of a neutron star resides mainly in its neutron superfluid core. There is a neutron superfluid also in the inner part of the crust lattice. This crust neutron superfluid carries of the stars's moment of inertia, and is coupled to the rest of the crust on very long time scales, typically extending to years (Alpar et al., 1981; Alpar et al., 1993). The crust superfluid moment of inertia is resolved in radio pulsars which are spinning down due to electromagnetic dipole radiation, as a sudden changes in rotation rate (or glitch events) with a magnitude and its time derivative (Lyne 1993; Shemar 1993). These kind of glitches can not be detected in the accretion powered X-ray binaries because the external torque noise in these systems dominates (Baykal & Swank 1996). Also the internal dynamics of radio pulsars spinning down may be quite different (Alpar 1993). In this work we will not be concerned with the crust superfluid. In the core of the neutron star, the rotating neutron superfluid is coupled to the stellar crust by interactions between the charged particles (electrons and protons) and the quantized vortices. In the case of any change in rotation velocity of the crust the whole charged component (protons, electrons) of the stellar interior is coupled to the crust on a short time scale, typically less than (Easson 1979). The angular velocity of the neutron superfluid is determined by the density of quantized vortices. The rotation velocity of the core neutron superfluid will not change unless the quantized vortex lines move radially (outward for spin-down and inward for spin-up). Any fluctuation on the crust creates a force on the vortices because of the relative velocity between the vortex line and charged component. Vortex lines relax with charged particles at dynamical (or at the crust core coupling time ) time scales, which is of the order of , where P is the rotation period of the neutron star, is the change in mass of proton as a result of its coupling with neutrons (Alpar et al., 1984; Alpar & Sauls 1988).

The above senario can be approximated with a two component neutron star model (Baym et al., 1969). In this model, one component is the crust-charge particle system, which consists of protons, electrons and the crust with an inertia with rotates with angular velocity . The second component is the core neutron superfluid, with moment of inertia , with rotates with angular velocity . Any external torque on the crust creates a lag between and . The two components are coupled by a crust core coupling time ,

Here is the external torque exerted on the star. From the power spectra in the previous section we found that fluctuations are consistent with white torque noise. We assume that external torque fluctuations are also white noise , where is the angular momentum added or subtracted from the star at . Then the fluctuations of the crust can be expressed (Baykal et al., 1991) as

where is the moment of inertia of the neutron star. If the neutron star response purely rigid or then the time series behave as a pure random walk in angular velocity (see Eq. 5).

By using the definition of the structure function for the time series above and defining the , then the mean square fluctuation of the white noise variable can be written as

where . At long time lag , angular accelerations fluctuations behaves as , while at short time lag behaves as . This means that in the long time lag limit, the core superfluid couples to the neutron star crust and responds to the external torque with total moment of intertia I. On the other hand in the short time lag neutron star responds to external fluctuations with crust moment of inertia (or with charged particle components) (see also Lamb et al., 1978a, b). In Fig. 5a, we plotted the angular acceleration fluctuations in the case of core superfluid mixture for a crust core coupling time day. In the plot we adapted the noise strength value (S) from the power spectrum as rad² /sec³. As it is seen from Fig. 5a, if there is a significant core superfluidity and even if the crust core coupling time is relatively short day, the crust response increases the angular acceleration fluctuations up to time lags of tens of days . In other words even if we do not see the torque events on shorter time scales we can resolve the crustal moment of inertia by measuring the angular accelerations and testing with a simple two component neutron star model. In Fig. 5b,c, we simulated 1000 independent time series in the form of Eq. 9, is sampled according to the OAO 1657-415 angular velocity history (see Fig. 2). Then we compared the angular acceleration fluctuations with observed fluctuations. In the simulation, we sampled the events uniformly with a rate days^-1 and used the input variance  rad² /sec². First, we simulated the time series with and day; the simulated angular frequency fluctuations (with larger error bars) are shown together with the observed fluctuations (with smaller error bars) in Fig. 5b. Clearly the simulated fluctuations are not compatible with the observed fluctuations. In Fig. 5c, a pure random walk time series (or a rigid body respose with ) is simulated. In this case, the simulated and observed angular frequency fluctuations agreed with each other at level at all time lags (see Fig. 5c). This indicates that the angular acceleration fluctuations seen in OAO 1657-415, are associated with external torques. The absence of a signature of core superfluidity is suggesting us that either crust-core coupling time is so short (i.e. days where sec is the pulse period) that all charged components and the core superfluid couple on the order of hours or the core superfluidity is not significant . If the latter possibility is correct, this is suggesting that either most of the neutrons in the core are too hot to be in the superfluid phase (Ainsworth et al., 1989) or the equation of state is stiff with higher crustal moment of inertia (Lamb 1991). The well studied high mass X ray binary Vela X-1 showed that less than of the moment of inertia of the star is weakly coupled to crust with coupling times in the range , P=283 sec (Boynton et al., 1984; Baykal et al., 1991) while radio observations constrain the crust core coupling time to be (Chau 1993).

Fig. 5a-c. a Theoretical angular accelerations for a crust core coupling time day and a ratio of core superfluid moment of inertia to crust moment of inertia . b Simulated angular accelerations (with larger error bars) with day and and observed angular accelerations. c Simulated angular accelerations for pure random walk model (larger error bars) and observed angular accelerations (see also text).

© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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