3. Comparison with observations
Several methods of estimation of the magnetc inclination have been suggested (LM88; R93; Candy & Blair 1986). Although some differences exit among these methods, their common aspects are based on the geometric relations of the polar cap model, and the inclination angle values evaluated by the different methods are not far apart. Lyne & Manchester(LM88) were the first to give the values of magnetic inclination angle for a sample of 106 pulsars based on polarization data. Rankin (1993) suggested a new method to estimate the value of the inclination angle based on the properties of the core width of the radio beam, and gave results for 0a larger sample of 151 pulsars (R93). However, there is an uncertainty in Rankin's method which relies on the core component of the radio beam, since, it is not certain that all those 151 pulsars in R93 actually contain the core component. In Sect. 3.1, we will use both the samples of LM88 and R93. In Sect. 3.2, we only use LM88, which we believe it to be less ambiguous. Also we will assume that the internal field is gauss for this section and that the dipolar field can differ from the internal field because of the observational evidence (Chanmugam 1992; Phinney &Kulkarni 1994).
3.1. Time evolution of pulsar inclination angle
The early statistical studies of observed data clearly support the decay of the inclination angle with time scales yrs(Proszyski 1979; Candy & Blair 1986). According to Eq. (18), the magnetic field strength strongly affects the evolution of this inclination angle. In order to inspect the theoretical prediction, we plot the inclination evolutionary curves in Figs. 2a and 2b, which show the distribution of inclination and apparent age, based on the data of LM88 and R93 respectively. The solid (empty) circles are pulsars with magnetic field stronger (weaker) than Gauss. It is clear that the solid circles are distributed rather randomly but the empty circles tend to concentrate at lower values of inclination angle. Table 1 shows that the mean inclination angle for stronger field pulsars () is about for LM88 and about for R93, respectively, but the mean inclination of weaker field pulsars () is about for LM88 and about for R93 respectively.
It is generally believed that the pulars at birth possess a distribution over the entire for the value of inclination angle (0), rather than all being perpendicular to the spin axis. It has been suggested (Gil & Han 1996) that the magnetic axis is randomly oriented with respect to the spin axis according to a constant probability density function (0))=2/ . It means that the number of pulsars with inclination angle values larger and smaller than the average values of should be equal, if the inclination angle does not decay. We compare our theoretical prediction (Eq. 18) with the observed data in Fig. 2a and Fig. 2b. The solid curve, the dashed curve and the dot-dashed curve are the theoretical curves corresponding to Gauss with =10ms, Gauss with =10ms and Gauss with =1ms, respectively. The theoretical curves indicate that the decay of the inclination angle of weak field pulsars is much faster than that of strong field. Some weak field pulsars occur above the long dashed curve; these can result from causes of longer initial periods, which are also expected to be randomly distributed.
3.2. Effect of magnetic field strength on the observational distribution of the inclination angle
It is important to know the initial value of the magnetic inclination in order to check our theoretical relation beteween inclination and field strength. A recent analysis of the evolution of the magnetic inclination angle based on the model of Candy & Blair(1986) and the polarization data of LM88 was done by Xu & Wu (1991) and the initial values (0) of pulsar's inclination were obtained. From this work, we choose those pulsars with as our statistical sample in order to make the evolution analysis more clear. We have also ignored one pulsar with a low field, LOG(B) =10.52, which is suspected to be recycled pulsar. The parameters of these pulsars are listed in Table 2, where the values of magnetic inclination are taken from LM88.
Table 2. The pulsars sample with large values of (0) from LM88 & XW91
Fig. 3 shows the dependence of the observed magnetic inclination angle on the magnetic field for the sample listed in Table 2. The open circles represent pulsars with period shorter than 1 second, and the solid circles represent the pulsars with the period longer than 1 second. The solid curve and the dashed curve are theoretical curves taking = with =3 and = with = 300, respectively. The observational distribution shows two important features which support our theoretical model. The first one is the inverse correlation between magnetic inclination angle and the field. The second one is that the period is a detemining factor influencing evolution of for pulsars with very low magnetic field. These two results are consistent with our theortical prediction given by Eq. (15). In this figure, it is more clearly seen that pulsars with higher magnetic field stength () have larger values of inclination angle, and those with the low values of magnetic field () have smaller values of inclination angles.
Fig. 4a shows the average of the inclination angle versus the product of the magnetic field and , which is the directly observable quantity, for the samples of LM88 when they are divided into four groups according to the field strength. Table 3 shows the number of pulsars, the average values of the magnetic field, the angle , and their corresponding standard deviations for each group. For the lowest field group, the average value of is significantly different from when the error bar is included. This fact indicates that the values of pulsars with low magnetic field (Log B = 11.0-11.5) are clearly decreased. The solid curve, the dashed curve and the dotted curve are theoretical curves with = 10, 50 and 150, respectively. Fig. 4b is similar to Fig. 4a except B is replaced by B corresponding to an average = . The solid curve, the dashed curve and the dot-dashed curve represent = 2, 5 and 10 respectively. Again, the decay of the inclination of weak field pulsars seems clear and is consistent with the theoretical prediction.
Table 3. The average values of and Log(B) from LM88
© European Southern Observatory (ESO) 1998
Online publication: March 23, 1998