TESTING MODELS OF MAGNETIC FIELD EVOLUTION OF NEUTRON STARS WITH THE STATISTICAL PROPERTIES OF THEIR SPIN EVOLUTIONS

Many studies on the possible magnetic field decay of neutron stars (NSs) have been done previously, based on the observed statistics of their periods (P) and period derivatives ($\dot{P}$). Some of these studies relied on a population synthesis of pulsars (Holt & Ramaty 1970; Bhattacharya et al. 1992; Han 1997; Regimbau & de Freitas Pacheco 2001; Tauris & Konar 2001; Gonthier et al. 2002; Guseinov et al. 2004; Aguilera et al. 2008; Popov et al. 2010); however, no firm conclusion can be drawn on if and how the magnetic fields of NSs decay (e.g., Harding & Lai 2006; Ridley & Lorimer 2010; Lorimer 2011). Alternatively some other studies used the spin-down or characteristics ages, $\tau _{\rm c}=(P-P_{0})/2\dot{P}$ (P₀ is the initial spin period of the given pulsar) or $\tau _{\rm c}=P/2\dot{P}$ if P ≫ P₀, as indicators of the true ages of NSs and found evidence for their dipole magnetic field decay (Pacini 1969; Ostriker & Gunn 1969; Gunn & Ostriker 1970). However, as we have shown recently (Zhang & Xie 2011), their spin-down ages are normally significantly larger than the ages of the supernova remnants physically associated with them, which in principle should be the unbiased age indicators of these NSs (Lyne et al. 1975; Geppert et al. 1999; Ruderman 2005). It has been shown that this age mismatch can be understood if the magnetic fields of these NSs decay significantly over their lifetimes, since the magnetic field decay alters the spin-down rate of an NS significantly, and thus causes the observed age mismatch (Lyne et al. 1975; Geppert et al. 1999; Ruderman 2005; Zhang & Xie 2011).

Pulsars are generally very stable natural clocks with observed steady pulses. However significant timing irregularities, i.e., unpredicted times of arrival of pulses, exist for most pulsars studied so far (see Hobbs et al. 2010 for an extensive review of many previous studies on timing irregularities of pulsars). The timing irregularities of the first type are "glitches," i.e., sudden increases in spin rate followed by a period of relaxation; it has been found that the timing irregularities of young pulsars with τ_c < 10⁵ yr are dominated by the recovery from previous glitch events (Hobbs et al. 2010). In many cases, the NS recovers to the spin rate prior to the glitch event and thus the glitch event can be removed from the data satisfactorily without causing significant residuals over model predictions. However, in some cases, glitches can cause permanent changes to both P and $\dot{P}$ of the NS, which cannot always be removed from the data satisfactorily, for instance, if the glitch has occurred before the start of the data taking and/or long-term recoveries from glitch events cannot be modeled satisfactorily. These changes can be modeled as a permanent increase of the surface dipole magnetic field of the NS; consequently, some of these NSs may grow their surface magnetic field strength gradually this way, and eventually have surface magnetic field strength comparable to that of magnetars over a lifetime of 10⁵–10⁶ yr (Lin & Zhang 2004). These are the first studies that linked the timing irregularities of pulsars with the long-term evolution of magnetic fields of NSs.

The timing irregularities of the second type have timescales of years to decades and thus are normally referred to as timing noise (Hobbs et al. 2004, 2010). Hobbs et al. (2004, 2010) carried out the most extensive study so far of the long-term timing irregularities of 366 pulsars. Besides ruling out some timing noise models in terms of observational imperfections, random walks, and planetary companions, Hobbs et al. (2004, 2010) concluded that the observed $\ddot{\nu }$ values for the majority of pulsars are not caused by magnetic dipole radiation or by any other systematic loss of rotational energy, but are dominated by the amount of timing noise present in the residuals and the data span. Some of their main results that are modeled and interpreted in this work are: (1) all young pulsars have $\ddot{\nu } > 0$, (2) approximately half of the older pulsars have $\ddot{\nu } > 0$ and the other half have $\ddot{\nu } < 0$; (3) quasi-oscillatory timing residuals are observed for many pulsars; and (4) the value of $\ddot{\nu }$ measured depends upon the data span and the exact choice of data processed.

Proper understanding of these important results can lead to better understanding of the internal structures and the magnetospheres of NSs. Since the spin evolution of pulsars cannot be caused by any systematic loss of rotational energy (Hobbs et al. 2004, 2010), it is natural to introduce some oscillatory parameters. Indeed, some of the observed periodicities have been suggested as being caused by free precession of the NS (Stairs et al. 2000). Some strong quasi-periodicities have been identified recently as due to abrupt changes of the magnetospheric regulation of NSs, perhaps due to varied particle emissions (Lyne et al. 2010). It was suggested that such varied magnetospheric particle emissions are also responsible for the observed long-term timing noise (Liu et al. 2011). Alternatively, it has been suggested that the propagation of Tkachenko waves in an NS may modulate its moment of inertia, and thus produce the observed periodic or quasi-periodic timing noise (Tkachenko 1966; Ruderman 1970; Haskell 2011). Because all of the above processes can only cause periodic or quasi-period modulations to the observed spin of NSs, all these modulations can be modeled phenomenologically as some sort of oscillations of the observed surface magnetic field strengths of NSs. It is thus reasonable to attribute the observed wide-spread quasi-oscillatory timing residuals to the oscillations of the magnetic fields of NSs. As shown in this work, the oscillating magnetic fields can also naturally explain the fact that approximately half of the older pulsars have $\ddot{\nu } > 0$ and the other half have $\ddot{\nu } < 0$ when the long-term magnetic field decay does not dominate the spin evolutions of pulsars. This is the main advantage of this model, and thus justifies invoking oscillating magnetic fields for modeling the long-term spin evolutions of pulsars.

In this work, we attempt to use the known statistical properties of the spin evolution of the pulsars to test some phenomenological models for the evolution of the dipole magnetic field of a NS. In Sections 2–5, we test the following three models: (1) the standard magnetic dipole radiation model with constant magnetic field, (2) long-term exponential decay with an oscillatory component, and (3) long-term power-law decay modulated with oscillations. We find the last model is the only one compatible with the data, with a power-law index α > 0.5 and an oscillation amplitude of around 5 × 10⁻⁵ for an oscillation period of several years. Finally, we discuss the physical implication of our results, summarize our results, and make conclusions and discussions, including predictions of the model for further studies. All timing data of pulsars used in this work are taken from Hobbs et al. (2010).

Assuming pure magnetic dipole radiation as the braking mechanism for a pulsar's spin-down, we have

$$\begin{equation} I\Omega \dot{\Omega }=-\frac{(BR^3)^2}{6c^3}{\Omega ^4}, \end{equation} \tag{ 1 }$$

or

$$\begin{equation} \dot{\nu }= -AB^2\nu ^3, \end{equation} \tag{ 2 }$$

where A = ((2πR³)²/6c³I) is normally assumed to be a constant, B is its dipole magnetic field at its magnetic pole, R is its radius, and I is its moment of inertia. Assuming $\dot{B}=0$, we have

$$\begin{equation} \ddot{\nu }= 3\dot{\nu }^2/\nu. \end{equation} \tag{ 3 }$$

This model thus predicts $\ddot{\nu }>0$, against the fact that many pulsars show $\ddot{\nu }<0$. This is a major failure of this model, or any other models with monotonic spin-down, as pointed out in Hobbs et al. (2010). In Figure 1, we plot the comparison between the observed $\ddot{\nu }$ and that predicted by Equation (3). It is clear that this model under-predicts the amplitudes of $\ddot{\nu }$ by several orders of magnitude for most pulsars.

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Assuming $\dot{B}\ne 0$, we have

$$\begin{equation} \ddot{\nu }= 3\dot{\nu }^2/\nu +2\dot{\nu }\dot{B}/B. \end{equation} \tag{ 4 }$$

From the above analysis, it is obvious that the second term in Equation (4) must dominate the observed $\ddot{\nu }$ (i.e., $3\dot{\nu }^2/\nu \ll |2\dot{\nu }\dot{B}/B|$), for both its magnitude and sign. To define K = AB², we have

$$\begin{equation} \dot{K}=\frac{3\dot{\nu }^2-\ddot{\nu }\nu }{\nu ^4}. \end{equation} \tag{ 5 }$$

To define the evolution timescale of K, $\tau _{K}=K/|{\dot{K}}|=B/2|{\dot{B}}|$, we have

$$\begin{equation} \tau _{K}=\left|\frac{\dot{\nu }\nu }{3\dot{\nu }^2-\ddot{\nu }\nu }\right|. \end{equation} \tag{ 6 }$$

Figure 2 shows the observed τ_K as a function of τ_c. Clearly, τ_K ≪ τ_c for most pulsars. This suggests that if the measured $\ddot{\nu }$ values are caused by magnetic field evolution, then the evolution must be significant for most pulsars in the sample. Figure 2 also shows that for all young pulsars with τ_c < 10⁵ yr, $\dot{K}<0$, i.e., their magnetic moment field decays with time. For the rest of the pulsars, there are almost equal numbers of pulsars with positive or negative $\dot{K}$. If one assumes that most pulsars are born with some common properties and they then evolve to have the observed statistical properties, it is then necessary that at least some pulsars should have both $\dot{K}<0$ and $\dot{K}>0$ at different times, i.e., K oscillates.

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We therefore model phenomenologically the evolution of the observed dipole magnetic field of a pulsar by a decay modulated by the dominated component of oscillations,

$$\begin{equation} B(t)= B_{\rm d}(t)\left(1+k\sin \left(\phi +2\pi \frac{t}{T}\right)\right), \end{equation} \tag{ 7 }$$

where t is the pulsar's age, and k, ϕ, and T are the amplitude, phase, and period of the oscillating magnetic field, respectively. Sometimes it might be necessary to take multiple oscillation components, since multiple peaks are often seen in the power spectra of the timing residuals of many pulsars (Hobbs et al. 2010). Assuming that the oscillation timescale is much less than the decay timescale (as will be discussed and justified later), i.e., $T\ll B_{\rm d}/\dot{B}_{\rm d}$, we have

$$\begin{equation} \dot{B}\simeq \dot{B}_{\rm d}+B_{\rm d} f\cos \left(\phi +2\pi \frac{t}{T}\right), \end{equation} \tag{ 8 }$$

where f = 2πk/T. For simplicity, here we only consider the amplitude of the oscillation term to model the statistical properties of the timing noise of pulsars, i.e.,

$$\begin{equation} \dot{B}= \dot{B}_{\rm d}\pm fB_{\rm d}. \end{equation} \tag{ 9 }$$

Since $\dot{B}<0$ can produce $\ddot{\nu }>0$, a viable model should at least have $|\dot{B}_{\rm d}|/B_{\rm d}>f$ for young pulsars and vice versa for old pulsars, in order to produce positive $\ddot{\nu }$ for young pulsars, but either positive or negative $\ddot{\nu }$ for old pulsars with almost equal chances, as discussed above.

Assuming B_d = B₀exp (− t/τ_d), Equation (9) becomes

$$\begin{equation} \dot{B}=B_{\rm d}(-1/\tau _{\rm d}\pm f). \end{equation} \tag{ 10 }$$

Substituting Equation (10) into Equation (4) and ignoring its first term (since $3\dot{\nu }^2/\nu \ll |2\dot{\nu }\dot{B}/B|$), we have

$$\begin{equation} \ddot{\nu }\simeq -2\dot{\nu }(1/\tau _{\rm d}\pm f). \end{equation} \tag{ 11 }$$

Since $\ddot{\nu }>0$ for most young pulsars, we assume 1/τ_d > f, i.e., $\ddot{\nu }\simeq -2\dot{\nu }/\tau _{\rm d}$, or $\tau _{\rm d}\simeq -2\dot{\nu }/\ddot{\nu }$ for young pulsars. In Figure 3, we show the observed $\ddot{\nu }$ and the inferred τ_d as a function of the observed $\dot{\nu }$, for young pulsars with τ_c ⩽ 10⁶ yr and $\ddot{\nu }>0$. Clearly, a single value or distribution of τ_d cannot describe the data adequately. Another difficulty of this model is that $\ddot{\nu }<0$ for nearly half of the pulsars cannot be explained with the same assumption of 1/τ_d > f. We thus conclude that a simple exponential decay model is not favored by the data.

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Assuming B_d = B₀(t/t₀)^−α, Equation (9) becomes

$$\begin{equation} \dot{B}=B_{\rm d}(-\alpha /t\pm f). \end{equation} \tag{ 12 }$$

Substituting Equation (12) into Equation (4) and ignoring its first term (since $3\dot{\nu }^2/\nu \ll |2\dot{\nu }\dot{B}/B|$), we have

$$\begin{equation} \ddot{\nu }\simeq -2\dot{\nu }(\alpha /t\pm f), \end{equation} \tag{ 13 }$$

where t is the real age of the pulsar. Without other information about t, here we replace it with the magnetic age of a pulsar t = t₀(B₀/B)^(1/α); we then have

$$\begin{equation} \ddot{\nu }\simeq \eta (-\dot{\nu })^{1+\beta }/\nu ^{3\beta }\pm 2\dot{\nu }f, \end{equation} \tag{ 14 }$$

where β = 1/2α, η = (3.3 × 10¹⁹/B₀)^2β/βt₀, and $B=3.3\times 10^{19}\sqrt{P\dot{P}}$ G is assumed.

Since $\ddot{\nu }>0$ for most young pulsars, we assume α/t > f, i.e.,

$$\begin{equation} \ddot{\nu }\simeq \eta (-\dot{\nu })^{1+\beta }/\nu ^{3\beta }, \end{equation} \tag{ 15 }$$

for young pulsars. For the ith pulsar, we have

$$\begin{equation} \eta _i=\ddot{\nu }_i/(-\dot{\nu })^{1+\beta }\nu ^{-3\beta }, \end{equation} \tag{ 16 }$$

We then calculate fit the root mean squares (rms) for each value of β = 1/2α,

$$\begin{equation} {\rm rms}=\frac{1}{n\eta _{\rm m}}\sqrt{\sum (\eta _i-\eta _{\rm m})^2}, \end{equation} \tag{ 17 }$$

where n is the number of pulsars and η_m is the median value of all η_i. In Figure 4(a), we show rms as a function of α, indicating that α ⩾ 0.5 is required, but its upper limit is unconstrained. In Figures 4(a) and (b), we also show the ranges of η_m and the corresponding t₀ and B₀.

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The uncertainty in determining the exact value of α is due to the unknown real ages of these pulsars; using the magnetic ages as their ages introduces a strong coupling between the parameters in the magnetic field decay model and their ages. This is manifested in the correlations of α with rms, η_m, t₀, and B₀. Nevertheless, their ranges are reasonable, with α ⩾ 0.5. In Figures 5(a) and (b), we show the comparison between the predictions of Equation (15) and data with α = 0.5 and α = 1.0, respectively. In both cases, the model can describe the data reasonably well. We thus conclude that a simple power-law decay model is favored by data.

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Equation (14) shows that the oscillation term f dominates both the signs and magnitudes of $\ddot{\nu }$ for old pulsars with t > α/f; this explains naturally why there are almost equal numbers of pulsars with $\ddot{\nu }>0$ or $\ddot{\nu }<0$. This agrees with the observation that a single old pulsar can be observed to show either $\ddot{\nu }>0$ or $\ddot{\nu }<0$, depending on which segment of data is used (Hobbs et al. 2010). We also confirmed this with Monte Carlo simulations in a preliminary study (Figures 5 and 6 in Zhang & Xie 2012); a more detailed study will be presented elsewhere (Y. Xie & S.-N. Zhang 2012, in preparation). Figure 4(a) shows that rms remains almost constant with α > 1; we thus restrict our analysis in the following for only two cases: α = 0.5 and α = 1.0.

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Equation (14) can be rewritten as

$$\begin{equation} \ddot{\nu }\nu ^{3\beta }\simeq \eta (-\dot{\nu })^{1+\beta }\pm 2\dot{\nu }\nu ^{3\beta }f, \end{equation} \tag{ 18 }$$

suggesting that f ≠ 0 causes deviations to the expected correlation between $\ddot{\nu }\nu ^{3\beta }$ and $\dot{\nu }$, as shown in Figure 6. Choosing the same range of f, the data for both positive and negative $\ddot{\nu }$ can be explained; η is taken as η_m determined from young pulsars with $\ddot{\nu }>0$ and τ_c ⩽ 2 × 10⁶ yr, for α = 0.5 and α = 1.0, respectively. For each pulsar in the sample, f can also be determined from Equation (14) or Equation (18), as shown in Figure 7; clearly, f is not correlated with either $\dot{\nu }$ or ν (we also find that f is not correlated with either $\ddot{\nu }$ or τ_c, which is not shown here for brevity). This indicates that the statistical properties of the timing noise of all kinds of pulsars can be explained with the same oscillation model of similar oscillation amplitudes.

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In Figure 8, the distributions of f/α, for all pulsars with positive or negative $\ddot{\nu }$, are shown in the cases of α = 0.5 and α = 1.0, respectively. All distributions show well-defined peaks that are consistent with each other and the peak value of f/α is around 3 × 10⁻¹²; we also find that choosing different values of α with 0.5 < α < 2.0 always gives similar f/α. Finally, in Figure 9, we show the comparisons between the observed $\ddot{\nu }$ and that predicted by Equation (14), with f/α = 3 × 10⁻¹² for α = 0.5 and α = 1.0, respectively. This simple model can describe the general properties of the data over about nine orders of magnitudes of $\ddot{\nu }$.

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6.1. The Braking Index

Equation (6) can be rewritten as

$$\begin{equation} \dot{K}=\frac{\ddot{\nu }}{\nu ^3}\left(\frac{3}{n_{\rm b}}-1\right), \end{equation} \tag{ 19 }$$

where $n_{\rm b}=\ddot{\nu }\nu /\dot{\nu }^2$ is the conventionally defined braking index. Equation (19) means that $\dot{K}>0$ when n_b < 3, including n_b < 0 when $\ddot{\nu }<0$. This agrees with many previous works reporting that n_b < 3 indicates magnetic field increase and vice versa (e.g., Blandford & Romani 1988; Chanmugam & Sang 1989; Manchester & Peterson 1989; Lyne et al. 1993, 1996; Johnston & Galloway 1999; Wu et al. 2003; Lyne 2004; Chen & Li 2006; Livingstone et al. 2006; Middleditch et al. 2006; Livingstone et al. 2007; Yue et al. 2007; Espinoza et al. 2011). However, these works have reported very few cases where n_b is measured reliably.

In Figure 10, we show the distribution of n_b in this sample. There is no clustering around n_b = 3 at all, and the values of positive n_b and the absolute values of negative n_b exhibit almost identical distributions. The numbers of positive or negative values of n_b are 193 and 173, respectively. Similarly, the numbers of n_b > 3 or n_b < 3 (including all negative n_b) are 192 and 174, respectively. Statistically, these numbers are indistinguishable. Therefore, about half of pulsars in this sample are observed with increasing or decreasing magnetic fields.

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6.2. Magnetic Field Decay

Several physical mechanisms have been proposed for magnetic field decay in NSs, e.g., ohmic dissipation, Hall effect, and ambipolar diffusion. We can therefore have (Goldreich & Reisenegger 1992; Heyl & Kulkarni 1998)

$$\begin{equation} \frac{dB}{dt} = -B\left(\frac{1}{t_{\rm ohmic}}+\frac{1}{t_{\rm ambip}}+\frac{1}{t_{\rm hall}}\right), \end{equation} \tag{ 20 }$$

where t_ohmic ∼ 2 × 10¹¹(L²₅/T²₈)(ρ/ρ_nuc)³ yr, t^s_ambip ∼ 3 × 10⁹(L₅²T²₈/B₁₂²) yr (for the solenoidal component), t^ir_ambip ∼ (5 × 10¹⁵/T₈⁶B²₁₂)(1 + 5 × 10⁻⁷L₅²T⁸₈) yr (for the irrotational component), and t_hall ∼ 5 × 10⁸(L²₅T²₈/B₁₂)(ρ/ρ_nuc) yr, where B₁₂ is the surface magnetic field strength in units of 10¹² G, T₈ denotes the core temperature in units of 10⁸ K, ρ_nuc ≡ 2.8 × 10¹⁴ g cm⁻³ is the nuclear density, and L₅ is a characteristic length scale of the flux loops through the outer core in units of 10⁵ cm.

Assuming T₈ does not vary or only varies slowly with time (Pons et al. 2009), the ohmic dissipation will produce an exponential decay of the magnetic field, $B= B_{0}e^{-t/\tau _{\rm d}}$. The other two mechanisms all produce power-law decay, B = B₀(t/t₀)^−α, where α = 1 or α = 0.5 for the Hall effect or ambipolar diffusion (for t ≪ 10¹² yr), respectively. We recently found that this model with constant core temperatures of NSs can explain well the observed age mismatch of NSs discussed above, and the ambipolar diffusion effect seems to be dominant, though the statistics are rather limited (Zhang & Xie 2011; Xie & Zhang 2011)

As shown in Figure 3 and discussed in Section 3, the simple exponential decay cannot describe the data for young pulsars with τ_c ⩽ 10⁶ yr. Therefore, a constant core temperature (T₈ = const) ohmic dissipation model for all magnetic field decay is ruled out. However, for a cooling NS, i.e., when T₈ decreases with time, the ohmic dissipation model would result in an increasing τ_d with time, in rough agreement with that shown in Figure 3(b). Indeed, an increasing τ_d with time would also make the first term in Equation (11) less important or negligible for old pulsars, thus explaining why there are almost equal numbers of pulsars with either positive or negative $\ddot{\nu }$. Of course, an increasing τ_d with time may be able to mimic a power-law decay mathematically.

As shown in Figures 4–9 and discussed in Sections 4 and 5, a power-law decay with α ⩾ 0.5 can explain all the observed statistical properties of the timing noise of pulsars. Unfortunately, the exact value or even the upper limit of α cannot be constrained by the data, though α ≈ 1 is preferred. For both the Hall effect and the ambipolar diffusion (for t ≪ 10¹² yr), a cooling NS implies faster magnetic field decay, and thus increasing α with time.

We therefore conclude that the current data do not conflict with the above discussed model of magnetic field decay of NSs, but also cannot constrain precisely how these effects regulate their magnetic field decay.

6.3. Magnetic Field Oscillations

In this work, we have tested models of magnetic field evolution of NSs with the statistical properties of their timing noise reported by Hobbs et al. (2010). In all models, the magnetic dipole radiation is assumed to dominate the spin-down of pulsars; therefore, different models of their magnetic field evolution lead to different properties of their spin-down. Our main results in this work are summarized as follows.

• 1.  
  We tested the standard model of pulsar's magnetosphere with constant magnetic field. We found that this model underpredicts the second derivatives of their spin frequencies ($\ddot{\nu }$) by several orders of magnitudes for most pulsars (Figure 1). This model also fails to predict $\ddot{\nu }<0$, which is observed in nearly half of the old pulsars. The standard model of a pulsar's magnetosphere with constant magnetic field is thus ruled out, and $\ddot{\nu }$ is found to be dominated by the varying magnetic field.
• 2.  
  Statistically, about half of the pulsars in this sample show increasing or decreasing magnetic fields. The magnetic field strength of most pulsars are found to either increase or decrease with timescales significantly shorter than their characteristic ages (Figure 2).
• 3.  
  We constructed a phenomenological model of the evolution of the magnetic fields of NSs, which is made of a long-term decay modulated by short-term oscillations (STOs).
• 4.  
  A simple exponential decay is not favored by the apparent correlation of the inferred decay constant with their characteristic ages for young pulsars (Figure 3(b)) and fails to explain the fact that $\ddot{\nu }$ is negative for roughly half of old pulsars.
• 5.  
  A simple power-law decay, however, can explain all the observed statistical properties of $\ddot{\nu }$ (Figures 5 and 6) and infer a narrow distribution of an oscillation amplitude of about 5 × 10⁻⁵ with a preferred decay index of about unity (Figures 7 and 8).

We thus conclude that long-term magnetic fields and STOs are ubiquitous in all NSs and that the data support the model consisting of a power-law decay modulated by STOs for the evolution of their magnetic fields. However, we can only constrain the decay index to be α ⩾ 0.5 (Figures 4 and 8(b)).

Our results are consistent with the models of magnetic field decay we discussed, in which ohmic dissipation, Hall effect, and ambipolar diffusion are possible effects causing magnetic field decay. However, the uncertainty in the decay index α makes it difficult to constrain further or distinguish between these effects. The problem in determining α is the unavailability of the true ages of pulsars. With the true age t available for some pulsars, α can be determined directly with Equation (13); f = 0 can be safely assumed for young pulsars. Unfortunately, only several pulsars in the sample of Hobbs et al. (2010) have the ages of their supernova remnants available, thus preventing a statistically meaningful determination of α this way. We thus suggest that reliable determination of the true ages of a large sample of young NSs is needed in order to constrain further the physical mechanisms of the magnetic field decay of NSs. This can also provide an independent test of our model.

The main prediction of this model is that the spin evolution of each pulsar should follow

$$\begin{equation} \ddot{\nu }(t)\simeq 2\dot{\nu }(t)\left(-\frac{\alpha }{t}+\sum f_i\cos \left(\phi _i+2\pi \frac{t}{T_i}\right)\right), \end{equation} \tag{ 21 }$$

where both $\ddot{\nu }(t)$ and $\dot{\nu }(t)$ should be time-resolved, rather than the averaged value conventionally reported over the whole data span. Hobbs et al. (2010) have shown that both the value and sign of the averaged $\ddot{\nu }$ depend on the time span used, which can be reproduced with our model presented here (Zhang & Xie 2012). However, a clear-cut and robust test of our model is to compare the prediction of the above equation. As discussed above, the first or the second term in the above equation may be ignored for old millisecond pulsars or young pulsars, respectively. The current ongoing programs of intensive long-term precise timing monitoring of pulsars should produce the required data for validating our model. In doing so, both the decay index and the oscillating properties of each individual pulsar can be determined, which in turn can provide important insights into the physics of NSs.

We appreciate discussions with and comments from E. P. J. van den Heuvel, S. Tsuruta, J. L. Han, W. W. Tian, X. P. Zheng, R. X. Xu, F. J. Lu, H. Tong, D. Chen, X. F. Wu, C. M. Zhang, and J. F. Liu. The anonymous referee is thanked for comments and suggestions. S.N.Z. acknowledges partial funding support by the National Natural Science Foundation of China under grant Nos. 11133002, 10821061, 10725313, and by 973 Program of China under grant No. 2009CB824800.