RADIO EFFICIENCY OF PULSARS

Andrzej Szary; Bing Zhang; George I. Melikidze; Janusz Gil; Ren-Xin Xu

doi:10.1088/0004-637X/784/1/59

1. INTRODUCTION

Despite the fact that pulsars were discovered almost half a century ago, the emission mechanism of the pulsed radio emission remains unresolved. The challenge lies in the difficulty to identify the correct "coherent" mechanism (Melrose 2006) that can power radio emission with similar emission characteristics over about 4 orders of magnitude in rotation period, P, and up to 11 orders of magnitude in the rate of increase of pulsar periods, $\dot{P} = {dP} / {dt}$ . In this paper, we perform a systematic analysis of radio emission efficiency for the current sample of pulsars⁵ in an effort to study global radio emission properties and to constrain the radio emission mechanism. In Section 2, we present the method to calculate radio luminosity L and radio emission efficiency, ξ, of pulsars. The dependences of both L and ξ on pulsar parameters are presented in Section 3. The results are summarized in Section 4 along with a discussion on the physical implications of the found correlations.

2. RADIO LUMINOSITY AND EFFICIENCY

2.1. Integrated Radio Luminosities

For a number of reasons, a correct estimate of a pulsar's radio luminosity is difficult to determine (see, e.g., Lorimer & Kramer 2012). A common practice is to assume that the intensity distribution along the observer's line-of-sight, cut through the emission region, is representative of the entire emission beam. Then, the luminosity of a pulsar can be calculated as follows:

$\begin{equation} L = \frac{4 \pi d^2}{\delta } \sin ^2 \left(\frac{\rho }{2} \right) \int _{\nu _{\rm min}}^{\nu _{\rm max}} S_{\rm mean}(\nu) {d}\nu, \end{equation} \tag{ 1 }$

where δ is the pulse duty cycle, d is a distance to the pulsar, ρ is the angular radius of the emitting cone, ν_min and ν_max bracket the radio frequency range in which the pulsar is detected, and S_mean(ν) is the mean flux density measured at a given frequency, ν. The pulse duty cycle can be calculated using the so-called equivalent width, W_eq, (i.e., the width of a top-hat pulse having the same area and peak flux as the true profile) as δ = W_eq/P. To derive ρ, we have to know the geometry of the pulsar emission beam: α, the inclination angle between the rotation and magnetic axes; β, the impact parameter; and W, the observed pulse width. These values are usually not known for most pulsars. In order to keep the sample of pulsars as large as possible, in this paper we calculate the luminosity, L, assuming some typical values for all pulsars: δ ≈ 0.04, ρ ≈ 6°, ν_min ≈ 10⁷ Hz, and ν_max ≈ 10¹¹ Hz (see Lorimer & Kramer 2012, for more details), so that

$\begin{equation} L \simeq 7.4 \times 10^{27} \left(\frac{d}{\rm kpc} \right)^2 \left(\frac{S_{1400}}{\rm mJy} \right) \,{\rm erg \, s^{-1}}. \end{equation} \tag{ 2 }$

Note that here, we neglect any dependence of δ⁻¹sin ²(ρ/2) on the spin parameters, P and $\dot{P}$ . In this way, we avoid any bias in the calculation of L and keep our analysis model-independent (see Section 4 for discussion). Additionally, such an approach allows us to compare the obtained results with X-ray and γ-ray observations.

2.2. Monochromatic Luminosities

Sometimes it is convenient to use monochromatic luminosities (often also called "pseudoluminosities"),

$\begin{equation} L_{\rm \nu } \equiv S_{\rm \nu } d^2, \end{equation} \tag{ 3 }$

to estimate how luminosity varies in frequency. Here, S_ν is the mean flux density measured at different frequencies (e.g., 400 MHz, 1400 MHz, and 2000 MHz). Note that L_ν has similar dependences on S_ν and d as L.

2.3. Radio Emission Efficiency

The radio emission efficiency of pulsars is defined as the fraction of rotational energy transformed into radio emission, i.e.,

$\begin{equation} \xi \equiv \frac{L}{\dot{E}}, \end{equation} \tag{ 4 }$

where

$\begin{eqnarray} &&\dot{E} = 4 \pi ^2 I \dot{P} P^{-3} \simeq 3.95\times 10^{31} I_{45}\left(\frac{\dot{P}}{10^{-15}} \right) \left(\frac{P}{s} \right)^{-3} \,{\rm erg \, s^{-1}} \nonumber\\ \end{eqnarray} \tag{ 5 }$

is the rate of loss of rotational energy (also called spin-down luminosity), and I = 10⁴⁵I₄₅ g cm² is the moment of inertia.

Note that L and $\dot{E}$ are two independent parameters that come from very different measurements. While L is derived from measuring both radio emission flux and the distance of a pulsar, $\dot{E}$ is derived from pulsar timing measurements. The L parameter depends on the unknown pulsar radio emission physics. There is no a priori reason to expect that ξ should depend on $\dot{E}$ (or other parameters defined by timing parameters) in any way.

3. CORRELATIONS

In the following, we investigate how L and ξ correlate with other pulsar parameters.

3.1. Spin-down

Figure 1(a) shows radio luminosity, L, as a function of the spin-down power, $\dot{E}$ . One immediately sees that there is essentially no dependence between the two parameters, with best fit $L\propto \dot{E}^{0.10}$ and $L \propto \dot{E}^{0.06}$ for normal and binary/millisecond pulsars, respectively. A similar conclusion is achieved when one replaces L with the pseuodoluminosities L_ν. The linear fits in different frequencies show that the exponents for $L_\nu \hbox{--}\dot{E}$ relations are all close to 0, i.e., 0.18, 0.1, and −0.08 for 400, 1400, and 2000 MHz, respectively, with a rough trend by which the exponent decreases with increasing frequency. Note, however, that the pulsar samples at 400 MHz (641 objects) and especially at 2000 MHz (27 objects) are considerably smaller than the pulsar sample at 1400 MHz (1436 objects).

The weak dependence of L on $\dot{E}$ suggests a near-linear inverse correlation between ξ and $\dot{E}$ . Figure 1(b) shows such an anti-correlation, with the best linear fits $\xi \propto \dot{E}^{-0.90}$ and $\xi \propto \dot{E}^{-0.94}$ for normal pulsars and binary/millisecond pulsars, respectively. Even though the spread in values of both radio luminosity and efficiency is high for a given spin-down luminosity, the $\xi - \dot{E}$ dependence is clearly visible, with low efficiency (e.g., ξ = 10⁻⁸–10⁻⁵) at the high spin-down rate (e.g., $\dot{E}=10^{36} \, {\rm erg \, s^{-1}}$ ) and high efficiency (e.g., ξ ≳ 10⁻³) at the low spin-down rate (e.g., $\dot{E}=10^{30} \, {\rm erg \, s^{-1}}$ ).

Such a near-linear inverse correlation between ξ and $\dot{E}$ is nontrivial. For comparison, in panels (c) and (d) in Figure 1, we also show how X-ray and γ-ray efficiencies depend on $\dot{E}$ . It is clearly seen that the X-ray efficiency is essentially independent on $\dot{E}$ , i.e., $\xi _{\rm x} \propto \dot{E}^{-0.08}$ (see Figure 1(c)), and the γ-ray efficiency only weakly depends on $\dot{E}$ , i.e., $\xi _{\gamma } \propto \dot{E}^{-0.5}$ for normal pulsars and as $\xi _{\gamma } \propto \dot{E}^{-0.24}$ for millisecond ones (see Figure 1(d)). When calculating both the X-ray and γ-ray efficiencies, we assumed a same solid angle for all pulsars, which is the same assumption made in calculating radio emission efficiencies.

3.2. Age

From the very beginning of pulsar astronomy it was suggested that the radio luminosity of pulsars must decline with age. Such an evolution could explain the rapid drop in pulsar distribution around the period of 1 s (Gunn & Ostriker 1970). As suggested by Taylor & Manchester (1977), many more pulsars would be observed if the luminosity was constant.

Since $\dot{E} \propto \dot{P} P^{-3}$ and $\tau \propto \dot{P}^{-1} P$ , the above negative linear $\xi \hbox{--}\dot{E}$ correlation would be translated as a positive linear ξ–τ correlation. Figure 2(a) shows that such a dependence is indeed there, with a best linear fit (for normal pulsars only) of $\xi \propto \dot{\tau }^{1.06}$ . Notice that millisecond pulsars are excluded in the analysis. Since they have experienced the recycling spin-up process, their τ is not the characteristic age since birth.

**Figure 2.** Dependence of the radio efficiency, ξ (panel (a)), X-ray efficiency, ξ_x (panel (b)), and γ-ray efficiency, ξ_γ (panel (c)) on pulsar age, τ. The dashed lines correspond to the linear fit for normal pulsars; see Figure 1 for a more detailed description.
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This interesting relation suggests a surprising result that as a pulsar ages, it somehow transforms its spin-down luminosity more efficiently into radiation. Such a relationship is nontrivial, and provides valuable information about the mechanism of radio emission.

Again, for comparison, we plot ξ_x and ξ_γ against τ in Figures 2(b) and (c), respectively. We find that unlike radio emission, there is essentially no obvious correlation in X-rays (with ξ_x∝τ^0.11), and there is only a weak and rather scattered correlation in γ-rays (with ξ_γ∝τ^0.63). This again suggests that the radio emission mechanism is different from those of high-energy radiation.

3.3. Selection Effect?

The lack of pulsars with high-efficiency, high- $\dot{E}$ , and young-age pulsars suggests that there is no significant selection effect for the two reported correlations above. One may still suspect that the two correlations in the low- $\dot{E}$ and old-age regime may be affected by an observational selection effect, which is against the detection of low-ξ pulsars due to the sensitivity limit of radio telescopes.

Figure 3 presents the dependence of the observed mean flux density, S₁₄₀₀, measured at 1400 MHz on the spin-down luminosity, $\dot{E}$ . As can be seen from the figure, the flux distribution of pulsars with relatively low $\dot{E}$ is not significantly different from the flux distribution of pulsars with relatively high $\dot{E}$ . There is no significant depletion of high-flux, low- $\dot{E}$ pulsars, nor is there an increase of low-flux, low- $\dot{E}$ pulsars. We argue that the relatively large sample of pulsars with $\dot{E} < 10^{31} \, {{\rm erg\, s^{-1}}}$ (143 objects), and the fact that they all have ξ > 10⁻⁴, implies that the lack of detection of pulsars with low efficiency is not due to the insufficient sensitivity of detectors. The two correlations reported above are therefore intrinsic.

3.4. $P\hbox{--}\dot{P}$

There have been efforts to determine how L depends on P and $\dot{P}$ (e.g., Ostriker & Gunn 1969; Lyne et al. 1975; Malov & Malov 2006). The results are affected by the selection effect, and have been inconclusive (see review paper by Bagchi 2013). We will show in this section that these dependences, if any, are rather weak.

We first show the distributions of radio efficiency in different locations of the $P\hbox{--}\dot{P}$ diagram and P–τ diagram as observed (see Figure 4). As can be clearly seen, the observed radio luminosity does not depend in any significant way on P or $\dot{P}$ .

It would be interesting to investigate the intrinsic underlying P and $\dot{P}$ dependences of L, which may be revealed through Monte Carlo (MC) simulations. (Faucher-Giguère & Kaspi 2006, hereafter FK06) argued that in the absence of torque decay (e.g., due to magnetic field decay) the radio luminosity of pulsars must be correlated with pulsar age, and hence with P and $\dot{P}$ . They found that the luminosity law (i.e., dependence of the radio luminosity on P and $\dot{P}$ ) should be close to $L\propto P^{-1.5} \dot{P}^{0.5}$ . In order to independently investigate an intrinsic luminosity distribution and the eventual dependence of L on P and $\dot{P}$ , we have performed some MC simulations based on the open-source package PsrPopPy (Bates et al. 2013). This package includes two methods to obtain a synthetic sample of pulsars, the snapshot and evolutionary methods. The snapshot method consists of the following steps: generating pulsar periods, modeling pulse widths, generating radio luminosities, distributing the pulsars in the Galaxy, modeling electron density, and generating spectral indices. The evolutionary method consists of the steps from the snapshot method but is also extended by additional ones that include generating pulsar period derivatives, generating magnetic fields, generating rotational alignment and modeling its time evolution, modeling pulsar spin-down, and finally, evolving pulsars through the Galactic potential (see Bates et al. 2013, for more details). Therefore, we used the evolutionary part of this code to reproduce the results of FK06. After the calculation of a synthetic population, each pulsar is run through the parameters of the Parkes Multibeam Pulsar Survey (PMPS; Manchester et al. 2001) in order to constrain the population based upon known detections. PMPS is the most successful survey to date with 1062 normal pulsars detected, which allows one to test a synthetic population using more than half of the total observed population of normal pulsars.

Table 1 summarizes the parameters used by FK06 to produce their Figures 7 and 14 calculated, as well as the best fit parameters we have obtained from our simulations (see more details below).

Table 1. Population Synthesis Parameters Used in MC Simulations Based on the Evolutionary Method

Parameter	Parameter value
Radial distribution model Yusifov & Kucuk (2004)
R₁	55 pc
a	1.64
b	4.01
Birth height distribution	Exponential
Initial Galactic z-scale height	50 pc
Birth velocity distribution	Exponential
<v_3D >	380 km s⁻¹
Pulsar spin-down model Faucher-Giguère & Kaspi (2006)
Beam alignment model	Orthogonal
Breaking index	3.0
Birth spin period distribution	Normal
<P₀ >	300 ms ^a, 200 ms ^b
$\sigma _{P_0}$	150 ms ^{a, b}
Scattering model Bhat et al. (2004)
Spectral index of the scattering model	−3.86
Maximum age of pulsars	1 Gyr
Spectral index distribution model	Normal
<α >	−1.4
σ_α	0.96
Magnetic field distribution model	Lognormal
〈log₁₀(B_d[G])〉	12.65
$\sigma _{\log _{10} B_{\rm d} }$	0.55
Random luminosity distribution model	Lognormal
<log₁₀(L₁₄₀₀(mJy kpc²)) >	−1.1 ^a, 0.5 ^b
$\sigma _{\log _{10} L_{1400}}$	0.9 ^a, 1.0 ^b
Number of pulsars detected in PMPS	1100

Notes. ^aPopulation model parameters used to reproduce results of FK06. ^bOptimal population model parameters.

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Figure 5(a) shows the reproduced $P\hbox{--}\dot{P}$ diagram for the random luminosity model, calculated using the theoretical death line and approximated by the equation B_d/P² = 0.17 × 10¹² G s⁻² (Bhattacharya et al. 1992; Figure 14 in FK06). The result is similar to that of FK06, who noted that, in the absence of magnetic field decay and using the random luminosity model, the results of MC-based population synthesis showed a clear pileup of observed objects near the death line in the $P\hbox{--}\dot{P}$ diagram. However, we note that the used approach did not take into account that a radio emission process (whatever it may be) should have an upper limit for its efficiency. In Figure 5(a), pulsars with derived radio efficiencies greater than 1% are marked by red dots. The efficiency of some of these pulsars approaches, and even exceeds 100%. This is physically unreasonable. More likely, the condition for pulsar radio emission may be such that a certain maximum efficiency is imposed. One should check whether the radio efficiency (see Equations (2) and (4)) does not exceed 100% (ξ < 1) for each independent draw of L₁₄₀₀, P, and B_d.

In Figure 5(b), we show the $P\hbox{--}\dot{P}$ diagram obtained by the MC simulation using our optimal random luminosity model (see Table 1) with the condition for pulsar death based only on the upper limit for radio efficiency ξ_max = 0.01. Using the efficiency limit not only allows one to avoid a pileup of pulsars near the theoretical death line, but also explains the existence of a few pulsars observed in the so-called graveyard region.

In Figure 6, we present a comparison of the observed distribution with synthetic distributions for our optimal model (presented in Figure 5(b)). We have found that the random luminosity model with proper input parameters can reproduce the observed sample much better than the models proposed by FK06. The Kolmogorov–Smirnov (K-S) test on the flux distribution for the power-law model gives the probability (the K-S P-value) 10⁻⁸ (see Figure 6 in FK06), while the K-S test on the flux distribution for our optimal random luminosity model results in a probability of 1%. Note that in our optimal model the distribution of the spin period at pulsar birth is centered at 200 ms. This allowed us to obtain the higher probability of reproducing the period distribution than the one obtained by FK06 (compare 4% with 0.7%), but negatively affected the probability of reproducing the period derivative distribution (0.1%), and hence, the B_d distribution (compare 2% with 15%). It is worth noting that one of the parameters that strongly affects both period and period derivative distributions is the maximum efficiency limit. Taking into account the relatively large number of parameters required by the evolutionary method and the fact that we arbitrarily choose ξ_max = 0.01, the parameters of the model can be further optimized in order to better reproduce the observed sample.

Calculations show that by using the condition for pulsar death based on the radio efficiency limit we can avoid a pileup of pulsars near the death line. As a result, we have found that neither the magnetic field decay nor the dependence of the luminosity on P and $\dot{P}$ are required to reproduce the observed sample.⁶

Figure 7 presents the $P\hbox{--}\dot{P}$ diagram for the whole sample of observed pulsars studied in this paper, with the color scheme denoting how radio efficiency, ξ, is distributed. The figure clearly shows that the radio emission graveyard in the $P \hbox{--}\dot{P}$ diagram corresponds to a region where ξ may exceed its upper limit (a few percent).

4. SUMMARY AND DISCUSSION

Using a large sample of pulsars from the ATNF Catalogue, we draw the following conclusions in this paper. (1) Radio pulsar luminosity has a very weak, if any, dependence on pulsar spin-down luminosity, $\dot{E}$ , over nearly eight orders of magnitude in $\dot{E}$ . There might be a slight variation of the exponent of the $L\hbox{--}\dot{E}$ correlation with frequency. (2) There is a near-linear inverse correlation between radio efficiency, ξ and $\dot{E}$ , for both normal and binary/millisecond pulsars. (3) The radio efficiency, ξ, is roughly linearly correlated with pulsar age, τ, for normal pulsars. (4) The two reported correlations are not due to an observational selection effect. (5) Since radio luminosity does not depend on P and $\dot{P}$ (or the dependence is very weak), the proposed mechanism of radio emission must explain the significant increase in radio efficiency for low- $\dot{E}$ pulsars.

A weak $L\hbox{--}\dot{E}$ correlation was hinted in previous studies (e.g., Lorimer et al. 1993; Ridley & Lorimer 2010). Malov & Malov (2006) found a weak positive dependence, $L \propto \dot{E}^{0.29}$ , with a smaller sample (338 objects), which was selected by requiring the pulsars to have well-determined spectra, and/or well-determined distance. We adopt a much larger sample, which has the following advantages. First, a larger sample size can insure more reliable correlations. Second, many uncertainties can be effectively canceled out, e.g., the unknown moment of inertia of neutron stars, the influence of scintillation on pulsar flux density, or how well the one-dimensional line-of-sight cut through an emission beam could represent the entire beam. With our enlarged sample, the index of the correlation, if any, is much flatter than the ones found in the previous studies (see, e.g., Arzoumanian et al. 2002; Malov & Malov 2006).

As discussed in Section 2.1, by using the simplified formula (Equation (2)), we have neglected some complicated pulsar geometry factors, such as α, β (which defines the duty cycle, δ, along with P and ρ). For individual pulsars, the derived L should have a large error. Including the entire sample would cancel out most of these geometric factors, so that our discovered $\xi \hbox{--}\dot{E}$ and ξ–τ correlations would be intrinsic. One possible factor that is not fully canceled is the P-dependence. This is because both ρ and δ depend on P. However, the explicit exponent of P-dependence is unknown. If we assume that the mechanism of radio emission is the same for both normal and millisecond pulsars, we find that by adding a dependence of ∼P^−0.5 to the integrated luminosity, the $\xi \hbox{--}\dot{E}$ correlations for both normal and binary/millisecond pulsars become consistent with each other. Adding this period dependence results in the following luminosity relationships: $L_{400}\propto \dot{E}^{0.27}$ , $L_{1400}\propto \dot{E}^{0.19}$ , and $L_{2000}\propto \dot{E}^0$ . The determined correlations are closer but still shallower than those found by Malov & Malov (2006). With such a correction, the $\xi \hbox{--}\dot{E}$ correlation is slightly shallower, i.e., $\xi \propto \dot{E}^{-0.81}$ , but is still very significant. It has an even smaller impact on the ξ–τ correlation, i.e., ξ∝τ^1.01 after correction. In a more general form, we can write that $\delta ^{-1} \sin ^2{(\rho /2)} \propto P^p \dot{P}^q$ . Since we are not using geometry information in our analysis, we cannot unambiguously define both p and q. However, assuming similar radio luminosities of both normal and millisecond pulsars, we can write that $\delta ^{-1} \sin ^2{(\rho /2)} \propto P^{p} \dot{P}^{-0.16(2p+1)}$ . Note that even the introduction of some model-dependent values of p does not change the general picture presented in this paper.

It would seem natural to assume that the spin-down parameters affect the radio luminosity (and hence the radio efficiency) of pulsars. Therefore, many authors (e.g., Ostriker & Gunn 1969; Lyne et al. 1975; Stollman 1986; Malov & Malov 2006; Faucher-Giguère & Kaspi 2006) tried to define the luminosity law, however, the results vary greatly (depending on pulsar samples and methods used) and were inconclusive (see, e.g., Bagchi 2013). The spin-down parameters determine the magnetic field strength at the light cylinder $B_{\rm LC} \propto P^{-2.5} \dot{P}^{0.5}$ . Then, assuming the dipolar configuration of the magnetic field, we can estimate the field strength at the stellar surface as $B_{\rm d} \propto (P \dot{P})^{0.5}$ and the vacuum potential drop as $V\propto B_{\rm d} P^{-1} \propto P^{-0.5} \dot{P}^{0.5}$ (Ruderman & Sutherland 1975). The main parameters that affect the properties of pulsar radio emission are the density and distribution of the electron–positron pair plasma. To estimate the plasma properties, we need to use a specific model of acceleration (e.g., the vacuum gap, the space-charged limited flow, or the partially screened gap models) and also take into account various emission processes (e.g., curvature radiation, inverse Compton scattering). It was shown that the plasma density and distribution are highly dependent on some factors that cannot be estimated by the spin-down parameters (see, e.g., Zhang & Harding 2000; Hibschman & Arons 2001a, 2001b). Thus, we cannot specify the properties of the plasma responsible for radio emission using only P and $\dot{P}$ . However, assuming the dipolar configuration of the pulsar magnetic field, some theoretical predictions suggest that the older the pulsar, the smaller the final pair multiplicity (Hibschman & Arons 2001a, 2001b). This would suggest that some kind of dependence of the radio luminosity on the pulsar age could also be found. As we have shown, however, this is not the case for the observed sample of radio pulsars (see Figure 4).

In order to reproduce the observed sample of pulsars, FK06 argued that in the absence of torque decay pulsar radio luminosity must depend on P and $\dot{P}$ as $L\propto P^{-1.5}\dot{P}^{0.5}$ (or $L\propto P^{-1.39}\dot{P}^{0.48}$ with exponents optimized by Bates et al. 2013). However, the power-law luminosity model results in a relatively low P-value for the flux distribution in the K-S test. We have found that by replacing the modeled death line (Bhattacharya et al. 1992) with the radio efficiency limit allows us to reproduce the observed sample even with the luminosity model previously excluded by FK06, namely the random luminosity model. Introducing the radio efficiency limit into MC simulations results in much higher probabilities from the K-S test, allows one to avoid a pileup of pulsars near the theoretical death line, and, furthermore, explains the existence of a few pulsars observed in the so-called graveyard region. We argue that the fact that the observed luminosity does not depend on both P and $\dot{P}$ and that the random luminosity model of the intrinsic luminosity allows for the reproduction of the observed sample is a strong indication in favor of the random model.

It is very difficult to explain the spread of up to four orders of magnitude for both L and ξ for a given spin-down luminosity only by statistical uncertainties. As we have mentioned above, there must be some parameter/parameters other than P, $\dot{P}$ that influence the radio efficiency of pulsars. Both, the clear $\xi \hbox{--}\dot{E}$ linear inverse correlation and the ξ–τ linear correlation for radio emission call for the following physical picture: the plasma responsible for generating pulsar radio emission must be produced under similar conditions regardless of pulsar age, dipolar magnetic field strength, and spin-down rate. One possibility is that the pulsar polar cap region may have dominant crust-anchored magnetic anomalies so that the near surface magnetic field configuration significantly deviates from the dipolar geometry (Gil et al. 2002a). X-ray observations of old pulsars indeed revealed hot spots that are significantly smaller than the conventional polar cap (Zhang et al. 2005; Gil et al. 2008; Szary 2013), which is consistent with having a strong, non-dipolar magnetic field at the surface, due to crust-anchored local anomalies.

The crust-anchored anomalies can be characterized by the parameter b = B_s/B_d = A_dp/A_bb, which describes the ratio of the actual value of the magnetic field at the surface, B_s, to the dipolar component of the magnetic field, B_d (see, e.g., Gil et al. 2002b). Here, A_dp ≈ 6.2 × 10⁴P⁻¹ m² is the conventional polar cap area (i.e., calculated assuming pure dipolar configuration of the magnetic field), and A_bb is the actual hot spot area (i.e., derived from X-ray observations). The sample of pulsars with an X-ray hot spot for which b can be estimated is small. Nevertheless, with this small sample (Szary 2013), we can clearly see a correlation between pulsar age and b (see Figure 8).

**Figure 8.** Dependence of the b parameter on pulsar age, τ (see the text for more details). The dashed line corresponds to the linear fit for all pulsars with detected radio emission. Note that for PSR J0108−1431 we used the single blackbody fit performed by Pavlov et al. (2009).
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The found correlation indicates that when a pulsar becomes older, its surface magnetic field becomes more dominated by the crust-anchored magnetic anomalies generated, e.g., by the Hall drift (see Geppert et al. 2013; Viganò et al. 2013). Such a crust-anchored anomaly does not depend on $\dot{E}$ and age, which allows pulsars to produce enough electron–positron pairs at an old age to power radio emission.

The fact that pulsars near the graveyard tend to have a very high ξ also has profound implications for the understanding of radio pulsar death. Traditionally, the pulsar death line (Ruderman & Sutherland 1975; Chen & Ruderman 1993) was defined by the condition of the production of electron–positron pairs, which depends on many uncertainties, including the near-surface γ-ray emission mechanism (Zhang et al. 2000) and magnetic field configurations (Gil & Mitra 2001). The results presented in this paper offer a simpler interpretation of radio pulsar death. It is possible that the required physical condition to power radio emission is similar among pulsars of all ages, which requires a certain minimum spin-down power. When pulsars spin down slightly below this threshold, the radio emission mechanism simply cannot operate. This explains why high ξ pulsars are located near the death line. A similar conclusion was drawn by Arzoumanian et al. (2002) based on the modeling "of the birth properties and rotational, kinematic, and luminosity evolution of a MC population of neutron stars," even though the luminosity and spin-down rate scaling presented in their work, $L \propto \dot{E}^{0.5}$ , does not reflect the actual $L\hbox{--}\dot{E}$ relationship (see Figure 1(a)).

This work is partially supported by a visitor program of Kavli Institute for Astronomy and Astrophysics, Peking University, China, and National Science Centre Poland under grants 2011/03/N/ST9/00669 and DEC-2012/05/B/ST9/03924. B.Z. acknowledges NASA NNX10AD48G for support, and R.X.X. acknowledges support by the National Science Foundation of China under grant No. 11225314. We thank the anonymous referee for constructive comments.

RADIO EFFICIENCY OF PULSARS

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ABSTRACT

1. INTRODUCTION