We present new analysis of the birth rate of AXPs and SGRS and their associated SNRs. Using
Kolmogorov-Smirnov statistics together with parametric fits based on a robust estimator, we find a birth rate of ~1/(1000 years) for AXPs/SGRs and their associated SNRs. These high rates suggest that all massive stars (greater
than ~(23–32)

Early studies of association of Anomalous X-ray Pulsars (AXPs) with supernova remnants (SNRs) suggested that 5% of core-collapse SN results in AXPs [

In this study we present an updated investigation of the birth rate of AXPs/SGRs and in addition, the birth rate of associated SNRs is given. Since AXPs/SGRs ages rely on spin-down age estimates whereas SNRs ages are based on shock expansion models, this constitutes two independent estimates for birth rates. We find here that both samples yield a high birth rate for AXPs/SGRs of (1/5)–(1/10) of all core-collapse SNe, higher than previously appreciated. (An independent study by Gill and Heyl [

We start by analyzing the age distribution of AXPs/SGRs and their associated SNRs. One can derive birth rates by fitting a linear trend to the observed cumulative number versus age relation. (The cumulative age distribution is the step-like function with unit step along the

SGRs/AXPs data mainly from Gaensler et al. [

Source | Assoc. | |||||||
---|---|---|---|---|---|---|---|---|

SGR1806 | 281 | 3500 | 30000 | 10300 | 100 | 3800 | ||

SGR1900 | 1050 | 9600 | 30000 | 17000 | 7 | 2000 | 32000 | |

SGR0525 | 1960 | 5000 | 16000 | 8940 | S (0.2–0.7%) | 50 | 800 | 3600 |

SGR1627 | N/A | 2600 | 30000 | 8830 | 11 | 460 | N/A | |

AXP1E1841 | 4510 | 500 | 2500 | 1120 | S (0.01%) | 7 | ||

AXP1845 | N/A | 600 | 30000 | 4240 | S (0.2%) | N/A | ||

AXP1E2259 | 228000 | 3000 | 17000 | 7140 | S (0.05%) | |||

AXP1E1048 | N/A | N/A | N/A | N/A | N/A | N/A | N/A | |

AXPRXS1708 | 8960 | N/A | N/A | N/A | N/A | N/A | N/A | N/A |

TAXPXTEJ1810 | N/A | N/A | N/A | N/A | N/A | N/A | N/A | |

TAXPJ0100 | 6760 | N/A | N/A | N/A | N/A | N/A | N/A | N/A |

AXP1547 | 1400 | N/A | N/A | N/A | N/A | N/A | N/A | N/A |

AXP4U0142 | 70200 | N/A | N/A | N/A | N/A | N/A | N/A | N/A |

From Table

We apply the Kolmogorov-Smirnov (KS) statistical test [

(a) shows the cumulative probability distribution for ages of SNRs associated with AXPs/SGRs (diamonds). Also shown is the expected probability distributions for the cases of constant birth rate (solid line) and of constant age (dashed line). (b) shows the cumulative probability distribution for spin-down ages of AXPs/SGRs (diamonds) and expected probability distributions for constant birth rate (solid line) and of constant age (dashed line) models. The ages for the constant age models were chosen to give minimum KS statistic (and lowest rejection probability) for both SNR and spin-down ages.

In order to derive a birth rate from the data we go beyond the nonparametric KS test and carry out model fitting: one set of fits assumes normal statistics and another set uses a robust estimator (e.g., Press et al. [

Figure

(a) shows the cumulative age distribution for SNRs associated with AXPs and SGRs (diamonds and dot-dashed line). The solid line is the expected distribution for constant birth rate of 1/(1700 years). The triangles indicate the upper and lower SNR age limits corresponding roughly to

The birth rate derived from associated SNRs is

One effect is incompleteness of either sample, which would increase the birth rate of that sample; in this case incompleteness of the SNR sample could increase the birth rate to match the birth rate from spin-down. As can be seen from Figure

The second effect is that SNRs or AXPs/SGRs spin-down ages could be systematically off by a factor of

However, for 3 of the objects listed (SGR1806

To summarize this section, the AXPs/SGRs birth rate from spin-down is about 1/(500 years) (

We consider the implications of the high birth rate. A birth rate of 1/(500 years) to 1/(1000 years) implies

An alternate possibility is that

This raises the following questions.

How do all progenitors with

Why is there a sudden jump in the magnetic field strength between compact remnants from progenitors with mass greater than

In regards to point (1), observations of OB stars [

One natural mechanism would be dynamo generation during neutron star formation [^{51} erg) conservatively limiting the birth periods to

In summary, our reasoning is as follows: (i) high birth rates of magnetars imply that all massive stars with

We offer an alternate explanation which allows normal magnetic fields for neutron stars born from progenitors with mass _{low}, in addition to lower mass progenitors. In this picture, the magnetic field amplification occurs long after the neutron star formation, but only for neutron stars born from massive progenitors. The amplification occurs during the conversion from baryonic matter to quark matter which happens after the neutron star core reaches quark deconfinement density.

Amplification of the magnetic field up to

Deconfinement can be delayed from the time of formation of the neutron star by two mechanisms. First, there is a delay between neutron star birth and the time deconfinement density is reached. The proto-neutron star cools rapidly slightly increasing its density but a much more effective mechanism to increase the central density exists for rotating neutron stars [

Staff et al. [^{12} G, experience deconfinement with typical delays of several hundred years (see Table 2 and discussion by Staff et al. [

The energy loss rate from the young neutron star is given by
^{12} G) in our model, there is no over-energetic central pulsar in the SNR. This is consistent with observations of Vink and Kuiper [^{7}–10^{8} larger than in our model.

To represent the general idea of a delayed amplification of the magnetic field, we write the time since SN explosion as
^{12} G thus spin-down is slow during this period. The delay time,

From Table

The only object in Table

We have argued above that standard spin-down age estimates do not represent true ages. This affects the birth rate estimate for SGRs/AXPs given above in two ways. Firstly, the spin-down era is shorter by a factor of

Alpar [^{12} G). (Instead Dar and DeRújula [

For AXPs 1E2259

Our study of the birth rate of AXPs and SGRS and their associated SNRs suggests that about

This raises these issues: (i) how do all progenitors with _{low} (

In this study, we introduce the notion of delayed magnetic field amplification as a plausible solution to these issues. We propose that neutron stars from progenitors with mass ^{12} G) magnetic fields. A neutron star from a progenitor with an approximate mass in range

The authors thank the referee (A. D. Kaminker) for helpful suggestions. This research is supported by grants from the Natural Science and Engineering Research Council of Canada (NSERC).