On the diversity of compact objects within supernova remnants – I. A parametric model for magnetic field evolution Free

Abstract

1 INTRODUCTION

Thanks to the advance of modern X-ray telescopes such as Chandra and XMM–Newton, and the synergy with radio observations, we now know that isolated neutron stars (NSs) can manifest themselves as pulsars (PSRs) with a surface dipole magnetic field spanning more than five orders of magnitude, in the ∼10¹⁰–10¹⁵ G range.¹ Observationally, this has led to their organization into different classes, including (1) the rotationally powered radio and X-ray bright objects, like the Vela pulsar with B ∼ 10¹¹–10¹³ G, (2) the magnetically powered pulsars (or magnetars) with B ∼ 10¹⁴–10¹⁵ G, exceeding the quantum electrodynamics (QED) limit of 4.4 × 10¹³ G and observed primarily at high energies, (3) the highly magnetized pulsars (HBPs) with magnetic fields intermediate between the classical pulsars and magnetars, but still exceeding the QED limit, and (4) the central compact objects (CCOs) observed only in X-rays (so far), near the centres of supernova remnants (SNRs) and with inferred low magnetic fields, B ∼ 10¹⁰–10¹¹ G. This diversity led several authors to attempt a unification through evolutionary models of NSs with their properties dictated primarily by a continuum of magnetic field strengths (see e.g. Kaspi 2010; Dall'Osso, Granot & Piran 2012; Mereghetti 2013; Perna et al. 2013; Vigano et al. 2013; Safi-Harb 2015, and references therein).

The magnetic field is estimated using a standard model of NS evolution which assumes energy loss due to the emission of radiation from a point-like rotating magnetic dipole in vacuum, providing a spin-down torque with a braking index n = 3 (Gunn & Ostriker 1969). This picture assumes rapid rotation of the NS after birth, so the observed period (P) differs from the initial period (P₀) by a large amount (i.e. P₀ ≪ P) due to the constant torque acting to slow the NS spin (Burrows & Lattimer 1988). However, the braking index has been measured for a small sample of young pulsars, and all so far differ from the prediction of the standard model with n < 3 (Espinoza et al. 2011). A lower braking index can come about from a variety of mechanisms including a change in the moment of inertia of the star over time (Ho & Anderson 2012), alignment of the magnetic field and rotation axis (Macy 1974; Lyne et al. 2013), the emission of a particle wind (Harding, Contopoulos & Kazanas 1999; Tong et al. 2013), magnetospheric effects (Michel 1991; Spitkovsky 2006) and environmental interactions (Menou, Perna & Hernquist 2001). Another serious problem with the standard picture concerns the NSs that are associated with SNRs. Generally, pulsar ages found from their ‘characteristic age’ (⁠|$\tau _{\rm PSR}=P/2 \dot{P}$|⁠) by assuming dipole radiation and the independently measured SNR ages are in disagreement, sometimes by orders of magnitude (in particular for the CCOs).² The observed braking index and age discrepancy arise from the standard, and commonly adopted, assumption of a constant torque acting to brake the NS over its life span.

The growing evidence for NSs with X-ray luminosity in excess of their spin-down energy presents another difficulty for the standard scenario. Some of these objects are thought to be powered by the dissipation of magnetic energy rather than spin-down losses, examples of which include the anomalous X-ray pulsars (AXPs) and some soft gamma-ray repeaters (SGRs), unified under the class of ‘magnetars’ (Duncan & Thompson 1992; Paczynski 1992; Usov 1992). These are X-ray bright objects that are slowly rotating pulsars with exceptionally high magnetic fields and normally discovered through their bursting activity. However, a neat classification scheme for these objects proves elusive in the light of the discovery of ‘low-B magnetars’ (e.g. Rea et al. 2010), and an HBP having behaved like a magnetar, yet thought to be a rotation-powered pulsar powering a bright pulsar wind nebula (Gavriil et al. 2008; Kumar & Safi-Harb 2008). The situation is further complicated by the CCOs with extremely low fields, dubbed as ‘anti-magnetars’ (Gotthelf & Halpern 2013), yet still show an X-ray luminosity in excess of their spin-down energy. One recent interpretation for these objects is the suppression of their external field through magnetic field burial (Bernal, Lee & Page 2010; Ho 2011; Vigano & Pons 2012; Bogdanov 2014), also implied by spectroscopic models of these objects (Ho & Heinke 2009). In this scenario, the accretion of supernova fall-back material occurs following the birth of the NS. This period of vigorous accretion has the effect of burying the dipole magnetic field component within the NS crust, reducing the spin-down energy loss and making the NS appear significantly older than its associated SNR. In this alternative model of NS evolution, field growth is needed to explain the initially small braking index and low surface fields, while a decaying toroidal component is invoked to explain the excess X-ray luminosity (Ho 2012). The field burial scenario has been most recently described in significant detail by Ho (2015), who performed detailed calculations of the fall-back accretion process, including the inner structure of the NS, conductivity of the NS crust and a realistic equation of state. Besides an internal decaying toroidal component, the dipole field component also decays on large time-scales (Dall'Osso et al. 2012; Vigano et al. 2013). The decaying external field is described by a parametrized model given by Colpi, Geppert & Page (2000), expanded on by Dall'Osso et al. (2012) and used to describe the evolution of the AXP 1E 2259+586 by Nakano et al. (2015).

In this paper we present a phenomenological parametrized family of models for magnetic field evolution in the NS population. Our model unifies the description of magnetic field growth and decay by making use of variations on the parametric forms from Dall'Osso et al. (2012) and Colpi et al. (2000), which we derive in Sec-tion 2. This model also reproduces the results of Negreiros & Bernal (2015) for exponential field growth and replicates the findings of Dall'Osso et al. (2012) for decaying fields. We fit our model to the observations of various NSs in Section 3, testing our model against the detailed physical predictions found by Ho (2015). We discuss the results of our fits in Section 4. Finally, our conclusions are summarized in Section 5.

2 THEORY AND PARAMETER SPACE EXPLORATION

The model as stated has six parameters: the field index α, growth factor ϵ, time-scale τ_m, asymptotic field B_G, the initial period P₀ and the model time t. The model time can be treated as a free parameter to vary between the lower and upper SNR age limits, τ_SNR− and τ_{SNR +}, respectively, or can be fixed before beginning the optimization. The model then outputs the quantities we want to fit to the observed values: the period, period derivative and braking index (at the present time). The standard fitting problem is to vary the input parameters to produce a match with the output and the observed values. Thus, the problem is under constrained, in that there are fewer fit quantities than parameters, leading to a family of solutions. However, a closer inspection shows that the previously stated input parameters are not truly independent. In fact, a simplification can be achieved by changing our modelling approach.

As an example, in Fig. 1, we show the parameter space for the Vela pulsar, PSR B0833−45, which has a measured braking index and associated SNR age. On the left, we show the (τ_m, B_G) parameter space using only the age constraint from the associated SNR. The lower limit τ_{SNR −} is the dashed blue line, and the upper limit τ_SNR+ is the solid blue line. The region between the curves is the parameter space area that obeys the age constraint, and is coloured red. On the right we show the same region of parameter space, but include the braking index contours, denoted as green lines. Since the braking index is known to high accuracy, the red coloured regions satisfying both constraints simultaneously is significantly reduced. The lower group in both panels has α = 0.1, ϵ = 0.1, and the upper group α = 0.9, ϵ = 0.001. Note that there is degeneracy between the groups of solutions for a given τ_m depending on the chosen field index α. Thus, while this method does not provide a unique solution, it allows us to quantify the regions of parameter space that contain physical solutions given a constant α and ϵ pair as input. We investigate the differences between the low- and high-α cases to study the limiting behaviour of the field growth model. Generally, the solutions that have observed fields close to the asymptotic field B_G are near the end of their evolution. The α = 0.1, ϵ = 0.1 solutions with the lowest asymptotic fields have nearly finished their evolution and will grow by only ≈1 per cent over the next few kyr. The braking index of these solutions will rapidly grow to the dipole value n = 3 over this span of time. The solutions with α = 0.9 and ϵ = 0.001 have significantly larger asymptotic fields that are more than an order of magnitude higher than the low-α solutions. These NSs will take a significantly longer span of time for their fields to evolve to the final state and reach braking index n = 3. For these solutions, the field growth significantly outlasts the observable life of the SNR and will appear as a highly magnetized, isolated NS with no apparent SNR association.

Figure 1.

Parameter constraint plots for the Vela pulsar, PSR B0833−45. The left-hand panel shows the parameter space region that satisfies the age constraint (red). The lower SNR age limit is the blue dashed line, and the solid blue line is the upper SNR age limit. On the right, the same region of parameter space is shown but we include the braking index constraints (green curves). The red region shows the intersecting set, which satisfies both the age constraint and the braking index constraint. The lower group in both panels has α = 0.1, ϵ = 0.1, and the upper group α = 0.9, ϵ = 0.001. The horizontal black line in both panels marks the observed dipole field.

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3 MODEL FITTING AND NS EVOLUTION

Let us investigate the consequences of field growth in NSs using two initial approaches. First, we vary the field index α to demonstrate how this parameter affects the time evolution. Secondly, since this field growth model is phenomenological in nature, we investigate how well it can reproduce the results of numerical simulations, such as the detailed modelling of the burial and emergence of the magnetic fields in young accreting NSs explored recently by Ho (2015). That study focused on the young NSs with braking indices n < 2, in particular the rotation-powered pulsars PSR J0537−6910 associated with the LMC SNR N157B, the Vela pulsar B0833−45, and the HBP J1734−3333 which has a proposed association with G354.8−0.8 (Manchester et al. 2002). We note that the relationship between this SNR and HBP J1734−3333 is tenuous and may be the result of a coincidental alignment. We list these systems in Table 2, along with a carefully selected list of other NSs that are (1) securely associated with SNRs (thus providing an independent estimate of the true age), and (2) with ‘extremal’ fields, namely from the class of magnetars, HBPs and CCOs.

For the purpose of illustrating the effect that changing α has on the field evolution, we consider the HBP J1734−3333 as an example and use the age derived in Ho (2015), t = 2.07 kyr. We arbitrarily set the initial field to a typical NS field strength, B₀ = 10¹¹ G, which fixes ϵ for a given B_G by equation (13). We generate a family of curves using a constant field index that spans the full range 0 ≤ α < 1. For each value of α, we follow the standard model fitting approach, treating the time-scale τ_m, asymptotic field B_G and initial period P₀ as fit parameters, which are varied numerically. The results are shown in Fig. 2, where we plot the period, period derivative, characteristic age, magnetic field, braking index and luminosity as functions of time for each of the α values, matching to the observed P, dP/dt and n. The horizontal dashed lines represent the observed values and the vertical dashed line is the adopted current age t. In the luminosity panel, we show the spin-down luminosity (⁠|$\dot{E}$|⁠) as a horizontal dotted line and the 2–10 keV X-ray luminosity (L_x) as a dashed line. We call attention to two important features of this figure. First, the characteristic age decays rapidly from an initially high value regardless of α. This general behaviour provides an explanation for young NSs that have a characteristic age larger than the corresponding SNR age. Secondly, the large characteristic age at early times gives a negative braking index at early times through equation (8), which allows the field growth scenario to explain the observations of objects with n < 0, such as PSR J0537−6910 with n = −1.5 (Middleditch et al. 2006). The field index α smoothly controls how quickly the braking index reaches the asymptotic value n = 3.

Figure 2.

Fits to HBP J1734−3333 for a variety of field index α. The vertical dashed line marks the adopted age t = 2.07 kyr, chosen to facilitate comparison with the result from Ho (2015). The horizontal dashed lines mark the observed quantities. In the luminosity plot (lower right-hand panel), the horizontal dotted line marks the spin-down energy loss rate and the horizontal dashed line marks the observed X-ray luminosity.

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Next, we use our field growth model and contour approach to recover the quantitative behaviour of the detailed simulation performed by Ho (2015). We assume the value of the asymptotic field and age derived in that work a priori, and then we treat the time-scale 0.1 ≤ τ_m ≤ 10 kyr and field index 0 ≤ α ≤ 0.99 as free parameters. We construct a grid of parameter values (τ_m, α) and calculate t and n for each. Then we find the level sets of these functions at the derived age and mean braking index. The intersection of the two curves gives a unique τ_m and α pair. With ϵ = 0.0021, we find P₀ = 1.0597 s, compared to 1.06 s given by Ho (2015). The initial period does not change significantly for lower ϵ. The discrepancy grows slowly as ϵ is increased. Using the contour approach, we find the parameters necessary for our model to reproduce the evolutionary trajectories of PSR J1734−3333, PSR B0833−45 and PSR J0537−6910 from Ho (2015). The details of the fits are given in Table 3 and marked with an asterisk. It is a testament to the flexibility and usefulness of the parametric form that we were able to recover the behaviour of a simulation involving detailed and complex physical processes.

Finally, we apply the contour method to the remaining HBPs and PSRs listed in Table 3, including the braking index when possible. We follow the prescription outlined for contouring in Section 2, holding α and ϵ constant and finding the level sets of t and n as functions of the time-scale τ_m and asymptotic field B_G. We do not consider any asymptotic field strength greater than the maximum observed magnetar field, B_G = 2.4 × 10¹⁵ G, of SGR 1806−20 (Nakagawa et al. 2009). For each system shown in Table 1, we provide example solutions with large and small asymptotic fields in Table 2, and plot the trajectories of these example solutions in the |$P{\rm -}\dot{P}$| phase space in Fig. 3. In this plot, the evolutionary trajectories of the SGRs are given as green lines, the HBPs as red lines, the PSRs yellow lines and the CCOs as blue lines. The HBP J1846−0258 is marked by a light grey diamond, HBP J1119−6127 is a dark grey diamond and HBP J1734−3333 is a white diamond. The PSRs J0537−6910 and B0833−45 are marked by grey and white stars, respectively. Markers that are black represent objects with X-ray luminosity in excess of spin-down luminosity. The parameters that describe the trajectories shown in this figure are likewise given in Table 3. Note that we do not provide an example trajectory for HBP J1119−6127, which will be discussed in the next section.

Figure 3.

Phase space plot of evolutionary trajectories P–|$\dot{P}$|⁠. The thin black dashed diagonal lines denote constant characteristic age from 100 yr (upper left) to 1 Gyr (lower right). The dotted black diagonal lines represent an increasing magnetic dipole field, from 10¹¹ G (lower left) to 10¹⁵ G (upper right). Black symbols represent sources that have X-ray luminosity in excess of spin-down luminosity. The low asymptotic field trajectories are marked as solid lines, and the high field solutions from Table 3 are dashed lines. The evolutionary tracks for HBPs are red, SGRs are green, PSRs are yellow and the CCOs denoted with blue. HBP J1846−0258 is marked by a light grey diamond and the post-outburst trajectory is shown. HBP J1734−3333 is a white diamond and HBP J1119−6127 is a dark grey diamond (note that this object is not accompanied with a trajectory). The PSRs J0537−6910 and B0833−45 are marked by grey and white stars.

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In Fig. 4, we plot the characteristic age against the model time, following the same conventions as Fig. 3. Despite the apparent similarity of many of the trajectories in the P–|$\dot{P}$| phase space, the τ–t plot clearly shows the difference between these objects as a function of time. It is worth noting that the time evolution of the CCOs characteristic age explains the apparent large discrepancy between the pulsars adopted ages (appearing very old) and their associated young SNRs. In particular, for the three systems shown, the PSR and SNR ages match at times equal to or exceeding ∼10^4.5 yr, by which time the SNR would have mostly dissipated. Therefore, the characteristic age for these objects considered will not reflect their true age as long as they are within their SNRs. This feature, along with the low asymptotic field strength (see Table 3), also leads to the suggestion that CCOs could be ancestors of ‘old’ isolated radio pulsars as long as they overcome the accretion or field-growth phase (which would explain their X-ray dominant emission) and their surface field grows to the critical limit required for radio emission. The late time evolution of the CCOs may also link them to the class of objects known as X-ray dim isolated NSs (XDINS; Haberl 2004). These are radio-quiet X-ray pulsars with long periods (3.45–11.37 s) and no apparent SNR associations. Some of these objects are believed to have high magnetic fields in excess of 10¹³ G.

Figure 4.

Evolutionary trajectories plotted by NS characteristic age against model time for the systems shown in Table 3. The SNR ages are represented as horizontal lines. The markers are placed at the mean value, or at the extreme when no upper or lower limit exists. The low asymptotic field trajectories are marked as solid lines, and the high field solutions from Table 3 are dashed lines. Colours are the same as in Fig. 3. The thick black line is τ_PSR = t.

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4 DISCUSSION

The P–|$\dot{P}$| phase space trajectories shown in Fig. 3 demonstrate possible evolutionary links between the apparently diverse set of NSs shown in Table 3. As the NS fields grow, they evolve from the bottom of the figure upward, passing through the region of the phase space inhabited by the CCOs. Thus the PSRs J0537−6910 and B0833−45, and the HBPs J1846−0258 and J1734−3333 may have undergone a similar CCO stage during their evolution. The trajectories of these HBPs carry them towards the current position of AXP 1E2259+586. If HBPs and AXPs are related through evolution then field decay must begin once the buried field has emerged, raising the braking index to values n > 3. We have also fit the CCOs to explore their potential future behaviour, and note that RX J0822.0−4300 has asymptotic behaviour for both large and small fields which is very close to PSR B0833−45. However, the time evolution of these objects is dramatically different as seen in Fig. 4. Thus, objects like the HBPs may pass through the CCO stage relatively quickly, whereas objects such as CCO 1E 1207.4−5209 spend a more significant portion of their lives in this state. Finally, we note that CCO CXOU J185238.6+004020 requires an extremely low asymptotic field, with B_G < 6.9 × 10¹¹ G. Thus, even after the field emerges from this NS, it remains relatively low.

Since the SGRs 0527−66 and 1627−41 have characteristic ages less than the upper SNR age limit, we have also examined these objects using the growth model. However, the field growth mechanism is not generally expected to play a role in the evolution of the SGRs, since their characteristic ages are only smaller than the upper limit of the associated SNR age, and no lower limits are known. Moreover, these systems do not have a measured braking index, which is crucial in making the case for field growth (n < 3) or field decay (n > 3). Additionally, field decay has been proposed to explain the SGRs energetics as it has for AXPs, although their X-ray luminosity is not consistently larger than their spin-down energy. Due to the lack of a lower age limit, we consider solutions that produce τ_PSR < t ≤ τ_{SNR +}. Generally solutions that satisfy this condition with large B_G require a longer growth time-scale for a given α and ϵ pair, so we can find large fields B_G > 2.0 × 10¹⁵ G, provided we consider sufficiently large τ_m. We plot the evolution of two example solutions in Figs 3 and 4. Interestingly, both SGR 1627−41 and SGR 0527−66 reach similar states in the limit of large asymptotic fields shown in Fig. 3, and the trajectories imply the SGR fields are still evolving. A lower limit for the SNR age would significantly constrain these results, provided that τ_{SNR −} > τ_PSR. For completion, we attempted fits to the AXPs as well, but these required unrealistically high initial spin periods. We stress that despite these interesting fits, field decay is necessary to reconcile the characteristic and SNR ages of the AXPs. This conclusion is supported by results from the literature (e.g. Nakano et al. 2015), and is implied for the SGRs evolution as well (Dall'Osso et al. 2012).

HBP J1846−0258 presents an interesting case since the braking index has been observed to decrease from n = 2.64 to 2.16 (Livingstone et al. 2006, 2011) following an outburst and spectral changes in 2008 (Gavriil et al. 2008; Kumar & Safi-Harb 2008). This braking index change was not accompanied by a change in luminosity or pulse profile which is difficult to explain on such short time-scales, but may represent a re-organization of the magnetosphere (Archibald et al. 2015). We fit the pre- and post-outburst configurations of the system which we label as A (n = 2.64) and B (n = 2.16). However, we were not able to find any region of parameter space through our contour methods that could simultaneously satisfy both pre- and post-outburst configurations. This may not be the case if the field index were allowed to vary in time, but with constant α, field growth cannot neatly explain the behaviour of this NS. The HBP J1846−0258 is a complicated case, particularly because of the presence of a bright pulsar wind nebula powered by this object. This nebula implies wind-braking likely plays an important role in the evolution of this NS. For α = 0.99 and minimum ϵ = 0.019, we find a maximum field B_G = 6.5 × 10¹⁴ G on a growth time-scale 9.2 kyr. Changing α and ϵ results in a lower field on shorter time-scales. For a given pair of α and ϵ, the system can be well constrained by the period condition and the SNR age, though low remnant ages are generally favoured.

Finally, there was a problem fitting to HBP J1119−6127. For this system, we were not able to simultaneously fit both the age of the associated SNR G292.2−0.5 (Kumar, Safi-Harb & Gonzalez 2012), and the observed braking index n = 2.684 (Weltevrede, Johnston & Espinoza 2011). With the SNR age constraint, the derived braking index is n ≈ 1.8 assuming t = 4.2 kyr. Alternatively, with the braking index constraint in place, the derived age was found to be close to the characteristic age, and cannot be reconciled with the observed SNR age. Intriguingly, a low value for the braking index of J1119−6127 was also proposed by Kumar et al. (2012), who suggested that the braking index may have recently changed from a lower value n < 2. Since neither of these scenarios satisfies the constraints, we have not included an evolutionary track for HBP J1119−6127 in Figs 3 and 4, and also exclude it from our summary of solutions in Table 3. We plan to investigate HBP J1119−6127 with other emission mechanisms in future work.

It is also relevant that many of the systems included in Table 3 have large initial spin periods (i.e. P₀ ≈ 1 s, and approaching P for many systems), which are higher than expected for the traditional magnetar model (Duncan & Thompson 1992; Igoshev & Popov 2013). This is notable because a problem with the magnetar model is the lack of superenergetic SNRs which would be expected from an SNR hosting a rapidly spinning proto-NS (for example, Vink & Kuiper 2006; Safi-Harb & Kumar 2013).

5 CONCLUSIONS

We have devised a flexible and conveniently parametrized model for a growing magnetic field, which is based on the parametric forms used by Colpi et al. (2000) and refined by Dall'Osso et al. (2012). This parametrization can accommodate a variety of field time-dependence in addition to the exponential model suggested by Negreiros & Bernal (2015), and the interpretation of the parameters is straightforward. By including the observationally measured period and period derivative and assuming the field index and growth parameter ϵ constant, we are able to study the portions of the parameter space containing solutions which reproduce the observables, without the need for an external optimization routine. We have shown that this phenomenological model is able to reproduce the detailed simulations of field growth by Ho (2015) to high accuracy. By fitting the HBPs securely associated with SNRs with known ages and measured braking indices, we found interesting evolutionary trajectories for the systems in phase space. We conclude that if field growth is significant in the life cycle of HBPs, then they may be closely related to the CCOs early in their evolutionary histories. The end result of the field growth in CCOs may connect these objects to the HBPs and XDINSs. We also investigate the possibility of field growth in SGRs, however, the behaviour of these systems are largely unconstrained due to the absence of a lower SNR age limit and lack of measured braking index.

Field growth is not applicable to the AXPs, which require field decay to explain the observed difference in PSR and SNR ages, and a growing field is not necessary to explain the SGRs, provided that the characteristic age is larger than the true SNR age. Thus, in the context of magnetic field evolution, we conclude that both field growth and decay processes are required to explain the diverse population of NSs. Once the field has reached its asymptotic value, field decay may begin, increasing the braking index to values n > 3 later in life. Thus, the time-dependence of magnetic fields provides an interesting avenue to unify the population of NSs, and in particular, explain the apparently large characteristic ages for systems associated with relatively young SNRs.

While this evolutionary picture is simple and based on a phenomenological model, there are many emission mechanisms which have been proposed in the literature to solve the braking index and PSR–SNR age discrepancy problems. The field growth model was shown to be ineffective in explaining the constraints present in the HBP J1119−6127 and the time evolution of HBP J1846−0258, both of which are associated with pulsar wind nebulae. In a follow-up paper, we will thoroughly investigate alternatives to the physical emission mechanisms at work in these and various other classes of objects and the subsequent implications for the PSR–SNR association and evolution.

This work was primarily supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) through the Canada Research Chairs Programme. SSH is also supported by an NSERC Discovery grant and the Canadian Space Agency. This research made use of NASA's Astrophysics Data System, McGill's magnetars catalogue and the U. of Manitoba's high-energy SNR catalogue (SNRcat). We also thank the referee for the thoughtful comments that helped significantly improve the clarity of this work.

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