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A&A
Volume 684, April 2024
Article Number L1
Number of page(s) 7
Section Letters to the Editor
DOI https://doi.org/10.1051/0004-6361/202449382
Published online 28 March 2024

© The Authors 2024

Licence Creative CommonsOpen Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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1. Introduction

An increasing number of bow shock pulsar wind nebulae (BSPWNe) shows evidence of filamentary X-ray structures (Hui & Becker 2007; Pavan et al. 2014; Temim et al. 2015; Klingler et al. 2016, 2018; Medvedev et al. 2019; Marelli et al. 2019; Bordas & Zhang 2020; Zhang et al. 2020; Klingler et al. 2020; de Vries & Romani 2022) protruding in the interstellar medium (ISM). The best characterized examples are the Guitar Nebula (Hui & Becker 2007; de Vries et al. 2022), PSR J2030+4415 (de Vries & Romani 2020; de Vries et al. 2022), and the Lighthouse Nebula (Pavan et al. 2014, 2016; Klingler et al. 2023). The filaments appear elongated in one direction for a distance ranging from 0.6 pc in the case of the Guitar Nebula to ∼15 pc in the case of the Lighthouse, with relatively mild morphological fluctuations. The filaments’ thickness is very small (from less than 1% of the length to ∼10%). The spectrum and morphology of the X-ray emission are compatible with synchrotron radiation of very high-energy leptons in a magnetic field that is substantially larger than the typical interstellar magnetic field (typically by a factor of ∼10), a possible indication (Bandiera 2008) that some type of instability is excited by particles escaping the nebula (Olmi 2023), leading to magnetic field amplification. In turn, this can account for the thickness of the filaments as a consequence of severe synchrotron energy losses. This qualitatively appealing picture needs a quantitative connection with known instabilities and with the actual energetics of the particles leaving the BSPWN.

Here we show that the length and thickness of the filaments require specific conditions that can only be satisfied if (1) electrons and positrons with sufficiently high energies are spatially charge-separated when leaving the BSPWN (Olmi & Bucciantini 2019, 2023), and (2) they are numerous enough to excite the nonresonant hybrid instability Bell (2004; NRI). Neither of these conditions is trivial: charge separation can only occur for pairs with energy sufficiently close to the maximum potential drop (MPD) of the pulsar, a condition that severely constrains the current density available to excite the instability.

We make the case that the electrons (or positrons) that escape the BSPWN may be focused in a narrow angle when leaving the source, and propagate along the local Galactic magnetic field (GMF) lines. The angular collimation is important in that it confines particles in a region with a small cross section, which in turn increases the current density. The excitation requires that the energy density associated with the particles dominating the current be larger than that of the preexisting magnetic field (Bell 2004; in this case the GMF).

The nonresonant nature of the instability is a crucial ingredient in this context: the lack of scattering allows the current-carrying particles to stream away from the BSPWN at speed close to that of light (c), thereby filling a filament for a length of order ∼CR, where τCR is the time needed for the saturation of the instability. During this phase, the perturbed magnetic field keeps growing, possibly beyond the average Galactic value, on scales much smaller than the Larmor radius of the particles dominating the current. At saturation, or close to it, the nonlinear evolution of the instability drives power on large scales (Bell 2004), and eventually on scales comparable to the Larmor radius of the particles in the amplified field. At this point, particles start scattering efficiently and rapidly isotropize. The synchrotron emission of these electrons or positrons in the amplified field is expected to produce the X-rays that we observe from a region of length ∼CR.

The proper motion of the pulsar is characterized by the pulsar velocity, Vpsr, and the (typically large) angle, θf, that it forms with the local GMF (which is assumed to coincide in direction with the filament elongation). This would shape the emission region as a wide stripe, unless the emitting particles suffer synchrotron losses on a timescale, τcross = wf/(Vpsr sin θf), with wf being the filament width.

Imposing τloss = τcross returns the strength of the magnetic field in the filament, which must be compared with the value inferred from the saturation of the NRI. We show that the picture we are proposing is able to describe in a coherent way the length, thickness, and luminosity of the observed filaments.

2. Physical model of the filament

The discussion that follows aims at a quantitative description of three observables: (i) the length of the feature, Lf, (ii) its transverse thickness, wf ≪ Lf, and (iii) the observed X-ray luminosity in a given energy band, LX1 − X2. To date, the filaments have only been reliably measured in a handful of sources: the Guitar Nebula, the Lighthouse, and J2030+4415. These will be our benchmark cases. As was discussed early on (Bandiera 2008), the small filament widths require that the magnetic field inside these structures be larger than the GMF (typically by a factor of ∼10) so that synchrotron losses are fast enough. At the same time, however, the radiating particles must reach distances from the pulsar much larger than the thickness of the filaments. This latter requirement suggests that the instability responsible for magnetic amplification is nonresonant. A resonant instability would in fact limit the particle motion to a few Larmor radii from the pulsar, due to effective resonant scattering, which is incompatible with the observed length. Here we show that the excitation of the current-driven NRI (Bell 2004) satisfies all these conditions and provides a suitable explanation of the observed phenomena.

While BSPWNe are expected to produce an equal number of electrons and positrons, it has been shown that particles with different charges escape the nebula along different paths (Olmi & Bucciantini 2019) at energies ≲mc2γMPD, where the Lorentz factor corresponding to the MPD is

(1)

m and e are the electron mass and charge, respectively, and Ė is the spin down luminosity of the pulsar. The current associated with the escaping particles can be expressed as >jp = e ϵĖ/(m c2 γesc A), where ϵ is the fraction of Ė carried by the escaping particles, γesc is the minimum energy of the escaping particle, and A = πR2 is the area at the base of the filament. The current can then be written as

(2)

The NRI excites perturbations that in the linear and early nonlinear phases grow on small scales, with the maximum growth occurring at

(3)

at a rate,

(4)

where is the Alfvén speed in the ISM with mass density ρISM and Ωc = eB0/(mc) is the electron cyclotron frequency in the unperturbed ambient field, B0.

The condition for these modes to be excited is kmaxRL > 1, with RL = mc2γesc/(eB0) the Larmor radius of the particles with γesc. The inequality can be rewritten as

(5)

where RL,  MPD = mc2γMPD/(eB0). Excitement of the NRI requires the energy density carried by the electric current to locally exceed that of the ambient magnetic field. This is reflected in a lower limit on the fraction of Ė that needs to be channelled into the tube: once ϵ > ϵlim, the NRI is excited. The saturation of the NRI has been the subject of much literature: as was already found in the seminal work on this instability (Bell 2004), the growth of the unstable modes, associated with scales much smaller than RL, is also accompanied by power deposition at larger scales, until there is power on scales comparable to the Larmor radius of the particles dominating the current calculated in the amplified field, ΔB. At that point, scattering becomes important and the current is disrupted. This condition reads as , where and , and it leads to

(6)

It is important to keep in mind that during the exponential growth of the instability the perturbations remain on scales much smaller than the resonant scale, so that scattering is inhibited and, in the first approximation, the motion of the particles can be pictured as quasi-ballistic. We see a posteriori that a rather small collimation angle, α, of the injected particles is required and the longitudinal speed of the particles in the quasi-ballistic phase is ∼c. This phase lasts for a few e-folds of the instability, ∼5 (Bell 2004), so that particles move along the magnetic field for a length,

(7)

where in the last equality we have introduced γ 7 = γesc/107, B3 = B0/3 μG, and n1 = ρISM/(1 mp cm−3), with mp the proton mass.

Within the assumed scenario, one can use Eqs. (6) and (7) to relate the strength of the amplified magnetic field to the measured length of the filament. The result is

(8)

which, as expected for the NRI, is independent of the initial value of the field, B0.

A crucial ingredient in building a physical interpretation of the filaments is the motion of the pulsar with velocity Vpsr. In the synchrotron loss time the pulsar has travelled a distance,

(9)

In our scenario, dsync(γX) is interpreted as the measured thickness of the filament, wf, and it allows one to derive γesc:

(10)

which turns out to depend only on directly observed quantities, except for the density. Lacking better constraints, the latter can be derived by measuring the standoff distance, d0, defined by the balance between the pulsar wind energy flux and the ISM ram pressure: . We can then express the Lorentz factor of the current-carrying particles as a function of observed quantities alone. This closes the system of equations.

Finally, we can make use of the observed X-ray luminosity to prove that the global picture presented above is meaningful. If the spectrum of the escaping particles reflects the power law behaviour that is typically observed in PWNe (Torres et al. 2014), one expects that if γesc ≥ γb ∼ 5 × 105, then the number of injected particles per unit energy and time interval can be written as Q(γ) = Q0(γ/γb)p, where p ≃ 2.2 − 2.5 is the high-energy particle injection index and Q0 is a normalization constant that can be obtained by assuming that a fraction, η, of the power emitted by the pulsar goes into particles with γ ≥ γb:

(11)

Similarly, the energy injected in the form of particles with γ ≥ γesc, namely the particles that escape the nebula and end up in the feature, is

(12)

where we used Eq. (11) for Q0. Under our assumptions, these particles will radiate all their energy in the feature, so that the X-ray luminosity in the energy range EX1 − EX2 can be written as

(13)

where γXi ≥ γesc is the Lorentz factor of a lepton that, in the amplified magnetic field, ΔB, emits synchrotron radiation in the observed energy interval. The assumption underlying Eq. (13) is that the life of the emitting particles is limited by synchrotron losses. From the previous equation we then express our last unknown parameter, ϵ, in terms of measured quantities alone:

(14)

This estimate assumes that the velocity distribution of particles is isotropic. As is shown in Appendix A, this is a reasonable approximation.

Once saturation of the NRI has been reached (after a time of ∼5τCR), the motion of the particles with Lorentz factor γesc is no longer ballistic, since scattering causes the particles to diffuse, with a diffusion coefficient that can be estimated as

(15)

For γ > γesc, namely for more energetic particles that we assume to be responsible for the X-ray emission, the diffusion occurs in the small-scale turbulence regime, in which D(γ) = D(γesc)(γ/γesc)2 (Subedi et al. 2017). One can easily check that for the X-ray emitting particles the timescale for diffusive escape from the filament of length Lf is much longer than the timescale for synchrotron losses:

(16)

where σT is the Thomson cross section and γX the Lorentz factor of the X-ray emitting particle. In the second equality, we used the relation between γ and peak synchrotron photon energy in the amplified field, and expressed the latter in keV as .

3. Testing the scenario

We could now test the proposed scenario on some filaments for which all of the quantities listed above were measured with sufficient accuracy. We used the measured parameters of the system, listed in Table 1, to derive γesc from Eq. (10). We could then use γesc in Eq. (8) and estimate the value of ΔB. This also set the ratio, ϵ/ϵlim, through Eq. (6) and, once ϵ was estimated from X-ray measurements through Eq. (14), we could finally constrain the size of the particle injection region, R, via Eq. (5). The results of this procedure are reported in Table 2.

Table 1.

Relevant observational data of our benchmark systems.

Table 2.

Summary of the results for the three filaments considered.

The Guitar Nebula and its filament represent a prototypical case: this was the first system discovered (Hui & Becker 2007), and to date it is the best characterized, thanks to multi-epoch monitoring (de Vries et al. 2022). The Guitar Nebula is produced by the radio pulsar PSR B2224+65. Its bow shock is only visible in Hα emission (Chatterjee & Cordes 2002), and it has a peculiar guitar-like shape: in the ∼80″-long tail one can distinguish a thin, elongated head, followed by a wider body made by at least two bubbles. This peculiar shape has been interpreted as the result of mass loading of neutral atoms into the bow shock from a dense ambient medium (Morlino et al. 2015; Olmi et al. 2018). As was expected, given its age, the pulsar has a rather low Ė, corresponding to a maximum Lorentz factor, γMPD ≃ 1.2 × 108. The filament is ∼0.6 pc long and shows a sharp leading edge (in the direction of motion of the pulsar) and a smoother trailing one, with hints of variation in the photon index (Johnson & Wang 2010) suggestive of synchrotron cooling (de Vries et al. 2022).

When we apply our model to the Guitar Nebula feature, we find that the amplified magnetic field, ΔB ≈ 80 μG, is produced by a current of particles, with γesc ≈ 0.1γMPD, carrying a negligibly small fraction of the pulsar spin-down energy, ϵ ≈ 6 × 10−3. The amplified field also ensures that the Larmor radius of X-ray emitting particles, which we find to be a factor of ∼3 more energetic than the current-carrying ones, is much smaller than the feature width, RL(γ X)/wf ≈ 3 × 10−3, which is consistent with our picture in which the latter is determined by synchrotron losses.

A final consistency check concerns the relative size of the injection region and the Larmor radius of escaping particles in the unperturbed magnetic field. If RL(γesc, B0)≫R, the particles will occupy a much larger region of space than assumed in Eq. (2) and the current density will decrease proportionally. In fact, RL(γesc, B0) = mc2γesc/(eB0)≫R, but based on numerical simulations of the particle escape from these systems (Olmi & Bucciantini 2019), the flux of particles leaving the nebula is well collimated, and limited to a surface area of . Consistently, we find that the necessary value of R is R ≈ d0/2 for the Guitar Nebula, implying a highly collimated initial particle flow, within an angle, α ∼ R/RL(γesc, B0)≈3°. This initial anisotropy is quickly destroyed as the instability grows and full isotropization is also achieved for the more energetic X-ray emitting particles on a timescale, tiso ≪ τsync (see Appendix B), ensuring the validity of the assumptions underlying our calculations. In Fig. 1 we show how γesc/γMPD and ϵ vary with nISM and Vpsr, both depending on the uncertain estimate of the source distance.

thumbnail Fig. 1.

Guitar Nebula: Color map of the ratio, γesc/γMPD (base-10 logarithm, left panel), and the efficiency, ϵ (right panel), as a function of the pulsar velocity, Vpsr (within the estimated uncertainty; de Vries et al. 2022), and the ambient number density, nISM. The black curve shows the relation we used to estimate nISM (pressure equilibrium at the bow shock standoff distance, d0). The black points indicate the position of the system for the best estimate of Vpsr (and nISM).

In light of the encouraging results obtained for the Guitar, we applied our model to the other two systems with well-surveyed X-ray filaments, the Lighthouse Nebula (Pavan et al. 2016; Klingler et al. 2023) and PSR J2030+4415 (de Vries & Romani 2022). The filament associated with the powerful Lighthouse Nebula is the only one also observed in hard X-rays (Klingler et al. 2023), thanks to a recent NuSTAR campaign. The feature remains clearly visible up to ∼25 keV, with a width and length compatible with those inferred from the higher-resolution Chandra data. The measurement of wf along the feature is complicated by the observed striped morphology. Moreover, there is a large uncertainty on the pulsar speed, inferred from the association with a young (10–30 kyr) supernova remnant in the vicinity (García et al. 2012). The Lighthouse feature is the longest one currently known and the only one showing a bent morphology. In our model, this bending is to be understood as a result of the GMF structure, being the length of the feature not much smaller than the presumed GMF correlation length of ∼ tens pc. Focusing on the Chandra band, we find our model to be consistent with observations of the Lighthouse only if Vpsr ≤ 1300 km s−1, corresponding to ΔB ≈ 26 μG and γesc ≈ 8 × 107(≈γX1). A higher pulsar proper motion implies a higher magnetic field ( from Eqs. (8)–(10)), and hence a lower γ X (). At the same time, our method for estimating the density based on the standoff distance implies a number density of the ISM, , and (Eq. (10)), so that, as Vpsr increases, our assumption that γ X ≥ γesc is soon violated. If, rather than estimating nISM as described above, we assume that (Pavan et al. 2014) nISM = 0.1 cm−3, then and our model will work for1 Vpsr ≤ 1600 km s−1. The range of validity of our model and the resulting value of γesc/γMPD is shown in Fig. 2.

thumbnail Fig. 2.

Left panel: Color map of the ratio γesc/γMPD (base-10 logarithm) for the Lighthouse Nebula, as a function of the pulsar velocity, Vpsr, and ambient number density, nISM. The solid black curve shows the relation we used to estimate nISM (pressure equilibrium at the bow shock standoff distance, d0). The black point indicates the position of the system based on the best estimate (Pavan et al. 2014) of nISM: nISM = 0.1 cm−3, Vpsr = 1300 km s−1. The vertical lines mark the maximum value of Vpsr that makes our model viable for nISM = 0.1 cm−3 (Vpsr, max = 1300 km s−1, dashed line) and nISM determined from Vpsr and d0 (Vpsr, max = 1600 km s−1, dash-dotted line). Right panel: Color map of ΔB as a function of Vpsr and wf (uncertainties on the source distance reflect on both quantities). The best values for the Guitar and the Lighthouse Nebula are represented as magenta and yellow circles, respectively. The upper and lower limits obtained for J2030+4415 are shown as blue downward and upward arrows, respectively. The horizontal blue line shows the current filament thickness.

The very thin feature in J2030+4415 is powered by a γ-ray pulsar, with a spin-down luminosity corresponding to γMPD = 5 × 108. Multi-epoch Hα observations of the bow shock (de Vries & Romani 2022) clearly indicate an important rearrangement of the apex around tevent ≃ 32 yr ago, possibly due to a sudden variation in the ambient medium density, causing a compression of d0. If the observed wf were determined by the synchrotron lifetime of the particles then, based on our model, we would estimate ΔB ≈ 120 μG and γesc ≈ 108, which would imply that τsync ∼ 60 yr > tevent. However, it is difficult to believe that the feature was formed before tevent, so in this case we think that we are seeing all the particles that have been injected during tevent and wf = Vpsr sin θftevent. In this scenario, we can only estimate boundaries for ΔB: the condition tevent < tsync implies that ΔB < 181 μG, while the condition that X-ray emission must come from particles with γX < γMPD implies that ΔB > 40 μG. These boundaries, reported in Table 2, clearly show that the field must also be amplified by at least a factor of ∼10 in this case. The right panel of Fig. 2 shows the relation between the amplified magnetic field, the pulsar speed, and the filament thickness of all three sources.

4. Discussion and conclusions

We claim that the filaments of non-thermal X-ray emission emerging from selected BSPWNe may be the first clear indication of the excitation of NRI due to a pencil beam current of charge-separated electrons (or positrons) leaving the parent nebulae. In two out of three cases, the length, thickness, and X-ray luminosity of the filaments can be well accounted for. In the third case, due to the complex history of the source, it is only possible to derive boundaries on the magnetic field, which however imply efficient amplification, possibly explained by particles leaving the source at an energy close to the pulsar potential drop.

The amplified field should not however hinder the ballistic motion of the leptons, at least in the beginning, so as to allow the particles to populate the whole length of the filament. This is exactly what is expected to happen when the beam excites the NRI (Bell 2004). In order for the mechanism to work, particles’ injection into the ISM needs to be collimated within a narrow range of pitch angles. The filaments are predicted to follow the structure of the large scale GMF at the location of the BSPWN.

The interpretation of the filaments in terms of excitation of the NRI by charge-separated electrons or positrons released by BSPWNe into the ISM is rich in implications. The amplified turbulence level can lead to extended confinement of the particles in selected regions around the sources, with implications for the pulsar’s contribution to cosmic ray (CR) leptons (Schroer et al. 2023). In addition, it is tempting to speculate on a relation between the processes discussed here and the recently discovered phenomenon of TeV pulsar halos, regions of very high-energy (≳TeV) gamma-ray emission extending for tens of pc around a few pulsars (Abeysekara et al. 2017). In the halos, too, the particle transport seems to be suppressed by two to three orders of magnitude (López-Coto et al. 2022) compared with the Galactic values, as has been inferred from CR secondary and primary ratios (Schroer et al. 2023; Evoli et al. 2020), and the most obvious source of turbulence to explain the suppression seems to be the escaping particles themselves.

The recently discovered radio filaments around some bow shock nebulae (see e.g., Khabibullin et al. 2024) or the radio filaments observed in the Galactic center region (Yusef-Zadeh et al. 1984; Morris & Yusef-Zadeh 1985; Yusef-Zadeh & Morris 1987; Goedhart et al. 2023) may appear to be reminiscent of a similar process. However, the observed radio emission requires particles of much lower energy than the X-ray one (γ ∼ 104 − 105), and no need to amplify the ambient magnetic field. Moreover, those particles can more efficiently escape the bow shock from the tail rather than the head (due to their smaller Larmor radii, Olmi & Bucciantini 2019), and then illuminate the preexisting structures of the magnetic field (Barkov & Lyutikov 2019). Hence, we argue that these phenomena are most likely of a different origin than the filaments around BSPWNe.


1

Notice however that values of Vpsr > 1600 km s−1 are beyond 5σ of the peak velocity of the pulsar distribution (Faucher-Giguère & Kaspi 2006) and are only suggested based on the possible association of PSR J1101–6101 with SNR G290.0–0.8.

Acknowledgments

This work has been partially funded by the European Union – Next Generation EU, through PRIN-MUR 2022TJW4EJ. B. Olmi, E. Amato and R. Bandiera also acknowledge support from the Italian National Institute for Astrophysics with PRIN-INAF 2019.

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Appendix A: Initial collimation of the particle beam

In the main text, we have shown that a good collimation – that is, small pitch angles – is one of the requirements for the formation of the observed filaments. A good collimation is also a natural outcome of the fact that the particles move from a region with a high magnetic field (the bow shock head),

(A.1)

to a typical ISM magnetic field, around 3 μG in magnitude. It is well known that, for particles with a small Larmor radius compared to the scale length of B variation, the enclosed magnetic flux is an adiabatic invariant and leads to the constancy of Bsin2α. Let us assume that, originally, the pitch angles of the escaping particles uniformly cover a 2π solid angle, equivalent to a mean value of sin α = π/4. This would imply, for the reference values given above, an average sin α ≃ 0.07. For the considered cases, the “small Larmor radius” approximation is valid, being the ratio of the gyro radius of the particles with energy γesc in the magnetic field of the bow shock head, 0.1 − 0.3 d0 (Bhead/100 μG)−1, where the bow shock standoff distance, d0, gives the scale of the variation in the field from the inside to the outside.

Appendix B: Delayed isotropization of X-ray emitting particles in the feature

Once the instability is saturated, the particles with energy, γesc, start to diffuse in the amplified magnetic field. One might wonder how much later this will also happen to the particles responsible for the emission at the X-rays, due to the fact that they have a larger Lorentz factor.

During its motion in the perturbed magnetic field, ΔB, a particle with Lorentz factor γ X > γesc will be deflected by an angle, , on the interaction scale and where is the Larmor radius of the particle in the amplified magnetic field. In a distance, L, the number of interactions (or deviations) that the particles experience is L/l. Then the total deflection can be estimated as . The isotropization length (Liso), or the isotropization timescale (tiso ≃ Liso/c) is obtained by requiring that Δθ = 1, so that

(B.1)

With the typical lifetime of X-ray emitting particles in the amplified field being much longer than tiso – namely, τsynch ≃ 220 yr (ΔB/50 μG)−3/2 (Eph, keV/0.5 keV)−1/2 – the distribution of particles can be safely assumed to be isotropic for the sake of computing the observed X-ray emission.

All Tables

Table 1.

Relevant observational data of our benchmark systems.

Table 2.

Summary of the results for the three filaments considered.

All Figures

thumbnail Fig. 1.

Guitar Nebula: Color map of the ratio, γesc/γMPD (base-10 logarithm, left panel), and the efficiency, ϵ (right panel), as a function of the pulsar velocity, Vpsr (within the estimated uncertainty; de Vries et al. 2022), and the ambient number density, nISM. The black curve shows the relation we used to estimate nISM (pressure equilibrium at the bow shock standoff distance, d0). The black points indicate the position of the system for the best estimate of Vpsr (and nISM).

In the text
thumbnail Fig. 2.

Left panel: Color map of the ratio γesc/γMPD (base-10 logarithm) for the Lighthouse Nebula, as a function of the pulsar velocity, Vpsr, and ambient number density, nISM. The solid black curve shows the relation we used to estimate nISM (pressure equilibrium at the bow shock standoff distance, d0). The black point indicates the position of the system based on the best estimate (Pavan et al. 2014) of nISM: nISM = 0.1 cm−3, Vpsr = 1300 km s−1. The vertical lines mark the maximum value of Vpsr that makes our model viable for nISM = 0.1 cm−3 (Vpsr, max = 1300 km s−1, dashed line) and nISM determined from Vpsr and d0 (Vpsr, max = 1600 km s−1, dash-dotted line). Right panel: Color map of ΔB as a function of Vpsr and wf (uncertainties on the source distance reflect on both quantities). The best values for the Guitar and the Lighthouse Nebula are represented as magenta and yellow circles, respectively. The upper and lower limits obtained for J2030+4415 are shown as blue downward and upward arrows, respectively. The horizontal blue line shows the current filament thickness.

In the text

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