As discussed before, a fraction of PSRs is bound to emerge from the progenitor SNR before our fiducial final time tend = 105yr, due to the high average kick velocity that characterizes the pulsar population. The typical escape time can be estimated by matching the PSR displacement due to its kick velocity (Vpsr) with the size of the SNR in the Sedov–Taylor phase:
(14)$$\begin{eqnarray}
t_{\rm esc} \simeq 725\,\, \mbox{kyr} \left[\left(\frac{E_{\rm sn}}{10^{51} \, \mbox{erg}}\right)\! \left(\frac{\rho _0}{1\, \mbox{part/cm^{3}}}\right)^{-1}\! \left(\frac{V_{\rm psr}}{100\, \mbox{km/s}}\right)^{-5} \right]^{\, 1/3}\!\!\! . \\
\end{eqnarray}$$Considering the mean (median) value of the PSR velocity distribution of 380 (330) km s−1 and of the ISM number density of 0.7 (0.25) particles cm−3, we obtain a mean (median) escape time of tesc ≃ 88 (160) kyr. Even taking into account that transition of SNRs to the radiative phase is expected at 35 (60) kyr, the escape time only slightly reduces to tesc ≃ 77 (120) kyr. Since tend = 100 kyr, only a fraction of the sources will then escape the SNR by the end of the simulation. For those systems with tesc tend, the runaway PSR will give rise to the formation of a bow shock nebula. These nebulae, whose first examples were detected in Hα (Chevalier, Kirshner & Raymond 1980; Kulkarni & Hester 1988; Cordes, Romani & Lundgren 1993; Bell et al. 1995; van Kerkwijk & Kulkarni 2001; Jones, Stappers & Gaensler 2002; Brownsberger & Romani 2014; Dolch et al. 2016; Romani, Slane & Green 2017), more recently have been discovered and observed in X-rays and sometimes in radio (Gaensler et al. 2002, 2004; Arzoumanian et al. 2004; Chatterjee et al. 2005; Yusef-Zadeh & Gaensler 2005; Hui & Becker 2007; Hui & Becker 2008; Kargaltsev & Pavlov 2008; Misanovic, Pavlov & Garmire 2008; de Rosa et al. 2009; Ng et al. 2010; De Luca et al. 2011; Ng et al. 2012; Marelli et al. 2013; Jakobsen et al. 2014; Auchettl et al. 2015; Klingler et al. 2016; Kargaltsev et al. 2017; Posselt et al. 2017; Kim et al. 2020). They are characterized by a cometary shape, with a tiny head typically of the order of 1016 cm, whose size is set by ram pressure balance between the PSR wind and the incoming (in the PSR frame) ISM, followed by a long tail opposite to the PSR motion, which can extend for very long distances up to a few pc. Given their limited spatial extension and low residual luminosity (Kargaltsev et al. 2017), bow shock nebulae will not probably be statistically relevant in γ-rays and so far have not been detected (Abdalla et al. 2018c). Runaway PSRs, however, have recently been associated to extended TeV haloes (Abeysekara 2017; Sudoh et al. 2019), most likely due to escaping pairs (Bykov et al. 2017; Evoli, Linden & Morlino 2018; Olmi & Bucciantini 2019c; Di Mauro, Manconi & Donato 2020; Evoli et al. 2021). However, the formation and properties of γ-ray haloes are still poorly understood, and different interpretations lead to very different expectations in terms of the possible detection of these sources in the next future (Sudoh et al. 2019; Giacinti et al. 2020). The modelling of these complex sources is outside the scopes of the present work, so we simply keep trace of the position of escaped PSRs for possible future implementations. The fraction of PWNe escaped from their parent SNR at the end of the simulation is represented in the right-hand panel of Fig. 1, where evolved PWNe are shown on top of the initial distribution of PWNe + SNRs.
Citation: Gaensler et al. 2002
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