an almost *unique* solution to the pre-SN progenitor binary of PSR J0737−3039 (Figure 25), an argument that was first made by Piran & Shaviv (2005). This can be understood from the observations of the small proper motion of this system, which results in ($w/v_{rel}$) ≪ 1, and thus a firm solution is obtained from the observed values of $P_{\text{orb}}$ and $e$; see Equations (19) and (20). Similarly, it automatically follows that the misalignment angle in this case must be small (Equation (16)), in agreement with observations. For 94% of our solutions, the pre-SN binary had an orbital period of $P_{\text{orb,i}} = 0.085 \pm 0.005$ days, and the mass of the (ultra-stripped) exploding star must have been $M_{\text{He,f}} = 1.56 \pm 0.06 M_{\odot}$. Interestingly enough, the very first calculation of a helium star–NS binary system leading to an ultra-stripped SN (Tauris et al. 2013) had pre-SN values of $P_{\text{orb,i}} = 0.070$ days and $M_{\text{He,f}} = 1.50 M_{\odot}$, and is thus basically a *solution* to the immediate progenitor of PSR J0737−3039. For this particular system, Tauris et al. (2013) found that the nature of the SN would be an Fe CCSN of Type Ic. From their binary stellar evolution modeling, we can therefore also deduce that the pre-SN system is the descendant of a helium star–NS binary with an initial (post-CE) orbital period of ~0.1 days and a helium star ZAMS mass of ~2.9 $M_{\odot}$.

In the discussion above, we have neglected any influence from gravitational damping on this system. The fact that we also observe the young pulsar in this system means that we can set an upper limit to the age of this binary. The characteristic spin-down age of pulsar B (the young NS) is about 50 Myr, which is also a typical radio pulsar lifetime for a non-recycled pulsar (Lorimer & Kramer 2004). Unless the braking index of this pulsar is significantly less than 3, this will serve as a proxy for the upper limit of its true age. However, in this particular binary, the magnetosphere of pulsar B, and thus its spin-down activity, is likely to be affected by the relativistic wind from pulsar A (the recycled pulsar, releasing spin-down energy at a rate of 3600 times more than pulsar B), as also indicated by observations (Lyutikov 2004; McLaughlin et al. 2004; Breton et al. 2012). Hence, the true age of pulsar B could possibly be more than 100 Myr (Lorimer et al. 2007).

For a DNS age of 100 Myr, we calculate corresponding post-SN parameters of $P_{\text{orb}} = 0.138$ days and $e = 0.119$ (see location of red star), and the solution would be that the pre-SN binary had an orbital period of $P_{\text{orb,i}} = 0.11 \pm 0.01$ days and that the mass of the exploding star must have been $M_{\text{He,f}} = 1.59 \pm 0.09 M_{\odot}$. The distribution of valid kick velocities remains roughly similar to the one shown in Figure 25.

Since the exact age of PSR J0737−3039 remains unknown and could in principle take any value between 0 and 100 Myr (or even more), we conclude that this DNS system formed from a pre-SN binary with $P_{\text{orb,i}} = 0.10 \pm 0.02$ days and $M_{\text{He,f}} = 1.59 \pm 0.09 M_{\odot}$, as well as a small imparted kick, $w \lesssim 70$ km s⁻¹.

A final note on this double pulsar system is that it is often being used to constrain the empirical Galactic DNS merger rate. The outcome is dependent on the beaming geometry of both pulsars A and B. As an example, Kim et al. (2015) derived a value of $21_{-14}^{+28}$ Myr⁻¹.

### 8.3. PSR J1518+4904

Since the discovery of this DNS system (Nice et al. 1996), its component masses have remained difficult to constrain (Janssen et al. 2008). The total mass of the system ($M = 2.72 M_{\odot}$) remains a firm result, and we use that for our SN simulations of this system. Assuming that the mass of the

second-born NS $M_{\text{NS,2}}$ falls within the range of such measured masses in other DNS systems (1.17–1.39 $M_{\odot}$, Table 1), and using MC simulations with a flat probability distribution for $M_{\text{NS,2}}$ in this range, the mass of the first-formed NS is then simply evaluated from $M_{\text{NS,1}} = M - M_{\text{NS,2}}$. The results of our SN simulations are shown in Figure 27. The solutions yield values of the pre-SN orbital period between 4 and 12 days. The solutions for the mass of the exploding star in the second SN is restricted to <2.8 $M_{\odot}$, with a preference toward smaller values. Similarly, we only find solutions for a relatively small kick velocity ($w < 100$ km s⁻¹) with a peak<fnref n="21" /> near $w \simeq 40$ km s⁻¹.

Ongoing intensive data analysis, including data obtained in recent observation campaigns, yields preliminary values of $M_{\text{NS,1}} = 1.41 M_{\odot}$ and $M_{\text{NS,2}} = 1.31 M_{\odot}$ for the (mildly) recycled radio pulsar and its young NS companion, respectively (G. Janssen 2017, private communication). The reported error bars are still rather large, roughly ±0.08 $M_{\odot}$. Redoing our MC simulations with these mass constraints does not introduce any significant changes to our results presented in Figure 27. Any corrections for gravitational damping in this system, during the time interval between its formation and now, are negligible due to its wide orbit ($P_{\text{orb}} = 8.63$ days).

### 8.4. PSR B1534+12

PSR B1534+12 was discovered by Wolszczan (1991) and studied further in detail by e.g., Stairs et al. (2002) and Fonseca et al. (2014). The system parameters are well-constrained (Table 2), not only because of its precisely measured masses, but also via its determined misalignment angle, $\delta = 27 \pm 3^\circ$, and its proper motion. The results from our SN simulations are shown in Figure 28. The mass of the exploding star is constrained to be $M_{\text{He,f}} < 3.7 M_{\odot}$. More noteworthy, the kick in the second SN is shown to have been large: $w \simeq 175–300$ km s⁻¹. This makes this binary special in the sense that it represents one out of only two DNS systems (besides PSR B1913+16) for which there is evidence for a large kick in an ultra-stripped SN.

The reason that a large kick is required in this system is the relatively large measured value of $\delta$. Without any constraints on this angle, solutions are found for a wide range of kick values, $50 \lesssim w \lesssim 425$ km s⁻¹, illustrating again the importance of measuring $\delta$. The small error bar on the measurement of $\delta$ explains why the direction of this kick is seen to be confined to a relatively small area on the sphere surrounding the exploding star (lower-right panel). Corrections for gravitational damping in this system (since its formation) is rather limited due to the relatively small characteristic age of the recycled pulsar ($\tau = 248$ Myr) and somewhat small eccentricity ($e = 0.274$); see the red solid star in the upper-left panel.

### 8.5. PSR J1753−2240

There are no mass constraints from observations of this DNS system (Keith et al. 2009). Assuming that the total mass of the system is within the interval of the total masses of all other known DNS systems ($M = 2.57–2.88 M_{\odot}$), and similarly that the mass of the second-born NS, $M_{\text{NS,2}}$, is also within the range (1.17–1.39 $M_{\odot}$) measured in other DNS systems, we simulated the possible solutions for the pre-SN parameters and the NS

<footnote n="21">²¹ We repeat that this peak is based on flat input distributions of various pre-SN parameters, neglecting previous binary stellar interactions. A full study requires population synthesis.</footnote>
kick. The results are shown in Figure 29. As can be seen, we cannot derive interesting constraints for the origin of this DNS system. The kick is likely to have been small ($<$100 km s⁻¹), but we also find solutions for kicks up to $w = 400$ km s⁻¹. The wide orbit of this system excludes any gravitational damping of significance.

## 8.6. PSR J1755−2550

This interesting DNS candidate was discovered by Ng et al. (2015). The nature of the system is still unclear. The spin period of the radio pulsar is 315 ms, which makes it quite likely to be a young pulsar. The young nature of this pulsar was recently confirmed by a measurement of $\dot{P} = 2.43 \times 10^{-15}$ s s⁻¹ (C. Ng et al. 2017, in preparation). Indeed, if the NS had been even mildly recycled, we would have expected a smaller spin period, given its orbital period of 9.7 days (see Section 5.1). The compact object composition of the companion star remains unknown. It could be either a DNS system or a WD+NS system where the NS formed after the WD (Tauris & Sennels 2000). From the mass function, it is known that $M_2 > 0.40 M_\odot$, and given the unknown orbital inclination angle, the probability of a WD+NS binary is somewhat larger than that of a DNS system; see discussions in C. Ng et al. (2017, in preparation). On the other hand, the eccentricity of this system ($e = 0.089$) is similar to many DNS systems. The wide orbit of the system excludes any gravitational damping of significance, so this cannot be the explanation for its small eccentricity. Assuming that the system is indeed a DNS system, we show in Figure 30 our results from the simulations of the second SN. As in the case of PSR J1753−2240 discussed above, we have no constraints on the total mass nor on the value of $M_{NS,2}$, and we follow the same procedure as for PSR J1753−2240. The kick is likely to have been small ($<$50 km s⁻¹), but we also find solutions for kicks up to almost $w = 400$ km s⁻¹.

## 8.7. PSR J1756−2251

PSR J1756−2251 (Faulkner et al. 2005) is very well constrained from precise measurements of its NS masses ($M_{NS,1} = 1.341 M_\odot$, $M_{NS,2} = 1.230 M_\odot$) and the misalignment angle ($\delta < 34^\circ$) of the recycled pulsar (Ferdman et al. 2014). The proper motion measurement in declination, however, has been difficult to measure for this pulsar, since it is located near the ecliptic plane. Ferdman et al. (2014) derived a best estimate of $\mu_\delta = 5.5 \pm 7.0$ mas yr⁻¹, which we combine with its proper motion in right ascension, $\mu_\alpha = -2.42 \pm 0.08$ mas yr⁻¹, to estimate its transverse velocity and a determination of its 3D systemic velocity of $v^{LSR} = 41$ km s⁻¹ ($v_{1\sigma}^{LSR} = 21–71$ km s⁻¹; see Table 3).

This DNS system shares many properties with the double pulsar (PSR J0737−3039) and can be seen as a wider-orbit version of it. The SN simulations reveal a similar solution for the origin of the second-born NS as an ultra-stripped exploding star with a mass of $1.72 \pm 0.22 M_\odot$. The kick velocity is found to be $w < 130$ km s⁻¹, with a main peak in solutions toward $w \to 0$. It is interesting to note that this DNS system is the only one for which half of the solutions require a kick in a forward direction, i.e., $\theta < 90^\circ$. This is partly due to the relatively small 3D systemic velocity inferred for this system.

Using the NE2001 electron distribution model yields a DM distance of $d = 0.73$ kpc, whereas applying the model of Yao et al. (2016) results in a distance four times larger ($d = 2.81$ kpc).

Such a large distance would significantly increase our derived interval of possible kick velocities for this system. However, a firm upper value for the distance of 1.2 kpc can be derived using the measured orbital period decay (Ferdman et al. 2014). This example illustrates again the need for precise distance determinations via parallax measurements.

The orbital period of this system of $P_{orb} = 0.320$ days, and its eccentricity of $e = 0.181$, means that some gravitational damping may have been at work since its formation. Assuming an age of $t = \tau = 443$ Myr yields initial values of $P_{orb} = 0.353$ days and $e = 0.198$ right after the second SN (see the red solid star in the upper-left panel in Figure 31). However, this potential slight change in initial values does not impose any effects on the constraints of the pre-SN parameters and the kick velocity derived here.

## 8.8. PSR J1811−1736

PSR J1811−1736 (Corongiu et al. 2007) is a wide-orbit, highly eccentric DNS system with $P_{orb} = 18.8$ days and $e = 0.828$. Its total mass is $2.57 M_\odot$, whereas its individual NS mass components remain unknown. Assuming that the mass of the second-born NS, $M_{NS,2}$, is within the range of observed masses in other DNS systems (1.17–1.39 $M_\odot$), we simulated the possible solutions for the pre-SN parameters and the NS kick. The results are shown in Figure 32. As can be seen, there is a large range of possible values for all pre-SN parameters and the kick, and we conclude that we cannot derive interesting constraints for the origin of this DNS system. The wide orbit of PSR J1811−1736 excludes any gravitational damping of significance.

## 8.9. PSR J1829+2456

This DNS system was discovered by Champion et al. (2004). The total mass of the system is $2.59 M_\odot$ and the second-born NS has a mass $M_{NS,2} > 1.22 M_\odot$. Assuming as usual $M_{NS,2} \le 1.39 M_\odot$, our SN simulations yield the results shown in Figure 33. Also in this case, there is a large range of possible values for all pre-SN parameters and the kick, and thus we cannot derive interesting constraints for the origin of this DNS system. The relatively wide orbit ($P_{orb} = 1.18$ days), and the small eccentricity ($e = 0.139$), of the system means that its orbital parameters have hardly changed since its formation.

## 8.10. PSR J1906+0746

This is a remarkable binary since the observed radio pulsar is the young, second-born NS (Lorimer et al. 2006). The evidence is a combination of a large B-field (i.e., $\dot{P} = 2.03 \times 10^{-14}$ s s⁻¹) and a relatively slow spin period ($P = 144$ ms); see Table 2 and Figure 2. These values are typical for a young, non-recycled NS. The NS component masses are measured to be $M_{NS,1} = 1.322 M_\odot$ and $M_{NS,2} = 1.291 M_\odot$, respectively (van Leeuwen et al. 2015). Since the companion star remains unseen, this system could, in principle, also be a WD+NS system. ONeMg WD masses of $1.322 M_\odot$ are certainly possible. We show our simulations of the (second) SN in Figure 34. However, we cannot make any interesting constraints on the pre-SN parameters of the system. A remarkable result though is that this binary could, in principle, have survived a NS kick velocity of about $w = 1500$ km s⁻¹. A proper motion measurement would be able to constrain the kick velocity. Since we treat the first-formed compact object as a point mass in
these simulations, the results are equally valid for a WD+NS system where the NS forms after the WD (Tauris & Sennels 2000).

The close orbit of this system ($P_{\text{orb}}$ = 0.166 days) implies that gravitational damping since birth could be of significance. However, this NS must be quite young given its high value of $\dot{P} = 2.03 \times 10^{-14}$ s s$^{-1}$ ($\tau$ = 114 kyr), and therefore its present orbital parameters resembles very well its birth parameters. O'Shaughnessy & Kim (2010) argued that this DNS system makes quite a significant contribution to the empirical estimate of the Galactic DNS merger rate.

## 8.11. PSR J1913+1102

Lazarus et al. (2016) recently announced the discovery of this DNS system with $P_{\text{orb}}$ = 0.206 days and $e$ = 0.090. The total mass of the system is reported to be $M = 2.88 M_{\odot}$. The NS masses are expected to be measured soon, but until then we assume that the mass of the second-born NS, $M_{\text{NS,2}}$, is within the usual range of 1.17–1.39 $M_{\odot}$, as measured in other DNS systems. In Figure 35, we show our simulated possible solutions for the pre-SN parameters and the NS kick. As can be seen, we cannot derive interesting constraints for the origin of this DNS system. The orbit of the binary could have been affected significantly by gravitational damping since its formation. The main reason for this is the large characteristic age of the system ($\tau$ = 2690 Myr), which potentially allows for a large orbital decay on a gigayear timescale. In the upper-left panel, we have plotted its birth location with a red solid star ($P_{\text{orb}}$ = 0.438 days and $e$ = 0.189) for an assumed age $t = \tau$. These initial values are almost independent of the exact NS masses (here we simply assumed $M_{\text{NS,1}} = 1.58 M_{\odot}$ and $M_{\text{NS,2}} = 1.30 M_{\odot}$, motivated by typical values of $M_{\text{NS,2}}$; see Table 1).

## 8.12. PSR B1913+16

The Hulse–Taylor pulsar (Hulse & Taylor 1975; Kramer 1998; Weisberg et al. 2010) is an unusual case since this DNS system must have been accompanied by a rather large kick during the second SN explosion (e.g., Cordes & Wasserman 1984; Burrows & Woosley 1986; Wex et al. 2000; Willems et al. 2004). We confirm this result from our analysis of its 3D systemic velocity (Section 2.2) and the SN simulations presented in Figure 36. We find kick solutions for the entire interval of $w = 185–465$ km s$^{-1}$. This result (as well as that for the other DNS system with a large kick, PSR B1534+12) is in excellent agreement with the findings of Wong et al. (2010). For the mass of the exploding star, however, we cannot constrain its mass to better than $M_{\text{He,f}} < 4.4 M_{\odot}$. Indeed, the possibility of a somewhat more massive exploding star, compared to some of the DNS systems that experienced small kicks (e.g., PSR J0737−3039 and PSR J1756−2251), may be a clue to understanding the large kick in this system (see discussions in Section 6.6).

A caveat to this conclusion is that the distance to PSR B1913+16 is uncertain. The Yao et al. (2016) DM distance of 5.25 kpc is only half the distance of 9.8 kpc provided by Weisberg et al. (2008) and Weisberg & Huang (2016), based on the Galactic model of Reid et al. (2014). Adapting this smaller distance results in typical kick velocities that are about 50–100 km s$^{-1}$ smaller than those quoted above, although with a minimum kick velocity remaining at a relatively large value of $w = 165$ km s$^{-1}$. Perhaps more interesting and important,

the mass of the exploding star can in this case be constrained to be $M_{\text{He,f}} < 2.6 M_{\odot}$.

The torus of possible kick angles in the $(\phi, \theta)$ plane is a result of the constraint from the measured misalignment angle, $\delta = 18 \pm 6^\circ$ (Ferdman et al. 2013). The combination of a small orbital period of $P_{\text{orb}}$ = 0.323 days and a large eccentricity of $e$ = 0.617 means that gravitational damping is significant in this system, as discussed in Section 7. We illustrate this by marking the birth location of PSR B1913+16 in the $(P_{\text{orb}}, e)$ plane (upper-left panel in Figure 36) with a red solid star for a DNS age of $t$ = 108.7 Myr. The SN simulations leading to these birth properties are shown in Figure 37. The differences in the solutions for the pre-SN parameters and the second NS kick are very limited compared to the solutions shown in Figure 36. For an age of $t$ = 108.7 Myr, we do find solutions for a slightly more massive exploding star (up to $4.8 M_{\odot}$). However, the most likely kick remains at roughly 200–450 km s$^{-1}$.

## 8.13. PSR J1930−1852

This important very wide-orbit DNS system was discovered by Swiggum et al. (2015). The binary has $P_{\text{orb}}$ = 45 days and a marginally recycled pulsar spinning at $P$ = 185 ms. The total mass of this system is 2.59 $M_{\odot}$ and the second-born NS has a mass $M_{\text{NS,2}} > 1.30 M_{\odot}$. Assuming as usual $M_{\text{NS,2}} \le 1.39 M_{\odot}$, our SN simulations yield the results shown in Figure 38. Also in this case, there is a large range of possible values for all pre-SN parameters and the kick, and thus we cannot derive interesting constraints for the origin of the DNS system. The kick was most likely rather small ($w < 100$ km s$^{-1}$), although we find a small tail of possible solutions up to 290 km s$^{-1}$. Gravitational damping is not relevant for this system. Assuming an age of 50 Myr or 100 Myr reveals an initial spin period of the marginally recycled pulsar of 154 ms or 115 ms, respectively, assuming a spin evolution with a braking index of $n = 3$ (see Equation (29)). The orbital parameters and the spin of this marginally recycled pulsar are discussed further in Tauris et al. (2015).

# 9. Ramifications from SN Simulations

## 9.1. Resulting NS and Pre-SN Properties

The extreme stripping of helium stars during Case BB RLO, following the CE phase, leads to relatively low masses of the second exploding star in DNS systems—typically of values $M_{\text{He,f}} \le 2.5 M_{\odot}$ for a broad range of initial values of orbital periods and helium star masses (Tauris et al. 2015). The solutions for the second SNe presented in Figures 24–38 are based purely on kinematics. Many of the solutions with massive exploding stars (up to $\sim 7 M_{\odot}$) are therefore not physical, since binary stellar evolution would, in most cases, prevent such massive exploding stars. Only in a few cases is it possible that very massive helium stars (WR stars) may produce exploding stars with such large masses if their stellar wind mass-loss rate is small (e.g., Yoon et al. 2010 and references therein). To illustrate the strong dependence on stellar wind mass-loss rate, Tauris et al. (2015) calculated that the outcome of a binary system with a NS and a 10 $M_{\odot}$ helium star could be an exploding star with any mass between 3–7 $M_{\odot}$. However, if such a binary system is in a close orbit, GW radiation will force the system into RLO while the helium star
is still massive, causing the system to merge upon RLO. Alternatively, for a wide-orbit system, the binary may expand significantly as a result of the strong wind such that the NS escapes being recycled via Case BB RLO.

Since some of the quantities plotted in Figures 24–38 are correlated with the mass of the exploding star (e.g., the magnitude and required direction of the kick velocities leading to a given DNS solution), care must be taken when drawing conclusions from these SN simulations. Full population synthesis is needed to disentangle cross-dependent quantities and enable more trustworthy probability distributions of various parameters at play.

## 9.2. Direction of the NS Kick

In Figure 23, we have plotted the directions of all kick velocities leading to a solution for the simulations of the second SN of the DNS systems shown in Figures 24–38. We restricted the number of solutions to 3000 for each system and stacked together the ($\phi, \theta$) solution planes on top of each other. The darker the blue-scale color (log scale), the more solutions in a given direction. A remarkable apparent preference is seen for a kick direction out of the plane of the pre-SN binary. The pre-SN orbital plane is along $\phi = 0^{\circ}$ and $\phi = \pm 180^{\circ}$, and the kick solutions clearly cluster in directions out of the plane ($\phi \approx 90^{\circ}$ and $\phi \approx -90^{\circ}$). Furthermore, the kicks leading to the solutions show a significant preference for a direction backwards ($\theta > 90^{\circ}$). This latter result is perhaps not surprising since the probability of the system remaining bound is larger for a backward kick; see Section 6. For a verification of an isotropic input (trial) distribution of kick velocity directions, see Figure 39 in the Appendix.

Given the above discussion on kinematic solutions versus realistic (from a binary stellar evolution point of view, Section 9.1) masses of the exploding stars ($M_{He,f}$), we performed an additional simulation of kick distributions in the observed DNS systems, in which we only kept solutions with $M_{He,f} < 2.5 M_{\odot}$. The outcome of this experiment remains fairly identical to the result shown in Figure 23. The only minor difference is that the peak of the kick directions is located at a slightly smaller value of $\theta \approx 125^{\circ}$, and that more solutions allow for kicks in a forward direction ($\theta < 90^{\circ}$). The latter result is expected for smaller masses of the exploding stars.

Further motivated by the puzzle of an apparent NS kick anisotropy, we performed additional investigations and found that this anisotropy can be understood from a combination of an input prior and a selection effect.

### 9.2.1. Kick Anisotropies and Input Priors

First, we discuss how our application of unconstrained input distributions of kick magnitudes (i.e., without upper limits) can affect the solutions of kick angles for those DNS systems in which there are no observational constraints on the proper motion. These DNS systems include PSRs J1753−2240, J1755−2550, J1829+2456, J1906+0746, J1913+1102, and J1930−1852, which have small ($e \approx 0.1$) or intermediate ($e \approx 0.3–0.4$) eccentricities, and for which our SN simulations revealed kick solutions up to several 100 km s⁻¹ (in a few cases, even above 1000 km s⁻¹). In order for the simulated progenitor systems to remain bound and reproduce the observed DNS systems, the kick directions must preferentially

<figure><img src="image_25.png" /><figcaption>**Figure 23.** Distribution of kick directions for all simulated DNS systems shown in Figures 24–38. In this plot, 3000 kick directions ($\theta, \phi$) were simulated for the second SN of each system and stacked together. The darker the blue-scale color (log-10 scale, spanning 3 dex), the more solutions in a given direction. The red circles indicate kick directions parallel ($\phi = 90^{\circ}, \theta = 90^{\circ}$) and anti-parallel ($\phi = -90^{\circ}, \theta = 90^{\circ}$) to the pre-SN orbital angular momentum vector. The yellow crosses mark $\theta = 135^{\circ}$, and $\phi = 90^{\circ}$ or $\phi = -90^{\circ}$. The central and bottom panels show the distribution of the $\phi$ and $\theta$ angles, projected from the upper panel. An apparent preference is seen for a kick direction out of the orbital plane of the pre-SN binary, as well as in a backward direction (yellow crosses). However, this anisotropy may well be a combined artifact of input priors and a selection effect—see the text. Figure 39 gives a verification of the isotropy for the input (trial) kicks used in our SN simulations.</figcaption></figure>

be in the regions near the yellow crosses in Figure 23. The reason for this is that to compensate for the large applied kick magnitudes, the kick directions are limited in order to reproduce the small post-SN eccentricities (see Figures 11 and 40), which are needed to match most of the observed DNS systems. If instead we impose limitations on the maximum kick magnitude in the second SN in these systems, i.e., treating the kick magnitudes as elicited priors, and only allow kicks up to some limited maximum, say $w = 50–60$ km s⁻¹, then the apparent kick anisotropy along $\phi$ disappears.
If assuming that nature produces an isotropic distribution of kick directions, we can therefore reverse the above argument and claim further evidence for small kicks in ultra-stripped SNe in general (but not always, as demonstrated by PSRs B1913+16 and B1534+12).

### 9.2.2. Kick Anisotropies and Selection Effects

Second, we must beware of the observational selection effect (Chaurasia & Bailes 2005) wherein high-eccentricity and tight-orbit DNS systems will be removed from the observable sample shortly after their birth due to their small merger timescales. From Equation (26), we find that any DNS system with $e > 0.694$ will merge at least 10 times faster than circular DNS systems with the same post-SN $P_{\text{orb}}$ and NS masses. This effect also gives an explanation for the apparent kick anisotropy since DNS systems with small eccentricities remain observable on a longer timescale and, importantly, since they have small eccentricities, their kick angles are restricted to a certain area in the $(\phi, \theta)$ plane near the yellow crosses in Figure 23 (see Figure 11, and Figure 40 in the Appendix for the case of an ultra-stripped SN in a tight binary).

Any finding of a preference for a certain kick direction would otherwise have been interesting and should be kept in mind when comparing to other studies indicating, for example, a preference for a kick along the spin axis of isolated radio pulsars (Postnov & Kuranov 2008; Kuranov et al. 2009; Noutsos et al. 2013 and references therein). We find that for DNS systems the kick direction is not particularly favored along the pre-SN orbital angular momentum vector (marked by red circles in Figures 23 and 24–38), which is likely to be parallel to the spin axis of the exploding star. When comparing to measurements of post-SN binaries, it is uncertain, however, whether or not the spin axis of the exploding star is tossed in a new (possibly random) direction as a result of the SN (Spruit & Phinney 1998; Farr et al. 2011), in which case all past memory will be lost. This seems to be the case, for example, for both of the known young pulsars in DNS systems: PSR J0737−3039 (Breton et al. 2008) and J1906+0746 (G. Desvignes et al. 2017, in preparation).

Finally, it should be noted that if such tossing of the spin axis during core collapse also applies to the formation of BHs, then this might explain the small effective inspiral spin parameters, $\chi_{\text{eff}}$, inferred for the first four LIGO events (Abbott et al. 2017) without the need for a dynamical formation channel to explain their origin.

To summarize, we find that the apparent kick anisotropy (in a diagonal backward direction; see Figure 23), obtained from all of the kinematic solutions to the known DNS systems, is most likely caused by allowing for (unrealistic?) large kicks in the second SN. Furthermore, an observational selection effect favoring small eccentricities seems to be at work, which also affects the apparent kick angles. Additional investigations and statistical analysis is needed to confirm our preliminary findings. We caution that we are still dealing with small number statistics in terms of known DNS systems and that it is unclear how exactly to disentangle the obtained results from the probability bias for survival and observability.

# 10. DNS Mergers

In the previous sections, we have investigated in detail the formation of DNS systems and also discussed their further evolution. We end this paper with a short discussion of the final fate of close DNS binaries.

## 10.1. Short GRBs

Merging DNS and NS+BH binaries are considered the most promising sources to explain the phenomenology of short GRBs (Paczynski 1986; Eichler et al. 1989; Narayan et al. 1992; Fox et al. 2005; Rezzolla et al. 2011; Berger 2014). These short GRBs (sGRBs) are possibly followed by a so-called “kilonova” or “macronova” (Li & Paczyński 1998; Kulkarni 2005; Metzger et al. 2010; Barnes & Kasen 2013; Berger et al. 2013; Tanaka & Hotokezaka 2013; Tanvir et al. 2013; Fernández & Metzger 2016; Metzger 2017; Gottlieb et al. 2017). According to the standard scenario for the generation of sGRBs, an accretion-powered jet from a remnant BH torus system is formed within 10–100 ms after the merger event (e.g., Narayan et al. 1992; Janka et al. 1999; Rezzolla et al. 2011; Just et al. 2016). Moreover, the energy release should cease once the torus has been accreted on a timescale of $\le 1$ s, which is consistent with the duration of the sGRB prompt emission of less than $\sim 2$ s.

Observations by the Swift satellite, however, have recently revealed long-lasting ($10^2–10^5$ s) X-ray activity in a large fraction of sGRB events, which is indicative of a continuous energy release from a central engine on timescales orders of magnitude larger than the typical torus accretion timescale (Rowlinson et al. 2010, 2013; Gompertz et al. 2013, 2014; Lü et al. 2015). This late X-ray activity is difficult to explain with the accretion of the torus or the prolonged afterglow radiation produced by the interaction of the jet with the ambient medium (Kumar & Zhang 2015). Rezzolla & Kumar (2015) and Siegel & Ciolfi (2016) recently suggested a solution to this issue. If a large fraction of DNS mergers result in the formation of a metastable or long-lived NS rather than a BH, extraction of rotational energy via magnetic spin-down from such an object (typically a millisecond magnetar; Usov 1992) could power the observed long-lasting X-ray activity. Such a model will not work for the NS+BH scenario for sGRBs and would therefore exclude a large fraction of sGRBs from originating from NS+BH mergers.

## 10.2. Heavy r-process Elements

The merger process of DNS systems is a vital source of chemical enrichment of the interstellar medium with heavy r-process elements such as, for example, gold (e.g., Eichler et al. 1989; Freiburghaus et al. 1999; Wanajo et al. 2014; Just et al. 2015; Rosswog 2015; Hirai et al. 2015; Beniamini et al. 2016b; Wu et al. 2016). These elements were previously thought to be synthesized primarily in SNe. The total mass of the ejected baryonic matter from DNS mergers, including the yield of heavy r-process elements, is typically of the order $0.01 M_\odot$, depending on the mass ratio between the two merging NSs. Until recently, all observed DNS systems had a mass ratio $q \simeq 1$. The discovery of PSR J0453+1559 (Martinez et al. 2015) revealed the first known DNS system with a more asymmetric pair of NS masses ($1.56$ and $1.17 M_\odot$) and $q > 1.3$. This mass asymmetry is very important as it may possibly
increase (see below) the yield of expelled heavy *r*-process elements, when the lighter NS fills its Roche lobe and becomes tidally disrupted.

In Tauris et al. (2015), the modeling of DNS progenitor systems resulted in second-born NS masses in the interval 1.1–1.8 $M_\odot$ (±0.1 $M_\odot$). The fact that the lightest observed first-born NSs in DNS systems have masses <1.2 $M_\odot$ means that, in principle, we might even expect a future detection of a DNS merger with a mass ratio $q \ge 1.5$. Such a DNS merger would produce *r*-process rich ejecta with larger asymmetry and a different direction-dependent composition than those expected for a DNS merger with $q \simeq 1$ (Just et al. 2015), and possibly, but not necessarily, a larger yield (depending on the NS masses and the NS EoS; Just et al. 2015; Lehner et al. 2016).

### 10.3. Merger Rates

The Galactic merger rate of DNS systems has been estimated for almost four decades (e.g., Clark et al. 1979). As we have discussed in the previous sections, the standard formation scenario of DNS binaries involves a number of highly uncertain aspects of binary interactions (e.g., Voss & Tauris 2003; Tauris & van den Heuvel 2006; Belczynski et al. 2008; Dominik et al. 2012). The main uncertainties include, in particular, the treatment of CE evolution (Ivanova et al. 2013; Kruckow et al. 2016), SN kicks (Janka 2012, 2017) and the efficiency of accretion and spin-up from mass transfer (e.g., Tauris et al. 2015 and this paper). Together, these processes lead to large uncertainties in the expected merger rates of several orders of magnitude. Currently, the DNS merger rate is predicted to be about 1–100 Myr⁻¹ per Milky Way Equivalent Galaxy (Abadie et al. 2010).

The vast majority,<fnref n="22" /> almost all, of DNS systems that merge within a Hubble time due to GW radiation are produced by ultra-stripped SNe (Tauris et al. 2015). And because some relatively wide-orbit DNS systems are also produced via ultra-stripped SNe, the observed rate of ultra-stripped SNe is expected to be larger than the rate of DNS mergers. Furthermore, ultra-stripped SNe may also originate in binary systems with a BH or a WD accretor, and thus the total rate of ultra-stripped SNe is estimated to be one in every 100–1000 SNe (Tauris et al. 2013).

After the recent success in detecting GW signals from merging BHs by advanced LIGO (Abbott et al. 2016a, 2016b, 2017), it is also expected that GWs from merging DNS systems will be detected in the near future with LIGO and other detectors like VIRGO and KAGRA. Thus, it will soon be possible to obtain DNS merger rate constraints from GW observations. This should place a lower limit on the rate of ultra-stripped SNe. Comparing the rates of ultra-stripped SNe inferred from high-cadence optical surveys and DNS mergers from GW detectors, we will be able to test whether the ultra-stripped SN channel is actually the major path for forming DNS systems, and thus constrain binary stellar evolution using GWs.

## 11. Summary of DNS Formation and Conclusions

DNS systems are produced in the Galactic disk from the interaction of two OB-stars in a close binary system. To produce a DNS system, the minimum ZAMS mass of the primary star must be about $10 \pm 2 M_\odot$ (depending on metallicity and orbital period), whereas the secondary star in some cases initially can have a ZAMS mass as low as $5 \pm 1 M_\odot$ (e.g., Tauris & Sennels 2000; Voss & Tauris 2003; K. Belczynski 2017, private communication), depending on its efficiency in accreting material from the primary star (which will bring it above the SN threshold mass).

To form a DNS system, the binary system must survive several phases of mass transfer without coalescing and remain bound after both SN explosions (Figure 1). Furthermore, to produce a DNS merger (sGRB), the evolution has to lead to a final DNS system with an orbital period short enough (and/or eccentricity large enough) to merge within a Hubble time as a result of orbital damping from GW radiation.

One obstacle that hinders the formation of DNS systems is that many of the initially close-orbit systems are expected to coalesce early during their evolution (e.g., Sana et al. 2012), or slightly later when the more massive star evolves off the main sequence, expands, and starts to transfer mass to its companion star (Wellstein et al. 2001). To avoid this problem, it has been common practice to model the formation of DNS systems starting from relatively wide-orbit binaries and first let the systems evolve through a stable RLO and then, after the formation of the first NS, continue through a CE, following the HMXB phase (see Figure 1). This practice has become the standard scenario for modeling the production of DNS systems (Tauris & van den Heuvel 2006 and references therein). However, there are currently no self-consistent hydrodynamical simulations for successfully modeling the spiral-in of NSs inside a massive envelope, leading to envelope ejection. As a result, this leads to large uncertainties in both the estimated number of post-CE systems and their orbital separations (Ivanova et al. 2013; Kruckow et al. 2016). Another issue that is often ignored is that the fate of a massive star in a binary depends on when it loses its hydrogen-rich envelope (Brown et al. 1999, 2001; Wellstein & Langer 1999).

As an alternative formation channel, the so-called double core scenario has been suggested (Brown 1995; Dewi et al. 2006). In this scenario, both NS progenitors evolve beyond helium core burning before experiencing a CE phase in which the cores of both stars spiral-in, thereby producing a close binary of two evolved helium/carbon-oxygen stars that subsequently collapse to form NSs. However, this scenario requires significant fine-tuning since the initial masses of the two stars have to be very close to each other (with an initial ZAMS mass ratio of $0.96 < q < 1$).

Finally, there is a third formation channel for DNS systems which involves dynamical interactions in dense cluster environments (Phinney & Sigurdsson 1991; Grindlay et al. 2006). This dynamical channel, however, is expected to be significantly less frequent compared to the standard formation channel of DNS systems. Some simulations predict that this channel will account for less than 1% of all detected DNS mergers (Bae et al. 2014). One reason is that dense cluster simulations show that the segregation of NSs (and hence the formation of DNS systems via encounters) is not efficient on a Hubble timescale, especially at low metallicities (Banerjee 2017;

<footnote n="22">²² It is possible that a few merging DNS systems originate from wide binaries in which the NS is shot directly into a tight/eccentric orbit, i.e., similarly to the direct-SN mechanism discussed by Kalogera (1998). Voss & Tauris (2003) find that this route may account for up to 5% of all merging DNS systems.</footnote>
see Figures 6 and 7). In the following, we continue summarizing our results for the standard formation channel.

The extreme stripping of helium donor stars, via Case BB RLO in post-HMXB/post-CE systems, prior to the second SN, leads to almost naked metal cores that will produce EC or Fe CCSNe with little ejecta mass. Tauris et al. (2013) demonstrated that in extreme cases, post-CE mass stripping in a helium star-NS binary can produce an Fe CCSN progenitor star with a total mass of only ~1.50 $M_\odot$ prior to collapse and an envelope mass of barely 0.05 $M_\odot$. The resulting explosion leads to a SN Ic with an ejecta mass in the range 0.05–0.20 $M_\odot$. Through synthetic light-curve calculations, they found that SN 2005ek (Drout et al. 2013) is a viable candidate for such an event. Tauris et al. (2015) followed up with systematic calculations of helium star-NS binaries leading to ultra-stripped pre-SN stars with masses of 1.45–3.5 $M_\odot$. They argued that it is expected that the second-born NSs left behind will have gravitational masses in the range 1.1–1.8 $M_\odot$ (±0.1 $M_\odot$), depending on the mass cut during the explosion and the yet unknown EoS of NS matter and its associated release of gravitational binding energy (Lattimer & Yahil 1989). Tauris et al. (2015) also concluded that the observed spectral classes of ultra-stripped SNe can be either Type Ib or Type Ic, depending on the amount of helium left after the stripping of the envelope, as well as the amount and mixing of synthesized nickel in the ejecta.

From light-curve and spectral modeling of ultra-stripped SNe, Moriya et al. (2017) recently argued that, in addition to SN 2005ek, SN 2010X and PTF10iuv are also good candidates for ultra-stripped SNe. Therefore, each of these SNe could potentially be an event that gave birth to a DNS system. Given the current ambitious observational efforts to search for peculiar and weak optical transients, it seems probable that many more ultra-stripped SNe with diminutive ejecta will be detected within the coming years.

Our main conclusions from the present investigation of DNS formation can be summarized as follows:

1. With a focus on the combination of the latest observational data and recent progress in the physics and qualitative understanding of processes related to the production of DNS systems, we have investigated various aspects of massive binary evolution from the pre-HMXB stages to accretion, CE evolution, and SN explosions. A main uncertainty in our formation picture is related to the transition from the different subpopulations of HMXBs to DNS systems (Figure 4)—for example, related to their inspiral in a CE and the spin-up of the first-born NS before the second SN explosion. We argued that few, if any, of the presently known Galactic HMXBs will survive their subsequent CE evolution and eventually produce a DNS system. This result is not in conflict with population statistics given that the active radio lifetime of mildly recycled pulsars in DNS systems is often a factor of >100 larger than the X-ray lifetime of their HMXB progenitors.

2. From our analysis of five potential phases of accretion onto the NS in HMXB systems (the first-born NS in DNS systems), we conclude that by the time these NSs are observable as mildly recycled radio pulsars, they have

accreted at most ~0.02 $M_\odot$. This means that their present masses closely resemble their birth masses. This is important for extrapolating the pre-SN stellar properties and probing the explosions leading to the first-born NSs, in comparison to the second-born (young) NSs in DNS systems. The first-born NSs are noted to have larger (birth) masses by an amount of the order ~0.1 $M_\odot$ (Figure 3). An explanation for this difference in masses is that in the second SN, the progenitor of the exploding star is being stripped significantly deeper by its NS companion.

3. The span of precisely measured NS masses in DNS systems is at present 1.17–1.56 $M_\odot$ (±0.01 $M_\odot$). Compared to masses of MSPs with WD companions (Antoniadis et al. 2017), the observed NS mass distribution in DNS systems is lacking the tail of massive NSs (≥1.7 $M_\odot$). Such masses are, however, under certain circumstances, even possible for the second-born NS (the one often formed in an ultra-stripped SN), depending on the mass of its helium star progenitor (Tauris et al. 2015). Furthermore, some HMXBs (the potential progenitors of DNS systems) like Vela X-1 have NS masses of about 1.9–2.1 $M_\odot$ (Barziv et al. 2001; Falanga et al. 2015). Therefore, the lack of massive NSs in presently known DNS systems might simply be a matter of small number statistics or it could reflect the fact that sufficiently massive helium stars rarely form via this formation channel. A full population synthesis is needed to investigate this further, or we simply await the recorded mass spectrum from near-future LIGO/VIRGO detections of DNS mergers.

4. The observed DNS systems show a clear trend between the $P_{\text{orb}}$ and $P$ of the first-born (recycled) pulsars (Figures 4 and 6). We find that the empirical data can approximately be fitted to the following birth relation: $P \approx 36 \pm 14 \text{ ms} (P_{\text{orb}}/\text{days})^{0.40 \pm 0.10}$ (Equation (7)), although the spread in data points is large, especially for small $P_{\text{orb}}$. The fact that a correlation is present at all is another indication that kicks imparted on the second-born NSs cannot be too large in general. To further test the validity of such a correlation, it is important to find more wide-orbit DNS systems like PSR J1930−1852 (Swiggum et al. 2015) which has $P_{\text{orb}} = 45 \text{ days}$, by far the widest DNS system known. As expected from binary stellar evolution, this radio pulsar has the slowest spin period (185 ms), which proves its past history with only marginal recycling (Tauris et al. 2015).

5. Using binary stellar evolution arguments, we find that general correlations between $P_{\text{orb}}$, $P$, and eccentricity should, in principle, be present in DNS systems in the case of *symmetric* SNe and for systems evolving from similar helium star donors. We calculated such correlations on the basis of previous theoretical modeling (Tauris et al. 2015) of progenitors of ultra-stripped SNe. However, the combination of different masses of the exploding stars and, in particular, the addition of (even small) kicks in random directions will easily tend to camouflage any such correlations (see Figures 9, 10, and 12), although some trends are still present in the observational data.

6. We discussed kicks added to newborn NSs and revisited the arguments to distinguish between the following three
cases: (i) isolated NSs (or very wide-orbit NS binaries), (ii) the first SN explosion in a close binary from a stripped star, and finally, (iii) the second SN explosion in a close binary from an ultra-stripped star. We find that the following picture is emerging where the first case often seems to produce large kicks (on average, $\langle w \rangle \simeq 400-500 \text{ km s}^{-1}$), whereas the first SN in a close binary (leading to HMXBs) results in significantly smaller average kicks, and the second SN (ultra-stripped SNe) is often, but not always, accompanied by kicks of even less than 50 km s⁻¹. Indeed, the majority of the observed DNS systems can best be explained with small kicks in the second SN (e.g., Beniamini & Piran 2016), although in a couple of cases a larger kick (200–400 km s⁻¹) must have been at work (see also Fryer & Kalogera 1997; Wex et al. 2000). The explanation for this diversity in kick magnitudes among DNS systems could either reflect some stochasticity involved in the development of the explosion asymmetries (Wongwathanarat et al. 2013), or be related to the mass of the metal core of the exploding star (see below).

7. From current DNS data, we find that all second-born NSs with a mass $\gtrsim 1.33 M_\odot$ seem to be associated with rather large kicks, whereas below this mass limit they can be related to small kick events ($w \lesssim 50 \text{ km s}^{-1}$; see Table 7 and Figure 16). We have presented theoretical arguments in favor of such a possible correlation between NS mass and kick magnitude (Section 6.6). The nature of the SNe leading to the latter (low-mass, small kick) NSs is either EC SNe or Fe CCSNe from stars with small iron cores. Given the somewhat narrow mass window for producing EC SNe (Podsiadlowski et al. 2004), it is likely that Fe CCSNe dominate the formation rate of low-mass NSs. Detailed population synthesis studies are needed to answer this (M. U. Kruckow et al. 2017, in preparation).

8. Based on observational constraints of a few key parameters (orbital periods, eccentricities, NS masses, systemic velocities, and misalignment angles), we have simulated the kinematic effects of the second SN in all observed DNS systems. As a result, we have been able to probe the solutions to place limits on, e.g., the masses of the exploding stars and the kick velocity distribution. For DNS systems that are well constrained from observations, e.g., PSR J0737-3039, we are able to find an almost *unique* solution for the pre-SN progenitor system. We find that for this particular system, the second-born NS (pulsar B) originates from an exploding star with a mass of $1.59 \pm 0.09 M_\odot$ and with an orbital period of $0.10 \pm 0.02 \text{ days}$. A similar small progenitor mass was derived by Piran & Shaviv (2005) based on kinematic arguments. Binary stellar evolution models by Tauris et al. (2013) have confirmed that indeed such small progenitor star masses (down to $\sim 1.50 M_\odot$) are possible from ultra-stripping via Case BB RLO. For PSR J1756-2251, we can constrain the mass of the second exploding star to be $1.72 \pm 0.22 M_\odot$. However, for most of the other DNS systems, we can only produce a possible range of the solution parameters. The reason for this is large differences in the outcome of SNe due to kicks with

different directions. In particular, it would be desirable to measure more transverse velocities (i.e., proper motions and distances) and misalignment angles to help constrain these systems better. The inferred transverse velocities still suffer from (in some cases relatively large) uncertainties in the DM distance estimates. Improved distances of nearby DNS systems can also be derived from future determinations of $\dot{P}_{\text{orb}}$.

9. One outcome of our SN simulations reproducing the observed DNS systems is an apparent preference for a NS kick direction out of the orbital plane of the pre-SN binary (Figure 23). However, we argue that this anisotropy is caused by a combination of considering all valid kinematic solutions, irrespective of any physical limits on the applied kick magnitudes of ultra-stripped SNe in general, and an observational selection bias in favor of low-eccentricity DNS systems. Further investigations and statistical analysis on this topic are needed.

10. In the following decade, the completion of the SKA and the newly constructed FAST radio telescopes are anticipated to boost the number of known DNS systems —for the SKA, possibly by a factor of 5 to 10 (Keane et al. 2015). Such an increase in sources will greatly improve the population statistics and thus our knowledge of DNS formation. SKA will also improve the pulsar timing significantly, which is important for deriving NS masses, distances, and proper motions. In addition, and also from near-future observations, the rate of ultra-stripped SNe can be compared to the rate at which GW detectors (LIGO, VIRGO, KAGRA, IndIGO) will detect DNS mergers. This will help us to verify, or falsify, our hypothesis that ultra-stripped SNe occur in (almost) all DNS mergers originally produced in the Galactic disk. Finally, it will be interesting to compare the mass distribution of DNS mergers from future LIGO observations with our theoretical predictions.

We thank the organizers of and all participants in the MIAPP 2015 workshop on “The Many Faces of Neutron Stars” for many stimulating discussions. T.M.T. and M.U.K. acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG) Grant No. TA 964/1-1. At Garching, the work was supported by the DFG through the Excellence Cluster “Universe” EXC 153 and by the European Research Council through grant ERC-AdG No. 341157-COCO2CASA.

## Appendix

### Simulated Second SN of Individual DNS Systems

Figures 24–38 show our simulations of the second SN in the progenitor binaries of the individual DNS systems discussed in Section 8.

Figure 39 demonstrates the isotropy of the trial kick velocity directions used as input for the simulations shown in Figure 23.

Figure 40 is similar to Figure 11, but shows the case for an ultra-stripped SN in a tight binary and with a large kick.
<figure><img src="image_26.png" /></figure>

<figure><img src="image_27.png" /></figure>

<figure><img src="image_28.png" /></figure>

<figure><img src="image_29.png" /></figure>

<figure><img src="image_30.png" /></figure>

<figure><img src="image_31.png" /><figcaption>**Figure 24.** Properties and constraints on the formation of PSR J0453+1559. See Table 2 and discussions in Section 8.1.</figcaption></figure>
<figure><img src="image_32.png" /></figure>

<figure><img src="image_33.png" /></figure>

<figure><img src="image_34.png" /></figure>

<figure><img src="image_35.png" /></figure>

<figure><img src="image_36.png" /></figure>

<figure><img src="image_37.png" /></figure>

**Figure 25.** Properties and constraints on the formation of PSR J0737-3039. See Table 2 and discussions in Section 8.2. The red star shows the initial location of this DNS system if it had formed 100 Myr ago.
<figure><img src="image_38.png" /></figure>

<figure><img src="image_39.png" /></figure>

<figure><img src="image_40.png" /></figure>

<figure><img src="image_41.png" /></figure>

<figure><img src="image_42.png" /></figure>

<figure><img src="image_43.png" /></figure>

**Figure 26.** Properties and constraints on the formation of PSR J0737-3039. See Table 2 and discussions in Section 8.2. In this simulation, the constraints on the systemic velocity have been ignored. The lower-right panel displays the distribution of misalignment angles.
<figure><img src="image_44.png" /></figure>

<figure><img src="image_45.png" /></figure>

<figure><img src="image_46.png" /></figure>

<figure><img src="image_47.png" /></figure>

<figure><img src="image_48.png" /></figure>

<figure><img src="image_49.png" /><figcaption>**Figure 27.** Properties and constraints on the formation of PSR J1518+4904. See Table 2 and discussions in Section 8.3.</figcaption></figure>
<figure><img src="image_50.png" /></figure>

<figure><img src="image_51.png" /></figure>

<figure><img src="image_52.png" /></figure>

<figure><img src="image_53.png" /></figure>

<figure><img src="image_54.png" /></figure>

<figure><img src="image_55.png" /></figure>

**Figure 28.** Properties and constraints on the formation of PSR B1534+12. See Table 2 and discussions in Section 8.4. The red star shows the initial location of this DNS system for an age of *t* = *τ* = 248 Myr.
<figure><img src="image_56.png" /></figure>

<figure><img src="image_57.png" /></figure>

<figure><img src="image_58.png" /></figure>

<figure><img src="image_59.png" /></figure>

<figure><img src="image_60.png" /></figure>

<figure><img src="image_61.png" /><figcaption>**Figure 29.** Properties and constraints on the formation of PSR J1753-2240. See Table 2 and discussions in Section 8.5.</figcaption></figure>
<figure><img src="image_62.png" /></figure>

<figure><img src="image_63.png" /></figure>

<figure><img src="image_64.png" /></figure>

<figure><img src="image_65.png" /></figure>

<figure><img src="image_66.png" /></figure>

<figure><img src="image_67.png" /><figcaption>**Figure 30.** Properties and constraints on the formation of PSR J1755−2550. See Table 2 and discussions in Section 8.6.</figcaption></figure>
<figure><img src="image_68.png" /></figure>

<figure><img src="image_69.png" /></figure>

<figure><img src="image_70.png" /></figure>

<figure><img src="image_71.png" /></figure>

<figure><img src="image_72.png" /></figure>

<figure><img src="image_73.png" /></figure>

**Figure 31.** Properties and constraints on the formation of PSR J1756−2251. See Table 2 and discussions in Section 8.7. The red star shows the initial location of this DNS system for an age of *t* = *τ* = 443 Myr.
<figure><img src="image_74.png" /></figure>

<figure><img src="image_75.png" /></figure>

<figure><img src="image_76.png" /></figure>

<figure><img src="image_77.png" /></figure>

<figure><img src="image_78.png" /></figure>

<figure><img src="image_79.png" /><figcaption>**Figure 32.** Properties and constraints on the formation of PSR J1811−1736. See Table 2 and discussions in Section 8.8.</figcaption></figure>
<figure><img src="image_80.png" /></figure>

<figure><img src="image_81.png" /></figure>

<figure><img src="image_82.png" /></figure>

<figure><img src="image_83.png" /></figure>

<figure><img src="image_84.png" /></figure>

<figure><img src="image_85.png" /><figcaption>**Figure 33.** Properties and constraints on the formation of PSR J1829+2456. See Table 2 and discussions in Section 8.9.</figcaption></figure>
<figure><img src="image_86.png" /></figure>

<figure><img src="image_87.png" /></figure>

<figure><img src="image_88.png" /></figure>

<figure><img src="image_89.png" /></figure>

<figure><img src="image_90.png" /></figure>

<figure><img src="image_91.png" /><figcaption>**Figure 34.** Properties and constraints on the formation of PSR J1906+0746. See Table 2 and discussions in Section 8.10.</figcaption></figure>
<figure><img src="image_92.png" /></figure>

<figure><img src="image_93.png" /></figure>

<figure><img src="image_94.png" /></figure>

<figure><img src="image_95.png" /></figure>

<figure><img src="image_96.png" /></figure>

<figure><img src="image_97.png" /></figure>

**Figure 35.** Properties and constraints on the formation of PSR J1913+1102. See Table 2 and discussions in Section 8.11. The red star shows the initial location of this DNS system for an age of *t* = *τ* = 2690 Myr.
<figure><img src="image_98.png" /></figure>

<figure><img src="image_99.png" /></figure>

<figure><img src="image_100.png" /></figure>

<figure><img src="image_101.png" /></figure>

<figure><img src="image_102.png" /></figure>

<figure><img src="image_103.png" /><figcaption>**Figure 36.** Properties and constraints on the formation of PSR B1913+16. See Table 2 and discussions in Section 8.12. The red star shows the initial location of this DNS system after its formation; see Section 7 for discussions and the simulations in Figure 37.</figcaption></figure>
<figure><img src="image_104.png" /></figure>

<figure><img src="image_105.png" /></figure>

<figure><img src="image_106.png" /></figure>

<figure><img src="image_107.png" /></figure>

<figure><img src="image_108.png" /></figure>

<figure><img src="image_109.png" /><figcaption>**Figure 37.** Properties and constraints on the formation of PSR B1913+16, assuming the system formed 108.7 Myr ago. This is its maximum age if assuming evolution with a braking index of *n* = 3.</figcaption></figure>
<figure><img src="image_110.png" /></figure>

<figure><img src="image_111.png" /></figure>

<figure><img src="image_112.png" /></figure>

<figure><img src="image_113.png" /></figure>

<figure><img src="image_114.png" /></figure>

<figure><img src="image_115.png" /><figcaption>**Figure 38.** Properties and constraints on the formation of PSR J1930−1852. See Table 2 and discussions in Section 8.13.</figcaption></figure>
<figure><img src="image_116.png" /><figcaption>**Figure 39.** Distribution of the trial kick velocity directions that are used for our SN simulations shown in Figure 23. This plot verifies the random directions of input kick velocities (i.e., isotropy on the kick sphere). To obtain isotropy, the probability for the directions of $\theta$ and $\phi$ are weighted by $P(\theta) = \sin(\theta)$ and $P(\phi) = \text{constant}$, respectively. The statistics of this plot is based on a total of 39,000 SNe (similar number as in Figure 23 for simulations of 13 observed DNS systems). The resulting statistical noise is very minor.</figcaption></figure>
<figure><img src="image_117.png" /><figcaption>**Figure 40.** Dependence of the post-SN eccentricity on the direction of the kick for an ultra-stripped SN with a large kick of $w = 300$ km s$^{-1}$ (see Figure 11 for a comparison). The assumed pre-SN parameters are $P_{\text{orb,i}} = 0.5$ days and $M_{\text{He,f}} = 1.80 M_{\odot}$ and the companion star (first-born NS) is assumed to have a mass of $M_{\text{NS,1}} = 1.40 M_{\odot}$. The resulting NS is produced with a mass of $M_{\text{NS,2}} = 1.50 M_{\odot}$. Systems with $\theta \rightarrow 180^{\circ}$ will have very tight and eccentric post-SN orbits, and some of them will merge in less than 1 Myr as a result of GW radiation.</figcaption></figure>

## ORCID iDs

T. M. Tauris https://orcid.org/0000-0002-3865-7265

M. Kramer https://orcid.org/0000-0002-4175-2271

P. C. C. Freire https://orcid.org/0000-0003-1307-9435

H.-T. Janka https://orcid.org/0000-0002-0831-3330

E. Bozzo https://orcid.org/0000-0002-7504-7423

S. Chaty https://orcid.org/0000-0002-5769-8601

J. Antoniadis https://orcid.org/0000-0003-4453-3776

R. P. Breton https://orcid.org/0000-0001-8522-4983

D. J. Champion https://orcid.org/0000-0003-1361-7723

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