3 A binary supernova in Upper Scorpius

  
3.1 $\zeta$ Oph and PSR J1932+1059

$\zeta$ Oph is a single O9.5Vnn star, and was first identified as a runaway originating in the Sco OB2 association by Blaauw (1952b). Based on its proper motion, which points away from the association, its radial velocity, and the large space velocity ($\sim$30 km s^-1), Blaauw suggested that $\zeta$ Oph might have formed in the center of the association $\sim$3 Myr ago. Later investigations (e.g., Paper I; Blaauw 1993; van Rensbergen et al. 1996) showed that $\zeta$ Oph either became a runaway $\sim$1 Myr ago in the Upper Scorpius subgroup of Sco OB2, or 2-3 Myr ago in the Upper Centaurus Lupus subgroup (cf. de Zeeuw et al. 1999).

 

 

Table 3: Data for the nearby runaway stars and pulsars discussed in this paper. The stars which have an $^\ast$ appended to their HIP identifier are the classical runaways (i.e., a parent group was already known before this study, see Paper I). Unless indicated otherwise, the position ($\alpha$, $\delta$), proper motion ( $\mu _{\alpha \ast }$, $\mu _\delta$), and parallax ($\pi$) were taken from the Hipparcos Catalogue (ESA 1997), the radial velocity ( $v_{\rm rad}$) from the Hipparcos Input Catalogue (Turon et al. 1992), the space velocity ( $v_{\rm space}$) with respect to the standard of rest of the runaway, the rotational velocity ( $v_{\rm rot} \sin i$) for the runaways from Penny (1996) and the period P for the pulsars in seconds, the spectral type from Mason et al. (1998) or the Hipparcos Catalogue for the runaways and the characteristic age ( $\tau = P/(2\dot{P})$) for the pulsars, and the helium abundance ($\epsilon$) from Herrero et al. (1992), defined as the number of He atoms relative to H. The mass $M_{\rm SK}$ has been derived from the Schmidt-Kaler (1982) calibration, using interpolation. The mass $M_{\rm BB}$ is taken from Vanbeveren et al. (1998). The last column (N) indicates the number of the runaway/pulsar in Fig. 2. The proper motion and radial velocity are not corrected for Solar motion and Galactic rotation. The astrometric data ($\alpha$, $\delta$, $\pi$, $\mu _{\alpha \ast }$, and $\mu _\delta$) for the pulsars (the last five lines) are taken from the Taylor et al. (1993) catalogue. Abbreviations used: mas = milli-arcsec; $\mu_{\alpha\ast} = \mu_\alpha \cos \delta$

HIP	HD	Name	$\alpha$ ( $^{\rm h~m~s}$)	$\delta$ ( $^{\circ}~'~''$)	$\pi$	$\mu _{\alpha \ast }$	$\mu _\delta$	$v_{\rm rad}$	$v_{\rm space}$	$v_{\rm rot} \sin i$	SpT	$M_{\rm SK}$	$M_{\rm BB}$	$\epsilon$	N
			[J1991.25]	[J1991.25]	[mas]	[mas yr-1]	[mas yr-1]	[km s-1]	[km s-1]	[km s-1]		[$M_\odot$]	[$M_\odot$]	[#]
3881	4727	$\nu$ And	0 49 48.83	+41 04 44.2	$4.80\pm0.75$	$22.68\pm0.53$	$-18.05\pm0.48$	$-23.9\pm\phantom{1}1.2$	32.1	80a	B5V+F8V	6.9b			1
14514$^\ast$	19374	53 Ari	3 07 25.69	+17 52 47.9	$4.32\pm0.98$	$-23.54\pm0.93$	$9.30\pm0.95$	$21.2\pm\phantom{1}1.2$c	39.4	10d	B1.5V	10.4	8.5		2
18614$^\ast$	24912	$\xi$ Per	3 58 57.90	+35 47 27.7	$1.84\pm0.70$	$1.92\pm0.74$	$2.30\pm0.62$	$58.8\pm\phantom{1}5.0$e	64.9	204	O7.5III	33.8	33.5	0.18	3
22061	30112		4 44 42.16	+0 34 05.4	$2.94\pm0.86$	$-44.89\pm0.77$	$-29.28\pm0.67$	$6.0\pm\phantom{1}5.0$	86.5		B2.5V	8.6	7.5		4
24575$^\ast$	34078	AE Aur	5 16 18.15	+34 18 44.0	$2.24\pm0.74$	$-4.05\pm0.66$	$43.22\pm0.44$	$57.5\pm\phantom{1}1.2$	113.3	25	O9.5V	15.9	21.1	0.09	5
26241	37043	$\iota$ Ori	5 35 25.98	-05 54 35.6	$2.46\pm0.77$	$2.27\pm0.65$	$-0.62\pm0.47$	$28.7\pm\phantom{1}1.1$f	8.0	71g	O9III+B1IIIh	37.8i	38.6
27204$^\ast$	38666	$\mu$ Col	5 45 59.89	-32 18 23.0	$2.52\pm0.55$	$3.01\pm0.52$	$-22.62\pm0.50$	$109.0\pm\phantom{1}2.5$	107.8	111	O9.5V	15.9	21.1		6
29678	43112		6 15 08.46	+13 51 03.9	$2.38\pm0.72$	$24.21\pm0.76$	$10.65\pm0.49$	$36.0\pm\phantom{1}5.0$	63.0	<25j	B1V	11.5	12.0		7
38455	64503		7 52 38.65	-38 51 46.2	$5.09\pm0.52$	$-9.49\pm0.43$	$4.02\pm0.42$	$-31.0\pm\phantom{1}5.0$	41.4	212k	B2V	9.4	8.0		8
38518	64760		7 53 18.16	-48 06 10.6	$1.68\pm0.50$	$-4.90\pm0.53$	$5.89\pm0.38$	$41.0\pm\phantom{1}5.0$	31.1	220d	B0.5Iab	25.0	35.1		9
39429	66811	$\zeta$ Pup	8 03 35.07	-40 00 11.5	$2.33\pm0.51$	$-30.82\pm0.44$	$16.77\pm0.41$	$-23.9\pm\phantom{1}1.2$	62.4	203	O4I		67.5	0.14l	10
42038	73105		8 34 09.60	-53 04 17.5	$2.87\pm0.47$	$-12.14\pm0.54$	$10.13\pm0.48$	$37.0\pm10.0$	31.3		B3V	7.9	7.0		11
46950	83058		9 34 08.80	-51 15 19.0	$3.50\pm0.53$	$-8.50\pm0.49$	$6.39\pm0.48$	$35.0\pm10.0$	32.1		B1.5IV	10.4m	9.0		12
48943	86612		9 59 06.32	-23 57 02.8	$5.19\pm0.77$	$-23.22\pm0.70$	$5.30\pm0.70$	$39.0\pm\phantom{1}5.0$	35.2	230d	B5V	5.8			13
49934	88661		10 11 46.47	-58 03 38.0	$2.52\pm0.50$	$-10.71\pm0.49$	$6.63\pm0.45$	$31.0\pm10.0$	31.2	280d	B2IVnpe	9.4m	8.0		14
57669	102776		11 49 41.09	-63 47 18.6	$7.10\pm0.69$	$-17.93\pm0.95$	$4.44\pm0.63$	$29.0\pm\phantom{1}2.5$	31.1	251n	B3V	7.9	7.0		15
69491	124195		14 13 39.84	-54 37 32.2	$2.96\pm0.63$	$-18.03\pm0.41$	$-11.15\pm0.41$	$66.0\pm10.0$	77.2		B5V	5.8			16
76013	137387	$\kappa^1$ Aps	15 31 30.82	-73 23 22.4	$3.20\pm0.59$	$0.38\pm0.48$	$-18.28\pm0.55$	$62.0\pm\phantom{1}5.0$	69.0		B1npe				17
81377$^\ast$	149757	$\zeta$ Oph	16 37 09.53	-10 34 01.7	$7.12\pm0.71$	$13.07\pm0.85$	$25.44\pm0.72$	$-9.0\pm\phantom{1}5.5$	23.5	348	O9.5Vnn	15.9	21.1	0.16	18
82868	152478		16 56 08.85	-50 40 29.2	$4.34\pm0.82$	$-10.21\pm0.84$	$-9.55\pm0.62$	$19.0\pm\phantom{1}5.0$	30.3		B3Vnpe	7.9	7.0		19
91599	172488		18 40 48.06	-08 43 07.5	$3.61\pm1.16$	$-9.64\pm1.13$	$-22.64\pm0.79$	$34.1\pm\phantom{1}1.2$o	44.7		B0.5V	12.7	13.5		20
102274	197911		20 43 21.62	+63 12 32.9	$1.42\pm0.62$	$-13.72\pm0.53$	$-3.66\pm0.53$	$-3.8\pm\phantom{1}5.0$	46.1		B5				21
109556$^\ast$	210839	$\lambda$ Cep	22 11 30.58	+59 24 52.3	$1.98\pm0.46$	$-7.22\pm0.44$	$-11.06\pm0.39$	$-75.1\pm\phantom{1}1.2$	74.0	214	O6I	40.0	64.6	0.17l	22
J0826+2637			8 26 51.31	+26 37 25.6	$2.6\phantom{\pm11.7}$	$61\pm3$	$-90\pm2$			${\it P} =~0.53$	$\tau = 4.92$	1.4p			1
J0835-4510			8 35 20.68	-45 10 35.8	$2.0\phantom{\pm11.7}$	$-48\pm2$	$35\pm1$			0.09	0.01	1.4p			2
J1115+5030			11 15 38.35	+50 30 13.6	$1.9\phantom{\pm11.7}$	$22\pm3$	$-51\pm3$			1.65	10.53	1.4p			4
J1932+1059			19 32 13.87	+10 59 31.8	$5.9\phantom{\pm11.7}$	$99\pm6$	$39\pm4$			0.22	3.10	1.4p			8
Gemingaq			6 33 54.15	+17 46 12.9	$6.4\pm1.7$	$138\pm4$	$97\pm4$					1.4p			9

Notes: a: $v_{\rm rot} \sin i$ from Slettebak et al. (1997). b: Total mass for the binary: B5V ( $5.8~M_\odot$) + F8V ( $1.1~M_\odot$). c: $v_{\rm rad}$ from Duflot et al. (1995). The Hipparcos Input Catalogue radial velocity is incorrect. d: $v_{\rm rot} \sin i$ from Bernacca & Perinotto (1970). e: We took the average $v_{\rm rad}$ from Bohannan & Garmany (1978), Garmany et al. (1980), Stone (1982), and Gies & Bolton (1986). f: $v_{\rm rad}$, i.e., center of mass velocity, from Stickland et al. (1987). g: $v_{\rm rot} \sin i$ from Gies (1987). h: Spectral type of the secondary of $\iota$ Ori from Stickland et al. i: Total mass for the binary $\iota$ Ori. The individual masses are $22.9~M_\odot$ for the primary and $14.9~M_\odot$ for the secondary. j: $v_{\rm rot} \sin i$ from Morse et al. (1991). k: $v_{\rm rot} \sin i$ from Uesugi & Fukuda (1970). l: $\epsilon$ from Kudritzki & Hummer (1990). m: Assumed mass for main-sequence star instead of luminosity class IV. n: $v_{\rm rot} \sin i$ from Brown & Verschueren (1997). o: $v_{\rm rad}$ from Gies & Bolton (1986). p: We take the characteristic mass for a neutron star. q: Position from Caraveo et al. (1998); parallax and proper motion from Caraveo et al. (1996).

If $\zeta$ Oph is a BSS runaway, as suggested by its high helium abundance ( $\epsilon=0.16$, corresponding to a mass fraction X = 0.577 of H) and large rotational velocity (348 km s^-1), and if the binary dissociated after the supernova explosion, we might be able to identify the associated neutron star. None of the pulsars in Fig. 2 was ever inside the Upper Centaurus Lupus subgroup, but two could have originated from the Upper Scorpius subgroup: PSR J1239+2453 and PSR J1932+1059.

We first consider PSR J1239+2453. Its estimated distance is $\sim$560 pc. It passed within about 20 pc of the Upper Scorpius region $\sim$1 Myr ago if and only if its (unknown) radial velocity is large and positive ($\sim$650 km s^-1). With a tangential velocity of $\sim$300 km s^-1 (the proper motion is 114 mas yr^-1), the space velocity would have to be over 700  km s^-1, which is uncomfortably large. Furthermore, while 1 Myr is consistent with the kinematic age for $\zeta$ Oph, it is in conflict with the characteristic age ( $P/(2\dot{P}) = 23$ Myr) of the pulsar. The latter is an uncertain age indicator, but the difference between the two times is so large that we consider it unlikely that PSR J1239+2453 was associated with $\zeta$ Oph. The pulsar is currently at a Galactic latitude of $86^\circ$, i.e, at $z\sim 560$ pc above the Galactic plane. Typical z-oscillation periods of pulsars are of order 100 Myr (e.g., Blaauw & Ramachandran 1998), so that maximum height is reached after $T_{1/4}\sim25$ Myr. Taking the characteristic age at face value suggests the pulsar is near its maximum height above the plane, had a z-velocity of about 30 km s^-1, and was not formed in the Upper Scorpius association (age $\sim$5 Myr), but was born $\sim$25 Myr ago in the Galactic plane outside the Solar neighbourhood.

The path of the other pulsar, PSR J1932+1059 (earlier designation PSR B1929+10), also passed the Upper Scorpius association some 1-2 Myr ago. The characteristic age of this pulsar is only $\sim$3 Myr, consistent with the kinematic age of $\zeta$ Oph within the uncertainties. The present z-velocity of the pulsar ($\sim$40 km s^-1 away from the Galactic plane) predicts a maximum distance away from the plane of 680 pc and $T_{1/4} \sim 28$ Myr. The pulsar is presently located only $\sim$10 pc below the plane. Since it presumably formed close to the plane, this means that PSR J1932+1059 either formed recently or well over 50 Myr ago. Considering that both the characteristic age and the typical pulsar ages (up to $\sim$50 Myr) (Blaauw & Ramachandran 1998) are significantly smaller than $\sim$50 Myr, we conclude that the pulsar formed recently. Upper Scorpius is the only site of star formation along the past trajectory of the pulsar. We thus consider PSR J1932+1059 a good candidate for the remnant of the supernova which caused the runaway nature of $\zeta$ Oph.

3.2 Data

Table 3 summarizes the data for $\zeta$ Oph and PSR J1932+1059. The radial velocity of the pulsar is unknown. The pulsar proper motion listed by Taylor et al. (1993) was calculated from timing measurements (Downs & Reichley 1983). More accurate proper motions can be obtained from VLBI observations; Campbell (1995) measured a provisional proper motion and parallax of PSR J1932+1059 of $(\mu_{\alpha\ast},\mu_\delta) = (96.7\pm1.7,41.3\pm3.5)$ mas yr^-1 and $\pi = 5\pm1.5$ mas, respectively, including a full covariance matrix. These measurements are in good agreement with those of Taylor et al. (1993; see Table 3 and Fig. 5).

  
3.3 Simulations

Our hypothesis is that $\zeta$ Oph and PSR J1932+1059 are the remains of a binary system in Upper Scorpius which became unbound when one of the components exploded as a supernova. Support for this hypothesis would be to find both objects at the same position at the same time in the past. Our approach is to calculate their past orbits and simultaneously determine the separation between the two objects, $D_{\rm min}(\tau)$, as a function of time, $\tau$. We define $D_{\rm min}(\tau)$ as $\vert\vec{x}_{\zeta~{\rm Oph}} - \vec{x}_{\rm pulsar}\vert$, where $\vec{x}_j$ is the position of object j. We consider the time $\tau_0$ at which $D_{\rm min}(\tau)$reaches a minimum to be the kinematic age. To take the errors in the observables into account we calculate a large set of orbits, sampling the parameter space defined by the errors. We use the Taylor et al. (1993) proper motion for the pulsar. The errors in the positions of the runaway and the pulsar are negligible, and those in the proper motions of the two objects and in the parallax of the runaway are modest ($\le 10$%). However, the radial-velocity error of $\zeta$ Oph is considerable (5 km s^-1). The distance to the pulsar has a significant error, and its radial velocity is unknown. Accordingly, we first determine the region in the $(\pi_{\rm pulsar}, v_{\rm rad,pulsar})$ parameter space for which the pulsar approaches the runaway when we retrace both orbits. Sampling a grid in $(\pi_{\rm pulsar}, v_{\rm rad,pulsar})$ while keeping the other parameters fixed, we find that, for  mas and  km s^-1, the motions of the pulsar and the runaway are such that their separation decreases as one goes back in time.

Adopting $\pi_{\rm pulsar} = 4\pm 2$ mas and $v_{\rm rad,pulsar} = 200\pm50$ km s^-1, we calculate three million orbits for the pulsar and the runaway. Considering that the pulsar proper-motion errors might be underestimated (Campbell et al. 1996; Hartman 1997), we increased them by a factor of two. For each run we create a set of positions and velocities for the runaway and the pulsar consistent with the (modified 3$\sigma$) errors on the observables. We also calculated the orbit of the Upper Scorpius association back in time, using the mean position and velocity derived by de Zeeuw et al. (1999, their Table 2). $30\,822$ of these simulations resulted in a minimum separation between the pulsar and the runaway of less than 10 pc. In $4\,214$ simulations the pulsar and the runaway had a minimum separation less than 10 pc and were both situated within 10 pc of the center of the association (the smallest minimum separation found was 0.35 pc). Thus, only a small fraction (0.14%) of the simulations is consistent with the hypothesis that the pulsar and the runaway were once, $\sim$1 Myr ago, close together within the Upper Scorpius association. We now show that given the measurement uncertainties, this low fraction is perfectly consistent with the two objects being in one location in the past.

Figure 3: Left: Distribution of minimum separations, $D_{\rm min}(\tau_0)$, between $\zeta$ Oph and PSR J1932+1059. The solid line denotes the expected distribution of $D_{\rm min}$, see Sect. 3.3. Right: Distribution of the times $\tau_0$ at which the minimum separation was reached

Figure 3 shows the distribution of the minimum separations, $D_{\rm min}(\tau_0)$, and the kinematic ages, $\tau_0$, of the $4\,214$ simulations mentioned above. The lack of simulations which yield very small minimum absolute separations is due to the three-dimensional nature of the problem. Consider the following case: two objects are located at exactly the same position in space, e.g., the binary containing the pulsar progenitor and the runaway. However, the position measurement of each of these objects has an associated typical error. The distribution of the absolute separation between the objects, obtained from repeated measurements of the positions of both objects, can be calculated analytically for a Gaussian distribution of errors (see Appendix A). The solid line in Fig. 3 shows the result for an adopted distance measurement error of 2.5 pc, which agrees very well with our simulations. The peculiar statistical properties of the sample of successful simulations make it difficult to give a simple argument to derive the value of 2.5 pc from the uncertainties in the kinematic properties of the runaway and the pulsar. We suspect that the disagreement between the solid line and histogram for separations >6 pc is most likely due to a slight mismatch between the model and the actual situation. Even so, Fig. 3 shows that due to measurement errors, very few simulations will produce a small observed minimum separation, even when the intrinsic separation is zero.

Figure 4: Astrometric parameters of the runaway $\zeta$ Oph and the pulsar PSR J1932+1059 at the start of each of the $4\,214$ simulations for which the minimum separation between the runaway and the pulsar was less than 10 pc, and both of them were within 10 pc of the center of Upper Scorpius sometime in the past

Figure 4 shows the astrometric parameters of the pulsar and the runaway at the start of the orbit integration, i.e., the "present'' observables, for the simulations which result in a minimum separation less than 10 pc occurring within Upper Scorpius. The parameters of $\zeta$ Oph show no correlations except between the parallax and the radial velocity. This is expected due to the degeneracy of these two quantities (a change in stellar distance, depending on whether it increases or decreases the separation between the star and the association, can be compensated for by a larger or smaller radial velocity, respectively).

The parameters of the pulsar behave very differently. In addition to the $\pi$ vs. $v_{\rm rad}$ correlation, we find that the parallax is also correlated with both of the proper motion components. As a result, the proper motion components are correlated with each other. This means that only a subset of the full parameter space defined by the six-dimensional error ellipsoid of the pulsar fulfills the requirement that the pulsar and the runaway meet. Furthermore, if they met, then we know the radial velocity of the pulsar for each assumed value of its distance. A reliable distance determination would thus yield an astrometric radial velocity. The current best distance estimate of the pulsar derived from VLBI measurements, $\pi = 5\pm1.5$ mas (Campbell 1995), predicts a radial velocity of 100-200 km s^-1. This radial velocity is comparable to the tangential velocity: $\sim$100 km s^-1 (for $\pi = 5$ mas).

The pulsar proper motions of the $4\,214$ successful simulations are shown in Fig. 5, together with the proper-motion measurements of Lyne et al. (1982) (dot-dash line), Taylor et al. (1993) (solid line), and Campbell (1995) (dashed line). The measurements show a reasonable spread, reflecting the difficulty in obtaining pulsar proper motions, but are consistent, within 3$\sigma$, with the proper motions predicted by the simulations.

3.4 Interpretation

The observed astrometric and spectroscopic parameters of $\zeta$ Oph and PSR J1932+1059 are consistent with the assumption that these objects were very close together $\sim$1 Myr ago (Fig. 3). At that time both were within the boundary of Upper Scorpius (Fig. 6), which has a nuclear age of $\sim$5 Myr (de Geus et al. 1989).

Several characteristics of $\zeta$ Oph and Upper Scorpius support the interpretation that the runaway and the pulsar must have been produced by the BSS. (i) The HI distribution in the direction of Upper Scorpius shows an expanding shell-like structure. De Geus (1992) argued that the energy output of the stellar winds of the massive stars in Upper Scorpius is two orders of magnitudes too low to account for the kinematics of the HI shell, and proposed that a supernova explosion created the Upper Scorpius HI shell. (ii) Based on the present-day mass function, de Geus showed that the initial population of Upper Scorpius most likely contained one star or binary more than the present population. The estimated mass of this additional object is $\sim$40 $M_\odot$. Stars of this mass have main-sequence lifetimes of $\sim$4 Myr, and end their lives in a supernova explosion. Since the association has an age of $\sim$5 Myr, the supernova explosion might have taken place $\sim$1 Myr ago. (iii) The characteristics of $\zeta$ Oph are indicative of close binary evolution. The helium abundance is large, and the star has a large rotational velocity (see Sect. 1). These facts make it very likely that the same supernova event in Upper Scorpius created PSR J1932+1059 and endowed $\zeta$ Oph with its large velocity.

Figure 5: Proper motions of the pulsar PSR J1932+1059 at the start of the $4\,214$ successful simulations (dots; see Sect. 3.3), and the proper-motion measurements of the pulsar: dot-dash line denotes Lyne et al. (1982), solid line denotes Taylor et al. (1993), and the dashed line denotes Campbell (1995). The contours indicate the 1, 2, and 3$\sigma$ confidence levels

It could be that $\zeta$ Oph and PSR J1932+1059 are not related. This would imply that the pulsar originated in Upper Scorpius $\sim$1 Myr ago and that $\zeta$ Oph obtained its large velocity either in a separate BSS event in Upper Scorpius, or in Upper Centaurus Lupus, $\sim$3 Myr ago, as suggested by van Rensbergen et al. (1996). In the latter case, it would also have to be formed by the BSS, because of its high helium abundance, large rotational velocity, and the $\sim$10 Myr difference between the age of Upper Centaurus Lupus and the kinematic age of $\zeta$ Oph (Sect. 1). Given the small probability of finding a runaway star and a pulsar with orbits that cross, and with both objects at the point of intersection at the same time, we conclude that $\zeta$ Oph and PSR J1932+1059 were once part of the same close binary in Upper Scorpius, providing the first direct evidence for the generation of a single runaway star by the BSS.

Figure 6: The orbits of $\zeta$ Oph, PSR J1932+1059, and Upper Scorpius. The present positions are denoted by a star for the runaway, a filled circle for the pulsar, and by a filled square for the association. The top panel shows the distance vs. Galactic longitude of the stars. The bottom panel shows the orbits projected on the sky in Galactic coordinates. The small open circles in the bottom panel denote the present-day positions of the O, B, and A-type members of Upper Scorpius, taken from de Zeeuw et al. (1999). The large circle denotes the position of the association at the time of the supernova explosion, and has a 10 pc radius. This figure assumes a set of space motions consistent with the common origin hypothesis

3.5 Pulsar kick velocity

If $\zeta$ Oph and PSR J1932+1059 were once part of a binary, then we can derive a number of properties of this system. For example, the true age of the pulsar must be the kinematic age of 1 Myr (as compared to the characteristic age estimate of 3 Myr). It follows that, if no glitches occurred, the pulsar had a period of 0.18 s at birth, as compared to the current period of 0.22 s.

The velocity distribution of the pulsar population is much broader (a few$\times$100 km s^-1) than that of the pulsar progenitors (a few$\times$10 km s^-1). The mechanism responsible for this additional velocity (the "kick velocity'' $\vec{v}_{\rm kick}$) is not well understood (e.g., Lai 1999). The kick velocity is most likely due to asymmetries in the core of a star just before, or during, the supernova explosion.

The simulations described in Sect. 3.3 provide the velocities of the runaway star, the pulsar, and the association at the present time. This makes it possible to determine $\vec{v}_{\rm kick}$of the neutron star. For the $4\,214$ successful runs we find that the average velocities with respect to Upper Scorpius are: $\vec{v}_{\zeta~{\rm Oph}} = (-6.4\pm 4.2,$ $33.8\pm1.4$, $5.8\pm2.0)$ km s^-1 and $\vec{v}_{\rm pulsar} = (48.6 \pm 21.7$, $222.9\pm 36.1$, $-70.7 \pm 8.4)$ km s^-1 in Galactic Cartesian coordinates (U,V,W).

To derive $\vec{v}_{\rm kick}$ we consider a binary with components of mass M₁ and M₂ in a circular orbit, in which the first component (star₁) explodes as a supernova and creates a neutron star. At the time of the explosion, star₁ is the least massive component of the binary, due to the prior mass transfer phase, and is most likely a helium star. The rapidly expanding supernova shell, with mass $\Delta M = M_1 - M_{\rm n}$, where $M_{\rm n} = 1.4~M_\odot$ is the typical mass a neutron star, will quickly leave the binary system. The shell has a net velocity equal to the orbital velocity of star₁ at the moment of the explosion (v₁). A net amount of momentum ( $\Delta M \times v_1$) is thus extracted from the system and the binary reacts by moving in the opposite direction with a velocity $v = -(\Delta M\times v_1) / (M_2 + M_n)$, the so-called "recoil velocity''. The binary will remain bound after the explosion because less than half of the total mass of the system is expelled (M₁ < M₂; cf. Paper I). However, if the neutron star receives a kick in the supernova explosion the binary might dissociate, depending on the direction and magnitude of the kick velocity. We simulate this by using a simple orbit integrator for two bodies. We determine the semi-major axis and orbital velocities assuming the binary has a circular orbit and masses $M_1 = 5~M_\odot$and $M_2 = 15~M_\odot$ ($\zeta$ Oph). We then change the mass of star₁ to $M_{\rm n}$ and add a kick-velocity to its orbital velocity. We start the integration at this point and try to reproduce the observed velocity of $\zeta$ Oph, the pulsar, and the angle between the two velocity vectors (35$^\circ$). It turns out that a kick velocity of order 350 km s^-1 is needed in a direction almost opposite to the orbital velocity of star₁'s prior to the explosion. This value is in good agreement with the average pulsar kick velocity found by Hartman (1997) and Hansen & Phinney (1997). The current velocity of the pulsar, $\sim$240 km s^-1, is more than 100 km s^-1 smaller than the kick it acquired. Our simulations show that this deceleration is due to the gravitational pull of $\zeta$ Oph on the pulsar.

The mass of $\zeta$ Oph used in the above estimate is consistent with the calibration of Schmidt-Kaler (1982). The more recent mass calibration of Vanbeveren, Van Rensbergen & De Loore (1998) suggests 21 $M_\odot$ (Table 3). This would increase the inferred kick velocity to $\sim$400 km s^-1.