SPIN TILTS IN THE DOUBLE PULSAR REVEAL SUPERNOVA SPIN ANGULAR-MOMENTUM PRODUCTION

The radio-pulsar system PSR J0737−3039 is the only binary pulsar known to consist of two radio pulsars: PSR J0737−3039 A (Burgay et al. 2003) and PSR J0737−3039 B (Lyne et al. 2004). Table 1 gives the parameters of this system. This unique configuration has permitted measurements of spin orientation for both pulsars (Ferdman et al. 2008; Lyutikov & Thompson 2005; Breton et al. 2008): pulsar A's spin is tilted from the orbital angular momentum vector by no more than 14 deg at 95% confidence (Ferdman et al. 2008); pulsar B's by 130.0^+1.4_−1.2 deg at 99.7% confidence. Here we argue that this large difference between the two pulsar spin tilts requires that the origin of most of B's spin is connected to its supernova (SN) explosion; the spin of B's progenitor, expected to be aligned with the pre-SN orbit due to tidal interactions, cannot be invoked to explain the present-day misaligned spin of pulsar B. PSR J0737−3039 B is currently believed to have formed from an electron-capture SN triggered in a massive O-Ne-Mg white dwarf (van den Heuvel 2004; Willems & Kalogera 2004; Piran & Shaviv 2005; Stairs et al. 2006; Wang et al. 2006; Willems et al. 2006; van den Heuvel 2007; Breton et al. 2008; Wong et al. 2010). Our results demonstrate that, whatever the details of its formation mechanism, the SN that formed PSR J0737−3039 B produced the majority of its current spin. If the source of the present-day spin of pulsar B is a single, impulsive kick, then this kick must be off-center so that it tumbles the pulsar to its current orientation. Using constraints on the SN kick magnitude derived from the orbital and kinematic parameters of the system (Wong et al. 2010) we find that this kick must have been displaced from the center of mass of the exploding star by at least 1 km and probably 5–10 km. Such offset distances are a significant fraction of the expected radii of neutron stars. Off-center kicks were first suggested in Spruit & Phinney (1998) on purely theoretical grounds.

PSR J0737−3039 likely evolved from two stars originally massive enough to undergo SN explosions and form two neutron stars (Tauris & van den Heuvel 2010) at the end of their nuclear lifetimes. Given the measured spin magnitudes and inferred magnetic fields (Burgay et al. 2003; Lyne et al. 2004; Ferdman et al. 2008; Lyutikov & Thompson 2005; Breton et al. 2008), pulsar A was the first-born neutron star, while pulsar B formed in a second SN. After the first SN the system passed through a high-mass X-ray binary phase. In this phase, pulsar A accreted matter from its companion, leading to some spin-up. Eventually, pulsar A's companion evolved off the main sequence and its expanding hydrogen envelope enveloped pulsar A. In this common-envelope phase, tidal interactions between the stars circularized the orbit and are expected to have aligned the spins of pulsar A and of pulsar B's progenitor with the orbital angular momentum axis (perpendicular to the orbital plane). The transfer of orbital kinetic energy to the envelope eventually removed the outer layers of pulsar B's progenitor, leaving pulsar A in a tight orbit with the exposed helium-rich core of B's progenitor. After another brief period of mass transfer onto pulsar A (Dewi & van den Heuvel 2004; Willems & Kalogera 2004), the helium star exploded in the second SN, forming pulsar B. As a result of the multiple mass-transfer phases between the two SN events, just before pulsar B's SN the system was in a close, circular orbit with both stars' spins aligned with the orbital angular momentum vector.

Due to asymmetries associated with the SN ejecta (matter and/or neutrinos) SNe are thought to be capable of imparting a significant recoil impulse, a "kick," to any remnant surviving the explosion (see, e.g., Janka et al. 2008 and references therein). When an SN occurs in a binary system, these kicks can significantly alter the orbital parameters or even disrupt the binary. The kick component directed parallel to the pre-SN orbital plane causes a change in the eccentricity and semimajor axis of the orbit; the component perpendicular to the pre-SN orbital plane can also cause a change in the inclination of the orbital plane. In the PSR J0737−3039 system, pulsar A's small spin-tilt angle (less than 14 deg at 95% confidence using a two-pole emission model; Burgay et al. 2003; Lyne et al. 2004; Ferdman et al. 2008) is indicative of a relatively small out-of-plane kick from the SN that formed pulsar B (Wong et al. 2010). Pulsar A's spin-orbit misalignment occurs only because the orbital plane is tilted by the SN kick, while pulsar A's spin remains fixed in the inertial frame aligned with the pre-SN orbital angular momentum axis (Figure 1). Such a spin tilt for pulsar A occurs independently of the effects of the second SN on pulsar B's spin. In other words, the observed tilt of pulsar A's spin by itself does not require any change in the spin angular momentum of pulsar B relative to its progenitor. However, unless the SN contributes significant amounts of angular momentum to the nascent pulsar, the orientation of pulsar B's spin will be the same as its progenitor's spin, i.e., aligned with the pre-SN orbital plane and pulsar A's spin. Surprisingly, pulsar B's spin is in fact retrograde: tilted by 130.0^+1.4_−1.2 deg (99.7% confidence; Ferdman et al. 2008) relative to the current orbital angular momentum vector (see Figure 1).

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To produce pulsar B's retrograde spin, the SN must have significantly torqued pulsar B, causing it to tumble to the currently observed spin-orbit orientation. The pre-SN spin, S₀, the angular momentum produced by the SN ejecta, ΔS, and the post-SN spin,³ S_SN are related by the conservation of angular momentum

$$\begin{equation} {\bf S} _{{\rm SN}} = {\bf S} _0 + \Delta {\bf S}. \end{equation} \tag{ 1 }$$

To determine ΔS, we must know S₀ and S_SN, but we only know the direction, not the magnitude, of S₀ and the relationship between S_SN and the spin measured today is complicated by relativistic precession (Breton et al. 2008). However, we can still place constraints on ΔS. Relativistic precession causes the individual pulsar spins to precess about the total angular momentum of the system, which is approximately parallel to the orbital angular momentum. Such precession preserves the angle between the total angular momentum and the spin (which is the spin colatitude), but not the azimuthal orientation. Thus, the colatitude of S_SN relative to the normal to the current orbital plane is equal to the colatitude of the current spin—130 deg. Based on the spin of pulsar A, the current orbital plane could be tilted at most 14 deg relative to the pre-SN orbital plane. Therefore, the colatitude of S_SN relative to the pre-SN orbital plane—and therefore relative to S₀—is at least 116 deg. This is also the minimum angle between S₀ and ΔS. The angular momentum produced by the SN must be significantly misaligned with the progenitor spin. To date, most SN simulations have focused on non-rotating progenitors (for example, see Blondin & Mezzacappa 2007; Rantsiou et al. 2011; Wongwathanarat et al. 2010); it remains to be seen whether the spin produced by the SN from the collapse of a rotating progenitor can be so significantly misaligned with the progenitor's rotation axis.

The typical moment of inertia (Spruit & Phinney 1998) of a neutron star is 0.36MR²; using pulsar B's measured mass of 1.25 M_☉ (see Table 1) and a radius of 10 km, its current spin angular momentum is 2 × 10⁴⁵ g cm² s⁻¹. Since this spin is retrograde whereas the pre-SN spin is roughly aligned (within 14°, given pulsar A's small spin tilt) with the current orbital plane, we can place a lower limit on the change of angular momentum needed to explain pulsar B's large and retrograde spin tilt,

$$\begin{equation} \Delta S \ge 2 \times 10^{45}\, {\rm g} \,{\rm cm}^2 \,{\rm s}^{-1}, \end{equation} \tag{ 2 }$$

where equality holds when S₀ = 0. Because the angle between S₀ and S_SN is greater than 90 deg, any progenitor spin only increases the amount of angular momentum that must be added to the pulsar by the kick. This is demonstrated geometrically in Figure 2(c).

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The above discussion has been fully general. To extract more constraints from the observed spin–spin misalignment, we must make some assumptions about the origin of the pulsar spin. As a simplified model to elucidate the scales involved in this scenario, let us assume that the same impulsive kick (i.e., linear momentum) that changes the orbit of the system is also offset from the center of mass of pulsar B, and therefore applies a torque sufficient to produce the observed spin angular momentum. The kick and offset vectors must lie in the plane perpendicular to ΔS (see Equation (3)). The kick velocity, v_K, the offset vector relative to the center of mass, r, and the change in B's spin vector are related by

$$\begin{equation} \Delta {\bf S} = {\bf r} \times \Delta {\bf p} = {\bf r} \times M_B {\bf v} _K, \end{equation} \tag{ 3 }$$

where Δp = M_Bv_K is the change in linear momentum induced by a change in velocity of v_K in an object with mass M_B. The offset length r and kick velocity magnitude v_K must then satisfy the inequality

$$\begin{equation} r \ge \frac{\Delta S}{M_B v_K}, \end{equation} \tag{ 4 }$$

where ΔS is the magnitude of the change of B's spin; the equality holds only when the kick and offset are perpendicular to each other.

The relative orientation of the current spin provides a constraint on the kick direction in this scenario. Let the colatitude of ΔS relative to the pre-SN orbital plane be θ_Δ; because the angle between S₀ and S_SN is greater than 90 deg, no matter the magnitude of S₀ we must have θ_Δ ⩾ 116 deg. Let the plane perpendicular to ΔS make an angle ψ with respect to the pre-SN orbital plane. Then we have ψ = 180 − θ_Δ ⩽ 64 deg. The kick, v_K, lies in this plane and therefore must have colatitude θ_K that satisfies

$$\begin{equation} 90-\psi = 26 \le \theta _K \le 154 = 90 +\psi. \end{equation} \tag{ 5 }$$

This geometry is illustrated in Figure 2. The constraint in Equation (5) is consistent with the constraint on kick colatitude in Figure 7 of Wong et al. (2010), but tighter.

The constraints on the magnitude of the spin change, Equation (2), together with Equation (4), imply a lower limit on the offset distance

$$\begin{equation} r \ge 3.2 \left(\frac{25 \,{\rm km\,s}^{-1}}{v_K} \right) \,{\rm km}. \end{equation} \tag{ 6 }$$

If we assume that the core of pulsar B's progenitor was in synchronous, rigid-body rotation just before the SN then S^SR₀ ≃ 2 × 10⁴⁵ g cm² s⁻¹ and the limit on the offset distance rises by a factor of 1.8:

$$\begin{equation} r_{{\rm SR}} \ge 5.8 \left(\frac{25 \,{\rm km\,s}^{-1}}{v_K} \right) \,{\rm km}. \end{equation} \tag{ 7 }$$

Wong et al. (2010) used the measured semimajor axis, eccentricity, and proper motion of the J0737−3039 system (see Table 1) to constrain the kick imparted to the system by the second SN. The Wong et al. (2010) analysis assumed that the pre-SN orbit was circular and that the system came from a progenitor population with number density

$$\begin{equation} n(R,z) = n_0 \exp \left(-\frac{R}{h_R} \right) \exp \left(- \frac{|z|}{h_z} \right), \end{equation} \tag{ 8 }$$

with h_R = 2.8 kpc and h_z = 0.07 kpc the galactic scale length and height, respectively, moving with the local galactic rotation velocity. The SN kick and mass loss must then induce the current eccentricity and semimajor axis in the orbit and give the system as a whole a velocity such that it moves in the galactic potential to its current location in the 100–200 Myr since the second SN (Lorimer et al. 2007). Because of the uncertainty in the amount of mass loss, pre-SN semimajor axis, pre-SN galactic location, the age of the system, and the measurement uncertainty in the current orbital parameters, a range of kick magnitudes between 0 and 60 km s⁻¹ is allowed in the Wong et al. (2010) analysis (Wong et al. 2010, Figure 5). In Figure 3 we show the probability distribution of minimum offset distances implied by this distribution of kick velocities. Even for large kick velocities, the minimum offset distance exceeds ∼1 km. For the smallest allowed kicks, the minimum offset distances exceed the current ∼10 km radius of the neutron star.

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The magnitude of the offset, r, required to produce the needed ΔS depends on the relative orientation of the offset and kick, v_K. If the angle between r and v_K in the plane perpendicular to ΔS is θ_rK, then

$$\begin{equation} r = \frac{\Delta S}{M_Bv_K\sin \theta _{rK}}, \end{equation} \tag{ 9 }$$

which is larger than the minimum offset distance by a factor of (sin θ_rK)⁻¹ (see Equation (3)). Kicks that are nearly aligned with the radial vector (small θ_rK) require arbitrarily large off-center distances to match the current spin orientation. Even modest misalignments of 40 deg give an enhancement factor of sin ⁻¹40 ∼ 1.6 over the offset for perpendicular r and v_K.

Multi-dimensional simulations (Blondin & Mezzacappa 2007) of core-collapse SNe have shown that an instability in a stationary accretion shock (SASI) may provide a method of depositing a substantial amount of spin angular momentum (2 × 10⁴⁷ g cm² s⁻¹) onto a proto-neutron star as part of the collapse process itself and separate from any rotation of the progenitor. (In fact, the simulations of Blondin & Mezzacappa 2007 used non-rotating progenitors.) This would be more than enough angular momentum to account for the observed spin angular momentum of pulsar B (2 × 10⁴⁵ g cm² s⁻¹). The general consensus from recent evolutionary studies (van den Heuvel 2004; Willems & Kalogera 2004; Podsiadlowski et al. 2005; Stairs et al. 2006; Wang et al. 2006; Willems et al. 2006; van den Heuvel 2007; Breton et al. 2008; Wong et al. 2010) is that pulsar B was formed in an electron-capture SN, where an iron core is never formed and instead core collapse is initiated through electron captures onto Ne/Mg nuclei (Dessart et al. 2006; Kitaura et al. 2006; Podsiadlowski et al. 2004; Miyaji et al. 1980; Nomoto 1984, 1987). During the course of such a collapse, the proto-neutron star shrinks from a radius of ∼100 km to ∼10 km.

We emphasize that the assumption of the foregoing discussion that the kick is applied at a single location is simplistic (see, e.g., Spruit & Phinney 1998); in a real SN, both linear and angular momentum will be accumulated by the proto-neutron star throughout its formation at many different locations. In this more realistic context, the constraints above on offset distances should be interpreted instead as constraints on the offset scale at which the bulk of the linear and angular momentum is accumulated. More precise constraints will require detailed modeling of the hydrodynamic process of momentum accumulation in the SN that formed PSR J0737−3039B. Nevertheless, it is interesting that the location of kicks inferred from such a simple model is consistent with kick origins in the bulk of the shrinking proto-neutron star during the SN (see Figure 3). Some recent SN modeling suggests that the processes that produce the kick and those that impart rotation to the resulting neutron star produce independent kicks and spins, and therefore there is little correlation between the kick magnitude and direction and the rotation imparted to the post-SN compact object (Wongwathanarat et al. 2010; Rantsiou et al. 2011). In this case the offset length scale inferred above from the dynamical constraints on the kick would not be relevant. We conclude that only if pulsar B's spin is actually linked to the torque induced by the physical mechanism producing the kick it must be offset from the center of mass of the collapsing neutron star progenitor.

Regardless of the specifics of the collapse process, however, the expected alignment of the spin of pulsar B's SN progenitor with the pre-SN orbital angular momentum and the observed misalignment of pulsar B's spin and orbit at present uniquely imply that pulsar B's spin is dominated by angular momentum produced during the SN process, not angular momentum provided by the progenitor. The realization of this empirical constraint on angular momentum production in SNe presented here is uniquely enabled by the spin–spin misalignment in the PSR J0737−3039 system (Lyne et al. 2004; Ferdman et al. 2008; Lyutikov & Thompson 2005) and can be used to guide core-collapse simulations and the quest for the understanding of compact object spins and kicks.

We thank Tsing Wai Wong for providing data on the distribution of allowed pulsar kicks from the current orbital constraints from Wong et al. (2010, Figure 5). We also thank Ingrid Stairs, Hans-Thomas Janka, Adam Burrows, and Rodrigo Fernandez for helpful comments on drafts of this manuscript. W.M.F. and V.K. are partially supported by NSF grant AST-0908930. K.K. acknowledges support from a NASA summer research grant through the Illinois Space Grant NNG05G381H. M.L. is supported by NASA grant NNX09AH37G.