Neutrino-heated winds from millisecond proto-magnetars as sources of the weak r-process

TL;DR

This study investigates how neutrino-driven winds from rapidly rotating, magnetized proto-neutron stars can produce heavy elements, potentially explaining observed abundance variations and contributing to ultra-high energy cosmic rays.

Contribution

It demonstrates that proto-magnetars with millisecond periods can synthesize heavier elements than spherical winds, highlighting the role of magnetic fields and rotation in nucleosynthesis diversity.

Findings

Proto-magnetars with P ~ 1-5 ms produce extended heavy element distributions.

Outflows near the closed zone and equatorial plane synthesize the heaviest elements.

Variations in magnetic field and rotation influence nucleosynthesis patterns.

Abstract

We explore heavy element nucleosynthesis in neutrino-driven winds from rapidly-rotating, strongly magnetized proto-neutron stars for which the magnetic dipole is aligned with the rotation axis, and the field is assumed to be a static force-free configuration. We process the proto-magnetar wind trajectories calculated by Vlasov et al 2014 through the r-process nuclear reaction network SkyNet using contemporary models for the evolution of the wind electron fraction during the proto-neutron star cooling phase. Although we do not find a successful second or third peak r-process for any rotation period P, we show that proto-magnetars with P around 1-5 ms produce heavy element abundance distributions that extend to higher nuclear mass number than from otherwise equivalent spherical winds (with the mass fractions of some elements enhanced by factors of 100-1000). The heaviest elements are synthesized by outflows emerging along flux tubes which graze the closed zone and pass near the equatorial plane outside the light cylinder. Due to dependence of the nucleosynthesis pattern on the magnetic field strength and rotation rate of the proto-neutron star, natural variations in these quantities between core collapse events could contribute to the observed diversity of the abundances of weak r-process nuclei in metal-poor stars. Further diversity, including possibly even a successful third-peak r-process, could be achieved for misaligned rotators with non-zero magnetic inclination with respect to the rotation axis. If proto-magnetars are central engines for GRBs, their relativistic jets should contain a high mass fraction of heavy nuclei of characteristic average mass number A of order 100, providing a possible source for ultra-high energy cosmic rays comprised of heavy nuclei with an energy spectrum that extends beyond the nominal GZK cut-off for protons or iron nuclei.

Tables

Table 1. Table 1: Summary of Time-integrated Models

P, ms	${\overline{A}}^{\left(a\right)}$	${\overline{Z}}^{\left(a\right)}$	$X_{\alpha }$	$X_n^{†}$	$X_p^{†}$	$M_{\mathrm{ej}},M_{\odot }$
-∗	87.34	37.71	0.561	$9.24\cdot {10}^{-7}$	$3.65\cdot {10}^{-7}$	$9.98\cdot {10}^{-4}$
1	98.94	42.61	0.429	$4.83\cdot {10}^{-4}$	$5.58\cdot {10}^{-5}$	$3.45\cdot {10}^{-3}$
2	98.73	42.53	0.685	$3.42\cdot {10}^{-4}$	$4.07\cdot {10}^{-5}$	$1.79\cdot {10}^{-4}$
3	97.81	42.13	0.719	$2.18\cdot {10}^{-4}$	$2.55\cdot {10}^{-5}$	$1.06\cdot {10}^{-4}$
4	97.36	41.94	0.714	$1.90\cdot {10}^{-4}$	$2.22\cdot {10}^{-5}$	$7.80\cdot {10}^{-5}$
5	95.07	40.96	0.688	$1.62\cdot {10}^{-4}$	$1.90\cdot {10}^{-5}$	$6.64\cdot {10}^{-5}$
10	94.07	40.53	0.675	$9.34\cdot {10}^{-5}$	$1.08\cdot {10}^{-5}$	$2.52\cdot {10}^{-5}$

Equations

$$\frac{R_{\mathrm{rs}}}{R_{\mathrm{L}}}\approx 37{\left(\frac{v_{\mathrm{ej}}}{10^4\mathrm{km}{\mathrm{s}}^{-1}}\right)}^{3/2}\left(\frac{t}{1\mathrm{s}}\right){\left(\frac{P}{1\mathrm{ms}}\right)}^{-1}.$$

$$s=s_0+X_{\alpha }(s,\rho ,Y_e)\cdot \Delta s_{\alpha },$$

$${\nu }_e+n\leftrightarrow p+e^{-}{\bar{\nu }}_e+p\leftrightarrow n+e^{+}.$$

$$Y_{\mathrm{e},\mathrm{eq}}={\left(1+\frac{L_{{\bar{\nu }}_e}}{L_{{\nu }_e}}\frac{⟨{\epsilon }_{{\bar{\nu }}_e}⟩-2\Delta +1.2{\Delta }^2/⟨{\epsilon }_{{\bar{\nu }}_e}⟩}{⟨{\epsilon }_{{\nu }_e}⟩+2\Delta +1.2{\Delta }^2/⟨{\epsilon }_{{\nu }_e}⟩}\right)}^{-1},$$

$$f_{\mathrm{open}}\approx \frac{2\pi {\theta }_{\max }^2}{4\pi }\approx \frac{1}{2}\frac{R_{\mathrm{ns}}}{R_{\mathrm{Y}}}\approx 0.13{\left(\frac{P}{1\mathrm{ms}}\right)}^{-1},$$

$$R\le \min \frac{X_{\mathrm{i},\odot }{M}_{\mathrm{gas}}}{M_{\mathrm{ej},\mathrm{i}}t}=\frac{X_{\mathrm{i},\odot }{M}_{\mathrm{gas}}}{X_{\mathrm{i},\mathrm{wind}}{M}_{\mathrm{ej}}{t}_{\mathrm{gal}}},$$

$${\chi }_{75}=170E_{20}^{-1.5}{A}_{56}^{1.3}{\bar{n}}_{\mathrm{ej}}^{-1}\mathrm{Mpc}$$

Full text

††pagerange: Neutrino-heated winds from millisecond proto-magnetars as sources of the weak $r$-process–3.2††pubyear: 2014

1 Introduction

The astrophysical sites responsible for synthesizing the heaviest elements in the Universe via the rapid neutron capture process ($r$-process, Burbidge et al. 1957; Cameron 1957) have been debated for decades (for reviews, see Qian & Wasserburg 2007; Arnould et al. 2007; Sneden et al. 2008; Thielemann et al. 2011).

Core collapse supernovae (SNe) have long been considered potential $r$-process sites. This is in part due to their short delays following star formation, which allows the earliest generations of metal-poor stars in our Galaxy (e.g. Mathews et al. 1992, Sneden et al. 2008), or satellite dwarf galaxies (e.g. Roederer 2016), to be polluted with $r$-process elements prior to significant iron enrichment. Throughout the 1990s, the high entropy neutrino-heated winds from proto-neutron stars (PNS) (Duncan et al. 1986; Qian & Woosley 1996), which emerge on a timescale of seconds after a successful explosion as the PNS deleptonizes, were considered the most likely $r$-process site111 Another $r$-process mechanism in the core collapse environment results from $\nu-$induced spallation in the He shell (e.g., Banerjee et al. 2011). This channel is limited to very low metallicity $Z\lesssim 10^{-3}$ and thus cannot represent the dominant $r$-process source over the age of the galaxy, though it could be important for the first generations of stars. within the core collapse environment (e.g., Meyer et al. 1992; Woosley et al. 1994). A high entropy, or correspondingly low density, results in an $\alpha$-rich freeze-out of the 3- and effective 4-body reactions responsible for forming seed nuclei in the wind (Woosley & Hoffman 1992). The resulting higher ratio of neutrons to seed nuclei then allows neutron captures to proceed to heavier elements than if the protons were instead entirely trapped in heavy seeds.

However, some contemporary (e.g., Takahashi et al. 1994) and many subsequent calculations of the wind properties (e.g., Qian & Woosley 1996; Kajino et al. 2000; Sumiyoshi et al. 2000; Otsuki et al. 2000; Thompson et al. 2001; Arcones et al. 2007; Roberts et al. 2010; Martínez-Pinedo et al. 2012; Roberts et al. 2012; Fischer et al. 2012) showed that the requisite combination of low electron fraction $Y_{e}\lesssim 0.5$ and high entropy needed to reach the second or third $r$-process peaks (Hoffman et al. 1997) were unlikely to be reached. Possible exceptions include a very massive PNS (Cardall & Fuller 1997), or given the presence of additional wind heating by damping of convectively-excited acoustic or Alfvén waves (Suzuki & Nagataki 2005; Metzger et al. 2007) or of non-standard physics, such as an eV-mass sterile neutrino (e.g., Tamborra et al. 2012; Wu et al. 2014).

If all SNe produced the $r$-process in equal quantities, the required mass of $r$-process elements per event to explain Galactic abundances is relatively low, $\sim 10^{-5}M_{\odot}$ (e.g. Macias & Ramirez-Ruiz 2016). However, several lines of evidence instead support much ‘higher yield’ $r$-process events being common in our Galaxy, both now and in its early history. These include the detection of 244Pu on the ocean floor at abundances roughly 2 orders lower than that expected if the source were frequent, low-yield events like those predicted from PNS winds in normal SNe (Wallner et al., 2015; Hotokezaka et al., 2015). A fraction of the stars in the dwarf galaxy Reticulum II are highly enriched in $r$-process elements, indicating that this galaxy was polluted early in its history by a single $r$-process event with a yield much higher than the standard neutrino-driven wind (Ji et al., 2016). Based on an analysis of the mass swept up by the SN blast wave, Macias & Ramirez-Ruiz (2016) argue that, in order to explain the largest enhancements in [Eu/Fe] at low metallicites, individual $r$-process events must synthesize at least $10^{-3.5}M_{\odot}$ of $r$-process material, again significantly higher than predicted by standard PNS wind models. Such high abundances, if present in young Galactic SN remnants, would be detectable by their radioactive gamma-ray or X-ray decay lines (Qian et al. 1998; Ripley et al. 2014).

Increasingly, the mergers of compact binary neutron star systems are seen as promising alternative sites for at least the heaviest $r$-process elements (Lattimer & Schramm 1974; Eichler et al. 1989; Freiburghaus et al. 1999). General relativistic hydrodynamical simulations of the merger events show that some of the matter ejected dynamically during the merger retains a sufficiently low electron fraction ($Y_{e}\lesssim 0.1$) to form the heavy $r$-process elements extending beyond the third $r$-process peak at atomic number Z\mathrel{\hbox{\hbox to0.0pt{\hbox{\lower 4.0pt\hbox{\sim}}\hss}\hbox{>}}}78 (e.g., Goriely et al. 2011; Wanajo et al. 2014).

Less clear in the merger framework is the origin of the charged particle process nuclei222These elements are sometimes also referred to as light element primary process nuclei (Arcones & Montes 2011). ($Z=38-40$) and of the ‘light’ or ‘weak’ $r$-process nuclei ($Z\approx 41-55$) nuclei. It is well known that $r$-process nuclei with $Z\lesssim 56$ in metal-poor stars show greater star-to-star variation in their abundance patterns than the heavier $r$-process nuclei (e.g. Honda et al. 2006; Roederer et al. 2010), despite showing an apparently robust pattern similar to the solar abundances in the heavier range $56\lesssim Z\lesssim 76$ (e.g. Sneden et al. 2008). Such abundances variations are again difficult to explain from standard PNS winds, which, other than due to usually modest variations in the PNS mass, should produce broadly similar yields for each supernova. This has motivated considering alternative sources for the light $r$-process nuclei in the environments of neutron star mergers, such as accretion disk winds (Fernández & Metzger 2013; Perego et al. 2014; Just et al. 2015; Martin et al. 2015; Wu et al. 2016) or components of the dynamical ejecta with higher electron fractions (Wanajo et al. 2014; Goriely et al. 2015).

This paper focuses on another variation on the canonical picture of neutrino-driven wind that occurs if the PNS is formed rapidly-rotating, with an ultra-strong magnetic field B\mathrel{\hbox{\hbox to0.0pt{\hbox{\lower 4.0pt\hbox{\sim}}\hss}\hbox{>}}}10^{14}-10^{15} G, similar to those of Galactic magnetars (a so-called “millisecond proto-magnetar”; Thompson 2003). If magnetic fields are dynamically-important during the SN explosion itself, magnetic acceleration of matter away from the PNS can act to lower the asymptotic electron fraction of the unbound ejecta by preventing the outflowing matter from coming into equilibrium with neutrino absorption reactions (Metzger et al. 2008), possibly facilitating a heavy $r$-process (Metzger et al., 2008; Winteler et al., 2012; Nishimura et al., 2015, 2016). Although such models have great potential, quantitative studies of MHD-SNe are still in their infancy (e.g. Takiwaki et al. 2016). High resolution three-dimensional simulations are needed to resolve the dynamo responsible for tapping into the shear kinetic energy to generate a large-scale magnetic field (Guilet & Müller 2015; Mösta et al. 2015; Sawai & Yamada 2016). They are also needed to capture the growth of non-axisymmetric (e.g. magnetic kink and sausage mode) instabilities, which can disrupt MHD jet-like structures that otherwise are stable in axisymmetric simulations (e.g., Mösta et al. 2014). The observed rate of presumably MHD-powered, hyper-energetic SNe is also low compared to the total core collapse rate (e.g. Podsiadlowski et al. 2004), thus requiring a high $r$-process yield per event to explain the Galactic abundances through this channel alone.

Even if rotation and magnetic fields are not dynamically important during the explosion phase itself, they will become so during the subsequent neutrino-wind phase (Thompson 2003; Thompson et al. 2004; Metzger et al. 2007). Indeed, such a situation is potentially much more common than a full-blown MHD-powered SN, because the angular momentum of the progenitor stellar core needed to produce a PNS rotating with a period of P\mathrel{\hbox{\hbox to0.0pt{\hbox{\lower 4.0pt\hbox{\sim}}\hss}\hbox{>}}}3-4 ms is probably lower than that needed to explode the star (although see Thompson et al. 2005). Furthermore, the impact of such a moderately rapidly-rotating magnetar on the properties of the explosion, such as its total energy and 56Ni yield, could be comparatively modest (e.g. Suwa & Tominaga 2015). The total birth rate of Galactic magnetars is estimated to exceed 10% of the total core collapse rate (Woods & Thompson 2006), but the birth rotation rates are largely unconstrained observationally. A small population of core collapse supernovae with extremely high optical luminosities may result from millisecond magnetar birth (e.g. Kasen & Bildsten 2010; Woosley 2010; Metzger et al. 2014). However, the limited range of rotation periods and surface dipole magnetic field strengths which result in greatly enhanced SN emission imply that only a fraction of all magnetar births manifests this way.

Vlasov et al. (2014, hereafter V14) solved for the steady-state structure of neutrino-heated winds from rotating, magnetized PNS, building on the previous one-dimensional equatorial monopole calculations of Metzger et al. (2007). V14 assumed that the magnetic field structure was that of an axisymmetric aligned dipole under the force-free approximation (Timokhin, 2006). This is valid provided that the energy density of the magnetic field greatly exceeds that of gas pressure and the kinetic energy density, as they find to be valid throughout the radii where the most important nucleosynthesis occurs for surface dipole field strengths of B_{\rm d}\mathrel{\hbox{\hbox to0.0pt{\hbox{\lower 4.0pt\hbox{\sim}}\hss}\hbox{>}}}10^{14}-10^{15} G, depending on the neutrino luminosity (V14, their Fig. 3). V14 obtained a series of one-dimensional solutions calculated along flux tubes corresponding to different polar field lines. They stitched together these flux tube solutions to determine the global wind properties across the entire open magnetosphere at a fixed neutrino luminosity and rotation period.

V14 found that proto-magnetars with rotation periods of $P\sim 2-3$ ms produce outflows more favorable for the production of third-peak $r$-process nuclei. This is due to their much shorter expansion times $\tau_{\rm dyn}$ through the seed nucleus formation region, yet only moderately lower entropies $s$, as compared to spherical non-rotating PNS winds. They found that the critical ratio of $s^{3}/\tau_{\rm dyn}$ was higher than for spherical winds, but not sufficiently so that nucleosynthesis would proceed to the third $r$-process peak at $A\sim 200$, based on the analytic criteria of Hoffman et al. (1997). Except in the case of extremely rapid rotation, near the centrifugal break-up period of $P\sim 1$ ms, magnetic acceleration was not found to significantly enhance the mass outflow rate per unit surface area. In fact, the total mass loss rate $\dot{M}$ is in most cases substantially lower than in an otherwise equivalent spherical wind because outflows occur only along open magnetic field lines, which thread only a small fraction ($\lesssim 10\%$, depending on the rotation period) of the total PNS surface.

The present paper explores the nucleosynthetic yield of magnetar birth in greater detail by processing the wind trajectories from V14 through the nuclear reaction network SkyNet (Lippuner & Roberts 2015) in order to determine the detailed wind abundance pattern ($\S\ref{sec:methods}$). Even in the non-rotating spherical case, several critical wind properties remain uncertain; most notably the electron fraction $Y_{e}$ depends sensitively on the differences between the luminosities and mean energies of the electron neutrinos and anti-neutrinos diffusing out from the PNS interior. The accuracy of these luminosities and neutrino energies is currently limited by uncertainties in the radiation transport and microphysics. Thus, in most cases we focus on comparing our calculated yields to those of otherwise equivalent spherical winds, in order to isolate the diversity in the nucleosynthesis imprinted exclusively by magneto-rotational effects ($\S\ref{sec:results}$). Although we find that proto-magnetars are probably incapable of synthesizing the heaviest $r$-process elements, at least during the neutrino wind phase, their winds may still contribute to the inferred Galactic diversity of weak $r$-process sources ($\S\ref{sec:weak}$).

Beyond their role as possible sources of $r$-process nucleosynthesis and engines for powering luminous supernovae, millisecond magnetars are contenders for the central engines powering gamma-ray bursts (Usov 1992; Wheeler et al. 2000; Thompson et al. 2004; Metzger et al. 2011a). If GRBs are indeed powered by the rotational energy of a magnetar, then the nucleosynthesis products of their winds will be directly entrained in the relativistic jet which escapes from the star and powers the prompt gamma-ray emission. As we describe in $\S\ref{sec:uhecr}$, this unique jet composition could have implications for the composition of ultra-high energy cosmic rays if they are accelerated in GRB jets (Metzger et al. 2011b).

2 Nuclear Reaction Network Calculations

2.1 Thermodynamic Trajectories

In the limit of force-free electrodynamics, the geometric structure of the proto-magnetar wind from aligned rotator is fully specified by the magnetar rotation period $P$ and the Y point radius $R_{\rm Y}$, which defines the intersection between the last open field lines of the polar cap and the equatorial plane. We operate under the assumption that $R_{\rm Y}$ equals the radius of the light cylinder, $R_{\rm L}=c/\Omega=2\pi cP$, where $\Omega=2\pi/P$ is the magnetar rotational angular frequency (Fig. 1). Unbound outflows occur along open magnetic flux bundles, with the outflow geometry varying with latitude $\theta$ from the pole at $\theta=0$ to the last open field line at $\theta=\theta_{\rm max}\simeq\sin^{-1}\left[R_{\rm ns}/R_{\rm L}\right]$, where $R_{\rm ns}=12$ km is the assumed NS radius.

Our nucleosynthesis calculations are performed along steady-state, one-dimensional wind trajectories for different field lines $\theta$, as calculated by V14 for different values of $P$ and electron neutrino luminosity $L_{\nu}\equiv(L_{\nu_{e}}+L_{\bar{\nu}_{e}})/2$, where $L_{\nu_{e}}$ and $L_{\bar{\nu}_{e}}$ are the neutrino and anti-neutrino luminosities, respectively. As described below, the results for different flux tubes are combined by integrating across the entire open magnetosphere, $\int_{0}^{\theta_{\rm max}}\langle...\rangle d\Omega$, to quantify the total nucleosynthesis of the wind as a function of $P$ and $L_{\nu}$. The steady-state approximation we adopt is valid if the magnetosphere structure does not change with rotation or due to changes in the convective structure of the star. The timescale for the outflow to pass through the nucleosynthesis region is much shorter than the timescale over which $L_{\nu}$ is decreasing due to the Kelvin-Helmholtz cooling evolution of the NS, or the timescale over which $P$ is increasing due to angular momentum losses due to magnetic dipole spin-down.

We start our reaction network calculations just after free nucleons recombine into $\alpha$-particles at a temperature of $T\simeq 7\times 10^{9}$ K. At small radii, where the magnetic field is dynamically strong compared to the thermal or kinetic energy densities, the force-free approximation is valid and we employ density trajectories from V14. At sufficiently large radii, matter inertia comes to dominate the energy density of the magnetic field, and the outflows should approach a spherical outflow. The density profile of a steady-state wind which has reached a constant asymptotic velocity, should approach $\propto 1/r^{2}\propto 1/t^{2}$, while at larger radii where internal velocity gradients become important the profile will approach $\rho\propto 1/r^{3}\propto 1/t^{3}$ appropriate for a freely expanding homologous outflow. As shown in Figure 3, we interpolate directly between the V14 trajectories at small radii and the asymptotic $1/t^{3}$ dependence at large radii. This simplification is motivated by the fact that the qualitative features of the abundance patterns are robust to the detailed density trajectory outside of the radii where charged particle processes cease and the neutron-to-seed ratio has been determined.

Another simplification is that we neglect heating or deceleration of the wind by the reverse shock, which is produced as the wind interacts with the surrounding supernova ejecta. This is justified in part333In addition, highly magnetized winds experience much weaker compressional heating than unmagnetized winds due to the additional support from magnetic pressure. because the radius of the pulsar wind termination shock $R_{\rm rs}\simeq(v_{\rm ej}/c)^{1/2}R_{\rm ej}$ (e.g. Gaensler & Slane 2006), where $v_{\rm ej}$ is the mean velocity of the SN ejecta, is generally much larger than the radius where the formation of seed nuclei occurs. The latter generally occurs close to the light cylinder radius, which determines the radial scale of outflow divergence. At time $t$ since explosion we have $R_{\rm ej}\simeq v_{\rm ej}t$ and thus

[TABLE]

We thus have $R_{\rm rs}\gg R_{\rm L}$ for times t\mathrel{\hbox{\hbox to0.0pt{\hbox{\lower 4.0pt\hbox{\sim}}\hss}\hbox{>}}}1 s and rotation periods $P\lesssim 10$ ms of interest, implying that seed formation occurs well inside the termination shock.

We start the nucleosynthesis network after the formation of $\alpha$-particles. By this time, most of the key parameters of the outflow (entropy $s$, expansion timescale $\tau_{\rm dyn}$, electron fraction $Y_{e}$ and mass loss rate $\dot{M}$) have already been determined in the free nucleon zone at small radii, where nuclear statistical equilibrium (NSE) provides a good approximation for equation of state. Furthermore, after $\alpha$-particle formation, neutrino heating and cooling have become negligible and hence are not included in the network reactions. Figure 2 shows that our calculations begin at radii $5-7$ times larger than the location where the neutrino heating rate reaches its maximum just above the NS surface. The locations of maximum neutrino heating are marked by stars in this figure, while the locations where $T=0.5$ MeV, which approximately corresponds to the locations of $\alpha$-formation, are marked by circles.

V14 do not account for entropy gain from $\alpha$-particle formation, which we therefore increase by hand from its initial value $s_{0}$ from V14 according to

[TABLE]

where $X_{\alpha}(s,\rho)$ is mass fraction of $\alpha$ under NSE with given entropy, $\Delta s_{\alpha}=Q_{\alpha}/T_{\alpha}$ is entropy gain from $\alpha$ formation, $Q_{\alpha}$ is the heat of $\alpha$-formation, and $T_{\alpha}$ is the temperature of $\alpha$ formation. Since the latter depends weakly on other wind parameters, we adopt a fixed value of $\Delta s_{\alpha}=10k_{B}$ nucleon*-1*, corresponding to $T_{\alpha}=8$ GK. The starting temperature for the reaction network is determined from the density and the entropy according to equation (2), with subsequent evolution tracked self-consistently from the density trajectory and radioactive heating.

Finally, we must specify the initial outflow electron fraction used in our network calculations, which equals the “final” electron fraction $Y_{e,\rm final}$ set by processes near the PNS surface. In standard thermally-driven PNS winds, this value is determined by the competition between electron neutrino and electron anti-neutrino capture reactions on free nucleons,

[TABLE]

These reactions have frozen-out (become slow compared to the expansion rate) at the large radii where our calculations would begin, resulting in an electron fraction close to the equilibrium value set by neutrino absorption reactions (Qian & Woosley 1996)

[TABLE]

where $\Delta=1.293$ MeV is proton-neutron mass difference and $\langle\varepsilon_{\bar{\nu}_{e}}\rangle/\langle\varepsilon_{\nu_{e}}\rangle$ are the mean energies of electron neutrinos and antineutrinos, respectively.

For relatively slowly rotating proto-magnetars, $P>2$ ms, the final electron fraction is very similar to the normal thermally-driven case, i.e. $Y_{\rm e,final}\simeq Y_{\rm e,eq}$. However, in the most rapidly spinning cases, $P\lesssim 2$ ms, we can have $Y_{\rm e,final}<Y_{\rm e,eq}$ due to rapid magnetocentrifugal acceleration (Thompson et al. 2004; Metzger et al. 2007, 2008, V14), which causes a premature freeze-out of the neutrino absorption processes before $Y_{e}$ is raised completely from its low value near the PNS surface. More quantitatively, introducing the definition ($\Delta Y_{e})_{\rm cent}\equiv Y_{\rm e,final}-Y_{\rm e,eq}$, we find that maximum ($\Delta Y_{e})_{\rm cent}\simeq-0.14$ for $P=1$ ms, $L_{\nu}=6\cdot 10^{51}$ ergs s${}^{-1},\theta=\theta_{\rm max}$ and ($\Delta Y_{e})_{\rm cent}\simeq-0.011$ for $P=2$ ms, $L_{\nu}=10^{52}$ ergs s${}^{-1},\theta=\theta_{\rm max}$. We keep the dependence of $(\Delta Y_{e})_{\rm cent}$ on $L_{\nu}=0.5(L_{\nu_{e}}+L_{\bar{\nu}_{e}})$ and $\theta$ and we neglect the dependence on the other parameters ($Y_{e,eq},L_{\nu_{e}},\varepsilon_{\nu_{e}},\varepsilon_{\bar{\nu}_{e}}$). As the initial conditions for our network calculations we therefore take $Y_{\rm e,final}=Y_{\rm e,eq}(t)+(\Delta Y_{e})_{\rm cent}[L_{\nu},\theta]$ for $P=1,2$ ms and $Y_{e}=Y_{e}^{\rm eq}(t)$ for all other periods.

The value of $Y_{\rm e,eq}(t)$ is calculated from equation (4) using the time evolution of $L_{\nu_{e}},L_{\bar{\nu}_{e}},\langle\varepsilon_{\bar{\nu}_{e}}\rangle,\langle\varepsilon_{\nu_{e}}\rangle$ from PNS cooling calculations of Roberts et al. (2012). Given the uncertainties in neutrino radiation transport (e.g. Fischer et al. 2012; Roberts et al. 2012; Martínez-Pinedo et al. 2012), especially in the essentially unexplored case of very rapid rotation (however, see Thompson et al. 2005; Thompson 2007) and given the effects of ultra-strong magnetic fields on the microphysics (Lai & Qian 1998; Duan & Qian 2004), we concentrate on comparing the properties rotating proto-magnetar winds to the conventional unmagnetized spherical wind case.

2.2 Reaction Network Calculations

We start our nucleosynthesis calculations at the point where V14 trajectory reaches $T=7\times 10^{9}$ K, corresponding to the approximate temperature of $\alpha-$particle formation. However, our actual starting temperature is slightly larger because of the entropy enhancement from $\alpha$-formation (eq. 2). We use the nuclear reaction network SkyNet (Lippuner & Roberts, 2015) for the nucleosynthesis calculation. The composition starts out in nuclear statistical equilibrium (NSE), a good approximation for the initial temperature T\mathrel{\hbox{\hbox to0.0pt{\hbox{\lower 4.0pt\hbox{\sim}}\hss}\hbox{>}}}7\times 10^{9} K. Given the extrapolated density as a function of time (see §2.1), SkyNet then evolves the abundances of 7843 nuclear species under the influence of over 140,000 nuclear reactions. The evolved species range from free neutrons and protons to 337Cn ($Z=112$). SkyNet also evolves the entropy and temperature, which change due to the expansion of the material and energy released by the nuclear reactions. SkyNet uses a modified version of the Helmholtz equation of state (Timmes & Swesty, 2000), which treats every nuclear species as a separate Boltzmann gas, including also electron-positron gas with arbitrary degree of relativism and photon gas. In some trajectories the initial temperature after taking into account the entropy gain from $\alpha$-formation is above $10^{10}$ K. In these cases, SkyNet uses nuclear statistical equilibrium for evolution of the network up to $10^{10}$ K, below which it switches to full network evolution with all reactions.

The rates of the strong reactions come from the JINA REACLIB database (Cyburt et al., 2010), but only the forward rates are used and the inverse rates are computed from detailed balance. This is to ensure consistency with NSE, which depends on the nuclear masses. Spontaneous and neutron-induced fission rates are taken from Frankel & Metropolis (1947), Panov et al. (2010), Mamdouh et al. (2001), and Wahl (2002). Most of the weak rates come from Fuller et al. (1982), Oda et al. (1994), and Langanke & Martínez-Pinedo (2000) whenever they are available, and otherwise the REACLIB weak rates are used. We used the nuclear masses and partition functions from the WebNucleo XML file distributed with REACLIB, which contains experimental data where available and finite-range droplet macroscopic model (FRDM, see e.g. Moller et al., 2015) data otherwise.

For $Y_{e}>0.5$, charged particle process nuclei ($Z\simeq 38-40$) can also be formed through $\nu p$ process (e.g. Fröhlich et al. 2006; Arcones & Montes 2011); however, even though SkyNet supports including neutrino interactions, we do not include these in the present calculations. We leave an exploration of charged particle process nuclei formation in proton-rich proto-magnetar winds to future work.

2.3 Classes of Abundance Models

The nucleosynthesis products of proto-magnetar winds vary as a function of the latitude of the open flux tube at a fixed time. They also vary in a global sense, integrated over the entire magnetosphere at a fixed time, or averaged over all times in the PNS cooling evolution. We cover the range of possible diagnostics of the wind nucleosynthesis by calculating the abundance patterns in three general cases:

Individual flux tubes along different latitudes $\theta$ at a fixed time (or equivalently, neutrino luminosity), producing abundance yields as a function of $(\theta,L_{\nu},Y_{e},P$). 2. 2.

The entire wind at a fixed time, by integrating individual flux tubes over the solid angle of the open magnetosphere, producing abundance yields as a function of ($L_{\nu},Y_{e},P$). 3. 3.

The entire wind (integrated over the open magnetosphere) averaged over the Kelvin-Helmholtz cooling epoch $L_{\nu}(t)$ (Fig. 4), producing abundance yields as a function of just the rotation period $P$.

In the angle-integrated cases, we weight the abundances by the mass loss rate along each given open flux tube. In the time-integrated case, they are weighted also by the total ejected mass from the entire magnetosphere at each epoch $t$. In the latter case, by fixing the rotation period, we have assumed that the magnetic spin-down time of the pulsar is longer than Kelvin-Helmholtz cooling timescale of seconds; this approximately is justified for surface magnetic dipole field strengths of $B_{\rm d}\lesssim 10^{16}$ G.

In the time-integrated case, we also consider two methods for treating the time evolution of the electron fraction $Y_{e}$. In one case, we fix $Y_{e}$ at a constant value throughout the cooling epoch. In the second case we derive its value as described in $\S\ref{sec:traj}$ using the equilibrium electron fraction (eq. 4) from the time evolution of $L_{\nu},L_{\bar{\nu}},\langle\varepsilon_{\bar{\nu}_{e}}\rangle,\langle\varepsilon_{\nu_{e}}\rangle$ from the PNS cooling calculations of Roberts et al. (2012), as shown on Figure 4. Since we are focused on the $r$-process, we integrate only over epochs when $Y_{e,eq}<0.5$, i.e. at times $0.1\lesssim t\lesssim 5$ s, and all reported wind properties (e.g., ejecta mass, mass fractions, etc. - all the data in Table 3.2) refer to just this time period. This is a reasonable approximation to the total yield of the wind, since the bulk of the total mass loss is occurs at early times $t\sim 1-2$ s.

3 Results

3.1 Variation in wind properties across the magnetosphere

Based on the $s^{3}/\tau_{\rm exp}$ threshold criterion of Hoffman et al. (1997), V14 concluded that rotating proto-magnetars are more suitable for producing heavy $r$-process nuclei than in the spherical wind case, but not to the degree necessary to reach the third abundance peak (Z\mathrel{\hbox{\hbox to0.0pt{\hbox{\lower 4.0pt\hbox{\sim}}\hss}\hbox{>}}}78) for currently favored values of Y_{e}\mathrel{\hbox{\hbox to0.0pt{\hbox{\lower 4.0pt\hbox{\sim}}\hss}\hbox{>}}}0.4.444For some trajectories in the most rapidly spinning $P=1$ ms case for which $Y_{e}\lesssim 0.38$, the value of $s^{3}/(\tau_{\rm exp}Y_{e}^{3})$ did exceed the modified Hoffman et al. (1997) criterion for reaching $A=200$. Although our nucleosynthesis calculations presented here confirm this broad conclusion, we find some quantitative differences. While V14 found that the most promising rotation period for producing heavy elements was $P=2$ ms, here we find that $P=1$ ms is instead optimal. $P=1$ ms is a special case because of centrifugal effects are particularly pronounced, and the values of $s$ and $Y_{\rm e,final}$ thus differ significantly from the spherical case.

We focus our analysis in this section on models with $P=1$ and $3$ ms. The latter case plays important role because the rotational energy of the magnetar with $P=3$ ms, $E_{\rm rot}=I\Omega^{2}/2\approx 10^{51}$ ergs, is comparable to the kinetic energy of a normal SN. Hence magnetars born with P\mathrel{\hbox{\hbox to0.0pt{\hbox{\lower 4.0pt\hbox{\sim}}\hss}\hbox{>}}}3 ms could in principle be ‘hidden’ among the normal population of normal SNe without violating constraints on the observed kinetic energies of their ejecta from SN spectra and the total energies of their remnants (as already discussed, hypernovae are known to be rare; e.g. Podsiadlowski et al. 2004; Woosley & Bloom 2006).

Figure 5 shows the mass fraction as a function of nuclear charge $Z$ for different outflow latitudes $\theta$, calculated for $P=1$ ms (top panel) and $P=3$ ms (lower panel). For all values of $\theta$, the abundance distribution extends to higher masses $Z$ than the otherwise equivalent spherical wind with the same $L_{\nu}$ and $Y_{e}$, which is shown for comparison with a purple line. Part of this effect is due to the shorter expansion time through the seed formation region caused by the faster diverging outflow areal function $\propto 1/r^{3}$ in the dipole magnetic field, as compared to the spherical wind case (areal function $\propto 1/r^{2}$), as well as centrifugal force from rotation.

Figure 5 also shows that the abundance distribution proceeds to heavier elements with increasing $\theta$, due to the additional acceleration caused by magneto-centrifugal acceleration along field lines inclined with respect to the rotation axis. The heaviest nuclei are synthesized in those outflows which graze the closed zone and pass near the equatorial plane outside the light cylinder.

Shown for comparison with a dashed blue line are the abundances integrated over the open zone of the entire magnetosphere, from which it is apparent that the flow properties near the last open field line ($\theta_{\rm max}$) also dominate the total abundance of the wind. This is expected because outflows with larger $\theta$ contribute a greater fraction of the total open solid angle of the magnetosphere and, to a lesser extent, because the mass loss rate per unit surface area is enhanced by magneto-centrifugal acceleration for larger $\theta$ (Thompson et al. 2004; Metzger et al. 2007).

Most of our calculations employ Newtonian gravity for a NS of mass $M=1.4M_{\odot}$. However, Figure 6 shows the results for a more massive NS with $M=2M_{\odot}$ NS, as well as for a model with $M=1.4M_{\odot}$ but using the Paczyński-Wiita potential to mimic the effects of general relativity (GR). The effect of a higher NS mass, or the effectively higher mass due to the deeper Paczyński-Wiita potential, is also to increase the maximum mass nuclei synthesized, extending it up to $Z\approx 54$ (Xenon), near the peak of the 2nd $r$-process peak. This well known effect results because of the additional heating, and hence higher asymptotic entropy, achieved by the winds to escape from the deeper potential well (Cardall & Fuller 1997). Note, however, that we have not included the GR-induced gravitational redshift on the mean neutrino energies, which somewhat reduces the neutrino heating rate and acts to mitigate this effect (e.g. Thompson et al. 2001).

3.2 Time-integrated models