Proto-Neutron Star Winds, Magnetar Birth, and Gamma-Ray Bursts

TL;DR

This paper reviews proto-neutron star wind theory, extends it to include magnetic fields and rotation, and argues that rapidly rotating proto-magnetars can produce relativistic winds suitable for long gamma-ray bursts.

Contribution

It provides new analytic models for magnetically-driven proto-neutron star winds considering multi-dimensional effects and links these to gamma-ray burst production.

Findings

Proto-magnetars with millisecond periods produce relativistic winds shortly after collapse.

Magnetically-driven winds have properties conducive to gamma-ray burst generation.

Analytic scalings connect proto-magnetar parameters to observable burst characteristics.

Abstract

We begin by reviewing the theory of thermal, neutrino-driven proto-neutron star (PNS) winds. Including the effects of magnetic fields and rotation, we then derive the mass and energy loss from magnetically-driven PNS winds for both relativistic and non-relativistic outflows, including important multi-dimensional considerations. With these simple analytic scalings we argue that proto-magnetars born with ~ millisecond rotation periods produce relativistic winds just a few seconds after core collapse with luminosities, timescales, mass-loading, and internal shock efficiencies favorable for producing long-duration gamma-ray bursts.

Equations

$$\frac{GM}{R_{\nu }}\approx {\int }_{R_{\nu }}^{\infty }\dot{q}\frac{dr}{v_r},$$

$$\frac{GM}{R_{\nu }}\propto \frac{L_{{\nu }_e}{ϵ}_{{\nu }_e}^2}{{\dot{M}}_{\mathrm{th}}}{\int }_{R_{\nu }}^{\infty }\rho dr\approx \frac{L_{{\nu }_e}{ϵ}_{{\nu }_e}^2}{{\dot{M}}_{\mathrm{th}}}{\rho }_{\nu }{H}_{\nu },$$

$${\dot{M}}_{\mathrm{th}}\approx 10^{-4}{L}_{52}^{5/3}{ϵ}_{10}^{10/3}{M}_{1.4}^{-2}{R}_{10}^{5/3}{M}_{\odot }/s,$$

$$\eta \equiv \frac{{\dot{E}}_{\mathrm{th}}}{L_{\nu }}\sim 10^{-5}{L}_{52}^{5/6}{ϵ}_{10}^{11/3}{R}_{10}^{4/3}{M}_{1.4}^{-2}.$$

$${\omega }_{\mathrm{A}}/R_{\nu }\simeq B_{15}^{2/3}{P}_{\mathrm{ms}}^{-2/3}{\dot{M}}_{\Omega ,-4}^{-1/3}{R}_{10}^{4/3}({\omega }_{\mathrm{Y}}/{\omega }_{\mathrm{L}}{)}^{-1},$$

$$\sigma \approx \frac{{\dot{E}}_{\mathrm{mag}}^{\mathrm{REL}}}{2\pi {\theta }_{\mathrm{LCFL}}^2{\dot{M}}_{\Omega }{c}^2}\approx 3B_{15}^2{P}_{\mathrm{ms}}^{-3}{L}_{52}^{-5/3}{ϵ}_{10}^{-10/3}{R}_{10}^{10/3}{M}_{1.4}^2\exp \left[-{\left(\frac{P}{P_{\mathrm{cf}}}\right)}^{-3}\right]{\left(\frac{{\omega }_{\mathrm{Y}}}{{\omega }_{\mathrm{L}}}\right)}^{-1}$$

Full text

1 Neutrino-Driven PNS Winds

After a successful core-collapse supernova (SN), a hot proto-neutron star (PNS) cools and deleptonizes, releasing the majority of its gravitational binding energy ($\sim 3\times 10^{53}$ ergs) in neutrinos. With initial core temperature $T>10$ MeV, a PNS is born optically-thick to neutrinos of all flavors because the relevant neutrino-matter cross sections scale as $\sigma_{\nu n}\propto\epsilon_{\nu}^{2}\propto T^{2}$, where $\epsilon_{\nu}$ is a typical neutrino energy. Indeed, because neutrinos are trapped, a PNS’s neutrino luminosity $L_{\nu}$ remains substantial and quasi-thermal for a time after bounce $\tau_{{\rm KH}}\sim 10-100$ s, as roughly verified by the 19 neutrinos detected from SN1987A 20 years ago [1],[2]. Although this Kelvin-Helmholtz (KH) cooling epoch is short compared to the time required for the shock, once successful and moving outward at $\sim 10^{4}$ km/s, to traverse the progenitor stellar mantle, $\tau_{{\rm KH}}$ is still significantly longer than the time over which the initial explosion must be successful. While the specific shock launching mechanism is presently unknown, it must occur in a time $t<$ 1 s $\ll\tau_{{\rm KH}}$ after bounce for the PNS to avoid accreting too much matter.

Thus, even after the SN shock has cleared a cavity of relatively low density material around the PNS, $L_{\nu}$ remains substantial. Detailed PNS cooling calculations [3] show that the electron neutrino(antineutrino) luminosity $L_{\nu_{e}(\bar{\nu}_{e})}$ is $\sim 10^{52}$ erg/s at $t\sim 1$ s and declines as $\propto t^{-1}$ until $t\simeq\tau_{\rm KH}$, after which $L_{\nu_{e}(\bar{\nu}_{e})}$ decreases exponentially as the PNS becomes optically thin. This persistent neutrino flux $F_{\nu_{e}(\bar{\nu}_{e})}$ continues to heat the PNS atmosphere, primarily through electron neutrino(antineutrino) absorption on nuclei ($\nu_{e}+n\rightarrow p+e^{-}$ and $\bar{\nu}_{e}+p\rightarrow n+e^{+}$). Because the inverse, pair capture rates dominate the cooling, which declines rapidly with temperature ($\dot{q}^{-}\propto T^{6}$) and hence with spherical radius $r$, a region of significant net positive heating ($\dot{q}\equiv\dot{q}^{+}-\dot{q}^{-}>0$) develops above the neutrinosphere radius $R_{\nu}$. This heating drives mass-loss from the PNS in the form of a thermally-driven wind [4]. To estimate the dependence of the resultant mass-loss rate ($\dot{M}_{\rm th}$) on the PNS properties explicitly, consider that in steady state the change in gravitational potential required for a unit mass element to escape the PNS ($GM/R_{\nu}$) must be provided by the total heating it receives accelerating outwards from the PNS surface:

[TABLE]

where $M$ is the PNS mass, $v_{r}$ is the outward wind velocity, and $\dot{q}$ is per unit mass. Because $\dot{q}$ is quickly dominated by heating from neutrino absorption, which scales as $\dot{q}^{+}\propto F_{\nu_{e}}\sigma_{n\nu}\propto L_{\nu_{e}}\epsilon_{\nu_{e}}^{2}/4\pi r^{2}$, we see that equation (1) implies that

[TABLE]

where we have used $\dot{M}_{\rm th}=4\pi\rho r^{2}v_{r}$ for a spherical wind, $\rho$ is the mass density, $H$ is the PNS’s density scale height, $\epsilon_{\nu_{e}}$ crudely defines a mean electron neutrino or antineutrino energy, and a subscript “$\nu$” denotes evaluation near $R_{\nu}$. Neglecting rotational support and assuming that the thermal pressure $P$ is dominated by photons and relativistic pairs (which also becomes an excellent approximation as the density plummets abruptly above the PNS surface), we have that $H_{\nu}\sim P_{\nu}/\rho_{\nu}g_{\nu}\propto T_{\nu}^{4}R_{\nu}^{2}/M\rho_{\nu}$, where $g_{\nu}\propto M/R_{\nu}^{2}$ is the PNS surface gravity and $T_{\nu}\propto(L_{\nu_{e}}\epsilon_{\nu_{e}}^{2}/R_{\nu}^{2})^{1/6}$ is the PNS surface temperature. $T_{\nu}$ is set by the balance between heating and cooling at the PNS surface ($T_{\nu}^{6}\propto\dot{q}^{-}=\dot{q}^{+}\propto L_{\nu_{e}}\epsilon_{\nu_{e}}^{2}/R_{\nu}^{2}$). Inserting these results into equation (2) and including the correct normalization from the relevant weak cross sections, one finds the expression for $\dot{M}_{\rm th}$ first obtained by ref [4]:

[TABLE]

where $L_{52}\equiv L_{\nu_{e}}\times 10^{52}$ erg/s, $\epsilon_{10}\equiv 10\epsilon_{\nu_{e}}$MeV, $R_{\nu}\equiv 10R_{10}$ km, and $M\equiv 1.4M_{1.4}M_{\odot}$.

Endowed with an enormous gravitational binding energy and a means, through this neutrino-driven outflow, for communicating a fraction of this energy to the outgoing shock, a newly-born PNS seems capable of affecting the properties of the SN that we observe. However, a purely thermal, neutrino-driven PNS wind is only accelerated to an asymptotic speed of order the surface sound speed: $v^{\infty}_{\rm th}\sim c_{s,\nu}\approx\sqrt{2kT_{\nu}/m_{p}}\approx 0.1L_{52}^{1/12}\epsilon_{10}^{1/6}R_{10}^{-1/6}$ c. Thus, the efficiency $\eta$ relating wind power $\dot{E}_{\rm th}\approx\dot{M}_{\rm th}(v^{\infty}_{\rm th})^{2}/2$ to total neutrino luminosity ($L_{\nu}\sim 6L_{\nu_{e}}$) is quite low:

[TABLE]

In particular, although neutrino energy deposited in a similar manner may be responsible for initiating the SN explosion itself at early times (i.e., the neutrino SN mechanism [5]), $\eta$ drops rapidly as the PNS cools. Quasi-spherical winds of this type are therefore not expected to affect the SN’s nucleosynthesis or morphology (although the wind itself is considered a promising $r$-process source [4]).

2 Magnetically-Driven PNS Winds

Some PNSs may possess a more readily extractable form of energy in rotation. A PNS born with a period $P=P_{\rm ms}$ ms is endowed with a rotational energy $E_{\rm rot}\simeq 2\times 10^{52}P_{\rm ms}^{-2}R_{10}^{2}M_{1.4}\,{\rm ergs}$, which, for $P<4$ ms, exceeds the energy of a typical SN shock ($\sim 10^{51}$ ergs). Given a mass loss rate $\dot{M}$ and torquing lever arm $\omega_{\tau}$, a wind extracts angular momentum $J$ from the PNS at a rate $\dot{J}\simeq\Omega\omega_{\tau}^{2}\dot{M}$, where $\Omega=2\pi/P$ is the PNS rotation rate. With the PNS’s radius $R_{\nu}$ as a lever arm and the modest thermally-driven mass-loss rate given by equation (3), the timescale for removal of the PNS’s rotational energy, $\tau_{J}\equiv J/\dot{J}\sim MR_{\nu}^{2}/\dot{M}\omega_{\tau}^{2}\sim M/\dot{M}_{\rm th}$, is much longer than $\tau_{{\rm KH}}$. However, if the PNS is rapidly rotating and possesses a dynamically-important poloidal magnetic field $B_{p}$ (through either flux-freezing or generated via dynamo action [6]), then both $\dot{M}$ and $\omega_{\tau}$ can be substantially increased; this reduces $\tau_{J}$, allowing efficient extraction of $E_{\rm rot}$.

For magnetized winds $\omega_{\tau}$ is the Alfvén radius $\omega_{A}$, defined as the cylindrical radius where $\rho v_{r}^{2}/2$ first exceeds $B_{p}^{2}/8\pi$ [7]. The magnetosphere of a PNS is most likely dominated by its dipole component, with a total (positive-definite) surface magnetic flux given by $\Phi_{\rm B}=2\pi B_{\nu}R_{\nu}^{2}$, where $B_{\nu}$ is the polar surface field. To estimate $\omega_{A}$ for magnetized PNS outflows recognize that mass and angular momentum are primarily extracted from a PNS along open magnetic flux. For an axisymmetric dipole rotator this represents only a fraction $\approx 2(\pi\theta_{\rm LCFL}^{2})/4\pi\simeq R_{\nu}/2\omega_{\rm Y}$ of $\Phi_{\rm B}$, where $\theta_{\rm LCFL}\approx\sqrt{R_{\nu}/\omega_{\rm Y}}$ is the latitude (measured from the pole) at the PNS surface of the last closed field line (LCFL), $\omega_{\rm Y}$ is the radius where the LCFL intersects the equator (the “Y point”), and we have assumed that $\omega_{\rm Y}\gg R_{\nu}$ ($\theta_{\rm LCFL}\ll 1$). Plasma necessarily threads a PNS’s closed magnetosphere and cannot be forced to corotate superluminally; thus $\omega_{\rm Y}$ cannot exceed the light cylinder radius $\omega_{\rm L}\equiv c/\Omega=48P_{\rm ms}$ km, making it useful to write the PNS magnetosphere’s total open magnetic flux as $\Phi_{\rm B,open}\approx\pi B_{\nu}R_{\nu}^{2}(R_{\nu}/\omega_{\rm L})(\omega_{\rm Y}/\omega_{\rm L})^{-1}$. Now, the overall latitudinal structure of a PNS magnetosphere (i.e., the allocation of open and closed magnetic flux, and the value of $\omega_{\rm Y}/\omega_{\rm L}$) is primarily dominated by the dipolar closed zone. However, recent numerical simulations [8] show that where the field is open it behaves as a “split monopole”. In this case the poloidal field scales as $B_{p}\sim\Phi_{\rm B,open}/r^{2}\approx 0.2B_{\nu}P_{\rm ms}^{-1}R_{10}(\omega_{\rm Y}/\omega_{\rm L})^{-1}(R_{\nu}/r)^{2}$, rather than the dipole scaling $\propto(R_{\nu}/r)^{3}$. The constant of proportionality is chosen to assure that $B_{p}(R_{\nu})\rightarrow B_{\nu}$ in the limit of vanishing closed zone ($\omega_{\rm L},\omega_{\rm Y}\rightarrow R_{\nu}$) and is in agreement with numerical results (see eq. [28] of ref [8]).

2.1 Non-Relativistic Winds and Asymmetric Supernovae

Non-relativistic (NR) magnetically-driven winds reach an equipartition between kinetic and magnetic energy outside $\omega_{A}$ such that the kinetic energy flux at $\omega_{A}$ ($\dot{M}v_{r}(\omega_{A})^{2}/2$) carries a sizeable fraction of the rotational energy loss extracted by the wind’s surface torque $\dot{E}_{\rm rot}=\dot{J}\Omega=\dot{M}\Omega^{2}\omega_{A}^{2}$; thus, we have that $v_{r}(\omega_{A})\sim\Omega\omega_{A}$. Combining this with the modified monopole scaling for $B_{p}$ motivated above and mass conservation $\dot{M}_{\Omega}\equiv\rho r^{2}v_{r}$ ($\dot{M}_{\Omega}$ is the mass flux per solid angle) we find that:

[TABLE]

where $\dot{M}_{\Omega}\equiv\dot{M}_{\Omega,-4}\times 10^{-4}M_{\odot}$s*-1sr-1*, $B_{\nu}\equiv B_{15}\times 10^{15}$ G, and we have concentrated on the open magnetic flux that emerges nearest the closed zone (polar latitude $\approx\theta_{\rm LCFL}$) and which thereby dominates the spin-down torque.

From equation (5) we see that winds from rapidly rotating PNSs with surface magnetic fields typical of Galactic “magnetars” ($B_{\nu}\sim 10^{14}-10^{15}$ G) possess enhanced lever arms for extracting rotational energy [9]. Furthermore, their total outflow power $\dot{E}_{\rm mag}^{\rm NR}\approx\dot{E}_{\rm rot}\approx 2\pi\theta_{\rm LCFL}^{2}\dot{M}_{\Omega}\Omega^{2}\omega_{A}^{2}\approx 10^{49}B_{15}^{4/3}P_{\rm ms}^{-13/3}\dot{M}_{\Omega,-4}^{1/3}R_{10}^{17/3}(\omega_{\rm Y}/\omega_{\rm L})^{-3}$ ergs/s dominates thermal acceleration ($\dot{E}_{\rm mag}^{\rm NR}>\dot{E}_{\rm th}$) for $B_{15}>0.4P_{\rm ms}^{13/4}L_{52}^{23/24}\epsilon_{10}^{23/12}R_{10}^{-11/3}M_{1.4}^{-1}(\omega_{\rm Y}/\omega_{\rm L})^{9/4}$. This condition becomes easier to satisfy as the PNS cools, allowing magnetized winds to dominate later stages of the KH epoch for PNSs with even relatively modest $B_{\nu}$ and $\Omega$. NR magnetically-driven winds, in addition to being more powerful than spherical, thermally-driven outflows, are efficiently hoop-stress collimated along the PNS rotation axis [8]. The power they deposit along the poles may produce asymmetry in SN ejecta distinct from the shock-launching process itself.

Strong magnetic fields and rapid rotation can also increase the outflow’s power through enhanced mass-loss because $\dot{E}_{\rm mag}^{\rm NR}\propto\dot{M}_{\Omega}^{1/3}$. When the PNS’s hydrostatic atmosphere is forced to co-rotate to the outflow’s sonic radius $\omega_{\rm s}=(GM\sin[\theta_{\rm LCFL}]/\Omega^{2})^{1/3}$ then $\dot{M}_{\Omega}$ is enhanced by a factor $\phi_{\rm cf}\sim\exp[(v_{\phi,\nu}/c_{s,\nu})^{2}]$ over $\dot{M}_{\rm th}/4\pi$ due to centrifugal (“cf”) slinging [9], where $v_{\phi,\nu}\approx R_{\nu}\Omega\sin[\theta_{\rm LCFL}]\approx R_{\nu}\Omega\sqrt{R_{\nu}/\omega_{\rm Y}}$ is the PNS rotation speed at the base of the open flux. Using our estimate for $c_{s,\nu}$ from $\S$ 1, we see that enhanced mass loss becomes important for $P_{\rm ms}<P_{\rm cf,ms}\equiv L_{52}^{-1/18}\epsilon_{10}^{-1/9}R_{10}^{10/9}(\omega_{\rm Y}/\omega_{\rm L})^{-1/3}$ (i.e., only for PNSs with considerable rotational energy $E_{\rm rot}>10^{52}$ ergs). Fully enhanced mass loss ($\dot{M}_{\Omega}=\dot{M}_{\rm th}\phi_{\rm cf}/4\pi$) requires $\omega_{A}>\omega_{\rm s}$, which in turn requires that $B_{15}>B_{\rm cf,15}\equiv P_{\rm ms}^{7/4}R_{10}^{-13/4}\dot{M}_{\Omega,-4}^{1/2}M_{1.4}^{1/2}(\omega_{\rm Y}/\omega_{\rm L})^{5/4}\simeq 0.3P_{\rm ms}^{7/4}L_{52}^{5/6}\,\epsilon_{10}^{5/3}M_{1.4}^{-1/2}R_{10}^{-29/12}\exp[0.5(P/P_{\rm cf})^{-3}](\omega_{\rm Y}/\omega_{\rm L})^{5/4}$, where we have taken $\dot{M}_{\rm th}$ from $\S$ 1. For cases with $B_{\nu}<B_{\rm cf}$ but $P<P_{\rm cf}$, $\dot{M}_{\Omega}$ lies somewhere between $\dot{M}_{\rm th}/4\pi$ and $\phi_{\rm cf}\dot{M}_{\rm th}/4\pi$ (see [10] for numerical results). Millisecond proto-magnetars generally attain $\phi_{\rm cf}$, except perhaps at early times when the PNS is quite hot.

2.2 Relativistic Winds and Gamma-Ray Bursts

As the PNS cools, eventually $\omega_{A}\rightarrow\omega_{\rm L}$ and the PNS outflow becomes relativistic (REL). This transition occurs after $\tau_{\rm KH}$ for most PNSs (they become pulsars), but rapidly rotating proto-magnetar winds become relativistic during the KH epoch itself. Similar to normal pulsars, PNSs of this type lose energy at the force-free, “vacuum dipole” rate: $\dot{E}^{\rm REL}_{\rm mag}\approx 6\times 10^{49}B_{15}^{2}P_{\rm ms}^{-4}R_{10}^{6}(\omega_{\rm Y}/\omega_{\rm L})^{-2}\,{\rm ergs/s}$ (again modulo corrections for excess open magnetic flux $\dot{E}_{\rm mag}^{\rm REL}\propto\Phi_{\rm B,open}^{2}\propto(\omega_{\rm Y}/\omega_{\rm L})^{-2}$ [8]), which gives a familiar spin-down timescale $\tau_{\rm J}=E_{\rm rot}/\dot{E}^{\rm REL}_{\rm mag}\approx 300B_{15}^{-2}P_{\rm ms}^{2}R_{10}^{-4}M_{1.4}(\omega_{\rm Y}/\omega_{\rm L})^{2}$ s. On the other hand, the mass loading on a PNS’s open magnetic flux is set by neutrino heating, a process totally different from the way that matter is extracted from a normal pulsar’s surface. In fact, a proto-magnetar outflow’s energy-to-mass ratio $\sigma$ is given by

[TABLE]

From equation (6) we see that because a PNS’s mass-loss rate drops so precipitously as it cools, $\sigma\propto L_{\nu_{e}}^{-5/3}\epsilon_{\nu_{e}}^{-10/3}$ rises rapidly with time, easily reaching $\sim 10-1000$ during the KH epoch for typical magnetar parameters [9],[10]. Detailed evolution calculations indicate that $E_{\rm rot}$ is extracted roughly uniformly in log($\sigma$) [10].

To conclude with a concrete example, consider a proto-magnetar with $B_{\nu}=10^{16}$ G and $P_{\rm ms}=3$ at $t=10$ seconds after core collapse. From the cooling calculations of ref [3] we have $L_{52}(10$ s$)\approx 0.1$ and $\epsilon_{10}(10$ s$)\approx 1$ (see Figs. [14] and [18]) and so, under the conservative estimate that $\omega_{\rm Y}=\omega_{\rm L}$, equation (6) gives $\sigma\approx 500$. Because $\sigma$ represents the potential Lorentz factor of the outflow (assuming efficient conversion of magnetic to kinetic energy), we observe that millisecond proto-magnetar birth provides the right mass-loading to explain gamma-ray bursts (GRBs). Further, the power at $t=10$ s is still $\dot{E}^{\rm REL}_{\rm mag}\approx 10^{50}$ erg/s with a spin-down time $\tau_{\rm J}\approx 30$ s, both reasonable values to explain typical luminosities and durations, respectively, of long-duration GRBs. Lastly, because $\sigma$ rises so rapidly with time as the PNS cools, in the context of GRB internal shock models a cooling proto-magnetar outflow’s kinetic-to-$\gamma$-ray efficiency can be quite high; our calculations indicate that values of $10-50\%$ are plausible. We conclude that magnetar birth accompanied by rapid rotation (but requiring less angular momentum than collapsar models) represents a viable long-duration GRB central engine.