Proto-Neutron Star Winds, Magnetar Birth, and Gamma-Ray Bursts
Proto-Neutron Star Winds, Magnetar
Birth, and Gamma-Ray Bursts
Brian D. Metzger∗,†, Todd A. Thompson∗∗ and Eliot Quataert∗
∗Astronomy Department and Theoretical Astrophysics Center, 601 Campbell Hall,
Berkeley, CA 94720; bmetzger@astro.berkeley.edu, eliot@astro.berkeley.edu
†Department of Physics, 366 LeConte Hall, University of California, Berkeley, CA 94720
∗∗Department of Astrophysical Sciences, Peyton Hall-Ivy Lane, Princeton University,
Princeton, NJ 08544; thomp@astro.princeton.edu
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 luminosi-
ties, timescales, mass-loading, and internal shock efficiencies favorable for producing long-
duration gamma-ray bursts.
Keywords: neutron stars, stellar winds, supernovae, gamma ray bursts, magnetic fields
PACS: 97.60.Bw, 97.60.Gb; 97.10.Me
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
(∼ 3×1053 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 σνn ∝ ǫ
ν ∝ T
2, where ǫν is a typical neutrino energy.
Indeed, because neutrinos are trapped, a PNS’s neutrino luminosity Lν remains
substantial and quasi-thermal for a time after bounce τKH ∼ 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 ∼ 104 km/s, to traverse
the progenitor stellar mantle, τKH 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 ≪ τKH 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 mate-
rial around the PNS, Lν remains substantial. Detailed PNS cooling calculations [3]
show that the electron neutrino(antineutrino) luminosity Lνe(ν̄e) is ∼ 10
52 erg/s at
t∼ 1 s and declines as∝ t−1 until t≃ τKH, after which Lνe(ν̄e) decreases exponentially
as the PNS becomes optically thin. This persistent neutrino flux Fνe(ν̄e) continues
to heat the PNS atmosphere, primarily through electron neutrino(antineutrino)
http://arxiv.org/abs/0704.0675v1
absorption on nuclei (νe+n → p+ e
− and ν̄e+ p → n+ e
+). Because the inverse,
pair capture rates dominate the cooling, which declines rapidly with temperature
(q̇− ∝ T 6) and hence with spherical radius r, a region of significant net positive
heating (q̇ ≡ q̇+− q̇− > 0) develops above the neutrinosphere radius Rν . This heat-
ing 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 (Ṁth) on the PNS proper-
ties explicitly, consider that in steady state the change in gravitational potential
required for a unit mass element to escape the PNS (GM/Rν) must be provided
by the total heating it receives accelerating outwards from the PNS surface:
, (1)
where M is the PNS mass, vr is the outward wind velocity, and q̇ is per unit mass.
Because q̇ is quickly dominated by heating from neutrino absorption, which scales
as q̇+ ∝ Fνeσnν ∝ Lνeǫ
/4πr2, we see that equation (1) implies that
ρdr ≈
ρνHν , (2)
where we have used Ṁth = 4πρr
2vr for a spherical wind, ρ is the mass density, H
is the PNS’s density scale height, ǫνe crudely defines a mean electron neutrino or
antineutrino energy, and a subscript “ν” denotes evaluation near Rν . 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 Hν ∼
Pν/ρνgν ∝ T
ν/Mρν , where gν ∝ M/R
ν is the PNS surface gravity and Tν ∝
(Lνeǫ
/R2ν)
1/6 is the PNS surface temperature. Tν is set by the balance between
heating and cooling at the PNS surface (T 6ν ∝ q̇
− = q̇+ ∝Lνeǫ
/R2ν). Inserting these
results into equation (2) and including the correct normalization from the relevant
weak cross sections, one finds the expression for Ṁth first obtained by ref [4]:
Ṁth ≈ 10
10 M⊙/s, (3)
where L52 ≡Lνe×10
52 erg/s, ǫ10 ≡ 10ǫνeMeV, Rν ≡ 10R10 km, andM ≡ 1.4M1.4M⊙.
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∞th ∼ cs,ν ≈
2kTν/mp ≈ 0.1L
10 c. Thus, the efficiency η relating wind
power Ėth ≈ Ṁth(v
2/2 to total neutrino luminosity (Lν ∼ 6Lνe) is quite low:
∼ 10−5L
1.4 . (4)
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]), η 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 = Pms ms is endowed with a rotational energy
Erot ≃ 2×10
52P−2ms R
10M1.4 ergs, which, for P < 4 ms, exceeds the energy of a typical
SN shock (∼ 1051 ergs). Given a mass loss rate Ṁ and torquing lever arm ωτ , a wind
extracts angular momentum J from the PNS at a rate J̇ ≃Ωω2τṀ , where Ω= 2π/P
is the PNS rotation rate. With the PNS’s radius Rν 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, τJ ≡ J/J̇ ∼MR
ν/Ṁω
τ ∼M/Ṁth, is much longer than
τKH. However, if the PNS is rapidly rotating and possesses a dynamically-important
poloidal magnetic field Bp (through either flux-freezing or generated via dynamo
action [6]), then both Ṁ and ωτ can be substantially increased; this reduces τJ ,
allowing efficient extraction of Erot.
For magnetized winds ωτ is the Alfvén radius ωA, defined as the cylindrical
radius where ρv2r/2 first exceeds B
p/8π [7]. The magnetosphere of a PNS is
most likely dominated by its dipole component, with a total (positive-definite)
surface magnetic flux given by ΦB = 2πBνR
ν , where Bν is the polar surface field.
To estimate ωA 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 ≈ 2(πθ2LCFL)/4π ≃
Rν/2ωY of ΦB, where θLCFL ≈
Rν/ωY is the latitude (measured from the pole)
at the PNS surface of the last closed field line (LCFL), ωY is the radius where the
LCFL intersects the equator (the “Y point”), and we have assumed that ωY ≫Rν
(θLCFL ≪ 1). Plasma necessarily threads a PNS’s closed magnetosphere and cannot
be forced to corotate superluminally; thus ωY cannot exceed the light cylinder
radius ωL ≡ c/Ω = 48Pms km, making it useful to write the PNS magnetosphere’s
total open magnetic flux as ΦB,open ≈ πBνR
ν(Rν/ωL)(ωY/ωL)
−1. Now, the overall
latitudinal structure of a PNS magnetosphere (i.e., the allocation of open and closed
magnetic flux, and the value of ωY/ωL) 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
Bp ∼ ΦB,open/r
2 ≈ 0.2BνP
msR10(ωY/ωL)
−1(Rν/r)
2, rather than the dipole scaling
∝ (Rν/r)
3. The constant of proportionality is chosen to assure that Bp(Rν)→Bν in
the limit of vanishing closed zone (ωL,ωY →Rν) 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 ωA such that the kinetic energy flux at ωA
(Ṁvr(ωA)
2/2) carries a sizeable fraction of the rotational energy loss extracted by
the wind’s surface torque Ėrot = J̇Ω = ṀΩ
2ω2A; thus, we have that vr(ωA)∼ ΩωA.
Combining this with the modified monopole scaling for Bp motivated above and
mass conservation ṀΩ ≡ ρr
2vr (ṀΩ is the mass flux per solid angle) we find that:
ωA/Rν ≃B
ms Ṁ
Ω,−4R
10 (ωY/ωL)
−1, (5)
where ṀΩ ≡ ṀΩ,−4×10
−4M⊙s
−1sr−1, Bν ≡B15×10
15 G, and we have concentrated
on the open magnetic flux that emerges nearest the closed zone (polar latitude
≈ θLCFL) 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” (Bν ∼ 10
14 − 1015
G) possess enhanced lever arms for extracting rotational energy [9]. Fur-
thermore, their total outflow power ĖNRmag ≈ Ėrot ≈ 2πθ
LCFLṀΩΩ
2ω2A ≈
1049B
−13/3
ms Ṁ
Ω,−4R
10 (ωY/ωL)
−3 ergs/s dominates thermal acceleration
(ĖNRmag > Ėth) for B15 > 0.4P
23/24
23/12
−11/3
1.4 (ωY/ωL)
9/4. This condi-
tion 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 Bν
and Ω. 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 out-
flow’s power through enhanced mass-loss because ĖNRmag ∝ Ṁ
Ω . When
the PNS’s hydrostatic atmosphere is forced to co-rotate to the outflow’s
sonic radius ωs = (GM sin[θLCFL]/Ω
2)1/3 then ṀΩ is enhanced by a factor
φcf ∼ exp[(vφ,ν/cs,ν)
2] over Ṁth/4π due to centrifugal (“cf”) slinging [9], where
vφ,ν ≈ RνΩsin[θLCFL] ≈ RνΩ
Rν/ωY is the PNS rotation speed at the base
of the open flux. Using our estimate for cs,ν from § 1, we see that enhanced
mass loss becomes important for Pms < Pcf,ms ≡ L
−1/18
10 (ωY/ωL)
(i.e., only for PNSs with considerable rotational energy Erot > 10
52 ergs).
Fully enhanced mass loss (ṀΩ = Ṁthφcf/4π) requires ωA > ωs, which
in turn requires that B15 > Bcf,15 ≡ P
−13/4
10 Ṁ
Ω,−4M
1.4 (ωY/ωL)
5/4 ≃
0.3P 7/4ms L
1.4 R
−29/12
10 exp[0.5(P/Pcf)
−3](ωY/ωL)
5/4, where we have taken
Ṁth from § 1. For cases with Bν < Bcf but P < Pcf , ṀΩ lies somewhere between
Ṁth/4π and φcfṀth/4π (see [10] for numerical results). Millisecond proto-magnetars
generally attain φcf , except perhaps at early times when the PNS is quite hot.
2.2. Relativistic Winds and Gamma-Ray Bursts
As the PNS cools, eventually ωA → ωL and the PNS outflow becomes relativistic
(REL). This transition occurs after τKH 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: ĖRELmag ≈ 6×10
49B215P
10(ωY/ωL)
−2 ergs/s (again modulo
corrections for excess open magnetic flux ĖRELmag ∝ Φ
B,open ∝ (ωY/ωL)
−2 [8]), which
gives a familiar spin-down timescale τJ =Erot/Ė
mag ≈ 300B
10 M1.4(ωY/ωL)
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 σ is given by
ĖRELmag
2πθ2LCFLṀΩc
≈ 3B215P
−10/3
1.4 exp
From equation (6) we see that because a PNS’s mass-loss rate drops so precipitously
as it cools, σ ∝ L−5/3νe ǫ
−10/3
rises rapidly with time, easily reaching ∼ 10− 1000
during the KH epoch for typical magnetar parameters [9],[10]. Detailed evolution
calculations indicate that Erot is extracted roughly uniformly in log(σ) [10].
To conclude with a concrete example, consider a proto-magnetar with Bν = 10
G and Pms = 3 at t= 10 seconds after core collapse. From the cooling calculations
of ref [3] we have L52(10 s)≈ 0.1 and ǫ10(10 s)≈ 1 (see Figs. [14] and [18]) and so,
under the conservative estimate that ωY =ωL, equation (6) gives σ≈ 500. Because σ
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 ĖRELmag ≈ 10
50 erg/s with a spin-down time τJ ≈ 30 s, both
reasonable values to explain typical luminosities and durations, respectively, of
long-duration GRBs. Lastly, because σ 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-γ-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.
REFERENCES
1. Bionta, R. M., Blewitt, G., Bratton, C. B., Caspere, D., & Ciocio, A. 1987, Physical Review
Letters, 58, 1494
2. Hirata, K. S., et al. 1988, Phys. Rev. D, 38, 448
3. Pons, J. A., Reddy, S., Prakash, M., Lattimer, J. M., & Miralles, J. A. 1999, ApJ, 513, 780
4. Qian, Y.-Z., & Woosley, S. E. 1996, ApJ, 471, 331
5. Bethe, H. A., & Wilson, J. R. 1985, ApJ, 295, 14
6. Thompson, C., & Duncan, R. C. 1993, ApJ, 408, 194
7. Weber, E. J., & Davis, L. J. 1967, ApJ, 148, 217
8. Bucciantini, N., Thompson, T. A., Arons, J., Quataert, E., & Del Zanna, L. 2006, MNRAS,
368, 1717
9. Thompson, T. A., Chang, P., & Quataert, E. 2004, ApJ, 611, 380
10. Metzger, B. D., Thompson, T.A., & Quataert, E. 2007, ApJ in press
	Neutrino-Driven PNS Winds
	Magnetically-Driven PNS Winds
	Non-Relativistic Winds and Asymmetric Supernovae
	Relativistic Winds and Gamma-Ray Bursts
