Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 4 November 2018 (MN LATEX style file v2.2)
Magnetohydrodynamic Rebound Shocks of Supernovae
Yu-Qing Lou1,2,3 ⋆ and Wei-Gang Wang1
1Physics Department and Tsinghua Centre for Astrophysics (THCA), Tsinghua University, Beijing, 100084, China;
2Department of Astronomy and Astrophysics, the University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA;
3National Astronomical Observatories, Chinese Academy of Sciences, A20, Datun Road, Beijing 100012, China.
4 November 2018
ABSTRACT
We construct magnetohydrodynamic (MHD) similarity rebound shocks joining ‘quasi-
static’ asymptotic solutions around the central degenerate core to explore an MHD
model for the evolution of random magnetic field in supernova explosions. This pro-
vides a theoretical basis for further studying synchrotron diagnostics, MHD shock ac-
celeration of cosmic rays, and the nature of intense magnetic field in compact objects.
The magnetic field strength in space approaches a limiting ratio, that is comparable
to the ratio of the ejecta mass driven out versus the progenitor mass, during this
self-similar rebound MHD shock evolution. The intense magnetic field of the remnant
compact star as compared to that of the progenitor star is mainly attributed to both
the gravitational core collapse and the radial distribution of magnetic field.
Key words: magnetohydrodynamics (MHD) – shock waves – stars: neutron – stars:
winds, outflows – supernova remnants – white dwarfs
1 INTRODUCTION
Self-similar evolution of a spherical gas flow under self-
gravity and thermal pressure has been studied over past
four decades: from simulations and the discovery of Larson-
Penston (L-P) type solutions (Bodenheimer & Sweigart
1968; Larson 1969a, b; Penston 1969a, b), to the construc-
tion of the expansion-wave collapse solution (EWCS) using
the central free-fall asymptotic solution (Shu 1977) as well as
to the application of phase-match techniques for construct-
ing infinite series of discrete global solutions including L-P
type solutions (Hunter 1977) and solutions for envelope ex-
pansion with core collapse (EECC; Lou & Shen 2004). Prop-
erties of eigensolutions crossing the sonic critical line were
examined (Jordan & Smith 1977; Shu 1977; Whitworth &
Summers 1985; Hunter 1986). Self-similar shocks were stud-
ied and applied to various astrophysical settings by Tsai &
Hsu (1995), Shu et al. (2002), Shen & Lou (2004), and Bian
& Lou (2005). While these major results were obtained for
an isothermal gas, the counterpart problem with a poly-
tropic equation of state (EoS) was also studied by Cheng
(1978), Goldreich & Weber (1980), Yahil (1983), Suto &
Silk (1988), McLaughlin & Pudritz (1997), Fatuzzo et al.
(2004) and Lou & Gao (2006). In most cases, the polytropic
results share a feature that by setting the polytropic index
γ = 1 in the isothermal limit, all asymptotic behaviours ap-
⋆ E-mail: louyq@tsinghua.edu.cn and lou@oddjob.uchicago.edu;
wwg03@mails.tsinghua.edu.cn
proach the isothermal counterpart solutions. However, Lou
& Wang (2006) reported new ‘quasi-static’ asymptotic so-
lutions unique to a polytropic gas with γ > 1.2 and con-
structed self-similar rebound shocks for supernovae (SNe).
Chiueh & Chou (1994) studied a self-similar MHD
problem by including the magnetic pressure gradient force
in the momentum equation. Yu & Lou (2005) improved
their formulation and provided a more detailed analysis (see
Zel’dovich & Novikov 1971 for a discussion of random mag-
netic field). Wang & Lou (2006) studied this MHD problem
for a polytropic gas and derived the ‘quasi-static’ asymp-
totic solutions. Self-similar MHD shocks were explored by
Yu et al. (2006). As magnetic field is inevitably involved in
SNe and is crucial for synchrotron radiation and cosmic ray
acceleration, we construct here rebound MHD shocks with
‘quasi-static’ asymptotic solutions to model magnetic field
evolution in SN explosions.
Type II, Ib, Ic SNe are thought to be caused by grav-
itational core collapse due to an insufficient nuclear fuel;
such collapse creates an over-dense core, which rebounds
abruptly initiating a powerful rebound shock. The energetics
of sustaining such a rebound shock has been an outstanding
problem. We approach this issue in the following perspec-
tive. Triggered by such a core collapse, the rebound shock
is essentially supported by the neutrino-driven mechanism,
and several complicated physical processes are involved in
the stellar interior: all four elementary forces and the cou-
pling of various fluids and matters such as baryons, neutri-
nos, photons etc. (e.g., Janka et al. 2006). We approximate
c© 0000 RAS
http://arxiv.org/abs/0704.0223v1
2 Y.-Q. Lou & W.-G. Wang
such a dynamic system in terms of a single fluid with a
polytropic EoS, and treat the shock as an energy-conserved
self-similar shock. Conceptually, the ‘rebound shock’ here
refers to a neutrino-driven shock, as opposed to the ‘prompt
shock’ mentioned in Janka et al. (2006). We constructed
such a rebound shock (Lou & Wang 2006) to model a SN
explosion followed by a self-similar evolution leading to a
quasi-static configuration. In reference to the hydrodynamic
model of Lou & Wang (2006), the main thrust of this Let-
ter is to construct approximately a self-similar model of a
quasi-spherically symmetric rebound MHD shock for a SN
explosion, providing the profile and evolution of magnetic
field to facilitate future studies of synchrotron radiation and
MHD shock acceleration of cosmic rays, and to probe the
nature of intense magnetic field of compact stellar objects
left behind.
2 FORMULATION AND ANALYSIS
2.1 The Self-Similar MHD Formulation
A quasi-spherical similarity MHD flow embedded with a
completely random magnetic field on small scales is formu-
lated the same as in Yu & Lou (2005) and Yu et al. (2006);
the key difference here is the polytropic EoS p = κργ instead
of an isothermal gas, where p is the pressure, ρ is the mass
density, and κ is constant. Using the magnetic flux frozen-in
condition, the ideal MHD equations, viz., the mass conserva-
tion equation, the radial momentum equation, the magnetic
induction equation and the polytropic EoS, can be reduced
to two nonlinear ordinary differential equations (ODEs)
(n− 1)v +
nx− v
3n− 2
2(x− v)(nx− v)
α(nx− v)2 − (γαγ + hα2x2)
, (1)
(n− 1)
αv(nx− v) + 2hα2x2
(nx− v)2
(3n− 2)
− 2γαγ
(x− v)
α(nx− v)2 − (γαγ + hα2x2)
along with a useful relation m = αx2(nx−v) by the following
MHD self-similar transformation in a polytropic gas flow
r = k
x, u = k
v, ρ =
4πGt2
, p =
kt2n−4
k3/2t3n−2
(3n− 2)G
(nx−v) , < B2t >=
kt2n−4
, (3)
where G is the gravitational constant, M is the enclosed
mass at time t within radius r, u is the radial flow speed,
< B2t > is the mean square of random transverse magnetic
field Bt, x is the independent self-similar variable, v(x) is
the reduced flow speed, α(x) is the reduced density, m(x)
is the reduced enclosed mass, the prime ′ stands for the
first derivative d/dx, k and n are two parameters, and h
is a parameter for the strength of < B2t >
1/2. We expedi-
ently take γ = 2 − n for a polytropic EoS with a constant
κ ≡ k(4πG)γ−1 = pρ−γ . The magnetosonic critical curve is
determined by the simultaneous vanishing of the numerator
and denominator on the RHS of eq (1) or (2). The two eigen-
solutions of v′ across the magnetosonic critical curve can be
derived by using the L’Hôspital rule (Lou & Wang 2006; Yu
& Lou 2005; Yu et al. 2006). The solutions are obtained for
v(x) and α(x), and the magnetic field < B2t >
1/2 is then
known from transformation (3).
2.2 Analytic Asymptotic MHD Solutions
For h < hc ≡ n2/[2(1− n)(3n− 2)], eqs (1) and (2) give the
magnetostatic solution of a magnetized singular polytropic
sphere (MSPS) with v = 0 and
2γ(4− 3γ)
(1− γ)
]−1/n
, (4)
t >= h
2γ(4− 3γ)
(1− γ)
]−2/n
2−4/n
. (5)
There exists an asymptotic MHD solution approaching this
limiting form at small x (referred to as the type I ‘quasi-
static’ asymptotic MHD solution), viz., v = LxK and
2γ(4− 3γ)
(1− γ)
]−1/n
(K + 2− 2/n)L
n(K − 1)
n2/(2γ)
4− 3γ
]−1/n
K−1−2/n
, (6)
where K is the root of quadratic equation
/2 + n(3n− 2)h]K2 − (4− 3n)[n/2 + (3n− 2)h]K
+ γ(2/n− 2)(3n− 2)h = 0 . (7)
When 12 − 8
2 < n < 0.8 and h0 < h < hc, where h0 ≡
(3 + 2
2)n − 4
4 − (3 − 2
/[2n(3n − 2)], or when
2/3 < n < 12−8
2 for h < hc, eq (7) gives two roots K > 1,
corresponding to two possible ‘quasi-static’ solutions.
The asymptotic MHD solution at large x is α =
−2/n + · · · and
v = B0x
1−1/n −
(3n− 2)
2h(n− 1)
1−2/n
n[2(n+ γ)− 3]
(2−2γ−n)/n
+ · · · , (8)
where A0 and B0 are two constants. Solution (8) at large x
can be connected to ‘quasi-static’ asymptotic MHD solution
(6) and (7) at small x by a Runge-Kutta integration (Press
et al. 1986), crossing the magnetosonic critical curve either
smoothly or with an MHD shock (Yu et al. 2006).
2.3 MHD Shock Jump Conditions
MHD shock conditions (Yu et al. 2006; Lou & Wang 2006)
include conservations of mass, momentum, energy and mag-
netic flux, and in self-similar forms, they appear as
= 0 , (9)
= 0 , (10)
c© 0000 RAS, MNRAS 000, 000–000
MHD Rebound Shocks in Supernovae 3
(γ − 1)
αγ−1s
+ 2hαs
= 0 , (11)
where quantities in square brackets with superscript ‘1’ (up-
stream) and subscript ‘2’ (downstream) remain conserved
across the MHD shock front indicated by a subscript s. The
parameter k changes according to k2 = k1x
s2 on two
sides of a shock. For the specific entropy to increase from
upstream to downstream sides, xs1 > xs2 is necessary. MHD
shock conditions (9)−(11) lead to a quadratic equation (Lou
& Wang 2006); once we specify physical conditions on one
side of a chosen shock location, the corresponding quantities
α, v, x on the other side are readily computed.
3 REBOUND MHD SHOCKS IN SUPERNOVA
EXPLOSIONS
Various rebound MHD shocks are constructed numerically,
parallel to Lou & Wang (2006). With chosen inner and outer
radii, e.g., ri = 10
6cm and ro = 10
12cm for neutron star
formation, and when the k parameter in transformation (3)
is specified, we apply our solutions to a physical rebound
MHD shock scenario for SNe (Lou & Wang 2006).
3.1 Final and Initial Configurations
Similar to the hydrodynamic rebound shock model of Lou
& Wang (2006), the final configuration (small x) of our re-
bound MHD shock solutions gradually evolves to a MSPS
and is regarded as a remnant compact object after the re-
bound MHD shock ploughing through stellar ejecta; the
initial configuration (large x) marks the onset of gravity-
induced core collapse with outer inflows or outflows such as
stellar winds or stellar oscillations.
We define the outer initial mass Mo,ini and the inner ul-
timate mass Mi,ult the same way as in Lou & Wang (2006)
and regard them as rough estimates for the masses of the
progenitor star and the remnant compact object. The ratio
of the two masses isMo,ini/Mi,ult = λ1(ro/ri)
(3−2/n) where
λ1 ≡ A0(k1/k2)1/n
n2/[2γ(4−3γ)]+(1−γ)h/γ
involves
parameters of the rebound MHD shock and is equal to the
ratio of enclosed masses at the same r. Similar to the result
of Lou & Wang (2006), we find numerically that λ1 > 1 de-
pends on the choice of solutions, clearly indicating that a
rebound MHD shock drives out stellar materials.
By eq (3), the final magnetostatic configuration gives
t,ult >
k2γ(4− 3γ)
]−1/n
1−2/n
The ratio of initial to final magnetic fields at the same r is
t,ini >
1/2 / < B2
t,ult >
1/2= λ1, where λ1 > 1 by numer-
ical exploration. Thus a rebound MHD shock breakout pro-
cess reduces the magnetic field by the same ratio of enclosed
masses at the same r; yet this decrease in magnetic field is
insignificant as compared to the radial variation of magnetic
field, i.e., the r1−2/n dependence. As γ approaches 4/3 or
n → 2/3, this scaling approaches r−2, while the dependence
of enclosed mass on r goes to r0. For a ∼ 10G (0.1G) surface
magnetic field at ro = 10
12cm, we estimate a magnetic field
in the interior of the final configuration (ri = 10
6cm) to be
∼ 1013G (1011G), sensible for magnetized neutron stars; if
we take ri = 10
9cm, then the final interior magnetic field
is estimated to be ∼ 107G (105G), fairly close to relevant
magnetic field strengths of white dwarfs (e.g., Euchner et al.
2005, 2006; Schmidt et al. 2003).
3.2 Evolution of Rebound MHD Shocks
Time evolution of density, velocity and enclosed mass are
similar to those described by Lou & Wang (2006). We focus
here on the magnetic field evolution. Figure 1 shows a typical
time evolution of < B2t >
1/2 / < B2
t,ult
>1/2 to complement
the r1−2/n behaviour. Magnetic field increases at first, and
gradually decreases until reaching the magnetostatic config-
uration much smaller than the initial configuration in size. In
short, magnetic field changes moderately. The crucial point
is that the magnetic field varies significantly in r within a
star. If we take the magnetic field at the outer boundary
to be the surface magnetic field of the progenitor star and
take the magnetic field at the inner boundary as the surface
magnetic field of the remnant compact star, then a large
ratio of ∼ 1012 appears in forming a neutron star (Lou &
Wang 2006). This model feature may explain the intense
magnetic field of neutron stars inferred from spin-down ob-
servations of radio pulsars. In our scenario, after the passage
of such a rebound MHD shock, stellar ejecta detach from
the central degenerate neutron star which is thus exposed
with a surface magnetic field of 1013∼11G. In the same spirit
of Lou & Wang (2006), we also suggest the formation of
magnetic white dwarfs from the end of main-sequence stars
with 6 ∼ 8M⊙; in this scenario, the surface magnetic field
of an exposed central white dwarf is in a plausible range of
∼ 107∼5G (e.g., Euchner et al. 2005, 2006; Schmidt et al.
2003).
The major point to be emphasized is that random mag-
netic field preexists inside progenitor stars through various
dynamo processes. We detect magnetic field strengths of or-
der 10−2∼3G on stellar surface and this corresponds to a
much stronger magnetic field in the stellar interior with a
scaling of ∼ r1−2/n shortly after the initiation of core col-
lapse. In addition, the interior magnetic field can be consid-
erably strengthened by the free-fall core collapse preceding
the emergence of a rebound MHD shock (see Lou & Wang
2006 for descriptions of the rebound shock scenario and the
core collapse process), according to the frozen-in flux and
accretion shock conditions. In reality, these two processes
happen concurrently to produce the resultant self-similar
distribution of magnetic field. In short, the interior magnetic
field would be much stronger than the surface magnetic field
and can be further enhanced to reach a high-field regime.
The origin of stellar magnetic field was argued by
several authors to come from various processes, includ-
ing dynamo effects and thermomagnetic instabilities (e.g.,
Reisenegger et al. 2005). Our MHD scenario of interior core
collapse and rebound shock appears to grossly match with
observational facts. From our MSPS configuration with a
random magnetic field strength scaled as Bt ∝ r1−2/n in
a polytropic gas, we see a real possibility that the interior
magnetic field can be actually much stronger than the sur-
c© 0000 RAS, MNRAS 000, 000–000
4 Y.-Q. Lou & W.-G. Wang
r(cm)
t = ∞
Typical Magnetic Field Evolution During Shock Breakout
Figure 1. The ratio Bt/Bt,ult ≡< B
1/2 / < B2
t,ult
>1/2 is
the rms magnetic field strength divided by the corresponding rms
magnetic field strength of the final magnetostatic configuration
at the same r. This example is constructed by integrating inward
from (x0 , v0 , α0) on the magnetosonic critical curve and using an
eigensolution to match with a quasi-static solution as x → 0+; we
use the solution portion within xs2 < x0 for the downstream. We
then obtain the upstream point (xs1, vs1, αs1) by the MHD shock
jump conditions from the values of (xs2, vs2, αs2) obtained in the
former integration and further integrate outward to determine
the upstream solution. The relevant parameters are γ = 1.32,
n = 0.68, h = 0.01, k1 = 7.7 × 10
16 cgs units, k2 = 4 × 10
cgs units, x0 = 1.778 , v0 = 0.4620 , α0 = 0.067, and xs2 = 1.1.
Here, t1 = 6.61× 10
−5s is the time when the MHD shock crosses
the inner boundary and is the initial time of application; t2 =
4.40 × 104s is the time when the MHD shock crosses the outer
boundary; tm1 = 0.1s and tm2 = 1 × 10
8s are two intermediate
times between t1 and t2 and t2 and t = ∞.
face magnetic field. Once the onset of a gravitational col-
lapse has been initiated within a magnetized progenitor star
and following subsequent free-fall core collapse and accretion
shock, an eventual emergence of a rebound MHD shock can
evolve in a quasi-spherical self-similar manner and can end
up to a MSPS configuration with a high-density compact
degenerate object left behind.
4 CONCLUSIONS AND DISCUSSION
We outline and propose the model scenario of a quasi-
spherical rebound MHD shock to form high-density com-
pact stars after a gravity-induced collapse in the core of a
progenitor star that runs out of nuclear fuels. The stellar in-
terior magnetic field is expected to be enhanced during the
core collapse before the eventual emergence of a rebound
MHD shock; also the interior magnetic field should be much
stronger than the stellar surface magnetic field prior to the
onset of a core collapse and during the outward propagation
of the rebound MHD shock. Once the magnetostatic config-
uration of a remnant degenerate star appears, stellar ejecta
gradually detach from the compact object, exposing intense
surface magnetic fields of ∼ 1013∼11G for neutron stars or
∼ 107∼5G for magnetic white dwarfs.
Formally, MSPS solution (4) for density diverges as x →
0+. Conceptually, this can be readily reconciled by the onset
of degeneracy in core materials at a nuclear mass density.
In our model, there are two parameters for magnetic
field: index n for radial variation and ratio h. While it ap-
pears in this Letter that n depends on the stiffness (i.e., γ)
of EoS, as discussed below it is in fact a parameter free from
the stiffness (i.e., γ). Meanwhile, ratio h represents an ideal
MHD approximation that dictates the magnitude variation
of random transverse magnetic field; other factors, such as
metalicity, differential rotation, convective motions, buoy-
ancy etc. (Janka 2006), are important in generating random
magnetic fields inside a star prior to the onset of the core
collapse.
In contexts of SN explosions, two-shock models, i.e.,
models involving a ‘forward shock’ for the SN remnant shock
after the powerful rebound shock crashing into the inter-
stellar medium and a ‘reverse shock’ produced by the same
impact process (see, e.g., Chevalier et al. 1992 and Truelove
& McKee 1999), have been studied earlier. The major for-
mulation difference between these earlier works and ours is
that they ignored self-gravity of stellar ejecta. By estimates,
the self-gravity cannot be obviously dropped and thus these
models including forward and reverse shocks would be ap-
plicable in the limit of extremely strong shocks in order to
ignore self-gravity. Another major difference is that these
earlier models focus on circumstellar interactions, while we
focus on a rebound MHD shock as it travels within the mag-
netized stellar interior.
Our polytropic model is currently restricted to γ = 2−n
for a constant κ merely for expediency. This constraint can
be actually removed if we consistently allow the reduced
pressure to be ∝ αγmq where index parameter q ≡ 2(n +
γ − 2)/(3n − 2) 6= 0 in general and m = αx2(nx− v) is the
reduced enclosed mass. It is then possible for 1 < γ < 2
while n → 2/3. This more general case will be reported
separately (Wang & Lou 2006).
Numerical MHD simulations and observations are
needed to further test our scenario for rebound MHD shocks
in SNe, such as direct or indirect observation of density and
flow speed profiles (Lou & Wang 2006) as well as diagnostics
of synchrotron emissions caused by relativistic electrons in
random magnetic field generated by MHD shocks.
ACKNOWLEDGMENTS
This research has been supported in part by the ASCI
Center for Astrophysical Thermonuclear Flashes at the
Univ. of Chicago, by THCA, by the NSFC grants 10373009
and 10533020 at the Tsinghua Univ., and by the SRFDP
20050003088 and the Yangtze Endowment from the Min-
istry of Education at the Tsinghua Univ.
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	INTRODUCTION
	Formulation and Analysis

  We construct magnetohydrodynamic (MHD) similarity rebound shocks joining
`quasi-static' asymptotic solutions around the central degenerate core to
explore an MHD model for the evolution of random magnetic field in supernova
explosions. This provides a theoretical basis for further studying synchrotron
diagnostics, MHD shock acceleration of cosmic rays, and the nature of intense
magnetic field in compact objects. The magnetic field strength in space
approaches a limiting ratio, that is comparable to the ratio of the ejecta mass
driven out versus the progenitor mass, during this self-similar rebound MHD
shock evolution. The intense magnetic field of the remnant compact star as
compared to that of the progenitor star is mainly attributed to both the
gravitational core collapse and the radial distribution of magnetic field.

Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 4 November 2018 (MN LATEX style file v2.2)
Magnetohydrodynamic Rebound Shocks of Supernovae
Yu-Qing Lou1,2,3 ⋆ and Wei-Gang Wang1
1Physics Department and Tsinghua Centre for Astrophysics (THCA), Tsinghua University, Beijing, 100084, China;
2Department of Astronomy and Astrophysics, the University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA;
3National Astronomical Observatories, Chinese Academy of Sciences, A20, Datun Road, Beijing 100012, China.
4 November 2018
ABSTRACT
We construct magnetohydrodynamic (MHD) similarity rebound shocks joining ‘quasi-
static’ asymptotic solutions around the central degenerate core to explore an MHD
model for the evolution of random magnetic field in supernova explosions. This pro-
vides a theoretical basis for further studying synchrotron diagnostics, MHD shock ac-
celeration of cosmic rays, and the nature of intense magnetic field in compact objects.
The magnetic field strength in space approaches a limiting ratio, that is comparable
to the ratio of the ejecta mass driven out versus the progenitor mass, during this
self-similar rebound MHD shock evolution. The intense magnetic field of the remnant
compact star as compared to that of the progenitor star is mainly attributed to both
the gravitational core collapse and the radial distribution of magnetic field.
Key words: magnetohydrodynamics (MHD) – shock waves – stars: neutron – stars:
winds, outflows – supernova remnants – white dwarfs
1 INTRODUCTION
Self-similar evolution of a spherical gas flow under self-
gravity and thermal pressure has been studied over past
four decades: from simulations and the discovery of Larson-
Penston (L-P) type solutions (Bodenheimer & Sweigart
1968; Larson 1969a, b; Penston 1969a, b), to the construc-
tion of the expansion-wave collapse solution (EWCS) using
the central free-fall asymptotic solution (Shu 1977) as well as
to the application of phase-match techniques for construct-
ing infinite series of discrete global solutions including L-P
type solutions (Hunter 1977) and solutions for envelope ex-
pansion with core collapse (EECC; Lou & Shen 2004). Prop-
erties of eigensolutions crossing the sonic critical line were
examined (Jordan & Smith 1977; Shu 1977; Whitworth &
Summers 1985; Hunter 1986). Self-similar shocks were stud-
ied and applied to various astrophysical settings by Tsai &
Hsu (1995), Shu et al. (2002), Shen & Lou (2004), and Bian
& Lou (2005). While these major results were obtained for
an isothermal gas, the counterpart problem with a poly-
tropic equation of state (EoS) was also studied by Cheng
(1978), Goldreich & Weber (1980), Yahil (1983), Suto &
Silk (1988), McLaughlin & Pudritz (1997), Fatuzzo et al.
(2004) and Lou & Gao (2006). In most cases, the polytropic
results share a feature that by setting the polytropic index
γ = 1 in the isothermal limit, all asymptotic behaviours ap-
⋆ E-mail: louyq@tsinghua.edu.cn and lou@oddjob.uchicago.edu;
wwg03@mails.tsinghua.edu.cn
proach the isothermal counterpart solutions. However, Lou
& Wang (2006) reported new ‘quasi-static’ asymptotic so-
lutions unique to a polytropic gas with γ > 1.2 and con-
structed self-similar rebound shocks for supernovae (SNe).
Chiueh & Chou (1994) studied a self-similar MHD
problem by including the magnetic pressure gradient force
in the momentum equation. Yu & Lou (2005) improved
their formulation and provided a more detailed analysis (see
Zel’dovich & Novikov 1971 for a discussion of random mag-
netic field). Wang & Lou (2006) studied this MHD problem
for a polytropic gas and derived the ‘quasi-static’ asymp-
totic solutions. Self-similar MHD shocks were explored by
Yu et al. (2006). As magnetic field is inevitably involved in
SNe and is crucial for synchrotron radiation and cosmic ray
acceleration, we construct here rebound MHD shocks with
‘quasi-static’ asymptotic solutions to model magnetic field
evolution in SN explosions.
Type II, Ib, Ic SNe are thought to be caused by grav-
itational core collapse due to an insufficient nuclear fuel;
such collapse creates an over-dense core, which rebounds
abruptly initiating a powerful rebound shock. The energetics
of sustaining such a rebound shock has been an outstanding
problem. We approach this issue in the following perspec-
tive. Triggered by such a core collapse, the rebound shock
is essentially supported by the neutrino-driven mechanism,
and several complicated physical processes are involved in
the stellar interior: all four elementary forces and the cou-
pling of various fluids and matters such as baryons, neutri-
nos, photons etc. (e.g., Janka et al. 2006). We approximate
c© 0000 RAS
http://arxiv.org/abs/0704.0223v1
2 Y.-Q. Lou & W.-G. Wang
such a dynamic system in terms of a single fluid with a
polytropic EoS, and treat the shock as an energy-conserved
self-similar shock. Conceptually, the ‘rebound shock’ here
refers to a neutrino-driven shock, as opposed to the ‘prompt
shock’ mentioned in Janka et al. (2006). We constructed
such a rebound shock (Lou & Wang 2006) to model a SN
explosion followed by a self-similar evolution leading to a
quasi-static configuration. In reference to the hydrodynamic
model of Lou & Wang (2006), the main thrust of this Let-
ter is to construct approximately a self-similar model of a
quasi-spherically symmetric rebound MHD shock for a SN
explosion, providing the profile and evolution of magnetic
field to facilitate future studies of synchrotron radiation and
MHD shock acceleration of cosmic rays, and to probe the
nature of intense magnetic field of compact stellar objects
left behind.
2 FORMULATION AND ANALYSIS
2.1 The Self-Similar MHD Formulation
A quasi-spherical similarity MHD flow embedded with a
completely random magnetic field on small scales is formu-
lated the same as in Yu & Lou (2005) and Yu et al. (2006);
the key difference here is the polytropic EoS p = κργ instead
of an isothermal gas, where p is the pressure, ρ is the mass
density, and κ is constant. Using the magnetic flux frozen-in
condition, the ideal MHD equations, viz., the mass conserva-
tion equation, the radial momentum equation, the magnetic
induction equation and the polytropic EoS, can be reduced
to two nonlinear ordinary differential equations (ODEs)
(n− 1)v +
nx− v
3n− 2
2(x− v)(nx− v)
α(nx− v)2 − (γαγ + hα2x2)
, (1)
(n− 1)
αv(nx− v) + 2hα2x2
(nx− v)2
(3n− 2)
− 2γαγ
(x− v)
α(nx− v)2 − (γαγ + hα2x2)
along with a useful relation m = αx2(nx−v) by the following
MHD self-similar transformation in a polytropic gas flow
r = k
x, u = k
v, ρ =
4πGt2
, p =
kt2n−4
k3/2t3n−2
(3n− 2)G
(nx−v) , < B2t >=
kt2n−4
, (3)
where G is the gravitational constant, M is the enclosed
mass at time t within radius r, u is the radial flow speed,
< B2t > is the mean square of random transverse magnetic
field Bt, x is the independent self-similar variable, v(x) is
the reduced flow speed, α(x) is the reduced density, m(x)
is the reduced enclosed mass, the prime ′ stands for the
first derivative d/dx, k and n are two parameters, and h
is a parameter for the strength of < B2t >
1/2. We expedi-
ently take γ = 2 − n for a polytropic EoS with a constant
κ ≡ k(4πG)γ−1 = pρ−γ . The magnetosonic critical curve is
determined by the simultaneous vanishing of the numerator
and denominator on the RHS of eq (1) or (2). The two eigen-
solutions of v′ across the magnetosonic critical curve can be
derived by using the L’Hôspital rule (Lou & Wang 2006; Yu
& Lou 2005; Yu et al. 2006). The solutions are obtained for
v(x) and α(x), and the magnetic field < B2t >
1/2 is then
known from transformation (3).
2.2 Analytic Asymptotic MHD Solutions
For h < hc ≡ n2/[2(1− n)(3n− 2)], eqs (1) and (2) give the
magnetostatic solution of a magnetized singular polytropic
sphere (MSPS) with v = 0 and
2γ(4− 3γ)
(1− γ)
]−1/n
, (4)
t >= h
2γ(4− 3γ)
(1− γ)
]−2/n
2−4/n
. (5)
There exists an asymptotic MHD solution approaching this
limiting form at small x (referred to as the type I ‘quasi-
static’ asymptotic MHD solution), viz., v = LxK and
2γ(4− 3γ)
(1− γ)
]−1/n
(K + 2− 2/n)L
n(K − 1)
n2/(2γ)
4− 3γ
]−1/n
K−1−2/n
, (6)
where K is the root of quadratic equation
/2 + n(3n− 2)h]K2 − (4− 3n)[n/2 + (3n− 2)h]K
+ γ(2/n− 2)(3n− 2)h = 0 . (7)
When 12 − 8
2 < n < 0.8 and h0 < h < hc, where h0 ≡
(3 + 2
2)n − 4
4 − (3 − 2
/[2n(3n − 2)], or when
2/3 < n < 12−8
2 for h < hc, eq (7) gives two roots K > 1,
corresponding to two possible ‘quasi-static’ solutions.
The asymptotic MHD solution at large x is α =
−2/n + · · · and
v = B0x
1−1/n −
(3n− 2)
2h(n− 1)
1−2/n
n[2(n+ γ)− 3]
(2−2γ−n)/n
+ · · · , (8)
where A0 and B0 are two constants. Solution (8) at large x
can be connected to ‘quasi-static’ asymptotic MHD solution
(6) and (7) at small x by a Runge-Kutta integration (Press
et al. 1986), crossing the magnetosonic critical curve either
smoothly or with an MHD shock (Yu et al. 2006).
2.3 MHD Shock Jump Conditions
MHD shock conditions (Yu et al. 2006; Lou & Wang 2006)
include conservations of mass, momentum, energy and mag-
netic flux, and in self-similar forms, they appear as
= 0 , (9)
= 0 , (10)
c© 0000 RAS, MNRAS 000, 000–000
MHD Rebound Shocks in Supernovae 3
(γ − 1)
αγ−1s
+ 2hαs
= 0 , (11)
where quantities in square brackets with superscript ‘1’ (up-
stream) and subscript ‘2’ (downstream) remain conserved
across the MHD shock front indicated by a subscript s. The
parameter k changes according to k2 = k1x
s2 on two
sides of a shock. For the specific entropy to increase from
upstream to downstream sides, xs1 > xs2 is necessary. MHD
shock conditions (9)−(11) lead to a quadratic equation (Lou
& Wang 2006); once we specify physical conditions on one
side of a chosen shock location, the corresponding quantities
α, v, x on the other side are readily computed.
3 REBOUND MHD SHOCKS IN SUPERNOVA
EXPLOSIONS
Various rebound MHD shocks are constructed numerically,
parallel to Lou & Wang (2006). With chosen inner and outer
radii, e.g., ri = 10
6cm and ro = 10
12cm for neutron star
formation, and when the k parameter in transformation (3)
is specified, we apply our solutions to a physical rebound
MHD shock scenario for SNe (Lou & Wang 2006).
3.1 Final and Initial Configurations
Similar to the hydrodynamic rebound shock model of Lou
& Wang (2006), the final configuration (small x) of our re-
bound MHD shock solutions gradually evolves to a MSPS
and is regarded as a remnant compact object after the re-
bound MHD shock ploughing through stellar ejecta; the
initial configuration (large x) marks the onset of gravity-
induced core collapse with outer inflows or outflows such as
stellar winds or stellar oscillations.
We define the outer initial mass Mo,ini and the inner ul-
timate mass Mi,ult the same way as in Lou & Wang (2006)
and regard them as rough estimates for the masses of the
progenitor star and the remnant compact object. The ratio
of the two masses isMo,ini/Mi,ult = λ1(ro/ri)
(3−2/n) where
λ1 ≡ A0(k1/k2)1/n
n2/[2γ(4−3γ)]+(1−γ)h/γ
involves
parameters of the rebound MHD shock and is equal to the
ratio of enclosed masses at the same r. Similar to the result
of Lou & Wang (2006), we find numerically that λ1 > 1 de-
pends on the choice of solutions, clearly indicating that a
rebound MHD shock drives out stellar materials.
By eq (3), the final magnetostatic configuration gives
t,ult >
k2γ(4− 3γ)
]−1/n
1−2/n
The ratio of initial to final magnetic fields at the same r is
t,ini >
1/2 / < B2
t,ult >
1/2= λ1, where λ1 > 1 by numer-
ical exploration. Thus a rebound MHD shock breakout pro-
cess reduces the magnetic field by the same ratio of enclosed
masses at the same r; yet this decrease in magnetic field is
insignificant as compared to the radial variation of magnetic
field, i.e., the r1−2/n dependence. As γ approaches 4/3 or
n → 2/3, this scaling approaches r−2, while the dependence
of enclosed mass on r goes to r0. For a ∼ 10G (0.1G) surface
magnetic field at ro = 10
12cm, we estimate a magnetic field
in the interior of the final configuration (ri = 10
6cm) to be
∼ 1013G (1011G), sensible for magnetized neutron stars; if
we take ri = 10
9cm, then the final interior magnetic field
is estimated to be ∼ 107G (105G), fairly close to relevant
magnetic field strengths of white dwarfs (e.g., Euchner et al.
2005, 2006; Schmidt et al. 2003).
3.2 Evolution of Rebound MHD Shocks
Time evolution of density, velocity and enclosed mass are
similar to those described by Lou & Wang (2006). We focus
here on the magnetic field evolution. Figure 1 shows a typical
time evolution of < B2t >
1/2 / < B2
t,ult
>1/2 to complement
the r1−2/n behaviour. Magnetic field increases at first, and
gradually decreases until reaching the magnetostatic config-
uration much smaller than the initial configuration in size. In
short, magnetic field changes moderately. The crucial point
is that the magnetic field varies significantly in r within a
star. If we take the magnetic field at the outer boundary
to be the surface magnetic field of the progenitor star and
take the magnetic field at the inner boundary as the surface
magnetic field of the remnant compact star, then a large
ratio of ∼ 1012 appears in forming a neutron star (Lou &
Wang 2006). This model feature may explain the intense
magnetic field of neutron stars inferred from spin-down ob-
servations of radio pulsars. In our scenario, after the passage
of such a rebound MHD shock, stellar ejecta detach from
the central degenerate neutron star which is thus exposed
with a surface magnetic field of 1013∼11G. In the same spirit
of Lou & Wang (2006), we also suggest the formation of
magnetic white dwarfs from the end of main-sequence stars
with 6 ∼ 8M⊙; in this scenario, the surface magnetic field
of an exposed central white dwarf is in a plausible range of
∼ 107∼5G (e.g., Euchner et al. 2005, 2006; Schmidt et al.
2003).
The major point to be emphasized is that random mag-
netic field preexists inside progenitor stars through various
dynamo processes. We detect magnetic field strengths of or-
der 10−2∼3G on stellar surface and this corresponds to a
much stronger magnetic field in the stellar interior with a
scaling of ∼ r1−2/n shortly after the initiation of core col-
lapse. In addition, the interior magnetic field can be consid-
erably strengthened by the free-fall core collapse preceding
the emergence of a rebound MHD shock (see Lou & Wang
2006 for descriptions of the rebound shock scenario and the
core collapse process), according to the frozen-in flux and
accretion shock conditions. In reality, these two processes
happen concurrently to produce the resultant self-similar
distribution of magnetic field. In short, the interior magnetic
field would be much stronger than the surface magnetic field
and can be further enhanced to reach a high-field regime.
The origin of stellar magnetic field was argued by
several authors to come from various processes, includ-
ing dynamo effects and thermomagnetic instabilities (e.g.,
Reisenegger et al. 2005). Our MHD scenario of interior core
collapse and rebound shock appears to grossly match with
observational facts. From our MSPS configuration with a
random magnetic field strength scaled as Bt ∝ r1−2/n in
a polytropic gas, we see a real possibility that the interior
magnetic field can be actually much stronger than the sur-
c© 0000 RAS, MNRAS 000, 000–000
4 Y.-Q. Lou & W.-G. Wang
r(cm)
t = ∞
Typical Magnetic Field Evolution During Shock Breakout
Figure 1. The ratio Bt/Bt,ult ≡< B
1/2 / < B2
t,ult
>1/2 is
the rms magnetic field strength divided by the corresponding rms
magnetic field strength of the final magnetostatic configuration
at the same r. This example is constructed by integrating inward
from (x0 , v0 , α0) on the magnetosonic critical curve and using an
eigensolution to match with a quasi-static solution as x → 0+; we
use the solution portion within xs2 < x0 for the downstream. We
then obtain the upstream point (xs1, vs1, αs1) by the MHD shock
jump conditions from the values of (xs2, vs2, αs2) obtained in the
former integration and further integrate outward to determine
the upstream solution. The relevant parameters are γ = 1.32,
n = 0.68, h = 0.01, k1 = 7.7 × 10
16 cgs units, k2 = 4 × 10
cgs units, x0 = 1.778 , v0 = 0.4620 , α0 = 0.067, and xs2 = 1.1.
Here, t1 = 6.61× 10
−5s is the time when the MHD shock crosses
the inner boundary and is the initial time of application; t2 =
4.40 × 104s is the time when the MHD shock crosses the outer
boundary; tm1 = 0.1s and tm2 = 1 × 10
8s are two intermediate
times between t1 and t2 and t2 and t = ∞.
face magnetic field. Once the onset of a gravitational col-
lapse has been initiated within a magnetized progenitor star
and following subsequent free-fall core collapse and accretion
shock, an eventual emergence of a rebound MHD shock can
evolve in a quasi-spherical self-similar manner and can end
up to a MSPS configuration with a high-density compact
degenerate object left behind.
4 CONCLUSIONS AND DISCUSSION
We outline and propose the model scenario of a quasi-
spherical rebound MHD shock to form high-density com-
pact stars after a gravity-induced collapse in the core of a
progenitor star that runs out of nuclear fuels. The stellar in-
terior magnetic field is expected to be enhanced during the
core collapse before the eventual emergence of a rebound
MHD shock; also the interior magnetic field should be much
stronger than the stellar surface magnetic field prior to the
onset of a core collapse and during the outward propagation
of the rebound MHD shock. Once the magnetostatic config-
uration of a remnant degenerate star appears, stellar ejecta
gradually detach from the compact object, exposing intense
surface magnetic fields of ∼ 1013∼11G for neutron stars or
∼ 107∼5G for magnetic white dwarfs.
Formally, MSPS solution (4) for density diverges as x →
0+. Conceptually, this can be readily reconciled by the onset
of degeneracy in core materials at a nuclear mass density.
In our model, there are two parameters for magnetic
field: index n for radial variation and ratio h. While it ap-
pears in this Letter that n depends on the stiffness (i.e., γ)
of EoS, as discussed below it is in fact a parameter free from
the stiffness (i.e., γ). Meanwhile, ratio h represents an ideal
MHD approximation that dictates the magnitude variation
of random transverse magnetic field; other factors, such as
metalicity, differential rotation, convective motions, buoy-
ancy etc. (Janka 2006), are important in generating random
magnetic fields inside a star prior to the onset of the core
collapse.
In contexts of SN explosions, two-shock models, i.e.,
models involving a ‘forward shock’ for the SN remnant shock
after the powerful rebound shock crashing into the inter-
stellar medium and a ‘reverse shock’ produced by the same
impact process (see, e.g., Chevalier et al. 1992 and Truelove
& McKee 1999), have been studied earlier. The major for-
mulation difference between these earlier works and ours is
that they ignored self-gravity of stellar ejecta. By estimates,
the self-gravity cannot be obviously dropped and thus these
models including forward and reverse shocks would be ap-
plicable in the limit of extremely strong shocks in order to
ignore self-gravity. Another major difference is that these
earlier models focus on circumstellar interactions, while we
focus on a rebound MHD shock as it travels within the mag-
netized stellar interior.
Our polytropic model is currently restricted to γ = 2−n
for a constant κ merely for expediency. This constraint can
be actually removed if we consistently allow the reduced
pressure to be ∝ αγmq where index parameter q ≡ 2(n +
γ − 2)/(3n − 2) 6= 0 in general and m = αx2(nx− v) is the
reduced enclosed mass. It is then possible for 1 < γ < 2
while n → 2/3. This more general case will be reported
separately (Wang & Lou 2006).
Numerical MHD simulations and observations are
needed to further test our scenario for rebound MHD shocks
in SNe, such as direct or indirect observation of density and
flow speed profiles (Lou & Wang 2006) as well as diagnostics
of synchrotron emissions caused by relativistic electrons in
random magnetic field generated by MHD shocks.
ACKNOWLEDGMENTS
This research has been supported in part by the ASCI
Center for Astrophysical Thermonuclear Flashes at the
Univ. of Chicago, by THCA, by the NSFC grants 10373009
and 10533020 at the Tsinghua Univ., and by the SRFDP
20050003088 and the Yangtze Endowment from the Min-
istry of Education at the Tsinghua Univ.
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http://arxiv.org/abs/astro-ph/0604261
	INTRODUCTION
	Formulation and Analysis
