byung-il jun- interaction of a pulsar wind with the expanding supernova remnant

8/3/2019 Byung-Il Jun- Interaction of a Pulsar Wind with the Expanding Supernova Remnant

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THE ASTROPHYSICAL JOURNAL, 499: 282293, 1998 May 201998.The American AstronomicalSociety. All rightsreserved.Printed in U.S.A.(

INTERACTION OF A PULSAR WIND WITH THE EXPANDING SUPERNOVA REMNANT

BYUNG-IL JUN1Department of Astronomy, University of Minnesota, 116 Church Street SE, Minneapolis, MN 55455

Received 1997 September 5; accepted 1998 January 2

ABSTRACT

Recent Hubble Space Telescope observations of the Crab Nebula show lamentary structures thatappear to originate from the Rayleigh-Taylor (R-T) instability operating on the supernova ejecta acceler-ated by the pulsar-driven wind. In order to understand the origin and formation of the laments in theCrab Nebula, we study the interaction of a pulsar wind with the uniformly expanding supernovaremnant by means of numerical simulation. We derive the self-similar solution of this model for ageneral power-law density prole of supernova ejecta.

By performing two-dimensional numerical simulations, we nd three independent instabilities in theinteraction region between the pulsar wind and the expanding supernova remnant. The rst weak insta-bility occurs in the very beginning and is caused by the impulsive acceleration of supernova ejecta by thepulsar wind. The second instability occurs in the postshock ow (shock wave driven by pulsar bubble)during the intermediate stage. This second instability develops briey while the gradients of density andpressure are of opposite signs (satisfying the criterion of the R-T instability). The third and most impor-

tant instability develops as the shock driven by the pulsar bubble becomes accelerated (rP t6@5). This isthe strongest instability and produces pronounced lamentary structures that resemble the observed la-ments in the Crab Nebula. Our numerical simulations can reproduce important observational features ofthe Crab Nebula. The high-density heads in the R-T ngertips are produced because of the compress-ibility of the gas. The density of these heads is found to be about 10 times higher than other regions inthe ngers. The mass contained in the R-T ngers is found to be 60%75% of the total shocked mass,and the kinetic energy within the R-T ngers is 55%72% of the total kinetic energy of the shockedow. The R-T ngers are found to accelerate with a slower rate than the shock front, which is consistentwith the observations. By comparing our simulations and the observations, we infer that the some nger-like laments (region F or G in Hesters observations) started to develop about 657 yr ago.

Subject headings: hydrodynamics instabilities ISM: individual (Crab Nebula) pulsars: general shock waves supernova remnants

1. INTRODUCTION

Pulsars release their rotational energy in the forms of arelativistic wind and electromagnetic waves & Gunn(Rees

and this energy generates the nebula that expands1974),into the surrounding supernova remnant. In the CrabNebula, this spin-down power of the pulsar is sufficient tosupport the synchrotron radiation of the nebula, althoughthe complete model linking the pulsar to the nebula is yet tobe given. The pulsar wind is also expected to interact withthe expanding supernova remnant and to produce someinteresting observational results.

The study of the interaction between the pulsar wind andthe enveloping supernova ejecta has been motivated sincethe recent observations of the Crab Nebula by the Hubble

Space Telescope (HST) revealed the detailed morphologyand ionization structures of the helium-rich laments. Thecomplex emission-line laments would appear to originatefrom the Rayleigh-Taylor (R-T) instability operating on theejecta accelerated by the pulsar wind et al.(Hester 1996).These nger-like structures seem to grow inward (towardthe center of the remnant) and to be connected to each otherat their origins by a faint, thin tangential structure (the skin ). Some of the long nger-like structures terminate indense head regions. The interface between the synchro-tron nebula and an ejecta shell has been thought to be R-T

1 Current address: Lawrence Livermore National Laboratory, P.O.Box 808, L-630, Livermore, CA 94551; jun2=llnl.gov.

unstable & Gull since(Chevalier 1975; Chevalier 1977),motions of the laments and the synchro-(Trimble 1968)tron nebula itself et al. revealed their(Bietenholtz 1991)outward postexplosion acceleration. The magnetic eld inthe Crab has been inferred to be strong (a few hundredmicrogauss) and seems to play an important role in theformation of the laments. Polarimetric VLA observationsof the Crab show a radially aligned magnetic eld orienta-tion & Kronberg although the rotating(Bietenholz 1990),pulsar is likely to produce toroidal (tangential) magneticelds & Gunn The radial magnetic eld can be(Rees 1974).produced as R-T ngers stretch the existing eld lines.

In the Crab Nebula, the observed mass and kineticenergy are much smaller than expected from the typicalsupernova (see review by & Fesen SeveralDavidson 1985).authors have proposed that the Crab formed from a normalType II supernova remnant and that an outer fast-movingshell contains most the of mass and kinetic energy

& Coroniti despite the(Chevalier 1977 ; Kennel 1984),absence of convincing detection of the surrounding shell(e.g., et al. In this model for the Crab, the pulsarFrail 1995).wind pushes into the uniformly expanding supernova ejecta.Therefore, the model includes four shocks from the outsideinward : a supernova blast wave, a reverse shock movinginto the supernova ejecta, an outward-facing shock wavedriven by expanding pulsar bubble, and a wind terminationshock Note that the forward shock front driven by(Fig. 1).the pulsar bubble is actually disturbed by the instability (see

The signature of the shock front driven by the pulsarFig. 5).

282


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Contact Discontinuity

R-T Finger

Forward Shock Driven by Pulsar Bubble

Relativistic Wind

Supernova Blast Wave

Supernova Reverse Shock

Wind Termination Shock

Pulsar

Supernova Ejecta

PULSAR WIND WITH SUPERNOVA REMNANT 283

FIG. 1.Schematic representation of the normal Type II supernovaremnant model for the simulation. The gure shows the pulsar windblowing into the uniformly expanding supernova remnant. The pulsar islocated in the center. The small dotted circle outside of the pulsar is thewind termination shock. This shock heats up the wind gas and produces ahot bubble expanding into the supernova remnant. The contact discontin-uity between the pulsar bubble and the shocked supernova gas is shown tobe Rayleigh-Taylor unstable. The R-T ngers are connected to each otherby a thin shell conned by the forward shock. Note that this forward shockfront is actually distorted by the development of the instability in thenumerical simulation (see Outside the forward shock are theFig. 6).expanding supernova ejecta, the reverse shock, and the blast wave of thesupernova.

bubble has been found near the boundary of the CrabNebula recently by & Hester although theSankrit (1997),existence of the shock is debated because the steepening ofthe radio spectral index near the outer boundary of theCrab Nebula has not been found et al.(Frail 1995 ;

et al. Sankrit & Hester nd that theBietenholtz 1997).shock should be expanding with a velocity about v \ 150km s~1 into the freely expanding supernova ejecta.

The normal Type II supernova remnant model is particu-larly attractive because there is no need for a peculiar low-energy supernova event, and it can be applied to otherpulsar-powered remnants. Also, the mechanism for theobserved acceleration is naturally explained because thepulsars wind shock expands with the law rP t6@5 in theself-similar stage if the pulsars luminosity is assumed to beconstant In fact, the low-energy event(Chevalier 1977).model itself, which is considered because of the discourag-ing observational results on the absence of a fast-movingshell in the Crab Nebula, has many problems reconcilingthe observed features of the Crab Nebula. First, even low-energy supernovae should produce an outer shock since thesupersonically moving gas generates a shock ahead of it.This shock should have a currently expanding velocity,about km s~1. Therefore, the fact that nov

shockD 1400

steepening of the spectral index near the boundary of theCrab Nebula has been found cannot support the low-energy event model. Actually, this low-energy event modelcan generate a stronger shock than that of the normal TypeII supernova model. Second, the acceleration of both theline-emitting laments and the synchrotron nebula and thewell-resolved Rayleigh-Taylor nger-like laments pointinginward have not been explained by the low-energy eventmodel. Currently, the only mechanism that can accelerateboth the synchrotron nebula and the laments is the pulsarwind blowing into the freely expanding supernova ejectaaround the Crab Nebula.

In this paper, we will present the results of our numerical

investigation of a Type II supernova remnant model on theorigin of the lamentary structures in the Crab Nebula. In

we study the self-similar solution for our model. In 2,our numerical methods and initial conditions for the 3,

simulation is described. presents our numericalSection 4results on the hydrodynamic evolution and uid insta-bilities of the interaction region. In we will discuss the 5,evolution of Rayleigh-Taylor ngers in terms of mass and

kinetic energy. We discuss the related issues to the CrabNebula further in and summarize our main conclusions 6in 7.

2. SELF-SIMILAR SOLUTION

Chevalier has studied the self-similar solu-(1977, 1984)tion for the current model of the pulsar bubble expandinginto a uniformly moving medium of constant density. Inorder to understand the dynamics of the system and ournumerical results, we generalize Chevaliers self-similarsolution for a power-law density medium. We only considerthe region between the contact discontinuity and theforward shock driven by the pulsar bubble. Using dimen-sional analysis, the expansion law can be obtained readily,rP t(6~n~l)@(5~n), where n and l are the power-law indicesfor the moving ejecta density (oP r~n) and the pulsar lumi-nosity (LP t~l). For a constant pulsar luminosity anduniform ejecta density, the bubble radius expands with thelaw, rP t1.2 ; that is, it is accelerating. It should be notedthat the solution is only applicable for n\3 because of thenite mass requirement [MP tn~3r3~n/(3 [ n)], as pointedout by & Fransson Also, the bubble mustChevalier (1992).expand at least with constant velocity (no deceleration isallowed) because the surrounding medium is freely expand-ing. In actual evolution, the bubble can decelerate at earlystages while the bubble velocity is much higher than theejecta velocity because the medium motion can be negligi-ble (see the one-dimensional numerical results in 4).However, as the bubble expansion approaches the self-similar stage, the eect of the moving medium is not negligi-ble, and the bubble expansion cannot decelerate. Therefore,the self-similar solution is only applicable for l 1.

In order to derive a self-similar solution for the generalpower-law density prole of supernova ejecta, we can denethe similarity variables:

f\r

Ata, (1)

o\ tb~naD(f) , (2)

v \ aAta~1V(f) , (3)

p \ a2A2tb~na`2a~2P(f) , (4)

where A is constant, a \ (6 [ n [ l)/(5 [ n), and b is anexpansion factor dened as b \ n [ 3. The expansion factorb is 0 if the surrounding medium is at rest. The eect of aconstant expansion of the supernova ejecta is to change thedensity with the law oP tn~3 at the same radius.

The one-dimensional uid equations for a spherical coor-dinate system are

Lo

Lt] v

Lo

Lr]o

Lv

Lr]

2ov

r\ 0 , (5)

Lv

Lt] v

Lv

Lr]

1

o

Lp

Lr\ 0 , (6)


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0.985 0.990 0.995 1.000

1

10

(c) n = 0, l = 0.5, a = 1.1, = 5/3

0.980 0.985 0.990 0.995 1.000

1

10

(a) n = 0, l = 0, a = 1.2, = 5/3

0.9940 0.9960 0.9980 1.0000

100

101

(d) n = 0, l = 0, a = 1.2, = 1.1

0.960 0.970 0.980 0.990 1.000

1

10

(b) n = 3, l = 0, a = 1.5, = 5/3

D

P

V

D

P

V

D

P

V

D

P

V

284 JUN Vol. 499

FIG. 2.Self-similar solutions for the model described in D, P, and V represent the density, gas pressure, and the velocity, respectively. Each plotFig. 1.shows thephysical variables in the region between thecontact discontinuity andthe shock front (f\ 1).

Lp

Lt] v

Lp

Lr] cp

Lv

Lr]

2cpv

r\ 0 , (7)

where c is the adiabatic index.Using the similarity variables, we can transform the uid

equations into the ordinary dierential equations:

Aba

[ n ]2V

f

BD ] (V[ f)

dD

df] D

dV

df\ 0 , (8)

(a [ 1)

aV] (V[ f)

dV

df]

1

D

dP

df\ 0 , (9)

Ab [ 2a

[ n ] 2 ]2cV

f

BP ] (V[ f)

dP

df] cP

dV

df\ 0 .

(10)

These equations can be integrated from the shock front(driven by the expanding bubble) using the followingboundary conditions :

D(1) \c] 1

c[ 1, (11)

V(1) \ 1 ]c[ 1

c] 1(c [ 1) , (12)

P(1) \2

c] 1(1 [ c)2 , (13)

where c is 1/a for a moving medium and 0 for a stationarymedium.

Self-similar solutions for several dierent cases are shownin The adiabatic index is chosen to be 5/3 exceptFigure 2.for because the shocked supernova ejecta regionFigure 2d


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0.0e+00 2.0e+180.0e+00

5.0e07

1.0e06

0.0e+00

5.0e+08

1.0e+09

1.5e+09

2.0e+09

1025

1024

1023

1022

(a) t = 100yrs

0.0e+00 5.0e+180.0e+00

1.0e08

2.0e08

3.0e08

0.0e+00

5.0e+08

1.0e+09

1.5e+09

2.0e+09

1026

1025

1024

1023

1022

(b) t = 500yrs

0.0e+00 1.0e+190.0e+00

2.0e09

4.0e09

6.0e09

8.0e09

0.0e+00

5.0e+08

1.0e+09

1.5e+09

2.0e+09

1027

1026

1025

1024

1023

1022

(c) t = 1000yrs

W.T.

C.D.

F.S.

No. 1, 1998 PULSAR WIND WITH SUPERNOVA REMNANT 285

FIG. 3.One-dimensional numerical simulation of the normal Type II supernova remnant model. shows the result at the stationary medium stage.Fig. 3aFrom inside outward, the wind termination shock (W.T.), contact discontinuity (C.D.), and forward shock (F.S.) are seen at the designated locations. Eachplot in the same column represents the density, velocity, and gas pressure from top to bottom, respectively. and are the results at the intermediateFig. 3b 3cstage andthe moving mediumstage, respectively (see thetext for details).

is considered to be nonrelativistic, while the pulsar wind isrelativistic. Each plot shows density, gas pressure, andvelocity proles in the region between the contact discon-tinuity and the shock front. In general, the ow within theshell (shocked supernova ejecta) is found to be stable underthe Rayleigh-Taylor instability criterion in all cases, sincethe gas pressure gradient has the same sign as the densitygradient. But the contact discontinuity is Rayleigh-Taylorunstable for a[1 because the shell is denser than theregion inside of the contact discontinuity. is caseFigure 2aIII in The shell thickness is aboutChevalier (1984).

Comparing Figures and the shell0.02rshock

. 2a, 2b, 2c,becomes thicker as a increases, as also found by Chevalier(1984). We nd that the case in shows a thickerFigure 2bshell than the case for a \ 1.5, n \ 0. shows theFigure 2dsame case as except for c\ 1.1. This case is con-Figure 2asidered to illustrate the eect of radiative cooling. Asexpected, the shell is found to be thinner in the case forc\ 1.1 than c\ 5/3.

3. NUMERICAL METHODS AND INITIAL CONDITIONS

Understanding the dynamical interaction between thepulsar wind and the ejecta requires multidimensionalnumerical simulations with high resolution because thedevelopment of the instability is highly nonlinear. High-resolution simulations are particularly necessary to resolve

the thin shell and small regions containing R-T ngers. Forthis problem, we utilize the ZEUS-3D code, developed andtested at the National Center for Supercomputing Applica-tion & Norman ZEUS-3D is a three-(Clarke 1994).dimensional Eulerian nite dierence code for solving theideal MHD equations. The grid velocity is also allowed tochange every time step so that the grid can follow theexpanding system. This is necessary in modeling the CrabNebula in order to follow the expanding supernovaremnant accurately and to keep adequately high resolutionat the interaction region between the pulsar wind and thesupernova ejecta. For the detailed numerical algorithms ofthis code, readers are referredto & NormanStone (1992).

The model includes the pulsar wind, moving supernovaejecta, and supernova shell conned by the supernova blastwave and reverse shock. Since we are interested in the inter-action region between the pulsar wind and the supernovaejecta, our simulation excludes the supernova shell. Readersare referred to Jun & Norman for detailed(1996a; 1996b)study of the dynamics of supernova shell. The pulsar wind isgenerated by the constant mechanical luminosity,L\ 1.0] 1040 ergs s~1. The luminosity of the pulsar isgiven by L\ 2.0nr2ov3, where r, o, and v are the initialradius of the wind, the density of the wind gas, and theinitial wind velocity, respectively. This luminosity is muchhigher than the current luminosity of the Crab pulsar,


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108

109

1010

1011

time

1017

1018

1019

1020

1021

R_

cd

lu=100,di=1.0,do=100.0,vo=500

286 JUN Vol. 499

LD 1.0] 1038 ergs s~1, and is determined by accountingfor the observed rate of decrease in the luminosity. Theinitial radius of the wind is chosen to be 0.1 pc, and thedensity of the pulsar wind gas is assumed to beo\ 1.67]10~24 g cm~3. The initial wind velocity is deter-mined from the given luminosity accordingly. The super-nova ejecta is initialized to be uniformly moving withv \ 500 km s~1 at 0.1 pc, and the velocity at larger radius is

linearly proportional to the radius. The density prole ofthe supernova ejecta is usually modeled by a constantdensity in the inner core region and the power-law densityprole (oP r~n) in the outer region, although the mass ratiobetween the two regions can change depending on themodel. According to & Fransson theChevalier (1992),forward shock driven by the pulsar bubble is likely to bestill in the inner core region during the constant luminosityphase. In our simulation, we assume the constant density ofsupernova ejecta to be o\ 1.67] 10~22 g cm~3 over theentire ejecta. Note that the ejecta density decreases withtime because of ejecta expansion. We assume the entire uidto be nonrelativistic gas with c\ 5/3, for simplicity. Wedefer a more general simulation with varying luminosityand dierent density proles of the ejecta including detailedphysics such as magnetic elds and cooling to a futureproject. Because of our idealized simple initial conditions,we do not attempt to compare the evolution with exacttimescale to the Crab Nebula. We focus our investigationon the development and the formation of the lamentarystructures by uid instabilities.

The simulation is carried out in a spherical coordinatesystem with the resolution 800] 500 in the r-/ plane. Thethin shell is resolved by about 16 grid cells in the r-directionwith this mesh size because the thickness of the thin shell isapproximately 0.02 times the shock radius. This resolutionin the shell remains same throughout the evolution becausegrids are expanding as the forward shock moves out. Weupdate the outer boundary condition in the r-direction totake into account the condition of the moving supernovaejecta in every time step. The boundary conditions in the/-direction are assumed to be periodic. The density eld isperturbed with the amplitude of 1% in the entire region.Incoming ow from the outer r-boundary also includes thedensity perturbation in order to trigger the instabilitywithin the postshock ow (see next section for theinstability). We evolve a Lagrangian invariant passivequantity (mass fraction function) to follow the contact dis-continuity between the supernova ejecta and the pulsarwind. The mass fractions are assigned to be 0.0 in the pulsarwind and 1.0 in the supernova ejecta.

4. HYDRODYNAMIC EVOLUTION AND INSTABILITIES

4.1. One-dimensional Results

First, the global evolution of the ow is studied by one-dimensional numerical simulations In the early(Fig. 3).stage, the shock wave driven by the expanding pulsarbubble propagates much faster than the moving supernovaejecta, and the density of supernova ejecta can be con-sidered to be roughly constant (stationary medium stage).

plots density, velocity, and gas pressure (from topFigure 3ato bottom) at t \ 100 yr. Three typical hydrodynamical fea-tures, namely a wind termination shock, a contact discon-tinuity between the wind bubble and the shocked ambientmedium, and a forward shock driven by pulsar bubble, are

FIG. 4.The radius of the contact discontinuity as a function of time.The thick solid line represents the one-dimensional numerical result, whileother thin solid lines show constant expansion rate, rP ta, a \ 2/5, 3/5, 1,

6/5 from thelowest to the highest.

clearly seen from inside outward. These ow structuresresemble the wind solution in a stationary medium asdescribed by et al. The shock expansion canWeaver (1977).be approximated as As the shock velocityr

shockP t3@5.

decreases and becomes comparable to the ejecta velocity,the ejecta can be no longer considered as a stationarymedium, and the eect of decreasing density of the ejectabecomes important (moving medium stage). Then the shockaccelerates and enters the self-similar stage, which isdescribed by and (Chevaliero

ejectaP t~3 r

shockP t6@5 1977,

This stage is illustrated in shows1984). Figure 3c. Figure 3bthe intermediate stage evolving from the stationary mediumstage to the moving medium stage. One noticeable feature isthe thickness of the shell between the contact discontinuityand the forward shock. This shell thickness decreases withtime and becomes considerably thinner, about 0.02] r

shock,

in the moving medium stage. This is a feature predicted bythe self-similar solution. The expansion rate of the contactdiscontinuity as a function of time is shown in TheFigure 4.thick solid line represents the one-dimensional numericalresult, while thin solid lines show constant expansion rates,rP ta, a \ 2/5, 3/5, 1, 6/5 from the attest to the steepestcurve for comparison. At the end of the simulation, theexpansion rate of the contact discontinuity reaches abouta \ 1.189, which is close to the predicted value by the self-


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FIG. 5.(a) The density of the two-dimensional numerical result at t \ 100 yr. From the center, the circles correspond to the contact discontinuity, theforward shock driven by the pulsar bubble, and computational boundary. The computational boundary is round because the simulation is carried out in thespherical geometry and remapped onto the Cartesian grids. The wind termination shock is located inside the contact discontinuity. It is not visible in thisimage because of low density (see The development of a very weak instability near the contact discontinuity is seen. (b) The result at t \ 600 yr. TheFig. 3).forward shock is seen near the outer computational boundary. A number of small ngers are developed in the postshock ow because of the R-T instability inthe ow. Lighter color represents higher density.

similar solution for the constant luminosity and uniformejecta density. Recall that the expansion rate derived in theself-similar solution for varying luminosity and power-lawdensity prole is rP ta, a \ (6 [ n [ l)/(5 [ n), where n isthe power-law index of the ejecta density prole (oP r~n),and l is the power-law index for the pulsar luminosity(LP t~l). We nd that the timescale for the system to reachthe self-similar stage is sensitive to the physical variablessuch as the pulsar wind and background ejecta ow. Thesystem evolves faster for the higher pulsar luminosity butslower for the higher supernova ejecta density. Also, theevolution takes less time for higher ejecta velocity andhigher density of pulsar wind gas for a given luminosity.The faster evolution for the higher density of pulsar wind ina given pulsar luminosity is likely because the highermomentum is allowed by choosing a higher density for agiven luminosity (recall that the pulsar mechanical lumi-nosity is L\ 2.0nr2ov3).

4.2. Hydrodynamic Instabilities : T wo-dimensionalNumerical Simulation

Numerically, we found three independent instabilities inthe interaction region between the pulsar wind and thesupernova ejecta. The rst weak instability occurs in thevery beginning and is caused by the impulsive accelerationof dense ejecta by the low-density pulsar wind. But this is avery weak instability and does not grow signicantly at theinterface (see The second instability develops in theFig. 5a).postshock ow during the intermediate stage This(Fig. 5b).second instability develops briey while the gradients ofdensity and pressure are of opposite signs (satisfying thecriterion of the R-T instability). This unstable ow developsin the transition between the stationary medium stage

(Weavers self-similar stage) and the moving medium stage(Chevaliers self-similar stage). In the stationary mediumstage, both pressure and density proles in the postshockregion show positive gradients (see On the otherFig. 3a).hand, in the moving medium stage, both pressure anddensity in the postshock region show negative gradients (see

In each stage, the postshock ow is stable.Fig. 3c).However, while the stationary medium stage evolves intothe moving medium stage, the postshock pressure anddensity evolve by PP v2t~3 and oP t~3, respectively.Therefore, the pressure decreases more rapidly with timethan the density if the power-law index of the shock expan-sion (rP ta) is smaller than 1.0. The dierent evolutions ofdensity and pressure result in the unstable ow that shows apositive gradient in density prole but a negative gradientin pressure prole (see This unstable ow disap-Fig. 3b).pears when the shock expansion accelerates. Small ngersin are the results of this second instability. WeFigure 5bnote that the ngers are conned within the shell, and theouter shock front is not aected by this instability. The rstand second instabilities do not result in the formation ofsignicant long laments, but they play a role as seed per-turbation for the next strong instability.

The third and most important instability develops in thecontact discontinuity between the pulsar wind and theshocked supernova ejecta as the pulsar bubble becomesaccelerated (rP ta, a[1.0). This is the strongest instabilityand produces pronounced lamentary structures (Fig. 6).The thin layer between the shock and the contact discontin-uity is severely distorted because of the instability (note thatthe shock front is not aected by the rst and secondinstabilities). shows a number of thin ngers point-Figure 6ing toward the center. At later times, most of these thin


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288 JUN Vol. 499

FIG. 6.Density images of two-dimensional numerical simulation at t \ 1000 2000 3000 and 4000 yr Note that the(Fig. 6a), (Fig. 6b), (Fig. 6c), (Fig. 6d).forward shock front is always located near the outer computational boundary because the grid is moving with the shock velocity. Lighter color representshigher density.

ngers become unstable, while some ngers maintain theirstability longer. Some unstable ngers are found to bedetached from the shell. The disruption of long thin ngersis caused by a secondary Kelvin-Helmholtz (K-H) insta-bility that develops along the shear layer between thepulsars low-density nebula and the R-T nger. The growthrate of the K-H instability decreases as the relative density

ratio between two gases increases. The formation of thesestable long ngers is possible because of the large densitydierence between the shocked thin layer and the low-density pulsar bubble. Therefore, ngers will be able tomaintain their stable long structure for longer evolution ifthe density in the pulsar nebula is lower. Actually, this iswhat we expect to happen in the Crab Nebula. The stability


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0 2e+19 4e+19 6e+19

Radius (cm)

1028

1027

1026

1025

1024

Density(g/c.c.)

One dimensional simulation

Two dimensional simulation


FIG. 7.Comparison between the one-dimensional numerical resultand the two-dimensional result at t \ 4000 yr. The two-dimensional resultis averaged along the angle direction.

of ngers should also be aected by the strong magneticeld in the Crab Nebula et al.(Hester 1996).

As the R-T ngers grow, the mixing layer becomesthicker at later times. The normalized thickness of themixing layer is expected to increase with time as long as theshell acceleration goes on. The angle-averaged density ofthe two-dimensional numerical simulation at t \ 4000 yr iscompared to the one-dimensional result in TheFigure 7.shell becomes much thicker, and the average density in the

shell is decreased as results of mixing in the two-dimensional numerical simulation compared to the one-dimensional result.

What is most noticeable is the formation of dense headson the R-T ngers, which can be compared to regions F orG in the HST image of et al. These denseHester (1996).nger heads are attributed to the compressibility of the gas.The density in dense nger heads is found to be about 10times higher than in other regions of the ngers. The densityis expected to increase if the cooling is important. Thegeneral morphology of our numerical results is compared tothe HST observations of the laments in the Crab Nebula(see Hester et al. 1996) and found to resemble the obser-vations.

5. MASS AND ACCELERATION OF R-T FINGERS

shows the mass distribution of the shockedFigure 8supernova ejecta as functions of radius (left column) anddensity (right column). The histograms in the left panelsillustrate the mass distribution contained within a shellbounded by two radii The mass is normalized by(r

1, r

2).

total mass. The radius is also normalized by the outerboundary of the computational plane. Recall that the outerboundary in the r-direction expands with the shock veloc-ity, and the shock front is distorted because of the stronginstability. The right panels show the normalized mass dis-tribution as a function of the density normalized by theunshocked ejecta density. The mass distribution of shocked

supernova material extends farther inside with an increas-ing evolution time as seen from top (t \ 1000 yr) to bottom(t \ 4000 yr) due to the growth of R-T ngers. Also, thehigher density ngers are found to form at later times(compare the mass distributions as a function of density at1000 and 2000 yr or later). This result conrms that the highdensity at the tip of R-T nger is actually generated by thecompressible ow pouring down to the ngers from the

shell rather than simple drifting material from the shell.The total mass in the shell changes with time as the

supernova material ows in from the outer shock and somemass drains away through the formation of the R-T ngers.

shows the relative amount of mass and kineticTable 1energy conned within the shell and R-T ngers at a giventime. The mass is normalized by the total shocked super-nova mass. The thickness of the shell approaches to about

in the self-similar stage (rP r1.2), where is0.02rshock

rshock

the radius of outer shock. Our two-dimensional numericalsolution shows that the outer shock is severely distorted bythe instability, and the thickness of thin shell varies as afunction of azimuthal angle. In order to estimate the masswithin the shell approximately, we rst take the shell thick-ness as where is the shock radius at0.02r

shock(/), r

shock(/)

/. As a comparison, we also listed each mass in the shell andthe R-T ngers by taking the shell thickness as 0.03r

shockin (bottom portion). Our results show that the rela-Table 1tive mass amount in the shell decreases from t \ 1000 yr tot \ 2000 yr and then increases later. This is because the R-Tngers, due to the acceleration, are not fully grown, andsmall R-T ngers still exist close to the shell. Besides, theshell thickness at t \ 1000 yr is greater than 0.02r

shock.

By looking at the trend from t \ 2000 yr to t \ 4000 yr, wecan see that the relative mass in the shell increases withtime, while the relative mass in R-T ngers decreases. Thismeans that incoming mass ux from the outer shock islarger than outgoing mass ux through the formation ofR-T ngers. Once the long R-T ngers are fully developed,the mass conned in R-T ngers is roughly about60%75% of total shocked mass and larger than containedin the shell (25%40%). Kinetic energy also shows a ten-dency similar to that of the mass. Considering that kineticenergy changes only between t \ 2000 yr and t \ 4000 yr,about 55%72% of total shocked kinetic energy is in theR-T ngers, and 28%45% is in the shell. Also, the kineticenergy in the shell increases with time, while the kineticenergy in the R-T ngers decreases with time. The slightly

TABLE 1

MASS AND KINETIC ENERGY IN A THIN SHELL AND

R-T FINGERS

Mshell

Mfingers

Eshell

Efingers

Year (%) (%) (%) (%)

Thickness of Thin Shell \ 0.02rshock

1000 .. .. .. 35.8 64.2 36.7 63.32000 .. .. .. 24.9 75.1 28.0 72.03000 .. .. .. 27.0 73.0 30.8 69.24000 .. .. .. 32.2 67.8 37.4 62.6

Thickness of Thin Shell \ 0.03rshock

1000 .. .. .. 62.8 37.2 63.8 36.32000 .. .. .. 32.5 67.5 36.1 63.93000 .. .. .. 35.1 64.9 39.5 60.54000 .. .. .. 38.4 61.6 44.3 55.7


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0.7 0.8 0.9 1.0

radius

0.00

0.05

0.10

0.15

0.00

0.05

0.10

0.15

mass

0.00

0.10

0.20

0.00

0.10

0.20

0.30

0.40

0.50

1 11 21 31 41 51

density

0.00

0.10

0.20

0.30

0.40

0.00

0.10

0.20

0.30

0.40

mass

0.00

0.10

0.20

0.30

0.00

0.10

0.20

0.30

0.40

290 JUN Vol. 499

FIG. 8.Mass distribution of the shocked supernova ejecta as a function of radius (left) and density (right). The plots correspond to the mass distributionat t \ 1000, 2000, 3000, and4000 yr, from topto bottom.

higher percentage of kinetic energy than mass in the shellcan be explained because the ow velocity in the shell ishigher than the velocity of the R-T nger.

shows the power-law indices of expansions ofFigure 9the R-T ngers and of the forward shock front for the dier-ent periods of time. The radius of R-T ngers is measured atthe tip of the ngers. This point is detected by averaging themass fraction function and taking the location where theangle-averaged mass fraction is 0.01 (recall that the mass

fraction in the pulsar wind was 0.0, and the supernovaejecta material had mass fraction 1.0 in the beginning). Firstof all, the expansion rates for both R-T ngers and shockfront increase with time. After about t \ 1000 yr, both theR-T ngers and shock front start to accelerate, but theshock front achieves a higher expansion rate than the R-Tngers. The expansion of the R-T ngers reaches an asymp-totic limit at about a \ 1.041.05 (rP ta). This expansionrate is a little higher than the free expansion. In earlier


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0.0 1000.0 2000.0 3000.0 4000.0

Time (yrs)

0.70

0.80

0.90

1.00

1.10

1.20

1.30

PowerLaw

IndexofExpansi

on

RT Fingers

Shock Front


FIG. 9.The expansion rate of the R-T ngers and shock front asfunction of dierent period of time. The gure plots power-law index ofexpansion, The power-law indices for thea \ [log (r

1/r

2)]/[log (t

1/t

2)].

shock front and the R-T ngers are represented by circles and rectangles,respectively.

stages (t \ 100500 yr), the expansion rates of the R-Tngers and the shock front are close to each other because

the R-T instability is not fully developed. Once the R-Tngers grow due to the instability, they are decoupled fromthe shell, and they experience smaller acceleration than thethin shell. On the other hand, the shock front keeps acceler-ating until the expansion rate reaches the self-similar value(a \ 1.2). In our simulation, the power-law index for theshock front expansion reached about 1.157 duringt \ 30004000 yr. We expect that the power-law index ofthe expansion rate of the shock front will reach its self-similar value as we evolve the system further. This resulttells us about the important history of R-T ngers. Thesupernova material is rst accelerated impulsively by thepulsar wind and decelerates until the expansion of thin shellbetween the contact discontinuity and the forward shock

starts to accelerate as the system approaches the self-similarstage. Once the thin shell accelerates, the R-T instabilitydevelops, and the R-T ngers experience less accelerationby decoupling from the shell and falling down along theeective gravity vector. According to our numerical results,we can infer that the accelerated laments in the CrabNebula should have been accelerated more efficiently rightbefore the R-T ngers were completely decoupled from thethin shell. We will discuss this issue further in the nextsection.

6. APPLICATION TO THE CRAB NEBULA AND DISCUSSION

Our simulation is simple in terms of relevant physics andinitial conditions. For example, we have not considered the

eects of magnetic elds and radiative cooling. Strong mag-netic elds are known to aect the stability and growth ofR-T ngers (see Norman, & Stone We expectJun, 1995).that the inclusion of a magnetic eld in our numerical simu-lation may generate more stable long ngers since the sec-ondary K-H instability can be suppressed by the magneticeld. Also, tangential magnetic eld may enhance couplingbetween the thin shell and the R-T ngers and slow down

the inward velocity of the R-T nger. As a result, the expan-sion power-law index of the R-T ngers may be increased.Radiative cooling is expected to increase the density in thethin shell and R-T ngers and to aect the growth of theR-T instability. The cooling process may change the massdistribution of the shocked supernova ejecta. Therefore, theinclusion of magnetic eld and radiative cooling in themodel is important in order to explain the detailed structureof the laments. Besides, due to the lack of good informa-tion about initial conditions such as the pulsar wind, theexact quantitative comparison to the Crab Nebula shouldnot be made at the moment. Nevertheless, some qualitativeresults can be applied to understand the Crab Nebulabetter.

et al. found that the synchrotron com-Bietenholz (1991)ponent of the Crab Nebula has accelerated since the super-nova explosion, and the acceleration of the synchrotroncomponent may be larger than that of the emission-linelaments. This observation is explained well by our numeri-cal simulation since the forward shock front and thin shellaccelerates with a higher rate than the R-T ngers. FromTable 2 of et al. we can derive the power-Bietenholz (1991),law index of the expansion for the outer edge of the syn-chrotron nebula because the observations were carried outin two dierent years, 1982 and 1987. Using the relationfor the expansion power-law index, a \ [log (r

1/r

2)]/

where and are the radii at two dierent[log (t1/t

2)], r

1r

2ages, and we calculate the expansion power-law index,t

1

t

2

,a \ 1.245D 1.328. This power-law index is somewhatlarger than that predicted (1.2) from the self-similar solutionfor the constant luminosity and uniform ejecta density. Thediscrepancy could have two possible origins. First, theobserved expansion parameter in et al. isBietenholz (1991)only an approximate value because of the difficulty in com-paring two radio images. Also, the exact date of the obser-vations used can change the result. Second, the density inthe supernova ejecta may be a decreasing function of theradius. For example, the self-similar solution predicts thepower-law index to be 1.25 for the power-law index of theejecta density of 1.0 and constant pulsar luminosity. Inorder to calculate the expansion power-law index for theemission-line laments, we use Trimbles result of the con-vergence date, t \ 1140 yr, using two sets of data in 1950and 1966. Therefore, we can derive the expansion param-eter, e \ 0.9806, using the relation (seet

e\ dt/(1 [ e)

et al. And we calculate the expansionBietenholz 1991).power-law index, a \ 1.107. This value is also larger thanour numerical result, a \ 1.04D 1.05. In general, theobserved expansion power-law indices of synchrotronnebula and laments are reasonably close to our numericalsimulation. This is an encouraging result that supports thecurrent model for the Crab Nebula. More accurate futureobservations of the expansion rate of each component arehighly desirable.

With two known expansion power-law indices, we canderive when the R-T instability due to the acceleration


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292 JUN Vol. 499

began to develop in the Crab Nebula. We assume that theexpansion rate has been constant since the development ofthe R-T instability. We can now write a relation among thelength of the R-T nger, an initial time of the R-T instabilitydevelopment, and the expansion power-law indices for(a

fthe laments and for the shock front) of the R-T ngers:a

s

dr

rs

\1

[At

t0Baf~as

, (14)

where dr is the length of the R-T nger, is the radius of thers

forward shock front, t is the current age of the Crab Nebula,and is the initial time of the R-T instability development.t

0Now, let us use Hesters observations to obtain the length ofthe R-T nger. By choosing the regions F or G in Figure 1of et al. we take the length of the R-T nger asHester (1996),about 0.2 times the radius of the forward shock front. Herewe also assume that the shock front is located near the topof the R-T nger, and we use the expansion power-lawindex for the outer edge of the Crab Nebula as the expan-sion power-law index for the shock front. Then, we derivethe initial age of the R-T instability development at the

regions F or G, yr. This means that the R-T ngerst0 \ 266in region F or G began growing 657 yr ago. This is aninteresting result with regard to the history of the line-emitting laments. The laments in the Crab Nebulastarted to form around the age of 266 yr, and the acceler-ation of the laments should had been strongest at thattime. We should note that this initial time for the R-T insta-bility is an approximate estimate because of our assump-tions including constant pulsar luminosity. Other smallerngers in region D imply two dierent interpretations forthe timescale of the formation. First, those ngers may havestarted developing more recently compared to the ngers inregions F or G if the shell in region D experiences sameexpansion rate as region F or G. Second, if the ngers in

region D have developed at the same time as regions F orG, then the expansion rate, that is acceleration of the shell,must have been slower in that region than region F or G.This spherically asymmetric expansion rate could be causedby the asymmetric pulsar wind. Observational measure-ments of the expansion rate in dierent azimuthal regions isrequired to check these ideas. One also needs to rememberthat we have not considered the eect of magnetic elds incontrolling the growth of the R-T ngers. Our simulationcannot explain long, large lamentary structures extendingcontinuously from the skin to near the center. We speculatethat these structures could be formed as the pulsars windsweeps the supernova ejecta which contain a large degree ofinhomogeneity in density.

7. CONCLUSIONS

Our numerical simulation of the interacting pulsarbubble with expanding supernova ejecta produces well-developed lamentary structures that resemble the la-ments in the Crab Nebula. Three instabilities are found todevelop independently in dierent evolutionary stages. The

rst weak instability develops near the contact discontin-uity at early stages of the evolution due to the impulsiveacceleration of the dense ejecta by the pulsar wind. Thesecond instability develops within the postshock owregion between the shock front and the contact discontin-uity. The second instability is a time-transient phenomenonbefore the system develops into the self-similar ow. Thethird Rayleigh-Taylor instability driven by the acceleration

of the thin shell, which develops in the later stage, is foundto be the strongest and to provide a main mechanism forthe origin of the laments.

As a result of the third R-T instability, the forward shockfront is severely distorted, while the rst two instabilitiescould not disturb the shock front. A number of long R-Tngers are generated by the instability, and these ngersbecome unstable (Kelvin-Helmholtz instability) due to therelative motion between the ngers and background ow.However, the development of the K-H instability can bedelayed by a higher density ratio between two uids and astrong magnetic eld. The ngers produce dense heads attheir tips, and these dense nger heads are attributed to thecompressibility of the gas. The density of these heads isabout 10 times higher than other parts of the ngers and isexpected to increase in the presence of the cooling eect.After the long R-T ngers are fully developed, the masscontained in the R-T ngers is found to be roughly60%75% of the total shocked mass. Kinetic energy withinthe R-T ngers is about 55%72% of the total kineticenergy in the shocked ow. The R-T ngers are found toaccelerate with a slower rate than the shock front as theyare decoupled from the shell. In our simulation, the expan-sion power-law index (a) for the tip of R-T ngers is about1.04D 1.05, while the index for the shell is about 1.157.These results are close to the observed values in the CrabNebula (1.107 for the line laments and 1.245D 1.328 forthe outer edge of the synchrotron nebula). The small dier-ence between our numerical results and the observationscould be improved if the expanding supernova ejecta hasthe density prole ofoP r~1 or r~2. Considering the dier-ent expansion rates between the outer edge of the synchro-tron nebula and the line-emitting laments, we can inferthat the several long ngers (region F or G in Hestersobservations) in the Crab Nebula started to grow about 657yr ago.

In summary, our model can explain important obser-vational features of the Crab Nebula. They are the acceler-ations of the laments and the synchrotron nebula,formation of a thin skin connecting the laments, formationof the laments pointing toward the center, and high-density nger heads.

I am grateful to Roger Chevalier, Je Hester, Tom Jones,Mike Norman, and Jim Stone for useful discussions andencouragement. The work reported here is supported byNSF grant AST 93-18959 and by the Minnesota Super-computer Institute.

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byung-il jun- interaction of a pulsar wind with the expanding supernova remnant

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