Time-Dependent Models for a decade of SN 1993J

0.56 MB·

by  Lorenzo Zaninetti

AstrophysicsandSpaceScience
DOI10.1007/s•••••-•••-••••-•
Time-Dependent Models for a decade of SN 1993J
L. Zaninetti 1
1
1
0
2

n
a
J
(cid:13)c Springer-Verlag••••
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1
Abstract A classical and a relativistic law of motion yrassumesaninitialfreeexpansioninwhichR t un-
∝
A] for a supernova remnant (SNR) are deduced assuming til the surrounding mass is of the order of 1 M⊙ and
aninversepowerlawbehaviorforthe density ofthe in- a second phase characterized by the energy conserva-
G
terstellar medium and applying the thin layer approx- tioninwhichaccordingtothe SedovsolutionR t2/5,
h. imation. A third equation of motion is found in the see McCray and Layzer (1987). A third phase c∝harac-
p frameworkofrelativistichydrodynamicswith pressure, terized by an adiabatic expansion with R t2/7 starts
∝
- applying momentum conservation. These new formu- after 104 yr, see McCray and Layzer (1987). A more
o
r las are calibrated against a decade of observations of sophisticatedapproachgivenby Chevalier (1982,?) an-
st SN1993J. The existing knowledgeof the diffusive pro- alyzes self-similar solutions with varying inverse power
a cesses of ultrarelativistic electrons is reviewed in order law exponents for the density profile of the advanc-
[
to explain the behavior of the ‘U’ shaped profile of in- ing matter, R−n, and ambient medium, R−s. The
1 tensity versus distance from the center of SN1993J. previous assumptions give a law of motion R tnn−−3s
v ∝
when n > 5. Another example is an analytical so-
9 Keywords supernovae: general supernovae: individ-
lution suggested by Truelove and McKee (1999) where
1
ual (SN 1993J ) ISM : supernova remnants
4 the radius–time relationship is regulated by the de-
3 creaseindensity: asanexample,adensityproportional
1. 1 Introduction toR−9givesR t2/3. Withregardtoobservations,the
∝
0 radius–timerelationshipwasclarifiedwhena decadeof
1
The study of the supernova remnant (SNR) started very-long-baseline interferometry (VLBI) observations
1
with Oort (1946) where an on ongoing collisional ex- ofSN1993Jatwavelengthsof3.6,6,and18cmbecame
:
v citationasaresultofapost-explosionexpansionofthe available, see Marcaide et al. (2009). As a first exam-
i
X SNR against the ambient medium was suggested. The ple, these observations collected over a 10 year period
r nextsixdecadeswherededicatedtothedeductionofan canbeapproximatedbyapowerlawdependenceofthe
a analyticalor numericallaw of expansion. The targetis type R t0.82. This observational fact rules out the
∝
arelationshipfortheinstantaneousradiusofexpansion, Sedov model and the momentum conservation model.
R,ofthetype tm wheretistimeandmisaparame- Theobservedradius–timerelationshipleavesaseriesof
∝
terthatdependsonthechosenmodel. Onadoptingthis questions unanswered or merely partially answered:
point of view, the Sedov expansion predicts R t0.4,
∝ Isitpossibletodeduceaclassicalequationofmotion
see Sedov (1959), and the thin layer approximation •
for the SNR with an adjustable parameter that can
in the presence of a constant density medium predicts
befoundfromanumericalanalysisoftheradius–time
R t0.25,seeDyson, J. E. and Williams, D. A.(1997).
∝ relationship?
AsimpleapproachtotheSNRevolutioninthefirst104
Isitpossibletodeducearelativistic-mechanicsequa-
•
tion of motion for the SNR, since the initial velocity
L.Zaninetti ofthe SNRcanbe onthe orderof1/3ofthe velocity
DipartimentodiFisicaGenerale, of light?
Universit`adegliStudidiTorino Is it possible to deduce a relativistic-hydrodynamics
ViaPietroGiuria1, •
equationofmotionfortheSNRapplyingmomentum
I-10125Torino,Italy
conservation?
2
Can we build a diffusive model which explains the 3.1 The constant expansion velocity
•
behavior of the emission intensity of the SNR?
Canasimplemodelforthetimeevolutionofthetotal The SNR expands at a constant velocity until the sur-
•
mapped flux densities of SN1993J be built? rounding mass is of the order of the solar mass. This
time , t , is
Inordertoanswerthesequestions,Section2reports M
thedataonSN1993J.Section3.3reportsanewclassi-
3 M⊙
cdaelnlcaewfoorfmthoetidoennsaistsyumofinthgeamneindviuermse. pSoewcteiornla4wrdeeppoernts- tM =186.45 √3np0v10000 yr , (2)
the evolution of SN1993J on the basis of four models. where M⊙ is the number of solarmasses in the volume
Section 5 contains two new relativistic laws of motion occupiedbytheSNR,n ,thenumberdensityexpressed
0
as well as a fit to the data. Section 6 reviews the exist- in particles cm−3, and v the initial velocity ex-
10000
ing situation with the radiative transport equation as pressedinunitsof10000km/s,seeMcCray and Layzer
wellasthreedifferentprocessesofdiffusionforultrarel- (1987).
ativisticelectrons. Section6alsocontainsanewscaling
lawforthetemporalevolutionofthefluxofSN1993Jat 3.2 Two solutions
6 cm.
AfirstlawofmotionfortheSNRistheSedovsolution
2 A spherical SNR 25 Et2 1/5
R(t)= , (3)
4 πρ
(cid:18) (cid:19)
The supernova SN1993J started to be visible in M81
where E is the energy injected into the process and t
in 1993, see Ripero et al. (1993), and presented a
is time, see Sedov (1959); McCray and Layzer (1987).
circular symmetry for 4000 days, see Marcaide et al.
Our astrophysical units are: time, (t ), which is ex-
(2009). Itsdistanceis3.63Mpc (thesameasM81),see 1
pressed in years; E , the energy in 1051 erg; n ,
Freedman et al.(1994). TheexpansionofSN1993Jhas 51 0
the number density expressed in particles cm−3 (den-
beenmonitoredinvariousbandsoveradecadeandFig.
sityρ=n m,wherem=1.4m ). Intheseunits,equa-
1 reports its temporal evolution. 0 H
tion (3) becomes
The instantaneous velocity of expansion can be de-
duced from the following formula
E t 2
R(t) 0.313 5 51 1 pc . (4)
r r
v = j+1− j , (1) ≈ s n0
j
t t
j+1 j
−
The Sedov solution scales as t0.4. We are now ready to
where r and t denote the radius and the time at
j j couple the Sedov phase with the free expansion phase
the position j. The uncertainty in the instantaneous
velocity is found by implementing the error prop- 0.0157tpc if t 2.5yr
R(t)= ≤ (5)
agation equation (often called the law of errors of 0.0273√5t2pc if t>2.5yr .
(cid:26)
Gauss) when the covariant terms are neglected, see
Bevington, P. R. and Robinson, D. K. (2003)). Fig. 2 This twophasessolutionis obtainedwiththe following
reports the instantaneous velocity aw well the relative parameters M⊙ =1 , n0 =1.127105 , E51 =0.567 and
uncertainty. In particular, the observed instantaneous Fig. 3 reports it’s temporal behavior as well the data.
velocity decreases from v = 15437 km at t = 0.052 yr The quality of the simulation , ǫq, can be obtained by
s
to v =8474 km at t = 10.53 yr. We briefly recall that a comparison of observed and simulated quantities:
s
Fransson et al. (2005) quote an inner velocity from the
(R R )
shapes of the lines of 7000km and an outer velocity ǫ =(1 | pc,obs− pc,num |) 100, (6)
of 10000km. ≈ s q − Rpc,obs ·
≈ s
where R is the observed radius, in parsec and
pc,obs
R is the radius from our simulation in parsec. In
pc,num
3 Classical case
thecaseofthetwo-phasessimulationwehaveǫ =65%
q
whichisalowvalueincomparisonwiththeothermod-
This section reviews the free expansion , two simple
els here considered see Table 1.
lawsofmotionforthe SNRandanew lawofmotionin
A second solution is connected with momentum
the light of the classical physics.
conservation in the presence of a constant density
ModelsforSN1993J 3
Fig. 1 Radius in pc versus year of SN1993J with vertical error bars. The em bands are λ=3.6cm and λ=6cm. The
data are extracted from Table 1 in Marcaide et al. 2009. The dotted line represents an expansion at a constant velocity.
Fig. 2 Instantaneous velocity of SN1993J with uncertainty.
Fig. 3 Theoretical radius as given by the two-phases solution with data as in Table 1 (full line), and astronomical data
of SN1993J with vertical error bars.
4
medium, see Dyson, J. E. and Williams, D. A. (1997); variablescan be separatedand anintegrationterm-by-
Padmanabhan (2001); Zaninetti (2009). The astro- term gives the following nonlinear equation
NL
F
physical radius in pc as a function of time is
= 4R 3d R 3d2 R 3R dR4−d+R 4d2
NL 0 0 0 0
F − −
R(t)= 4 R03(4.0810−6v1 (t1−t0)+R0)pc , (7) (cid:0) +1(cid:1)2R03v0t+3R04−4R04d
q +7R03v0dt0 +R03v0d2t
where t and t are times in years, R is the radius in
1 0 0 7R 3v dt 12R 3v t R 3v d2t =0 . (13)
pc when t = t and v is the velocity in 1km units − 0 0 − 0 0 0 − 0 0 0
1 0 1 s
when t1 =t0. The momentum solution in the presence An approximate solution of NL(r) can be obtained
of a constant density medium scales as t0.25. assuming that 3RdR4−d F(4R3d R3d2)R
0 ≫ − 0 − 0
3.3 Momentum conservation with variable density R(t)=
1
(R 4−d dR 4−d(4 d)
We assume that aroundthe SNR the density of the in- 0 − 3 0 −
terstellarmedium(ISM)hasthefollowingtwopiecewise +1(4 d)v0R03−d(3 d)(t t0))4−1d . (14)
dependencies 3 − − −
Up to now, the physical units have not been specified,
ρ if R R
ρ(R)= 0 ≤ 0 (8) pc for length and yr for time are perhaps an accept-
ρ (R0)d if R>R .
(cid:26) 0 R 0 able choice. With these units, the initial velocity v
0
In this framework, the density decreases as an inverse is expressed in pc and should be converted into km;
yr s
powerlawwithanexponentdthatcanbefixedfromthe this meansthatv0 =1.0210−6v1 wherev1 isthe initial
observed temporal evolution of the radius, with d = 0 velocity expressed in km.
s
meaning constant radius. The mass swept, M , in the The astrophysical version of the above equation in
0
interval 0 r R is pc is
0
≤ ≤
4 R(t)=
M = ρ πR 3 . (9)
0 0 0
3 1
(R 4−d dR 4−d(4 d)+3.40210−7
0 0
− 3 − ×
The mass swept, M, in the interval 0 r R with
r ≥R0 is ≤ ≤ ×(4−d)v1R03−d(3−d)(t1−t0))4−1d pc , (15)
d where t and t are times in years, R is the radius in
M =−4r3ρ0π Rr0 (d−3)−1 pc at t11=t0 an0d v1 is the velocity at0t1 =t0 in ksm.
(cid:18) (cid:19)
ρ πR 3 4
+4 0 0 + ρ πR 3 . (10)
0 0
d 3 3 4 Classical fits
−
Momentum conservation requires that
Thequalityofthefitsismeasuredbythemeritfunction
χ2
Mv =M v , (11)
0 0
(R R )2
where v is the velocity at t and v0 is the velocity at χ2 = th− obs , (16)
σ2
t = t0. This formula is not invariant under Lorentz Xj obs
transformations and the initial velocity, v , can be
0
where R , R and σ are the theoretical radius,
greater than the velocity of light. The previous ex- th obs obs
the observed radius and the observed uncertainty re-
pression as a function of the radius is
spectively.
r 3β (3 d) A first numerical analysis of the observed radius–
0 0
β = − , (12)
3r dR3−d r 3d time relationshipofSN1993Jcanbe doneby assuming
0 0
− a power law dependence of the type
where β =v /c, β=v/c and c is the velocity of light.
0 0 l l l
Here,wehaveintroducedarelativisticnotationforlater R(t)=rptαp . (17)
use. In this differentialequationoffirstorderin R,the
The two parameters r and α can be found from the
p p
following logarithmic transformation
ln(R(t))=ln(r )+α ln(t) , (18)
p p
ModelsforSN1993J 5
which can be written as
y =a +b x . (19)
LS LS
Theapplicationoftheleastsquaremethodthroughthe
FORTRAN subroutine LFIT from Press et al. (1992)
allows to find a ,b and the errors σ and σ ,
LS LS a b
see numerical values in Table 1. The error on r is
p
found by implementing the error propagation equa-
tion (often called law of errors of Gauss) when the Table 1 Numerical values of the parameters of the fits
covariant terms are neglected (see equation (3.14) andχ2. N representsthenumberoffreeparametersandǫq
in Bevington, P. R. and Robinson, D. K. (2003)), thequality of the simulation.
σ =exp(a)σ , (20)
rp a N values χ2 ǫ
q
and σ = σ . In this case, the velocity is
αp b
power law
V(t)=r α t(αp−1) . (21)
p p
2 α =0.82 0.0048 6364 98.54%
p
±
A second numerical analysis can be done by assum-
r =(0.015 0.00011)
ing a piecewise function as in Fig. 4 of Marcaide et al. p ±
(2009) piecewise
R(t)=(cid:26) rrbbrr((ttbbttrr))αα12 iiff tt>≤ttbbrr. (22) 4 αα21==0.07.883±±0.00.00717; 32 97.76%
r =0.05 pc;t =4.10 yr
This type of fit requires the determination of four pa- br br
rameters, i.e., tbr, the break time, rbr the radius of ex- approximate radius
pansion at t=t and the exponents of the two phases
br
4 d=2.54;r =0.019 pc; 7186 96.95%
arealpha andalpha . Thetwo-regimefitcanbe visu- 0
1 2
alizedinFig. 4andTable1reportsthefourparameters. t =0.249 yr;v =100000km
0 0 s
A third type of fit can be done by adopting the ap-
nonlinear radius
proximate radius as given by equation (15); Fig. 5 re-
ports a fit of this type. 4 d=2.93;r =0.019 pc; 276 93.3%
0
A fourth type of fit implements the nonlinear equa-
t =0.249 yr;v =100000km
tion (13); the four roots can be found with the FOR- 0 0 s
TRAN subroutine ZRIDDR from Press et al. (1992); relativistic radius
Fig. 6 reports a fit of this type.
4 d=2.54;r =0.0045 pc; 5557 93.05%
0
t =0.249 yr;β =0.333
0 0
5 Relativistic case
relativistic pressure radius
The relativistic analysis is split in two.
5 d=0.89;r =0.00072 pc 1844 99.93%
0
1. Therelativisticmechanicscaseinwhichthetemper- f =10−5
ature effects are ignoredand the thin layer approxi-
mation is used. t0 =0.249 yr;β0 =0.333
2. The case of relativistic hydrodynamics in which the
pressureisconsideredandthevelocityisfoundfrom
momentum conservation.
5.1 Relativistic mechanics
Newton’slawinspecialrelativity,afterEinstein(1905),
is:
dp d
F = = (mV) , (23)
dt dt
6
Fig. 4 Theoretical radius as given by thetwo-regime fit represented by equation (22) with data as in Table 1 (full line),
and astronomical data of SN1993J with vertical error bars.
Fig. 5 Theoretical radius as given by the approximate radius, equation (15), with data as in Table 1 (full line). The
astronomical data of SN1993J are represented with vertical error bars.
Fig. 6 Theoretical radius as obtained by the solution of the nonlinear equation (13) (full line), data as in Table 1. The
astronomical data of SN1993J are represented with vertical error bars.
ModelsforSN1993J 7
with 5.2 Relativistic hydrodynamics
m
r
m= , (24) Arelativisticflowonflatspacetimeisdescribedbythe
1 v2
− c2l energy–momentum tensor, Tµν,
q
whereF isthe force,pistherelativisticmomentum, m Tµν =wuµuν pgµν , (29)
−
isthe relativisticmass,m isthe rest-mass,v is theve-
r
locity andc is the velocity oflight, see equation(7.16) where uµ is the 4-velocity, and the Greek index
l
in French, A.P. (1968). In the case of the relativistic varies from 0 to 3, w is the enthalpy for unit vol-
expansion of a shell in which all the swept material re- ume, p is the pressure and gµν the inverse met-
sides at two different points, denoted by radius R and ric of the manifold Weinberg (1972); Landau (1987);
radius R , the previous equation gives: Hidalgo and Mendoza (2005); Gourgoulhon (2006).
0
Momentum conservation in the presence of velocity,
β β
M =M 0 , (25) v, along the radial direction states that
0
1 β2 1 β2
− − 0 v 1
twwhepeerne0β0a=ndv0R/cal,nβdp=Mv/cisl,tMherisestthme aressstswmeapstsbswetewpetebne0- (w(cl)21− vc2l2 +p)A=cost , (30)
0
and R0. This formulais invariantunder Lorentz trans- whereA(r)isthe consideredsurfacearea,whichisper-
formationsandtheinitialvelocity,v0,cannotbegreater pendicular to the motion. The enthalpy per unit vol-
than the velocity of light. Assuming a spatial depen- ume is
dence of the ISM as given by formula (8), relativistic
conservation of momentum gives w=c2ρ+p , (31)
l
−4ρπ 3r3 rr0 d−r03d β = 4ρπr03β0 . (26) wrehaedreerρmisaythbeedpeunzszitleyd,abnydtchleisγt2hfeacvteolorcinityeqoufalitgiohnt.(T30h)e,
3 ((cid:16)3+(cid:0)d) (cid:1) 1 β2 (cid:17) 3 1 β02 where γ2 = 1 . However it should be remembered
− − − 1−v2
According to thpe previous equatiopn, β is c2l
that w is not an enthalpy, but an enthalpy per unit
volume: the extra γ factor arises from ‘length contrac-
( 3+d)β r 3
0 0
β = − − (27) tion’ in the direction of motion Gourgoulhon (2006).
D
β We continue assuming
with Dβ =9R06β02 p6R06β02d
− 1
+9r6−2dR02d p= 3fρcl2 , (32)
6r3−dR d+3d+R 6d2
0 0
− where f is a parameter which has the range 0 f 1
9β 2r6−2dR 2d+6β 2r3−dR d+3d . ≤ ≤
− 0 0 0 0 and is supposed to be constant during the expansion,
see formula (2.10.26) in Weinberg (1972) for a hot ex-
In this differential equation of first order in r, the vari-
tremely relativistic gas. The previous equation (30)
ables can be separated and the integration can be ex-
becomes
pressed as
ZRR0 DβdR=c(3−d)β0R03(t−t0) . (28) (cid:0)cl2ρ+11−3fβρ2cl2(cid:1)β2 + 31fρcl2!A=constant. (33)
p
Theintegralofthe previousequationcanbeperformed The density is supposed to vary during the expansion
analytically only in the cases d = 0, d = 1 and d = 2, as
but we do not report the result because we are inter- R
ρ=ρ ( 0)d , (34)
ested in a variable value of d. The integral can be eas- 0
R
ily evaluated froma theoreticalpoint of view using the
subroutine QROMB from Press et al. (1992). The nu-
merical result is reported in Fig. 7 and the input data
in Table 1. The behavior of relativistic and classical
velocities are reported in Fig. 8.
8
Fig. 7 Theoretical radius as obtained by the solution of the relativistic-mechanics equation (28) (full line), data as in
Table 1. The astronomical data of SN1993J are represented with vertical error bars.
Fig. 8 Theoreticalrelativistic-mechanics velocityasgivenbyequation(27)(dashedline),theoreticalclassical velocityas
given byequation (12) (full line) and instantaneous velocity of SN1993J with uncertainty.
ModelsforSN1993J 9
where ρ is the density at R = R . In two surfaces of where E is the energy and v is the divergence of
0 0
∇·
the expansion we have: the expansion velocity, see formula (11.27) in Longair
(1994). Asimpleexpressionfor vcanbefoundfrom
R0 dR2β2 1+β2 −1 the power law model, see equati∇on·s (17) and (21)
− R −
1/3(cid:18) R0(cid:19) dR2f(cid:0) 1+β2(cid:1)−1 v= Rαα−1rpα1 (3α−1) , (39)
∇· R
− R −
(cid:18) (cid:19)
R 2β 2 R(cid:0)2f (cid:1) where R is the temporary radius of the expansion and
0 0 0
+ +1/3 =0 . (35)
1+β 2 1+β 2 rp and α are reported in Table 1. The lifetime, τad, of
0 0
− − an ultrarelativistic electron for adiabatic losses is
The positive solution of the second degree equation is:
E
τ = =344.39R1.22yr , (40)
N ad dE
β = (36)
dt
D
N = ( 9R d+2R−d+2β 4+6R 2dR−2d+4β 2f when the radius R is expressed in pc. During the ten
0 0 0 0
− −
3R 2dR−2d+4β 4f yearsofobservedexpansion,theradiusofSN1993Jhas
0 0
− grownfrom 0.01pcto 0.1pcandthereforethetime
6R d+2R−d+2β 2f +9R d+2R−d+2β 2 ≈ ≈
− 0 0 0 0 scale of adiabatic losses has increased from 1.25 yr
3R 2dR−2d+4f +3R d+2R−d+2f ≈
0 0 to 20.7 yr.
−
+9R 4β 4+3R d+2β 4R−d+2f +6R 4β 2f ≈
0 0 0 0 0 0
R0d+2f2R−d+2+R0d+2f2R−d+2β02+R04f2)12 6.1.2 Synchrotron losses
−
D =3R dR−d+2β 2
0 0
An electron which loses its energy due to synchrotron
3R dR−d+2 3R 2β 2 R 2f .
− 0 − 0 0 − 0 radiation has a lifetime τr, where
The equation of motion is
E
τ 500E−1H−2sec , (41)
r
R D ≈ Pr ≈
dR=c(t t ) . (37)
0
ZR0 N − E is the energy in ergs, H the magnetic field in Gauss,
and P is the total radiated power,see formula (1.157)
TheintegralisevaluatedusingthesubroutineQROMB r
in Lang (1999).
from Press et al. (1992). The numerical results are re-
Theenergyisconnectedtothecriticalfrequency,see
ported in Figs 9 and 10. Fig. 10 reports the decrease
formula (1.154) in Lang (1999), as
of the relativistic velocity.
ν =6.266 1018HE2 Hz . (42)
c
×
6 How the image is formed
The lifetime, τ , for synchrotron losses is
syn
In this section, the existing knowledge about adiabatic 1
τ =39660 yr . (43)
and synchrotron losses, the acceleration of particles syn
H√Hν
by the Fermi II mechanism, , 3D mathematical diffu-
sions with constant diffusion coefficients and the rim Thetime-scaleofsynchrotronlossesisshorterthanthat
model with constant density are reviewed and applied ofthe adiabatic losses if the followinginequality is ver-
toSN1993J.Anewexampleofa1Drandomwalkwith ified
a step length equal to the relativistic electron gyro-
1.572109
radius is also reported. A new simple model for the ν > Hz , (44)
H3τ2
temporal evolution of the flux densities is introduced. ad
where H is expressed in Gauss and τ in years. The
6.1 Acceleration and losses ad
previous equation can also be expressed as
6.1.1 Adiabatic losses 1
H >1162.98 Gauss , (45)
√3νt 2/3
Anultrarelativisticgaswhichexperiencesanexpansion ad
loses energy at the rate that is the magnetic field at which the synchrotron
losses prevails on the adiabatic losses and Fig. 11 re-
dE 1
( )= ( v)E , (38) ports the numerical values of the transition.
− dt 3 ∇·
10
Fig. 9 Theoretical radiusasobtainedbythesolution oftherelativistic-hydrodynamicsequation(37)(fullline) anddata
as in Table 1 with uncertainty.
Fig. 10 Theoretical relativistic-hydrodynamicsvelocity with pressureas givenbyequation (36)(full line),and instanta-
neous velocity of SN1993J with uncertainty.
Fig. 11 Magnetic field in Gauss overwhich thesynchrotron losses prevail on the adiabatic losses.