dynamics of massive shells ejected in a supernova explosion
TRANSCRIPT
ELSEVIEK
20 May 1996
Physics Letters A 2 14 ( 1996) 227-23 I
PHYSICS LETTERS A
Dynamics of massive shells ejected in a supernova explosion
Dario Ntiiiez a,b,l, H.P. de Oliveira a~c,2 a Theoretical Physics Insrirute. University of Alberta, Edmonton, Alberta, Canada T6G 2JI
h Insri/uto de Ciencias Nucleares, CINAM, Circuiro Exterior CU, A.P: 70-543, Mexico, D.E 04510. Mexico
’ Universidade do Estado do Rio de Janeiro, Institute de Fisica.
R. Sao Francisco Xavier, 524, Maracana, CEP 20550-013, Rio de Janeiro, Brazil
Received 16 October 1995; revised manuscript received 5 February 1996; accepted for publication I March 1996
Communicated by J.P Vigier
Abstract
We set up the general relativistic equations of motion of a shell that models the matter ejected in supernova explosions. The non-relativistic Ostriker-Gunn equation is obtained properly. It is shown that in general the equation of motion admits a first integral and the evolution of the shell is reduced to a dynamical system of four equations.
1. Introduction
In the present work we deal with the dynamics of
the matter ejected in supernova explosions which is modeled as thin shells. We use the matching conditions in general relativity to describe the motion of a thin
shell of matter. Then, we show that these equations admit a first integral. From it we obtain an expression
that generalizes the one derived from the Newtonian
formalism giving not only the generalization of the “classical” equation of motion of a thin shell, but also showing that it admits a first integral in the general
case. In this way, the general problem becomes more simplified with respect to numerical or algebraic ma- nipulation, and it also allows a better understanding of the physical quantities involved. Applications to con- crete cases will be given elsewhere.
The study of the dynamics of a supernova remnant, a plerion, has been greatly developed since the works
’ E-mail: [email protected].
’ E-mail: [email protected].
of Ostriker and Gunn [ 11, where they modeled the dynamics of this remnant as a thin shell moving in
a background that has a radiating mass inside and is
dust on the outside. Supposing spherical symmetry, they started with Newton’s second law for describing the acceleration of the shell. The force terms were introduced essentially by hand. Thus, the following equation of motion is obtained,
M$$ = -G (MN + +S)MS
R2 + 4rR2(Pc - Ss), (1)
where MS is the mass of the shell, MN the mass of the neutron star, R is the radius of the shell, PC is the pres- sure in the cavity related to the radiation emitted by the star, and PIS is the pressure due to the contact with
the interstellar medium. This quantity, defined as PIS = pls ( dR/dt)2, plays the role of a friction term known as the “snow plow” effect [ 21. In this way, the first term represents the gravitational and self-gravitational force, the second a driving force due to the radiation and the third a decelerating one due to the interaction
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228 D. N&z. H.P. de Oliueira / Physics Letters A 214 (1996) 227-231
with the medium. The dynamical study is completed with an equation for the change of the mass of the
shell (usually due only to the dust that the shell col- lects while moving in the medium) and, finally, an equation for the change of the shell’s internal energy, proportional to the radiation emitted by the star.
This system of equations has been analyzed in many
ways. Chevalier [ 31 and Ostriker and McKee [ 41 have analyzed the kinetic equation of the shell to show the
different stages of the evolution when each of the terms
in turn becomes dominant, while Sato [5] and Sato and Yamada [6] have applied this set to the study of the SN1987A in particular. Most of the studies use numerical methods to obtain the change in luminosity
or in mass with respect to the change in radius of the remnant.
2. Shells in general relativity. First integral of the equation of motion
The study of the dynamics of a shell separating two backgrounds in the context of general relativity has resulted in a powerful and direct formalism since
the first works of Israel [7]. It has been applied in cosmology mainly to inflation (see, for instance, Ref. [ 8]), and to modeling the dynamics of the border
between two regions in different states, like bubbles in
water, or between two given spaces [ 93. Nevertheless, it has not been so widely used to study the dynamics of the plerion in the context of a nova explosion.
In the formalism mentioned above, the equation of motion for the shell is obtained from Einstein’s equa- tions and from the junction conditions for the metric tensor and its first derivative. The geometric quantities of each spacetime are decomposed into projections on the shell and a projection on a vector normal to the shell. Then, the continuity conditions are imposed.
We will work in spherically symmetric spacetime
described by
ds2 = -e2* f do2 - 2be* dr dv + r2 da’, (2)
where f and Cc, are in general functions of the null co- ordinate u and of the proper radius r, and b = f 1, de- pending on whether the null coordinate is advanced or retarded. The metric coefficients are related with the matter and energy distributions in spacetime through
the Einstein equations which, for this case, are sim-
PlY [lOI
m3,, = 41rr=T,,‘, m3r = -4rr2T,.“,
#,r = 4n-rT,,, (3)
where we have introduced the mass function m = m( v, r) defined by f( u, r) = 1 - 2m(v, r) /I-, and Tp,, is the stress energy tensor, specified by the distribu-
tion of matter and energy in the chosen background spacetime.
In this case, the kinetic equations for the shell are s
=$+8,.
p(F+ + F-) -[Tapn”nPl + R ,
(4)
where n, is the normal vector to the shell, U” is the velocity vector, r is the proper time, M = 47rR2a is the proper mass of the shell, CT being the surface en- ergy density of the shell, and R the radius. A func- tion in square brackets stands for the difference of that function in the region outside, denoted by +, with the
region inside the shell, denoted by -, and evaluated on the shell. Finally, p is the pressure of the shell, and F* is given by
F* = j/f* + I?=.
The dot stands for differentiation regarding the proper time, 7. In the general relativistic context, the dynam- ical study is completed by the conservation equation, which is an equation for the rate of change of the proper mass,
&f+pA =47rR2[TapuanP], (7)
where A = 47rR2 is the area of the shell. For a deriva- tion of the kinetic equations in some particular cases, see for instance Ref. [ 121.
For the line element given by Eq. (2), we have that the velocity and the normal vector, already orthonor- malized, at the shell, are given by
Ua = (ti,R,O,O), n, = beti(-l?,Lj,O,O), (8)
3 We are following the conventions of Ref. [ 111.
D. N&ez, H.P. de Oliveira /Physics Letters A 214 (1996) 227-231 229
where i! is related to B as follows,
The equation of motion (4) is then rewritten as
(10)
where B = if,r + @,,F + ibf,,.e@ti2. Now, from the definition of F, Eq. (6), we see that
Thus, after some manipulations, the equation of mo-
tion ( 10) can be expressed as
$(,F]+;)-;+/‘“:,i] -87rl?p=O.
(12)
The equation for the rate of change of the mass, Eq. ( 7), can in the spherical case be expressed as
+ (CI_,Fl? I
- 8&p. (13)
Substitutingthis last equation into Eq. ( 12), we obtain
&(lFl+;) + Af,(.e*fi2 - $,,Fd
2 1 = 0. ( 14)
Finally, using the formula for ti, Eq. (9), it can be shown that the expression inside the square brackets
is zero! Then
$ (IF, +;) =O,
which implies that
[F] +; =O.
(15)
The value of the integration constant, C, is zero, as can be shown by substituting back [F] + M/R = C in the second equation of motion (5). In this way, we have shown that Eq. ( 16) is the first integral of the
general equation of motion of a spherically symmetric thin shell.
It will be useful to rewrite Eq. (16) taking into account the definition of the mass function and Eq.
(6),
F+2 - F_2 = -$, (17)
where the function m is given by m = m, -m_, which in some cases can be identified with the gravitational
mass4. Combining the first integral equation ( 16), with Eq. ( 17)) we obtain an expression for F on either side of the shell in terms of the proper mass and the
gravitational mass
F*=;$. (18)
The square of this last equation, recalling Eq. ( 17), gives
( 19)
which is the equation of motion obtained by Lake [ 141
from the Lanczos equation, using the jump condition for the theta-theta component of the extrinsic curva-
ture. Notwithstanding that, we found it instructive to perform the direct integration of the kinetic equations.
3. Discussion
The fact that we can obtain the first integral of the equation of motion of the shell for the general case
is quite remarkable. Such an expression is valid for any matter distribution on either side of the shell, as well as for any type of shell, since there was no need
of any specification of the equation of state. Further- more, the integration is valid for any way in which the shell interacts with the two backgrounds! The only requirements have been spherical symmetry, that the backgrounds satisfy the Einstein equations, and the conditions of continuity on the shell.
4 In the cases studied by de la Cruz and Israel 1 I2 I or Chase I I3 1,
VI+ = w11,2-e:,~/2r, where ~~1.2 and rl,l stand for the gravitational
masses and electrical charges in both sides of the shell. They
defined MI = VIZ -ml = const as the gravitational mass of the shell
and identified this quantity as the conserved total energy of the
shell.
230 D. NGez, H.P. de Oliveira/Physics Letters A 214 (1996) 227-231
To have a better understanding of the meaning of the first integral of the equation of motion, it might help to
see its form in the Newtonian limit. For this purpose it proves better to take the form of the equation of
motion given by Eq. ( 18)) considering the cases when
the masses and the velocity, !?, are smaller than the unit. Then, performing a Taylor expansion, we obtain
m=M+iMd2- M(m_ + $I)
R .
The above equation simply states that the sum of re-
spectively the rest energy, the kinetic energy, the mu- tual potential energy and the gravitational self-energy of the shell, on the right-hand side, is not constant, but varies accordingly with the quantity m.
In this way, the dynamic of a shell moving be-
tween any two backgrounds, with spherical symmetry
reduces to a set of first-order equations, namely, the radial equation of motion ( 19), the balance equation for the proper mass, Eq. ( 13)) the equation of motion for the null coordinate, Eq. (9)) an equation of state for the shell, giving a relation between the pressure p
and the energy density, (T, and for specifying the kind of backgrounds between which the shell moves, we
have the Einstein equations, Eqs. (3). Despite the above characteristic concerning the dy-
namics of massive shells, it will be worthwhile to write down conveniently the second-order equation of mo-
tion to obtain the Newtonian limit, and show how Eq.
( 1) is recovered. After a direct calculation starting
from Eq. (19), we get
M#=- M(m_ + ;MF_) kl_M i+lF_
R2 +- RR +-
R MF_ F+ _~
R . (21)
Note that the first term on the right-hand side is nothing else than the relativistic generalization of the gravita-
tional interaction, and the remaining terms are related with the variation of m_, m and M. Hence, we have obtained the relativistic version of the second-order equation analyzed by Ostriker and Gunn [ I] in study- ing the dynamics of remnants of supernovas modeled by thin spherical shells. To show in a clear way the role played by the three last terms of the above equa- tion, we consider that the shell separates two Vaidya spacetimes: the interior filled with outgoing radiation
and the exterior with ingoing radiation. Thus, after taking the Newtonian limit of Eq. (21), we arrive at
the following expression,
Mji _ _M(m- + ;M) R2
f81~Rp+4~R~(q_ -q+), __
(22)
where q- and q+ are the energy density of the radia- tion at each side of the shell and p is the pressure of the shell. Therefore, we have recovered, in this simple model, the terms that describe the effect of the pres- sure of radiation inside the cavity and the snow plow effect (compare the last term of this equation with the
corresponding one of Ostriker’s equation >. Also, we have an extra term related to the pressure of the shell,
since Ostriker has apparently considered a pressure-
less shell. However, a more realistic model is realized in considering the exterior spacetime characterized by dust, like in the Friedman-Walker universe, to take into account more properly the interstellar medium outside the shell.
We want to stress the fact that Eq. (21), obtained from the first-order one, Eq. (19), contains the same information as the two second-order equations ob-
tained from the junction conditions, Eqs. (4), (5)) but has a more tractable form and the different terms are
easier to identify. Finally, instead of the proper time of the shell, r, the
time of an exterior or an interior observer can be used,
u+, u-. The radius of the shell becomes a function of u, R = R(u), so that I? = R,cti, and, after some manipulations, the equation of motion is given by
R .I’%
(23)
with r] = *l, for the two possible solutions to the algebraic equation and with
(24)
Usually, a problem could be posed as follows: given two backgrounds separated by a shell, i.e., two spher- ically symmetric solutions to the Einstein equations and an equation of state for the matter in the shell, de- termine R, u+, u_ and M from the equations of motion which are all first order. In this way we have a set of
D. Nliiiez. H.P. de Oliveiru / Physics Letters A 214 (1996) 227-231
four first-order coupled differential equations for four References
231
unknown functions, so the problem is solvable up to
a quadrature. Nevertheless, the analytical integration might prove
to be very hard except for the most simple cases, so
numerical methods are still needed. In a future work we will present the analysis of this set of equations in
various specific backgrounds.
Acknowledgement
It is a pleasure to thank Werner Israel for fruitful dis-
cussions and comments on the present work, as well as for warm hospitality. D.N. thanks the Direction Gen-
eral de Asuntos de1 Personal Academico, UNAM, for
partial support. HP. de Oliveira would like to acknowl- edge CAPES and CNPq for the financial support.
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I41 151
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