In radiation coordinates (Bondi, 1964), the metric takes the form d82-- - e2~ ( V d u 2 - 2 d u d r ) .... /b 2L + (1 - F)(f~ - 1). = A. (33) ... radiation Pn = .T" = T' = ~ will ..... u. 2e-05. 0. -2e-05. -4e-05. -6e-05. -8e-05. -02001. -0.00012. Energy Flux .2. ~ -.
R A D I A T I O N H Y D R O D Y N A M I C S A N D RADIATING S P H E R E S IN GENERAL RELATIVITY E AGUIRRE, H. HERNANDEZ and L.A. N01qEZ* Laboratorio de Ffsica Teorica, Departamento de Fisica, Faeultad de Ciencias, Universidad de Los Andes, Merida, Venezuela
(Received 1 February, 1994; accepted 14 March, 1994) Abstract. We comment on a methodproposedto studythe evolutionof General RelativisticRadiating Spheres in both radiation limits, i.e. free streaming out and diffusion,extendingit to handle any general radiating spherically symmetric distribution of matter. It is also shown that several dynamic models may emerge from a sole static equation of state. Previous erroneous calculations concerning this method are also commented. 1. Introduction In 1980, L. Herrera, J. Jimrnez and G. Ruggeri (Herrera et al., 1980), using Bondi,s formalism (Bondi, 1964) to study radiating spheres, proposed a general method (HJR method, for short) to obtain non-static solutions to Einstein Equations. Models are matched to Vaidya exterior metric mad, besides regularity conditions, they are only restricted by a heuristic assumption relating density, pressure and radial matter velocity. This ansatz, guided by solid physical principles, reduces the problem of solving Einstein Equations to a numerical integration ordinary differential equations for quantities evaluated at the surfaces (shocks and/or boundaries). This method has been used to model collapsing radiating configurations, starting from well known static equations of state in both radiation limits (free streaming out (Herrera, 1980) and diffusion (Herrera, Jimgnez and Esculpi, 1987)) and fruitfully applied to study several interesting astrophysical scenarios (Herrera and Ndfiez, 1990 and references therein). The main point of the present work is twofold. First, we would like to rephrase the HJR method in the Radiation Hydrodynamic language. Within this general approach we shall comment on two previous papers (Herrera et al., 1980, 1987). Secondly, we shall show how different dynamic models can be obtained from a same starting static solution. We will also comment in passing some early reported erroneous calculations. This work is organized as follows. In Section 2 conventions used, Einstein Field Equations, junction conditions and an outline of the method in terms of the Radiation Hydrodynamics framework are sketched. Section 3 contains a discussion of the strategies to obtain different dynamic models and the descriptions of the models * Postal Address: Apartado 54, Merida 5101A, Venezuela Astrophysics and Space Science 219: 153-170, 1994. (~) 1994 Kluwer Academic Publishers. Printed in Belgium.
154
E AGUIRREET AL.
worked out. Finally, conclusions are presented in Section 4. Correct calculations concerning previously reported models are presented in the Appendix. 2. Radiation Hydrodynamics, Field Equations, Junction Conditions and the H JR Method
The first comment is how the two previous schemes, concerning radiating spheres under the HJR approach (Herrera et al., 1980, 1987), are particular limits for the radiation field. In order to accomplish this task we are going to present the HJR method, using the Radiation-Hydrodynamics framework, in terms of the specific intensity of the radiation field, I(x, t; n, u). As in classical radiative transfer theory (Mihalas and Mihalas, 1984), the specific intensity of the radiation field, I (x, t; n, u) at the position x and time t, traveling in the direction n with a frequency u, is defined in order to get the amount of energy,
(1)
d£ = I(x, t; n, u) dS cos a dO du dr,
across a surface element dS, into solid angle dO around n (c~ is the angle between n and the normal to dS), transported by radiation of frequencies (u, u + du), in time dr. 2.1. THE METRIC AND THE ENERGY-MOMENTUMTENSOR Let us consider a non-static and spherically symmetric distribution of matter, formed by a fluid and radiation. In radiation coordinates (Bondi, 1964), the metric takes the form d 8 2 - --
e2~ ( V d u 2 - 2 d u d r ) - r 2 ( d O 2 + s i n 2 O d ( p 2 ) ,
(2)
where/3 and V are functions of u and r. Here u = z ° is a time like coordinate, r = x ~ is the null coordinate and 0 = x 2 and ~ = x 3 are the usual angle coordinates. The u-coordinate is the retarded time in flat space-time and, therefore, u-constant surfaces are null cones open to the future. This last fact can be readily noticed from the relationship between the usual Schwarzschild coordinates, (T, R, O, (I)), and Bondi's radiation coordinates: ~z=T-fVdr,
0=O,
r=R
and ¢ = ( I ) .
(3)
In radiation coordinates the velocity of matter is given by dr V a; du r l-a;
(4)
RADIATION HYDRODYNAMICS AND RADIATING SPHERES
155
and the function rh(u, r) can be defined by
This function ~ ( u , r) is the generalization, inside of the distribution, of the "mass aspect" defined by Bondi and collaborators (Bondi et al., 1962) which in the static limit coincides with the Schwarzschild mass. It is assumed that, for a local observer co-moving with a fluid having a radial velocity ~, the space-time contains: an isotropic fluid of density ~ and pressure/5, a radiation field of specific intensity I(x, t; n, u). The moments of the specific intensity of radiation for a planar geometry can be written as (Mihalas and Mihalas, 1984): -
-
1
PR = "~
du 0
d# I(x, t; n, u),
(6)
-1
where # = cos 0. Physically, PR, .~ and 79, represent the radiation contribution to the: energy density, energy flux density and radial pressure, respectively. For this moving observer the covariant energy momentum tensor can be written as
where the material part, T~u, ^M is
and the energy momentum tensor for the radiation field, ~ R #u, can be written as (Mihalas and Mihalas, 1984; Lindquist, 1966):
156
^R T/,,=
(% _
0 o
F. AGUIRRE ET AL.
-79.T 0 o
o
0 ½(pR - 79) o
o )
0 0
(11)
"
½(p -791
Notice the anisotropy induced by the radiation field. Once Minkowskian co-moving energy momentum tensor is built in terms of physical observables on a local frame (f3,/5, PR, ~ and 79), it can be transformed as (12)
Ox '~ OxZ Lv(-w)L)~ (-w)T"~"
where L ~ ( - w ) is a Lorentz boost in the radial direction (co is the radial velocity of the fluid for a Minkowskian observer), and O~:'r/Ox~ are coordinate transformations connecting Minkowskian coordinates (t, x, y, z) to Bondi coordinates (u, r, 0, ¢) (see Herrera and Nfifiez, 1990 for details). Thus, Einstein Field Equations can be written as e
-2~ r
V Ttt
1
- - 1 - CO2 (fi -t- P R -t- CO2(/5 _+_79 ) + 2co3v)
- rh0e -2~ + ( 1 -
--
1 (P e-2flTtr = 1 +------w
e -2/~ Trr=
~ 1
7"
~ml
+ DR -- CO([:) + 79) -- (1 -- CO).~C') =
(~+pR+/5+P-2f')--
,(
-TO =/5 + "~(PR - 79) = ~
1-
,
(13)
?nl 47rr2,
(14)
r -~2- ~r h _ ill,
_)
(15)
2/311 + 4/32 - /31r -
417r/301 e -2~ -I- 3/31(1 - 2~7%1) - r~ll 87rr
(16)
Differentiation with respect to u and r are denoted by subscripts 0 and 1, respectively. Observe that only four of the six local physical variables (co, ~,/5, PR, .~ and 79), can be algebraically obtained from field equations (13) to (16) in terms of the metric functions/3(u, r) and ~ ( u , r) and their derivatives. Therefore, more information has to be provided to this system. 2.2. LIMITS FOR THE RADIATION FIELD
In order to obtain four of the above physical variables, Herrera and collaborators have considered collapsing radiating configurations in the two limits for the radiation field: free streaming out (Herrera et al., 1980) and diffusion (Herrera et
RADIATION HYDRODYNAMICS AND RADIATING SPHERES
157
al., 1987). Barreto and Ntifiez (1991) have also studied general relativistic spheres where diffusion and free streaming processes coexist. It is clear that in order to deal with realistic physical scenarios, a relativistic Boltzmann Transport Equation should be considered (Lindquist, 1966) to describe the evolution of the radiation through the matter configuration. In this way the above radiation moments (6), (7) and (8) are related to the physical properties of the medium (absorption and/or emission). In spite of this, it is possible to consider several other physically interesting situations within the above mentioned limits (Mihalas and Mihalas, 1984). The free streaming out limit assumes that radiation (neutrinos and/or photons) mean free path is of the order of the dimension of the sphere. With this assumption it is obtained that
PR = .~" = 5p = ~.
(17)
On the other hand, in the diffusion limit approximation radiation is considered to have a mean free path much smaller than the characteristic length of the system. Within this limit, radiation is locally isotropic and we have
pR=3P
and U = q .
(18)
For the above two limits, Einstein Field Equations (13)-(16) reduce to those previously obtained (Herrera and Ntifiez, 1990). 2.3. JUNCTION CONDITIONS To complete the H JR method outlined below, it is necessary to match the internal solution to the Vaidya metric at the boundary surface of the distribution. This matching can be carried out either by using Darmois-Lichnerowicz Conditions or by demanding the continuity of the functions/3 and ~ across the boundary and requiring (Herrera and Jim6nez, 1983) -/30a + (1
2~a)/31a
mla 2a - 0'
(19)
Using that/3 is continuous and/3 = 0 for the Vaidya metric, we may expand it near the boundary r = a(u) /30a + h/31a = 0,
(20)
where h = da/du. Substituting (20) into (19) and using field equations (14) and (15),
& = (CVa(fia + PRa-- .Ua) -- Pa - Pa + .'~a)(1- (2r~a/a)) (fia + PR~ + [~ + 7va - 29ra)(1 - coa)
(21)
158
F. AGUIRRE ET AL.
On the other hand, it follows from (4) that &=(1
2~)
1 -w~coa
(22)
Now from (21) and (22) it is obtained that (1 + wa) (/5~ + 79~ - .T'a) = 0. If free streaming out limit is assumed, PR boundary,/5 should vanish; otherwise
(23) = .T" = 79,
hydrodynamic pressure at the
/ha + 79a = .T'a,
(24)
is obtained. Therefore, in the local comoving frame, the total (hydrodynamic + radiation) pressure at the boundary surface is balanced by $'a, the energy flux density. 2 . 4 . SURFACE EQUATIONS AND THE H J R M E T H O D
Defining two auxiliary functions: F
I (t3 + P R - co(P + T') - (1 - w)5r) l+w
(25)
and
P=
1
l+w
(/5 + 79 _
w(~ + PR) -- (1 -- w))v).
(26)
Notice that both (25) and (26), in the corresponding limits for the radiation field, reduce to HJR (Herrera et al., 1980, 1987; Herrera and Ntifiez, 1990) effective density and pressure, respectively. Again, field equations (14) and (15) can be integrated yielding: r
rh =
a
dg47rgEt5 and /3 = 0
g - 2rh (/5 + t5) d?.
(27)
r
Thus, if the r dependence of/5 and/5 are known we can get the metric functions rh and/3 up to some functions of u related to the boundary conditions. The crucial point of the H JR method is the system of ordinary differential equations for quantities evaluated at the surface (the System of Surface Equations) which provides the unknown dependence in the timelike coordinate. Scaling the radius a, the total mass rha = m and the timelike coordinate u by the total initial mass, rn(u = 0) = rn(0), i.e.
RADIATION HYDRODYNAMICS AND RADIATING SPHERES a
A-
fib
M---
re(O)'
u - -
re(o)'
U
159 (28)
and defining F=I---
2M
A
and f t -
1
(29)
1 - Wa
Equation (22) (the first surface equation) can be written as A = F ( a - 1).
(30)
Again, the dot over the variable represents the derivative with respect to the timelike coordinate. The second surface equation emerges from the evaluation of Equation (13) at r = a+0, taking the form
f/I = -FL,
(31)
where L, representing the total luminosity, can be written as L = 47rAZ.T'a(2f2 - 1).
(32)
Now, using above Equation (30) and definitions (28) and (29); we can re-state Equation (31) as
/b
=
2L + (1 - F)(f~ - 1) A
(33)
Finally, after some straightforward manipulations, starting from field equations (14), (15) and (16), it is obtained e2~
1--2~/r
,o
Or
1--2~/r
41rrP +-~
=
= - - -' ~ 2 ( P + ~r ( P / t - P ) - / 5 )
(34)
which is the generalization of Tolman-Oppenheimer-Volkov (TOV) equation for any dynamic radiative situation. The third surface equation can be obtained evaluating (34) at r = a+0, and it takes the form of:
f2
F
F
(pa)'o Ff22[~ 2Ff2 (
+
+ ( f ~ - 1)\(Fa~la-~a
Pa+
47rA(1 3ft)fiaft -
1
(Vro-Pa)
32~F) =0,
) +
(35)
160
E AGUIRRE ET AL.
where /~ =
~
+ 1:
2~-/r
47rrP + ~-~
.
(36)
Equations (30), (33) and (35) conform the System of Surface Equations (SSE). This system may be integrated (mostly numerically) for any given radial dependence of the effective variables. For completeness, we outline here a brief resum6 of the H JR method for isotropic radiating fluid spheres (see Herrera and Ntifiez, 1990, for details): (1) Take a static interior solution of Einstein Equations for a fluid with spherical symmetry, P s t a t i c = p(r) and Pstatic : P(r). (2) Assume that the r dependence of/5 and ~3are the same as in Pstatic and Pstatic, respectively. Be aware of the boundary condition: /Sa = -watSa.
(37)
(3) With the r dependence of/5 and/5 and using (27), we get metric elements and/3 up to some functions of u. (4) In order to obtain these unknown functions of u, we integrate SSE: (30), (33) and (35). The first two, Equations (30) and (33), are model independent, and the third one, Equation (35), depends of the particular choice of the equation of state. (5) One has four unknown functions of u for the SSE. These functions are: boundary radius A, the velocity of the boundary surface (related to f~), the total mass M (related to F ) and the "total luminosity" L. Providing one of these functions the SSE can be integrated for any particular set of initial data. (6) By substituting the result of the integration in the expressions for ~ and/3, these metric functions become completely determined. (7) The complete set of (four) matter variables can be algebraically found for any part of the sphere by using the field equations (13)-(16).
3. One Static Equation of State and Several Dynamic Models The second comment is that HJR method does not relate an static equation of state to a single dynamic configuration. In this section we shall present how it is possible to obtain several dynamic models relying in the way effective variables are built up from a particular static equation of state (point 2 in the above resumr). We shall present three different ways to construct these variables. For simplicity and in order to refer previous calculations, only models in the free streaming out limit for the radiation Pn = .T" = T' = ~ will be considered below. To illustrate this point we shall use Tolman V static solution (Tolman, 1939), i.e.:
16 l
RADIATION HYDRODYNAMICS AND RADIATING SPHERES
87rp = ~
+ ~
and 8rrP = 7r 2 + ~-g
(38)
where R is the static exterior radius of the configuration. 3.1. MODEL 1 In 1983, A. Patifio and H. Rago (1983), using the HJR method, developed a model starting from Tolman V static solution (38). The effective variables for PR model are written as
= -~ ~ r2 + z(u)r 1/3
and fi = -~ \ ~
z(u)r 1/3 .
(39)
Two u-functions, w(u) and z(u), replace the constant factor and, because of the junction condition (37), can be written in terms of the surface variables as 1 0 ( ( 1 - F)(4f~ - 3 ) )
w(u) = ~2(1 - F)(5 - 2a) and z(u) = ~
A7/3
(40)
The third surface equation (35) is: f2 - 3 + 2 F4
12L ']. 41-~(3 + 2F) 2 - 2 ( 2 F + 7 +l---2--ffj
(41)
Notice that [2 has been algebraically solved from Equation (35), therefore, only the two model-independent differential equations (30) and (33) are left to form the SSE. The luminosity profile L have to be provided in order to get the value of f2 and to integrate the SSE. 3.2. MODEL2 Starting from the same Toiman-V static solution described above by Equation (38), Krori et al. (1985) in 1985 incorrectly worked out a model (see Appendix). They assume effective variables as:
= ~
~r 2 +
g(u)r 1/3
and /5
=
~
~ r2
_
g(u)rl/3
.
(42)
Notice that g(u), is a single function replacing the constant of integration in the corresponding static solution. Because of the boundary condition (37)
g(u) =
3(4f2- 3) 7(5 - 2f~)AT/3 '
The correct metric elements (27), for these effective variables are:
(43)
162
E AGUIRRE ET AL.
~ ( u , r) = ~--~ (6r
_
79(u)r7/3)
(44)
'
and
/3(u,r) = l l n r ( 79(u)J/3The third Surface Equations (35) is ~ = (7 - 4 f ~ ) ( 4 a 2 - 1 9 a + 1 8 ) ( a - 1) 6(2a - 5)A
(46)
Because there is a single unknown function 9(u) in this model, the surface variables (A, F, f~ and L) are not independent. An algebraic relation between two of these variables can be found: 7 - 10F a -- 4(1 - F) "
(47)
This can be translated into a restriction in the total mass-radius of the configuration 3 M 1 1---6< -A- < 2"
(48)
In addition, using (28), (29) and (47), the luminosity profile can be solved (see Appendix): L -
3(a - 1) 4(2f2 - 5)
12fiA 4(4f2 - 7)(2f~ - 5) "
3.3. MODEL 3 A third set of effective variables (MO3) can be built as ¢5 -
k(u)( 3 10 (_~)1/3) 87r ~ r 2 + ~ and ]5
Again, two u-functions, (37) now implies
_
d(u)(
1
871"
7~ 2
2 (_~) 1/3) A2 .(49)
k(u) and d(u), have been used. The junction condition
k(u) = 7 ( 1 - F) and d(u) = -553(1 - F)(1 - a) 390f2
(50)
Metric coefficients (27) can be written as (51)
RADIATION HYDRODYNAMICS AND RADIATING SPHERES
163
and
13(u'r)=lnl(r)6(3F+7-7(1--F)(r/a)7/3"~7'lOF
]
(52)
where 6 = (196~ - 79)(1 - F ) and 7 = 2 8 0 F ~ - 5 5 3 F - 1027 156Q(3F + 7) 364Q(3F + 7) Finally, the third
fi_
(53)
surface equation takes the form of
-f~ P 1 -FF
(54)
-~- 7"3Q3 -]- 7"2~ 2 + T1Q -4- TO
with 7- 3
--
TI - -
1372 F _ 3081 A ' 491 - F
49 7 4 7 F - 1027 _ _ 30810 A -79 1 - F and T o - - - - 60 A
_
20
-7- 2
A
--
(55)
3.4. MODELLING In HJR approach, models are only restricted by several minimum reasonable physical requirements:
p(~, ~) > 0,
1
p(~, ~) > P(~, r),
~(~, u) > ~ ,
and - 1 < w(r,u) < 1.
(56)
Consequently, these requisites suggest the running time for performing simulations. We have considered a common time interval for the simulations: 0 < u < 16.0. In this work, when luminosity profiles have been used, they are assumed to be proportional to a gaussian pulse, i.e.,
-l(u-peak) 2
mrad
L - Fa~
exp ~
a
'
(57)
where mrad represents the total radiated mass; c~ is the pulse width (variance); and peak stands as the time for the maximum of the radiation pulse. In order to compare the dynamics of these models, we have selected initial data oriented to make gravitationally equivalent configurations, i.e., equal initial exterior radius and mass. We have chosen the following values: A(0) = 5.3, and m(0) = 1.0, for the three cases above and f~ = 0.752 for Model 3. Integration of PR model and Model 3, have been carried out using: m r a d = 0.00Ira(0), a = 1.5 andpeak = 8.0 as a set of parameters for the luminosity pulse.
164
E AGUIRREET AL.
Radii
5.5
I
t
I
I
I
]
" 7 - - -
P.R.5
A(u)
.-
.
.
.
--
4.5 4 3.5
0
I
I
I
I
I
I
I
2
4
6
8
I0
12
14
16
U
Fig. 1. Evolution of the boundary surface for the different models. Curves labeled as ER., K.B.S. and MO3 represent the evolution of the boundary surfaces for Model 1, Model 2 and Model 3, respectively. Figure 1 displays the evolution of the boundary of the distribution for the three models considered. The evolution for the physical variables (w, p = tg, P = / 5 and ~) for the models worked out, are sketched in Figures 2, 3 and 4. This recording has been made at fixed r/ao positions: 0.2, 0.4, 0.6, and 0.8 (a0 = a(0) is the initial position of the surface boundary). 4. Conclusions We have shown how starting with the most general case for a radiating sphere, is possible to recover previously HJR calculation schemes developed for two particular limits of the radiation field: free streaming out (PR = .~ = 79 = e) (Herrera et al., 1980) and diffusion (PR = 379 and U = ~) (Herrera et al., 1987). In the above mentioned general case: energy density, PR, energy flux density, U, and pressure density, 79, for the radiation field can be related, solving the system (6), (7) and (8) together with a general relativistic form of the Boltzmann equation. This work is underway. We have also exhibited one of the most interesting features of the method: its non-uniqueness. It can be appreciated from Figure 1 how different are the evolutions of the boundary surfaces despite that equivalent initial conditions have been considered. The main consequence of selecting one (KBS Model) or two u-functions (PR or MO3) in writing down the effective variables is that one-function-models are closed. All surface variables should emerge from
RADIATION HYDRODYNAMICS AND RADIATING SPHERES
165
the integration of the SSE because there will be an algebraic relation among these variables. On the other hand, in two-function-models one of the surface variables has to be provided. Since the only observable quantity entering into a "real" gravitational collapse is the luminosity profile, it seems reasonable to adopt the second approach providing such profiles. The Radiation-Hydrodynamic picture also differs from model to model (see Figures 2, 3 and 4). As it was reported earlier (Patifio and Rago, 1983), in Model 1 inner shells bounce earlier than the outer ones (see Figure 2). This model has a "cut off" whenever the square root of Equation (41) becomes imaginary. This is point reached as the boundary surface bounces. This particular time has been selected as the interval of evolution. It is also observed from Figure 2 that inner shells first emit and at the end of the evolution they absorb radiation. Contrary to previously obtained conclusions, correct evolution of Model 2, tends to hydrostatic equilibrium (see Figure 3). Luminosity Profile, L, is obtained from the integration of the System of Surface Equations (Equations (30), (33), (46)). As it is apparent from Figure 3 inner mass shells (curves 0.2, 0.4 and 0.6 in the energy flux display) always emit but outer shells (specially the boundary surface) absorb in early times of the evolution and later they become radiant. Finally, Model 3, which is only presented to illustrate a third alternative way to build up effective variables, starting from the same static solution, has some other interesting physical situations. First, observe from (49) and (50) that, when f2 = 1 (static situation) effective pressure vanishes and therefore, by definition, hydrodynamic pressure also becomes zero, i.e. /5 = 0. Fluid in these particular mass shells change to be incoherent (dust). Secondly, negative pressures, associated to metastable states (Landau and Lifschtz, 1959), are recorded in curves 0.4 and 0.6 of Figures 4. Finally in Model 3, there is some kind of "bifurcation" in the radiation flux profiles between curves 0.2 and 0.4. At this point it is worthy to consider some of the strategies followed to build up effective variables. In the first method, it is possible to obtain the effective density generalizing the static density with some functions of u (one or two functions); next, solve hydrostatic-equilibrium-like/~ = 0 (Equation (36) vanished) for the effective pressure P. Therefore,/~ vanishes for these models and they have proven to have a hydrostatic limit when u ---, ~ . This is the case for Schwarzschild-like model (Herrera et al., 1980) and both Tolman-like models (Krori et al., 1985) worked in the Appendix. The second procedure is to generalize simultaneously the static density and pressure with one or two functions of u. This has been used in a Tolman-VI-like (Herrera et al., 1980) and in the above Model 1 (Tolman-V-like, Patifio and Rago, 1983) and in several others (Herrera and Ntifiez, 1983). For these models/~ ~ 0 and the hydrostatic regime is not reached.
166
F. AGUIRREET AL. Density
Pressure
0.016 0.012
p(u)
0.6 0.8 . . . .
0.008 0.004
P(u)
0.006 0.005 0.004 0.003 0.002 0.001
I
I
I
-0.001
0 2 4 6 8 10 12 14 16
0.6 0.8"
I
,
,
0246
'
'
I
I
I
d...Lr
-0.06
•"l
..i
0.6 i
10'81
I
0 2 4 6 8 10 12 14 16 U
I
I
Energy Flux
~ J ~
-0.02
_
I
U
Velocity of matter 0.02 [
I
8 10 12 14 16
U
-0.1
I/,~_
........ --t...l...I.-t''F''l''~
du
I
0.001 0.0008 0.0006 0.0004 0.0002 0 -0.0002 -0.0004
~ I
I
I
i
I
I
I
t
i
0.2 - - 0.4" 0.6 . . . . _ 0.8 . . . .
i
I
~
2 4 6 8 10 12 14 16 U
Fig. 2. Evolution of the physical variables for Model 1. Density, Pressure, Velocity of matter and Energy Flux Density are monitored at four fixed mass shells. Curves are labeled by the corresponding ratios r/ao = 0.2, 0.4, 0.6, 0.8
Finally, we would like to end this work with the following comment. Because ultradense matter is not "available" in any earth laboratory, all " k n o w n " equations of state, independently of how "elaborated" is the micro-physics they use, emerge from not very well justified extrapolations and speculations. In any case, the most plausible situation for this "micro-physical description", if any, is static and nonradiating, Then, it is plausible to explore several alternative descriptions, emerging from this "reliable" ground, for dynamic and radiating distributions o f matter only restricted by mild physical assumptions (50). It is in this sense that, we believe, the non-uniqueness o f the method, instead of being a drawback, can be considered an advantage.
167
RADIATION HYDRODYNAMICS AND RADIATING SPHERES
Density
Pressure 0.006 0.005 0.004
0.012 0.2 0.4 0.6 0.8.-'
0.008 0.004
P(u) 0.003 0.002 0.001 0 -0.001
.'TT--- ~i: • "I" • 'I ' " I " "f " "f '
I
I
I
..'"""
_
0.8"'I" 0.6
I
I
I
-0.1
.... z _
. :
0.2 0.4 "' 0.6
.:
I
I
]
Energy Flux
D
.""
"
0.4
11
Velocity of matter
-0.06
|
0.2
u
i
I
2 4 6 8 10 12 14 16
0 2 4 6 8 10 12 14 16
-0.02
I
_
-0.14
2e-05 0 -2e-05 flu) -4e-05 -6e-05 -8e-05 -02001 -0.00012
0 2 4 6 8 1012 14 16
.2
~ -
0.4 " - _ 0.6 0.8 . . . . I
l
I
t
I
i
I
2 4 6 8 10 12 14 16 u
u
Fig. 3. Evolutionof the physical variables for Model 2. Density,Pressure, Velocityof matter and EnergyFlux Densityare monitoredat four fixedmassshells. Curvesare labeledby the corresponding ratios flag = 0.2, 0.4, 0.6, 0.8
Acknowledgements This work has been partially supported by the Consejo de Desarrollo Cientffico, Humanfstico y Tecnol6gico de la Universidad de Los Andes and the Programa de Formaci6n e Intercambio Cientffico (Plan II) de la Universidad de los Andes. Algebraic calculations of the present work has been checked with REDUCE. The authors wish to thank the staff of the SUMA, the computational facility of the Faculty of Science (Universidad de Los Andes), making this work possible.
Appendix Krori et al. (1985) worked out two models using Tolman IV and Tolman V static solutions (Tolman, 1939). They assume effective variables for these solutions as:
168
F. AGUIRRE ET AL.
Density I
0.012
p(u)
I
I
I
Pressure
/4
0.008
0.002
~
I
0.2 0.4
0.0015
J
I
0.001
I
I
I
0 -0.0005 -0.001 t I I I I I I I 0 2 4 6 8 10 12 14 16
2 4 6 8 10 12 14 16
U
U
Velocity of matter
Energy Flux 0.001 0.0005 0 -0.0005 ~(u) -0.001 -0.0015 -0.002 -0.0025 -0.003
0.1 0.06
0.2
-
0.02 -0.02
I
-
P(u) 0.0005
P
0.004
& a-ff
I
0.6
_
-0.06 -0.1 -0.14
0.4 I
IV'Vl
I
I
I
I
"1
0 2 4 6 8 10 12 14 16
0 2 4 6 8 10 12 14 16
U
U
Fig. 4. Evolution of the physical variables for Model 3. Density, Pressure, Velocity of matter and Energy Flux Density are monitored at four fixed mass shells. Curves are labeled by the corresponding ratios = 0.2, 0.4, 0.6, 0.8.
r/ao
1 - f(u)r 2 "~ 1 {1 + 3(a 2 + r 2 ) f ( u ) + 2(1 + 2(r/a)2) 2 ¢5 -- 87ra 2 ~ 1+
)
2(r/a)2
(58)
and
/51 87ra 2
(1- (a2 + 3r2)f(u)) 1 + 2(r/a)2
(59)
for a Tolman IV like model; and
, "9" f i = ~, ( 3~r2+-~g(u)r
and / 5 =
8-7
6o,
~
f(u)
g(u), are
corresponding to a Tolman V like model. Both and functions replacing terms containing constants of integration in the corresponding static
RADIATION HYDRODYNAMICS AND RADIATING SPHERES
169
solutions. These functions, because of the boundary condition (37), can be written in terms of the surface variables as: 8f~ - 5
3(4f~ - 3)
f(u) -- (16 - 4 ~ ) a 2 and 9(u) = 7(5 - 2f~)a7/3'
(61)
respectively. The correct metric elements, rh,/3 (27) and the third surface equations (35) for these effective variables are: v 3 (1 +_(a2_+r2)f(u)~
~h(u, r) = ~a 2
1 + 2(r/a) 2
fl(u,r)=~ln
3a ff
Ji
] ,
(62)
f ( - ~ ) r2
(63)
and = (16f~ - 19)(4f~ - 7)(f2 - 1) 12(f~ - 4)A
(64)
for Tolman IV type model; and, on the other hand, r ( 6 - 79(u)r7/3),
(65)
1 (Tg u)oT,3 k , ~ -
(66)
/3(u,r) = ~ l n and
h = (7 - 4f~)(4a 2 - 1912 + 18)(Q - 1) 6 ( 2 a - 5)A
(67)
in the Tolman V case. For both models,/) (Equation (36)) vanishes. It is also clear that for these models, as a consequence of the junction conditions (61), surface variables (A, F, f2 and L) are not independent. This situation leads two important consequences. First, for these models it is possible to find algebraic relations such as 7 - 8F f~- 2(2-F)
7 - 10F and f / - 4 ( l _ F ) '
respectively. Therefore, matter configuration are restricted to:
(68)
170
E AGUIRREET AL. 1
M
1
3
7 < - - A < 2 and ~
M
< ~
