- Email: [email protected]
Heat waves and thermohaline instability in a fluid
__ __ I!!!2
15 May 1995
&
ELSEVIER
PHYSICS LETTERS
A
Physics Letters A 201 (1995) 33-37
Heat waves and thermohaline instability in a fluid L. Herrera ‘, N. Falc6n Depariamento de Fisica. Facultad de Ciencias. Ukersidad
Central de Venezuela, Caracas, Venezuela
Received 18 May 1994; revised manuscript received 30 January 1995; accepted for publication 14 March 1995 Communicated by J.P. Vigier
Abstract The concept of thermohaline instability is revised for times shorter than the effective time of relaxation into diffusion. Using the Cattaneo equation for the heat flux, we show that before the establishment of the steady state resistive flow, any secularly unstable element of fluid will describe a damped oscillatory motion.
In a fluid distribution, layers of higher molecular weight above a colder region of lower molecular weight may be dynamically stable if the specific weight of the former is reduced, because of of its higher temperature, below that of the underlying layer. Then if a blob of the upper layer is pushed downward, buoyancy will push it back. However, on the time scale by which the blob loses its excess temperature, the buoyancy decreases and the blob sinks. This secular instability is controlled by the heat leakage of the blob [l]. The situation is similar to that of thermohaline convection, in which a layer of warm salt water is above a layer of fresh cold water. As far as the warmth of the salt water is sufficient to reduce its specific weight to below that of the fresh water the system is dynamically stable. However, as the warm salt water cools off, its density increases, and eventually small blobs of salty
water sink down [2]. Instabilities of this kind can occur in stars. Thus, for example, in a close binary system, helium-rich material may be transfered to a main sequence star. Then a helium-rich outer layer is formed, leading thereby to the occurrence of a thermohaline instability at the interphase between that layer and the original stellar material [3,4]. Also, in stars of about one solar mass, helium burning may ignite off centre, creating a carbon enriched shell which has a higher molecular weight than the region below [51. As a result, carbon “fingers” will grow and sink inward. Let us now consider a blob of matter situated in surroundings of different but homogeneous composition, i.e. Dp#O,
t=O,
where p is the molecular weight, Dp denotes the difference between p,, (the molecular weight of the blob) and CL,(the molecular weight of the surroundings), and
’ E-mail: [email protected]. 0375-9601/95/$09.50 0 1995 Elsevier Science B.V. At1 rights reserved SSDZ 0375-9601(95)00226-X
34
L. Herrera. N. Falcbn /Physics
where as before subscripts s indicate that the quantity is evaluated in the surroundings and P denotes the pressure. If the blob is supposed to be in mechanical equilibrium with its surroundings (DP = Dp = 01, then
Letters A 201 (1995133-37
spherical), the radiation density constant X 10~” erg crnp3 K-“) and the speed of Fron the requirement that d(DT)/dT = locity of the blob can now be evaluated, (see Ref. [ll for details) HP
(1)
H,,
cpDp
l/“=-(o,,-v)Td~=-
co,,-
(a = 7.57 light. 0, the veto obtain DT
‘IT,
T
’ (4)
(at constant
where subscripts d indicate that the quantity is calculated using the Maxwell-Fourier law (see Eq. (5) below), the scale height of pressure H, is defined as
P and p)
and HP = -P; (at constant
P and T). and for DT the following
Then, if DF > 0, the blob is hotter and radiates towards the surroundings. Under the above assumption (DP = 0) this loss of energy leads to an increased density and the blob sinks until Dp = 0. Since (1) is still valid and Dp is constant, we get again DT > 0. Thus the blob will sink (or rise for DE.L< O), so that DT remains constant, according to (1). While sinking or rising the temperature of the blob changes, both because of its adiabatic compression (or expansion) and because of the radiation. Thus, the total rate of change of DT can be written as [l]
(2) with
(at constant and the thermal adjustment 7d = Kp2d’T-3 16ac
cP,
entropy)
time 7,, is given by
(3)
where K, cp, d, a and c denotes respectively the mean absorption coefficient, the specific heat at constant pressure, the diameter of the blob (assumed
DT = DT(0)
expression
is obtained,
exp( -t/~~).
In the derivation of Eq. (2) and the expression for DT it has been explicitly assumed that the energy flux of the radiation is given by (Maxwell-Fourier) F=
-kVT,
(5)
where k=-
4acT3 3Kp
’
(5’)
represents the coefficient of conduction for the diffusion of radiative energy. This approximation is justified by the fact that particles carrying the energy of radiation, usually have very small mean free paths, in stellar interiors. Now, it is well khown that a Fourier-Maxwell law as Eq. (5) leads to a parabolic equation for T according to which perturbations propagate whith infinite speed (see Refs. [6,7] and references therein). The origin of this non-causal behaviour is to be found in Eq. (5) where it is assumed that the energy flux appears at the same time the temperature gradient is switched on. This neglecting of the relaxation time is in general very sensible because for most materials it is very small (of the order of lo-” s for the phonon-electron interaction and of the order lOpi3 s for the phonon-phonon and free electron interaction, at room temperature) [8]. There are, however, situations where the relaxation time may not be negligible. Thus, for example,
L. Herrera, N. Fal&n/Physics
for superfluid helium II the relaxation time is of the order of lop3 s for a temperature 1.2 IS [9]. Also in dense degenerate regions where the thermal conductivity is dominated by electrons, one expects the relaxation time to increase significantly, due to the larger mean free paths of electrons. The problem of heat propagation for times shorter than the relaxation time has been the subject of lengthy discussions since early work of Maxwell [lo] (see Refs [6,7], and references therein). A heat flux equation leading to a hyperbolic equation (telegraph equation) is aF ratiF=
-kVT,
(6)
where T denotes the relaxation time. Eq. (6) has been derived for the first time by Cattaneo [ll] from the kinetic theory of gases, and was reobtained afterward by many researchers (see Ref. [6] and references therein). The telegraph equation resulting from (6) has been shown to account very well for the experiment results on propagation of heat pulses in He II [12]. Eq. (6) may be written as an integral over the history of the temperature gradient F=
-*/’ exp[-(t-t’)/r]VT(x, 7 --03
whereas in the other extreme (Q = const) the resulting equation is the usual wave equation without attenuation. Somewhere in the middle of these extremes is the Cattaneo equation, with the thermal history of material weighted by (9) and leading to the telegraph equation. We shall now obtain an expression for the velocity of the blob, assuming Eq. (6) instead of the Fourier law (5). Let us first obtain the corresponding expression for DT. The absolute value F of the radiative flux from the blob due to its excess of temperature is now, because of (6’), F=k ./_ t IVT(x, m
Assuming the blob to be spherical with diameter and aproximating the temperature gradient by
t’) dt’,
1VT I= 2DT/d,
A=SF=kz/I
we recover the Fourier-Maxwell Q=iexp[-(r-fi)/r]
DT(t’) cc
exp[-(t-t’)/71
dt’,
(11) On the other hand, the rate by which the thermal energy of the blob of volumen V changes is given by
-pc/:.
(12)
(7) Equating
different choices of Q give rise to different constitutive models. Thus for Q=ka(t-t’)
d,
we obtain that the radiative loss h per unit of time from the whole surface S of the blob is
A=
Q(t-t’)VT(x, --ac
dt’. (10)
In general, if we write -1’
t’)lexp[-(t-t’)/71
t’) dt’. (6’)
F=
35
Letters A 201 (1995) 33-37
(11) and (12) we get 12kd-2
3(DT) -=-at
pcp7
(8) law, whereas
f DT( t’) I --cc
Xexp[-(t-t’)/71
dt’,
(13)
where the fact has been used that (9)
yields the Cattaneo equation. From the above it is clear that Q measures the “memory” of the material with respect to the history of the temperature gradient. At one extreme we have the Fourier law, which corresponds to a complete absence of memory (Eq. (811, leading to the well known diffusion equation,
Eq. (13) can be integrated
to obtain
DT( t) = DT(O)e-”
-
(1+DT;&J sin(:xq,(14)
L. Herrera, N. Falc&/Physics
36
with x=t/27,
d”p 7, = ECP.
6J = \1’47/Q - 1 ,
(15)
as it follows from (3) and (5’), and x-
” DT(t’) /-z
exp(t’/r)
dt’.
From now on we shall approximate mum value, namely
( 16) x by its maxi-
Sup[ x] = rDT( 0).
(17)
With (14) we can now obtain an expression for the velocity of the blob, retracing the same steps leading to (4) we obtain after some simple calculations (observe that now the last term in Eq. (2) has to be replaced by the right-hand side of Eq. (13)) V= Vd(rd/2r)
+sin(ox)
exp[ x(2r/rd
[
w--
L (
1-t
- l)]
DT:Z)rd)]]’
(l’)
or, using (15) and (17) V= V, exp[ x(2r/rd X [ (1 + rJr) + (1 - rJr)
- l)] cos( Wx) sin( ~x)/m]
(19)
In order to bring out explicitly the oscillatory behaviour of the blob, it is useful to consider some special cases: (i) 7 y rd. In this case, we obtain V = 2V, exp( t/27)
cos(J?;t/2r).
(20)
Thus the blob will oscillate with a period equal to 4rrr/&, and the amplitude damped by a factor exp(- t/27) (note the damping factor exp( - t/q,) in V,). (ii) r>> rd. In this case one obtains V= V, exp( t/r,)
cos( t/iq.
(21)
Then the exp( t/rd) factor cancels exp( - t/rd) in V,, and we obtain a purely oscillatory motion with period much smaller than the relaxation time. Therefore for times shorter than, or of the order
Letters A 201 (1995) 33-37
of, the relaxation time the blob will oscillate before sinking due to the thermohaline instability. This quasiperiodic behaviour of the blob is clearly related to the oscillations of DT described by Eq. (14). In the case of helium II, r is of the order of 10-j s (for a temperature 1.2 K) [9]. Oscillations with a period of this order of magnitude might be observable in a bath of He II for blobs of a heavier element. Another scenario where we could expect this effect to play an important role is in neutron stars. Indeed, not only are neutron stars formed in an important proportion by superfluid matter (due to neutron pairing), and therefore the relaxation time might not be negligible, but also, as mentioned above, one expects the relaxation time to be large, due to the larger mean free path of electrons. In the specific case of a compact X-ray source in a binary system [13], accreted matter may form an outer layer, creating conditions for thermohaline instability. Then blobs oscillating in compact X-ray sources will induce oscillations in the rotation rate and thereby oscillations in the emission rate. We recall that compact X-ray sources are known to present fluctuations in emission, on every time scale from seconds to months 1141. Along the same lines, particular attention deserves the quasiperiodic microstructure in radio pulsar emissions, recently reported by Cordes et al. [15]. Analysing the data obtained from five radiopulsars, they find a periodic microstructure with quasiperiods ranging from 0.5 to 5 ms. Without elaborating further on the modelling of this microstructure in terms of the instability discussed in this paper, let us just mention that the order of magnitude of the observed quasiperiods is about the same as the order of magnitude expected for the relaxation time in a neutron star. It is, however, useless to go deeper into the explanation of these these fluctuations in terms of blob oscillations until uncertainties pertaining the numerical values of r, rd, and the relaxation time for the transfer of the shift in the angular velocity, are dissipated. Finally, it is worth mentioning that thermohaline instability is also present in the collapsed core of a supernovae progenitor [16], playing thereby a potential important role in core collapse supernovae [17]
L. Herrera, N. Falc6n /Physics
References [l] R. Kippenhahn and A. Weigert, Stellar structure and evolution (Springer, Berlin, 1990) p. 44. [2] G. Veronis, Marine R-es. 21 (1965) 1; M. Stern, Tellus 12 (1960) 2. [3] S. Kato, Publ. Astron. Sot. Japan 18 (1966) 374. [4] R. Kippenhahn, G. Ruschenplatt and H.C. Thomas, Astron. Astrophys. 91 (1980) 175. [5] H.C. Thomas, Z. Astrophys. 67 (1967) 420. [6] D. Joseph and L. Preziosi. Rev. Mod. Phys. 61 (1989) 41. [l] D. Jou, J. Casas-Vazquez and G. Lebon. Rep. Prog. Phys. 51 (1988) 1105. [8] R. Peierls. Quantum theory of solids (Oxford Univ. Press, Oxford, 1956). [9] W. Band and L. Meyer, Phys. Rev. 73 (1948) 226.
Leiters A 201 (1995) 33-37
37
[lo] J.C. Maxwell, Philos. Trans. R. Sot. 157 (1867) 49. [ll] C. Cattaneo, Atti. Semin. Mat. Fis. Univ. Modena 3 (1948) 3. [12] J. Brown, D. Chung and P. Matthews, Phys. Lett. 21 (1966) 241. [13] C. Hansen and H. van Horn, Astrophys. J. 195 (1975) 735. 1141 F. Lamb, in: Proc. 7th Texas Symposium on Relativistic astrophysics Ann. NY Acad. Sci. 262 (1975) 331; J. Shaham, D. Pines and M. Ruderman, in: Proc. 6th Texas Symposium on Relativistic astrophysics, Ann. NY Acad. Sci. 224 (1973) 190. [151 J.M. Cordes, J. Weisberg and T. Hankins, Astron. J. 100 (1990) 1882. [16] L Smarr, J. Wilson, R. Barton and R. Bowers, Astrophys. J. 246 (1981) 515. [17] J. Wilson and R. Mayle, Phys. Rep. 163 (1988) 63.
15 May 1995
&
ELSEVIER
PHYSICS LETTERS
A
Physics Letters A 201 (1995) 33-37
Heat waves and thermohaline instability in a fluid L. Herrera ‘, N. Falc6n Depariamento de Fisica. Facultad de Ciencias. Ukersidad
Central de Venezuela, Caracas, Venezuela
Received 18 May 1994; revised manuscript received 30 January 1995; accepted for publication 14 March 1995 Communicated by J.P. Vigier
Abstract The concept of thermohaline instability is revised for times shorter than the effective time of relaxation into diffusion. Using the Cattaneo equation for the heat flux, we show that before the establishment of the steady state resistive flow, any secularly unstable element of fluid will describe a damped oscillatory motion.
In a fluid distribution, layers of higher molecular weight above a colder region of lower molecular weight may be dynamically stable if the specific weight of the former is reduced, because of of its higher temperature, below that of the underlying layer. Then if a blob of the upper layer is pushed downward, buoyancy will push it back. However, on the time scale by which the blob loses its excess temperature, the buoyancy decreases and the blob sinks. This secular instability is controlled by the heat leakage of the blob [l]. The situation is similar to that of thermohaline convection, in which a layer of warm salt water is above a layer of fresh cold water. As far as the warmth of the salt water is sufficient to reduce its specific weight to below that of the fresh water the system is dynamically stable. However, as the warm salt water cools off, its density increases, and eventually small blobs of salty
water sink down [2]. Instabilities of this kind can occur in stars. Thus, for example, in a close binary system, helium-rich material may be transfered to a main sequence star. Then a helium-rich outer layer is formed, leading thereby to the occurrence of a thermohaline instability at the interphase between that layer and the original stellar material [3,4]. Also, in stars of about one solar mass, helium burning may ignite off centre, creating a carbon enriched shell which has a higher molecular weight than the region below [51. As a result, carbon “fingers” will grow and sink inward. Let us now consider a blob of matter situated in surroundings of different but homogeneous composition, i.e. Dp#O,
t=O,
where p is the molecular weight, Dp denotes the difference between p,, (the molecular weight of the blob) and CL,(the molecular weight of the surroundings), and
’ E-mail: [email protected]. 0375-9601/95/$09.50 0 1995 Elsevier Science B.V. At1 rights reserved SSDZ 0375-9601(95)00226-X
34
L. Herrera. N. Falcbn /Physics
where as before subscripts s indicate that the quantity is evaluated in the surroundings and P denotes the pressure. If the blob is supposed to be in mechanical equilibrium with its surroundings (DP = Dp = 01, then
Letters A 201 (1995133-37
spherical), the radiation density constant X 10~” erg crnp3 K-“) and the speed of Fron the requirement that d(DT)/dT = locity of the blob can now be evaluated, (see Ref. [ll for details) HP
(1)
H,,
cpDp
l/“=-(o,,-v)Td~=-
co,,-
(a = 7.57 light. 0, the veto obtain DT
‘IT,
T
’ (4)
(at constant
where subscripts d indicate that the quantity is calculated using the Maxwell-Fourier law (see Eq. (5) below), the scale height of pressure H, is defined as
P and p)
and HP = -P; (at constant
P and T). and for DT the following
Then, if DF > 0, the blob is hotter and radiates towards the surroundings. Under the above assumption (DP = 0) this loss of energy leads to an increased density and the blob sinks until Dp = 0. Since (1) is still valid and Dp is constant, we get again DT > 0. Thus the blob will sink (or rise for DE.L< O), so that DT remains constant, according to (1). While sinking or rising the temperature of the blob changes, both because of its adiabatic compression (or expansion) and because of the radiation. Thus, the total rate of change of DT can be written as [l]
(2) with
(at constant and the thermal adjustment 7d = Kp2d’T-3 16ac
cP,
entropy)
time 7,, is given by
(3)
where K, cp, d, a and c denotes respectively the mean absorption coefficient, the specific heat at constant pressure, the diameter of the blob (assumed
DT = DT(0)
expression
is obtained,
exp( -t/~~).
In the derivation of Eq. (2) and the expression for DT it has been explicitly assumed that the energy flux of the radiation is given by (Maxwell-Fourier) F=
-kVT,
(5)
where k=-
4acT3 3Kp
’
(5’)
represents the coefficient of conduction for the diffusion of radiative energy. This approximation is justified by the fact that particles carrying the energy of radiation, usually have very small mean free paths, in stellar interiors. Now, it is well khown that a Fourier-Maxwell law as Eq. (5) leads to a parabolic equation for T according to which perturbations propagate whith infinite speed (see Refs. [6,7] and references therein). The origin of this non-causal behaviour is to be found in Eq. (5) where it is assumed that the energy flux appears at the same time the temperature gradient is switched on. This neglecting of the relaxation time is in general very sensible because for most materials it is very small (of the order of lo-” s for the phonon-electron interaction and of the order lOpi3 s for the phonon-phonon and free electron interaction, at room temperature) [8]. There are, however, situations where the relaxation time may not be negligible. Thus, for example,
L. Herrera, N. Fal&n/Physics
for superfluid helium II the relaxation time is of the order of lop3 s for a temperature 1.2 IS [9]. Also in dense degenerate regions where the thermal conductivity is dominated by electrons, one expects the relaxation time to increase significantly, due to the larger mean free paths of electrons. The problem of heat propagation for times shorter than the relaxation time has been the subject of lengthy discussions since early work of Maxwell [lo] (see Refs [6,7], and references therein). A heat flux equation leading to a hyperbolic equation (telegraph equation) is aF ratiF=
-kVT,
(6)
where T denotes the relaxation time. Eq. (6) has been derived for the first time by Cattaneo [ll] from the kinetic theory of gases, and was reobtained afterward by many researchers (see Ref. [6] and references therein). The telegraph equation resulting from (6) has been shown to account very well for the experiment results on propagation of heat pulses in He II [12]. Eq. (6) may be written as an integral over the history of the temperature gradient F=
-*/’ exp[-(t-t’)/r]VT(x, 7 --03
whereas in the other extreme (Q = const) the resulting equation is the usual wave equation without attenuation. Somewhere in the middle of these extremes is the Cattaneo equation, with the thermal history of material weighted by (9) and leading to the telegraph equation. We shall now obtain an expression for the velocity of the blob, assuming Eq. (6) instead of the Fourier law (5). Let us first obtain the corresponding expression for DT. The absolute value F of the radiative flux from the blob due to its excess of temperature is now, because of (6’), F=k ./_ t IVT(x, m
Assuming the blob to be spherical with diameter and aproximating the temperature gradient by
t’) dt’,
1VT I= 2DT/d,
A=SF=kz/I
we recover the Fourier-Maxwell Q=iexp[-(r-fi)/r]
DT(t’) cc
exp[-(t-t’)/71
dt’,
(11) On the other hand, the rate by which the thermal energy of the blob of volumen V changes is given by
-pc/:.
(12)
(7) Equating
different choices of Q give rise to different constitutive models. Thus for Q=ka(t-t’)
d,
we obtain that the radiative loss h per unit of time from the whole surface S of the blob is
A=
Q(t-t’)VT(x, --ac
dt’. (10)
In general, if we write -1’
t’)lexp[-(t-t’)/71
t’) dt’. (6’)
F=
35
Letters A 201 (1995) 33-37
(11) and (12) we get 12kd-2
3(DT) -=-at
pcp7
(8) law, whereas
f DT( t’) I --cc
Xexp[-(t-t’)/71
dt’,
(13)
where the fact has been used that (9)
yields the Cattaneo equation. From the above it is clear that Q measures the “memory” of the material with respect to the history of the temperature gradient. At one extreme we have the Fourier law, which corresponds to a complete absence of memory (Eq. (811, leading to the well known diffusion equation,
Eq. (13) can be integrated
to obtain
DT( t) = DT(O)e-”
-
(1+DT;&J sin(:xq,(14)
L. Herrera, N. Falc&/Physics
36
with x=t/27,
d”p 7, = ECP.
6J = \1’47/Q - 1 ,
(15)
as it follows from (3) and (5’), and x-
” DT(t’) /-z
exp(t’/r)
dt’.
From now on we shall approximate mum value, namely
( 16) x by its maxi-
Sup[ x] = rDT( 0).
(17)
With (14) we can now obtain an expression for the velocity of the blob, retracing the same steps leading to (4) we obtain after some simple calculations (observe that now the last term in Eq. (2) has to be replaced by the right-hand side of Eq. (13)) V= Vd(rd/2r)
+sin(ox)
exp[ x(2r/rd
[
w--
L (
1-t
- l)]
DT:Z)rd)]]’
(l’)
or, using (15) and (17) V= V, exp[ x(2r/rd X [ (1 + rJr) + (1 - rJr)
- l)] cos( Wx) sin( ~x)/m]
(19)
In order to bring out explicitly the oscillatory behaviour of the blob, it is useful to consider some special cases: (i) 7 y rd. In this case, we obtain V = 2V, exp( t/27)
cos(J?;t/2r).
(20)
Thus the blob will oscillate with a period equal to 4rrr/&, and the amplitude damped by a factor exp(- t/27) (note the damping factor exp( - t/q,) in V,). (ii) r>> rd. In this case one obtains V= V, exp( t/r,)
cos( t/iq.
(21)
Then the exp( t/rd) factor cancels exp( - t/rd) in V,, and we obtain a purely oscillatory motion with period much smaller than the relaxation time. Therefore for times shorter than, or of the order
Letters A 201 (1995) 33-37
of, the relaxation time the blob will oscillate before sinking due to the thermohaline instability. This quasiperiodic behaviour of the blob is clearly related to the oscillations of DT described by Eq. (14). In the case of helium II, r is of the order of 10-j s (for a temperature 1.2 K) [9]. Oscillations with a period of this order of magnitude might be observable in a bath of He II for blobs of a heavier element. Another scenario where we could expect this effect to play an important role is in neutron stars. Indeed, not only are neutron stars formed in an important proportion by superfluid matter (due to neutron pairing), and therefore the relaxation time might not be negligible, but also, as mentioned above, one expects the relaxation time to be large, due to the larger mean free path of electrons. In the specific case of a compact X-ray source in a binary system [13], accreted matter may form an outer layer, creating conditions for thermohaline instability. Then blobs oscillating in compact X-ray sources will induce oscillations in the rotation rate and thereby oscillations in the emission rate. We recall that compact X-ray sources are known to present fluctuations in emission, on every time scale from seconds to months 1141. Along the same lines, particular attention deserves the quasiperiodic microstructure in radio pulsar emissions, recently reported by Cordes et al. [15]. Analysing the data obtained from five radiopulsars, they find a periodic microstructure with quasiperiods ranging from 0.5 to 5 ms. Without elaborating further on the modelling of this microstructure in terms of the instability discussed in this paper, let us just mention that the order of magnitude of the observed quasiperiods is about the same as the order of magnitude expected for the relaxation time in a neutron star. It is, however, useless to go deeper into the explanation of these these fluctuations in terms of blob oscillations until uncertainties pertaining the numerical values of r, rd, and the relaxation time for the transfer of the shift in the angular velocity, are dissipated. Finally, it is worth mentioning that thermohaline instability is also present in the collapsed core of a supernovae progenitor [16], playing thereby a potential important role in core collapse supernovae [17]
L. Herrera, N. Falc6n /Physics
References [l] R. Kippenhahn and A. Weigert, Stellar structure and evolution (Springer, Berlin, 1990) p. 44. [2] G. Veronis, Marine R-es. 21 (1965) 1; M. Stern, Tellus 12 (1960) 2. [3] S. Kato, Publ. Astron. Sot. Japan 18 (1966) 374. [4] R. Kippenhahn, G. Ruschenplatt and H.C. Thomas, Astron. Astrophys. 91 (1980) 175. [5] H.C. Thomas, Z. Astrophys. 67 (1967) 420. [6] D. Joseph and L. Preziosi. Rev. Mod. Phys. 61 (1989) 41. [l] D. Jou, J. Casas-Vazquez and G. Lebon. Rep. Prog. Phys. 51 (1988) 1105. [8] R. Peierls. Quantum theory of solids (Oxford Univ. Press, Oxford, 1956). [9] W. Band and L. Meyer, Phys. Rev. 73 (1948) 226.
Leiters A 201 (1995) 33-37
37
[lo] J.C. Maxwell, Philos. Trans. R. Sot. 157 (1867) 49. [ll] C. Cattaneo, Atti. Semin. Mat. Fis. Univ. Modena 3 (1948) 3. [12] J. Brown, D. Chung and P. Matthews, Phys. Lett. 21 (1966) 241. [13] C. Hansen and H. van Horn, Astrophys. J. 195 (1975) 735. 1141 F. Lamb, in: Proc. 7th Texas Symposium on Relativistic astrophysics Ann. NY Acad. Sci. 262 (1975) 331; J. Shaham, D. Pines and M. Ruderman, in: Proc. 6th Texas Symposium on Relativistic astrophysics, Ann. NY Acad. Sci. 224 (1973) 190. [151 J.M. Cordes, J. Weisberg and T. Hankins, Astron. J. 100 (1990) 1882. [16] L Smarr, J. Wilson, R. Barton and R. Bowers, Astrophys. J. 246 (1981) 515. [17] J. Wilson and R. Mayle, Phys. Rep. 163 (1988) 63.