Binding energy and stability of spherically symmetric masses in general relativit

Bardeen, J. M.

English

NASA Technical Reports Server

JAMES M. BAIulEEao 
California Institute of Technology, Pasadena, California 
GPO PRICE 5; 
CFSTI PRICE($) S 
Hard copy (He) / d 0 
* 
Supported in part by the Office of Naval Research [Nonr-220(47fl and 
the National A e ~ ~ u t i c s  and Space AdmhietratiOn [lTG€l-05-002428~. -I 
https://ntrs.nasa.gov/search.jsp?R=19660007930 2020-03-16T23:02:27+00:00Z
9 
1-72 I 
ABsllRAcT 
An analysis of non-adiabatic perturbation8 in a sequence 
c 
~ 
I -  
of quad-etatic general relativistic stellar models is used to * 
prove wlat under certain essumptions about the entropy distribu- 
tion amaxinnam or minimum in the bindlng sera bpUee ~ip~ie 
radial mode of osci77ntian i s  neutrd equilibrium, a s 0  dis- 
cussed is the derivation of a test for e tabmty  of radial modes 
of oscillation by considering v i r tua l  changes in the total energy 
under adiabatic perturbtiane, 
, 
\ 
I .  
. .  
z -1- 
I. IlJTROlluerIam. 
General relativist ic equilibrium mcdels of gas spheres have been 
calculated by several authors for varioue equations of state. 
here t h e  recent work of Ben (1963) , Tooper (1964a,b) 8 and Taub (1965). 
A given model ie characterized by several parameters, such ae the t&Ll.  
energy M, thk total  baryon mss or %st mass" Mo, the central baryon or 
"rest mass" density 
density (W)c 
the value of the "gravitational potential" 2M/R at the surface. To 
completely specify a model, given the equation of state 
We mention 
the central ra t io  of pressure t o  "rest mass" 
(proportional t o  the temperature for an ideal gas), and 
the matter, it 
is necessaryto know the values of two of the abwe parameters, say Mo end 
(l/V)c, and also the distribution of entropy aa a function of radius. 
A t  zero tenq?erature (degenerate mtte r )  there i e  only me indepen- 
dent parameter, which may be taken t o  be Moo 
been considered by Wheeler (1963), Misner and Zapolslry (19&), and Harrison 
(1965). 
be constructed by le t t ing the c e n t r b  density vary monotonically but 
keeping the numb& of baryons (or Mo) constant and at each central density 
choosing an e n t w  dlstrlbutlon. 
mass undergoing a generalized Xelvln-Eelrpholtz contraction in quasi-static 
equilibrium in which increase as well as decrease of the total energy is 
allowed. 
energy, has a number of maxima and minima as a f'unction of central density. 
Models of th i s  sort have 
A t  non-zero temperatures a one-parameter sequence af models may 
Such a sequence might correspond t o  a 
@e finds that the t o t a l  energy, or equivalently the binding 
The s tabi l i ty  of equilibrium models general relativity has been 
discussed by Chsndrasebhar (1964a,b)- in terme of a n d  mode analysis of 
radial. Osclllatiom. Chandraeekhar'e etabfiity criterion has been applied 
-2- 
t o  a series of ieentropic models, assuming a gas-plus-radiation pressure 
equation of state, by !lboper (1961cb) with the nUmeriCd result that the 
-66 b@cOmes unstable at the first meximum of the blndhg energy. Fowler 
(1964) and W r i g h t  (1964) have sham this m d y t i c a l l y  In the ~ 6 t - ~ e w t ~ ~  
approximation. The main purpose of Wlie pper is to show that for any 
equation of 'state a sequence of general re ia t ivis t ic  equilibrium models 
with constant number of baryons and constant nm-thermal or "rest mass" 
energy per baryon at each maes element has the property that  at a stationary 
point In the binding energy aa a function of central density one and d y  
one radial. m o d e  of Osci&tiOn has eigenfrequency Ut2 - 0, if the change 
in specific entropy from one model t o  the next hae the same si@ at  every 
mas43 element. The condition on the .entropy change 1s satisfied, for -le , 
! if each model is iS@ntxq)icJ so the specific e n t r w  hae the same value a t  
every ma68 point in a given model. 
The last part of the peper is  devoted t o  an analysis of the s tabi l i ty  
of an equilibrium model in terms of virtwd. changes af the to ta l  energy 
under adiabatic perturbations, extehaing t o  general relativity the work af 
Dyson (1961). The interesting point iS tha t  the correct s tabi l i ty  integral 
is obtained only if the virtual changes in  the energy are summed a t  infinity 
as %F they had arrived there by some process of themid. diffusion and 
radiation. 
11. DEFIIJITIOIJS. 
Our metric, in coordinates such that the baryon number Inside a given 
r is Wepeadent of time and unite such that 0 - c = 1, is 
2 D8 - 2 2  2 2  Y-2 dt2 - X dr _- R dQ . 
The components of the energy-momentum tensor are taken to be 
= T~ = T? 'i -P, (Y/x) T: "44 = Ti = (A+W)/V, T1 
volume V i e  conveniently n d i z e d  80 the non-thennal energy per u n i t  
2 K. me specific 1 
maee", A, i e  e q d  or cloee t o  one. If any nuclear reactions are laking If 
place, A will be a f'unction af t i m e .  W i e  the thermodyrnzmic internal 
energy per un'it **ma~8~', 
that except at  extremely hi& temperatures or densitiee, t he  atomic caapo- 
sition Of the matter  is not in l o c a l  thermodynamic equilibrium, 
nuclear m e  and binding energies m e t  be excluded fromthe thermal energy 
if simple thermodyaamice is t o  be ueed. 
in order t o  be able to consider the transformation of one model In a 
seqyence to another, and m i g b t  I n  particular be due t o  thennal diff'usion. 
The partic- me&aniem a r ~  energy io68 is unimportant because even if it 
doe6 involve non-tbermal contributime t o  E and P# these will be negli- 
me juetificatian for splitting E i n  t h i s  way IS 
Thus, 
The energy flux K is  introduced 
gible compared t o  the thermal energies and pre88ures and even negligible 
in their effects compared t o  K, in quasi-static models. 
The quantity l /V  is an "inkrlsnt  m e n  density t ied to the baryon 
deIWity, a d  the'tOtd e&OllDt Of " b V B l ? h 3 l t  ma88w bSide a radiUS 
will be denoted by Mo, 
vx 
0 
Mo - 
Mo is a function only of r, and one may choose 
definition of a %inding energyA for  my purpose8 is as 
total. energy, 
r - Moo 
The true binding energy is $ AdMo - M, 
The most convenient 
Mo-M8 where M is the 
but eince the rela- 
I 
t ion of binding energy t o  s tabi l i ty  holds cmly if no nuclear reaction6 are 
-4- 
taking place and in t h i s  case V can be chosen so A P 1 through the whole 
star, tke difference is Unmortant. 
The only way the total. energy of the m s s  can change is thraugh 
the flux K, since 
2 E - 4% Et2 k - 4rt R2 KZ/Y 
k 
and P P 0 at the surf'ace of the mass. 2 = R'X = (l+$-NR)1'2, and 
U = Y k  Le t  
PiTne uv T a o equation may be written as v; Ir 
. 
(3) 
If S is the specific entropy and T the temperature, by thermodyaaou * cs 
men (5) becomes 
6 is the rate at  which energy (18 measured by a local observer is added to 
a mass element per unit  invariant mass. The significance Of (7)  becanes 
clearer when it is compared w i t h  the integral fo r  M; neglecting explicit 
t ime  derivatives in the s?L-..2 of a quasi-static approximation, 
M = 1 (A+V) 2 dMo . 
-5 - 
b a qwsi-static m o h e l  Z = /l-WJR, 
thermal, and gravitation& pdcentid energy per unit h v a r h l t  rnass Of a 
mss element. mer  a s- time intern 6t, C ~ Q  = 4 &t is  t n e  non-adiabatic 
chcmge in  the internal energy per unit No and 
and (AiW)Z is the res t  ma68 , 
6QZ i a  t h e  net lion-adiabatic 
change i n  "local" energy, which lneludes %he potentfa1 energy, per unit No. 
Equation ( 7 )  'gives the non-adiabatic change in t h e  total energy inside r, 
. (9) 
Tne non-adiabatic change in local energy at a radius 1 is red-sh-ifted by 
the amount (YZ)2/(n), i n its contribution t o  the non-adiabatic change of 
the t o t a l  energy inside a larger radius 2. 
XII. ,D~,OUF AHD inscussox OF   OR EM. 
Tne eqyilibrium equation, in  a fora  convenient fo r  mahing perturba- 
tions, is 
' To first order In a, w i t h  r = R of the initial m o d e l ,  
-6- 
(14) and (l5) are the result of 
J 
The w t i t y  Y'/Y = P'/(P+E) 
Tne hydrodynamic equations behind (10) - (13) are discussed by Misner and 
evaluated for t he  i n i t i a l  eciuilibrium model, 
(U&, 1965) a 
A sinusoidal, adiabatic perturbation with 6R(r,t) - 6R(r) e* 
mi 6~ = 6s = o f o i  r gives t h e  equation 
" R ( l  - 6 1 RY' 7) + & "1 IP+E) 
z2 y32 n2 
o2 IP-tE) R2 = P t E  
P - -  
YZ3 n2 y2 z3 R~ 
in w h i c h  g = R% 6R(r), 0 P R3/3, and 6QI is  the first order variation 
of (10). (18) w i t h  boundary conditions g/Q f in i t e  at R = 0 and g 
f i n i t e  at P = 0 is not @te a standard Sturm-Liowille problem, but as 
discussed by Ledoux and Walraven (1958) f o r  the Newtonian case, the eigen- 
values and eigenfunctions have the sane properties. 
NOW consider two neighboring models in same continuous sequence w i t h  
the same number of baryons, or the  same Moo Oae can define a displacement 
. 
/ .  . .  
-7 - 
6R( r) 
tine same to ta l  baryon number inside them. 
md thus in the l imit  that  the models converge, or that  
difference between the left-hand side6 Of (10) for  the t w o  models, expanded 
between e@3rd.ent points in the two models, equivalent points having 
Both models satisfy equation (10) 
SR( r) -, 0, the 
t o  first mer in 6R, m e t  vanieh. That $6, 6(0 for one of the models, under 
a perturbation having the shape of 6R(r) in this i i m i t ,  must vanish. 
In order t o  connect up this result w i t h  equation (la), the perturba- 
tion must be an adiabatic one. 
at the surface of the mass , we see that the dL”ference of total energies Of 
two adjacent m o d e l s ,  or e q y i d e n t l y  the difference of the birding energies, 
is given t o  first order by 
From equation (9) and the fact that 6N = S% 
. 
r 
energyt 6Yh is zero. 
only possible if SQ is zero a t  every mass element. 
SQ = SA + T6S does have the saute sign everywhere is if 6A is zero (no 
nuclear reactions) and if the  models are isentropic, so for  any pair of 
models 6.5 has a constant value throughout the plass. 
SQ and SS are zero at every mass element means that 
mass element and SP is adiabatic. 
(11) - (15) that all the perturbations =e adiabatic i n  the s t r i c t  sense 
needed t o  apply (18) 
is the value of r at the suzface. At a maximwn or minimum in the binding 
3cf SQ has the same sign at every mass elemat, th i s  io 
S 
A 
One case in which 
The fact, then, that 
8% is zero at every 
Furthermore, it is clear from equations 
1l the displacement between the t w o  neighboring models is completely 
adiabatic, then the function Si(r) 
value w2 = 0. 
is an eigenfunction of (18) with zigen- 
 his means some radial m o d e  of oscillation is  i n  neutral 
\ 
. 
-0- 
esuilibrT’m a t  a stationary point i n  t he  binding e n e r a  as a function of 
central density f o r  a series of isentropic models. 
Calculations w i t h  equation of state 
L 
8 
T 
density increases the binding energy increases t o  a maximum at w h i c h  the 
a constan* inciepe~ent oz v ana P, show cia= if rl > 4/3, as vie cerltrd 1 
i‘undamental mode becomes unstable, decreases t o  a minimum a t  which the first 
harmonic becovnes unstable, and continues fn rapidly damped 
successively higher m o d e s  becoming unstable. 
w i t h  
The fact  tha t  displacements between neighboring models take on the 
shape of a given radiaL mode of oscillation near the p o i n t  where it becomes 
unstable is reflected in how the radius of the mass changes along the 
sequence of models, oscillating out of‘ phase With the binding energy. 
the central density increases, the radius decreases at  the instabflity point 
of an even mode (even number of nodes) and increases a t  the ins-bflity point 
of an odd moiie. 
a t  the s i i a c e  fo r  the above m o d e l s ,  the curve is a lopsided spiral which 
rapidly converges t o  a limiting model when 
for  the case .D 2. 
As 
If the binding energy is plotted against the value of 2M/R 
PV >> 1, as is shown in Figure 1 
The above behavior, I expect, i s  typical a6 long 86 does not 
increase sharply with density, which; may cause an unstable mode to become 
stable again, 
binding energy w i l l  have a m i n i m u m  when the f’undamatal becomes stable and 
a maximum when it becomes unstable again. 
curve w i i l  oscil late once before entering the spiral. 
In  this event, if just the fundmental w a s  unstable, the 
In the plot against 2k!!R8 %he 
-9- 
The requirement of iseAt;.opy iney be relaxed somewhat without affec- 
t ing  the validity O f  the theorem as long as it is not possible for the 
entropy t o  increase in part Of the mas and decrease in another p a r t  euch 
that 8% is zero. However, the condition 6A P 0 would  seem t o  be a 
necessary condition for .the s t a b i l i t y  to have anything t o  do w i t h  tbe 
binding ener&. 
The converse to the above theorem, that  the marginal instabil i ty of 
a radial m o d e  of oscillation of a model implies a maximum or minimum in the 
binding energy, i s  not true. 
sequences of models containing t h i s  particular m o d e l  which are s t a tb“y  
in the binding energy at this  m o d e l ,  There also exist  sequences in which 
at the instabil i ty of the f’undamental mode, say, the binding energy is s t i l l  
increasing as the central density increases. 
the subsequent maximum in the  binding energy is not associated w i t h  the 
h s t a b i l i t y  of any mode  of oscillation. 
A l l  that  one can say is that there exis t  
Zn the latter type of sequence, 
of course, only the instabi l i ty  of the fundamental mode is of direct  
physical interest, since unstable equilibrium models cannot be realized in 
nature. 
Tv. VIRTUAL C”m IN TVX!€UJ ZXERGY. 
A t e s t  for the s tabi l i ty  of a mass may be obtained frm equation (18) 
in the form of the hte& 
E, for  all functions g(R) satisfying the baunda;ry conditions g/n fini%e 
at R .D 0 and g finite when P E 0, the integral (21) is greatxr than zero, 
\ 
-10- 
tae mass is stable; if for any such g the integral is  less than zero the 
mass is unstable. 
difference of (21) from zero is second order In the deviation of g from 
the eigenfunction of the fimb.mental mode. 
In addition, for a -6s neutral equilibrium, the 
To see how the integral (21) and a variations1 t e s t  for stability 
may be obtaihed frcm the virtual change in the total  energy of the mass, 
6vM, under a Virtual adia'batic disphcernerit, consider the htegrd (a), 
with V normalized so A = 1. 
second order in 6R 
Mo is constsnt under the vaxiations and to 
r 
Tne second-order variations in (22) 'w?e evaluated 
in a quantity as 
6f = blf 
by expanding the change 
\ 
Z is taken equal t o  (1-2M/B)li2 in the variatione, and 
me second-order v&iaticm i n  W is 
G2V i is evaluated from 
. 
The result is that the integrand of (22) can be wri t ten in the form 
L 
J 
6=M and &2M are the first and 6eCCmd Order actual, aa opposed 
t o  virtual, adiabatic changes i n  33 a t  a given mas8 point and are zero at 
the limits of integration. The fir& term in (26) is the actual adiabatic 
& a g e  of (~+w)z in the perturbation. The second term is the virtual 
I- -i 
change in (l+W)Z, 6,, (1+W)Z , and 5s second order in 6R.  Frw the fact t 1  
that (18) is essentially a Sturm-LiouVufR problem, the eigeaf'unctuns 
g,(n) 
the w e i g t t t  function (P+E)R /YZ n The extra factor of YZ in 6v (~+w)z 
means that the integral. (22) does & have the property that it is positive 
of the radial modes of oscillation axe o r t b o g O d  with respect. t o  
[ I  2 3 2  
definite for  all stable models. This is  because (22) Contains terms first 
order in the deviation of g from the eigenfunction of a radial m o d e  Of 
oscillation. The proper sturm-Liouville integral (21) is, within a con- 
stant factor, 
The interesting point is the similarit3 with the integral. (9). The 
' correct evaluation of -the virtual change i n  the total energy seem t o  be 
\ 
-12- 
instead of (22). This suggest6 the interpretation that it is the 6um Of 
the virtual changes in energy after they are carried t o  infinity, including ' 
the gravltatiozd red shifts associated with this,  that properly determines 
the stabilfty of a mass. O r  alternatively one might say that the Virtual 
I- 1 
change in the local energy, aV (1i-U)Z , being a nan-adiabatic change, bse 1 1  
an effect oh the to t a l  energy in the same way as a ncm-aiiia'oa.i;ic t d r n d .  
change , whose effect is given by (9) 
A variational test for stabil i ty may be obtained directly fropa (271, 
following the method of Dyson (1961) for the lbwtonian case. !be equation 
w i t h  the boundary condition3 g/Q f in i te  at R P 0 and g f b i t e  at P = 0 
is a StUrm-Liotnrille problem with a lowest eigenvalue It is easy 
to verif'y that a necessary and sufficient cmditian that (21) , equivalent 
t o  (27), be pogitive definite is t ha t  A vaxiational e s t a t e  
y = yo. 
yo > 413. 
of yo can be obtained from the f a c t t h a t  yo is the miniPnun Over a3.l g of 
Bowever, the direct  vaxbtional estimate 09 UD," obtained from (le) (see 
Chandrasekhax (L964b)) is in most cases easier t o  evaluate accurately 
nmerically. 
v. coNcLusIoN. 
Except for  the appearance of the g r a d t a t i d  red-shift factor YZ, 
which is just  a detail in the argument relating binding energy t o  stabil i ty,  
-w- 
the result6 of this paper are arrived at in the same way as they might be 
classically. This emphasizes what 8 great simplification spherical 
symmetry is in general relativity. 
The author would like to acknowledge etimulating diecussions With 
Professor Wi.U . iam A. Fowler and Profeeear Richard P. Fey~lan. The numerical 
cdcuhtiontr were done on the IBM 7094 computer at the Wifarnia instime 
of Technology. - 
, 
-14 - 
Ydsner, C. W. and Z a P d ~ k y j  E, So 1964, phys. Rev. Letters 12, 6350 
Taub, A. E. 1965, Proceedings of Second Texas Symposium on Relativistic 
Astrophysics (University of Chicago &e66 J chicago) , to be published. 
Tooper ,  R .  F. 196ka, Ap. J, a, 434. 
-15- 
FIGURE CAPTION 
a 
. .  
Figure 1: The fractional binding energy l e  plotted against the surface 
w u e  of ~M/R, or 2 4 c %  in ordinary unite, for a sequence 
of isentropic model6 with = 2 (see text). The central value 
Of rapBFI8 froPr 0.1 at &Ft - 0.25 Up to 250 at'** h8t 
pfoint plotted. 
(W)= - 0.520, the first minimum at 
second maxionrm at { P V ) ~  = 30. 
me first maximum ~f the binding energy i e  at 
( P V ) ~  = 5.07, and the 
0 
, . .  
b 
. 
.,. . 
a 
. .  


Binding energy and stability of spherically symmetric masses in general relativity

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server