PHYSICAL REVIEW LETTERS

VOLUME 15 11 OCTOBER 1965 NUMBER 15

VIBRATIONAL ENERGY OF NEUTRON STARS AND THE EXPONENTIAL LIGHT CURVES OF TYPE-I SUPERNOVAE

Arrigo Finzi

Istituto di Fisica, Universita di Roma, Rome, Italy, and IV Sezione del Centro Nazionale di Astrofisica del Centro Nazionale di Ricerche, Frascati, Rome, Italy

(Received 14 July 1965)

About one and one-half years ago, the hypoth­esis of the creation of neutron s tars in super­nova outbursts seemed to have been confirmed by the discovery of an x-ray source in the Crab Nebula. The source was accordingly interpret­ed as the thermal radiation of the neutron star. Since then, it has been recognized that this in­terpretation was not correct.1""3

For type-I supernovae, an alternative con­firmation of the neutron-star hypothesis may be based on the possibility it seems to offer for explaining their exponential light curves. The radial vibrations of the neutron stars seem to be the only possible energy source for the observed light; the energy released exponen­tially by the supernovae is approximately equal to the maximum vibrational energy available to the neutron s tars , when the energy losses due to beta reactions are taken into account.

A type-I supernova reaches a maximum ab­solute photographic magnitude of about -19 a few weeks after the explosion, and declines by approximately two magnitudes during the following month. After that, during a period of perhaps two years, the luminosity declines exponentially; assuming that the mechanism responsible for the exponential decay starts to operate soon after the explosion, and neglect­ing the bolometric correction, the total ener­gy released exponentially would be about 2X1049

erg. In order to account for a moderate bolo­

metric correction we shall take the energy r e ­leased exponentially to be 4X1049 erg.

It was generally assumed4 that this energy was released in the decay of some radioactive element (fissionable Cf254) present in the ex­panding envelope of the supernova; the assump­tion was based on the belief that the half-val­ue time tl/2 for the luminosity was about 54 days for all type-I supernovae. It must be said, however, that Burbidge et al,4 who had advanced this hypothesis, had themselves pointed out that the reproduction of the exponential radio­active decay in the exponential decline of the visible light was very difficult to explain.

Recent investigations have confirmed the ex­ponential character of the light curves, but have indicated at the same time that the half-value time is not always the same, ranging from 34 to 68 days.5 Since it is clearly impossible to assume that in each supernova a different r a ­dioactive element is providing the main source of energy, we are forced to look for another explanation of the phenomenon. On the other hand, once radioactive energy is ruled out, it is very difficult to imagine a different process taking place in a rarefied gas and causing an exponential energy release.

We suggest therefore that the energy released exponentially originates in the neutron star. There seem to be only two ways in which 4 x 1049 erg can be stored in a neutron star: in

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VOLUME 15, NUMBER 15 P H Y S I C A L R E V I E W L E T T E R S 11 OCTOBER 1965

the form of t h e r m a l energy, or in the form of vibra t ional energy. 6 Soon after i t s c rea t ion in a supernova outburs t , the neutron s t a r i s l ike­ly to be hot and i ts modes of vibrat ion a r e l ike­ly to be excited. However, t h e r m a l energy i s easi ly ruled out: An energy of 4X104 9 e rg c o r ­responds to an in terna l t e m p e r a t u r e of around 1 .4x i0 l o °K; at this t e m p e r a t u r e t h e r m a l e n e r ­gy d e c r e a s e s with t ime t approximately like t~1/3 and is r e l ea sed mainly in the form of neutr inos . 2

We a s s u m e , the re fo re , that the energy of the observed radiat ion is r e l e a s e d in the d i s ­sipation of the vibrat ional energy W of the s t a r . Neglecting the effects of genera l re la t iv i ty ,

JM(r-D«[£ ]2, av

where ft- 1053 e rg is the potential energy of the s t a r , and | = bit/it i s the ra t io of the rad ia l d isplacement bi\ of a point during an expansion, to the dis tance n of the point from the center of the s t a r . T is the adiabatic exponent defined by 6P/P = rdp/p; in the core of the s t a r the radia t ion p r e s s u r e is negligible in compar ison with the gas p r e s s u r e , while the degenera te gas i s a lmost completely nonre la t iv i s t ic ; t h e r e ­fore r is only slightly l e s s than 5 /3 , 7 if nucle­a r forces a r e not taken into account. In con­clusion, the value of | a v leading to W = 4x 1049

e r g should be about 0.05. Taking for defini teness the half-value t ime

tV2 for the luminosity to be 5x 106 s e c , the d i s ­sipation of vibrat ional energy needed to ensure the exponential light curve i s , on the other hand,

(ln2/tl/2)W ( l n 2 A l / 2 ) x i ( r - | ) Q [ ^ a v ] 2 =2 .3X10 4 i% ]2 e r g s ec \

We want to show that an energy W = 4x 1049

e r g i s approximately the maximum energy which can be s to red for a sufficiently long t ime in the s t a r in the form of v ibra t ions . To show th i s , we point out that the equi l ibr ium concentrat ion of neu t rons , p ro tons , and e lec t rons in the d e ­genera te gas of the neutron s t a r i s a function of the densi ty. In a pulsat ing s t a r the density v a r i e s per iodical ly ; therefore the concent ra ­tion is most of the t ime different from the equi­l ibr ium concentrat ion. For densi t ies slightly higher than nuclear densi ty, like those in the core of the s t a r , the equi l ibr ium concentrat ion of e lec t rons and protons i n c r e a s e s with i n c r e a s ­ing density. The re fo re , when the gas is com­p r e s s e d , the reac t ion 2 ' 3 n + n-~n+p + e+v will

t r ans fo rm some neutrons into protons and e l ec ­t r o n s , plus ant ineutr inos which will escape . Converse ly , when the gas is expanded the r e ­action n+p + e -~n+n+ v will t r ans fo rm some protons and e lec t rons into neutrons plus neu­t r inos which will e scape . It mus t be pointed out that the re laxat ion t imes for these r e a c ­tions a r e always many o r d e r s of magnitude longer than the per iod of vibrat ion of the s t a r , so that the energy r e l e a s e d in each reac t ion is lost as far as the vibra t ions a r e concerned.

We have calculated the diss ipat ion of v ib ra ­tional energy, descr ib ing the core of the s t a r as a F e r m i gas of noninteract ing pa r t i c l e s of density p = 6x 1014 g cm""3. The dissipat ion per g ram per second is

9-^-"k-«c-»Pn\%) 3 ( f ) ^ 2 K P v i r t , « 2 l / > , n ) l V 2 ' A ^ 2 , r c o s ^ , E=E E=E+

where

~E„ + AE

JE„ €ni*E„

2 £ „ + A £ - e , , xde

f€n1+€n2 + -Er\-E„—Ej.

(€ + c +E-E -E - c )de / (€ - e )2de = nx n2 0 n p \ \J% K \ e e

A£ 420

and AE * ( §£ w -§££-3 -E , g) l a v = 2 9 . 3 | a v MeV is the energy available for the react ion n + n~n + p + e + v

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at maximum compress ion , while the other s y m ­bols have the meaning and numer ica l va lues given in re fe rence 2.

Noting that

f 2 7 r c o s 8 ^ = 1.72,

we find for the diss ipat ion the value 8.38 x IO18

x [ | a y ] 8 e r g g - 1 s e c ' 1 , or 1.01 x 1 0 5 2 U a y ] 8 e r g sec""1 for a neutron s t a r of m a s s 1.2 x 1033 g. As a consequence of the high exponential , this diss ipat ion is very l a rge in compar ison with the energy lost in the exponential decay, 2.3 x 10 4 5[£ a v] 2 e r g sec" - 1 , until £ a v i s reduced to ( 2 . 3 x i 0 4 5 / l . 0 l x i 0 5 2 ) 1 / 6 = 0.078; after that it becomes quite negligible. This means that dur ing the f i r s t few days after the outburs t , the reac t ion just desc r ibed will r educe the v i ­bra t ional energy to the value | ( r - f ) f t x o . 0 7 8 2

= 10.1 x lo 4 9 e rg ; subsequently, the mechanism respons ib le for the exponential decay will dom­inate . The agreement between the theore t ica l value 10. l x 1049 e r g and the value 4x io 4 9 e r g deduced above from observat ion could be i m ­proved by taking into account the t e m p e r a t u r e dependence of the diss ipat ion of vibrat ional en­ergy be beta r eac t ions .

The probable mechan ism by which the v ib ra ­tional energy is converted into vis ible light will be desc r ibed in this Let ter only very brief­ly. An outgoing shock wave accompanying each

vibrat ion 8 ' 9 will t r ans fe r energy to the surface of the s t a r . He re , the energy will be d i s s ipa ­ted and emit ted in the form of t he rma l r a d i a ­tion consis t ing of hard x r a y s . These x r ays will be absorbed by the expanding envelope of the supernova and r e - e m i t t e d in the vis ible region. The ra t io of the per iod of vibrat ion of the neutron s t a r (which is l e s s than IO""3

sec) to the half-value t ime t1/2 of the luminos­ity is about 10~"10, the re fore the observed ex ­ponential decay would be accounted for by a s ­suming that the coefficient of reflect ion at the surface of the s t a r differs from unity by about io-10.

1S. Bowyer, E. T. B y r a m , T. A. Chubb, and H. F r i e d ­man, Science 146, 912 (1964).

2A. F inz i , Phys . Rev. 137, B472 (1965). 3 J . N. Bahcall and R. A. Wolf, Phys . Rev. Le t t e r s 14,

343 (1965). 4E. M. Burbidge, G. R. Burbidge, W. A. Fowler , and

F . Hoyle, Rev. Mod. Phys . 29, 547 (1957). 5C. Ber taud, Ann. As t rophys . 27, 548 (1964). 6A. G. W. Cameron , Nature 205, 787 (1965); 206,

1342 (1965). 7 L. D. Landau and E. M. Lifshi tz , Stat is t ical Phys ics

(Pergamon P r e s s , New York, 1958), p . 158. 8 P . Ledoux and T. Wal raven , Handbuch de r Physik

(Spr inger -Ver lag , Ber l in , 1958), Vol. 51 , p . 554. 9S. Rosse land, The Pulsa t ion Theory of Variable

S tars (Clarendon P r e s s , Oxford, England, 1949).

LORENTZ-COVARIANT GRAVITATIONAL ENERGY-MOMENTUM LINKAGES

J . Winicour and L. Tamburino

Aerospace R e s e a r c h Labora to r i e s , W r i g h t - P a t t e r s on Air F o r c e Base , Ohio (Received 29 June 1965)

This Le t te r p r e s e n t s a gravi ta t ional energy-momentum express ion with the t rans format ion p rope r t i e s of a Lorentz f ree vector in a s y m p ­totically flat but radia t ive s p a c e s . This e x p r e s ­sion and its t ransformat ion p rope r t i e s apply to finite reg ions a s well as to the en t i re space .

In the Bondi-van de r B u r g - M e t z n e r 1 fo rmula­tion of the c h a r a c t e r i s t i c ini t ia l -value problem, ce r ta in of the field equations need only be a p ­plied on a world tube. For world tubes of t o ­pology S 2 xE 1 , we have cas t these equations in the form of conservat ion conditions

*{(WVB^r dS (1)

where

^Bi 'kf (2)

-«faiVV*te".««v- <3)

Here Zx and S 2 r e p r e s e n t spacel ike s l i ces of the world tube with topologies S2, T is the p o r ­tion of the world tube bounded by Zx and S2 , Tp is the ene rgy-momentum tensor of the coupled f ields, R is the Ricci s c a l a r , and £^

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