ON T H E E M I S S I O N OF N E U T R I N O S AND G R A V I T A T I O N A L

W A V E S IN T H E F O R M A T I O N OF N E U T R O N STARS

Y O J I K O N D O

Johnson Space Center, Houston, TX, U.S.A.

G E O R G E E. M c C L U S K E Y , JR.

Lehigh University, Bethlehem, PA, U.S.A.

and

S A B A T I N O S O F I A

National Aeronautics and Space Administration, Washington, DC, U.S.A.

(Received 14 March, 1977)

Abstract. The energetics involved in the formation of neutron stars in close binaries as a result of supernova explosions are considered. The gravitational binding energy of the neutron star mus t find proper outlets. The mass ejection and cosmic ray particles can carry away only a small fraction (up to a few per cent) of this energy. Most of the binding energy goes into rotational kinetic energy, gravitational radiation and neutrino emissions. A scenario is considered in which most of the gravitational binding energy goes into rotational kinetic energy and is, ultimately, radiated away as gravitational waves.

1. Introduction

Despite much theoretical and observational research on the physical mechanisms involved in supernova explosions, uncertainties remain regarding a number of important factors. It is generally accepted that the stellar remnant of a supernova explosion is a neutron star or a black hole, although there are some indications that possibly no such remnant at all is left behind in a small number of cases. The existence of neutron stars is nearly certain since the interpretation of pulsars as rotating magnetic neutron stars and the similar identification of the X-ray pulsars in close binaries. With regard to black holes the situation is not as definite although Einstein's general relativity theory predicts their existence. The X-ray binary Cyg X-1 is a primary candidate for the observational 'discovery' of a black hole. Unfortunately, black holes can only be' discovered' by showing that an ' observed' massive object can be nothing but a black hole - a difficult task!

The effects of one component of a close binary system undergoing a supernova explosion on the orbital parameters of the system have been examined by a number of investigators: Boersma (1961) discussed the problem from the view point of mass loss from the system; Sofia (1967) treated it in terms of the total energy involved; Colgate (1970) discussed the effects of impact and ablation on the non-exploding component; McCluskey and Kondo (1971) parametrized the mass loss, impact and ablation effects; Sutantyo (1974) followed evolutionary paths leading to supernova explosions to study

Astrophysics and Space Science 51 (1977) 187-196. All Rights Reserved Copyright �9 1977 by D. Reidel Publishing Company, Dordrecht-Holland

188 Y. KONDO ET AL.

the formation of X-ray binaries; a detailed study was made by Wheeler et al. (1975); several investigations of evolutionary scenarios for X-ray binaries are available, including those by van den Heuvel and de Loore (1973), Tananbaum and Tucker (1974) and Kraft (1975).

The objective of this paper is to discuss the various modes in which the energy, generated by the formation of a neutron star or black hole by the collapse of a stellar core, can appear and to place physical constraints on this energy distribution using the fact that neutron stars appear to have been found in at least six X-ray binaries and one binary pu!sar.

2. Discussion

Since the detailed circumstances of supernova explosions are not known, we take the view that they must be reconcilable with the existence of the X-ray binaries Cen X-3, Cyg X-l, Her X-l, SMC X-l, Vel X-l, 3U 1700-37, and with the pulsar binary PSR 1913+ 16. At present only Cen X-3, Her X-l, SMC X-l, Vel X-1 and PSR 1913+ 16 have shown pulsar properties. Cen X-3, Her X- 1 and SMC X-1 have essentially circular orbits while Vel X-1 has an eccentricity of about 0.13 (Rappaport et al., 1976) and PSR 1913 + 16 has an eccentricity of about 0.62 (Hulse and Taylor, 1975). Since a supernova explosion in a binary system either disrupts the system or produces an eccentric orbit, the orbits of Cen X-3, Her X-1 and SMC X-1 must have been circularized by tidal interaction and/or through encounters with the extended atmosphere of the optical companion at periastron passage. The moderate eccentricity of the orbit of Vel X-1 must be explained by any successful theory of orbital circularization.

The relatively large eccentricity of PSR 1913 + 16 poses an interesting problem. The orbital period of about 7.5 h is far shorter than that of any known X-ray binary. Hulse (1975) measured the rate of periastron precession to be 4.~24 per year. This precession may be due to tidal distortion of the companion by the pulsar, rotational distortiGn'of a rapidly rotating companion and the general relativistic periastron precession. As shown by Masters and Roberts (1975), a much greater effect would be observed if the companion were a hydrogen Main-Sequence star. They conclude that it is very likely that the companion is a helium Main-Sequence star, white dwarf, neutron star, or black hole. Roberts et al. (1975) show that the total mass of the system cannot be greater than about 2.85 solar masses (| or tess than 1.00 | Balbus and Brecher (1976) estimate that the rate of change of the eccentricity due to tidal forces if the companion is a white dwarf is currently about 10-15:per second. If the age of the pulsar is about 104 yr (Van Horn et al., 1975), the orbit cannot yet have undergone significant circularization.

We now consider the energy generated by a supernova caused by the collapse of the core to a neutron star and the manner in which this energy is distributed in various modes. The energy source is primarily the gravitational binding energy released by the collapsing core. Since the core radius decreases by a factor of from 100 to 1000 and the binding energy decreases by the same factor, the initial binding energy is only 0.1-1.0 )s

ON THE EMISSION OF NEUTRINOS AND GRAVITATIONAL WAVES 189

of the energy released and will be neglected.

The binding energy of a model neutron star depends on the adopted equation of state although this dependence is not critical to our discussion. The models of Bowers et al. (1976) give a binding energy of 5.6 x 105z erg for a mass of 0.6 Mo , 3.1 x 1053 erg for a mass of 1.5 M o and 5.4 x 1053 erg for a mass of 2.0 M o . As pointed out by Sofia (1967), this energy must be accounted for as the energy dissipated at the time of the supernova explosion. Burbidge et al. (1957) noted that a dynamical collapse would lead to a thermonuclear detonation in the outer parts of the core composed primarily of carbon and oxygen. Such a detonation could possibly disperse the entire star (Arnett, 1969; Wheeler and Hansen, 1971). Colgate (1975) discusses various possibi- lities that would still lead to neutron star formation, and noted that only a small fraction of supernovae can lead to total disintegration since otherwise the heavy element abundance in the interstellar medium would have to be much larger than is observed. In this paper, we shall not consider carbon detonation supernovae, especially since we are considering those cases that lead to formation of neutron stars.

There are a number of ways in which the available energy of about 1053 erg can be dispersed. These include the kinetic energy of the ejected mass, photon luminosity, cosmic rays, pulsational energy, thermal energy, rotational kinetic energy, magnetic energy, neutrinos and gravitational waves.

If we assume an ejection velocity of 2 x 109 cm s- 1, the kinetic energy of the ejecta is 4 x 1051 erg per solar mass. Thus while it is conceivable that a supernova might eject 20-30 solar masses or that several solar masses might be ejected at 5 x 109 cm s- 1 thus allowing the kinetic energy of the supernova shell to equal that of the binding energy of the neutron star formed in the event, it is by no means certain that such phenomenon can or do occur and, as wi!l be shown later, it should also be emphasized that the aforementioned ejection velocity is extreme and atypical; such a velocity has not been- reported observationally.

The photon luminosity is typically (Colgate, 1975) about 4 x 1049 erg and is consequently negligible compared with the gravitational binding energy released. Colgate (1975) estimates that 2 x 1050 erg in cosmic rays may be generated, again a

small fraction of the available energy. When a neutron star forms by core collapse, it is probable that some energy will be stored in oscillations, at least initially. Cameron (1967) calculated models with initial vibrational energies from 1049 to about 6 x 1050 erg. Initial temperatures in a neutron star may reach 1011-1012 K yielding

thermal energies of 105~ erg.

The collapse of rotating, magnetic stellar cores and the associated magnetohydrody- namic phenomena has been studied by Meier et al. (1976). They conclude that the collapsing core may undergo magnetohydrodynamic disruption with sufficient energy release to power the supernova or would undergo a buoyancy instability and eject matter into the envelope, but with insufficient energy to eject the envelope. The first occurs when the stellar core is initially rotating rapidly, while the latter occurs when it is rotating slowly.

190 Y. KONDO ET AL.

Consider a ' typical ' stellar core of mass M~ and radius R 0. For a homogeneous,

rigidly rotating sphere, the rotational angular momentum is L = IW, where I is the

moment of inertia and W is the angular rotational velocity. I f the rotating core has mass Mo and radius Ro, the moment of inertia is I 0 - 2 2 -~M~Ro. The rotational kinetic energy is ER=�89 W2 or ER=O.2MoR~W 2. Conservation of angular momentum

allows us to calculate E R for the core as it collapses. Assume that the neutron star

radius is RN and that its mass is the same as the initial core. Then the change in rotational kinetic energy, AER may be expressed as

~E~ = M o ( n o W o ) 2 ( k 2 - 1), (1)

where k -= Ro/RN.

The decrease in gravitational potential energy AE~ is given by

M z AE o = - G ~o (k - 1). (2)

Then, the absolute value of AEa, can be found from the equation

AER 2 3 w~n~ (k + 1). (3) AE~ GMr

Since the pre-collapse rotational kinetic energy and gravitational binding energy are almost insignificantly small compared with the post-collapse values, the above ratio

may be approximated by the ratio of the total rotational kinetic energy and the total gravitational binding energy of the newly-formed neutron star.

The pre-collapse core may be of white dwarf size or may be an iron core with a much smaller radius. The critical rotational angular velocity, W~, at which centrifugal effect

balances gravity, neglecting relativistic corrections, is given by

(GM'~ 1/2 w o = \ R 3 1 �9

For a 1.50 M o neutron star with a radius of 12.31 km, W c = 1.036 x 104. The rotational kinetic energy would be 9.76 x 1052 erg and the binding energy is 2.35 x 1053 erg. Thus,

nearly 42 }; of the binding energy could go into rotational kinetic energy. Such a neutron star would be sufficiently rotationally distorted to emit gravitational waves at the expense of this rotational kinetic energy. According to Ostriker and Gunn (1969), if the Crab Nebula pulsar was initially rotating with an angular velocity of We= 104, the initial gravitational radiation would have been about 3 x 1048 erg s-1. Most of the

initial rotational kinetic energy is lost in the form of gravitational waves in the first year

after the supernova explosion. If angular momentum is conserved during the collapse, a neutron star of 1.5 M o

rotating at We= 104, which had either an iron core or white dwarf progenitor, would

ON THE EMISSION OF NEUTRINOS AND GRAVITATIONAL WAVES 191

have been rotating with a period of about 4 s or 600 s, respectively. Fragmentation would occur if the initial rate of rotation were much higher than this. If we assume conservation of angular momentum from the Main Sequence stage through core collapse, very moderate initial rotational velocities are required to lead to rotational

instabilities (Sofia, 1971). Meier et al. (1976) found that if the initial rotational kinetic energy is greater than

10 -4 of the initial gravitational binding energy, the core will undergo buoyancy instability as it collapses and mixing with the envelope will occur. If in addition the initial magnetic energy is greater than about 10-5 of the initial gravitational binding energy, magnetohydrodynamic explosions may occur possibly disrupting the star.

Initial magnetic field energies are much lower than 10- 5 EG and unless enhanced by more than simple compression - e.g., conversion of rotational energy to magnetic energy - the magnetic energy of a neutron star is negligible compared to the gravitational binding energy released.

Neutrino emission provides an effective means of removing some of the gravi- tational binding energy released out of the corel In fact, the deposition of neutrino momentum onto the envelope in sufficient amounts to eject the envelope is of critical importance to the supernova phenomenon. Wilson (1971) found that 1.6-7.8 • 1052 erg will be emitted in the form of neutrinos during core collapse of stars with

about 1-4 M o . Since the loss of gravitational binding energy must be explained, it has been suggested that (e.g., Colgate, 1975) nearly all of it is emitted in the form of neutrinos.

Finally, the gravitational energy release may lead to gravitational radiation. If the core collapse were strictly spherically symmetric, no gravitational radiation would be emitted. Rapid rotation, strong magnetic fields and non-linear pulsations can distort the collapsing star significantly. As much as 1053 or more erg in gravitational waves

might be emitted (Misner et aI., 1973). If core disruption occurs, large amounts of gravitational radiation could be emitted causing the core to reimplode, particularly as gravitational radiation can carry away angular momentum. Friedman and Schutz (1975) estimate that a neutron star rotating above its critical angular velocity will spin- down due to the emission of gravitational waves in seconds or less.

Fowler (1964) suggested that energy transfer via gravitational radiation might lead to a polar explosion thus producing the typical radio galaxies with extended radio sources along the opposing polar axes. Cooperstock (1967), in a detailed analysis of energy transfer via gravitational radiation in Fowler's quasistellar source model, concluded that gravitational radiation can impart energy to an envelope although in the Fowler model this would tend to give equatorial, not polar ejection.

Ross (1975) has studied the interaction of gravitational waves with dense matter. He concludes that high-density matter absorbs much more gravitational radiation than low-density matter. The maximum absorption occurs at frequencies less than about 1000 Hz and can reach 1 ~ , 6~o, 13~0, and 28~0 for densities of 101~ 1012, 1013 and 10 ~4 g c m - 3 respectively.

192 Y. KONDO ET AL.

This could be of some importance as the deposition of energy to the envelope of a

collapsing core by gravitational waves could aid in blowing off the envelope and in

giving rise to the supernova phenomenon. Detailed calculations are required to test the efficiency of this mechanism in an actual collapse.

The gravitational binding energy of a binary system is given by

GM1M2 I E b l = - - ,

2a

where M 1 , M 2 a r e the stellar masses and a is the semi-major axis. For stars other than neutron stars, this orbital binding energy will be less than about 1051 erg. As discussed

by Sofia (1967) and McCluskey and Kondo (1971) we must explain how a supernova releasing 105a 1054 erg of energy can occur in a close binary system and yet not disrupt

the system. As noted by McCluskey and Kondo (1971), if the less massive component in a binary system becomes a supernova, the system may remain bound in many cases assuming that the momentum, kinetic energy and amount of mass ejected are not much

larger than the typical observed values. I f the more massive component explodes, the

system is disrupted unless the explosion is relatively mild. The various investigations into the effects of a supernova explosion on a close binary

system all lead to the conclusion that the kinetic energy of the ejected matter cannot exceed about 1052 erg without disrupting the system in essentially all cases. Thus, the

existence of neutron stars in close binaries is not only consistent with the observational

determination of the kinetic energy of supernova ejecta but requires that in the events in which they were created the ejecta energy cannot have exceeded 10 ~1~- 1052 erg. The

earlier discussion then leads to the conclusion that the bulk, more than 90 ~ , of the energy released during core collapse must appear as rotat ional kinetic energy,

neutrinos and gravitational waves. Let us apply these considerations to several X-ray

sources and the pulsar binary 1913+ 16. It should be noted that in the X-ray binaries Cen X-3, SMC X-l , Vela X-1 and

probably 3 U 1700-17 the X-ray component is much less massive than the massive O - or B-type optical component. A typical scenario for the evolution of a detached close binary into a system in which a supernova occurs leads to a pre-supernova component

mass of 4 M o and an optical companion mass of 15 M o with an orbital period of 1 53 days (Blumenthal and Tucker, 1974). The supernova then ejects about 3 M o and

becomes a 1 M o neutron s t a r The results of McCluskey and Kondo (1971) then show that if Vej were greater than about 21 000 km s -1, the system would have been

disrupted. Thus, the maximum kinetic energy which the ejecta could have had is 1.3 x 1052 erg while the binding energy of the neutron star is about 1.4 x l 0 s3 erg.

The components of the X-ray binary SMC X- 1 have masses of about 20 M o for the optical component and 2 M o for the X-ray star (Primini et al., 1976). The orbit is circular with a period of 3.89 days. Then the semi-major axis of the orbit of is 29.15 R o . Let us assume that the pre-supernova system contained a 20 M o Main Sequence star of radius 8.5 R o and a companion about to explode and having a mass of 4 M o.

ON THE EMISSION OF NEUTRINOS AND GRAVITATIONAL WAVES 193

Adopting the model neutron stars of Bowers et al. (1976), the binding energy of the neutron star is 5.33 x 1053 erg. Use of Equations (1)-(7) of McCluskey and Kondo

(1971) shows that if 2 M e is ejected, the maximum kinetic energy which the ejected matter can have without disrupting the binary system is 1.15 x 1052 erg.

Consider the pulsar binary PSR 1913 + 16. Smarr and Blandford (1976) give a mass of 1.42 M e for the unexploding component, a mass of 1.42 M e for the pulsar. This means that the disruption of the system is determined only by the amount of mass ejected irrespective of its velocity. The maximum initial mass of the pre-supernova component would be 4.26 M e . If we adopt 3 • 109 cm s- j as an upper limit for the ejection velocity, the maximum kinetic energy of the ejecta would be 3.8 • 1052 erg.

In the case of Her X-1 we adopt the values given by Middleditch and Nelson (1976). The optical component has a mass of 1.30 M e and the X-ray component has a mass of 2.18 M e . Assuming a mass of 3.0 M e for the pre-supernova component, a radius of 1.36 R e for the unexploding star and an initial semi-major axis of 6 R e , we find a maximum kinetic energy for the ejecta of 4 • 1051 erg. We may conclude that the ejecta in these systems could not have carried away more than 10 % of the binding energy of the neutron star.

Consequently, as the earlier discussion implies, almost all of the energy released must be in the form of rotational kinetic energy, neutrinos and gravitational waves.

3. Conclusions

We have seen that the binding energy of a neutron star, which must find outlets in its formation, is mostly converted into kinetic energy of rotation, arising from the conservation of angular momentum, or emitted as neutrinos and/or gravitational waves. The energy release through mass ejection and cosmic rays is believed to amount to only a small fraction (up to a few percent) of the total energy involved.

Is it possible to determine which of the three processes, i.e., rotational kinetic energy, neutrino emission and gravitational radiation, is likely to dominate ? According to the recent results for the evolution of a rotating star of 7 M o (Endal and Sofia, 1976) assuming a modest redistribution of the rotation in the stellar interior, the angular rate of rotation of the core reaches a value corresponding to the critical initial rotation rate immediately prior to the on-set of the carbon burning (the critical initial rotation rate is that which leads to balancing of the centrifugal effect and the gravitational force at the surface of the newly formed neutron star.) If the rotation rate increases beyond this value, as is anticipated in the above-mentioned work, then the collapsing core cannot become a neutron star in its entirety, at least not in one step.

The following scenario is conceivable if the initial rotation velocity of the collapsing core exceeds the critical limit. Such a limit is on the order of 10- 2 (rad s- 1 ). During the collapse when the centrifugal effect, due to the increasing rotation, matches the gravitational force at the surface of the core, the inner portion of the core that can complete the collapse without violating the critical rotation limit becomes a neutron

194 V. KONDO ET AL.

star, possibly leaving behind a significant portion of the core mass as a rapidly rotating 'disk' . Part of this disk may gradually be accreted to the neutron star as the rotation of the neutron star decreases through gravitational radiation; in such a process, the angular momentum of the infalling matter will be transferred to the outer portion of the disk ejecting the matter away from the neutron star and the disk. After such an accretion process is completed, the mass of the neutron star may have increased significantly f rom its initial value.

If such a scenario holds true, then most of the binding energy will first become the rotational kinetic energy of the neutron star and the disk. During this first phase of the collapse, some energy will be dissipated as gravitational waves since the collapse of the rotating core is probably asymmetric. The neutrino emission rate will likely be lower than what is estimated for the formation of a neutron star of a comparable mass because the temperature in the collapsing core will not become as high as some theoretical estimates. The binding energy, which has been first converted into the rotational kinetic energy will then be dissipated through gravitational radiation. In a nutshell, the ultimate outlet of the majority of the binding energy is gravitational

radiation.

The validity of this scenario clearly depends on certain conditions being fulfilled as stipulated in the text. It is not implied here that the above scenario is inevitable. Rather, it is our attempt to present a sequence of events that may accompany the formation of a neutron star based on rather limited current understanding of the phenomenon. It is hoped that future observations will enable us to discern among various possibilities. In particular, development of a high precision gravitational wave detector might enable confirmation or rejection of the scenario described above.

Acknowledgement

We acknowledge, with pleasure, comments by Drs J. L. Modisette and W. M. Sparks. We also wish to thank Professor Z. Kopal for his helpful advice.

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Addendum

D r W. S u t a n t y o k i n d l y p o i n t e d ou t to us an i n a d v e r t e n t e r ror in the ar t icle by

M c C l u s k e y a n d K o n d o (1971) re ferenced in the cu r r en t paper . The E q u a t i o n (6) in tha t pape r shou ld read

a A + B

a0 2 A + B - C - 1 (6)

As a result , the Tab le s Ib a n d Ic also change s l ight ly; the cor rec ted Tab le s Ib a n d Ic are

g iven . These co r rec t ions do n o t al ter o u r c o n c l u s i ons in tha t article. I n fact, as m a y be

seen, the cor rec ted resul ts are sl ightly m o r e in favor o f the b i n a r y not b r e a k i n g u p af ter the s u p e r n o v a explos ion .