24 December 1965, Volume 150, Number 3704 SCI:ENCE:
squLeezes the material in its core into a smaller and smialler Vxolume and to higher and higher temperatllres. The sUbsequent fate of the star de- penids Lipon hoxy massixve it is. For a Gravitational Collapse and tbie star of less than 1 .2 solai- masses [the 'ChandrasekhCar limlit" (1)], quLasi- Death of a Stvir static contraction is halted by rising internal pressure when a central den- sitx ol' I(),; gr(Irams per clbic centi- Relativity and nuclear theory together predict the fLite mleter is reachedl. The stair then settles down into its final resting, state. a of a star which has burned all its nuclear fu e. *x;white dwxxarf' configluration (2). On the other haind. in a star ot' more than S. 'I-hor 1 .2 solar mi.asses. the stellair core is Kip squLeezed to suIch high densities dUring quLsi-static contraction that catastroph- ic nuclea;r processes occLur before rising internlail pressuLre cain halt the contrac- \\Nhat is the fate of a star wxhen it Adclitional obserx ational tests ot' these tion. These processes caIse the star to has consumned all its nuclear fuel ancl predictions. bileh extr(emelxy diflicult explodce with such violence that its -an no longer maintain the nuclear atnd perhLaps imlpossible with presenit- I1,nminosity approaches that of a g.alaxv reactions which have sustained it dav technolo,% x c uld provide for a periodl of Ihoult I O) days since its birth'? This is a question valuLable tests of the zravit.ation and IsuLpernova explosion (3)1- Wshich observational astrononmy has nucleIar theory upon xswhich the pre- The precise physical processes wxhich cione but little to answxer. The time dictions rest. minitite andtaccomparny a supernova reqLlired for a star to COnISuImle its It should be emnphasi2'ed at the oLut- explosion are probablx different in nLuclear fuel is so long (manvn billions set that theoretical sttLudies o t' the stars of' between 1.2 and abouLt 5 solar of years in nmost cases) thtat only a deaths of st(ars are still far froml coIll- masses from those in more mnassive few stars die in our galasxy per cen- plete. Such complicatiions as stellar stars. I shall first describe the mechan- tury; and the evolution of a star froml rotation. deviations fromn Ispherical svm- ism by which the less mlassive super- the end point of thermonuclear burn- metrv. and stellar maginetic fields are novae are prodliced by tracing out the inog to its final dead state is so rapid largely uinstildied aind \Kill he irnored dceth of a representative 2-solar-niass that its death throes are observable for here. Rough estimaItes indicate that star, as preclictecd hy theor-etical cal- only! a few Nrears. these phenomena, when not too pro- culations. Then I shall tLurn yiv atten- Despite the paucity of' observation- notlnced, will probably have little ef- tion to the miorem-lassive stUpernovae. al data, theoreticians are now able to fect on the quLlitative FPictul-e oLutlined A stair of 2 solar i.masses realches discuss the deaths of stars in ever in- here; for the mlost palrt, onIN qUantita- the enid point of thermonucleCar- bUrning creasing detail and w,vith a fair degree tive details are expectecLi to change as wvhen it has converted all of the hvdro- of' certainty, thanks to recent advances theoretical stuLdies becorine mlore reills- gen in its inter-ior to Fe';. the milost in computer technology andt in our tic. tightly houLnd of all nuclei. At this to-derstanding of the physics of the point the star begins to contract quaLsi- atomic nucleus. of elementary par- statically, compressing the matter in its ticles, and of Einstein's geomletrical Evolution to the D)ead State center into a smaller and smiialler theory of gravitation (general relativity). VOl LIme. The puLrpose of this article is to re- WNVhen, after hundreds of millions or Now, we know from elementary viewv the different tv pes of death and billions of years of rnormlal nLuclear quLanttlm theory that the smaller the the final resting, states of various types bLirning, a star has exhatLsted its supply region to which we confine a particle, of stars, as deduced theoretically, and of nLlclear fuel, it has only one wvay the larger the particle's zero-point ki- to point out those few direct astro- to replenish the therma energy that it netic energy beconmes. In particuilar, nomical observations wxhich have bear- is radiating: quLasi-static graxvitaitional when a particle is confined to a re- ing on the theoretical predictions. contraction. As it con tracts, the star g,ion of the order of its Comiipton Nvaye- converts its oyravitation;.1L potential en- length. its zero-point energy becomles The auLthor is a National Science Fotunidation post- into -v rest doctoral fellow at Palmer Physical t.aboratory, crgy thermiial enerr and radiates of the order of its mass; and be- Priniceton University. Princeton, New Jersey. it away, and at the Same timile it yond this point the zero-point energy 24 DECEMBER 1965 1671 collapsing core is suddenly faced with a huge central pressure, which calls its collapse to a halt and sends a shock wave propagating outward through it. In this core shock front the huge ki- netic energy of collapse is converted+ into heat, and temperatures of over 10 billion degrees are reached. At such high temperatures and densities, ele- mentary particle transformations pro- ceed at a rapid rate, and the heat pro- C() lo' duced in the core shock front is con- :320% 10 5% verted into high-energy neutrinos. The < E \ 1 \ CORE SHOCK WAVE l mean free path of the neutrinos is less than 100 meters under these extreme CK00 t - t- .98 ME CORE conditions. Hence, instead of escaping freely from the star, the neutrinos dif-, fuse outward, depositing the energy re- leased by the core's collapse in the en- NEUTRINO DEPOSITION ON NEUTRINO DEPOSITION OFF velope of the star and thereby raising the envelope to temperatures as high as 200 billion degrees. At these enor- 1.84 1.85 1.86 1.87 1.88 1.89 1.90 mous temperatures explosive nuclearv burning is initiated in the envelope, TIME (sec) with a consequent release of additional Fig. 1. The dynamics of the collapse and reexplosion of a star of 2 solar masses. thermal energy. Because of the huge Each solid curve represents the radius as a function of time for a spherical shell of thermal energies generated by neutrino matter inside which a certain fraction of the star's mass lies. Each curve is labeled deposition and by nuclear burning, the by this "mass-fraction." [Based on calculations by Colgate and White (4)] envelope of the star suddenly becomes gravitationally unbound. An exploding shock wave forms and blows the en- rises very rapidly with additional com- Colgate and White (4) have used velope away from the core with speeds pression. Since the Compton wave- computers at Lawrence Radiation Lab- approaching the speed of light; and the length of an electron is 100,000 times oratory to study the details of this huge thermal energies of the expand-* that of an Fe56 nucleus, electrons will catastrophic collapse (5). The details ing envelope are converted to radia- resist being squeezed by the contracting of the collapse of the 2-solar-mass tion so intense that the luminosity of star much sooner than will iron nu- star, as worked out by Colgate and the exploding star approaches that of clei. As the contraction proceeds, our White, are shown in Fig. 1 and are a galaxy. In addition, nuclear par- 2-solar-mass star pushes the zero-point described below. ticles are accelerated in the exploding energies of its electrons higher and shock wave in such numbers and to' higher, while the zero-point energies such high velocities as to account for of its iron nuclei remain negligible. Supernova Explosions a significant fraction of the galactic Eventually a point is reached at cosmic rays observed at the earth. which the sum of the rest mass of an Catastrophic collapse is initiated by Stars of more than about 5 solar Fe56 nucleus plus the rest mass of an electron capture when the center of masses undergo supernova explosions electron plus the electron's rising zero- the slowly contracting star reaches a similar to that described above. How- point energy exceeds the rest mass of density of about 1011 grams per cubic ever, the collapse of a massive star's a Mn56 nucleus. No longer is a free centimeter. Within a fraction of a sec- core is initiated not by electron cap- Mn56 nucleus unstable against beta ond after initiation of collapse, nearly ture, but by the sudden breakup of decay into electron plus Fe56; rather, all the electrons and Fe56 nuclei in- its Fe56 nuclei into He4 nuclei. This electrons and Fe56 nuclei are unstable side the star's core have been trans- breakup occurs when the temperature# against combining to form Mn56 and formed into highly neutron-rich nuclei in the contracting core has become so other neutron-rich nuclei (electron cap- and free neutrons, and the core is in high that there are photons present ture); and such combination begins to free fall. At 1.86 seconds after initia- with sufficient energy to disintegrate occur. At this point in the evolution tion the core has acquired a kinetic the Fe56 nuclei. The photodisintegra- of the star the high-zero-point-energy energy of collapse equivalent to a siz- tion of Fe56 reduces the temperature electrons are providing essentially all able fraction of its rest mass, and the and hence also the pressure in the the pressure that sustains the weight neutrons in the core have been com- core of the massive star, and there- of the star. When electrons begin to pressed into regions of the order of by initiates collapse. Once collapse has disappear by combining with Fe56 nu- their Compton wavelength (density been initiated, the evolution of a mas- clei, the sustaining pressure begins to -l014 g/cm3). At this point the sive supernova is the same as that of disappear; and the star, which can no zero-point kinetic energy of the neu- supernovae of less than 5 solar masses longer support itself, begins to collapse trons-and with it their zero-point with one possible exception: Theoreti- catastrophically. pressure-begins to rise rapidly. The cal analyses by W. A. Fowler and 1672 SCIENCE, VOL. 150 F. Hoyle (with which Colgate and White disagree) (6) indicate this, that in a massive star the explosion may be caused, not by the gravitational po- tential energy released in the core's C"J cbllapse, but by nuclear energy re- leased in the detonation of 016 dur- IC104td - f 9i ing the collapse of the star's mantle. - Detailed observations of supernovae l-6 (3) are in all respects compatible with the above theoretical description of them but do not yet rule out other explanations. A more rigorous test of -a: io-8°;1010 this description will come from a study of the neutrinos emitted by superno- U) vae-if and when it is technologically possible to detect such neutrinos. So much for the details of the evolu- a.-14 tion of a star into its final dead state. 10 Let us now turn to the nature of the final state-to the fate of the core 10 102 104 106 108 1Q10 1012 1014 1016 1Q18 lO20 1022 left behind in a supernova explosion 1nd to the final forms of stars which, DENSITY (g/cm3) because their masses are less than the Fig. 2. The Harrison-Wheeler equation of state for matter at the end point of thermo- Chandrasekhar limit, never become nuclear evolution. supernovae. pressure-density relation shown by the hyperons become stable (- 1015 g/ flarrison-Wheeler Equation curve between c and d in Fig. 2. Be- cm3), but totally unknown beyond tween points d and e the clustering of there. Fortunately, the form of the In order to study the final states of electrons about Fe5" nuclei is negli- equation of state in the region of den- stars, we need an equation of state for gible, and the electrons form a rel- sity >: 1013- grams per cubic centi- the kind of matter from which dead ativistically degenerate Fermi gas. As meter is not crucial to my discussion. stars are made: matter at the end the Fermi electron gas and Fe.6 nuclei We can assume for simplicity that in point of thermonuclear evolution. The are compressed still further (region e the limit of extreme densities equation of state for such matter was to f), the Fermi energy of the elec- pressure = (1/3) x density x c2, calculated in 1958 by B. K. Harrison trons plus the mass of an Fe56 nu- and J. A. Wheeler (7, 8) from a knowl- cleus becomes greater than the mass where c is the speed of light. Large edge of the physics of the nucleus. of a Mn56 nucleus. Consequently, elec- but physically reasonable departures 'their calculations were carried to an trons are squeezed onto the Fe56 nu- from this limiting form of the equa- accuracy as great as present under- clei to form Mn5" and other neutron- tion of state have only small effects on standing of high-density nuclear phys- rich nuclei (electron capture). As com- the final states of cold, dead stars. ics allows. The resultant "Harrison- pression becomes greater and greater, Wheeler equation of state" is plotted the configuration of lowest energy is in Fig. 2. pushed further and further away from Uniform Density Approximation At low densities (below point b of Fe') toward nuclei which are more Fig. 2) matter at the end point of ther- and more neutron-rich. Eventually neu- The final, dead state of any star monuclear evolution is in the form of trons become so numerous that they will be a spherical configuration of Fe56, and its pressure is provided by begin to drip off the nuclei (point f), matter at or near the end point of ther- solid-state forces. As the iron is com- and the material is gradually converted monuclear evolution. How will this pressed to higher densities (region b from a mass of neutron-rich nuclei matter be distributed inside the star? to c), solid-state forces begin to con- to a dense Fermi gas which is 8/10 What will be the star's central den- tribute less to the pressure than do neutrons, 1/10 protons, and 1/ 10 sity? its radius? its mass? In answering orbital electrons of the iron nuclei, electrons (point g and above). At these questions it is useful to introduce which resist being compressed. At point still higher densities the matter at the the concept of the total number, A, of c solid-state forces are negligible, and end point of thermonuclear evolution baryons contained in a star. The orbital electrons, which provide all consists of a mixture of neutrons, Baryons are heavy nuclear particles the pressure, constitute a "degenerate protons, electrons, lambda hyperons, -neutrons, protons, lambda hyperons, Fermi gas," except that they tend to and other massive particles which, and so on. Although baryons can be cluster about the iron nuclei. Feyn- although highly unstable in the labora- changed from one form to another man, Metropolis, and Teller (9) have tory, are completely stable at extreme (for example, electron + proton -- used the Fermi-Thomas statistical densities. neutron + neutrino), the total num- model of the atom to correct for this The equation of state is quite well ber of baryons is conserved in any clustering effect, thereby obtaining the known up to the point at which heavy elementary particle transformation. 24 DECEMBER 1965 1673 c'J tion, its (negative) gravitational energy -Ii l I I l I l l I its in- Z -3 initially rises more rapidly than ternal energy of compression. Hence, 101 total mass-energy initially decreases. For stars with A/Ao w 1.2 (for ex- ample 1.7 in Fig. 3), general relativis- 10 tic, nonlinear growth of gravitational A/A =0O04 energy begins so soon that compres- sional energy can never take over. U) - Hence, for such stars, increasing com- -10 paction reduces the total mass-energy monotonically to zero. However, if A A=0.6 A/Ao z 1.2, rapidly rising pressure o -tO causes compressional energy to become more important than gravitational en- 01 ergy before general relativity comes into play. Hence, for such stars total z - mass-energy reaches a minimum and 0 M0 then rises with increasing compres- z sion, until the nonlinear gravitational -lo effects of Einstein's general relativity take over and cause a drop of the.. total mass-energy to zero. At least this is part of the story. Additional compli- DENSITY (g/cm3) cations arise as a result of a "quirk" Fig. 3. Spherical configurations of uniform density for the Harrison-Wheeler equation in the equation of state for cold mat- of state: Negative of the binding energy plotted against density. Aside from an additive ter at the end point of thermonuclear constant, the quantity plotted on the vertical scale is the total mass-energy of the evolution. When a density of -101'2 configuration as measured in units of the mass-energy of the sun. Each curve repre- grams per cubic centimeter is reached, sents a sequence of uniform density configurations containing a fixed number of baryons, A. (AO is the number of baryons in the sun.) The cross-hatched regions neutron drip begins to occur in such are barriers which separate the configuration of stable equilibrium of highest density matter, causing the star's pressure and from gravitational collapse to zero volume. its energy of compression to rise much less slowly with increasing compaction than at lower densities. (See depres- For this reason, the total number of distributed throughout a star. Fortu- sion in the curve for the equation of baryons inside a star is a measure of nately, however, much can be learned state in Fig. 2.) For a star with the amount of matter which the star from a comparison of configurations of A/Ao z 0.4 (for example 0.6 of Fig. contains. By contrast, the mass of a uniform density. 3), negative gravitational energy is ris- star is not a good measure of its mat- A comparison of uniform density ing rapidly enough at this point to- ter content because the mass depends configurations for matter obeying the dominate the diminished compressional upon the state of binding of the Harrison-Wheeler equation of state is energy and cause total mass-energy to baryons. made in Fig. 3. On the vertical scale fall temporarily. However, for A/Ao ' An important result valid both in of Fig. 3 is plotted the negative of the 0.4 (for example, 0.04 of Fig. 3) even the Newtonian theory of gravitation binding energy of each configuration the diminished compressional energy at and in Einstein's theory is that (8, 10) -that is, the difference between the -1012 grams per cubic centimeter' in any cold, static star containing total mass-energy of the configuration dominates gravitational energy, and no a certain number of baryons, the bar- and the mass-energy it would have if temporary drop occurs in the total yons are distributed in such a way its matter were dispersed to infinite mass-energy. as to minimize or maximize the star's dilution-and on the horizontal scale At each minimum in its mass-energy total mass-energy [rest mass-energy is plotted the density of the configura- curve, a star has-in the uniform den-, plus thermal energy-if any-plus in- tion. Each curve represents the uni- sity approximation-a configuration of ternal energy of compression plus form density configurations for a star stable equilibrium, and at each maxi- (negative) gravitational potential ener- containing a particular number of bar- mum it has a configuration of unstable gy]. Hence one way to determine all yons and can be thought of as a "po- equlibrium. Hence, for a star contain- possible final states for a star con- tential energy curve" for that star. ing 0.04 as many baryons as are in taining A baryons is to compare all The qualitative behaviors of the the sun there are two equilibrium con-' conceivable distributions of A baryons mass-energy curves of Fig. 3 are easily figurations: a stable one at the mini- at the end point of thermonuclear evolu- understood in terms of an interplay mum in the mass-energy curve, cor- tion and to pick out those which mini- between negative gravitational poten- responding to a cold white dwarf star; mize or maximize the total mass-ener- tial energy and positive internal ener- and an unstable one at the maximum. gy. This would be a difficult task in gy of compression. Regardless of the If a star at the maximum is distended general, since there are innumerable number of baryons in a star, as the slightly, it will explode; if it is com- ways in which the baryons can be star is compressed from infinite dilu- pressed slightly, it will collapse. To 1674 SCIENCE, VOL. 150 Nhat will it collapse? According to gen- ral relativity theory, it will collapse o a singularity-that is, into zero volume and to infinite density. z "Consider next a star containing 0.6 is many baryons as our sun contains. [t has two stable equilibrium configura- 0 1.0 -ions (minima of Fig. 3): a cold white (I)CD -Y lwarf configuration made of a degener- Co ite electron gas and Fe56 nuclei, and (n) l neutron star configuration made of Co l degenerate neutron gas. There are Co C] :wo unstable equilibrium configurations
Rather, after it has cooled to near- r only one HWW equilibrium configura- zero temperature, the core will col- tion; and different equilibrium configu- Laise catastrophically once again, this mi = w4rp dr. (2) rations have different masses, radii, and time to zero volume. We also con- 0 total numbers of baryons, as well as zlude that a less massive supernova different central densities. .ore or a cold star with mass less The nonrelativistic (Newtonian) form The form of Fig. 4 can be com- than the Chandrasekhar limit can be of the equation of hydrostatic equili- pletely understood on physical grounds induced to collapse to zero volume if brium, Eq. 1, is obtained by taking (8), but here I shall only remark that itis compressed sufficiently. These con- the speed of light, c, to be infinite: the first oscillation in the mass curve qlusions are so startling that we would dp/dr = -Gpm/r2. (3) of Fig. 4 is due to electron capture and like to see them spelled out not only neutron drip in the matter of which the in the approximation of uniform den- At very high densities and pressures stars are composed (see Fig. 2 and sity, but also in the exact theory where (p p/c2 ; 1013 g/cm3) the general associated discussion), while subsequent the variation of density throughout the relativity terms in Eq. 1 cause a mul- oscillations are due to general relativity star is taken into account. tiplicative regeneration of pressure: The (gravitation) effects. gravitational force acting on an ele- By comparing Fig. 3 (uniform den- ment of fluid becomes quadratic in sity approximation) with Fig. 4 (exact Harrison-Wakano-Wheeler its pressure. It is this regeneration of theory of HWW configurations), we Configurations pressure which enables gravitational can conclude the following: A star con- forces to overwhelm the internal pres- taining 0.6 times as many baryons as In the exact theory, the variation of sure of a star in the relativistic re- our sun will settle down into one of density from the center to the surface gime, regardless of how high its pres- two possible stable equilibrium configu- of an equilibrium star is such as to sure may be for a given density, and rations when it dies: a cold white make its total mass-energy a maximum forces excessively dense stars to gravi- dwarf configuration (first minimum in or a minimum. By analytically perform- tationally collapse to zero volume. A/Ao = 0.6 curve of Fig. 3; con- inj this extremization, we obtain the By integrating Eq. 1 coupled with figuration at central density 3 X 106o general relativity equation of hydrosta- Eq. 2 for the mass inside radius r g/cm3 in Fig. 4), or a neutron star tic equilibrium and with the Harrison-Wheeler equa- configuration (second minimum in dp G (p + p/cc) (m + 47rr p/c'). tion of state (Fig. 2), Wakano (7, 8) A/Ao = 0.6 curve of Fig. 3; con- dr r (r-2 Gm/c2) has calculated all possible equilibrium figuration at central density 2 X 105 for cold matter at the in a star a , (1) configurations g/cm3 Fig. 4). Alternatively, Here G is Newton's gravitation con- end point of thermonuclear evolution. with A/Ao = 0.6 might attempt to stant, c is the speed of light, r is Figure 4 shows the masses and radii of assume one of two possible unstable 24 DECEMBER 1965 1675 tions on one side of the peak or valley Table 1. Wheeler's criteria for determining to configurations on the change in stability of the critical mode of the other. radial oscillation at a critical point as cen- To determine whether the critical tral density increases. mode of radial oscillation becomes Behavior of radius Direction of stable or becomes unstable at a particu- at critical point stability chang' lar peak or valley, we consider in Fig. 5 a portion of the curve of mass Maximum mass plotted against total number of baryons Decreases Becomes unstable for the HWW equilibrium configura- Increases Becomes stable tions. To each peak or valley in the Minimum mass
Decreases Becomes stable . NUMBER OF BARYONS- mass curve of Fig. 4 there corresponds a cusp in Fig. 5. Near a given cusp, Increases Becomes unstable Fig. 5. Harrison-Wakano-Wheeler configu- rations: A portion of the curve of mass configurations on the higher branch of plotted against central density (schematic Fig. 5 will be less stable than those only). At each cusp the branch of the on the lower branch; in going from stable until a peak in the mass curve curve corresponding to configurations of the lower to the higher branch the larger radius is labeled "R>," and that of Fig. 4 is reached, all configurations corresponding to configurations of smaller critical acoustical mode of oscillation with central density less than 3 X 108 radius is labeled "R<." goes from stability to instability. Now, grams per cubic centimeter (white the slope of the curve in Fig. 5 is dwarf stars) are stable. At the first the change in mass, dM, of an equilib- maximum of the mass curve the radius equilibrium configurations when it dies: rium configuration when a single bar- of the star is decreasing. Hence (see one of central density 3 X 1012 grams yon, dA, is brought in from infinity Table 1) the first radial mode becomes per cubic centimeter (first maximum and gently deposited on the surface unstable there. We denote this instabili- in A/Ao = 0.6 curve of Fig. 3) or of the star; that is the slope is ty by blackening the lowest oval in one of central density 2 X 1016 grams dM/dA =Mb X ( + P) Fig. 4. At the first minimum the radius per cubic centimeter (second maximum is once again decreasing, so the low- in,, x ( - GM/c2R) est mode becomes stable of Fig. 3). There are no equilibrium Newtonian theory again, and configurations at all for a dead star we lighten the lowest oval of Fig. 4. =Mb x ( - 2GM/c'R)i containing more than - 1.2 times the Between the first minimum and second number of baryons in the sun; such a General relativity theory (4) maximum lies the regime of stable neu- dead star must inevitably collapse to a Here nil, is the mass of one baryon, tron stars. At the second maximum the singularity. P is the gravitational potential at the radius is again decreasing, so the first surface of the star, G is Newton's mode becomes unstable; at the secondj gravitation constant, c is the speed minimum the radius is increasing, so Stability of HWW Configurations of light, and M and R are the total a second mode becomes unstable; and mass and radius of the equilibrium so on. The above comparison of Figs. 3 configuration. Near a critical point It is possible to make this analysis and 4 enables us to conclude that cer- (cusp of Fig. 5, peak or valley of of stability quantitative by means of a tain of the HWW equilibrium con- mass curve in Fig. 4), M changes very variational principle originally formi.i figurations are stable against gravita- little with increasing density, but R lated by Chandrasekhar (11). Meltzer tional collapse or explosion, whereas changes rapidly. Consequently, at a and I (12) have used Chandrasekhar's others are unstable. However, this cusp of Fig. 5 the branch of larger- variational principle to determine the method for studying stability has the radius configurations has the larger values of the frequencies of the low- disadvantage of being nonrigorous (re- slope, dM/dA. If the cusp is a point est three modes of radial oscillation call that Fig. 3 is based upon the uni- of maximum mass, the configurations for the HWW configurations. The re'- form density approximation), and it is with larger radii and larger dM/dA sults of our calculations, shown in Fig. not applicable to all HWW configura- lie on the lower branch of Fig. 5 and 6, are in perfect agreement with tions. To determine precisely which of are thus more stable than the configu- Wheeler's qualitative results. the HWW configurations are stable and rations with smaller radii. If the cusp Thus far I have not discussed sta- which are unstable it is better to use is a point of minimum mass, the con- bility and instability of HWW config'l- completely rigorous and beautifully figurations with larger radii lie on the rations against nonradial perturbations. simple criteria formulated by Wheeler upper branch and are thus less stable It is generally assumed-though I do (8), a description of which I present than the configurations with smaller not know of any proof-that a cold here: radii. star can be dynamically unstable against A star lying at a maximum or a These conclusions are summarized in nonradial perturbations only if its low- minimum in the curve of mass plotted Table 1. Using this table, we can de- est radial mode is also unstable. If against central density must have a termine from Fig. 4 the precise num- this is true, then the analysis of purely zero frequency mode of radial oscilla- ber of unstable modes of radial oscil- radial oscillations is sufficient to reveal tion, for such a star can move back lation for each HWW configuration: the absolute stability or instability of and forth on the peak or valley of A sphere of iron the size of a golf all HWW configurations. According to the mass curve without changing its ball is stable against collapse or ex- this analysis there are only two regioiTs mass. This mode of oscillation changes plosion. Since the lowest mode of of stability: The white dwarf region stability as we pass from configura- radial oscillation cannot become un- (central density less than 3 X 108 1676 SCIENCE, VOL. 150 g/ cm3) and the neutron star region for observing collapse toward a singu- the collapsing star moves through (central density between 3 X 1013 larity are nil. spacetime along the indicated curve, g/cm3 and 6 X 101- g/cm3). The I shall conclude with a brief descrip- while an observer moving around the white dwarf region includes stars con- tion of the dynamics of the gravita- collapsing star in an orbit of fixed 'aining less than 1.2 times the num- tional collapse to zero volume, which radius moves along the indicated hyper- ber of baryons in the sun, while the -according to general relativity theory bola. The collapsing star generates a neutron star region includes stars con- -will be the fate of sufficiently mas- singularity (jagged hyperbola) in space- taining between - 0.25 and - 0.7 sive supernova cores. In Fig. 7, I use time which swallows the star up; and times the number of baryons in the a spacetime diagram to depict the dy- any other object which falls into this sun (13). Any supernova core which namics of the collapse. In this diagram "Schwarzschild singularity" also gets contains more than ~ 1.2 times the time is plotted vertically and radial dis- destroyed there. number of baryons in the sun-and tance is plotted horizontally; angular Suppose that a man on the surface also less massive cores which become directions are suppressed from the di- of the collapsing star sends out a series sufficiently compressed in the dynamics agram since the collapse is assumed of signals (wavy 45-degree lines in Fig. of supernova collapse-must gravita- to be spherically symmetric. The time 7) to the orbiting observer, informing lionally collapse to zero volume. There and distance scales chosen are such him of the progress of the collapse. As are no equilibrium configurations for that radial light rays move along 45- the star gets closer and closer to a such stars. degree lines; but because the gravita- certain critical radius, called its tional field of the star "warps" the "Schwarzschild radius" (intersection of geometry of spacetime, freely moving path of surface of star with dotted Comparison with Observations particles do not move along straight 45-degree line in Fig. 7), the signals, lines in the diagram. The surface of which are sent at regularly spaced in- After a white dwarf star is formed, it cools off slowly (cooling time, several billion years) by radiating away its thermal energy as light. Astronomical C-3t measurements of the masses and radii of radiating white dwarfs are in fairly agreement with the predictions good z 12 - I I I I of Fig. 4 (14) and are in excellent W +1010 agreement with other, more detailed stellar models which take into account eotation and varying chemical composi- tion. .U- +106 Neutron stars are not as easily ob- 4 served as white dwarfs. During the first few thousand years after a neutron star is formed it is so hot that es- 'sentially all its radiation is in the x- d ray region of the spectrum, which is +1.0 -STABLE-. -40- blacked out by the earth's atmosphere -UNSTABLE (15). By the time the star has become P- -I02 cool enough to radiate primarily in the z optical region, its luminosity is so low that the star could not be detected with the 200-inch telescope unless it 0 were within a light-year of the earth. Hence, the only hope for detection of o ilo neutron stars is with rocket- and satel- Lite-borne x-ray telescopes. Recent ob- CENIA DENSITY(/c3 servations (16) with rocket-borne x-ray 102 telescopes have revealed x-ray sources, 1-10 1~0 some of which may be neutron stars. However, the most promising of the CENTRAL DENSITY (g/cm3) sources-a source in the Crab nebula, -Which was formed by a supernova ex- Fig. 6. Frequencies of the lowest three modes of radial oscillation for the Harrison- Wakano-Wheeler configurations. In the stable region the amplitude of radial oscillation plosion observed in A.D. 1054-is now varies sinusoidally [amplitude = (function of radius) x cos wt]; while in the unstable known (17) not to be a neutron star. region the amplitude grows exponentially [amplitude = (function of radius) X eat]. Finally, what are the prospects for In this figure the square of the angular frequency, w, is plotted against central density observing a star as it gravitationally in regimes of stability, and the square of the growth constant, a, is plotted against central density in regions of instability. Note the correlation of points of changing .pollapses to a singularity? Given even stability (black dots) here with maxima and minima in the mass curve of Fig. 4; the most powerful technology conceiv- and note the agreement of the direction of change of stability with the results of able in the next century, the prospects Wheeler's analysis (text, Table 1, and blackened ovals of Fig. 4). 24 DECEMBER 1965 1677 like a rubber band and simultaneously These unaccounted-for factors un- compressed from the sides to infinite doubtedly have considerable effect on density and zero volume. the quantitative results reported here; At least, this is the picture accord- but it is doubtful that, except in ex- ing to classical general relativity theory. treme cases, they can change the quali- But when a density of - 104. grams tative picture. per cubic centimeter is reached, classi- cal general relativity breaks down. To References and Notes follow the collapse beyond this point, 1. S. Chandrasekhar, Astrophys. J. 74, 81 (1931); Monthly Notices Roy. Astron. Soc. one must quantize the gravitational 95, 207 (1935). Theoretical estimates of the field-that is, combine general relativi- value of the Chandrasekhar limit have been frequently revised, but only by small amounts. ty theory with quantum mechanics-a For rapidly rotating white dwarfs the Chan- drasekhar limit is ..1.4, rather than -.1.2, formidable task which, as yet, is far solar masses. from completion. 2. There is evidence that for stars as massive as 2.5 solar masses, nuclear explosions in the late stages of evolution may cast off large amounts of material. In this manner the mass of such a star may be reduced Summary below the Chandrasekhar limit of .-1.2 solar masses, and the star can die the gentle death of a white dwarf. See, for example, When a star has burned all its nu- L. H. Auer and N. J. Woolf, Astrophys. J. 142, 182 (1965). clear fuel, it enters a phase of slow, 3. For summaries of the observed character- Fig. 7. Spacetime diagram illustrating the quasi-static gravitational contraction. If istics of supernovae, see F. Zwicky, Handbuich dynamics of gravitational collapse to zero der Physik (Springer, Berlin, 1958), vol. 51, p. volume. The time and radial coordinates the mass of the star is less than the 766, and G. Gamow, Sci. Am. 181, 18 (Dec. Chandrasekhar limit (- 1.2 solar mass- 1949). are those of Kruskal (18), but for the 4. See S. A. Colgate and R. H. White in purposes of qualitative exposition they can es) this contraction is brought to a Proceedings of the Second Texas Symposium be thought of as the intuitive time and halt by the rising pressure of the on Relativistic Astrophysics (Univ. of Texas Press, Austin, 1966) and references cited radial coordinates of Newtonian theory. star's electrons, which resist being com- there. In independent calculations, W. D. Arnett (unpublished Ph.D. thesis, Yale Uni- pressed into a small volume; and the versity, 1965) has verified the results of star becomes a cold, white dwarf of Colgate and White. Some of the ideas" behind the calculations of Colgate and White tervals, are received by the observer at central density 1106 grams per cubic were discussed earlier by W. Baade and F. more and more widely spaced inter- centimeter and radius - 10,000 kilo- Zwicky, Proc. Nat. A cad. Sci. U.S. 20. 254 (1934); Phys. Rev. 45, 138 (1934); ibid. 46, vals. The observer does not receive a meters. If the star's mass exceeds the 76 (1934); by F. Zwicky, Proc. Astron. Soc. Chandrasekhar limit, con- Pacific 48, 191 (1936); Proc. Nat. Acad. signal emitted just before the Schwarzs- quasi-static Sci. U.S. 25, 338 (1939); by G. Gamow. child radius is reached, until after an traction leads to a state of instability Phys. Rev. 55, 718 (1939); by G. Gamow and E. Schoenberg, ibid. 59, 539 (1941); by P. infinite amount of time has elapsed ac- against gravitational collapse. Collapse Bouvier, Arch. Sci. Geneva 5, 211 (1952); cording to his clocks; and he never re- of the star's core proceeds, with the and by E. M. Burbidge, G. R. Burbidge, W. A. Fowler, F. Hoyle, Rev. Mod. PhYs. ceives a signal emitted after the subsequent release of the energy of col- 29, 547 (1957). Schwarzschild radius is passed. That lapse as neutrinos, which blow away 5. It is an interesting sidelight that without the help of a highly sophisticated computer pro- signal, like the man who sent it, gets the envelope of the star (supernova gram which was originally written and uised caught and out explosion). If the supernova core con- in the development of nuclear weapons the, crushed of existence Colgate-White analysis of the dynamics of in the singularity. tains less than - 1.2 times the number stellar collapse would have been nearly impossible. Hence, to the orbiting observer, the of baryons in the sun and if the core 6. W. A. Fowler and F. Hoyle, Astrophv s. J. collapsing star appears to slow down does not become too compressed dur- 132, 565 (1960); Agtrophys. J. Supp. Ser. 9, 201 (1965); S. A. Colgate and R. H. White, as it approaches its Schwarzschild ra- ing supernova collapse, then the core A strophys. J., in press. dius; light from the star becomes more will settle into the form of a cold, 7. B. K. Harrison, M. Wakano, J. A. Wheeler, in Onzieme Conseil de Physique Solvay,. and more red-shifted; and clocks on white dwarf star (central density - La structure et l'evolution de l'univers (Stoops, the star appear to run more and more 106 g/cm3, radius ~ 10,000 km) or Brussels, 1958), pp. 124-141. 8. B. K. Harrison, K. S. Thorne, M. Wakano, slowly. It takes an infinite time for the a neutron star (central density - 1014 J. A. Wheeler, Gravitation Theory and Giav i-
- tationial Collapse (Univ. of Chicago Press, star to reach the Schwarzschild radius, g/cm3, radius 10 km). However, if Chicago, 1965). and as seen by the orbiting observer, the core contains more than ~ 1.2 9. R. P. Feynman, N. Metropolis, E. Teller, Phys. Rev. 75, 1561 (1949). the star never gets beyond there. times the number of baryong in the 10. K. S. Thorne and J. A. Wheeler. Bull. Amn.* But to the man standing on the star sun, or if it becomes sufficiently com- Phys. Soc. 10, 15 (1965); W. J. Cocke, Ann. Inst. Hentri Poincare 2, 283 (1965). as it collapses into oblivion there is pressed during supernova collapse, then 11. S. Chandrasekhar. Phys. Rev. Letters 12, at all it will not reach a state of 114, 437 (1964). In (8) I exhibit the rela- nothing special about the cold, hydro- tionship between Chandrasekhar's variational Schwarzschild radius. He passes right static equilibrium; rather, it will gravita- principle and mass-energy consideration& such as are used here in the uniform- through it and on into the singularity tionally collapse to zero volume. density-approximation discussion of stability. in a fraction of a second. What hap- This is the story of the death of a 12. D. W. Meltzer and K. S. Thorne, paper in preparation. See also C. W. Mistier and pens to the man on the star as col- star as predicted by a combination of H. S. Zapolsky, Phys. Rev. Letters 12, 635 lapse nears completion? Tidal gravita- nuclear theory, elementary particle (1964), where Chandrasekhar's variational principle is used to study the stability of the tional forces squeeze him from the theory, and general relativity theory. fundamental mode. sides and stretch him between head and Not taken into account in this analysis 13. The precise value of the upper limit on the baryon content of a neutron star is in some foot. As the singularity approaches, are such complications as stellar rota- doubt because of uncertainty in the equation . of state of cold matter at densities _ 1t1:, these tidal forces become infinitely tion, deviation from spherical sym- g/cm:'. The sipper Irmit may be as large as 2 strong and the man's body is stretched metry, and effects of magnetic fields. times the number of baryons in the sun. 1678 SCIENCE, VOL. 150 14. See, for example, B. K. Harrison and J. A. 16. For a review, see R. Giacconi, H. Gursky, 17. S. Bowyer, E. T. Byram, T. A. Chubb, H. Wheeler, paper in preparation; and E. J. R. Waters, B. Rossi, G. Clark. G. Gar- Friedmann, Science 146, 912 (1964). Schatzman, Handbuch der Physik (Spriniger, mire, M. Oda, M. Wada, chapter in High- 18. M. D. Kruskal, Phys. Rev. 119, 1743 (1960). Berlin, 1958), vol. 51, p. 723. Eoeri, Astrophysics, proceedings of course 19. This article is based largely on reference (8). *15. H.-Y. Chiu, Ann. Phys. 26, 364 (1964); D. C. XXXV of the International School of Physics I thank the men who coauthored that book Morton, Astrophys. J. 140, 460 (1964); J. N. "Enrico Fermi" (Academic Press, New York, with me-B. Kent Harrison, Masami Wakano, Bachall and R. A. Wolf, Phys. Rev. Letters in press); see also R. Giacconi and H. and especially John A. Wheeler-for the 14, 343 (1965). Gursky, Space Science Rev. 4, 151 (1965). stimulating collaboration which we had.
fer their sex factor with high frequency to F- cells, converting them to the F+ type (8). The F factor itself is the only genetic material normally transferred in such matings. F+ cells Genetic Transfer do occasionally, however, give rise to stable Hfr derivatives capable of trans- in Bacterial Mating ferring the entire bacterial chromo- some (3). Genetic experiments, which have What mechanism insures the orderly transfer been reviewed extensively (3, 9), indi- cate that transfer of the bacterial chro- of DNA from donor to recipient cells? mosome by an Hfr is rarely complete. Instead, the majority of the F- cells Julian D. Gross and Lucien Caro receive only a segment of the Hfr chromosome. The frequency of trans. mission, for any chromosome determi- nant, decreases with the distance of the determinant from the origin of The genetic material controlling all strains in which the sex factor has be- transfer. The sequence of transfer of the essential functions of Escherichia come associated with the chromosome genetic markers can be precisely de- coli is organized into a single chromo- (Hfr cells), conjugation results in the termined by artificially interrupting the Asome which consists, as far as is progressive linear transfer of the entire mating at various times and assaying known, of a continuous double-strand- chromosome, at a rate such that trans- for the inheritance, by the F- cells, ed molecu,le of DNA, approximately fer is complete in about 90 minutes of a series of I{r determinants. Trans- 1100 microns long (1-3). Both genet- (7). A striking aspect of this process ferred markers are expressed as a result ic and microscopic evidence indicate is that, for any one Hfr strain, the of recombination between the Hfr chro- that th'is chromosome has a circular chromosome is transferred in precisely mosomal fragment and the F- chro- structure (3, 4). Most of the DNA the same sequence from all mating mosome. In interrupted matings, the constituting the chromosome is packed cells. The origin of the sequence is capacity to act as an Hfr donor is into a loosely defined nuclear region defined by the position at which the invariably the last character transferred. less than 0.1 cubic micron in volume. F factor had been inserted into the Hfr cells occasionally revert to the In a fast-growing culture the cells of circular bacterial chromosome (3). F + type or give rise to cells with .E. coli are 2 to 3 microns long and Various models have been proposed variant sex factors (F' factors) capa- 0.8 micron in diameter. They contain as to how the process of DNA trans- ble of transferring, in addition to F two to four chromosomes, in various fer in conjugation may be related to itself, a number of genetic. niarkers stages of replication. The cells grow the mechanisms which coordinate previously located on one or both sides by elongating, forming a constriction chromosome replication and cell of the origin of transfer on the circu- at the equator, and separating into two growth. In this article we describe these lar Hfr chromosome (10). 'daughter cells each containing two models and discuss experiments which The properties of Hfr cells may be chromosomes (1, 5). Each chromo- have a bearing on them. accounted for by postulating that they some replicates once during each gen- arise by integrating the F factor into eration, and the products are segregat- the continuity of the circular bacterial ed so that, at division, each daughter Conjugation in E. coli chromosome at any one of a number cell receives the appropriate number of possible points. This would be ac- of chromosomes. The most studied system of conjuga- complished by a pairing between the Cells of E. coli harboring the sex tion is the one, just mentioned, con- sex factor and the chromosome, fol- factor F or other similar elements, all trolled by the transmissible sex factor lowed by a reciprocal genetic exchange. of which are constituted entirely or F. There are three main mating types: The process could be reversed to pro- primarily of DNA (6), can form a F--, F+, and Hfr. F- cells lack F can act as The authors are on the staff of the Biology 8cellular connection with suitable recipi- entirely: they only recipients Division, Oak Ridge National Laboratory, Oak ent cells. DNA, corresponding to the in matings with donor cells, and they Ridge, Tennessee. Dr. Gross is on leave of ab- sence from the Microbial Genetics Research Unit, sex factor, is then efficiently trans- do so with much higher efficiency than Medical Research Council, Hammersmith Hospital, ferred from donor to recipient. In either F+ or Hfr cells. F+ cells trans- London, England. 24 DECEMBER 1965 1679