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PROPERTIES OF MATTER AT VERY HIGH DENSITY
CHAPTER
XI
PROPERTIES OF MATTER AT VERY HIGH DENSITY § 106. The equation of state of matter at high density T H E study of the properties of matter at extremely high density is of fundamental importance. Let us follow qualitatively the change in these properties as the density is gradually increased. W h e n the volume per atom becomes less than the usual size of the atom, the atoms lose their individuality, and so the substance is transformed into a highly compressed plasma of electrons a n d nuclei. If the temperature of the substance is not too high, the electron component of this plasma is a degenerate Fermi gas. A n unusual property of such a gas has been mentioned at the end of § 57 : it becomes more nearly ideal as the density increases. T h u s , when the substance is sufficiently compressed, the interaction of the electrons with the nuclei (and with one another) becomes unimportant, and the formulae for an ideal Fermi gas may be used. According to (57.9) this 2 2 3 2 occurs when ne ^> (mee /h ) Z holds, where ne is the number density of electrons, me the electron mass, and Ζ some mean atomic number of the substance. W e therefore find for the total mass density of the substance the inequality 2
2
ρ » (mee jh f
m'Z
2
2
3
~ 20Z g/cm ,
(106.1)
where rri is the mass per electron, so that ρ = nem'} The "gas of nuclei" m a y still be far from degeneracy, because of the large mass of the nucleus, b u t its contribution to the pressure of the substance, for example, is in any case entirely negligible in comparison with t h a t of the electron gas. Thus the thermodynamic quantities for a substance under the conditions
t In all the numerical estimates given in this section it is assumed that the mean atomic weight of the substance is twice its mean atomic number, so that m' is twice the nucléon mass. It may be mentioned that the2 degeneracy temperature of the electrons corre3 e 4 /3 sponding to a density ρ ~ 20Z g/cm is of the order of 1 0 Z degrees. 317
318
Properties of Matter at Very High
Density
in question are given by the formulae derived i n § 5 7 , applied to the electron component. In particular, for the pressure we have* (106.2)
The condition (106.1) on the density gives for the pressure the numerical 8 1 0 /3 bar. inequality Ρ » 5 Χ 1 0 Z In the above formulae the electron gas is assumed non-relativistic. This implies that the Fermi limiting m o m e n t u m pF is small compared with mc (see § 6 1 ) , giving the numerical inequalities e
3
ρ
Ρ «c 1 0
17
bar.
When the density and pressure of the gas become comparable with these values, the electron gas becomes relativistic, and when the opposite inequalities hold we have the extreme relativistic case, where the equation of state is determined by formula (61.4) :t 2
P =
\(3n )^hc(Q/my\
(106.3)
A further increase in density leads to states where nuclear reactions consisting in the capture of electrons by nuclei (with emission of neutrinos) are thermodynamically favoured. Such a reaction decreases the charge o n the nucleus (leaving its atomic weight constant), and this in general causes a decrease in the binding energy of the nucleus, i.e. a decrease in its mass defect. The energy required to bring a b o u t such a process is m o r e than counterbalanced at sufficiently high densities by the decrease in the energy of the degenerate electron gas because of the smaller number of electrons. It is not difficult to write down the thermodynamic conditions which govern the "chemical equilibrium" of the nuclear reaction mentioned, which may be symbolically written as Az + e~ = A z _ i + v, where A z denotes a nucleus of atomic weight A and charge Z , e~~ an electron and ν a neutrino. T h e neutrinos are not retained by matter and leave t Numerically, this formula gives 13
Ρ = 1.0X ΙΟ ( ρ Μ ' )
5 /3
2
7
dyn/cm = 1.0X ΙΟ ( ρ / Λ ' )
5 /3
bar,
(106.2a)
where A' = m\mn is the atomic weight of the substance per electron (jnn being the 3 nucléon mass); ρ is measured in g/cm . The corrections to (106.2) because of the Coulomb interaction of the particles have been discussed in § 80. t With the notation of (106.2a), P=
l^XlOOfeM'^bar.
(106.3a)
§106
The Equation of State of Matter at High Density
319
the b o d y ; such a process must lead to a steady cooling of the body. Thus thermal equilibrium can be meaningfully considered in these conditions only if the temperature of the substance is taken as zero. T h e chemical potential of the neutrinos will n o t then appear in the equation of equilibrium. The chemical potential of the nuclei is mainly governed by their internal energy, which we denote by — ε Λ Ζ (the term "binding energy" usually refers to the positive quantity εΑΖ). Finally, let μ€(η€) denote the chemical potential of the electron gas as a function of the number density ne of particles in it. T h e n the condition of chemical equilibrium takes the form — εΑζ+μ€(η€) ~ ε = A, ~ Α,ζ-ι> or, putting eA%z-*A.z-\ μ€(η€)
= Δ.
Using formula (61.2) for the chemical potential of an extreme relativistic degenerate gas, we thus find (106.4)
ne = Δηΐπ\οΚγ.
The equilibrium condition therefore gives a constant value of the electron density. This means that, as the density of the substance gradually increases, the nuclear reaction in question begins to occur when the electron density reaches the value (106.4). As the substance is further compressed, more a n d more nuclei will each capture an electron, so t h a t the total number of electrons will decrease b u t their density will remain constant. Together with the electron density the pressure of the substance will also remain constant, being again determined mainly by the pressure of the electron g a s : substitution of (106.4) in (106.3) gives (106.5)
P = A*fl2n%hcy.
This will continue until each nucleus has captured an electron. A t still higher densities and pressures the nuclei will capture further electrons, the nuclear charge being thus reduced further. Ultimately the n u clei will contain so many neutrons t h a t they become unstable and break u p . 11 3 24 A t a density ρ ~ 3 X 1 0 g / c m (and pressure Ρ ~ 1 0 bar) the n e u t r o n s 12 a begin to be more numerous than the electrons, and when ρ ~ 1 0 g / c m the pressure due to the neutrons begins to predominate ( F . H u n d , 1936). This is the beginning of a density region in which matter m a y be regarded as essentially a degenerate neutron F e r m i gas with a small number of electrons and various nuclei, whose concentrations are given by the equilibrium conditions for the corresponding nuclear reactions. T h e equation of state of matter in this range is Ρ =
( 3 ?2 ) 23 /
l
-ξ*- ρ
where mn is the neutron mass.
5 /3
3
5 3
= 5.5Χ ΙΟ ρ / bar,
(106.6)
320
Properties of Matter at Very High 15
Density
3
Finally, at densities ρ » 6 χ 1 0 g/cm , the degenerate neutron gas becomes extreme-relativistic, and the equation of state is (106.7) It should be remembered, however, that at densities of the order of that of nuclear matter the specifically nuclear forces (strong interaction of nucléons) become important. In this range of densities formula (106.7) can be only qualitative. In the present state of our knowledge concerning strong interactions we can draw no definite conclusions concerning the state of matter at densities considerably above the nuclear value. We shall merely mention that in this range other particles besides neutrons may be expected to appear. Since particles of each kind occupy a separate group of states, the conversion of neutrons into other particles may be thermodynamically favoured because of the decrease in the limiting energy of the Fermi distribution of neutrons.
§ 1 0 7 . Equilibrium of bodies of large mass Let us consider a body of very large mass, the parts of which are held together by gravitational attraction. Actual bodies of large mass that are known to us, namely the stars, continuously radiate energy and are certainly n o t in thermal equilibrium. It is, however, of fundamental interest to discuss a n equilibrium body of large mass. W e shall neglect the effect of temperature on the equation of state, i.e. consider a body at absolute zero—a "cold" body. Since in actual conditions the temperature of the outer surface is considerably lower than the internal temperature, a discussion of a body with a constant non-zero temperature is in any case devoid of physical meaning. W e shall further assume that the body is n o t rotating; then it will be spherical in equilibrium, and the density distribution will be symmetrical a b o u t the centre. The equilibrium distribution of density (and of the other thermodynamic quantities) in the body will be determined by the following equations. The Newtonian gravitational potential φ satisfies the differential equation Αφ = 4π(?ρ, where ρ is the density of the substance and G the Newtonian constant of gravitation. In the case of spherical symmetry, (107.1) Moreover, in thermal equilibrium the condition (25.2) must be satisfied. In t h e gravitational field the potential energy of a particle of the body of mass
§ 107
Equilibrium of Bodies of Large Mass
321
m' is m'<£, and so we have /x + ra'<£ = constant,
(107.2)
where for brevity the suffix zero is omitted from the chemical potential o f the substance in the absence of the field. Expressing φ in terms of μ by means of (107.2) a n d substituting in (107.1), we have
As the mass of the gravitating body increases, so of course does its mean density, as the following calculations will confirm. When the total mass M of the body is sufficiently large, therefore, we can, as shown i n § 106, regard the substance as a degenerate electron Fermi gas, initially non-relativistic and then at still greater masses relativistic. T h e chemical potential of a non-relativistic degenerate electron gas is related t o the density ρ of the body by
2
m
mem
see formula (57.3), with ρ = m'NIV (m' is t h e mass per electron a n d me the electron mass). Expressing ρ in terms of μ a n d substituting in (107.3), we have*
* It is easy to see that, for an electrically neutral gas consisting of electrons and atomic nuclei, the equilibrium condition can be written in the form (107.2) witih the electron chemical potential as μ and the mass per electron as m'. For the derivation of this equilibrium condition (§ 25) involves considering the transport of an infinitesimal amount of substance from one place to another. In a gas consisting of both positively and negatively charged particles, such transport must be regarded as that of a certain quantity of neutral matter (i.e. electrons and nuclei together). The separation of the positive and negative charges is energetically very unfavourable, because'of the resulting very large electric fields. We therefore obtain the equilibrium condition in the form ^nuc+^el+i^nuc+^el)^ =
0
(with Ζ electrons per nucleus). Owing to the large mass of the nuclei (compared with that of the electrons) their chemical potential is very small compared with / i d . Neglecting / * n uc and dividing the equation by Z , we obtain μ^+ηι'φ
= 0.
As in § 106, it will be assumed in the numerical estimates that rri is twice the nucléon mass (rri = 2m ).n
322
Properties of Matter at Very High
Density
The physically meaningful solutions of this equation must not have singularities at the origin: μ constant for r 0. This requirement necessarily imposes on the first derivative the condition άμ/άτ = 0
for
r = 0,
(107.6)
as follows immediately from equation (107.5) after integration over r : Γ
δ A number of important results can be derived from equation (107.5) by simple dimensional considerations. The solution of (107.5) contains only two constants, λ and (for instance) the radius R of the body, a knowledge of which uniquely defines the solution. F r o m these two quantities we can form only one quantity with the dimensions of length, the radius R itself, and one 2 4 with the dimensions of energy, 1/λ Α (the constant λ having dimensions 2 1/2 cm"" erg~" ). It is therefore clear that the function μ(τ) must have the form
where / is some function of the dimensionless ratio r/R only. Since the den8/2 sity ρ is proportional to μ , the density distribution must be of the form ,
constant
IT \ ρ
—ΒΤ- [Ί[}
Μ =
Thus, when the size of the sphere varies, the density distribution in it remains similar in form, the density at corresponding points being inversely proportional to R*. In particular, the mean density of the sphere is inversely 6 proportional to JR : e ρ oc 1/Ä . The total mass M of the body is therefore inversely proportional to the cube of the radius: 3
M oc l/R . These two relations may also be written 1 8
R oc M - ' ,
2
ρ oc M .
(107.8)
Thus the dimensions of an equilibrium sphere are inversely proportional to the cube root of its total mass, and the mean density is proportional to the square of the mass. The latter result confirms the assumption made above that the density of a gravitating body increases as its mass increases.
Equilibrium of Bodies of Large
§ 107
323
Mass
The fact that a gravitating sphere of non-relativistic degenerate Fermi gas can be in equilibrium for any total mass M can be seen a priori from the following qualitative argument. The total kinetic energy of the particles in such 2/Z a gas is proportional to N(N/V) (see (57.6)), or, what is the same thing, 5/3 2 to M /R 9 and the gravitational energy of the gas as a whole is negative and proportional to MP/R. The sum of two such expressions can have a minimum - 1 / 3 (as a function of R) for any M9 and at the minimum R oc M . Substituting (107.7) in (107.5) and using the dimensionless variable £ = r/R9 we find that the function / ( £ ) satisfies the equation (107.9) with the boundary conditions / ' ( 0 ) = 0 , / ( l ) = 0. This equation cannot be solved analytically, and must be integrated numerically. It may be mentioned t h a t as a result we find/(0) = 178.2, / ' ( l ) = - 1 3 2 . 4 . Using these numerical values it is easy to determine the value of the con3 2 stant MR . Multiplying equation (107.1) by r dr and integrating from 0 to R9 we obtain 2
2
3
GM = JP[d0/dr] r _ Ä = - ( * / m ' ) [d,i/dr] r_ Ä = - / ' ( l ) / m ' A * , whence
3
MR
= 9\.9ffil&mîm's
13
5
3
= 2.2X 1 0 ( m „ / m ' ) Θ k m ,
(107.10)
3 3
where Θ = 2 X l 0 g is the Sun's mass. Finally, the ratio of the central 3 density ρ(0) to the mean density ρ = 3M/4nR is easily found to be ρ(0)/ρ =
-/W(0)/3A1) =
5.99.
(107.11)
Curve 1 in Fig. 50 (p. 324) shows the ratio Q(f)/q(0) as a function of r/R? Let us now examine the equilibrium of a sphere consisting of a degenerate extreme-relativistic electron gas. T h e total kinetic energy of the particles of lß i/Z (see (61.3)), and hence to M /R ; the such a gas is proportional to N(N/V) gravitational energy is proportional to — AP/R. Thus the two quantities depend on R in the same manner, and their sum will also be of the form constant/ R. It follows that the body cannot be in equilibrium : if the constant is positive, the body will tend to expand until the gas becomes non-relativistic; if the constant is negative, a decrease of R to zero corresponds to a decreased total energy, i.e. the body will contract without limit. The body can be in * In § 106 we have seen that matter may be 3regarded as a non-relativistic degener2 ate electron gas at densities ρ : » 20Z g/cm . If this inequality is satisfied for the mean density of the sphere considered, its mass must satisfy4 theV3 condition M :» 3 5X ΙΟ"" Z © . The corresponding radii are less than 5 X 1 0 Z ~ km.
324
Properties of Matter at Very High
Density
FIG. 5 0
equilibrium only in the special case where the constant is zero, a n d the equilibrium is then neutral, the value of R being arbitrary. This qualitative argument is, of course, entirely confirmed by exact quantitative analysis. T h e chemical potential of the relativistic gas considered is related to the density by μ = (3πψ* hc(qlmyi*
(107.12)
(see (61.2)). Instead of (107.5) we n o w have
2
2
Since λ n o w h a s dimensions erg"" c m " , we find that the chemical potential as a function of r must be of the form
* >=*V(7f)' r
>
( 1 0 7 1 4
and the density distribution ,
constant _ / r \
x
M
=
—R*— [T} F
3
Thus the mean density is inversely proportional to Ä , a n d the total mass 3 M oc R Q is independent of R : ρ oc l/R\
M = constant =
M0.
(107.15)
Equilibrium of Bodies of Large
§ 107
Mass
325
MQ is the only value of the mass for which equilibrium is possible: for M > Mo the body will tend to contract indefinitely, and for M < M0 it will expand. F o r an exact calculation of the "critical m a s s " M 0 , it is necessary to integrate numerically the equation 3
/ ' ( 0 ) = 0,
ψ-^ξ(Ρ%)=-/ >
/ ( 1 ) = 0,
(107.16)
which is satisfied by the function / ( I ) in (107.14). T h e result i s / ( 0 ) = 6.897„ / ' ( l ) = - 2 . 0 1 8 . F o r the total mass we find 2
GMo = P [d
-f'(\)lm'y/λ,
whence Z12
M o
=
3 1 ihc\ W*~\G)
=
58
w
m
- ( */ ')
2
©·
(107.17)
f
Putting m — 2m„, we find M0 = 1.45 Θ . Finally, the ratio of the central density t o the mean density is ρ(0)/ρ = -f*(0)/3f'(l) = 54.2. Curve 2 in f Fig. 50 shows ρ(**)/ ρ(0) in the extreme relativistic case as a function of r/R. The results obtained above concerning the relation between the mass and the radius of a "cold" spherical body in equilibrium can b e represented by a single relation M = M(R) for all radii R. F o r large R (and therefore for small densities), the electron gas may be regarded as non-relativistic, and t h e 3 function M(R) decreases as 1/P . When R is sufficiently small, however, the density is so large that we have the extreme relativistic case, a n d the function 0. Figure 51 M(R) is almost a constant M0; strictly M(R) — Mo when R shows the curve M = M(R) calculated with m' — 2mnX It should be noted that the limiting value 1.45 Θ is reached only very gradually; this is because the density decreases rapidly away from the centre of the body, a n d so the extreme relativistic case may hold near the centre while the gas remains n o n relativistic in a considerable p a r t of the volume of the body. W e m a y also mention that the initial part of the curve (R small) has n o real physical significance : at sufficiently small radii the density becomes so large that n u clear reactions begin to occur. The pressure will then increase with density less f
The formal problem of the equilibrium of a gravitating gaseous sphere with a power-law dependence of Ρ on ρ was investigated by R. Emden (1907). The physical deduction of the existence of the limiting mass and its value (107.17) was made by S. Chandrasekhar (1931) and L. Landau (1932). + The intermediate part of the curve is constructed by numerical integration of equation (107.3) with the exact relativistic equation of state for a degenerate gas; see § 61, Problem 3.
326
Properties of Matter
at Very High
Density
473
rapidly than ρ , and for such an equation of state n o equilibrium is possible^ Finally, this curve also has no meaning for large values of R (and small M) : as has already been mentioned (see the second footnote to this section), in this range the equation of state used above becomes invalid. Here it should be pointed out that there is an upper limit to the possible size of a " c o l d " body,
1.4 1.2 1.0 5
0.8 0.6 0.4 0.2 0
2
4
6
8
10
12 8
I0 cm
R FIG. 51
since on the curve in Fig. 51 large dimensions of the body correspond to small masses and small densities, b u t when the density is sufficiently small the substance will be in the ordinary " a t o m i c " state and will be solid at the low temperatures here considered. The dimensions of a body consisting of such a substance will obviously decrease as its mass decreases further, and n o t increase as shown in Fig. 5 1 . The true curve R = R(M) must therefore have a maximum for some value of M. The order of magnitude of the maximum radius can easily be determined by noting that it must correspond to the density at which the interaction 2 2 3 2 between electrons and nuclei becomes important, i.e. for ρ ~ (mee /h ) m'Z (see 106.1)). Combining this with equation (107.10), we obtain 2
2
#max ~ h \GV emem'ZW
1
- WmJm'Z *
km.
(107.18)
n
* If the chemical potential is proportional to a power of the density, μ oc q Λ + 1 n+1 (and so Ρ oc ρ ) , the internal energy of the body is proportional to Vg , i.e. 1 4 1 2 to A f " " / ^ ; the gravitational energy is again proportional to —M /R. It is easy to see that for Λ < -§- the sum of two such expressions has an extremum as a function of R9 but this extremum is a maximum, not a minimum.
§ 108
The Energy of a Gravitating
Body
327
§ 108. The energy of a gravitating body T h e gravitational potential energy Ev
of a body is given by the integral
£gr = i/e
(108.1)
taken over the whole volume of the body. It will, however, be more convenient for us to start from a different expression for this quantity, which may be found as follows. Let us imagine the body to be gradually "built u p " from material brought from infinity. Let M(r) be the mass of substance within a sphere of radius r. Let us suppose that a mass Af(r) with a certain value of r has already been brought from infinity; then the work required to add a further mass dM(r) is equal to the potential energy of that mass (in the form o f a spherical shell of radius r and thickness dr) in the field of the mass M(r), i.e. —GM(r) dM(r)/r. The total gravitational energy of a sphere of radius R is therefore
^
=
G
| M W d M W .
. 2>
Differentiation of the equilibrium condition (107.2) gives dP dr
, Αφ dr
Λ
the differentiation must be at constant temperature, and (Θμ/ΘΡ) Γ = υ is t h e volume per particle. The derivative —άφ/dr is the force of attraction on unit 2 mass at a distance r from the centre, and equals —GM(r)/r . Using also the density ρ = m'I υ, we have
ρ
dr
r
2
Expressing GM(r)/r from this in terms of d P / d r and writing d M ( r ) = 2 q{r)*4nr dr, we can p u t (108.2) in the form 3
= 4π f r 4 — EgT —
dr,
and then integrate by parts (bearing in mind that at the boundary of the body P(R) = 0 and that r*P - 0 as r 0): R
2
EgT = - \2n J Pr dr = - 3 J Ρ dV.
(108.4)
( 1 0 8
328
Properties of Matter at Very High
Density
Thus the gravitational energy of an equilibrium body can be expressed as an integral of the pressure over the volume. Let us apply this formula to the degenerate Fermi gases considered in§ 107. We make the calculation for the general case, and take the chemical potential of the substance to be proportional to some power of its density : 1
(108.5)
μ = Kg '". Since άμ = ν dP = (mV ρ) d P , we have
—Λρ^
+ 1
.
(108.6) W + l /W ' In the equilibrium condition μ/m' +φ = constant, the constant is just the potential at the boundary of the body, where μ vanishes; this potential is —GM/R (M = M(R) being the total mass of the body), and so we can write P=
μ Φ = - rri
GM
R
W e substitute this expression in the integral (108.1) which gives the gravitational energy, and use formulae (108.5), (108.6), obtaining 2
GM 2R
Finally, expressing the integral on the right in terms of Ev by (108.4), we have Ε
3
-
G
*
M
Π08Τ»
Thus the gravitational energy of the body can be expressed by a simple formula in terms of its total mass and its radius. A similar formula can also be obtained for the internal energy Ε of the body. The internal energy per particle is μ—Ρ ν (for zero temperature and entropy); the energy per unit volume is therefore — (μ-Ρν) ν
=
m
^r-P=nP,
where (108.5) and (108.6) have been used to derive the second equation. T h e internal energy of the whole body is therefore E=n\PdV
- J
1 = - j nE
2
M
S I
=
^
1
GM .
^
(108.8)
Finally, the total energy of the body is Etot = E+Egt
=
^
.
(108.9)
§109
329
Equilibrium of a Neutron Sphere 1
F o r a non-relativistic degenerate gas η = |-, and so * 6
(7M
2
£ =-y^f-, gr
3 GAP
£
3
=γπΓ>
Ε
GM
™ = -ΎΠΓ·
2
( 11 0) 0 8
·
In the extreme relativistic case, η = 3, so that £ gr
3 GM = - £ = - _ — ,
2
£t
ot
= 0.
(108.11)
T h e total energy is zero in this case, in accordance with the qualitative arguments given in § 107 concerning the equilibrium of such a Y ~>dy.î
§ 109. Equilibrium of a neutron sphere F o r a body of large mass there are two possible equilibrium states. One corresponds to the state of matter consisting of electrons and nuclei, as assumed in the numerical estimates i n § 107. The other corresponds to the " n e u t r o n " state of matter, in which almost all the electrons have been captured by p r o tons and the substance may be regarded as a neutron gas. When the body is sufficiently massive, the second possibility must always become thermodynamically more favourable than the first (W. Baade a n d F . Z w i c k y , 1934). Although the transformation of nuclei and electrons into free neutrons involves a considerable expenditure of energy, when the total mass of the body is sufficiently great this is more than counterbalanced by the release of gravitational energy owing to the decrease in size and increase in density of the body. First of all, let us examine the conditions in which the neutron state of a body can correspond to any thermodynamic equilibrium (which may be metastable). T o do this, we start from the equilibrium condition μ +m^ = constant, where μ is the chemical potential (the thermodynamic potential per neutron), mn the neutron mass, and φ the gravitational potential. Since the pressure must be zero at the boundary of the body, it is clear t h a t in an outer layer of the body the substance will be at low pressure and t In this case IE = —E&y in agreement with the virial theorem of mechanics, applied to a system of particles interacting according to Newton's law; see Mechanics^ 1 0 . Î In order to avoid misunderstanding, it may be mentioned that the energy is (and therefore Etoi in ( 1 0 8 . 1 1 ) ) includes the rest energy of the particles (which produce the pressure P). If Etot is defined as the "binding energy" of the body (with the energy of the matter dissipated through space taken as zero), the rest energy of the particles must be subtracted from it.
330
Properties of Matter at Very High
Density
density and will therefore consist of electrons and nuclei. Although the thickness of this "shell" may be comparable with the radius of the dense inner neutron "core", the density of the outer layer is much lower, and so its total mass may be regarded as small compared with the mass of the core.* Let us compare the values of μ +mn
Hence, whatever the radius R\ the mass and radius of the neutron core must satisfy the inequality mnMG/R > Δ. (109.1) Applying the results of § 107 to a spherical body consisting of a degenerate (non-relativistic) neutron gas, we find that M and R are related by 3
MR
3
= 9\M«/G m*
3
= 3.6X 10 © k m
3
(109.2)
(formula (107.10) with me and m' replaced by mn). Hence expressing M m terms of R and substituting in (109.1), we obtain an inequality for Af, which in numerical form is M > ~ 0 . 2 © . F o r example, with Δ for oxygen we get M > 0 . 1 7 © , and for iron M > 0 . 1 8 © . These masses correspond to radii R < 26 km.t This inequality gives a lower limit of mass, beyond which the neutron state of the body cannot be stable. It does not, however, ensure complete stability; the state may be metastable. T o determine the limit of metastability, we must compare the total energies of the body in two states: the neutron state and the electron-nucleus state. The conversion of the whole mass M t There is, of course, no sharp boundary between the "core" and the "shell", and the transition between them is continuous. t It must be emphasised that no literal astrophysical significance is to be attached to the numerical estimates in this section, which are based on simple assumptions about the structure of the body.
§109
Equilibrium of a Neutron
Sphere
331
from the electron-nucleus state to the neutron state requires an expenditure of energy MA/mn to counterbalance the binding energy of the nuclei. In the process, energy is released because of the contraction of the b o d y ; according to formula (108.10), this gain of energy is 3GM 7
2
/_1 \R„
1_\ 9
Re)
where Rn is the radius of the body in the neutron state, given by formula (109.2), and Re its radius in the electron-nucleus state, given by (107.10). Since Re » Rn, the quantity l/Re may be neglected, and we obtain the following sufficient condition for complete stability of the neutron state of the body (omitting the suffix in Rn) : (109.3)
3GMm„PR>A.
Comparing this condition with (109.1) and using (109.2), we see that the lower limit of mass determined by the inequality (109.3) is greater by a factor 3 /4 ( 7 / 3 ) = 1.89 than that given by (109.2). Numerically, the limit of metastability of the neutron state is therefore at a mass M ^ | 0 (and radius 1 R s* 22 k m ) . Let us now consider the upper limit of the range of mass values for which a neutron body can be in equilibrium. If we were to use the results o f § 107 (formula (107.17), with mn in place of m\ the value obtained for this limit would be 6 Θ . In reality, however, these results are n o t applicable here, for the following reason. In a relativistic neutron gas, the kinetic energy of the particles is of the order of, or greater than, the rest energy, and the gravitational potential φ ~ c\t In consequence it is n o longer valid t o use the Newtonian gravitational theory, and the calculations must be based on the general theory of relativity; and, as we shall see later, we find that the extreme relativistic case is n o longer reached. T h e calculations must therefore make use of the exact equation of state of a degenerate F e r m i gas; see § 6 1 , P r o b lem 3. T h e calculations are effected by numerical integration of the equations of a spherically symmetric static gravitational field, and the results are as follows^ 13
3
t The mean density of the body is then 1.4X 1Ö g/cm , and so the neutron gas may in fact still be regarded as non-relativistic, and the formulae used here are still valid. t In the relativistic electron gas, the kinetic energy of the particles is comparable with the rest energy of the electrons, but is still small in comparison with the rest energy of the nuclei, which contribute most of the mass of the substance. 8 The details of the calculations are given in the original paper by J. R. Oppenheimer and G. M. Volkoff, Physical Review 5 5 , 374, 1939,
332
Properties of Matter at Very High
Density
The limiting mass of a neutron sphere in equilibrium is found to be only = ^max 0 . 7 6 Θ , a n d this value is reached a t a finite radius Rmin = 9 . 4 k m . Figure 5 2 shows a graph of the relation obtained between the mass M and the radius R. Stable neutron spheres of larger mass or smaller radius cannot, therefore, exist. It should be mentioned that the mass M here denotes the product M = Nmn, where Ν is the total number of particles (i.e. neutrons) in the sphere. This quantity is n o t equal to the gravitational mass Mv of the body, which determines the gravitational field created by it in t h e surrounding space. Because of the "gravitational mass defect", in stable states we f always have Mff < M (in particular, for R = Rttdn9 Mff = 0 . 9 5 M ) .
0
4
8
12
16
20
24
28
32
"min
R,
km
FIG. 5 2
The question arises of the behaviour of a spherical body of mass exceeding A f m a x. It is clear a priori that such a body must tend to contract indefinitely. T h e nature of this unrestrained gravitational collapse has been described in Fields,® 1 0 2 - 1 0 4 . It should be noted that the possibility in principle of gravitational collapse, which (for the model considered of a spherical body) is unavoidable for M > A f ^ , is n o t in fact restricted t o large masses. A "collapsing" state exists for any mass, b u t for M < A f m ax it is separated by a very high energy barrier from the static equilibrium state, t t The point R = Rmin in Fig. 52 is in fact a maximum on the curve M = M(R), This curve continues beyond the maximum as an inward spiral which asymptotically approaches a centre. The parameter which increases monotonically along the curve is the density at the centre of the sphere, which tends to infinity for a sphere corresponding to the limiting point of the spiral (N. A. Dmitriev and S. A. Kholin, 1963). However, no part of the curve for R < Rmin corresponds to a stable state of the sphere. For an account of the investigation see N . A. Dmitriev and S. A. Kholin, Voprosy kosmogonii 9 , 254, 1963; Β. K. Harrison, K. S. Thorne, M. Wakano and J. A. Wheeler, Gravitation Theory and Gravitational Collapse,, University of Chicago Press, Chicago, 1965. t See Ya. B. Zel'dovich, Soviet Physics JETP 15, 446, 1962.
XI
PROPERTIES OF MATTER AT VERY HIGH DENSITY § 106. The equation of state of matter at high density T H E study of the properties of matter at extremely high density is of fundamental importance. Let us follow qualitatively the change in these properties as the density is gradually increased. W h e n the volume per atom becomes less than the usual size of the atom, the atoms lose their individuality, and so the substance is transformed into a highly compressed plasma of electrons a n d nuclei. If the temperature of the substance is not too high, the electron component of this plasma is a degenerate Fermi gas. A n unusual property of such a gas has been mentioned at the end of § 57 : it becomes more nearly ideal as the density increases. T h u s , when the substance is sufficiently compressed, the interaction of the electrons with the nuclei (and with one another) becomes unimportant, and the formulae for an ideal Fermi gas may be used. According to (57.9) this 2 2 3 2 occurs when ne ^> (mee /h ) Z holds, where ne is the number density of electrons, me the electron mass, and Ζ some mean atomic number of the substance. W e therefore find for the total mass density of the substance the inequality 2
2
ρ » (mee jh f
m'Z
2
2
3
~ 20Z g/cm ,
(106.1)
where rri is the mass per electron, so that ρ = nem'} The "gas of nuclei" m a y still be far from degeneracy, because of the large mass of the nucleus, b u t its contribution to the pressure of the substance, for example, is in any case entirely negligible in comparison with t h a t of the electron gas. Thus the thermodynamic quantities for a substance under the conditions
t In all the numerical estimates given in this section it is assumed that the mean atomic weight of the substance is twice its mean atomic number, so that m' is twice the nucléon mass. It may be mentioned that the2 degeneracy temperature of the electrons corre3 e 4 /3 sponding to a density ρ ~ 20Z g/cm is of the order of 1 0 Z degrees. 317
318
Properties of Matter at Very High
Density
in question are given by the formulae derived i n § 5 7 , applied to the electron component. In particular, for the pressure we have* (106.2)
The condition (106.1) on the density gives for the pressure the numerical 8 1 0 /3 bar. inequality Ρ » 5 Χ 1 0 Z In the above formulae the electron gas is assumed non-relativistic. This implies that the Fermi limiting m o m e n t u m pF is small compared with mc (see § 6 1 ) , giving the numerical inequalities e
3
ρ
Ρ «c 1 0
17
bar.
When the density and pressure of the gas become comparable with these values, the electron gas becomes relativistic, and when the opposite inequalities hold we have the extreme relativistic case, where the equation of state is determined by formula (61.4) :t 2
P =
\(3n )^hc(Q/my\
(106.3)
A further increase in density leads to states where nuclear reactions consisting in the capture of electrons by nuclei (with emission of neutrinos) are thermodynamically favoured. Such a reaction decreases the charge o n the nucleus (leaving its atomic weight constant), and this in general causes a decrease in the binding energy of the nucleus, i.e. a decrease in its mass defect. The energy required to bring a b o u t such a process is m o r e than counterbalanced at sufficiently high densities by the decrease in the energy of the degenerate electron gas because of the smaller number of electrons. It is not difficult to write down the thermodynamic conditions which govern the "chemical equilibrium" of the nuclear reaction mentioned, which may be symbolically written as Az + e~ = A z _ i + v, where A z denotes a nucleus of atomic weight A and charge Z , e~~ an electron and ν a neutrino. T h e neutrinos are not retained by matter and leave t Numerically, this formula gives 13
Ρ = 1.0X ΙΟ ( ρ Μ ' )
5 /3
2
7
dyn/cm = 1.0X ΙΟ ( ρ / Λ ' )
5 /3
bar,
(106.2a)
where A' = m\mn is the atomic weight of the substance per electron (jnn being the 3 nucléon mass); ρ is measured in g/cm . The corrections to (106.2) because of the Coulomb interaction of the particles have been discussed in § 80. t With the notation of (106.2a), P=
l^XlOOfeM'^bar.
(106.3a)
§106
The Equation of State of Matter at High Density
319
the b o d y ; such a process must lead to a steady cooling of the body. Thus thermal equilibrium can be meaningfully considered in these conditions only if the temperature of the substance is taken as zero. T h e chemical potential of the neutrinos will n o t then appear in the equation of equilibrium. The chemical potential of the nuclei is mainly governed by their internal energy, which we denote by — ε Λ Ζ (the term "binding energy" usually refers to the positive quantity εΑΖ). Finally, let μ€(η€) denote the chemical potential of the electron gas as a function of the number density ne of particles in it. T h e n the condition of chemical equilibrium takes the form — εΑζ+μ€(η€) ~ ε = A, ~ Α,ζ-ι> or, putting eA%z-*A.z-\ μ€(η€)
= Δ.
Using formula (61.2) for the chemical potential of an extreme relativistic degenerate gas, we thus find (106.4)
ne = Δηΐπ\οΚγ.
The equilibrium condition therefore gives a constant value of the electron density. This means that, as the density of the substance gradually increases, the nuclear reaction in question begins to occur when the electron density reaches the value (106.4). As the substance is further compressed, more a n d more nuclei will each capture an electron, so t h a t the total number of electrons will decrease b u t their density will remain constant. Together with the electron density the pressure of the substance will also remain constant, being again determined mainly by the pressure of the electron g a s : substitution of (106.4) in (106.3) gives (106.5)
P = A*fl2n%hcy.
This will continue until each nucleus has captured an electron. A t still higher densities and pressures the nuclei will capture further electrons, the nuclear charge being thus reduced further. Ultimately the n u clei will contain so many neutrons t h a t they become unstable and break u p . 11 3 24 A t a density ρ ~ 3 X 1 0 g / c m (and pressure Ρ ~ 1 0 bar) the n e u t r o n s 12 a begin to be more numerous than the electrons, and when ρ ~ 1 0 g / c m the pressure due to the neutrons begins to predominate ( F . H u n d , 1936). This is the beginning of a density region in which matter m a y be regarded as essentially a degenerate neutron F e r m i gas with a small number of electrons and various nuclei, whose concentrations are given by the equilibrium conditions for the corresponding nuclear reactions. T h e equation of state of matter in this range is Ρ =
( 3 ?2 ) 23 /
l
-ξ*- ρ
where mn is the neutron mass.
5 /3
3
5 3
= 5.5Χ ΙΟ ρ / bar,
(106.6)
320
Properties of Matter at Very High 15
Density
3
Finally, at densities ρ » 6 χ 1 0 g/cm , the degenerate neutron gas becomes extreme-relativistic, and the equation of state is (106.7) It should be remembered, however, that at densities of the order of that of nuclear matter the specifically nuclear forces (strong interaction of nucléons) become important. In this range of densities formula (106.7) can be only qualitative. In the present state of our knowledge concerning strong interactions we can draw no definite conclusions concerning the state of matter at densities considerably above the nuclear value. We shall merely mention that in this range other particles besides neutrons may be expected to appear. Since particles of each kind occupy a separate group of states, the conversion of neutrons into other particles may be thermodynamically favoured because of the decrease in the limiting energy of the Fermi distribution of neutrons.
§ 1 0 7 . Equilibrium of bodies of large mass Let us consider a body of very large mass, the parts of which are held together by gravitational attraction. Actual bodies of large mass that are known to us, namely the stars, continuously radiate energy and are certainly n o t in thermal equilibrium. It is, however, of fundamental interest to discuss a n equilibrium body of large mass. W e shall neglect the effect of temperature on the equation of state, i.e. consider a body at absolute zero—a "cold" body. Since in actual conditions the temperature of the outer surface is considerably lower than the internal temperature, a discussion of a body with a constant non-zero temperature is in any case devoid of physical meaning. W e shall further assume that the body is n o t rotating; then it will be spherical in equilibrium, and the density distribution will be symmetrical a b o u t the centre. The equilibrium distribution of density (and of the other thermodynamic quantities) in the body will be determined by the following equations. The Newtonian gravitational potential φ satisfies the differential equation Αφ = 4π(?ρ, where ρ is the density of the substance and G the Newtonian constant of gravitation. In the case of spherical symmetry, (107.1) Moreover, in thermal equilibrium the condition (25.2) must be satisfied. In t h e gravitational field the potential energy of a particle of the body of mass
§ 107
Equilibrium of Bodies of Large Mass
321
m' is m'<£, and so we have /x + ra'<£ = constant,
(107.2)
where for brevity the suffix zero is omitted from the chemical potential o f the substance in the absence of the field. Expressing φ in terms of μ by means of (107.2) a n d substituting in (107.1), we have
As the mass of the gravitating body increases, so of course does its mean density, as the following calculations will confirm. When the total mass M of the body is sufficiently large, therefore, we can, as shown i n § 106, regard the substance as a degenerate electron Fermi gas, initially non-relativistic and then at still greater masses relativistic. T h e chemical potential of a non-relativistic degenerate electron gas is related t o the density ρ of the body by
2
m
mem
see formula (57.3), with ρ = m'NIV (m' is t h e mass per electron a n d me the electron mass). Expressing ρ in terms of μ a n d substituting in (107.3), we have*
* It is easy to see that, for an electrically neutral gas consisting of electrons and atomic nuclei, the equilibrium condition can be written in the form (107.2) witih the electron chemical potential as μ and the mass per electron as m'. For the derivation of this equilibrium condition (§ 25) involves considering the transport of an infinitesimal amount of substance from one place to another. In a gas consisting of both positively and negatively charged particles, such transport must be regarded as that of a certain quantity of neutral matter (i.e. electrons and nuclei together). The separation of the positive and negative charges is energetically very unfavourable, because'of the resulting very large electric fields. We therefore obtain the equilibrium condition in the form ^nuc+^el+i^nuc+^el)^ =
0
(with Ζ electrons per nucleus). Owing to the large mass of the nuclei (compared with that of the electrons) their chemical potential is very small compared with / i d . Neglecting / * n uc and dividing the equation by Z , we obtain μ^+ηι'φ
= 0.
As in § 106, it will be assumed in the numerical estimates that rri is twice the nucléon mass (rri = 2m ).n
322
Properties of Matter at Very High
Density
The physically meaningful solutions of this equation must not have singularities at the origin: μ constant for r 0. This requirement necessarily imposes on the first derivative the condition άμ/άτ = 0
for
r = 0,
(107.6)
as follows immediately from equation (107.5) after integration over r : Γ
δ A number of important results can be derived from equation (107.5) by simple dimensional considerations. The solution of (107.5) contains only two constants, λ and (for instance) the radius R of the body, a knowledge of which uniquely defines the solution. F r o m these two quantities we can form only one quantity with the dimensions of length, the radius R itself, and one 2 4 with the dimensions of energy, 1/λ Α (the constant λ having dimensions 2 1/2 cm"" erg~" ). It is therefore clear that the function μ(τ) must have the form
where / is some function of the dimensionless ratio r/R only. Since the den8/2 sity ρ is proportional to μ , the density distribution must be of the form ,
constant
IT \ ρ
—ΒΤ- [Ί[}
Μ =
Thus, when the size of the sphere varies, the density distribution in it remains similar in form, the density at corresponding points being inversely proportional to R*. In particular, the mean density of the sphere is inversely 6 proportional to JR : e ρ oc 1/Ä . The total mass M of the body is therefore inversely proportional to the cube of the radius: 3
M oc l/R . These two relations may also be written 1 8
R oc M - ' ,
2
ρ oc M .
(107.8)
Thus the dimensions of an equilibrium sphere are inversely proportional to the cube root of its total mass, and the mean density is proportional to the square of the mass. The latter result confirms the assumption made above that the density of a gravitating body increases as its mass increases.
Equilibrium of Bodies of Large
§ 107
323
Mass
The fact that a gravitating sphere of non-relativistic degenerate Fermi gas can be in equilibrium for any total mass M can be seen a priori from the following qualitative argument. The total kinetic energy of the particles in such 2/Z a gas is proportional to N(N/V) (see (57.6)), or, what is the same thing, 5/3 2 to M /R 9 and the gravitational energy of the gas as a whole is negative and proportional to MP/R. The sum of two such expressions can have a minimum - 1 / 3 (as a function of R) for any M9 and at the minimum R oc M . Substituting (107.7) in (107.5) and using the dimensionless variable £ = r/R9 we find that the function / ( £ ) satisfies the equation (107.9) with the boundary conditions / ' ( 0 ) = 0 , / ( l ) = 0. This equation cannot be solved analytically, and must be integrated numerically. It may be mentioned t h a t as a result we find/(0) = 178.2, / ' ( l ) = - 1 3 2 . 4 . Using these numerical values it is easy to determine the value of the con3 2 stant MR . Multiplying equation (107.1) by r dr and integrating from 0 to R9 we obtain 2
2
3
GM = JP[d0/dr] r _ Ä = - ( * / m ' ) [d,i/dr] r_ Ä = - / ' ( l ) / m ' A * , whence
3
MR
= 9\.9ffil&mîm's
13
5
3
= 2.2X 1 0 ( m „ / m ' ) Θ k m ,
(107.10)
3 3
where Θ = 2 X l 0 g is the Sun's mass. Finally, the ratio of the central 3 density ρ(0) to the mean density ρ = 3M/4nR is easily found to be ρ(0)/ρ =
-/W(0)/3A1) =
5.99.
(107.11)
Curve 1 in Fig. 50 (p. 324) shows the ratio Q(f)/q(0) as a function of r/R? Let us now examine the equilibrium of a sphere consisting of a degenerate extreme-relativistic electron gas. T h e total kinetic energy of the particles of lß i/Z (see (61.3)), and hence to M /R ; the such a gas is proportional to N(N/V) gravitational energy is proportional to — AP/R. Thus the two quantities depend on R in the same manner, and their sum will also be of the form constant/ R. It follows that the body cannot be in equilibrium : if the constant is positive, the body will tend to expand until the gas becomes non-relativistic; if the constant is negative, a decrease of R to zero corresponds to a decreased total energy, i.e. the body will contract without limit. The body can be in * In § 106 we have seen that matter may be 3regarded as a non-relativistic degener2 ate electron gas at densities ρ : » 20Z g/cm . If this inequality is satisfied for the mean density of the sphere considered, its mass must satisfy4 theV3 condition M :» 3 5X ΙΟ"" Z © . The corresponding radii are less than 5 X 1 0 Z ~ km.
324
Properties of Matter at Very High
Density
FIG. 5 0
equilibrium only in the special case where the constant is zero, a n d the equilibrium is then neutral, the value of R being arbitrary. This qualitative argument is, of course, entirely confirmed by exact quantitative analysis. T h e chemical potential of the relativistic gas considered is related to the density by μ = (3πψ* hc(qlmyi*
(107.12)
(see (61.2)). Instead of (107.5) we n o w have
2
2
Since λ n o w h a s dimensions erg"" c m " , we find that the chemical potential as a function of r must be of the form
* >=*V(7f)' r
>
( 1 0 7 1 4
and the density distribution ,
constant _ / r \
x
M
=
—R*— [T} F
3
Thus the mean density is inversely proportional to Ä , a n d the total mass 3 M oc R Q is independent of R : ρ oc l/R\
M = constant =
M0.
(107.15)
Equilibrium of Bodies of Large
§ 107
Mass
325
MQ is the only value of the mass for which equilibrium is possible: for M > Mo the body will tend to contract indefinitely, and for M < M0 it will expand. F o r an exact calculation of the "critical m a s s " M 0 , it is necessary to integrate numerically the equation 3
/ ' ( 0 ) = 0,
ψ-^ξ(Ρ%)=-/ >
/ ( 1 ) = 0,
(107.16)
which is satisfied by the function / ( I ) in (107.14). T h e result i s / ( 0 ) = 6.897„ / ' ( l ) = - 2 . 0 1 8 . F o r the total mass we find 2
GMo = P [d
-f'(\)lm'y/λ,
whence Z12
M o
=
3 1 ihc\ W*~\G)
=
58
w
m
- ( */ ')
2
©·
(107.17)
f
Putting m — 2m„, we find M0 = 1.45 Θ . Finally, the ratio of the central density t o the mean density is ρ(0)/ρ = -f*(0)/3f'(l) = 54.2. Curve 2 in f Fig. 50 shows ρ(**)/ ρ(0) in the extreme relativistic case as a function of r/R. The results obtained above concerning the relation between the mass and the radius of a "cold" spherical body in equilibrium can b e represented by a single relation M = M(R) for all radii R. F o r large R (and therefore for small densities), the electron gas may be regarded as non-relativistic, and t h e 3 function M(R) decreases as 1/P . When R is sufficiently small, however, the density is so large that we have the extreme relativistic case, a n d the function 0. Figure 51 M(R) is almost a constant M0; strictly M(R) — Mo when R shows the curve M = M(R) calculated with m' — 2mnX It should be noted that the limiting value 1.45 Θ is reached only very gradually; this is because the density decreases rapidly away from the centre of the body, a n d so the extreme relativistic case may hold near the centre while the gas remains n o n relativistic in a considerable p a r t of the volume of the body. W e m a y also mention that the initial part of the curve (R small) has n o real physical significance : at sufficiently small radii the density becomes so large that n u clear reactions begin to occur. The pressure will then increase with density less f
The formal problem of the equilibrium of a gravitating gaseous sphere with a power-law dependence of Ρ on ρ was investigated by R. Emden (1907). The physical deduction of the existence of the limiting mass and its value (107.17) was made by S. Chandrasekhar (1931) and L. Landau (1932). + The intermediate part of the curve is constructed by numerical integration of equation (107.3) with the exact relativistic equation of state for a degenerate gas; see § 61, Problem 3.
326
Properties of Matter
at Very High
Density
473
rapidly than ρ , and for such an equation of state n o equilibrium is possible^ Finally, this curve also has no meaning for large values of R (and small M) : as has already been mentioned (see the second footnote to this section), in this range the equation of state used above becomes invalid. Here it should be pointed out that there is an upper limit to the possible size of a " c o l d " body,
1.4 1.2 1.0 5
0.8 0.6 0.4 0.2 0
2
4
6
8
10
12 8
I0 cm
R FIG. 51
since on the curve in Fig. 51 large dimensions of the body correspond to small masses and small densities, b u t when the density is sufficiently small the substance will be in the ordinary " a t o m i c " state and will be solid at the low temperatures here considered. The dimensions of a body consisting of such a substance will obviously decrease as its mass decreases further, and n o t increase as shown in Fig. 5 1 . The true curve R = R(M) must therefore have a maximum for some value of M. The order of magnitude of the maximum radius can easily be determined by noting that it must correspond to the density at which the interaction 2 2 3 2 between electrons and nuclei becomes important, i.e. for ρ ~ (mee /h ) m'Z (see 106.1)). Combining this with equation (107.10), we obtain 2
2
#max ~ h \GV emem'ZW
1
- WmJm'Z *
km.
(107.18)
n
* If the chemical potential is proportional to a power of the density, μ oc q Λ + 1 n+1 (and so Ρ oc ρ ) , the internal energy of the body is proportional to Vg , i.e. 1 4 1 2 to A f " " / ^ ; the gravitational energy is again proportional to —M /R. It is easy to see that for Λ < -§- the sum of two such expressions has an extremum as a function of R9 but this extremum is a maximum, not a minimum.
§ 108
The Energy of a Gravitating
Body
327
§ 108. The energy of a gravitating body T h e gravitational potential energy Ev
of a body is given by the integral
£gr = i/e
(108.1)
taken over the whole volume of the body. It will, however, be more convenient for us to start from a different expression for this quantity, which may be found as follows. Let us imagine the body to be gradually "built u p " from material brought from infinity. Let M(r) be the mass of substance within a sphere of radius r. Let us suppose that a mass Af(r) with a certain value of r has already been brought from infinity; then the work required to add a further mass dM(r) is equal to the potential energy of that mass (in the form o f a spherical shell of radius r and thickness dr) in the field of the mass M(r), i.e. —GM(r) dM(r)/r. The total gravitational energy of a sphere of radius R is therefore
^
=
G
| M W d M W .
. 2>
Differentiation of the equilibrium condition (107.2) gives dP dr
, Αφ dr
Λ
the differentiation must be at constant temperature, and (Θμ/ΘΡ) Γ = υ is t h e volume per particle. The derivative —άφ/dr is the force of attraction on unit 2 mass at a distance r from the centre, and equals —GM(r)/r . Using also the density ρ = m'I υ, we have
ρ
dr
r
2
Expressing GM(r)/r from this in terms of d P / d r and writing d M ( r ) = 2 q{r)*4nr dr, we can p u t (108.2) in the form 3
= 4π f r 4 — EgT —
dr,
and then integrate by parts (bearing in mind that at the boundary of the body P(R) = 0 and that r*P - 0 as r 0): R
2
EgT = - \2n J Pr dr = - 3 J Ρ dV.
(108.4)
( 1 0 8
328
Properties of Matter at Very High
Density
Thus the gravitational energy of an equilibrium body can be expressed as an integral of the pressure over the volume. Let us apply this formula to the degenerate Fermi gases considered in§ 107. We make the calculation for the general case, and take the chemical potential of the substance to be proportional to some power of its density : 1
(108.5)
μ = Kg '". Since άμ = ν dP = (mV ρ) d P , we have
—Λρ^
+ 1
.
(108.6) W + l /W ' In the equilibrium condition μ/m' +φ = constant, the constant is just the potential at the boundary of the body, where μ vanishes; this potential is —GM/R (M = M(R) being the total mass of the body), and so we can write P=
μ Φ = - rri
GM
R
W e substitute this expression in the integral (108.1) which gives the gravitational energy, and use formulae (108.5), (108.6), obtaining 2
GM 2R
Finally, expressing the integral on the right in terms of Ev by (108.4), we have Ε
3
-
G
*
M
Π08Τ»
Thus the gravitational energy of the body can be expressed by a simple formula in terms of its total mass and its radius. A similar formula can also be obtained for the internal energy Ε of the body. The internal energy per particle is μ—Ρ ν (for zero temperature and entropy); the energy per unit volume is therefore — (μ-Ρν) ν
=
m
^r-P=nP,
where (108.5) and (108.6) have been used to derive the second equation. T h e internal energy of the whole body is therefore E=n\PdV
- J
1 = - j nE
2
M
S I
=
^
1
GM .
^
(108.8)
Finally, the total energy of the body is Etot = E+Egt
=
^
.
(108.9)
§109
329
Equilibrium of a Neutron Sphere 1
F o r a non-relativistic degenerate gas η = |-, and so * 6
(7M
2
£ =-y^f-, gr
3 GAP
£
3
=γπΓ>
Ε
GM
™ = -ΎΠΓ·
2
( 11 0) 0 8
·
In the extreme relativistic case, η = 3, so that £ gr
3 GM = - £ = - _ — ,
2
£t
ot
= 0.
(108.11)
T h e total energy is zero in this case, in accordance with the qualitative arguments given in § 107 concerning the equilibrium of such a Y ~>dy.î
§ 109. Equilibrium of a neutron sphere F o r a body of large mass there are two possible equilibrium states. One corresponds to the state of matter consisting of electrons and nuclei, as assumed in the numerical estimates i n § 107. The other corresponds to the " n e u t r o n " state of matter, in which almost all the electrons have been captured by p r o tons and the substance may be regarded as a neutron gas. When the body is sufficiently massive, the second possibility must always become thermodynamically more favourable than the first (W. Baade a n d F . Z w i c k y , 1934). Although the transformation of nuclei and electrons into free neutrons involves a considerable expenditure of energy, when the total mass of the body is sufficiently great this is more than counterbalanced by the release of gravitational energy owing to the decrease in size and increase in density of the body. First of all, let us examine the conditions in which the neutron state of a body can correspond to any thermodynamic equilibrium (which may be metastable). T o do this, we start from the equilibrium condition μ +m^ = constant, where μ is the chemical potential (the thermodynamic potential per neutron), mn the neutron mass, and φ the gravitational potential. Since the pressure must be zero at the boundary of the body, it is clear t h a t in an outer layer of the body the substance will be at low pressure and t In this case IE = —E&y in agreement with the virial theorem of mechanics, applied to a system of particles interacting according to Newton's law; see Mechanics^ 1 0 . Î In order to avoid misunderstanding, it may be mentioned that the energy is (and therefore Etoi in ( 1 0 8 . 1 1 ) ) includes the rest energy of the particles (which produce the pressure P). If Etot is defined as the "binding energy" of the body (with the energy of the matter dissipated through space taken as zero), the rest energy of the particles must be subtracted from it.
330
Properties of Matter at Very High
Density
density and will therefore consist of electrons and nuclei. Although the thickness of this "shell" may be comparable with the radius of the dense inner neutron "core", the density of the outer layer is much lower, and so its total mass may be regarded as small compared with the mass of the core.* Let us compare the values of μ +mn
Hence, whatever the radius R\ the mass and radius of the neutron core must satisfy the inequality mnMG/R > Δ. (109.1) Applying the results of § 107 to a spherical body consisting of a degenerate (non-relativistic) neutron gas, we find that M and R are related by 3
MR
3
= 9\M«/G m*
3
= 3.6X 10 © k m
3
(109.2)
(formula (107.10) with me and m' replaced by mn). Hence expressing M m terms of R and substituting in (109.1), we obtain an inequality for Af, which in numerical form is M > ~ 0 . 2 © . F o r example, with Δ for oxygen we get M > 0 . 1 7 © , and for iron M > 0 . 1 8 © . These masses correspond to radii R < 26 km.t This inequality gives a lower limit of mass, beyond which the neutron state of the body cannot be stable. It does not, however, ensure complete stability; the state may be metastable. T o determine the limit of metastability, we must compare the total energies of the body in two states: the neutron state and the electron-nucleus state. The conversion of the whole mass M t There is, of course, no sharp boundary between the "core" and the "shell", and the transition between them is continuous. t It must be emphasised that no literal astrophysical significance is to be attached to the numerical estimates in this section, which are based on simple assumptions about the structure of the body.
§109
Equilibrium of a Neutron
Sphere
331
from the electron-nucleus state to the neutron state requires an expenditure of energy MA/mn to counterbalance the binding energy of the nuclei. In the process, energy is released because of the contraction of the b o d y ; according to formula (108.10), this gain of energy is 3GM 7
2
/_1 \R„
1_\ 9
Re)
where Rn is the radius of the body in the neutron state, given by formula (109.2), and Re its radius in the electron-nucleus state, given by (107.10). Since Re » Rn, the quantity l/Re may be neglected, and we obtain the following sufficient condition for complete stability of the neutron state of the body (omitting the suffix in Rn) : (109.3)
3GMm„PR>A.
Comparing this condition with (109.1) and using (109.2), we see that the lower limit of mass determined by the inequality (109.3) is greater by a factor 3 /4 ( 7 / 3 ) = 1.89 than that given by (109.2). Numerically, the limit of metastability of the neutron state is therefore at a mass M ^ | 0 (and radius 1 R s* 22 k m ) . Let us now consider the upper limit of the range of mass values for which a neutron body can be in equilibrium. If we were to use the results o f § 107 (formula (107.17), with mn in place of m\ the value obtained for this limit would be 6 Θ . In reality, however, these results are n o t applicable here, for the following reason. In a relativistic neutron gas, the kinetic energy of the particles is of the order of, or greater than, the rest energy, and the gravitational potential φ ~ c\t In consequence it is n o longer valid t o use the Newtonian gravitational theory, and the calculations must be based on the general theory of relativity; and, as we shall see later, we find that the extreme relativistic case is n o longer reached. T h e calculations must therefore make use of the exact equation of state of a degenerate F e r m i gas; see § 6 1 , P r o b lem 3. T h e calculations are effected by numerical integration of the equations of a spherically symmetric static gravitational field, and the results are as follows^ 13
3
t The mean density of the body is then 1.4X 1Ö g/cm , and so the neutron gas may in fact still be regarded as non-relativistic, and the formulae used here are still valid. t In the relativistic electron gas, the kinetic energy of the particles is comparable with the rest energy of the electrons, but is still small in comparison with the rest energy of the nuclei, which contribute most of the mass of the substance. 8 The details of the calculations are given in the original paper by J. R. Oppenheimer and G. M. Volkoff, Physical Review 5 5 , 374, 1939,
332
Properties of Matter at Very High
Density
The limiting mass of a neutron sphere in equilibrium is found to be only = ^max 0 . 7 6 Θ , a n d this value is reached a t a finite radius Rmin = 9 . 4 k m . Figure 5 2 shows a graph of the relation obtained between the mass M and the radius R. Stable neutron spheres of larger mass or smaller radius cannot, therefore, exist. It should be mentioned that the mass M here denotes the product M = Nmn, where Ν is the total number of particles (i.e. neutrons) in the sphere. This quantity is n o t equal to the gravitational mass Mv of the body, which determines the gravitational field created by it in t h e surrounding space. Because of the "gravitational mass defect", in stable states we f always have Mff < M (in particular, for R = Rttdn9 Mff = 0 . 9 5 M ) .
0
4
8
12
16
20
24
28
32
"min
R,
km
FIG. 5 2
The question arises of the behaviour of a spherical body of mass exceeding A f m a x. It is clear a priori that such a body must tend to contract indefinitely. T h e nature of this unrestrained gravitational collapse has been described in Fields,® 1 0 2 - 1 0 4 . It should be noted that the possibility in principle of gravitational collapse, which (for the model considered of a spherical body) is unavoidable for M > A f ^ , is n o t in fact restricted t o large masses. A "collapsing" state exists for any mass, b u t for M < A f m ax it is separated by a very high energy barrier from the static equilibrium state, t t The point R = Rmin in Fig. 52 is in fact a maximum on the curve M = M(R), This curve continues beyond the maximum as an inward spiral which asymptotically approaches a centre. The parameter which increases monotonically along the curve is the density at the centre of the sphere, which tends to infinity for a sphere corresponding to the limiting point of the spiral (N. A. Dmitriev and S. A. Kholin, 1963). However, no part of the curve for R < Rmin corresponds to a stable state of the sphere. For an account of the investigation see N . A. Dmitriev and S. A. Kholin, Voprosy kosmogonii 9 , 254, 1963; Β. K. Harrison, K. S. Thorne, M. Wakano and J. A. Wheeler, Gravitation Theory and Gravitational Collapse,, University of Chicago Press, Chicago, 1965. t See Ya. B. Zel'dovich, Soviet Physics JETP 15, 446, 1962.