THE MYSTERY OF THE COSMIC HELIUM ABUNDANCE
By Prof. F. Hoyle, F.R.S., and Dr. R. J. Tayler
University of Cambridge (1964)
It is usually supposed that the original material of the Galaxy was pristine material. Even solar material is usually regarded as ‘uncooked’, apart from the small concentrations of heavy elements amounting to about 2 per cent by mass which are believed on good grounds to have been produced by nuclear reactions in stars. However, the presence of helium, in a ratio by mass to hydrogen of about 1:2, shows that this is not strictly the case. Granted this, it is still often assumed in astrophysics that the ‘cooking’ has been of a mild degree, involving temperatures of less than than 108 K, such as occurs inside main-sequence stars. However, if present observations of a uniformly high helium content in our Galaxy and its neighbours are correct, it is difficult to suppose that all the helium has been produced in ordinary stars.
It is the purpose of this article to suggest that mild ‘cooking’ is not enough and that most, if not all, of the material of our everyday world, of the Sun, of the stars in our Galaxy and probably of the whole local group of galaxies, if not the whole Universe, has been ‘cooked’ to a temperature in excess of 1010 K. The conclusion is reached that: (i) the Universe had a singular origin or is oscillatory, or (ii) the occurrence of massive objects has been more frequent than has hitherto been supposed.
The section on observations (Table 1) is omitted.
We begin our argument by noticing that helium production in ordinary stars is inadequate to explain the values in Table 1, if they are general throughout the Galaxy, by a factor of about 10. Multiplying the present-day optical emission of the Galaxy, ∼4×1043 ergs sec-1, by the age of the Galaxy, ∼3×1017 sec, and then dividing by the energy production per gram, ∼ 6×1018 ergs g-1, for the process H → He, gives ∼2×1042 g (109M⊙). This is the mass of hydrogen that must be converted to helium in order to supply the present-day optical output of the Galaxy for the whole of its lifetime. Allowance for emission in the ultra-violet and in the infra-red increases the required hydrogen-burning, but probably not by a factor more than ∼3. Since the total mass of the Galaxy is ∼1011 M⊙ the value of the He/H to be expected from H → He inside stars is only ∼0.01. While it is true that the Galaxy may have been much more luminous in the past than it is now, there is no evidence that this was the case.
Next, we shift the discussion to a ‘radiation origin’ of the Universe, in which the rest mass energy density is less than the energy density of radiation. The relation between the T10, measured in units of 1010 K, and the time t in seconds can be worked out from the equations of relativistic cosmology and is:
(1) T10 = 1.52 t-1/2
In the theory of Alpher, Bethe and Gamow the density was given by:
(2) ρ ≈ 10-4T103 g cm-3
a relation obtained from the following considerations. The material is taken at t=0 to be entirely neutrons. At t≃103 sec, T10≃0.05, approximately half the neutrons have decayed. If the density is too low the resulting protons do not combine with the remaining neutrons, and very little helium is formed. On the other hand, if the density is too high there is a complete combination of neutrons and protons, and with the further combination of the resulting deuterium into helium very little hydrogen remains as t increases. Thus only by a rather precise adjustment of the density, that is, by (2), can the situation be arranged so that hydrogen and helium emerge in approximately equal amounts.
It was pointed out by Hayashi and Alpher,Follin and Herman that the assumption of material initially composed wholly of neutrons is not correct. The radiation field generates electron-positron pairs by:
(3) γ + γ ⇌ e– + e+
and the pairs promote the following reactions:
(4) n + e+ ⇌ p + ν
(5) p + e– ⇌ n + ν
The situation evidently depends on the rates of these reactions. It turnes out that for sufficiently small t the balance of the reactions is thermodynamic. This meanes that not only are protons generated by (4) and (5) , but also that the energy densities of the pairs and of the neutrinos must be included in the cosmological equations. At T10≃102, even μ-neutrinos are produced and these too should be included. The effects of these new contributions to the energy density is to modify (1) to:
(6) T10≃1.04 t-1/2
The values of σv for (4), (5), read from left to right, are:
in which the well-known Coulomb factor has been taken as unity, and where the symbols have the following significance: m, electronic mass; W, the energy, including rest mass, of the positron or electron; W0, energy difference between the neutron and protron (2.54 mc2); ± apply to (4) and (5) respectively; (fτ)lab, the fτ-value for free neutron decay (1,175 sec).
To obtain the rates of reactions (4) and (5), read from left to right, multiply (7), using the appropriate sign, by the number of positrons/electrons with energies between W and W+dW and then integrate the product with respect to W from zero to infinity. The corresponding values for the rates of (4) and (5) read from right to left can then easily be obtained by noticing that in thermodynamic equilibrium:
(8) n/p = exp(-W0/kT)
where n, p represent the densities of neutrons and protons.
If, however, one is content with accuracy to within a few per cent it is sufficient simply to write W=mc2+qkT in (7) and to multiply by the total number density of pairs. The value then chosen for q is that which makes mc2+qkT equal to the average electron energy at temperature T; this is a slowly varying function of T and is given by Chandrasekhar. The pair density is:
where K2 is the modified Bessel function of the second kind and second order. For T10∼1 this expression can be closely approximated by the simple form 30aT3/π4k. The number density of neutrino-antineutrino pairs (of both kinds) is 15aT3/π4k.
Adopting this simplified procedure, the reaction rate for n+e+→p+ν is easily seen to be:
(10) 0.071T103(1+0.476qT10)2 per neutron per sec.
The effect of n+ν→p+e– is approximately to double the rate at which neutrons are converted to protons. Using equation (8), the rates of the inverse reactions are obtained by multiplying equation (10) by exp(-W0/kT)=exp(-1.506/T10). Hence the following differential equation determines the variation of n/(n+p) with time:
(11) d[n/(n+p)]/dt=0.142T103(1+0.476qT10)2 ×
To express this in a form convenient for numerical integration use T10 as the independent variable. With the aid of equation (6):
(12) d[n/(n+p)]/dT10=0.308(1+0.476qT10)2 ×
Equation (12) can be integrated from a sufficiently high temperature, at which the neutrons and protrons are almost in thermodynamic balance, down to the temperature at which the pairs disappear and deuterons are formed. The results are insensitive to starting temperature if it is chosen above T10=2.5. When the protons and neutrons are in thermodynamic balance the right-hand side of (12) is zero and the initial value of n/(n+p) is chosen to make this right-hand side zero.
An important question evidently arises as to the precise value of T10 down to which equation (12) should be integraded. Rather surprisingly, it appears the deuterium combines into helium, through D(D,n)He3(n,p)T(p,γ)He4, at a temperature as high as T10=0.3, in spite of the small binding energy of deuterium. (The concentration of deuterium used in establishing this conclusion was just that which exists for statistical equilibrium in n+p⇌D+γ.) Hence, equation (12) must not be integrated to T10 below 0.3. We estimate that integration down to T10=0.5 probably gives the most reliable result, because (12) overestimates the rate of conversion of neutrons to protons below T10=0.5.
Mr. J. Faulkner has solved the equation for several starting temperatures. Provided T10>2.5 initially, he finds n/(n+p)=0.18 at T10=0.5, giving:
a result in good agreement with the calculations of Alpher, Follin and Herman. Allowing for the approximation in our integration procedure we estimate that this value is not more uncertain than 0.14±0.02. It should be particularly noted that, unlike the result of Alpher, Bethe and Gamow, this value depends only slightly on the assumed material density; essentially this result is obtained provided the density is high enough for deuterons to be formed in a time short compared to the neutron half-life and low enough for the rest mass energy density of the nucleons to be neglected in comparison with the energy density of the radiation field.
Before comparing this result with observation we note that variations of the cosmological conditions which led to equation (6) all seem as if they would have the effect of increasing He/H. If the rest mass energy density were not less than the sum of the energy densities of radiation, pairs and neutrinos, the Universe would have to expand faster at a given temperature in order to overcome the increased gravity, the time-scale would be shorter and the coefficient on the right-hand side of equation (12) would be reduced. Similarly, if there were more than two kinds of neutrino the expansion would have to be faster in order to overcome the gravitational attraction of the extra neutrinos, and the time-scale would again be shorter; and the smaller the coefficient on the right-hand side of equation (12) the larger the ratio He/H turns out to be.
We can now say that if the Universe originated in a singular way the He/H ratio cannot be less than about 0.14. This value is of the same order of magnitude as the observed ratios although it is somewhat larger than most of them. However, if it can be established empirically that the ratio is appreciably less than this in any astronomical object in which diffusive separation is out of the question, we can assert that the Universe did not have a singular origin. The importance of the value 0.09 for the Sun is clear; should this value be confirmed by further investigations the cosmological implications will be profound. (A similar situation arises in the case of an oscillating universe. The maximum temperature, achieved at moments of maximum density, must be high enough for all nuclei to be disrupted, that is, T10>1. Otherwise, after a few oscillations all hydrogen would be converted into hheavier nuclei, and this is manifestly not the case.)
It is reasonable, however, to argue in an opposite way. The fact that observed He/H values never differ from 0.14 by more than a factor 2, combined with the fact that the observed values are of necessity subject to some uncertainty, could be interpreted as evidence that the Universe did have a singular origin (or that it is oscillatory). The difficulty of explaining the observed values in terms of hydrogen-burning in ordinary stars supports this point of view. So far as we are aware, there is only one strong counter to this argument, namely, that there is nothing really special to cosmology in the foregoing discussion. A similar result for the He/H ratio will always be obtained if matter is heated above T10=1, and if the time-scale of the process is similar to that given by equation (6). In this connexion it may well be important that the physical conditions inside massive objects or superstars simulate a radiation Universe. Hoyle, Fowler, Burbidge and Burbidge (1964) were led, for reasons independent of those of the present article, to consider temperatures exactly inthe region T10≃1. These authors give the following differential equation between the time t and the density ρ, in such a superstar:
(14) dt = (24πGρ)-1/2dρ/ρ
and also the following relation for an object of mass M:
(15) ρ = 2.8×106(M⊙/M)1/2T103 g cm-3
Eliminating ρ and dρ we have:
(16) dt = (M/2.44×104)1/4dT10/T102.5
whereas the differential form of equation (6) is:
(17) dt = -2.08dT10/T103
The difference of sign arises because equation (16) was given for a contracting object. For re-expansion of an object the sign must be reversed, so that the time-scales are are identical if M≃5×105 M⊙. It may be significant that this is about the largest mass in which the temperature T can be reached without the object being required to collapse inside the Schwarzschild critical radius. If collapse inside this radius followed by re-emergence be permitted, larger masses can be considered. The time-scale is then increased above equation (17) and this has the effect of giving a smaller He/H ratio than that calculated above. If an object is inside the Schwarzschild radius and neutrinos do not escape from it, the conditions are closely similar to the cosmological case. On the other hand, the calculation must be slightly changed for objects that do not enter the Schwarzschild radius, since neutrinos are certainly not contained within them. Thus if the same time-scale were used, that is, M≃105 – 106 M⊙, absence of neutrinos would reduce the right-hand side of equation (12) by a factor of 2. A corresponding calculation leads to n/(n+p)=0.22, also in reasonable agreement with observations, especially as all material need not have passed through massive objects. However, a more detailed discussion of massive objects will be required to decide whether the required amount of helium cannot only be produced but also ejected from them.
This brings us back to our opening remarks. There has always been difficulty in explaining the high helium content of cosmic material in terms of ordinary stellar processes. The mean luminosities of galaxies come out appreciably too high on such a hypothesis. The arguments presented here make it clear, we believe, that the helium was produced in a far more dramatic way. Either the Universe has had at least one high-temperature, high-density phase, or massive objects must play (or have played) a larger part in astrophysical evolution than has hitherto been supposed. Clearly the approximate calculations of this present article must be repeated more accurately, but we would stress two general points:
(1) the weak interaction cross-sections turn out to be just of the right order of magnitude for interesting effects to occur in the time-scale available;
(2) for a wide range of physical conditions (for example, nucleon density) roughly the observed amount of helium is produced.