BETA PROCESSES AND ASTROPHYSICS

D. A. Frank-Kamenetskii

INTRODUCTION

According to present-day ideas,^1 the most important source of stellar energy for most stars is nuclear reactions which ultimately lead to the transformation of four protons into an $\alpha$-particle, two positrons, and two neutrinos:

\[ 4p \to \alpha + 2\beta^{+} + 2\nu. \]

This process occurs, of course, not by the direct collision of four particles, which is statistically extremely improbable, but as the result of a chain of successive reactions of $\gamma$-capture and $\beta$-decay.

Until recently it seemed that the rate of energy release is determined by the processes of $\gamma$-capture, while the $\beta$-processes are secondary. In recent times the development of the theory has led to the conclusion that the constant of the $\beta$-process may enter as a factor in the total rate of energy release and that the value of this constant may affect the observed properties of a star. Thus a close connection has arisen between astrophysics and the theory of $\beta$-decay. Attempts have appeared to use the theory of $\beta$-decay to predict the properties of real stars^2 and even to draw conclusions about the constant of the $\beta$-interaction from astrophysical data.^3

PROTON REACTION

From the fact that a star is in hydrostatic equilibrium it follows that the temperature at the center of a normal star must be of the order of 10 million degrees, i.e., in energy units of the order of 1 kev. If pure hydrogen is heated to such a temperature, the proton reaction should occur in it:

\[ p + p \to d + \beta^{+} + \nu. \]

This is a peculiar β-process, occurring not from a bound state, but at the moment of collision. The deuteron formed is instantly transformed into an α-particle, if only in the following way\(^ {4,5}\):

\[ \mathrm{d}(p\gamma)\mathrm{He}^{3},\quad \mathrm{He}^{3}+\mathrm{He}^{3}\to \alpha+2p . \]

This chain of reactions is called the hydrogen cycle.

The rate of energy release is determined by the primary reaction. The expression for this rate is written down in an elementary way. Let \(b\) be the collision radius\(^6\). In the present case this is the radius of the deuteron:

\[ b=\frac{\hbar}{\sqrt{M\varepsilon}}, \tag{1} \]

where \(\varepsilon\) is the binding energy of the deuteron. The effective collision cross section is equal to \(4\pi b^{2}\), and the collision time is \(\frac{2b}{v}\), where \(v\) is the relative velocity of the colliding protons. To obtain the reaction cross section, the collision cross section must be multiplied by the barrier factor

\[ P=\frac{2\pi e^{2}}{\hbar v}\,e^{-\frac{2\pi e^{2}}{\hbar v}} \tag{2} \]

and by the probability that β-decay will have time to occur during the collision. This probability is equal to the product of the collision time \(\frac{2b}{v}\) and the probability of β-decay per unit time:

\[ \frac{1}{\tau}=M_{sp}^{2}\Lambda^{2}fg_{1}. \tag{3} \]

Here

\[ \Lambda=\frac{1}{\sqrt{8\pi f}}\cdot \frac{\int \psi^{*}\psi\,dV}{b^{3}} \]

is the dimensionless orbital nuclear matrix element; \(f\) is the usual statistical Fermi function, \(g_{1}\) is the constant of the triplet β-interaction, having the dimension of inverse time. The value of \(g_{1}\) can be found from another purely triplet β-process, as

\[ g_{1}=\frac{4\ln 2}{ftM_{1}^{2}}. \tag{4} \]

The factor \(\ln 2\) comes from the tradition of using, to characterize β-processes, the half-life \(t\), and not the exponential time; the factor 4 comes from the fact that the probability is normalized to a cell of phase space containing four states of the positron–neutrino pair.

By the Pauli principle, the spins of the protons in the collision are antiparallel. A stable state of the deuteron with parallel spins can arise only by a spin flip, according to the Gamow–Teller selection rules. Hence the factor in the matrix element is \(M_{sp}^{2}=6\) (as in the decay of \(\mathrm{He}^{6}\)). This is the square of the spin matrix element. It comes from the three possible orientations of the spin of the final state and from the fact that either of the two initial protons can decay\(^1\).

Collecting all the factors (the probabilities of collision, of penetration through the barrier, and of the \(\beta\)-process), we obtain the final formula for the reaction cross section

\[ \sigma=\frac{8\pi b^{3}}{v}\,\frac{2\pi e^{2}}{\hbar v}\, e^{-\frac{2\pi e^{2}}{\hbar v}}\cdot 6\Lambda^{2} f g_{1}. \tag{5} \]

By the Pauli principle, only protons with antiparallel spins can collide. The number of such pairs is \(1/4\) of all proton pairs. Therefore, for a concentration \(n\) of protons per unit volume, the reaction rate will not be \(n^{2}\sigma v/2\), as it would be in the case of Bose particles, but

\[ w=\frac{n^{2}\sigma v}{8}. \tag{6} \]

To find the reaction rate in a plasma with temperature \(T\), this expression must be averaged over the Maxwell distribution of relative velocities \(v\). In doing so, use is made of the fact that the most important exponential factor

\[ e^{-\frac{2\pi e^{2}}{\hbar v}-\frac{\mu v^{2}}{2kT}} \]

(\(\mu\) is the reduced mass) has a sharp maximum at the value of the relative velocity

\[ v_{m}=\sqrt[3]{\frac{2\pi e^{2}kT}{\mu\hbar}}. \tag{7} \]

\(^1\) In Salpeter’s original paper\(^6\), an old system of normalization of matrix elements is used—not per cell of phase space, but per one state of the pair of emitted light particles. In this system the spin factor for the proton reaction is not 6 but \(3/2\), since six states correspond to four combinations of the spins of the positron and neutrino. In this case

\[ g=\frac{\ln 2}{ftM^{2}}, \]

but all values of \(M^{2}\) and the universal time \(A\) are then four times smaller than with the normalization now adopted.

The reaction mainly occurs precisely at this optimal velocity. At lower velocities the probability of passing through the barrier is small; higher velocities are too rare. Hence the exponential factor in the reaction rate takes the form \(e^{-\chi}\), where the barrier exponent \(\chi\) is expressed as

\[ \chi=3\sqrt[3]{\frac{\pi^{2}e^{4}\mu}{2\hbar^{2}kT}} . \tag{8} \]

In the general case, if the reaction occurs not between protons, but between nuclei with charges \(Z_1\) and \(Z_2\), the quantity under the root must be multiplied by \(Z_1^2 Z_2^2\). Numerically,

\[ \chi=4250\sqrt[3]{\frac{\bar A Z_1^2 Z_2^2}{T}}, \tag{9} \]

where \(\bar A\) is the reduced mass in units of atomic weight, and \(T\) is the temperature in \({}^{\circ}\mathrm K\).

Approximate integration of (6) by the saddle-point method leads to the expression for the reaction rate averaged over Maxwell’s distribution,

\[ \overline{w}=\frac{1}{4}\cdot\frac{2}{\sqrt{3}}\,\sigma(v_m)n^2\frac{2\pi e^2}{\hbar}\,e^{-\chi/3}, \tag{10} \]

where \(\sigma(v_m)\) is the reaction cross section at the optimal velocity (7). Substitution of (5) gives Salpeter’s final formula\(^6\)

\[ \overline{w}=\frac{1}{4}\cdot\frac{16\pi}{3^{5/2}}\,b^3\cdot 6\Lambda^2 f g_1 n^2\chi^2 e^{-\chi}, \tag{11} \]

At present, to characterize the \(\beta\)-interaction, instead of the constant \(g\) it is customary to use\({}^{23}\) the universal time \(A\), which is directly connected with the experimental quantity of the reduced time \(ft\). For a purely triplet process,

\[ A_1=ftM_1^2 . \tag{12} \]

The constant \(g\) used by Salpeter is related to the universal time as follows:

\[ g=\frac{4\ln 2}{A}. \tag{13} \]

Hence the rate of the proton reaction is

\[ \overline{w}=\frac{16\pi}{3^{5/2}}\,b^3\cdot 6\Lambda^2 f\frac{\ln 2}{A_1}\,n^2\chi^2 e^{-\chi}. \tag{14} \]

The orbital nuclear matrix element \(\Lambda\) can be calculated with rather high accuracy by the same methods as are used in the theory of the deuteron or of \(p\)-\(p\) scattering. This is the only nuclear system for which the matrix element of a \(\beta\)-process can be calculated exactly theoretically. As Landau and Smorodinskii\({}^{7}\) showed, all such calculations are practically insensitive to the details of the shape of the potential well; the results are determined entirely by the binding energy of the deuteron and by the effective radius of action of the nuclear forces.

The calculation of the matrix element for the proton reaction is the subject of the works of Bethe and Critchfield\({}^{8}\), Friman and Motz\({}^{9}\), and Salpeter\({}^{6}\). They obtained the following values:

\[ \begin{aligned} \text{Bethe and Critchfield}\quad & \Lambda^{2}=8.1,\\ \text{Friman and Motz}\quad & \Lambda^{2}=5.05,\\ \text{Salpeter}\quad & \Lambda^{2}=6.82. \end{aligned} \]

The greatest accuracy was achieved in Salpeter’s work, who calculated by two independent methods and estimates the accuracy of the result as \(\pm 5\%\). But, as is clear from the comparison, even the preceding calculations were already practically sufficiently accurate. A major misunderstanding arose from the fact that Bethe and Critchfield omitted the spin matrix element in their formula. This was explicitly noted in the work and was justified by the fact that the accuracy of the other parts of the calculation did not warrant the introduction into the formula of dimensionless factors of order unity. In the present case the accuracy of the calculations proved higher than the authors had supposed, and the “dimensionless factor of order unity” is equal to 6. By omitting it, Bethe and Critchfield greatly underestimated the reaction rate, as a result of which for many years the hydrogen cycle was considered a completely inadequate source of energy for stars. This is an instructive example of how dangerous a frivolous attitude toward dimensionless factors of order unity can be!

POSSIBLE VARIANTS

The deuterium formed in the proton reaction under the conditions of the central zone of stars is practically instantaneously converted by the reaction \(p\gamma\) into helium-3. The subsequent fate of helium-3 may be different. Up to now we have considered the Shatzman\({}^{4}\)—Fowler\({}^{5}\) variant, generally accepted at present, in which two helium-3 nuclei react with each other with the formation of an \(\alpha\)-particle and the regeneration of two protons. In the original Bethe—Critchfield\({}^{8}\) variant another chain of reactions was proposed:

\[ \mathrm{He}^{3}(\alpha,\gamma)\mathrm{Be}^{7},\quad \mathrm{Be}^{7}(\text{capt. }\beta^{-})\mathrm{Li}^{7},\quad \mathrm{Li}^{7}(p,\gamma)2\,\mathrm{He}^{4}. \]

In both cases the total rate of the process is determined by the proton reaction; the subsequent stages of the cycle proceed much more

faster and adjust themselves to the primary reaction. With respect to the rate of energy release, the two variants of the hydrogen cycle differ only by a constant factor of 2: in the Bethe–Critchfield variant one $\alpha$-particle is formed for each act of the primary reaction, whereas in the Shatzman–Fowler variant it is formed for two acts. Therefore, at the same rate of the primary reaction the rate of energy release in the Shatzman–Fowler variant will be half that in the Bethe–Critchfield variant.

Substitution of the numerical values of the constants, taking into account that for the proton reaction $f=0.153$, $b^3=8.03\cdot 10^{-38}$, and $\Lambda^2=6.82$, leads to the following expression for the rate of energy release

\[ \mathcal{E}=4.05\cdot 10^{11}\frac{Q}{A_1}\chi^2 e^{-\chi}\rho X^2, \]

where $\mathcal{E}$ is the rate of energy release in $\mathrm{erg}/\mathrm{g\ sec}$, $\rho$ is the density in $\mathrm{g}/\mathrm{cm}^3$, $X$ is the mass fraction of hydrogen, and $Q$ is the heat of reaction in $\mathrm{erg}/\mathrm{act}$.

For the Shatzman–Fowler variant

\[ Q=2.09\cdot 10^{-5}. \]

For the Bethe–Critchfield variant

\[ Q=4.18\cdot 10^{-5}. \]

If the matter in the central zone of a star consists only of hydrogen and helium, then the hydrogen cycle is the only thermonuclear source of energy. In the presence of carbon, the catalytic process

\[ \mathrm{C}^{12}(p,\gamma)\mathrm{N}^{13};\quad \mathrm{N}^{13}(\beta^+)\mathrm{C}^{13}; \]

\[ \mathrm{C}^{13}(p,\gamma)\mathrm{N}^{14},\quad \mathrm{N}^{14}(p,\gamma)\mathrm{O}^{15},\quad \mathrm{O}^{15}(\beta^+)\mathrm{N}^{15},\quad \mathrm{N}^{15}(p,\alpha)\mathrm{C}^{12} \]

is possible (the carbon cycle[^1]). Here the barrier is much higher, but the $\beta$-processes occur from a bound state, and not at the moment of collision. This type of catalysis differs sharply from the catalysis that we observe in ordinary “terrestrial” chemistry. Terrestrial catalysis is associated with a lowering of the activation energy, with the reaction proceeding by bypassing a high potential barrier. Stellar catalysis is associated with the small probability of the $\beta$-process at the moment of collision. By stabilizing the state from which the $\beta$-process takes place, one can sharply increase the rate of the cycle. Such a peculiar type of catalysis may be called $\beta$-catalysis.

Complex nuclear systems take part in the carbon cycle and are inaccessible to theoretical calculation. The reaction rates can be estimated only by extrapolating experimental data from much higher energies. Moreover, the content of heavy elements in the matter of the central zone of the star is also unknown. For these reasons, one can speak of the carbon cycle only if the hydrogen cycle proves insufficient.

Hydrogen Curves on Diagrams of Stellar States

To determine whether the hydrogen cycle is sufficient to explain the release of energy in real stars, an elementary physical problem is solved. Let us consider a gas sphere of pure hydrogen, placed in cosmic space and having reached hydrostatic and thermal equilibrium. How similar will its properties be to those of a real star of the same mass? The results of solving this problem\(^2\) are shown in Figs. 1–4, where, on the diagrams of stellar states, “hydrogen curves” are drawn, representing the properties of purely hydrogen stars.

Between the three external characteristics of the state of a star—mass \(M\), luminosity \(L\), and radius \(r_1\)—there exist two independent relations determined by two physical laws: the law of heat generation and the law of heat transport. Analysis shows that the mass–radius relation is sensitive mainly to heat generation, whereas the mass–luminosity relation is sensitive to heat transport. One may construct a relation that is completely insensitive to heat generation and depends only on heat transport; such is\(^ {10}\) the relation between the parameters \(\dfrac{L}{M^3}\) and \(\dfrac{M^{2.5}}{\sqrt{r_1}}\). The radius enters this relation only very weakly, and therefore it is regarded as one form of the mass–luminosity relation.

Fig. 1. The mass–luminosity relation in the coordinates \(\dfrac{L}{M^3}\) and \(\dfrac{M^{2.5}}{\sqrt{r}}\).

In Fig. 1 the mass–luminosity relation is presented in the just indicated coordinates, and in Fig. 2 in natural coordinates. The data

observations in Fig. 1 are taken from the work of Parenago and Masevich\(^ {11}\) and are shown by ovals, whose axes correspond to the uncertainty in the determination of the masses. In Fig. 2 the points are taken from the most recent compilation by Strand and Hall,\(^ {12}\) which claims higher accuracy. The solid line represents the hydrogen curve, whose position in these diagrams does not depend on the value of the \(\beta\)-interaction constant, nor in general on the law of energy release, but is determined only by the heat removal. Dilution of hydrogen with helium should increase, and dilution with heavy atoms should decrease, the luminosity at a given mass.

Fig. 2. Mass–luminosity relation in \(M\)—\(L\) coordinates.

Fig. 3 presents the mass–radius relation. The observational data, taken from the work of Parenago and Masevich,\(^ {11}\) are represented by horizontal segments, whose lengths correspond to the uncertainty of the masses.

Fig. 3. Mass–radius relation.

The solid lines represent two limiting positions of the hydrogen curve, corresponding to the two limiting values of the \(\beta\)-process constant, which will be discussed below. The agreement of this curve with the data-

...observations shows directly how close the law of energy production in stars is to the law of the hydrogen cycle.

The radius–luminosity relation is obtained by eliminating the mass from the two relations considered and in principle gives nothing new. In practice, however, it makes it possible to greatly broaden the range of stars accessible to comparison with theory, since masses are known only for binary stars. The radii of ordinary stars are not measured directly, but are found from the luminosity and the effective temperature by means of the relation

\[ L = 4\pi r_1^{2}\sigma T_e^{4}. \tag{15} \]

The effective temperature, in turn, is estimated from the spectral class. Therefore the diagram \(L — r_1\) is usually replaced by the completely equivalent Hertzsprung–Russell diagram, in which along one axis the absolute stellar magnitude is plotted, i.e. the logarithm of the luminosity, and along the other—the effective temperature or directly the spectral class.

Fig. 4. Hertzsprung–Russell diagram.

In Fig. 4 two limiting positions of the hydrogen curve...

on such a diagram are compared with observational data taken from Bondi’s compilation \(^{13}\). The dots represent stars from the nearest neighborhoods of the Sun \(^{14}\), upright crosses—binary stars \(^{15}\), and oblique crosses—subdwarfs \(^{16}\).

The observational material here is much broader than in the preceding diagrams, where only binary stars were represented. On this diagram, dilution of hydrogen should lower the luminosity. All graphs are constructed in solar units and on a logarithmic scale.

From consideration of the diagrams one can draw quite definite conclusions. It turns out that for the middle part of the main sequence (classes \(F, G, K\)) the hydrogen curve bounds from below the region of concentration of stars on the mass–luminosity diagram and from above the region of their concentration on the Hertzsprung–Russell diagram. But this is exactly what should be the case if these stars consist of hydrogen diluted by various amounts of helium.

In the transition to giants (classes \(A, B, O\)) there is observed, in all diagrams, a sharp systematic departure from the hydrogen curve toward higher luminosity. Apparently some special sources of energy, different from the hydrogen cycle, are included in these stars. But the fact that the departure from the hydrogen curve is clearly noticeable also on the mass–luminosity diagram, which does not depend on the law of energy release, indicates that at the same time the structure of the star also changes radically. At present it is considered generally accepted that in giant stars a considerable part of the mass is gathered near the center in a dense core. According to some views \(^{17-20}\), this should be an isothermal core in which all the hydrogen has burned out; according to others—a neutron core, whose formation, according to Landau’s hypothesis \(^{21}\), is possible in stars with masses greater than several solar masses.

At the opposite end, in the region of red dwarfs of class \(M\), the hydrogen curve continues to bound from above the region of concentration of stars on the Hertzsprung–Russell diagram, but clearly departs from the region of concentration of stars on the \(M-L\) and \(M-r_1\) diagrams. The meaning of this contradiction can be clarified by a careful examination of Fig. 4. It is seen there that binary stars of class \(M\) systematically deviate downward from the hydrogen curve. Meanwhile, the \(M-L\) and \(M-r_1\) diagrams contain only binary stars, since masses are known only for them.

In earlier analyses of diagrams of the state of stars, it was not the absolute values of the parameters that were found from theory, but only the derivatives (the slopes of the curves). With this method of analysis the closeness of real stars to hydrogen stars remained unnoticed. The reason is clear from Fig. 3, which gives a dependence directly connected with the law of energy release. If a line is drawn here that best approximates the observational points over a wide range,

then its slope will have nothing in common with the slope of the hydrogen curve. It will correspond to a much stronger temperature dependence of the reaction rate, whence there arose the long-dominant opinion that the principal source of stellar energy is the carbon cycle. But from the diagram it is clearly seen that, in the region of the middle part of the main sequence, the absolute values of the luminosity are very close to those calculated for purely hydrogen stars. This, of course, is far more convincing than the “mean slope,” which in different portions may be determined by different factors. If it should turn out that in stars of the middle part of the main sequence the source of energy is in fact not the hydrogen cycle, then we would have to conclude that, by an astonishing coincidence, these stars are carefully masked as hydrogen ones.

DETERMINATION OF THE CONSTANT OF THE BETA PROCESS FROM THE PROPERTIES OF THE SUN

Up to now we have been concerned with comparing the theory with regions of condensation of a large number of stars. The inaccuracy in the value of the constant of the β-process in such a statistical comparison is immaterial.

But we can go further and try to make an exact calculation for a single star whose properties are known to us with great accuracy—for the Sun. If the Sun is regarded as a hydrogen-helium star and the energy liberated in it is attributed only to the hydrogen cycle, then a complete calculation of a model of the Sun can be made\(^{22}\), leaving as the only free parameter the universal time of the triplet β-interaction, \(A_1\). Agreement with the actual properties of the Sun is obtained if this constant is assigned the value

\[ A_1 = 2060\ \text{sec.} \tag{16} \]

Only a year ago we thought that the order of magnitude was correct, but that the numerical value was too low. At that time theories were widespread\(^{23,24}\) according to which the constants of the singlet and triplet processes should be equal; meanwhile, the value of \(A_0\) is certainly close to 6000 sec.\(^{25}\). It is now firmly established\(^{26}\) that \(A_1 < A_0\). The numerical value of their ratio is poorly known, and as yet there is no experiment that would make it possible to refine it. From the decay of tritium it follows without doubt\(^{27,3}\) that \(A_1\) cannot be greater than 3600 sec. If, in accordance with Robson’s measurements\(^{28}\), one accepts that the half-life of the free neutron is not less than 10 minutes, then, conversely, it follows that \(A_1\) is not less than 3600. Thus, from terrestrial measurements, at first glance one may draw the conclusion:

\[ A_1 = 3600\ \text{sec.} \tag{17} \]

This value is also the one usually adopted in the literature\({}^{26}\). Salpeter\({}^{6}\) also proceeds from it in deriving his formula for the rate of the proton reaction, now generally accepted in astrophysics.

In reality\({}^{3}\), the value (17) is an upper limit for \(A_1\), since it is obtained from the decay of tritium if one sets \(M^2 = 3\), which corresponds to complete overlap of the wave functions of the initial and final states; the true value of \(M^2\) must be smaller. Thus, there is a certain contradiction between the experimental data on the decay of tritium and of the neutron, which is subject to further clarification.

Let us note that the inequality of the constants of the singlet and triplet \(\beta\)-interactions is apparently explained by the influence of meson corrections\({}^{33,34}\). But then, in the field of nuclear forces, due to meson corrections the value of the \(\beta\)-interaction constant itself may change somewhat. Thus, the possibility is not excluded that the value of \(A_1\) for the decay of the free neutron is in fact somewhat higher than for the decay of tritium and for the proton reaction.

The following fact also speaks in favor of such an assumption. The quoted decay times of the mirror nuclei \(\mathrm{N}^{13}\) and \(\mathrm{O}^{15}\) differ from one another by more than 20%: for \(\mathrm{N}^{13}\), \(ft = 4700\) sec.; for \(\mathrm{O}^{15}\), \(ft = 3750\) sec. The original experimental data are measured here with enormous accuracy, so that the possible error in \(ft\) does not exceed 1.6%. In the decay of these nuclei, \(\mathrm{C}^{13}\) and \(\mathrm{N}^{15}\) are obtained, whose magnetic moments fit beautifully on Schmidt’s line; therefore it is very difficult to explain the difference in \(ft\) by an unequal admixture of other possible states, which would also have had to affect the magnetic moments to no lesser degree.

Meanwhile, in one case we have an excess, in the other a deficiency, of a nucleon in comparison with the symmetric even-even configuration, and this may affect the field of nuclear forces and the meson corrections.

But even if \(A_1\) is not exactly a universal constant, it is very difficult to suppose that its value could vary within such broad limits as the difference between (16) and (17).

“ASTROPHYSICAL” AND “TERRESTRIAL” VALUES OF THE CONSTANT OF THE TRIPLET BETA INTERACTION

As we see, all the properties of the Sun and of most other stars can be described quantitatively by a simple hydrogen-helium model, if only one assigns to the universal time of the triplet \(\beta\)-interaction the “astrophysical” value (16), which is approximately 1.8 times lower than the most probable “terrestrial” value (17). It is quite reasonable to use this astrophysical value as an effective one; its deviation from the true one must compensate for all inaccuracies

of the hydrogen-helium model, which, evidently, do not go beyond factors of order unity.

But it is necessary to estimate the possible errors of both values. At first glance the astrophysical value seems extremely unreliable because of the inaccuracy of our knowledge of the chemical composition of the matter in the central zone of stars. However, it turns out unexpectedly that, for a given mass and radius of a star, the rate of the proton reaction is extremely weakly sensitive to the chemical composition[^29].

From the condition of hydrostatic equilibrium the central temperature of a star is expressed as

\[ T_0=\frac{\mu GM}{\eta R r_1}, \tag{18} \]

where \(\mu\) is the mean molecular weight of the matter at the center of the star, \(M\) is the mass of the star, \(r_1\) its radius, \(G\) the gravitational constant, \(R\) the gas constant, and \(\eta\) a dimensionless factor depending on the structure, whose value varies very little and is always extremely close to unity. For a given mass and radius the central temperature is proportional to the molecular weight \(\mu\), which for mixtures of hydrogen with helium depends on the weight fraction of hydrogen \(X\), as

\[ \frac{1}{\mu}=1.25X+0.75. \tag{19} \]

Heavy atoms affect the opacity incomparably more strongly than they do the molecular weight. As we have seen, most stars lie in the state diagrams rather close to the hydrogen curves. Therefore, the content of heavy elements in them does not change the opacity in a cardinal way, and consequently their influence on the molecular weight is small.

At the temperatures of order 10–15 million degrees prevailing in the central zone of stars, the rate of the proton reaction is approximately proportional to the fourth power of the temperature. In addition, since the reaction occurs in the collision of two protons, its rate is proportional to \(X^2\). When the concentration of hydrogen decreases, the mean molecular weight and with it the central temperature increase, which compensates for the direct effect of the concentration on the reaction rate. For a given mass and radius the rate of the proton reaction is proportional to the quantity \(X^2(1.25X+0.75)^{-4}\), which has a flat maximum at \(X=0.6\) and over a broad range depends extremely weakly on \(X\):

Thus, the calculation of the β-interaction constant from astrophysical data proves to be much more reliable than one might have thought.

Let us consider other physical factors that could influence the course of the proton reaction in the interiors of stars. The densities there are so large that the screening action of the electrons on the electrostatic field of the nuclei could somewhat increase the probability of passing through the barrier. But an estimate shows[^30] that quantitatively this effect is small for the proton reaction. According to the self-consistent-field theory (analogously to the Debye and Hückel theory of electrolytes), the potential \(\varphi\) is found from the equation

\[ \Delta \varphi = 4\pi N_0 \Gamma \frac{e^2}{kT}\varphi, \tag{20} \]

where \(N_0\) is the number of particles per unit volume,

\[ \Gamma = \sum n_i Z_i^2, \tag{21} \]

\(n_i\) is the fraction of particles with charge \(Z_i\). Hence

\[ \varphi = \frac{Ze^2}{r} e^{-\chi r}, \tag{22} \]

where

\[ \chi = \sqrt{4\pi N_0 \Gamma \frac{e^2}{kT}}. \tag{23} \]

The reciprocal of \(\chi\) is the screening radius (Debye radius). Expanding (22) in a series, we conclude that as \(r \to 0\) the electrostatic interaction energy of charges \(Z_1\) and \(Z_2\) is decreased, in comparison with the Coulomb energy, by the amount

\[ \Delta E = Z_1 Z_2 e^2 \chi. \tag{24} \]

A measure of the effect of screening is the ratio of this quantity to the thermal energy \(kT\). In the first approximation one may assume that the barrier has not changed, but that the initial energy has increased by \(\Delta E\), i.e. the probability of passage through the barrier has increased by the factor \(\exp\left(\dfrac{\Delta E}{kT}\right)\). Numerically,

\[ \frac{\Delta E}{kT} = 0.186 \frac{Z_1 Z_2 \sqrt{\rho \Gamma}}{T^{3/2}}, \tag{25} \]

where the temperature \(T\) is expressed in millions of degrees.

For the proton reaction \(Z_1 = Z_2 = 1,\ \Gamma = 1\); for our hydrogen-helium model of the Sun \(\rho_0 = 67.4,\ T_0 = 12.4\), whence

\[ \frac{\Delta E}{kT}=0.03. \]

Thus, electrostatic screening increases the rate of the proton reaction at the center of the Sun by only 3%.

Salpeter\(^6\) analyzed in detail the influence of several other, less important factors on the rate of the proton reaction. Strict averaging over the Maxwellian velocity distribution gives a correction factor for the inaccuracy of the saddle-point method:

\[ F(\chi)=\sqrt{\frac{\chi}{\pi}}\, e^{\chi}\int_0^\infty e^{-\chi\frac{\frac{2}{x}+x^2}{3}}\,dx . \tag{26} \]

For large \(\chi\) this expression may be expanded in a series in inverse powers of \(\chi\):

\[ F(\chi)=1+\frac{5}{12\chi}+\frac{35}{288\chi^2}+\ldots \tag{27} \]

For the proton reaction under conditions at the center of the Sun, the correction for the inaccuracy of the saddle-point method increases the calculated reaction rate by only 3%.

The statistical function \(f\) depends on the initial energy and, consequently, must also be averaged over the Maxwellian distribution. Here the dependence on energy is rather weak, and with sufficient approximation one may take the initial energy at the optimum velocity \(v_m\), determined by formula (7). Then, using for \(f\) the quite accurate approximate formulas of Feynberg and Trigg\({}^{31}\), Salpeter obtains:

\[ f=0.145\left[1+0.054\left(\frac{T}{15\cdot 10^6}\right)^{2/3}\right], \tag{28} \]

where \(T\) is the temperature in \({}^\circ\mathrm{K}\). Salpeter estimates the accuracy of this formula as \(\pm 3\%\). The value \(f=0.153\) used above is obtained from this formula for \(T=15\cdot 10^6\).

Thus, an analysis of the possible corrections leads to the conclusion that the accuracy of the theoretical calculation of the proton-reaction rate is about 10%, and that the discrepancy between the “astrophysical” and “terrestrial” values of the \(\beta\)-process constant can in no way be explained by the inaccuracy of the calculation.

CENTRAL DENSITY AND STRUCTURE OF THE STAR

The most probable explanation of this discrepancy\({}^{29}\) is that the hydrogen–helium model underestimates the density of the central zone of the star.

A characteristic feature of all stellar models is the high degree of concentration not only of the energy release, but also of the mass, toward the center.

In the peripheral zone, which occupies a noticeable part of the radius, not only is no energy released, but only a negligibly small part of the mass is contained there. In this zone both the total heat flux and the gravitating mass may be regarded as independent of the radius. Then the equations of hydrostatic and thermal equilibrium take the form

\[ \frac{1}{\rho}\frac{dp}{dr}=-\frac{GM}{r^2}, \tag{29} \]

\[ k\frac{dT}{dr}=-\frac{L}{4\pi r^2}, \tag{30} \]

where \(k\) is the radiative thermal conductivity. If the matter obeys the equation of state of an ideal gas

\[ p=\frac{RT\rho}{\mu}, \tag{31} \]

then, dividing (29) by (30), we obtain:

\[ \frac{d\ln p}{d\ln T}=\frac{\mu GM}{RL}\,k=\text{const}\cdot k. \tag{32} \]

For any power-law dependence of the radiative thermal conductivity on temperature and density it follows from this that \(p\) and \(T\) must be connected by the relation

\[ k=\text{const}. \tag{33} \]

In other words, if

\[ k=\beta \frac{T^m}{\rho^s}, \tag{34} \]

then the density and temperature must be connected by the “polytropic relation”

\[ \rho\sim T^n, \tag{35} \]

where the “polytropic index”

\[ n=\frac{m}{s}. \tag{36} \]

If, over the entire range of variation of the parameters, the radiative thermal conductivity cannot be represented by a single power-law expression, then the relations given cease to be exact. However, as the analysis of numerical calculations shows\(^{29}\), the qualitative conclusions are preserved. The stationary distribution of densities and temperatures in a star is always close to the condition \(k=\text{const}\). Hence an increase in the opacity of the peripheral zone leads to an enhanced concentration of density (for with increasing density of the central zone its opacity increases).

For this reason, small additions of heavy elements to hydrogen increase the central density of the model. In the central zone the densities are so high that the photoeffect on heavy atoms proves to be of little significance in comparison with bremsstrahlung absorption on hydrogen. At the center of the Sun the mean free path of the radiation is approximately three times smaller than the Compton one. As one approaches the periphery, the temperature and density fall in such a way that the decrease of temperature proves to be more significant than the decrease of density, and for pure hydrogen the role of bremsstrahlung absorption slowly but steadily increases toward the periphery. In the presence of heavy atoms, at low temperatures the relative role of the photoeffect increases sharply, since bound electrons begin to play a large role.

The hydrogen-helium model corresponds to the least possible central density and, consequently, to the least rate of nuclear reactions. The “astrophysical” value of the universal time of the triplet $\beta$-interaction, $A_1$, obtained from this model may serve as a reliable lower limit for the value of this quantity. By adding small amounts of heavy elements to hydrogen and helium, P. Naur succeeded$^{32}$ in constructing a model with an increased density concentration, in which the proton reaction provides all the energy production at the values most probable from the “terrestrial” point of view, $A_1 \simeq 3600$ sec.

If it were possible to determine the central density of the Sun by an independent method, then from astrophysical data we could determine the constant of the triplet $\beta$-interaction with no less, and perhaps even greater, accuracy than from terrestrial experiments. Unfortunately, all astronomical methods now known for determining the central density of stars give only extremely low accuracy.

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