The CNO Cycles

When T6 > 17 for a standard composition, the H burning occurs mainly through the CNO cycles. CNO elements must be initially present, their sum XC + XN + XO does not change when the cycles operate, as easily verified from Table 25.2. However, the ratios of CNO elements, like XN/XC, are modified by the cycles. There is a basic cycle, the CN cycle (see Fig. 25.1), to which two ON loops are added, the relative importance of the ON loops increases for higher T. In addition, there is a rare loop from 18O to 16O through 19F. The reactions of the CNO cycles are given in Table 25.2. As for the pp chains, the total energies are given and the v energies must be subtracted for having the energy participating in the radiative energy transfer.

Figure 25.1 shows graphically the basic CN cycle and the loops. The astrophys-ical factor of the resonant reaction 12C(p,y)13N is illustrated in Fig. 9.6. The net result of the cycle and of the loops is to convert four 1H into one 4He. From the indicated timescales, we see that the elements 13N, 15O,17F,18F are rapidly destroyed and can be set to equilibrium. This reduces the number of equations and allows one to adopt not too short time steps in evolutionary calculations. In the CN cycle, the slowest reaction is that which destroys 14N, thus the CN cycle accumulates the elements in the form of 14N. This is a major consequence of the CNO cycle. Some 12C is also turned into 13C so that the 12C/13C is decreased (Table 25.3). The energy

Pipe Math Formulas
Fig. 25.1 Illustration of CN cycle (hexagon), the two ON loops and an additional branching with indication of the reactions. The whole forms the CNO cycles

Table 25.2 The CNO cycles with energies Q and approximate timescales at T = 25 x 106 K. The maximum v energies are given, the relevant average v energies are, respectively, 0.70, 1.00, 0.94 and 0.37 MeV for the four reactions concerned

Q Approx. T

Table 25.2 The CNO cycles with energies Q and approximate timescales at T = 25 x 106 K. The maximum v energies are given, the relevant average v energies are, respectively, 0.70, 1.00, 0.94 and 0.37 MeV for the four reactions concerned

Q Approx. T

CN

12C +

1H -

^ 13N + y

1.944

MeV

103 yr

13n

^ 13C + e+ + ve

2.211 (v < 1.20)

MeV

420 s

13C +

1H -

^ 14N +Y

7.550

MeV

2.9 x 102 yr

14n +

1H -

^ 15O +y

7.293

MeV

9.4 x 104 yr

15O

^ 15N + e+ + ve

2.761 (v < 1.73)

MeV

120 s

15n +

1H -

^ 12C +a

4.966

MeV

3.7 yr

First ON

15n +

1H -

^ 16O + Y

12.126

MeV

4.6 x 103 yr

16O +

1H -

^ 17F + y

0.601

MeV

5.0 x 106 yr

17f

^ 17O + e+ + ve

2.762 (v < 1.74)

MeV

66 s

17O +

1H -

^ 14N+ a

1.193

MeV

3.1 x 106 yr

Second ON

17O +

1H -

^ 18f + Y

5.609

MeV

4.2 x 103 yr

18f

^ 18O + e+ + ve

1.656 (v < 0.634) MeV

6.6 x 103 s

18O +

1H -

^ 15N +a

3.980

MeV

5 yr

Third ON

18O +

1H -

^ 19F + y

7.993

MeV

5 x 103 yr

19f + 1

H-

^ 16O + a

8.115

MeV

5 x 102yr

Table 25.3 Typical CNO abundance ratios (in mass fractions)

Ratios

Cosmic values

CNO equilibrium

12c/14n

- 3.5

0.025

12C/13C

- 62

3.3

16o/14n

- 8.7

0.10

liberated by the CN cycle without the neutrinos is 25.03 MeV, i.e., 6.258 MeV per nucleon corresponding to an energy of 6.04 x 1018 erg g-1 (cf. Appendix A.1).

The branching ratio of the first ON loop with respect to the CN cycle is given by the ratio of the rates of 15N(p,y)12C to 15N(p,a)16O which is of the order of the inverse ratio of the lifetimes of 15N for the two reactions. Thus, the branching ratio between the first loop and the CN cycle is — 10-3. The branching ratio between the second and the first loops is given by the ratio of the (p,y) to (p,a) reactions upon 17O, which is — 1 with some uncertainty [17]. 17O is produced at T6 < 30 and destroyed above.

The branching ratio of the loop going through 19F is given by the ratio of the (p,y) to (p,a) reactions upon 18O, it is of the order of 10-3. Thus, this rare loop is often omitted. Nevertheless, the rates indicate that some small accumulation of 19F is produced. The main result of the three ON loops is to slowly convert (at a rate increasing with T) some 16O into 14N, this conversion is much slower than for 12C into 14N.

As a result of CNO reactions, the abundances of all isotopes involved are changed with respect to their initial cosmic abundances, while the sum of their mass fractions is unchanged. Table 25.3 illustrates the differences of some important abundance ratios between the cosmic values and those of the cycles at equilibrium in a 20 Me star (see also Sect. 27.4).

The C/N and O/N ratios are reduced by about 2 orders of a magnitude at CNO equilibrium with respect to the cosmic values. During their evolution the intermediate and massive stars show almost all possible values between these two extremes. Values of C/N and O/N different from the cosmic values are a signature of surface enrichments.

The zero-order approximation of the energy generation rate is of the form e = eoQ Tv with v « 17 at T6 = 25 and v « 13 at T6 = 50 (see 9.34 and remarks below). A better approximation is to take the rate [194] of the slowest reaction (here 14N+1H for the CN cycle) and to associate to it the total energy (25.03 MeV) of the cycle. This yields e_15.228/T91/3 1

£cno « 8.7 x 1025" X1 Xcn--f114 inerggs_1 . (25.18)

At the equilibrium of the CN cycle, XCN « X12 + X14, while at full CNO equilibrium XCN « X12 + X14 + X16, this last sum is of the order of Z/2 where Z is the metallic-ity. Of course, the detailed energy contributions of the various reactions have to be accounted for in accurate models.

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