p-p chain vs. CNO

p-p chain vs. CNO#

We will explore the p-p chain and the CNO cycle to learn which rates are the slowest in each and when each dominates inside of stars.

[1]:

import pynucastro as pyna

We start by getting the rates from the JINA ReacLib database. This has 80,000+ reaction rates in a standardized format.

[2]:

rl = pyna.ReacLibLibrary()

Simple p-p chain#

We’ll focus on the main p-p chain and get the 3 rates involved in that.

[3]:

pp_rates = ["p(p,)d",
            "d(p,g)he3",
            "he3(he3,pp)he4"]
rates = rl.get_rate_by_name(pp_rates)

We can look at which rates it pulled in:

[4]:

for r in rates:
    print(r)

p + p ⟶ H2 + e⁺ + 𝜈
p + p + e⁻ ⟶ H2 + 𝜈
H2 + p ⟶ He3 + 𝛾
He3 + He3 ⟶ p + p + He4

Note that ReacLib actually splits the p+p rate into two. We will add them together.

We can look at the temperature dependence of one of these rates

[5]:

r = rates[0]
fig = r.plot(Tmin=1.e7, Tmax=1.e8)
../_images/examples_pp-cno_12_0.png

We see that it is very temperature dependent

Next we build our network (we’ll use a RateCollection) with these rates

[6]:

net_pp = pyna.RateCollection(rates=rates)

We can create a thermodynamic state approximating the center of the Sun and evaluate the rates

[7]:

T = 1.5e7
rho = 150
comp = pyna.Composition(net_pp.unique_nuclei)
comp.set_solar_like()

[8]:

net_pp.evaluate_rates(rho, T, comp)

[8]:

{p + p ⟶ H2 + e⁺ + 𝜈: 2.978374743269655e-18,
 p + p + e⁻ ⟶ H2 + 𝜈: 1.357778476769111e-20,
 H2 + p ⟶ He3 + 𝛾: 0.005874533290651895,
 He3 + He3 ⟶ p + p + He4: 1.849294558580789e-13}

We see from this that the first reaction, p + p, is by far the slowest.

[9]:

fig = net_pp.plot(rho=rho, T=T, comp=comp, hide_xp=True, always_show_alpha=True)
../_images/examples_pp-cno_20_0.png

We can therefore approximate the p-p chain based on the slowest rate. If we express the rate as:

\[r = r_0 \left (\frac{T}{T_0}\right )^\nu\]

then we can find the exponent \(\nu\) about some temperature \(T_0\)

[10]:

r_pp = net_pp.get_rate_by_name("p(p,)d")[0]
T0 = 1.5e7
r_pp.get_rate_exponent(T0)

[10]:

3.9735120801889896

Around the conditions in the Sun, the p-p chain has a temperature dependence of \(\sim T^4\)

CNO#

Now we consider the main CNO cycle. This involves 6 rates

[11]:

cno_rates = ["c12(p,g)n13",
             "c13(p,g)n14",
             "n13(,)c13",
             "n14(p,g)o15",
             "n15(p,a)c12",
             "o15(,)n15"]
rates = rl.get_rate_by_name(cno_rates)

[12]:

for r in rates:
    print(r)

C12 + p ⟶ N13 + 𝛾
C13 + p ⟶ N14 + 𝛾
N13 ⟶ C13 + e⁺ + 𝜈
N14 + p ⟶ O15 + 𝛾
N15 + p ⟶ He4 + C12
O15 ⟶ N15 + e⁺ + 𝜈

[13]:

net_cno = pyna.RateCollection(rates=rates)

[14]:

comp = pyna.Composition(net_cno.unique_nuclei)
comp.set_solar_like()

Again we can look evaluate each of the rates:

[15]:

net_cno.evaluate_rates(rho, T, comp)

[15]:

{C12 + p ⟶ N13 + 𝛾: 1.0467759242295144e-17,
 C13 + p ⟶ N14 + 𝛾: 3.176211597339995e-17,
 N13 ⟶ C13 + e⁺ + 𝜈: 2.9720852738029567e-07,
 N14 + p ⟶ O15 + 𝛾: 1.7295781984662848e-20,
 N15 + p ⟶ He4 + C12: 6.954822248932605e-16,
 O15 ⟶ N15 + e⁺ + 𝜈: 1.2625681768893481e-06}

Now we see that the \({}^{14}\mathrm{N}(p,\gamma){}^{15}\mathrm{O}\) rate is the slowest.

[16]:

r_cno = net_cno.get_rate_by_name("n14(p,g)o15")

This is a much steeper function of temperature. Let’s explore the temperature dependence around a few points

[17]:

for T in [1.e7, 2.e7, 3.e7, 4.e7]:
    print(f"T = {T}, nu = {r_cno.get_rate_exponent(T):8.5f}")

T = 10000000.0, nu = 22.82962
T = 20000000.0, nu = 17.96637
T = 30000000.0, nu = 15.60186
T = 40000000.0, nu = 14.11535

So it varies from \(T^{23}\) to \(T^{14}\) in a temperature range that includes the Sun’s core.

Here’s what the CNO cycle looks like

[18]:

fig = net_cno.plot(rho=rho, T=T, comp=comp, hide_xp=True)
../_images/examples_pp-cno_38_0.png

Comparing p-p vs. CNO#

The temperature dependence suggests that if the temperature were a little higher, then CNO would go faster then p-p. We can plot the two rate-limiting rates on the same axes

[19]:

import numpy as np
import matplotlib.pyplot as plt

[20]:

Ts = np.logspace(6.8, 8, 50)

[21]:

fig, ax = plt.subplots()

ax.loglog(Ts, [r_pp.eval(q) for q in Ts], label="p-p")
ax.loglog(Ts, [r_cno.eval(q) for q in Ts], label="CNO")
ax.set_xlabel("T")
ax.set_ylabel(r"$N_A\langle \sigma v \rangle$")
ax.grid(linestyle=":", which="both")
leg = ax.legend()
../_images/examples_pp-cno_43_0.png

Notice that they cross over at a temperature just hotter than the Sun’s core. This means that stars more massive than the Sun (actually, about \(1.5~M_\odot\)) are powered by CNO and less massive stars are powered by p-p. The strong \(T\) dependence of CNO will play a role in determining the structure of those more massive stars.