8th Week: Reaction flows, pp- and CNO-cycle & Solar Neutrinos B

pp-chain

The pp-chain is the major energy producer of stars less than three solar masses. This reaction chain takes four protons and produces helium. However these reactions do not take the most direct route. These reactions are as follows for the ppI chain.

$p\left(p,e^{+}\nu \right)d$

$d\left(p,\gamma \right)^{3}He$

$^{3}He\left(^{3}He,2p\right)^{4}He$

This reaction chain has a bottleneck which limits the overall rate of this chain. This reaction is the $p\left(p,e^{+}\nu \right)d$ reaction. This limiter is seen by looking at the lifetime of a proton before annihilation by another proton. This can be calculated as follows:

$\tau ={\frac {1}{\rho N_{A}<\sigma v>}}$

After calculating for the conditions in the sun, the lifetime of a proton is on the order of the lifetime of the sun. This is placed juxtapose to the lifetime of a proton before annihilation by deuterium (lifetime of 1.6 s) is a long long time.

CNO Cycle

This is the major energy producer for stars greater than three solar masses. Also this cycle is the preferred method of helium-4 production from protons for these stars. The CNO cycle has two major forms, one is known as the CNO cycle and the other is known as the hot CNO cycle.

The CNO cycle consists of the following network: $^{12}C(p,\gamma )^{13}N$

$^{13}N(-,e^{+}\nu _{e})^{13}C$

$^{13}C(p,\gamma )^{14}N$

$^{14}N(p,\gamma )^{15}O$

$^{15}O(-,e^{+}\nu _{e})^{15}N$

$^{15}N(p,\alpha )^{12}C$

Solar neutrinos

Every time we have neutrinos we can be sure that a weak interaction has occur. Neutrinos spin 1/2 particles with no charge, and almost no mass that come in three flavors: electron neutrino ($\nu _{e}$) , muon neutrino $\nu _{\mu }$ and tau neutrino $\nu _{\tau }$. According to the standard model we have the upper limits for the masses of neutrinos:

Flavor	Mass
$\nu _{e}$	<2.2 eV/$c^{2}$
$\nu _{\mu }$	<170 KeV/$c^{2}$
$\nu _{\tau }$	<15.5 MeV/$c^{2}$

In the Sun neutrinos are emitted in the following pp-chain reactions:

• $p+p,\rightarrow e^{+}+\nu +d$ (this is the first reaction in the pp-chain)

• $^{7}{\mbox{Be}}+e^{-}\rightarrow \nu +^{7}{\mbox{Li}}$ from pp-II

• $^{7}{\mbox{Be}}\rightarrow ^{7}{\mbox{B}}+e^{+}\nu +\gamma$ from pp-III

If reactions produce electrons, positrons, or $\gamma$-rays, then their energy is retained in the stellar plasma. Neutrinos, on the other hand, interact so weakly with the medium that they escape from the site of thermonuclear burning. Since the neutrino energy is usually not deposited in the star, it has to be subtracted from the Q-value when calculating the nuclear energy generation.

Properties of Neutrinos

Early attempts to study the sun via neutrinos were thwarted by an unexpected result. The Homestake experiment intended to measure the neutrinos emitted from the sun, but found only a fraction of the expected output. Over the subsequent years many experiments showed that neutrinos oscillate between flavors. The measurements also showed that there is a day/night effect in the number of neutrinos which results in another property of neutrinos, that they resonate in matter.

This resonance condition can be summed up as:

$V(n)={\sqrt {2}}G_{F}n_{e}={\frac {\delta m^{2}}{2E}}cos(2\theta )\;$