PX282 - D7 - PP1 fusion chain

considering the hydrogen-fusion reaction:

4_{1}^{1} H \to_{2}^{4} He

governed by conservation laws:
- electric charge
- nucleon number
- lepton number
- energy and momentum
it is detailed in the PP1 chain, which is the most common in the sun
starts with the formation of deuterium:

2_{1}^{1} H \to_{1}^{2} H + e^{+} + ν_{e}

here, a proton has decayed into a neutron:

p \to n + e^{+} + ν_{e}

next:

\begin{matrix} _{1}^{2} H +_{1}^{1} H \to_{2}^{3} He + γ \\ 2_{2}^{3} He \to_{2}^{4} He + 2_{1}^{1} H \end{matrix}

overall:

4_{1}^{1} H \to_{2}^{4} He + 2 e^{+} + 2 ν_{e} + 2 γ

energy released

the reaction rates are set by the slowest process
for PP1, the initial decay of $p \to n$ requires the weak nuclear force and collisions, and it takes $10^{10}$ years for a given proton
energy is needed to overcome the strong nuclear force and escape the coulomb potential well
comparing the thermal and the coulomb energy:

\frac{3}{2} k_{B} T = \frac{Z_{1} Z_{2} e^{2}}{4 π ϵ_{0} r}

the required temperature for this case would be $T \approx 1.1 \times 10^{10} K$ , this is too high, and it can be concluded that the thermal energy is not enough
the only way this can happen is due to quantum tunnelling
according to the uncertainty principle

Δ p Δ x > ℏ

$Δ x$ can be large enough to 'cross the gap'
reaction ratios:

R_{1, 2} = n_{1} n_{2} σ v

where, $n_{1}$ and $n_{2}$ are number densities; $σ$ is the cross-section for collisions; $v$ is the relative velocity

the probability of tunnelling:

P \propto \exp (- \frac{2 π^{2} U}{E})

the velocity:

v \propto \exp (- \frac{E}{k_{B} T})

image: livius trache

expanding the exponentials:

R_{p p} \propto ρ^{2} X_{H}^{2} T^{4}

where, $ρ =$ general mass density; $X =$ mass fraction of hydrogen

the energy generation rate per unit mass:

ϵ_{p p} = ρ X_{H}^{2} T^{4}

other reactions

CNO cycle
- $H \to He : ϵ_{C N O} \propto ρ X_{H} X_{C N O} T^{20}$ for $T \sim 10^{8} K$
- the core gets denser as the reaction progresses, causing it to shrink and get hotter
- at high enough temperature, $H e$ can fuse
triple $α$ reaction:

3_{2}^{4} He \to_{6}^{12} C + γ

- $ϵ_{3 α} \propto ρ^{2} Y^{3} T^{40}$ for $T >\sim 10^{8} K$

the energy gets progressively lower, thus less time is taken for subsequent fusions into heavier nuclei