5th Week: Reaction Rates, Nuclear Networks & NSE

So far we have studied the nuclear reactions and the main thermodynamic properties. Now we move on further to use this knowledge to predict the astrophysical plasma processes such as energy release and composition changes.

Thermonuclear reaction rates

From the definition of nuclear cross section we know that the nuclear cross section between target Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} and projectile Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} can be written as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma = \frac{r/n_i}{n_jv} \ , }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v} is the relative velocity between targets with the number density Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_i} and projectiles with the number density Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_j} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} is the number of reactions per Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle cm^3} per Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle sec} . From here one can find

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r = \sigma v n_i n_j \ , }

or more generally when targets and projectiles follow the specific distribution then

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_{i; j} = \int \sigma \cdot \left|\vec{v_i}-\vec{v_j}\right| dn_i dn_j \ . }

For the nuclei in astrophysical plasma that obey the Maxwell-Boltzmann distribution we can write

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle dn_j = n_j \phi \left(\vec{v_j}\right) d^3 v_j \ , }

where

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi \left(\vec{v_j}\right) = \left( \frac{m_j}{2 \pi k T} \right)^{3/2} \exp\left[-\frac{m_jv_j^2}{2kT}\right ] \ .}

Nuclear Networks

We saw that there could be two types of expressions for the reaction rates

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_{i,j}=\frac{1}{1+\delta_{ij}}n_in_j < \sigma v> \ ,}

where the delta factor is included to avoid double counting when we have identical particles. And,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_{i,j}=\lambda _i n_i \ .}

So if we have the reaction Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i+j \rightarrow o+m \,} the change densities is given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left( \frac{\part n_i}{\part t} \right)_{\rho}=\left( \frac{\part n_j}{\part t} \right)_{\rho}=-r_{i,j}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left( \frac{\part n_o}{\part t} \right)_{\rho}=\left( \frac{\part n_m}{\part t} \right)_{\rho}=+r_{i,j}}

Now recall that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y_i=\frac{n_i}{\rho N_A} }

Then taking the derivative with respect to time we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot Y_i=\frac{n_i}{\rho N_A} \left( \frac{\part n_i}{\part t} \right)_{\rho} }

Using the previous equations we can write it as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot Y_i=\frac{r_{i,j}}{\rho N_A}=-\frac{1}{1+\delta_{ij}}\rho N_A <\sigma v> _{i,j} Y_iY_j}

and in the case where we produce a nucleus m

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot Y_m=\frac{1}{1+\delta_{ij}}\rho N_A <\sigma v> _{i,j} Y_iY_j}