Star convection gradient dependencies

Question

In star convection you can work with radiative and adiabatic temperature gradients, that for example the radiative one can be defined as,

$$\nabla_{rad}=\frac{3\kappa L P}{16 \pi a c G m T^4}$$

This can be seen for example in this source. Here they put emphasis in the dependency of luminosity $L$ and mass $m$ with radius, i.e. $L=L(r)$ and $m=m(r)$. But otherwise for the other star variables is not specified,

• $\kappa$ the opacity
• $P$ the pressure
• $T$ the temperature

As is not specified, my doubt is if this variables are used as radius dependent ones or fixed values. In this case what variable should I use? The star-center values, or the surface values?

Answer 1

The opacity, pressure and temperature all vary as a function of position in a star. For a spherically symmetric star you can assume they vary only as a function of the radial coordinate $r$.

The reason that luminosity and mass are written as $L(r)$ and $m(r)$ is to remind us that these do vary with radial coordinate and avoid confusion with the luminosity and mass of the entire star, which are meaningful and observable quantities. $L_* = L(R)$ and $M_* = m(R)$ respectively, where $R$ is the radius of the star.