7th Week: Stellar Structure and Evolution

As we look at the sky in the night, we can see that the Universe has a very complicated structure: planets, stars, galaxies, clusters etc. To understand how this stellar structure formed we use our knowledge of nuclear physics combined with cosmology.

Stellar structure

Stars of different mass and age have varying internal structures. Stellar structure models describe the internal structure of a star in detail and make detailed predictions about the luminosity, the color and the future evolution of the star.

Equations of stellar structure

Pressure equilibrium: Pressure and gravity must balance or the star will expand or contract. The balance of the gravity force and the pressure gradient is known as the hydrostatic balance.

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 \frac{dP_r}{dr}=-\rho_r \frac {GM_r}{r^2} }

Conservation of mass

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 \frac {dM_r}{dr}= 4\pi r^2 \rho _r }

Energy generation: To keep the temperature constant everywhere luminosity 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 L } must be generated.Considering the energy leaving the spherical shell yields the energy equation

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 \frac {dL_r}{dr}=4\pi r^2 \rho _r \epsilon }

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 \epsilon } is the energy generation rate (sum of all energy sources and losses) per g and s

Energy transport: The temperature inside a star must increase progressively towards its center to make energy flow from the center to the surface. The energy transport equation takes differing forms depending upon the mode of energy transport. For the case of convective luminosity transport 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 \frac{dT_r}{dr}=\left( 1- \frac{1}{\gamma} \right) \frac{T_r}{P_r}\frac{dP_r}{dr} }

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 \gamma = c_p / c_v } is the adiabatic index, the ratio of specific heats in the gas

For radiative energy transport 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 \frac{dT}{dr}=-\frac{3 \kappa \rho}{4acT^3} \frac{L(r)}{4 \pi r^2} }

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 \kappa } is the opacity of the matter 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 a } is the radiation density constant equal to 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 7.56591 \times 10^{15} erg/cm^3/K4 }

For conductive luminosity transport (appropriate for a white dwarf), the energy equation is

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 \frac{dT}{dr}= - \frac {1}{k} \frac {1}{4 \pi r^2} }

where k is the thermal conductivity.