Stellar Structure

The internal structure of stars.

Equations of Stellar Structure

Mass of a shell inside the star: $$ m = \rho(r) * 4 \Pi r^2 \Delta r \tag{Eq 8.1} $$

For real stars, density peaks in the center. But we can approximate. Density of the Sun:

i.e. Density is Mass / Volume

$$ \bar{\rho} = \frac{M_\odot}{\frac{4 \Pi R_\odot^3}{3}} = 1400 {kg} m^{-3} $$

Density of water is \(1000 kg m^-3\)

Mean density of terrestrial planets is \(5000 kg m^{-3}\)

Central density of sun is \(1.6*10^5 kg m^{-3}\)

Hydrostatic Equilibrium

Stars are mostly in equilibrium. They are held together by:

  1. Self gravity. Inward force, eventual cause of collapse, other forces work against it.
  2. Pressure Forces due to hot gas.
  3. Rotating stars bulge around the equator, outward fictitious force.
  4. Magnetic field, which also helps to stop collapse inwards.

Self Gravitational

$$ g = - \frac{GM(r)}{r^2} \tag{Eq 8.3}$$

Force of gravity on shell mass M(r), for star radius r. $$ F_{gravity} = - \frac{GM(r)}{r^2}m \tag{Eq 8.4}$$

$$ F_{gravity} = -F_{pressure} \tag{Eq 8.5}$$

Pressure Estimation

$$ P_c = \frac{3GM^2}{8 \Pi R^4} \tag{Eq 8.6}$$

For the Sun, assuming constant density $$ = \frac{3 * 6.67E-11 * (2E30)^2}{8 \Pi * (7E8)^4} \equiv 1.3 * 10^{14} N m^{-2} $$

Real pressure at the center: \(3.4 * 10^{16} N m^{-2}\)

Thermal Equilibrium

Energy generation must equal radiation loss to achieve. True for the whole star and at any point within it.

$$ L(r) = Luminosity loss of energy at radius r $$

Luminosity increase through the stellar core, and then flat in the envelope.

$$ L_\odot = 4 * 10^{26} W $$

If energy generation is uniform throughout, then for a core mass:

$$ M_{core} \epsilon (J s^{-1}) $$ $$\epsilon \text{ is the generation rate } Js^{-1}kg^{-1} \tag{Eq. 8.7} $$

So can write: $$ M_{core} \epsilon (J s^{-1}) = L $$

For the Sun, energe generation rate: $$ \epsilon = \frac{L_\odot}{M_\odot} = 2*10^{-4} J Kg^{-1} s^{-1} $$

Energy Transport

  1. Conduction
  2. Convection
  3. Radiation

Thermal Conduction

Energy Flux. K is conductivity, dt/dr is temperature gradient: $$ F_{cond} = -K \frac{\delta T}{\delta r} J s^{-1} m^{-2} $$

Convection

Bulk motions in the medium.

Radiation

The dominant mechanism.
X-Ray and Gamma Ray photons near the center. Random walk of photons from the center of the sun to the outside. 300,000 Years to escape. $$ F_{rad} = -K_{rad} \frac{\delta T}{\delta r} J s^{-1} m^{-2} $$

Radiative Conductivity: $$ F_{rad} = -(\frac{4acT^3}{3\rho k}) \frac{dT}{dr} $$

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