Lecture 12

INTERNAL STRUCTURE OF THE STARS

Key Concepts

• Energy is generated in the star's hot core, then carried outward to the cooler surface.
• Inside a star, the inward force of gravity is balanced by the outward force of pressure.
• The star is stabilized by a pressure-temperature thermostat.

(1) Energy is generated in the star's hot core, then carried outward to the cooler surface.

Fusion of hydrogen into helium requires T > 10,000,000 Kelvin.
Temperatures at the surfaces of stars are much lower: T = 3000 to 50,000 Kelvin.

Implications:
The interior of a star must be much hotter than its surface.
Fusion occurs in the hot interior of a star, not at its surface.

In fact, the rate of energy production for fusion depends very strongly on temperature.

Proton-proton chain:
rate proportional to T⁴
Increase the temperature by 10%, you increase the rate of energy production by 46%.

CNO cycle: rate proportional to T¹⁶
(an astonishingly strong dependence)
Increase the temperature by 10%, you increase the rate of energy production by 350%.

The upshot is that energy production in a main sequence star is confined to a central core. In the Sun, about 95% of the fusion energy is produced in the central 20% of the Sun's radius.

The Sun is opaque. Thus, the gamma rays produced by fusion and by annihilation of positrons with electrons cannot go zipping straight out of the Sun. There must be some method of transporting energy from the central core, where it is produced, to the surface, where it is radiated away in the form of photons.

Three methods of transporting energy:

• Conduction: energy is transported by the random motions of ions, atoms, and molecules banging into each other.
• Radiation: energy is transported by photons.
• Convection: energy is transported by circular convection currents as hot fluid rises and cold fluid sinks.

Conduction doesn't occur inside the Sun. (It only works well in solids.)

Radiation is most effective in the Sun's inner region (R <0.7sun⁾

^{Convection is most effective in the Sun's outer region, which is too opaque for effective radiation transport.}

^{Other stars have different arrangements for transporting energy. Cool, low-mass main sequence stars are entirely convective. Hot, massive main sequence stars have convective interiors and radiative exteriors (opposite to the Sun's arrangement).}

^{Interesting note:
The radiative zone in the Sun is not totally transparent: photons do not go shooting directly from the core to the base of the convective zone. Instead, photons travel only about an inch (on average) before being absorbed by a nucleus or electron and being re-emitted at a lower energy in a random direction. The photons make a ``random walk'' through the radiative zone, and only stumble into the convective zone after a very long time, on average.}

^{If the photons made a direct flight from the core to the convective zone, it would only take about 1.7 seconds. The staggering random walks that they take, however, means that they require 1,000,000 YEARS to reach the convective zone.}

^{(2) Inside a star, the inward force of gravity is balanced by the outward force of pressure.}

^{Main sequence stars are stable in size, neither expanding nor contracting. At every point in the star, the inward force of gravity is precisely balanced by the outward force of pressure. This balance between gravity and pressure is called hydrostatic equilibrium.}

^{Pressure is proportional to density times temperature.}

^{At the center of a star, pressure is very high. It has to be, in order to support the crushing weight of all the matter above it. The pressure is high because the density and temperature are high.}

^{In the Sun:}

• ^{central density = 150 grams / cm3}
• ^{central temperature = 14,000,000 Kelvin}
• ^{central pressure = 300 trillion times air pressure at sea level on Earth.}

^{(3) The star is stabilized by a pressure-temperature thermostat.}

^{Interesting question: what prevents fusion in the center of a main sequence star from ``running away'', turning the star into a hydrogen bomb? After all, the energy released by fusion increases the temperature of the core; a higher temperature means a higher fusion rate, which means a higher temperature, which means....well, you get the picture.}

^{What prevents a fusion runaway is the fact that the relation among pressure, temperature, and fusion rate creates a natural thermostat. (A thermostat is a mechanism which maintains a constant temperature, like the gizmos in furnaces and hot water heaters.)
Let's see how the natural thermostat in a star's core works.}

^{Suppose we start by INCREASING the fusion rate slightly.}

• ^{(1) Temperature increases.}
• ^{(2) Pressure increases.}
• ^{(3) Core expands.}
• ^{(4) Density and temperature decrease.}
• ^{(5) Fusion rate decreases.}

^{So there's a feedback mechanism which prevents the fusion rate from skyrocketing upward.}

^{Now suppose we start by DECREASING the fusion rate slightly.}

• ^{(1) Temperature decreases.}
• ^{(2) Pressure decreases.}
• ^{(3) Core contracts.}
• ^{(4) Density and temperature increase.}
• ^{(5) Fusion rate increases.}

^{So the feedback mechanism also prevents the fusion rate from nosediving downward.}

^{Of course, a thermostat can only keep your furnace (or star) at a constant temperature as long as you have a supply of fuel. If you run out of fuel, the temperature will drop unless you find an alternate fuel source.}

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