Astronomy 162:
Introduction to Stars, Galaxies, & the Universe
Prof. Richard Pogge, MTWThF 9:30

Lecture 13: Energy Generation & Transport in Stars

Readings: none, but see Ch 18 section 18-2

Key Ideas

Energy generation in stars:: Nuclear Fusion in the core.; Controlled by a Hydrostatic "thermostat".
Energy is transported to the surface by:: Radiation (photons); Convection (gas motions); Conduction
Thermal Equilibrium in Stars

Putting Stars Together

Physics needed to describe stars:

Law of Gravity
Equation of State ("gas law")
Principle of Hydrostatic Equilibrium
Source of Energy (e.g., Nuclear Fusion)
Way to transport that energy to the surface.

Hydrostatic Equilibrium

Balance between Pressure & Gravity.

If Pressure dominates, the star expands.
If Gravity dominates, the star contracts.

Sets up a Core-Envelope Structure:

Hot, dense, compact core.
Cooler, low-density, extended envelope.

Energy Generation

Stars shine because they are hot.

To stay hot stars must make up for the energy lost by shining.

Two Energy sources are available to stars:

Gravitational Contraction (Kelvin-Helmholtz)
Nuclear Fusion in the hot core.

Without a source of internal energy, a star would eventually cool off and go out.

Main-Sequence Stars

Generate energy by fusion of 4 ¹H nuclei (protons) into 1 ⁴He nucleus.

There are two nuclear reaction paths by which a star might accomplish this fusion:

Proton-Proton Chain:

Relies on proton-proton reactions
Dominates at low core Temperatures (T_C<18M K)

CNO Cycle:

Carbon acts as a catalyst
Dominates at high core Temperatures (T_C>18M K)

Proton-Proton Chain:

Fuse Hydrogen into Helium via a three-step nuclear reaction chain:

Notice that the p-p chain uses 6 protons in all, and ends up with 1 ⁴He nucleus and 2 protons at the end, for a net conversion of 4 protons into 1 Helium, with the release of energy as gamma-ray photons, neutrinos, and positrons.

CNO Cycle:

Fuse Hydrogen into Helium via a multi-step nuclear reaction cycle catalyzed by Carbon:

Notice that the CNO cycle starts with one ¹²C nucleus in step one, and add 4 protons during each of steps 1, 3, 4, and 6. At the end, you get a ⁴He nucleus and the ¹²C nucleus. This ¹²C nucleus is ready to once again jump back into the cycle.^[13.1]

The result is a net conversion of 4 protons into 1 Helium nucleus, with a release of energy in the form of gamma-ray photons, neutrinos, and positrons.

Because ¹²C is not consumed by this process (it goes in at the top & comes out at the end), we say that it acts as a catalyst in this nuclear reaction.

Because Carbon and Nitrogen have 6 and 7 protons, respectively, in order to overcome the repulsion of all these positive charges the protons must be moving very fast. This is why the CNO cycle occurs at higher temperatures than the P-P chain. The Sun only gets about 2% of its energy from CNO, but in slightly larger stars, about 1.1M_sun, CNO accounts for 50%, and then dominates the energy production at all higher masses.^[13.2]

Controlled Nuclear Fusion

Nuclear fusion reactions are Temperature sensitive:

Higher Core Temperature = More Fusion

BUT,

More fusion makes the core hotter,
Hotter core leads to even more fusion, ...

Why don't stars explode like a Hydrogen Bombs?

Hydrostatic Thermostat

If the fusion reactions run too fast:

The core heats up, leading to higher Pressure.
Higher Pressure makes the core expand against gravity.
Expansion cools core, slowing the rate of fusion.

If the fusion reactions run too slow:

The core cools, leading to a lower Pressure.
Lower pressure makes the core contract under gravity.
Contraction heats core, increasing the rate of fusion.

Result is like a thermostat: an increase or decrease in the rate of fusion results in a compensating response by the gas in the core. This prevents either runaway heating and explosion or a runaway collapse.

Thermal Equilibrium

We need to start with a basic principle of thermodynamics:

Heat always flows from hotter regions into cooler regions.

In a star, heat must flow:

From the hot core where the fusion is.
Out through the cooler envelope,
To the surface where it is radiated away into space as light.

Energy Transport

There are 3 ways to transport energy away from a heat source:

Radiation: Energy is carried by photons
Convection: Energy carried by bulk motions of the gas
Conduction: Energy carried by particle motions

Radiation

Energy is carried by photons radiating away from the heat source. If we follow an average photon emitted in the core, its path outward to the surface is as follows:

Photon leaves the core
Hits an electron or atom within ~1-mm and gets scattered.
Slowly staggers to the surface in a "random walk"

On average, it takes about 200,000 years for a photon from the core to random walk its way to the surface.^[13.3]

Convection

Energy carried from hotter regions below to cooler regions above by bulk buoyant motions of the gas.

Everyday examples of convection are boiling water and hot air "rising" off of a candle flame or a radiator.

In the figure above

Hot water near the flame becomes buoyant and rises against gravity.
As it rises, it displaces cooler water downward
When it reachs the top, it gives up its heat to the surrounding cooler water
Cool water displaced downward heats and repeats the cycle

Result is to setup a circulating "convection flow" in the liquid ("boiling").

The analoguous process can occur in some stars. Convection becomes efficient when the gas in the hot layers of a star cannot pass radiation through it effectively.

Conduction

Heat is passed from atom-to-atom in a dense material from hot to cool regions.

Example: Hold the a spoon by the handle and put its bowl into a candle flame. Over time, heat will get conducted from the bowl up the handle and you will feel it heat up. If you hang on too long, it will become so hot you will burn your fingers.

Conduction is most efficient when the densities are very high (atoms or electrons packed in close proximity to each other).

Energy Transport in Stars

Normal Stars:

A mix of Radiation & Convection transports energy from the nuclear fusion site in the core to the surface.
Conduction is inefficient (the density is too low).

White Dwarfs:

Ultra-dense stars (~10⁵ g/cc) with no nuclear fusion in the core.
Conduction dominates energy transport.
Results in a nearly uniform temperature from core to surface

Thermal Equilibrium in Stars

A star will be in Thermal Equilibrium when the amount of energy generated in the core is balanced by the transport of that energy to the surface to be radiated away as starlight.

Energy Generation = Star's Luminosity

Thermal equilibrium is a delicate balance:

Generate more energy than required by the star's Luminosity, the excess energy makes the star expand.
Generate less energy than required, and the deficit is made up by the star contracting.

Thermal equilibrium plays a vital role in the evolution of stars.

Summary:

Energy generation in stars:

Nuclear Fusion in the core
Controlled by a Hydrostatic "thermostat"

Energy generated in the core is transported to the surface by:

Radiation & Convection in normal stars
Conduction in white dwarf stars

The balance between energy generation and energy transport is called Thermal Equlibrium.

Along with Hydrostatic Equilibrium, these determine the detailed structure and evolution of a star.

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