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# Stellar Structure

 Bright star, would I were steadfast as thou art --John Keats

## Overview

• Examine the Some of the equations that describe the structure of stars: equation of state, hydrostatic equilibrium, energy transfer, and energy generation.
• Gain a conceptual understanding of stellar structure.

## Introduction

Stars live for astonishingly long times. This implies great stability as the internal pressure of the star balances the inward pull of gravity. This internal pressure is maintained by the energy from thermonuclear fusion in the core, but the fusion first starts because the core gets hot. Why does it get hot in the first place?

### Equation of State

Recall that stars form from the collapse of gas clouds. These clouds start off as structures that are a parsec or two across, yet the stars themselves are a tiny fraction of this size. During the collapse, the gas becomes compressed. A well-known property of a gas – more specifically, of an ‘ideal gas' – is that it heats up when it is compressed. Since temperature is a measure of the motions of the atoms in the gas, the atoms move faster as the gas compresses. This behavior obeys the equation of state (also known as the ideal gas law), which can be written:

P ∝ ρT

where P is the gas pressure, T is the temperature, and ρ is the gas density, which is equal to the number of particles per volume.

Note that if a gas is compressed so the pressure goes up, either the temperature or density – or both – must rise to account for that pressure change.  In the case of a star, both of these things happen: the density of the matter increases, but so does its temperature. The gas originally has a density of a few thousand atoms per cubic cm and a temperature of 10-20 K, and gravity easily forces it to collapse.  By the time fusion starts in the stellar core, the density is about 90 g/cm3 (that's about 15 times denser than steel, but it is still a gas) and the temperature is about 15 million K!  This results in a pressure high enough to balance the gravity of the star.

### Hydrostatic Equilibrium

If the pressure did not balance gravity, the Sun would collapse in just a few minutes! This form of stability is known as hydrostatic equilibrium and tells us that the pressure must precisely balance the gravity everywhere in the star for it to be stable. We are all familiar with this concept: a glass of water is in hydrostatic equilibrium. If the internal pressure were too large, the water would explode. If too small, the water would compress downward further.

### Energy Transfer

The energy of a star is produced in its core, but that energy must travel to the surface before it can expand outward into space as the light we see. There are two basic ways a star can move its energy from inside out: convection and radiation. Convection involves the mass movement of matter in the star so that the hot material moves upwards towards the surface, then cools and sinks back down. The matter transfers its energy by cooling, moving the energy to higher and higher levels of the star until it eventually reaches the surface.  Radiation can also transfer energy within a star if the gas of the star is sufficiently transparent so that the radiation can travel a long distance before being absorbed. In many stars, both convection and radiation operate at different locations. For the Sun, the core is radiative while the outer layers are convective. On Earth, radiation from the hot ground can often escape into space by simply radiating away. But sometimes, this energy causes convection to set up resulting in cumulus and cumulonimbus clouds, especially in the summer. Thus, even in our atmosphere, both mechanisms can operate.

### Energy Generation

The energy from a mature star derives from the steady thermonuclear fusion of smaller atoms into larger ones. In main sequence stars, fusion involves the conversion of hydrogen into helium with the associated decrease of mass, and the corresponding generation of energy through Einstein's equation, E = mc2. The equation of state and the hydrostatic equilibrium condition ensure that the core is both sufficiently dense and hot for this fusion to occur. If the density is too low, atoms don't encounter one another.  If the temperature is too low, the atoms don't actually combine.

To actually determine how fusion generates energy requires not only knowing how much mass is converted to energy, but also how frequently the reaction occurs. This demands knowledge of what is known as the ‘cross section' for fusion. This is just the probability that two atoms will fuse as a function of temperature and density.