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

Lecture 11: The Internal Structure of Stars

Readings: none(!)

Key Ideas

Observational Clues to Stellar Structure:
- H-R Diagram
- Mass-Luminosity Relationship
Hydrostatic Equilibrium
- Balance between Gravity & Pressure
Core-Envelope Structure of Stars
- Hot, dense, compact core
- cooler, low-density, extended envelope

From Stellar Properties to Stellar Structure

Any successful theory of stellar structure must explain the observed properties of stars.

Seek clues in correlations among the observed properties, in particular:

Mass
Luminosity
Radius
Temperature

The Hertzsprung-Russell Diagram

Recall the Hertzsprung-Russell (H-R) Diagram introduced back in Unit 1. This is a plot of Luminosity vs. Temperature for stars:

H-R Diagram

The most prominent feature of the H-R diagram is the Main Sequence (M-S):

Strong correlation between Luminosity and Temperature.
Hotter stars are Brighter than cooler stars along the M-S.
About 85% of nearby stars, including the Sun, are on the M-S.

All other stars differ in size:

Giants & Supergiants:

Very large radius, but same masses as M-S stars

White Dwarfs:

Very compact stars: ~R_earth but with ~0.6 M_sun!

If we make a plot of Luminosity vs. Mass for the Sun and all Main Sequence stars in binary systems that have good mass measurements, we that a Main Sequence star's Luminosity is very strongly correlated with its Mass:

This strong correlation is called the Mass-Luminosity Relationship. It is expressed mathematically as:

In words:

"More massive M-S stars are more luminous."

This relation is only true for Main Sequence stars: Giants, Supergiants, and White Dwarfs do not follow the Mass-Luminosity relation.

The graph above shows data for the Sun and 121 binary stars for which there are reliable mass estimates (mostly eclipsing binaries with some nearby visual binaries, particularly at the low-mass end). Clicking on the plot will show a full-size version. The line drawn is the best-fit M-L relation.^[11.1]

Mean Stellar Density

The mean density^[11.2] of a star is:

Mean Density = Mass / Volume

Main Sequence: small range of mean densities

Sun (G2v): ~1.6 g/cc
O5v Star: ~0.005 g/cc
M0v Star: ~5 g/cc

Giants: Low-density stars: ~10^-7 g/cc (e.g., K5III)

Supergiants: Very low-density: ~10^-9 g/cc (e.g., M2I)

White Dwarfs: High-density stars: ~10⁵ g/cc

For reference, at sea level on Earth, water has a density of 1 g/cc, and air has a density of ~0.001 g/cc.

Interpreting the Observations:

For Main-Sequence Stars:

Strong L-T Relationship on H-R Diagram
Strong M-L Relationship

Implies that they have similar structures & governing laws.

Giants, Supergiants, and White Dwarfs must have very different internal structures from M-S stars of similar mass.

Laws of Stellar Structure I: The Ideal Gas Law

Most stars obey the Ideal Gas Law:^[11.3]

Pressure = Density x Temperature

In words:

Compressing a gas results in higher P & T
Expanding a gas results in lower P & T

Tells us how changes in the internal Temperature of a star affects its internal Pressure.

Laws of Stellar Structure II: The Law of Gravity

Stars are very massive & bound together by their Self-Gravity

Gravitational Binding Energy increases as 1/R

In words:

As a star contracts, it becomes more gravitationally bound.
As a star expands, it becomes less gravitationally bound.

Hydrostatic Equilibrium

Two opposing forces are at work within a star:

Gravity pulling inward wants to make the star contract.
Pressure pushing outwards wants to make star star expand.

Pressure & Gravity work on each other:

Gravity confines the gas in the star against Pressure expansion.
Pressure supports the star against Gravitational collapse.

When there is exact balance between the two, we have a condition of Hydrostatic Equilibrium.

In this condition, the star neither expands nor contracts.

Core-Envelope Structure

Outer layers press down on the inner layers.

The deeper you go into a star, the greater the pressure.

The Gas Law says:

Great pressure = hotter, denser gas

The consequence is that the star develops a Core-Envelope structure:

A hot, dense, compact central CORE surrounded by
A cooler, lower density, extended ENVELOPE

Example: The Sun

Core:

Radius = 0.25 R_sun
T = 15 Million K
Density = 150 g/cc

Envelope:

Radius = R_sun = 700,000 km
T = 5800 K
Density = 10^-7 g/cc

The Essential Tension

The life of a star is a constant tug-of-war between Gravity & Pressure.

Tip the internal balance either way, and it will change the star's outward appearance. This establishes an important principle for how stars work:

Internal Changes have External Consequences

Thus, changes in the interior conditions of stars will be reflected in their outward appearances, and hence affect where they will be located in the H-R Diagram.

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