Todd Thompson
Department of Astronomy
The Ohio State University

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

Key Equations

L ~ M⁴

U_G ~ GM²/R

P = n k_B T

From Stellar Properties to Stellar Structure

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

• Mass
• Luminosity
• Radius
• Temperature

The Hertzsprung-Russell 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.

Mass-Luminosity Relationship

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

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

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

Mean Density = Mass / Volume

V = (4/3) π R³

R ~ 6.95 x 10¹⁰ cm

M ~ 2 x 10³³ g

~1.4 g/cm³

• Sun (G2v): ~1.4 g/cm³
• O5v Star: ~0.005 g/cm³
• M0v Star: ~5 g/cm³

(Remember, "v" here stands for the main sequence.)

Giants: Low-density stars: ~10^-7 g/cm³ (e.g., K5III)

Supergiants: Very low-density: ~10^-9 g/cm³ (e.g., M2I)

White Dwarfs: High-density stars: ~10⁵ g/cm³

For reference, at sea level on Earth, water has a density of 1 g/cm³, and air has a density of ~0.001 g/cm³.

Note: In reality, stars are denser in their cores and become
less dense as you move outwards to the surface. The mean stellar
density is just a simple way to quantify how tightly packed the
matter is within a given star, and has the virtue that is easy to
compute (by assuming stars are spherical and the matter is uniformly
distributed throughout).

Interpreting the Observations:

• 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

Pressure = Density x Temperature

P = n k_B T or PV = N k_B T

In words:

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

Laws of Stellar Structure II: The Law of Gravity

Gravitational Binding Energy increases as the radius gets smaller

U_G ~ GM²/R

In words:

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

Hydrostatic Equilibrium

• 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

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

Great pressure = hotter, denser gas

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

Example: The Sun

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

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

The Essential Tension

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.