Gravity vs. Pressure

The Physics of Protostellar Collapse

Lesson 01 // Stellar Evolution

The Birthplace

Giant Molecular Clouds (GMCs)

Physical Conditions

• Temperature: \(T \approx 10 - 20\) K
• Density: \(n \approx 10^2 - 10^6\) cm\(^{-3}\)
• Mass: \(10^4 - 10^6 M_{\odot}\)

"The Pillars of Creation" demonstrate how radiation pressure from nearby OB stars sculpts these dense cold regions.

[ Visualization: Infrared vs Visible ]

At these temperatures, the peak of the Planck curve is in the far-IR.

The Jeans Instability

Collapse occurs when the magnitude of the gravitational potential energy exceeds the internal thermal energy.

The Jeans Mass (\(M_J\))

\[ M_J \approx \left( \frac{5 k T}{G \mu m_H} \right)^{3/2} \left( \frac{3}{4 \pi \rho} \right)^{1/2} \]

Cooler Clouds

Lower \(T\) reduces \(M_J\), allowing smaller masses to collapse.

Denser Clouds

Higher \(\rho\) reduces \(M_J\), triggering fragmentation.

Self-Gravity Wins

Triggering Events:

• Supernova shockwaves
• Galactic spiral arm compression
• Cloud-cloud collisions

The Virial Theorem

For a system in statistical equilibrium, the time-averaged kinetic energy \(K\) and potential energy \(U\) are related by:

\[ 2K + U = 0 \]

Gravitational Energy

\(U \approx - \frac{GM^2}{R}\)

Thermal Energy

\(K = \frac{3}{2} NkT\)

Implication for Collapse

As a cloud collapses (\(U\) becomes more negative), half of the released potential energy goes into heating the gas (\(K\)), and the other half is radiated away.

Temperature Regulation

The star has a negative heat capacity; losing energy makes it hotter.

Hydrostatic Equilibrium

GRAVITY vs PRESSURE

The fundamental equation of stellar structure:

\[ \frac{dP}{dr} = - \frac{G M(r) \rho(r)}{r^2} \]

Ensures the star neither expands nor contracts.
Relates pressure gradient to local gravitational pull.
Requires pressure to increase monotonically toward the center.

Stellar Evolution and Nucleosynthesis

Gravity Versus Pressure

The Stellar Map

Internal Engines

Evolutionary Tracks

Ultimate Remnants

Gravity vs Pressure Presentation Slides

Jeans Mass Calculation Worksheet

Gravity vs Pressure Teacher Guide Notes Key

HR Diagram Workshop Slides

Star Cluster Data Lab

HR Diagram Teacher Guide Notes

Internal Engines Slides

Energy Transport Problems

Fusion Cycles Cheat Sheet

Evolutionary Tracks Slides

Evolutionary Path Chart

Post-MS Facilitation Guide

Death of Stars Slides

Degeneracy and Limits Worksheet

Extreme States Summary

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