Lecture on Astrophysics: Stefan-Boltzmann Law and Stellar Radii

Introduction

Previously discussed Wien's Displacement Law and its relation to surface temperature and electromagnetic radiation.
Purpose: Understanding the temperature to estimate the size of stars using the Stefan-Boltzmann Law.

Connects intensity to temperature.
Electromagnetic Radiation: Emitted by heated bodies over a range of wavelengths.
Intensity (I): Proportional to the fourth power of absolute temperature (T).

$$ I \propto T^4 $$

$$ I = \sigma T^4 $$

$$ I = L / 4 \pi R^2 $$

$$ A = 4 \pi R^2 $$

$$ L = 4 \pi \sigma R^2 T^4 $$

Given: A black body with maximum intensity at 1450 nm and surface temperature 2000 K.
Star Beta Geese: Max intensity at 850 nm, luminosity = 3.1 × 10³¹ W.
Objective: Estimate radius (R).
Steps:
- Use Wien's Law: $$ \lambda_{max} \propto \frac {1}{T} $$
- Find T for Beta Geese: $$ \frac {1450}{850} = \frac {T_{1}}{T_{2}} $$
- Calculate T₂ = 3410 K.
- Use Stefan-Boltzmann Equation:

$$ 3.1 × 10^{31} W = 4 \pi (5.67 × 10⁻⁸)(R^2)(3410 K)^4 $$

Combining Laws: Wien's Displacement Law (temperature) and Stefan-Boltzmann Law (size) to estimate stellar radii.
Flow Chart/Overview: Observing EM Radiation using laws to estimate distance and size of stars.
- Radiant Flux Intensity: Luminosity/4π
- Standard Candles: Known luminosity for distance estimation.
- Measure Radiant Flux Intensity and λ_max.
- Wien's Displacement Law → Estimate Temperature (T).
- Stefan-Boltzmann Law → Find Radius (R).