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Luminosity and Flux in Astrophysics

Apr 8, 2025

Lecture Notes: Luminosity and Flux Calculations

Overview

  • The lecture discusses the luminosity flux equation focusing on the Sun and the star Sirius.
  • Key points include calculations at different distances and conditions, using particular equations and constants.

Introduction

  • Flux from the Sun at Earth's distance is about 1370 watts/m² above Earth's atmosphere.
  • Definition: 1 watt = 1 joule/second.
  • Flux from Sirius: 1 x 10⁻⁷ watts/m², despite its higher luminosity compared to the Sun, resulting from its greater distance.

Part A: Calculating Sun's Luminosity

  • Objective: Use the equation L = 4πd² x Flux to find the Sun's luminosity in watts.
  • Distance to Sun:
    • 1 Astronomical Unit (AU) = 1.496 x 10¹¹ meters.
  • Calculation Steps:
    • Apply the given flux of 1370 watts/m².
    • Ensure units of distance are in meters and flux in watts/m² for accurate unit cancellation (meter² cancels out).
    • Result: Approximately 3.85 x 10²⁶ watts.

Part B: Calculating Sirius's Luminosity

  • Objective: Use the same equation for Sirius.
  • Distance to Sirius:
    • 8.8 light-years.
    • 1 light-year = 9.46053 x 10¹⁵ meters.
  • Calculation Steps:
    • Convert light-years to meters.
    • Use the flux of 1 x 10⁻⁷ watts/m².
    • Ensure proper unit conversion and calculation for luminosity.

Part C: Hypothetical Scenario for Sirius

  • Objective: Calculate luminosity assuming Sirius has a surface temperature of 15,000K.
  • Equation Used: Stefan-Boltzmann Law: Flux = σT⁴.
    • Where σ (Stefan-Boltzmann constant) = 5.6697 x 10⁻⁸ W/m²K⁴.
    • T = Effective surface temperature.
  • Radius of Sirius: 8 x 10⁸ meters.
  • Steps:
    • Calculate surface flux with given σ and T.
    • Use this flux in L = 4πR² x Flux, where R is the radius of Sirius.
    • Result should show higher luminosity due to higher hypothetical temperature.

Key Points

  • Flux at the surface versus flux received at Earth is crucial for understanding luminosity calculations.
  • Proper unit conversion and careful calculation with a calculator are emphasized.
  • Calculations demonstrate the relationship between temperature, radius, and distance in determining luminosity.

Conclusion

  • Understanding these calculations is essential for working with astrophysical data and applications.
  • Practice is crucial for mastering these types of problems.

Full transcript