Inverse Square Law (Part 3)

Absolute Magnitudes and Luminosities

  • Definition: Absolute magnitude (M) is the brightness of a star as if it were at a distance of 10 parsecs (pc).

  • Relationship between Apparent and Absolute Magnitudes:

    • Apparent brightness decreases inversely with the square of the distance between the source and the observer.

    • Essential for comparing brightness of stars at different distances.

    • Two identical stars, where one is twice as far, result in the farther star appearing dimmer by a factor of 1/4.

  • Light spreads over larger areas, causing brightness to decrease with distance.

    • Inverse-square law leads to mathematical modeling of how brightness changes with distance.

Examples and Calculations

Example Scenario

  • Observational setup: two identical stars, one at distance d=1d=1 and the other at distance d=2d=2, twice as far as the first star.

  • Brightness relationship:

    • The brightness of the second star decreases to (12)2=14^{}\left(\frac12\right)^2=\frac14 of the first star.

Absolute Magnitude Calculation

  • Distance measuring and parallax angle can be used to determine absolute magnitudes.

  • Formula: M=m−5⋅log10(d/10)M=m-5\cdot{log}_{10}(d/10)

    • Where:

      • m = apparent magnitude

      • d = distance in parsecs.

      • Alternative formula to solve for the difference in apparent and absolute magnitude: m−M=5log⁡10(d)−5m-M=5\log_{10}\left(d\right)-5

      • Alternative formula to solve for d: d=10(m−M+55)d=10^{\left(\frac{m-M+5}{5}\right)} 

  • Example Calculation for Proxima Centauri:

    • Given:

      • d=1.3 pcd=1.3\space pc

      • m=+11.1m=+11.1

    • Applying formula:

      • M=11.1−5⋅log10(1.3/10)M=11.1-5\cdot{log}_{10}(1.3/10)

      • Resulting in M=11.1−5⋅log⁡10(1.310)=11.1−(−4.4)=+15.5M=11.1-5\cdot\log_{10}\left(\frac{1.3}{10}\right)=11.1-\left(-4.4\right)=+15.5

Brightness, Luminosity, and Energy Output

  • Luminosity: Total energy output of a star per second. Usually expressed in multiples of the Sun’s luminosity, denoted L☉L☉ where is approximately 3.83×1026W3.83\times10^{26}{ W}.

  • Absolute magnitudes:

    • Range from approximately M=−10M = -10 (very bright stars) to M=+17M = +17 (very dim stars).

    • Correlation: Smaller (more negative) absolute magnitude indicates greater luminosity.

Important Relationships Between Luminosity and Magnitude

  • When stars are nearer than 10 pc to us: L_{v}<l_{v}, and M_{v}>m_{v}

  • When stars are furtherthan 10 pc to us: L_{v}>l_{v}, and M_{v}<m_{v}