Distances---Inverse Square Law

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When the direct method of trigonometric parallax does not work for a star because it is too far away, an indirect method called the Inverse Square Law of Light Brightness is used. This method uses the fact that a given star will grow dimmer in a predictable way as the distance between you and the star increases. If you know how much energy the star emits (its luminosity), then you can derive how far away it must be to appear as dim as it does. Stars become fainter with increasing distance because their energy is spread out over a larger and larger surface.

A star's apparent brightness (its flux) decreases with the square of the distance. The flux is the amount of energy reaching each square centimeter of a detector (e.g., your eye, CCD, piece of the sphere) every second. Energy from any light source radiates out in a radial direction so concentric spheres (centered on the light source) have the same amount of energy pass through them every second. As light moves outward it spreads out to pass through each square centimeter of those spheres.

The same total amount of energy must pass through each sphere surface. Since a sphere has a surface area of 4× (its radius)², the flux of energy on sphere-1 = (the flux of energy on sphere #2) × [(sphere #2's radius)/(sphere #1's radius)]². Notice that the radius for the reference flux (sphere #2) is on the top of the fraction while the radius for the unknown flux (sphere #1) is on the bottom---this is an inverse square law! As the distance INcreases, the flux DEcreases with the square of the distance. In formula form, this means the star's flux = star's luminosity / (4× (star's distance)² ). See the math review appendix for help on when to multiply and when to divide the distance factor.

Put another way: As the flux DEcreases, the star's distance INcreases with the square root of the flux. If you know how much energy pours through the star's surface and you measure how much energy you detect here on the Earth, then you can derive the star's distance from you. The star's distance = Sqrt[ (star's luminosity) / (4× (star's flux)) ].

Vocabulary

Formulae

• Inverse Square Law: Brightness at distance A = (brightness at distance B) × [(distance B)/(distance A)]². Position (B) is the reference position.
• flux = luminosity / (4× distance²)
• Unknown distance = reference distance × Sqrt[(reference flux)/(measured flux)].
• Unknown distance = Sqrt[luminosity / (4× flux)].

Review Questions

1. Two identical stars have different apparent brightnesses (fluxes). One star is 10 parsecs away from us and the other is 30 parsecs away from you. Which star is brighter and by how many times?
2. Two identical stars have different fluxes. One star is 5 parsecs away from you and appears 81 times brighter than the other star. How far away is the dimmer star?
3. The Earth receives about 1380 Watts/meter² of energy from the Sun. How much energy does Saturn receive from the Sun (Saturn-Sun distance = 9.5 A.U.)? (A Watt is a unit for the amount of energy generated or received every second.)
4. What is the luminosity of star in Watts that has a flux of 2.7 x 10^-8 Watts/meter² and is 4.3 light years away from us? A light year is 9.461 trillion kilometers or 9461 trillion meters.

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last updated: June 18, 2022