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 (eg., 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)) ].

flux | Inverse Square Law of Light Brightness | luminosity |
---|

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

- 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*? - 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? - The Earth receives about 1380 Watts/meter
^{2}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.) - What is the
**luminosity**of star in Watts that has a**flux**of 2.7 x 10^{-8}Watts/meter^{2}and is 4.3 light years away from us? A light year is 9.461 trillion*kilo*meters or 9461 trillion meters.

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last updated: March 23, 2015

Author of original content: Nick Strobel