Brightness, Luminosity and Flux of Stars Explained

Luminosity and flux are a measure of the total energy radiated by a star, galaxy, or another object per unit time in joules/second or watts.

In astronomy, luminosity and flux are both measures of an object's energy output. Luminosity is the total amount of energy radiated by a star, galaxy, or another astronomical object per unit time. It is related to brightness, which is the luminosity of an object in a given spectral region. In SI units luminosity is measured in joules per second or watts.

What is Luminosity?

Luminosity, given the symbol L in equations, is the total outward flow of energy from a radiating body per unit of time, in all directions and over all wavelengths. The SI units of luminosity are Watts (W) which quantify the rate of energy transfer in joules per second.

Luminosity is the rate at which a star, or any other body, radiates its energy. It is the same as the classification of light bulbs. A 40W bulb radiates less energy than a 100W bulb. The same is true for stars, however, they radiate far greater quantities of energy. The energy output of our Sun is around 3.89x10²⁶ W (that's 389 followed by 24 zeros!), and our Sun isn't even a bright star! As stars go, our Sun is a 20W bulb.

Because of the large quantities involved with luminosity, astronomers prefer to use a more convenient unit called solar luminosity, given the symbol L_⊙. One solar luminosity is equal to the luminosity of our Sun, but even so, stars can be as high as 1x10⁶L_⊙ so very large numbers cannot be avoided!

A star which has a luminosity of 2L_⊙ is twice as luminous as our Sun, and a star of 0.5L_⊙ is half as luminous.

What is Flux?

Flux, F, is defined as the total flow of light energy perpendicularly crossing a unit area per unit of time, or the total energy from an object per unit area over time. Flux is independent of the direction of that energy. Flux has units of J s^-1 m^-2 or W ^-2 (that's Joules per second per metre squared or simply Watts per second).

Intensity is closely linked with Flux. Intensity is measured in Watts per Square meter per Steradian (square radian). That means Intensity is a measure of the amount of radiant energy passing through a given area, in the direction of a given solid angle. There's a lot more information contained in Intensity since it can also be a function of direction (theta).

What is the Difference between Luminosity and Flux?

Whereas flux is the energy received over a unit area, luminosity is the total energy output of the star. Since the star radiates in all directions (isotropically) we only receive a tiny fraction of the energy radiated which is how we observe flux and calculate apparent magnitude. It would be helpful to know the relationship between the flux observed at Earth and the star's luminosity (total energy output).

Imagine a star at a distance d radiating equally in all directions. Flux is measured with a detector (whatever type) and has a surface area of 1 m² and is perpendicular to the star. From the diagram above we can see that the star radiates in all directions and forms a sphere around the star.

From previous definitions, luminosity (L) is the total energy output per second and flux (F) is the total energy per second crossing a unit area of surface, so we can determine the relationship between luminosity and flux. The surface area of a sphere of radius d is:

Equation 21 - Sphere, Surface Area of

So the flux, F, measured at Earth by the detector of the unit area is given by:

Equation 22 - Luminosity and Flux

We can see from the equation that flux decreases as distance increases and we can also see that distance is squared. It follows from this that light obeys the inverse square law - the observed flux from a star is inversely proportional to the square of the distance between it and an observer. This is more clearly illustrated in the diagram below.

The brightness of a star seen from the Earth depends on its intrinsic luminosity and also on its distance from the Earth. The observed brightness of a given star decreases inversely proportionally to its distance away. The presence of interstellar gas will further decrease the observed brightness. From the diagram we can see that for every additional distance unit, r, the light is spread over an additional area, r².

What is the Ratio of Flux?

In the article about the magnitude scale we saw that Pogson devised a scale whereby a 1st magnitude star is 100 times brighter than a 5th magnitude star. This logarithmic scale states that a 1st magnitude star is 2.512 times brighter than a 2nd, which is 2.512 times brighter than a 3rd and so on...

We can use this constant ratio per magnitude to obtain a formula for the ratio of fluxes. Consider two stars that have apparent magnitudes m and n and measured fluxes of F_m and F_n, the ratio of the fluxes is given by:

Equation 18 - Ratio of Fluxes

If one star is 6th magnitude (n = 6) and another star is 1st magnitude (m = 1) then the magnitude difference is given by (n-m) = 6-1 = 5 We can use Equation 18 to calculate the ratio of these fluxes.

Equation 19 - Ratio of Fluxes

We can see that this equation has shown that a difference in 5 magnitudes affects magnitude by a factor of 100 as per Pogson's rule. We can further reinforce this relationship between flux and magnitude by showing that the magnitude difference between two objects can be expressed in terms of the logarithm of the flux ratio. This form is known as Pogson's relation and one form or another one of the most useful equations in the astronomer's toolbox.

Equation 20 - Pogsons Relation