This luminosity calculator is a handy tool that allows you to calculate the energy emitted by stars, as well as how bright they appear to be when seen from Earth. Thanks to this calculator, you will also be able to determine the absolute and apparent magnitudes of stars. But that's not all - we will also provide you with a handy luminosity equation that will make comparing any two stars a piece of cake!

What is luminosity?

Luminosity is a measure of the energy radiated by an object, for example a star or a galaxy. For the stars of the main sequence, luminosity is directly related to their temperature - the hotter a star is, the more luminous it is. On the other hand, cooler stars emit less energy - hence, it's more difficult to spot them in the night sky.

Luminosity equation

The formula for stellar luminosity can be derived directly from the Stefan-Boltzmann law. This law states that for a black body, the energy radiated per unit time is equal to

P = σ * A * T⁴

where

  • σ is the Stefan Boltzmann constant, equal to 5.670367 * 10⁻⁸;
  • A is the surface area of the body (equal to A = 4πR² for spherical objects);
  • T is the temperature of the body, expressed in Kelvins.

Our luminosity calculator, uses a simplified version of this formula. Instead of calculating the energy as an arbitrary value, we can compare any star to the Sun. Then, after canceling the constants, we arrive at the luminosity equation:

L / L☉ = (R / R☉)² * (T / T☉)⁴

where:

  • L is the luminosity of the star;
  • R is the star's radius;
  • T is the star's temperature, measured in Kelvins;
  • L☉ is the luminosity of the Sun, equal to 3.828 * 10²⁶ W;
  • R☉ is the Sun's radius, equal to 695700 km;
  • T☉ is the temperature of the Sun, equal to 5778 K.

Absolute and apparent magnitude

You can also use this tool as an absolute magnitude calculator. Absolute magnitude is a different way to measure the luminosity. Instead of expressing it in watts, it can be shown on a logarithmic scale.

The lower the absolute magnitude, the more luminous the star is - some very bright stars can even have negative magnitudes! For example, the absolute magnitude of the Sun is equal to 4.74, and of Bellatrix to −2.78.

Absolute magnitude and luminosity are related with the formula

M = -2.5 * log₁₀(L / L₀)

where:

  • M is the absolute magnitude of the star;
  • L is its luminosity;
  • L₀ is the zero-point luminosity, equal to 3.0128 * 10²⁸ W.

Apparent magnitude, on the other hand, is a measure of brightness when the star is seen from Earth - hence, it takes into account the distance between the star and the Earth. You can find it with the apparent magnitude calculator, using the following equation:

m = M - 5 + 5*log₁₀(D)

where

  • m is the apparent magnitude of the star;
  • M is the absolute magnitude of the star;
  • D is the distance between the star and Earth, measured in parsecs.

The absolute magnitude is defined as the apparent magnitude of an object seen from the distance of 10 parsecs. It means that for D = 10 parsecs, the apparent and absolute magnitudes are equal in value.

Calculating luminosity: an example

Let's analyze Sun with this luminosity calculator to investigate its absolute and apparent magnitude.

  1. Input the radius and temperature of the Sun into the calculator. The radius is equal to R☉ = 695700 km, and the temperature to T☉ = 5778 K.

  2. The luminosity calculator will automatically find the luminosity of the Sun. It is equal to 3.828 * 10²⁶ W.

  3. To determine the absolute magnitude of the Sun, you can use the following equation:

M = -2.5 * log₁₀(L / L₀)

M = -2.5 * log₁₀(3.828 * 10²⁶ / L₀)

M = 4.74

  1. Last but not least, you can find the apparent magnitude of the Sun. The distance between the Earth and Sun is equal to 4.848* 10⁻⁶ parsecs. After putting this value in the apparent magnitude formula, you will obtain

m = M - 5 + 5*log₁₀(D)

m = 4.74 - 5 + 5*log₁₀(4.848* 10⁻⁶)

m = -26.83

The apparent magnitude of the Sun is equal to -26.83.

Bogna Haponiuk

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