# Luminosity Calculator

Created by Luciano Mino
Last updated: Jun 27, 2022

The luminosity calculator can help you find the luminosity of a distant star based on its radius and temperature using the Stefan-Boltzmann law.

In the following short article, we will talk cover:

• How to calculate luminosity using the luminosity equation;
• How to calculate luminosity from absolute magnitude; and
• Give an example of finding the Sun's luminosity.

## What is and how to calculate luminosity

Luminosity measures a light source's radiant . The Steffan-Botzlmann law describes the relationship between this radiated power and two parameters for an idealized black body:

$L = \sigma × A × T^{4}$

where:

• $L$ is the radiant power or luminosity measured in watts;
• $\sigma$ is the Stefan–Boltzmann constant ∼5.670367 × 10⁻⁸ W⋅m⁻²⋅K⁻⁴;
• $A$ is the surface area of the light source in m²; and
• $T$ is the object's temperature in Kelvin.

We can easily calculate the surface area of a star from its radius $R$, turning this expression into the luminosity equation for a star:

$L = \sigma × 4 \pi R × T^{4}$

When we're describing the luminosity of a star, we generally give this value in terms of the luminosity of the Sun (L⊙, 3.828×10²⁶ W):

$\frac{L}{L⊙} = \frac{R}{R☉}(\frac{T}{T☉})^{4}$

where:

• $R☉$ is the Sun's radius; and
• $T☉$ is the Sun's temperature.

🙋 You can choose to express the luminosity in gigawatts or in terms of the luminosity of the Sun with the luminosity calculator.

## Absolute and apparent magnitude

If we use the luminosity equation, we'll find that there are a lot of stars brighter than the Sun in the universe. However, they don't appear as bright when we look at them from Earth. Why is that?

That's because we're measuring the apparent brightness or how bright an object appears to an observer.

On the other hand, absolute brightness ($M$) is similar to luminosity but differs from it in using a logarithmic scale:

$M = -2.5 × log_{10}(\frac{L}{L_{0}})$

where:

• $L$ is the light source's luminosity; and
• $L_{0}$ is the zero-point luminosity, L₀ = 3.0128 × 10²⁸ W.

We can calculate the apparent brightness of a star ($m$) given its distance from the source ($d$):

$m = M - 5 + 5 × log_{10}(d)$

where $d$ is the distance in parsecs.

## How to calculate luminosity from absolute magnitude

Converting absolute magnitude to luminosity is straightforward. We just need to rearrange the absolute magnitude equation:

$L = L_{0}×10^{\frac{-M}{2.5}}$

where again $L_{0}$ is the zero-point luminosity.

## Finding the Sun's luminosity

Let's use the luminosity calculator to find the Sun's luminosity.

Assuming:

• A radius of ~696,340 km; and
• A temperature of ~5,778 K.

We plug that information into the calculator to find L = 3.835 × 10²⁶ W, which is close enough to the actual value of L⊙, 3.828×10²⁶ W.

You can use this tool to find the luminosity of any star. Give it a try!

Luciano Mino
mi
Star temperature
K
Luminosity
L☉
Absolute magnitude
Distance
pcs
Apparent magnitude
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