# Star Brightness

When we talk about the brightness of a star, there are two different concepts of brightness to be aware of.

### Intrinsic Brightness

Intrinsic brightness or luminosity ( $L$ ) is the power emitted from an object in the form of electromagnetic radiation. This can be over the entire electromagnetic spectrum, or a constrained range, e.g. the visible spectrum.

### Apparent Brightness

The apparent brightness or true brightness ( $b$ ) of a star is how bright it appears to be from a viewpoint. This is dependent on both the intrinsic brightness of a star, how far away the viewpoint is from the star, and the size of the star. However, at astronomical scales, the size of the star becomes negligible in almost all cases.

The further away you are from a star, the dimmer it appears, in accordance with the inverse-square law.

This is closely related to Irradiance at Higher.

The formula to calculate apparent brightness from intrinsic brightness can be derived from the idea of electromagnetic radiation spreading out in a sphere around a source.

Formula

$b=\frac{L}{4\pi {d}^{2}}$

Variable Key

• $b$ is the apparent brightness of an object, in watts per square metre.
• $L$ is the luminosity (intrinsic brightness) of the object, in watts.
• $d$ is the distance from the object, in metres.

Tip

You can use any other unit of distance as long as you're consistent - at astronomical scales, metres become incredibly impractical and astronomical units, light years and parsecs become more reasonable.

## Stefan-Boltzmann Law

The Stefan-Boltzmann law relates the irradiance of a star to its surface temperature. It states that the irradiance of an ideal black body is directly proportional to the fourth power of its temperature.

Since stars can be considered approximate black bodies, we can use the Stefan-Boltzmann law to calculate the temperature of a star from its irradiance, or vice versa. This is very useful for getting a rough idea of the surface temperature of stars, which we obviously can't pop by for an afternoon visit.

Formula

$I=\sigma {T}^{4}$

Variable Key

• $I$ is the irradiance of the star, in watts per square metre.
• $\sigma$ is the Stefan-Boltzmann constant, about .
• $T$ is the surface temperature of the star, in kelvins.