# How is the brightness of the star related to its temperature and size?

## Introduction

The brightness of a star depends on two main factors – its temperature and its size. The hotter a star is, the brighter it will be. Similarly, the larger a star is, the more total light it emits, making it brighter. So temperature and size both contribute to making some stars brighter than others.

## How Temperature Affects Brightness

The temperature of a star determines how much energy it radiates per unit of surface area. A hotter star will emit more energy at each point on its surface, making it brighter overall. This relationship is described quantitatively by the Stefan-Boltzmann law which states that the energy flux emitted per unit area is proportional to the fourth power of the absolute temperature.

So a small increase in stellar temperature causes a large increase in energy output and brightness. For example, a star at 10000 K will emit over 6 times as much energy per unit area as a star at 6000 K. This is why hot, blue stars are so much brighter than cool, red ones. The blue color indicates temperatures over 10000 K, while red stars are cooler at under 6000 K.

## How Size Affects Brightness

While temperature determines the brightness at each point on the star’s surface, the size of the star determines how much total surface area there is to emit light. Doubling the radius of a star increases its surface area by a factor of 4. This means that if two stars have the same temperature, the larger star will have more total luminosity because it has more total emitting area.

Combining both temperature and size, we can see that the most luminous stars will be both very hot and very large. The intrinsically brightest stars are blue supergiants, which have temperatures around 20000 K and radii hundreds of times larger than our Sun. Smaller stars cannot emit as much total energy as the entire stellar surface emits less light. Cool red dwarfs are the smallest and faintest stars despite their abundance in the Universe.

## The Stefan-Boltzmann Law

The Stefan-Boltzmann law mathematically defines the relationship between temperature, surface area, and emitted energy flux (brightness). It states that the power emitted per unit area of a star’s surface is directly proportional to the fourth power of its absolute temperature:

 Luminosity per unit area = σT4 σ Stefan-Boltzmann constant T Absolute temperature in Kelvin

This T4 relationship explains why hotter stars are so much brighter than cooler ones.

To find the total luminosity of a star, we simply multiply the luminosity per unit area by the surface area:

 Total luminosity = 4πR2σT4 R Radius of the star

So both temperature (to the fourth power) and radius (squared) contribute to making large, hot stars the brightest.

## Brightness and Stellar Classification

Astronomers categorize stars by spectral type or surface temperature. From hottest to coolest, the major spectral classes are O, B, A, F, G, K, M. This OBAFGKM sequence is known as the Harvard spectral classification. It corresponds directly to luminosity, with O stars being the most luminous and M stars the least.

Spectral Type Temperature (K) Luminosity
O 28000 – 50000 Very High
B 10000 – 28000 High
A 7500 – 10000 Above Average
F 6000 – 7500 Above Average
G 5000 – 6000 Average
K 3500 – 5000 Below Average
M 2400 – 3500 Very Low

As this table shows, O stars are the hottest and most luminous, while M stars are coolest and dimmest. This sequence directly follows the Stefan-Boltzmann law – hotter temperature means higher luminosity.

## Hertzsprung-Russell Diagram

The Hertzsprung-Russell diagram (HR diagram) is a scatter plot that compares star luminosity to temperature. It clearly shows the relationship between brightness and temperature across all spectral types.

The most luminous stars cluster at the top left of the HR diagram. These are the blue O stars with very high temperatures and luminosities. The coolest and dimmest red M stars are at the bottom right. In between we see the sequence of spectral types running diagonally, with temperature decreasing left to right and luminosity decreasing top to bottom.

The HR diagram elegantly visualizes how the Stefan-Boltzmann law of physics relates stellar temperature, luminosity, and spectral classification.

## Main Sequence Stars

The prominent diagonal band running from top left to bottom right includes the majority of stars. This is called the main sequence, where stars fuse hydrogen into helium in their cores.

Main sequence stars follow a clear temperature-luminosity relationship. O, B, and A stars are hotter and more luminous than G stars like our Sun. K and M dwarfs are cooler and dimmer.

Our Sun is classified as a G2V main sequence star with an intermediate temperature of 5780 K and luminosity of 1 solar luminosity (L☉). It lies about halfway along the main sequence band, meaning it has an average brightness compared to other hydrogen-fusing stars.

## Giants and Supergiants

Above the main sequence are the giant and supergiant stars, which have higher luminosity than main sequence stars of the same temperature. After the hydrogen fuel in their cores is depleted, stars become larger and more luminous.

The increased size boosts their total luminosity according to the Stefan-Boltzmann law. Giants and supergiants have surface temperatures similar to main sequence stars, but their larger surface area makes them intrinsically brighter.

At the top of the HR diagram lie the hypergiants and the most luminous supergiants. These are extremely large, massive stars with very high luminosity. Their position above the main sequence reflects their enlarged radii and total energy output compared to normal main sequence stars.

## White Dwarfs

At the bottom left of the H-R diagram are the white dwarfs, originally main sequence stars that have exhausted their nuclear fuel. They are extremely hot but very small, so have low total luminosity. A white dwarf may have a high surface temperature of 10000-100000 K, but because its radius is roughly that of the Earth, its overall luminosity is low. Hence white dwarfs appear dim.

## The Stefan-Boltzmann Law in Action

The position of the various types of stars on the H-R diagram neatly follows the Stefan-Boltzmann law.

– Hot blue O stars at the top left are the most luminous due to very high temperatures.

– Cool red M stars at the bottom right are the least luminous due to low temperatures.

– White dwarfs are hot but small, so low total luminosity.

– Giants and supergiants have greater surface area, increasing total brightness.

So a star’s temperature, size, and classification directly determine where it will lie on the HR diagram relative to the Stefan-Boltzmann law. This law quantitatively links stellar parameters to the observable quality of brightness.

## Conclusion

The brightness of a star is dependent on both its surface temperature and its radius. Hotter stars emit more energy per unit area according to the Stefan-Boltzmann law. Larger stars have more total surface area over which to emit light.

The interplay between these two factors places stars with different temperatures and sizes in specific areas of the Hertzsprung-Russell diagram. O and B stars are hot, blue, and luminous. K and M stars are cool, red, and dim. White dwarfs are hot but small and faint. Giants and supergiants are enlarged versions of main sequence stars.

So the observable quality of stellar brightness is ultimately tied to the Stefan-Boltzmann law of physics, which relates a star’s intrinsic properties of temperature and size to the amount of light it emits across all wavelengths.