A star is a glowing ball of gas so far away that, even through the finest telescope, it is never more than a point of light. We can't land on one, weigh it, or push a thermometer into its surface. And yet astronomers will tell you, with confidence, that Betelgeuse is a cool red super-giant some 700 times the radius of the Sun, while Sirius B is a searingly hot white dwarf no bigger than the Earth. How can anyone possibly know such things about an object they can only ever see as a speck?

The astonishing answer is that a star's light is a message, and it carries almost everything we need. Two numbers unlock the rest: the star's colour (which fixes its surface temperature) and its total power output (its luminosity). This page shows how the two physical laws behind those numbers — Wien's displacement law and Stefan's law — let us read a star like a book, and how plotting temperature against luminosity for thousands of stars reveals one of the most important pictures in all of astronomy: the Hertzsprung–Russell diagram.

A star glows like a black body

Heat something up and it glows. A poker in a fire runs from dull red, to bright orange, to yellow-white as it gets hotter; the glowing element of a toaster is a deep cherry red. This is thermal radiation, and any hot object emits it across a whole spread of wavelengths at once. A star is a very good approximation to an idealised black body — a perfect absorber and emitter — so its radiation follows a smooth, characteristic curve whose shape depends on only one thing: the temperature of its surface.

The key feature of that curve is where it peaks — the wavelength \lambda_{\max} at which the star pours out the most energy. A cool star peaks far into the red; a hot star peaks in the blue or even the ultraviolet. So the colour your eye (or a spectrometer) sees is a direct read-out of the star's temperature — hotter means bluer, cooler means redder. That relationship has an exact form.

Wien's law: colour tells you temperature

The peak wavelength and the surface temperature are locked together by Wien's displacement law. As the temperature T rises, the peak wavelength \lambda_{\max} shifts (is "displaced") to shorter values — the two are inversely proportional:

Rearranged for whichever quantity you want:

Worked example — the Sun's temperature

The Sun's spectrum peaks at about \lambda_{\max} = 500\ \text{nm} (in the green-yellow). What is its surface temperature?

Just under 6000 K — which is indeed the temperature of the Sun's visible surface. Notice the power of this: we measured a temperature 150 million kilometres away purely from the colour of the light, and nothing else.

Worked example — a cool red giant

Betelgeuse peaks at roughly \lambda_{\max} = 830\ \text{nm}, out in the near-infrared. Its temperature is

Far cooler than the Sun — which is exactly why it glows a deep red. A longer peak wavelength always means a lower temperature.

Luminosity: how much power a star pours out

Temperature is only half the story. A star's luminosity L is the total power it radiates — the energy leaving its whole surface every second, measured in watts. (Don't confuse it with how bright a star merely appears from Earth: a very luminous star can look faint simply because it is far away. Luminosity is the true, intrinsic output.)

Because a star radiates like a black body, its luminosity is fixed by just two properties: its radius r (a bigger surface radiates from more area) and its surface temperature T (a hotter surface radiates far more fiercely per square metre). The exact relationship is Stefan's law:

where 4\pi r^2 is the surface area of the (spherical) star and \sigma = 5.67\times10^{-8}\ \text{W}\,\text{m}^{-2}\,\text{K}^{-4} is the Stefan constant. The single most important thing to notice is the T^4: luminosity depends only on the square of the radius, but on the fourth power of the temperature. Temperature is the dominant lever.

Worked example — the luminosity of the Sun

The Sun has radius r = 7.0\times10^{8}\ \text{m} and surface temperature T = 5800\ \text{K}. Then

Take it in pieces: 4\pi r^2 = 6.16\times10^{18}\ \text{m}^2, and (5800)^4 = 1.13\times10^{15}\ \text{K}^4. Multiplying,

Nearly 4\times10^{26} watts — the accepted value for the Sun. We call this one solar luminosity, L_\odot, and use it as a yardstick for other stars.

Turning it around: finding a star's size

Stefan's law is at its most powerful run backwards. If we already know a star's luminosity (from its brightness and distance) and its temperature (from Wien's law), we can rearrange for the one thing we can never measure directly — its radius:

It is often cleanest to compare with the Sun. Dividing a star's Stefan equation by the Sun's, the constants cancel and we get a tidy ratio:

Worked example — how big is a red giant?

A red giant has luminosity L = 10\,000\,L_\odot but a cool surface of T = 3500\ \text{K} (against the Sun's 5800 K). How large is it?

The giant is about 275 times the radius of the Sun. Here is the crucial lesson: this star is cooler than the Sun, so each square metre of it radiates less than the Sun's surface — yet it is dazzlingly luminous. The only way to reconcile those facts is that it must be enormous. A cool star that is still very luminous is telling you, through Stefan's law, that it is gigantic.

The Hertzsprung–Russell diagram

Around 1910, Ejnar Hertzsprung and Henry Norris Russell independently had the same brilliant idea: take thousands of stars, and for each one plot its luminosity up the vertical axis against its surface temperature along the horizontal axis. If stars were scattered randomly, the plot would be a shapeless smear. They are not. Instead the stars fall into a few sharply defined regions — a pattern so revealing that the Hertzsprung–Russell (H–R) diagram became the single most useful chart in stellar astronomy.

Two quirks of the axes trip up every newcomer, so fix them now:

• Temperature runs backwards. For historical reasons the horizontal axis has temperature increasing to the left — hot blue stars sit on the left, cool red stars on the right. It is the one axis in physics that is habitually drawn in reverse.
• Luminosity is on a logarithmic scale. Stars span an incredible range — from about 10^{-4} to 10^{6} times the Sun's output — so each step up the axis is a factor of ten, not a fixed amount.

Explore the diagram below. Use the selector to spotlight each family in turn, and find where our Sun sits.

Reading the regions

Three great groupings dominate the diagram, and — armed with Stefan's law — you can work out the story behind each just from where it lies:

• The main sequence. The broad diagonal band, sweeping from hot-and-luminous at the top-left down to cool-and-dim at the bottom-right, holds about 90% of all stars — including the Sun. Every star here is doing the same thing: steadily fusing hydrogen into helium in its core. A star's position along the band is set by its mass: the heaviest stars are the hot, brilliant O and B stars at the top-left, the lightest are the cool, dim red dwarfs at the bottom-right.
• Red giants and super-giants. Up in the top-right sit stars that are cool (so, to the right) yet hugely luminous (so, high up). As we saw, cool-but-luminous can only mean vast — these are stars that have swollen enormously late in life. Super-giants like Betelgeuse occupy the very top.
• White dwarfs. Down in the bottom-left lie stars that are hot (to the left) yet faint (low down). Hot-but-dim can only mean tiny — these are the exposed, Earth-sized dead cores of former Sun-like stars, radiating fiercely per square metre but with almost no surface area to do it from.

The diagram is not just a snapshot — it is a map a star moves across as it ages. A star spends most of its life sitting almost still on the main sequence. When its core hydrogen runs out it swells and cools, so it migrates up and to the right into the red-giant region. Finally, a Sun-like star sheds its outer layers and its hot little core drops down to the bottom-left to become a white dwarf, slowly cooling and fading along the way. The whole life cycle of a star is a journey traced out on the H–R diagram.