Astronomy & Celestial Mechanics
the sky, computed
Stars

The diagram that sorted the stars, by plotting two things against each other

Plot brightness against colour for a few thousand stars and they do not scatter. They fall on a narrow band with two islands off it, and explaining that structure is most of stellar astronomy.
11 min read 6 figures Measuring the unreachableScales that defeat intuition

There is no reason a scatter plot should have structure. Take two properties of a few thousand objects, plot one against the other, and the usual result is a cloud.

Plot stellar luminosity against surface temperature and the cloud does not appear. Almost every star lands on a single narrow diagonal band, with a sparse group above it to the right and another far below to the left. Nothing about the measurements forces that; it is a fact about stars, and working out what it means occupied the twentieth century.

The Hertzsprung–Russell diagramLuminosity against surface temperature, both in solar units and on logarithmic axes, with temperature increasing to the left. The main sequence is computed from the mass–luminosity and mass–radius relations; the dashed diagonals are lines of constant radius.R = 0.01R☉R = 0.1R☉R = R☉R = 10R☉R = 100R☉giantssupergiantswhite dwarfs1M☉3M☉20M☉40M☉the Sun10−410−21102104106surface temperature (K), increasing to the leftluminosity, in solar units
Fig. 1 Luminosity against surface temperature, both in solar units and on logarithmic axes, with temperature increasing to the left. The main sequence here is computed from the mass–luminosity and mass–radius relations; the dashed diagonals are lines of constant radius.

Two axes, one expensive

The horizontal axis is temperature, and colour supplies it from two brightness measurements. It is cheap.

The vertical axis is luminosity — total output — and getting it requires an absolute magnitude, which requires a distance. It is expensive, and in 1910 it was nearly impossible: parallaxes existed for a few dozen stars.

Hertzsprung and Russell got round it in different ways, which is why both names are on the diagram. Hertzsprung worked with clusters, where all the stars are at the same unknown distance, so apparent magnitude serves as a stand-in for absolute and the diagram’s shape appears with the vertical axis uncalibrated. Russell used the few stars with measured parallaxes and got the absolute version for a smaller sample. Both found the same structure by 1913.

The temperature axis runs backwards — hot on the left — because Russell plotted spectral type in its traditional order and the order turned out to be a temperature sequence in reverse. It has never been fixed, and every HR diagram since carries the reversal.

What the band is

Ninety percent of stars sit on the main sequence, and the temptation is to read a diagonal band as a track along which stars travel. They do not.

The main sequence is a mass sequence. A star fusing hydrogen in its core sits at a position fixed almost entirely by its mass: a 0.5 solar-mass star is cool and dim, the Sun sits in the middle, a 20 solar-mass star is hot and enormously bright. Stars do not slide along the band; they sit at their mass’s position for most of their existence and then leave it, sideways.

Luminosity against massMain-sequence luminosity against mass, both in solar units, on logarithmic axes. The slope is between three and four across most of the range, so a small spread in mass becomes an enormous spread in output.0.100.321.03.210320.011.0100100001000000mass (solar units)0.2M☉ → 0.01L☉1M☉ → 1L☉5M☉ → 391L☉20M☉ → 50,088L☉dashed: a pure slope of 3.5
Fig. 2 Luminosity against mass, both in solar units. The slope of roughly three and a half is what compresses a wide range of masses into the narrow band above, and it is the reason the main sequence is a line rather than a region.

The band is narrow because the mass–luminosity relation is tight. Two stars of the same mass have nearly the same luminosity and nearly the same temperature, whatever else is true of them, so the sequence has almost no width. Its residual thickness comes from composition and from age — a star brightens slowly as it converts hydrogen to helium, so the Sun is about 30% more luminous now than when it formed.

The band is also a lifetime statement. Stars are found overwhelmingly on the main sequence because that is where they spend nearly all their time, not because most stars are somehow main-sequence stars by type. The regions off the band are sparsely populated because passing through them is quick.

The diagonals that give sizes

Luminosity, temperature and radius are related by the Stefan–Boltzmann law:

L=4πR2σT4.L = 4\pi R^2 \sigma T^4.

Two of the three are the axes, so the third is fixed at every point of the diagram. Lines of constant radius run diagonally, and the dashed lines in the figures are those lines, exact.

That turns the diagram into a size chart, and the sizes are extreme. Betelgeuse is cool — about 3,500 K, so each square metre of it radiates poorly — yet it is 100,000 times more luminous than the Sun. The only way to reconcile those is an enormous surface: about 900 solar radii, which would swallow Jupiter’s orbit.

White dwarfs run the other way. Sirius B is hotter than the Sun, so it radiates fiercely per unit area, and yet it is ten thousand times fainter. It must be tiny — about the size of the Earth, containing a solar mass. That inference, made from position on the diagram alone, was the first evidence for degenerate matter, and it was so unwelcome that Eddington remarked the star was sending a message that seemed absurd.

The Hertzsprung–Russell diagramLuminosity against surface temperature, both in solar units and on logarithmic axes, with temperature increasing to the left. The main sequence is computed from the mass–luminosity and mass–radius relations; the dashed diagonals are lines of constant radius.R = 0.01R☉R = 0.1R☉R = R☉R = 10R☉R = 100R☉1M☉3M☉20M☉40M☉the Sun10−410−21102104106surface temperature (K), increasing to the leftluminosity, in solar units
Fig. 3 The main sequence alone, with the constant-radius diagonals. Reading a star’s position against them gives its size without ever resolving it — which for all but a handful of stars is the only way there is.

The islands

Two regions off the main sequence, and both are stars that have finished with core hydrogen.

Giants and supergiants, upper right: cool and very bright, so very large. When a star exhausts the hydrogen in its core, the core contracts and heats while the envelope expands and cools, and the star moves right and up. The Sun will do this in about five billion years, expanding to engulf Mercury and Venus and rendering the Earth uninhabitable well before it arrives.

White dwarfs, lower left: hot and very faint, so very small. These are the exposed cores left when a low-mass star sheds its envelope, supported not by fusion — there is none — but by electron degeneracy pressure, a quantum effect with no classical counterpart. They do not generate energy; they cool, sliding slowly down and to the right over billions of years.

Both islands are sparsely populated for the same reason, and it is a selection effect rather than a fact about how many exist. Giants are brief; white dwarfs are faint. The most common star in the galaxy is a red dwarf at the bottom right of the main sequence, and not one of them is visible to the naked eye.

Reading a cluster’s age off it

The diagram’s most powerful use is on a star cluster, where every star formed at about the same time from the same material and sits at the same distance.

Plot a young cluster and the main sequence is complete, from the hottest stars down. Plot an old one and the top is missing: the massive stars have already left. The point where the sequence bends away — the turn-off — is at the mass whose main-sequence lifetime equals the cluster’s age.

So the turn-off is a clock. Lifetime depends steeply on mass, so measuring where the sequence ends dates the cluster, and dating globular clusters this way gives ages around 12–13 billion years. That number was for a long time in tension with the estimated age of the universe — the clusters appeared older than everything — and resolving it required both better stellar models and a revised distance scale, which is a good example of how far an error at the bottom of the ladder propagates.

How long a star lasts, against its massMain-sequence lifetime against mass, on logarithmic axes. Fuel grows in proportion to mass and consumption grows as its three-and-a-half power, so the lifetime falls steeply — a star of thirty solar masses lives for a few million years.0.321.03.2103210^610^710^810^910^1010^1110^12mass (solar units)the age of the universe0.5M☉ — 80 Gyr1M☉ — 10 Gyr3M☉ — 458 Myr10M☉ — 23 Myr30M☉ — 1 Myrslope ≈ −2.5
Fig. 4 Main-sequence lifetime against mass. A cluster’s turn-off mass and its age are the same information read through this curve, which is why a photograph of a cluster can be converted into a number of years.

What each axis costs

The two axes of the diagram have wildly different price tags, and that asymmetry shaped how the diagram was built.

Blackbody curves at 3000, 5800, 10000 KThermal emission against wavelength, each curve scaled to its own peak so the shift can be seen on one plot. The peak moves to shorter wavelengths as the temperature rises, which is why colour is a thermometer.visible050010001500200000.20.40.60.811.2wavelength (nm)3000 K, peak 966 nm5800 K, peak 500 nm10000 K, peak 290 nmeach curve scaled to its own peak
Fig. 5 Thermal emission at three temperatures. Two filters sample this curve at two wavelengths, and the ratio gives the horizontal axis — a temperature from a subtraction, for every star in an image at once.

The vertical axis needs an absolute magnitude, which needs a distance.

The distance ladder, and its overlapsThe reach of each distance technique on a logarithmic scale in parsecs. Each rung is calibrated where it overlaps the one below it, so an error low on the ladder propagates all the way to the top.10^-610^-410^-210^010^210^410^610^810^10radar rangingdirect: a timed echoparallaxgeometry, and nothing assumedspectroscopic parallaxassumes a star like the calibratorsCepheid variablesassumes the period–luminosity relation holdsTully–Fisherassumes rotation tracks luminositytype Ia supernovaeassumes a standard explosionHubble's lawassumes the expansion rate is knowndistance (parsecs)each rung is calibrated on the one belowan error at the bottom moves everything above it
Fig. 6 The distance techniques and their reach. Every vertical position on the HR diagram is a distance in disguise, so the diagram inherits the whole ladder’s uncertainty — and the ladder’s foundation is geometric.

The dependency runs both ways, which is the elegant part. A cluster’s main sequence can be fitted to the calibrated one, and the vertical offset required is the distance modulus. So the diagram consumes distances and produces them: main-sequence fitting is a rung of the ladder, sitting between parallax and the Cepheids, and it exists only because the diagram has the shape it does.

Where the model stops

It is a snapshot, not a track. The single most common misreading. Stars move on this diagram, but not along the main sequence, and the band is a locus of positions rather than a path.

Distances are needed. Every vertical position depends on a distance, and errors there move stars vertically in a way that mimics real differences.

Reddening moves stars sideways. Dust makes a star look cooler and fainter, sliding it down and to the right along a direction that is uncomfortably close to the main sequence’s own slope.

Binaries. An unresolved pair looks like one star of the combined luminosity, which lifts it above the sequence by up to 0.75 magnitudes and produces the second faint band visible in good cluster diagrams — the same pairs that, resolved and timed, supply every stellar mass there is.

Not all stars. Neutron stars and black holes are nowhere on it. Brown dwarfs fall off the bottom. Objects that never fused hydrogen have no business on a diagram organised around stars that do.

The figures also share a limitation of logarithmic axes: they compress the extraordinary. The vertical axis spans ten decades — a factor of 101010^{10} between the faintest white dwarf and the brightest supergiant — and on a log plot that looks like a comfortable page. It is not comfortable. It means one object in the picture emits ten billion times more light than another, and the diagram’s readability is bought by making that unimaginable.

The ladder from here

Later rungs: the colour–magnitude diagram, which is what is actually plotted. Cluster diagrams and main-sequence fitting as a distance method. The turn-off as a clock. Evolutionary tracks, and why they cross the diagram rather than run along it. The instability strip and the pulsating variables in it. The Chandrasekhar limit at the bottom left. The Hayashi track for stars still contracting. The initial mass function, which says how many stars start where. And the diagram for other galaxies, where the same structure appears with different chemistry.

Hertzsprung published in a photographic journal in 1911 and was largely unread. Russell presented the same result to the Royal Astronomical Society in 1913 and it became the organising diagram of a whole science within a decade.

Where this goes next

0.100.321.03.210320.011.0100100001000000mass (solar units)0.2M☉ → 0.01L☉1M☉ → 1L☉5M☉ → 391L☉20M☉ → 50,088L☉dashed: a pure slope of 3.5 Mass decides everything, by a power of three and a half Two stars of the same mass are almost the same star. Double the mass and the output multiplies by eleven — which is why a modest range of masses produces a colossal range of stars. visible050010001500200000.20.40.60.811.2wavelength (nm)3000 K, peak 966 nm5800 K, peak 500 nm10000 K, peak 290 nmeach curve scaled to its own peak Colour is a thermometer, and it reads across the galaxy A star's colour gives its surface temperature, from two brightness measurements and no other information. It is the cheapest useful measurement in astronomy. 0.321.03.2103210^610^710^810^910^1010^1110^12mass (solar units)the age of the universe0.5M☉ — 80 Gyr1M☉ — 10 Gyr3M☉ — 458 Myr10M☉ — 23 Myr30M☉ — 1 Myrslope ≈ −2.5 Why the biggest stars die first, and take the galaxy with them A star thirty times the Sun's mass has thirty times the fuel and burns it forty thousand times faster. It lasts a few million years, and everything heavier than iron exists because of it.