Astronomy & Celestial Mechanics
the sky, computed
Concept

Main sequence — where it appears

The band on the temperature-luminosity diagram occupied by stars fusing hydrogen in their cores, which is most of them. It is a place rather than a track: a star does not move along it as it ages but sits at the position its mass dictates and then leaves.

Named by 8 essays across 2 fields — each of them below, with the objects they name alongside it.

The distance modulus. The difference between apparent and absolute magnitude against distance, on a logarithmic distance axis. It is a straight line of slope five per decade, passing through zero at ten parsecs — the definition of the absolute magnitude. Reading a distance off it requires the absolute magnitude, which is never measured and always inferred.

A brightness is a distance only if something is known

The inverse square law turns a brightness into a distance in one line. The line contains a quantity that has never been measured for any object outside the solar system.

starlight · Magnitudes The Hertzsprung–Russell diagram. Luminosity 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.

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.

stars · The HR diagram Luminosity against mass, against a slope of 3.5. Main-sequence luminosity against mass, both in solar units, on logarithmic axes, over the range 0.079 to 63 solar masses. The measured curve comes from the eclipsing binaries and is the same in every drawing of it; what changes here is what it is compared against. The dashed line is a pure power law of exponent 3.5, and the curve crosses it rather than following it — the local slope runs from about 2.3 at the bottom of the range, where the interiors are convective, through nearly 4 near a solar mass where bound-free opacity dominates, to about 3 among the massive stars where electron scattering does. Quoting one exponent across the whole sequence is a convenience and the places it fails are the places the interior physics changes. Because the slope is between three and four across most of the range, a small spread in mass becomes an enormous spread in output: the 63-solar-mass end is 3.0e+9 times brighter than the 0.079-solar-mass end.

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.

stars · The mass–luminosity relation How long a star lasts, against its mass. Main-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.

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.

stars · Stellar lifetimes The Hertzsprung–Russell diagram. Luminosity 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.

The main sequence is a place stars sit, not a track they travel

The commonest misreading of the Hertzsprung–Russell diagram is that stars slide down the band as they age. They do not. They sit at one point for ninety percent of their lives and then leave sideways.

stars · The HR diagram Inside a star that is holding itself up (polytrope n = 3). Temperature, density and pressure through a star, as fractions of their central values, against fractional radius. All three come from one numerical integration of the Lane–Emden equation at index 3, which is the hydrostatic balance written for a gas whose pressure is a power of its density. The inner half of the radius holds 90% of the mass, and the outer half is nearly weightless — which is why the load, and so the temperature, is concentrated where the burning is.

A star is held up by its own weight

A star has a central temperature because it has a central pressure, and it has a central pressure because everything above is pressing down. The nuclear reactions do not set that temperature — they obey it.

stars · Hydrostatic equilibrium Leavitt's law, through the Milky Way calibrators. Mean absolute magnitude against the logarithm of the pulsation period, for classical Cepheids with independently known distances, and RR Lyrae itself for comparison. The line is M = −2.81 log P − 1.43: a tenfold longer period is 2.81 magnitudes brighter, a factor of 13 in luminosity. The vertical axis runs the astronomers' way, with brighter upward.

A star that tells its distance by how slowly it blinks

Some stars pulsate, and the slow ones are the bright ones. That single correlation turns a clock into a ruler, and it is how the size of the universe was first measured.

stars · Variable stars An exponent between 2.3 and 4, not a number. The local logarithmic slope of the mass–luminosity relation against mass — the exponent that "L ∝ M^3.5" stands in for — read off the same fit to eclipsing binaries that the relation itself is drawn from. It is a step function with two corners, and the steps are 2.3 below 0.43 M☉, 4 0.43 to 2 M☉, 3.5 2 to 55 M☉. The horizontal lines are what homology predicts: a star in radiative equilibrium has L ∝ μ⁴M³/κ̄, which gives an exponent of 3 when the opacity is electron scattering and a constant, 5.5 when it is Kramers and depends on density and temperature, and 1 where radiation pressure holds the star up and the luminosity is pinned to the Eddington value. The measured steps sit between those predictions rather than on them, which is what a real star does — no star is homologous, the low-mass ones are convective throughout and the heavy ones have convective cores, and the fitted exponents are what is left after all of that. The useful statement is not that the exponent is 3.5 but that it is between 2 and 5 everywhere and 4 in the middle, and that the reason it moves is a change of which opacity is carrying the energy out.

An exponent that is a slope, not a law

The phrase “L goes as M to the three and a half” stands in for a curve with three straight pieces and two corners. The exponent is 2.3 below half a solar mass, 4 in the middle and falls towards 1 at the top — and each change of exponent is a change in which opacity is carrying the energy out.

stars · The mass–luminosity relation

Named alongside it

The objects these essays reach for when they reach for this one.

Absolute magnitudeEddington limitStellar radiusWhite dwarfDistance ladderGiant branchHydrostatic equilibriumInitial mass functionInstability stripMass luminosity relationMetallicitySupernova

All concepts