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 12 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 Brighter by mass, fainter by temperature. Main sequences from 0.5 to 1.3 solar masses in luminosity against effective temperature, hotter to the left, for solar composition, [Fe/H] = −1.5, from homology applied to the solar-composition relation. The dots join stars of the same mass: lowering the metals moves each one up and to the left, so a 0.8 solar-mass star goes from 0.41 to 0.66 solar luminosities and from 5049 to 5879 K. The main sequence is steep, so a move of that size to the left carries a star past solar stars much brighter than itself: at a fixed temperature of 5800 K the metal-poor sequences lie 0.59 magnitudes below the solar one. A metal-poor star is brighter than a solar one of the same mass and fainter than a solar one of the same temperature — which is why such stars were called subdwarfs.

Brighter for its mass and fainter for its colour

Two stars of the same mass are almost the same star only if they are made of the same material. Take most of the metals out and the opacity falls, the radiation leaks out faster and the star settles brighter and much hotter — so far hotter that, at a given colour, it sits below the ordinary main sequence. The relation between mass and light has composition inside it, and every mass, age and distance read off it inherits that.

stars · The mass–luminosity relation Isochrones that crowd on the main sequence and fan out above it. Isochrones for stars of solar metallicity at 1, 2, 4, 6, 8, 10, 12 billion years, luminosity against temperature with temperature falling to the right. On the main sequence they lie almost on top of one another — a star brightens and barely changes colour as it ages — and only where they turn off and cross towards the giants do they separate. Two boxes show the measurement errors of a star with a good parallax and spectrum (twice 70 K and 3 per cent): one around a Sun-like star of 4.6 billion years, whose age they allow anywhere in 4.53 (3.38–5.63) billion years; one around a subgiant of the same mass at 10.5, allowed only 10.5 (10.3–10.7).

A lone star is dated only as it leaves

A star in a cluster is dated by the bend its neighbours trace. A star alone has one position on the Hertzsprung–Russell diagram, and an age for it means finding which isochrone passes through that point. On the main sequence they almost all do: a Sun-like star measured as well as a parallax and a spectrum allow is anywhere from three and a half to five and a half billion years old. A star the same mass on its way off the main sequence is dated to two per cent.

stars · The HR diagram Four clocks for a 1 M☉ dwarf, and the best of them. The fractional age uncertainty each method reaches for an isolated 1 solar-mass main-sequence star, against its age, logarithmic, for isochrones, rotation, oscillations and chemical clock, with representative published precisions. Rotation is best while the star is young, once its spin has forgotten its start and before its braking weakens; isochrones and oscillations improve with age as the star moves further on the diagram. The combination (heavy line) is 14 per cent at 1 Gyr, 9 at 3 and 4 at 8. It is better than ten per cent (dashed) from 2.6 to 9.4 Gyr.

Four clocks and the stars none of them reads

An isolated dwarf star can be dated four ways — by its position on the Hertzsprung–Russell diagram, by how fast it spins, by its oscillations, and by the ratio of two elements its galaxy made at different rates. Each clock has an age range where it works and masses it cannot read. Laid side by side for a Sun-like star they cover most of its life to ten per cent or so. For a star of eight-tenths of the Sun's mass only one clock is left, it works only while the star is young, and the commonest stars in the Galaxy's disc are dated one at a time no better than fifteen per cent at any age.

stars · The HR diagram A lifetime that stops getting shorter. Main-sequence lifetime against stellar mass, both logarithmic, from a simple model in which luminosity rises as the 3.8th power of mass until radiation pressure caps it at 40% of the Eddington limit (solid), and without the cap (dashed). The Sun lives 10 billion years, a 10 M☉ star 36 million, a 40 M☉ star 3.2 million, and above about 60 M☉ the lifetime stops falling: 2.7 million at 100 M☉, 3.0 million at 300. Without the cap the steep law would give the heaviest stars lifetimes of tens of thousands of years; with it, every very massive star lives a few million years whatever its mass.

A lifetime that stops getting shorter

A star twice the Sun's mass lives a fifth as long, ten times the mass a few hundredths. Extrapolate the steep law that says so to the most massive stars known, a hundred and fifty or two hundred suns, and they should burn out in a few tens of thousands of years. They do not: every star above about sixty solar masses lives between two and a half and three million years, whatever its mass. Radiation pressure caps how bright a star can be per unit of its mass, and once luminosity and fuel both grow in proportion to mass, the mass cancels out of the lifetime.

stars · Stellar lifetimes

Named alongside it

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

Eddington limitMass luminosity relationMetallicityAbsolute magnitudeIsochroneMain-sequence turnoffStellar radiusWhite dwarfDistance ladderElectron scatteringGiant branchHertzsprung–Russell diagram

All concepts