Astronomy & Celestial Mechanics
the sky, computed
Concept

Brown dwarf — where it appears

An object below about eighty Jupiter masses, which never ignites hydrogen and therefore fades continuously rather than settling on a main sequence. It is distinguished from a low-mass star by the presence of lithium, which a real star destroys and a brown dwarf does not.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

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 One population, three integrals, two ends. The same mass function weighted three ways, each normalised so that the area under it is one, against mass on a logarithmic axis — so a share of the page is a share of the total. The light is a zero-age population's, every star still on the main sequence, which is the only age at which the top of the range is present at all. The steep curve on the left is the number of stars, which peaks at 0.08 M☉ and falls away because the function is steeper than m⁻¹; the middle one is the mass they carry; the one on the right is the light they emit, computed from this file's own main-sequence relation and peaking at 55.0 M☉ — 682 times further up the axis. The population that is counted and the population that is seen are two different populations. Above 8 M☉ there are 0.63% of the stars, 21% of the mass and 99% of the light. The slopes are α = 1.3 above 0.08 M☉, α = 2.3 above 0.5 M☉, and the low-mass end is where the honesty runs out: it has never been measured in another galaxy, and every mass inferred from a luminosity assumes it.

Most stars are small, and most of the light is not

The mass function is steep and the mass–luminosity relation is steeper, so counting stars and measuring their output are integrals of the same function dominated by opposite ends of it. Every mass inferred from a brightness passes through that mismatch.

stars · Initial mass function The least a star can average, whatever it is made of. The lower bound on the mass-averaged temperature of a main-sequence star of ideal gas, μmₕGM/6kR, against its mass, both logarithmic, with the radius taken from the main-sequence mass–radius relation, for mean molecular weights of 0.5 and 0.62 — 0.5 is fully ionised hydrogen, the lightest possible gas and so the lowest floor; 0.62 is the solar mixture. For a star of the Sun's mass and radius the floor is 1.9 million kelvin whatever the star is made of. The dashed curve is the average a centrally condensed star actually has, for the index-three polytrope that approximates the Sun: three times the floor, 7.2 million kelvin for the solar mixture. The floor rises slowly with mass, because radius grows almost as fast. Nothing in it refers to nuclear reactions: mechanics alone requires a star to be hot enough inside that hydrogen is ionised throughout and fusion is within reach.

Mechanics sets a star's temperature before fusion does

A ball of gas with the Sun's mass and radius must average at least two million kelvin inside, whatever it is made of and whatever powers it, because nothing else can hold it up. The bound comes from the virial theorem and one inequality about gravitational binding. Where it is met, a body can be a star; where it demands temperatures a body plainly does not have — the Earth, Jupiter, a white dwarf — it proves that something other than hot gas is holding that body up.

stars · Hydrostatic equilibrium Curvature turns a curve into an island. The companions — separation across, mass up, on edge-on circular orbits — that fit simulated velocities of a star with a 12 Jupiter-mass companion at 22 AU (period 103 years), within a Δχ² of 6.18 of the best fit, for observing spans of 6, 12, 24 years with 60 velocities of 2 m/s noise each. Over 6 years the data show a slope and the allowed region is a long band along M ∝ a², spanning 3.3 decades in mass. As the span grows the curvature of the orbit appears, and the band contracts — to 1.0 decades at 24 years — towards the true companion. Inclined orbits, which the fit leaves out, would extend every region to higher masses.

A slope that is a whole curve of companions

A companion on an orbit longer than the observations shows in a star's velocity only as a steady drift. A drift is one number, an acceleration, and an acceleration is a mass divided by the square of a distance — so a planet at ten astronomical units and a brown dwarf at fifty produce the same straight line. Turning the line into a companion needs the orbit to bend, a picture, or the other two components of the acceleration.

exoplanets · Reflex velocity One brightness, a mass for every age. Luminosity against age for young objects of 0.01, 0.02, 0.05, 0.08, 0.2, 0.5 solar masses contracting at nearly fixed surface temperature, logarithmic; the dashed line is one observed luminosity, 0.012 of the Sun's. Every object fades as it contracts, as the age to the power −2/3, so the same luminosity belongs to a different mass at every age: 38 Jupiter masses at 1 Myr, 52 Jupiter masses at 2 Myr, 80 Jupiter masses at 5 Myr, 111 Jupiter masses at 10 Myr. A factor of ten in assumed age is a factor of 2.9 in inferred mass. The boundary between stars and brown dwarfs, 0.075 solar masses, is crossed by an error in the age alone.

A turnover whose position is an age

The mass function of a young cluster is measured by counting its objects by brightness and converting each brightness into a mass. Below the hydrogen-burning limit nothing holds a brown dwarf's brightness steady: it fades as it contracts, so the conversion needs the cluster's age, and an age wrong by a factor of two moves every mass by about forty per cent. The shape of the substellar mass function survives the error; its position on the mass axis and the ratio of brown dwarfs to stars do not — and comparing those between clusters is how the universality of star formation is tested.

stars · Initial mass function An association that has been flying apart for ten million years. The velocity of each of 220 stars along one direction against its position along the same direction, for a model association whose stars were born 10 million years ago inside a region 1 pc across (one standard deviation), with random velocities of 1 km/s, and that began coasting apart at once. The fast stars have gone furthest, so position and velocity are correlated, and the slope of the line through them is the inverse of the time they have been moving: the fit gives 10.1 million years. A star at 10 pc moving at 1 km/s has been travelling for ten million years — which is what an expansion age is. Measuring it needs the stars' distances and their motions across the sky, both of which space astrometry now supplies for associations within a few hundred parsecs.

An expansion clock read from the stars' motions

The masses of a young cluster's brown dwarfs are read from their brightness, and a brown dwarf's brightness is a mass multiplied by an age. The age can now be measured without any model of a star: the cluster is flying apart, and the time since it began is its size divided by its speed. But the expansion clock counts only the coasting, not the stars' lives, and it runs fast when the stars began spread out — two biases of opposite sign, and nothing in the motions themselves to say which one wins.

stars · Initial mass function

Named alongside it

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

Initial mass functionAstrometryHydrostatic equilibriumMass luminosity relationProper motionStellar populationBarycentreCooling ageDegeneracyDirect imagingDynamical massEclipsing binary

All concepts