Astronomy & Celestial Mechanics
the sky, computed
Concept

Brightness temperature — where it appears

The temperature a blackbody would need in order to produce the observed intensity at a given frequency. It is a way of quoting a brightness rather than a claim about the source, and it exceeds any thermal value for coherent or non-thermal emission.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

Three slopes: 2.5, −0.75 and −1.25, and only the middle one is about the electrons. A radio source's spectrum across five decades of frequency, computed for an electron population with index p = 2.5 in a uniform field. Nothing in this curve is a temperature, because a power law has no scale and therefore nothing a thermometer could read. The straight section between the two bends has slope −0.75, measured here off the drawn curve, and the electron index follows from it and from nothing else: α = (p − 1)/2, so a flux ratio between two frequencies is a measurement of the energy distribution of particles in a place no detector will ever visit. The two bends are the other two measurements. Below 80 MHz the source is opaque to its own radiation and rises as ν^2.5 — a slope fixed at 5/2 by the geometry alone, whatever the electrons are doing — and the frequency at which that happens gives an angular size for a source nothing has resolved, because the turnover is where the brightness temperature meets the electrons' own. Above 12 GHz the spectrum steepens by 0.50: the electrons that radiate at high frequency lose their energy fastest, so the top of the distribution has already emptied, and the frequency of the break is a clock. What no part of this curve gives is the magnetic field. Only the field and the particle density together enter the emission, and every field strength ever quoted for a radio source comes from assuming the two share the energy equally.

A spectrum with no temperature in it

Every spectrum in this collection so far has been a thermometer. A radio lobe's is a power law, and a power law has no scale — so there is nothing for a thermometer to read, and what the shape carries instead is the energy distribution of the particles that made it.

starlight · Synchrotron radiation One whole orbit. The system's total brightness through one orbit: the transit at phase 0, the slow rise and fall of the planet's illuminated hemisphere between, and the secondary eclipse at phase 0.5 where the planet's own light is removed. The transit is 1.05%; the secondary eclipse is 1800 ppm, about 6 times shallower.

The planet is seen when it disappears

Half an orbit after the transit the planet passes behind its star, and the light that vanishes is the planet's own. Subtracting two brightnesses taken hours apart isolates a body nothing has ever resolved.

exoplanets · Exoplanet atmospheres The sign of one gradient decides which way the disc is shaded. The emergent intensity across a stellar disc, against the fractional radius, for four gradients of the source function with depth. Every curve is the Eddington–Barbier relation evaluated at the appropriate μ and checked against a numerical integration of the transfer equation. The contrast between centre and limb is the gradient, and its sign is the gradient's sign. A source function rising inward — the ordinary case, since temperature rises inward and the source function follows it — darkens the limb: a sight line at the edge leaves from higher up, where the gas is cooler. A flat source function produces a uniformly bright disc whatever the geometry. And a source function that falls inward brightens the limb, because now the shallower ray is reading a hotter layer. That last case is not hypothetical: above a star's photosphere the temperature stops falling and begins to rise, so at wavelengths that see those layers — millimetre continuum, the cores of strong lines, the ultraviolet — the Sun's limb is brighter than its centre. Limb darkening is therefore not a property of stars but of a gradient, and the same atmosphere shows both signs at different wavelengths.

A limb brighter than the middle

Limb darkening is not a property of stars. It is a property of a gradient, and the same relation that produces it produces the opposite whenever the source function falls inward — which it does above every stellar photosphere. The Sun's limb is darker in the visible and brighter at a millimetre, and both are the same equation.

starlight · Limb darkening The power a source loses to its own photons, against how bright it is. The ratio of the power a synchrotron source loses by inverse-Compton scattering of its own photons to the power it radiates as synchrotron emission, against its brightness temperature, both on logarithmic axes. To first order the ratio goes as the fifth power of the brightness temperature over 10¹² K — so at 10¹¹ K it is 10⁻⁵, negligible, and at 10¹² K it is one. The scattered photons are scattered again, and each order multiplies by the same factor, so the second-order curve rises as the tenth power: at twice the limit the total loss is 1056 times the synchrotron power, and the electrons radiate away their energy in hours as X-rays and gamma rays. A steady source cannot sit above the line where the ratio is one. That is the inverse-Compton catastrophe, and it makes 10¹² K a ceiling on the brightness temperature of any incoherent synchrotron source at rest.

A temperature that is a speed

No steady synchrotron source can be brighter than about a trillion kelvin, because above that its electrons lose their energy scattering their own photons faster than they can radiate it. So when a radio source is measured brighter — as the cores of quasars routinely are — the excess is not a hotter source. It is a source coming towards the observer, and the brightness temperature becomes a measurement of how fast.

starlight · Synchrotron radiation Brightness temperature across the infrared, for three profiles. The brightness temperature of the planet's emission against wavelength from 1 to 12 microns, for the three temperature profiles, with water and carbon monoxide bands in the opacity. In a band the opacity is high, so the light comes from higher, lower-pressure layers; in the windows between bands it comes from deeper. Over a profile falling outward the bands are cooler than the windows: at 6.3 microns 1457 K against 1570 K at 3.6. Over the inverted profile they are hotter: 1851 against 1538. Over the isothermal profile every wavelength has the same brightness temperature and the spectrum is a blackbody, whatever molecules are present.

Molecules seen only through a change of temperature

The heat a planet gives off, measured colour by colour as it disappears behind its star, carries the fingerprints of the molecules in its atmosphere — but only because the atmosphere's temperature changes with height. Water absorbs where the air cools upward and emits where it warms, and in an atmosphere at one temperature every molecule vanishes from the spectrum, however much of it there is. An emission spectrum is a sounding of a temperature gradient, and the molecules are what it is taken with.

exoplanets · Exoplanet atmospheres One sight line, two spectra: emission peaks at 46 K while the absorption reaches τ = 1.10. The 21-centimetre line along one sight line through a cold cloud (5·10²⁰ atoms cm⁻² at 60 K) embedded in warm gas (3·10²⁰ cm⁻² at 6000 K, half in front and half behind), computed by radiative transfer. Above: the emission spectrum, brightness temperature against velocity — a broad pedestal from the warm gas (dashed, alone) with the cold cloud's narrow peak on top, capped at 46 K because the cloud is nearly opaque at line centre. Below: the same gas seen in absorption against a bright radio source behind it, the fraction of the source's light transmitted. The absorption shows the cold cloud and almost nothing of the warm gas, whose optical depth is a hundred times smaller per atom. Read in the optically thin approximation, the emission gives a column of 6.26·10²⁰ cm⁻² against a true 8·10²⁰.

A temperature that belongs to none of the gas

The hydrogen between the stars announces itself at 21 centimetres twice — in emission, which counts atoms whatever their temperature, and in absorption against a radio source behind, which counts atoms divided by their temperature. The ratio of the two is a temperature. Along any real sight line it is the harmonic mean of a cold phase and a warm one, a number no parcel of the gas actually has — and where the cold gas is thick enough, the emission stops counting it at all.

galaxies · Interstellar medium

Named alongside it

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

Emission spectrumEquipartitionOptical depthRadiative transferSecondary eclipseSynchrotron radiationAbsorption lineAbundance degeneracyAlbedoChromosphereContribution functionCooling break

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