Astronomy & Celestial Mechanics
the sky, computed
Concept

Photometry — where it appears

The measurement of how much light a source delivers, in a stated passband. Absolute photometry against a standard network stops at about a per cent; differential photometry against comparison stars on the same image reaches a millimagnitude.

Named by 7 essays across 2 fields — each of them below, with the objects they name alongside it.

The magnitude scale, plotted. The logarithm of the received light against magnitude from -27 to 32, with the vertical scale left unlabelled because only its slope matters. The relation is a straight line of slope −0.4, which is what makes five magnitudes exactly a hundredfold — and the numbers run backwards, so brighter is smaller. 9 landmarks are marked, from the Sun to the deepest exposures, spanning a factor of 1.2·10²³ in received light.

A scale that runs backwards, multiplies, and works

Brighter stars have smaller magnitudes, and five steps is a factor of a hundred. A scale invented by eye in the second century BC turned out to be logarithmic, because eyes are.

starlight · Magnitudes The fractional distance error, rung by rung. Cumulative fractional uncertainty in a measured distance against the distance itself, both on logarithmic axes. Each rung of the ladder is calibrated against the one below it, so its scatter adds in quadrature to everything already inherited, and the total can only rise: radar to the planets 10⁻⁵%, parallax 1.0%, main-sequence fitting 5.1%, Cepheids 6.5%, Type Ia supernovae 8.2%.

Every distance is measured with the last one

No single method reaches from a planet to a distant galaxy. The ladder is built rung by rung, each calibrated on the one below, and the errors multiply all the way up.

starlight · Distance ladder 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 extinction law, for three kinds of dust. How much of a star's light dust removes, against inverse wavelength, normalised to one at the V band. Blue light is extinguished more than red, which is why reddening and extinction are the same measurement — and the hump at 4.6 inverse microns is a feature of the grains themselves, present on almost every sight line and still without an agreed carrier. Larger grains give a flatter law and a larger R_V.

Dust makes everything look further away

A star behind dust is fainter, so a distance taken from its brightness comes out too large. The same dust also makes it redder, and the reddening is measurable where the dimming is not — which is the only reason the correction can be applied at all.

starlight · Extinction B − V against temperature, computed and measured. B − V against effective temperature. The curve is the colour index of a blackbody, obtained by integrating Planck's law against B and V response functions and subtracting a constant so that the index is exactly zero at 9600 K — the convention that an A0V star has every colour index zero, which is a choice and not a measurement. The 15 points are the main sequence as it is actually measured, and they do not lie on the curve: at the Sun's temperature the blackbody gives 0.446 where the sky gives 0.653, and at M0V 0.995 against 1.40. The model is too blue almost everywhere, and least wrong near 9600 K — which is why the zero point is put where it is. A colour index is a difference of two magnitudes, so it is a difference of two integrals, and changing either filter changes the number.

A magnitude has to say which light

The same star is a different magnitude in every filter, and a colour index is a difference of two conventions rather than a property of the star. Both are integrals of a spectrum against a piece of glass, and the zero point is a choice somebody made in 1953.

starlight · Photometric systems A transit of a planet 0.103 of its star's radius. The star's brightness through one transit, computed by integrating the uniform stellar disc over the region the planet covers. The depth is 1.05%, which is exactly (Rp/R⋆)² = 0.01055. The four contact points are where the two discs are externally and internally tangent, at separations 1 ± 0.103 stellar radii.

A planet measured by the light it removes

A transit gives a depth, and the depth is a ratio of two radii rather than a size. Everything a transit says about a planet is said in units of a star nobody has visited either.

exoplanets · Transits A transit depth of 1.200 per cent for a planet of area 1.000 per cent. Three transits of the same planet across the same star, differing only in how the star's brightness falls toward its edge. A planet of radius ratio 0.1 covers 1.000 per cent of the stellar disc's area, and if the disc were uniformly bright that would be the depth. It is not uniformly bright: a sight line near the limb leaves the photosphere at a shallow angle and therefore from a cooler layer, so the edge is dimmer than the centre, and a planet crossing near the middle blocks light that is brighter than average. The transit drawn with realistic coefficients is 1.200 per cent deep — 20 per cent deeper than the area — and it is also rounder, because the covered brightness changes through the crossing instead of staying flat. The consequence is stated in the numbers beside the curves. Each is a least-squares fit of the radius ratio to the realistic curve, performed with a different assumed limb-darkening law, and the recovered radius moves by up to 3.6 per cent depending on which law is assumed. Fitting with the law the curve was made from returns the input to five figures, which is the control: the bias is the mis-specification and not the fitter. Since the coefficients come from a model atmosphere rather than from the light curve, every published planetary radius carries a systematic from stellar physics that no amount of photometric precision removes — and it is the dominant one for the best-measured planets. The picture holds the impact parameter fixed; a grazing transit is worse, because it samples only the limb, where the disagreement between laws is largest.

The depth is not the area

A planet covering one per cent of its star's disc does not make a transit one per cent deep. The star is brighter in the middle, so a planet crossing the middle blocks more than its share — and the correction depends on coefficients that come from a stellar atmosphere model rather than from the light curve.

starlight · Limb darkening

Named alongside it

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

Distance modulusApparent magnitudeColour indexParsecAbsolute magnitudeAstronomical unit (AU)Colour excessDistance ladderImpact parameterInverse square lawLimb darkeningMetallicity

All concepts