Astronomy & Celestial Mechanics
the sky, computed
Generator

The passband-set generator

B − V against temperature, computed and measured
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.

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.

4 essays call passband-set. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Every figure listed here is the same construction drawn at different numbers, so a correction to one is a correction to all of them.

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. Starlight

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.

 The Bouguer line, and the intercept nobody observed. Instrumental magnitude against airmass for one star of magnitude 10 outside the atmosphere, observed at 5 airmasses in 5 bands. Each slope is that band's extinction coefficient, computed from Rayleigh scattering, an aerosol term and ozone rather than assumed: U 0.493, B 0.246, V 0.126, R 0.061, I 0.027 magnitudes per airmass. Every line is fitted through its points and extended to X = 0, and the intercept there is the published magnitude — a measurement made at an airmass no observation is ever taken at, because the smallest airmass available is 1 and that is already a whole atmosphere. Two consequences follow and neither is a detail. The slope has to be re-measured every night, because the aerosol term changes with the weather and is not a property of the site. And the U-band line is 3.9 times steeper than the V-band one, so the extrapolation is 3.9 times longer in exactly the band where photons are scarcest — which is why ultraviolet photometry from the ground was always the least trustworthy part of a magnitude system. Starlight

A magnitude measured where nothing was measured

Every published brightness is an extrapolation off the end of a graph. A star is observed through one atmosphere at least, a line is fitted against airmass, and the number quoted is its intercept at zero — a place no observation is ever taken from.

 A colour term of -0.059 magnitudes per magnitude, and one star that will not obey it. Synthetic photometry of blackbodies from 3,000 to 42,000 K through two V bands: the standard one, and a natural system whose effective wavelength is 9 nm longer and whose width is 1.12 times as large. The vertical axis is the difference between the two magnitudes for the same star — not a constant, because a wider redder band collects a different fraction of a hot spectrum than of a cool one. Fitting a straight line against B − V gives a colour term of -0.0585 magnitudes per magnitude and leaves a residual of 3.1 millimagnitudes, which is why a linear transformation is the standard reduction and why it works. The mark off the line is a cool star with molecular absorption bands in the red, drawn from the same blackbody with three synthetic bites taken out of it: it sits 27 millimagnitudes from the fit, 9 times the blackbody scatter. A colour term knows one number about a star and a spectrum has a shape, and that gap is the reason all-sky photometry stops at a per cent while differential photometry on one field reaches a millimagnitude. Starlight

The same star through two telescopes

A magnitude is defined by a response curve, and no two telescopes have the same one. The difference between two observatories' measurements of one star is not a constant to be subtracted but a function of the star's colour — and for a star whose spectrum has structure, not even that.

 Four spectra, and 3.9 magnitudes between them at the same redshift. The K-correction — the magnitude that has to be added to compare a redshifted object with a nearby one through the same filter — against redshift, for four power-law spectra. At zero redshift every correction is zero by construction. Beyond that they diverge, because a filter at a fixed observed wavelength samples a different part of the source's own spectrum at every redshift, and how much flux is there depends on the spectrum. A flat spectrum needs no correction at all at any redshift, and the two extremes drawn differ by 3.9 magnitudes by z = 1.2. The circularity is the point: applying the correction requires the spectrum, and the spectrum is what a magnitude is being used to constrain. The dashed line is the way out — observe in a band chosen so that it lands on the rest-frame band of interest, and the spectral term cancels, leaving only the bandwidth stretch. Starlight

A magnitude in a band the source never had

A filter passes light at a fixed observed wavelength, and a redshifted source emitted that light at a shorter one. Comparing a distant galaxy with a nearby one through the same filter therefore compares two different parts of two spectra — and the correction between them needs the spectrum, which is what the magnitude was going to be used to find out.

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