Astronomy & Celestial Mechanics
the sky, computed
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Photometric systems — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · starlight
  2. 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.

    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.

    part 2 · starlight
  3. A dome flat that is 3.5 per cent wrong leaves 0.042 magnitudes across the field. Two things called the flat field. The left panel is the true illumination of the focal plane, falling by 12 per cent from centre to corner because of vignetting and the filter's own radial transmission. The middle panel is what a dome flat measures, which is the illumination produced by a screen at a finite distance lit by lamps — a different angular distribution, and therefore a different fall-off by a few per cent. The right-hand plot is what survives dividing one by the other: a smooth radial gradient of 0.042 magnitudes from centre to edge. Pixel-to-pixel response scatter, which is what most people mean by a flat field, is 1.8 per cent per pixel and averages down to 0.255 per cent inside a photometric aperture. The term everybody removes is the one that does not matter, and the term that matters is smooth, is different for every flat-fielding method, and looks exactly like a real gradient in the sky.

    A response measured pixel by pixel

    Two completely different quantities are called the flat field. One is the detector's pixel-to-pixel response, which everybody removes and which averages away anyway. The other is the illumination pattern of the optics, which is smooth, is different for every method of measuring it, and survives into every magnitude the instrument produces.

    part 3 · starlight

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