Astronomy & Celestial Mechanics
the sky, computed
Concept

The half-light radius — where it appears

The level in an atmosphere at which an occulted star has faded to half its unobstructed brightness. It is a pressure rather than a surface, near a microbar for the bodies it is quoted for, so a radius reported this way names a level and not a solid boundary.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

7 clocks, and a body 233 kilometres across. A stellar occultation reduced. Each horizontal segment is one observer's chord: the star vanished, the star came back, and the interval multiplied by the shadow's 21.4 km/s across the ground is the length drawn. The longest, at an offset of -28 km, is 252.3 kilometres. The dashed ellipse is the limb fitted to the chords by least squares in the half-length squared, and its equivalent-area diameter is 233.9 km against the silhouette's true 232.9 — the residual is the shape the fitted ellipse cannot hold, not an error in any timing. The two open marks are observers inside the predicted path who saw nothing, and they are measurements: they bound the limb inside their own offsets, which is what fixes the extent when the positive chords all fall on one side. At 0.12 s per contact the length precision is 2.6 km, 1.10 per cent of the body — a size measured with a clock rather than with an angle, on an object no telescope resolves.

A shape measured by the edge of a shadow

When a small body passes in front of a star the observable is two times. Multiply the interval by the body's speed across the sky and it becomes a chord across a silhouette no telescope can resolve — and enough chords give a profile to a kilometre or two, for an object a few hundred across.

sky · Occultations A shadow whose centre is brighter than no shadow at all. The flux an observer records against their distance from the centre of the shadow, for a body of half-light radius 1180 km with an isothermal atmosphere of scale height 55 km. Away from the centre the curve is the ordinary occultation light curve — the star fading as refraction spreads its light — and near it the two limbs' contributions both carry a geometric factor of the impact parameter over the shadow position, which grows without bound on the axis. The spherical atmosphere reaches 34.79 of the unocculted flux. The ray that arrives on the axis has impact parameter 1019 km, which is 161 km — 2.9 scale heights — below the half-light level, at a pressure 19 times higher. That is the only part of an occultation that reaches there. The peak is finite only because the star is not a point: the geometric factor is softened at 3 km, which is the star's own size projected to the shadow. The second curve is the same atmosphere flattened by 2.0 per cent, which spreads the focus over 20 km and drops the peak to 19.8. A real flattened body gives a caustic rather than a broad peak — several sharp spikes, spread in two dimensions rather than one — so the width here is right, the structure is not, and the height is an upper bound.

The brightest instant of an occultation is its middle

A spherical atmosphere is a lens with a focal length of astronomical units, and an observer standing at the exact centre of the shadow is standing at its focus. The star does not disappear there — it brightens, by more than it would have been unocculted, and the ray that arrives has come from far deeper than anything else in the event.

sky · Occultations The same disc measured smaller at every redshift. The isophotal radius, at a limit of 26.5 magnitudes per square arcsecond, of three exponential discs that do not change at all — central surface brightnesses 20.5, 21.65, 23, the same scale length throughout — against the redshift they are placed at. Surface brightness falls as (1+z)⁴, three magnitudes at z = 1 and six at z = 3, before anything about the galaxy evolves or any band shift is accounted for, so every isophote moves inward. The discs vanish from an isophotal catalogue entirely at z = 2.98, 2.05, 1.24. The dashed line is the half-light radius, 1.68 scale lengths, which is defined by the galaxy's own light and does not move. Galaxies at high redshift are genuinely smaller than their present-day counterparts; how much smaller depends on which of these two kinds of radius is quoted, and the isophotal kind would report a shrinkage for galaxies that have none.

A radius that is a measurement of the threshold

A galaxy has no edge. Its light fades exponentially for as far as anyone can measure, so every size ever quoted for one is the radius at which the fading crossed some line that somebody drew. Drawn at a fixed surface brightness, the line measures how bright the galaxy's centre is as much as how big the galaxy is — and at high redshift it measures the expansion of the universe as well.

galaxies · Surface brightness

Named alongside it

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

The central flashScale heightShadow pathAlbedoAngular-diameter distanceAstrometric predictionAtmospheric oblatenessCausticExponential discThe Fresnel scaleHazeIsophotal radius

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