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 6 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 Mass profiles that pinch together near the half-light radius. The enclosed mass of each of the 13 haloes that fit the same flat dispersion of 10 km/s, against radius in units of the stars' three-dimensional half-light radius, 391 parsecs. At a tenth of that radius they span a factor of 17, and at four times it a factor of 2.5; they pinch together near the half-light radius, closest at 1.47 of it, where they agree to 14 per cent. The dot is the estimate 3σ²r/G at the radius where the stars' density falls as r⁻³, 0.94 of the half-light radius; there the fitted masses lie between 0.65 and 1.03 times it. Each model trades a wrong guess about the orbits against a wrong guess about the mass, in opposite senses inside and outside.

The one radius where the orbits stop mattering

A velocity dispersion weighs a galaxy only if the shape of its stars' orbits is known, and it never is. Yet every model that fits a dwarf galaxy's flat dispersion profile — cored or cusped, radial orbits or tangential — agrees on the mass inside one particular radius, close to the radius that contains half the light. The degeneracy has a node, and the node is where the measurement should be quoted.

galaxies · Velocity dispersion Haloes that agree on the line of sight, and disagree across the sky. The dispersion of the tangential component of motion on the sky — across the radius vector, as a proper motion measures it — predicted by each of the 13 dark haloes that fit the same flat line-of-sight dispersion of 10 km/s, against projected radius in half-light radii, for stars in a Plummer profile. Coloured by the halo's inner slope; the dashed line is the line-of-sight value all of them reproduce. At the half-light radius the haloes' line-of-sight dispersions agree to 2.0 per cent, and their tangential dispersions range from 8.3 to 10.8 km/s, a factor of 1.29. The anisotropy each halo needed to match the line of sight is written into the motion across it, which no line-of-sight measurement sees.

Motions across the sky measure the orbits first

A dwarf galaxy's line-of-sight velocities leave the shape of its stars' orbits free, and that freedom hides whether its dark halo has a core or a cusp. Proper motions add the two components across the sky. They read the orbits' shape almost at once, with a few hundred stars — and they reach the slope of the halo only with ten thousand.

galaxies · Velocity dispersion A compact galaxy with a faint envelope, seen at redshift two. The surface-brightness profile of a model galaxy at redshift 2, in magnitudes per square arcsecond against radius in kiloparsecs on a logarithmic axis (solid): a compact de Vaucouleurs core of 1 kpc half-light radius and an exponential envelope of 10 kpc holding 30 per cent of the light, dimmed by 4.77 magnitudes by the expansion. The dotted curve is the core alone. Horizontal lines mark survey limits of 27 and 30 magnitudes per square arcsecond. A single Sérsic profile fitted to what is brighter than each limit (dashed) gives a half-light radius of 1.3 kpc and index 4.6 above 27, and a half-light radius of 1.5 kpc and index 5.1 above 30. The galaxy's true half-light radius, core and envelope together, is 2.1 kpc. The envelope is above the deeper limit out to 16 kpc, and the fit still does not follow it: a single profile weighted towards the bright centre cannot bend to two components, and more depth only nudges the compromise it settles on.

One profile for a galaxy that has two

The worry about the compact galaxies of ten billion years ago was that their growth since might be partly a faint envelope the sky hides. Modelled, the envelope does hide — but not mainly under the sky. A single profile fitted to a core with an envelope reports a compromise at any depth, too large for a tight envelope and too small for a wide one, and the growth it measures is an understatement.

galaxies · Surface brightness

Named alongside it

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

The central flashThe core–cusp problemDark matterDwarf spheroidalJeans equationOrbital anisotropyScale heightShadow pathSurface brightnessTolman dimmingVelocity dispersionAlbedo

All concepts