Astronomy & Celestial Mechanics
the sky, computed
Concept

Gravitational lensing — where it appears

The deflection of light by mass along the line of sight, which magnifies, distorts or multiplies the image of whatever lies behind. It measures mass regardless of whether the mass emits anything, which is why it is one of the four independent routes to dark matter.

Named by 11 essays across 6 fields — each of them below, with the objects they name alongside it.

A magnification, and a spike inside it. The brightness of a background star as a foreground one passes in front of it. The smooth curve is exact: a point mass magnifies a point source by (u² + 2)/(u√(u² + 4)), which peaks at 6.7 for this track. The spike is the planet, of mass ratio 0.001, lensing one of the two images. Its duration is the Einstein time scaled by √q — about 23 hours against 30 days — so the whole planetary signal is a few hours in an event lasting a month, and it never repeats. The deviation is drawn in the approximation that the planet lenses the image in isolation; a real caustic crossing has structure this smooths over, and its true height is set by the source's size rather than by the geometry.

A star magnified by a planet nobody will see again

Microlensing weighs a planet by the way its gravity bends light around it. The measurement lasts a few hours, cannot be repeated, and is the only one that does not require the planet's star to be visible at all.

exoplanets · Microlensing Why an Einstein radius is a mass. The geometry, drawn at an angle some ten thousand times larger than the real one so that anything is visible at all. Light from a source directly behind a lens reaches the observer along every path that passes the lens at the same distance, so the image is a ring rather than a point. The ring's angular radius is θ_E = √(4GM/c² · D_ls/D_l D_s), which for a lens of 1.0×10¹² M☉ at these distances is 2.52 arcseconds and encloses 10 kpc at the lens. Rearranged, it is a mass in terms of an angle and three distances — and the mass so obtained is inside a cylinder rather than a sphere, and assumes nothing whatever about the lens being in equilibrium, which is the assumption every other weighing in this collection makes.

Weighed by the light that bends past it

Every other mass in this collection is measured from something orbiting, which requires the system to have settled down. A gravitational lens weighs whatever is in the way with no such assumption — the light does not care whether the mass is in equilibrium.

galaxies · Strong lensing A cluster weighed three ways, and its stars weighed once. A cluster of galaxies with a velocity dispersion of 1000 km/s, gas at 8 keV and a strong-lensing Einstein radius of 25 arcseconds, each turned into a mass inside 1.5 Mpc by its own relation and nothing else: 7.0, 8.9 and 15.2 × 10¹⁴ M☉. The three assume, respectively, that the galaxies are in equilibrium, that the gas is, and nothing whatever — so their agreement to within a factor of 2.2 is not three restatements of one assumption. The lensing bar is the loosest of the three and is drawn that way deliberately: it measures the mass inside a cylinder of radius 109 kpc, 1.11×10¹⁴ M☉, and carrying that out to 1.5 Mpc as though it were a sphere overstates it. The stars are 2.9 per cent of it.

A cluster weighed three ways

The speeds of a cluster's galaxies, the temperature of its gas and the bending of light behind it are three measurements with almost nothing in common. They agree within a factor of two, and all three exceed the mass of the stars by about a hundred.

galaxies · Clusters Three mass models, one rotation curve. Rotation curves for three decompositions of the same galaxy, from a disc contributing 30 per cent of the outer rotation to one contributing 95. Each is the quadrature sum of a stellar disc, whose shape is fixed by the light distribution and whose amplitude is an unknown mass-to-light ratio, and a dark halo with parameters of its own. All three reproduce the same flat outer rotation, because the halo's amplitude is adjusted to make up whatever the disc does not supply. They differ in the inner few kiloparsecs, by 27 kilometres a second here — which is more than the measurement error and less than the uncertainty in the disc's own contribution, since that depends on a mass-to-light ratio nobody measures directly. The maximum-disc assumption picks the largest disc consistent with the data, and it is a convention rather than a result.

Three mass models that fit the same curve

A rotation curve is one function of radius and it is fitted with three components, only one of which is known. The stellar disc's contribution scales with a mass-to-light ratio nobody measures, and whatever the disc does not supply the dark halo does — so a family of models reproduces the same curve exactly, and choosing among them is a convention rather than a measurement.

galaxies · Dark matter A hole's shadow at 17° to its spin axis. The critical curve — the outline of a black hole's shadow on a distant observer's sky — for spins from 0 to 0.998, seen 17 degrees from the spin axis, in gravitational radii. Every curve is the projection of the unstable spherical photon orbits, and nothing about an accretion flow enters it. The sizes are 5.20, 5.12, 4.92, 4.84: a spread of 6.9 per cent across the entire range of spin, which is what makes an image of one of these a mass measurement with almost no model in it. What spin does instead is push the curve sideways and flatten one edge — the fast hole's outline sits 0.70 gravitational radii off centre, because photons co-rotating with the hole are dragged round and escape from closer in than counter-rotating ones can. That displacement is the spin signal, and it is measured against a centre nothing else marks.

A ring whose size is almost only a mass

The dark patch at the centre of an image of a black hole is the projection of the orbits light cannot escape from, and its size changes by seven per cent across the whole range of spin and inclination. That insensitivity is what makes an image a mass measurement with almost no model in it — and what leaves the spin in the part of the picture that barely moves.

gravitation · Black hole spin One hot spot, seen through more and more gravity. The X-ray flux from a small hot spot at colatitude 60° on a rotating neutron star seen from inclination 60°, over one rotation, each curve divided by its own mean, for no bending, 2GM/Rc² = 0.20, 2GM/Rc² = 0.34, 2GM/Rc² = 0.48 — the last two are a 1.4 M☉ star of 12 km and a 2.0 M☉ star of 12.4 km. Light bending is computed exactly; Doppler effects are left out. With no bending the spot is behind the star for 39 per cent of the turn and the pulsed fraction is 1.00. As the compactness rises the star's gravity bends light from the far side into view: the dark interval shrinks to 27 per cent, then vanishes, and the pulsed fraction falls to 1.00, 0.99, 0.64. The more compact the star, the more of its own back it shows and the smoother its pulse.

The pulse that shows the back of the star

A hot spot on a spinning neutron star flashes once a turn, and the flash is weaker than it would be on an ordinary star because the star's gravity bends light from its far side into view. How much weaker measures the ratio of mass to radius. But on its own the flash cannot tell a compact star from a tilted one — the two trade exactly — and what separates them is the lean the spin puts into the pulse, which measures the radius directly.

stars · Degeneracy The Sun bends light from every direction in the sky. The gravitational deflection of starlight by the Sun, as seen from the Earth, against the angle between the star and the Sun, as general relativity predicts it (solid) and as a Newtonian particle of light would be bent (dashed), exactly half. At the Sun's limb the deflection is 1.75 arcseconds, the value measured at the eclipse of 1919. It falls only as the cotangent of half the angle, so it is 4.07 milliarcseconds at right angles to the Sun and 9.8 at 45°, the closest a space astrometry mission looks; it reaches zero only directly opposite the Sun. Every star in a modern catalogue has been displaced by an amount far larger than the precision of its position, and the shading marks the angles such a mission actually observes.

Every star in the catalogue is bent by the Sun

The bending of starlight by the Sun is remembered as an eclipse measurement of 1.75 arcseconds at the solar limb. The same bending reaches every direction in the sky, falling only slowly away from the Sun, and at right angles to it is still four milliarcseconds — two hundred times the precision of a modern star position. Every catalogue corrects every star for it, and the correction has become the most precise test of how much space a mass curves.

sky · Aberration A cluster's mass written as a dipole on the microwave background. The change in microwave-background temperature caused by lensing behind a cluster of 3·10¹⁴ solar masses at redshift 0.5, where the unlensed background has a uniform gradient of 13 microkelvin per arcminute running left to right. Contours at ±0.5, ±1, ±1.5, ±2, ±2.5 μK: blue where the sky is warmer than it would have been, orange where it is cooler. The cluster bends light towards itself, so each point on the sky shows the temperature of a point a little closer to the cluster's centre: on the side towards which the gradient rises that point is colder, and the sky there is cooler than it would have been; on the downhill side it is warmer. The pattern is a dipole aligned with the gradient, peaking at 2.9 μK, and it carries the cluster's mass profile in its radial shape. Its orientation is set entirely by the background, not by the cluster, which is what lets it be told apart from the cluster's own emission.

A cluster weighed against the background behind it

Every cluster of galaxies has the same source behind it — the last-scattering surface, at a distance known to a fraction of a per cent — and bends that source's smooth gradient into a dipole of a few microkelvin. The dipole knows nothing about gas or galaxy shapes, so stacked over hundreds of clusters it is a mass scale that no hydrostatic assumption can bias.

cosmology · Sunyaev zeldovich One cluster weighed two ways, and the redshift where the two swap places. The fractional error on the mass of a single cluster of 3·10¹⁴ solar masses, on a logarithmic axis, against the cluster's redshift. Weighed by the shapes of the galaxies behind it (rising curves), the error is 50 per cent at redshift 0.3 and 750 per cent at 1.5 for a survey of today's depth, 10 galaxies per arcmin²; 26 per cent at redshift 0.3 and 187 per cent at 1.5 for a survey of the next depth, 30 galaxies per arcmin²: the galaxies far enough behind the cluster to be lensed by it run out. Weighed by the dipole it draws on the microwave background (nearly flat curves), the error is 461 per cent at 0.3 and 337 per cent at 1.5 for a map with 10 μK-arcmin of noise; 224 per cent at 0.3 and 153 per cent at 1.5 for a map with 3 μK-arcmin of noise — never good for one cluster, and almost the same at every distance, because the source is at redshift 1090. The curves cross at z = 1.22 (10 galaxies per arcmin², 10 μK-arcmin), z = 0.89 (10 galaxies per arcmin², 3 μK-arcmin), z = 1.77 (30 galaxies per arcmin², 10 μK-arcmin), z = 1.41 (30 galaxies per arcmin², 3 μK-arcmin). Below the crossing a galaxy survey weighs each cluster better; above it, only the background weighs it at all.

A mass scale checked near and carried far

A cluster can be weighed by the galaxies behind it or by the microwave background behind it, and the two share no systematic error. They do not share a redshift range either: the galaxies run out at about one, the background never does, and a comparison can only be made where both still work.

cosmology · Sunyaev zeldovich A ratio relativity predicts from one density. The ratio EG — written E with a subscript G in the literature — of the lensing around a set of galaxies to their clustering and infall, against redshift. In relativity it is Ωₘ₀/f(z) and depends on nothing else: with Ωₘ₀ = 0.3153, 0.598 today, 0.455 at z = 0.32 and 0.360 at z = 1, falling because f rises towards one as matter comes to dominate. The dashed curves keep light's bending as in relativity and change how hard matter pulls, by −10% and +10% today, growing with the dark energy's share: a weaker pull slows the infall and raises EG. The point is the first measurement, 0.39 ± 0.06 at z = 0.32 from galaxy–galaxy lensing in the Sloan survey.

A ratio in which the galaxies cancel

Matter falls in one gravitational potential and light is bent by the sum of two, and relativity says the two are equal. A survey that measures both the infall of galaxies and the lensing around them can divide one by the other so that the galaxies' bias and the fluctuations' amplitude both cancel — leaving a ratio that relativity predicts from the matter density alone, and that a gravity which bends light differently from the way it pulls matter cannot fake.

cosmology · Dark energy Where the quasar counts turn over, the signal changes sign. The local slope s of quasar number counts against limiting magnitude (solid), for counts that steepen to 0.90 at the bright end and flatten to 0.29 at the faint end with a turnover near magnitude 19, and the factor 2.5s − 1 that multiplies twice the convergence in the change of counts (dashed). The factor crosses zero at magnitude 20.1: brighter quasars are found in excess round foreground galaxies, fainter ones in deficit, and quasars near the crossing show no correlation at all. The predicted reversal, observed, is what identifies the correlation as lensing rather than as dust or as quasars and galaxies sitting in the same structures.

A lens counted instead of measured

A foreground mass makes the sources behind it brighter and spreads them over a larger patch of sky. Whether the patch then holds more of them or fewer depends only on how steeply their numbers rise towards faint fluxes, and at one slope the two effects cancel exactly. So a lens can be weighed without measuring a single shape — by counting — and the signal changes sign from bright quasars to faint ones in a way that nothing but lensing can imitate.

galaxies · Weak lensing

Named alongside it

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

MagnificationCluster mass functionConvergenceCritical surface densityDark matterDegeneracyEinstein radiusGeneral relativityHydrostatic biasThe mass-sheet degeneracyMatched filterStacking

All concepts