Illustrated Physics
physics, drawn
Concept

Einstein radius — where it appears

The angular radius of the ring into which a gravitational lens turns a source lying exactly behind it. It grows as the square root of the lens's mass and sets the scale of image separations, magnifications and caustics for any lens.

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

Also named here as gravitational lens — the same set of essays touches all of them, so they are one junction rather than several.

A lens with no focal length. Where a ray crosses the axis, against how far off the axis it passed the deflecting body, for 1 solar mass of radius 1 solar radius. A glass lens deflects a ray by an angle proportional to its distance off axis, which is precisely the condition for every ray to arrive at one point — the flat dashed line. Gravity deflects by 4GM/c²b, which grows smaller further out, so the crossing distance goes as b² and each ray has its own focus. The grazing ray crosses at 548 astronomical units and a ray passing at 12 radii crosses at 78857; the square law is verified on the drawn curve to 1.5e-16. So there is no image plane at all, only a half-line of foci beginning at the first of those and running outward for ever. Anything placed on that line sees not an image but a ring, and moving along it does not refocus anything — it selects which rays are being seen.

The lens with no focal length

A glass lens bends a ray by an angle proportional to how far off the axis it passes, which is precisely the condition for every ray to arrive at one point. Gravity bends by an angle that falls with distance off the axis, so every ray has its own focus and there is no image plane anywhere — only a half-line of foci, beginning 548 astronomical units from the Sun and running outward for ever.

astrophysics · Light deflection Five images of one source, and the curves where they are made. The sky around a lens with a core of 0.10 and an external shear of 0.15, in Einstein radii, with its two critical curves: an outer one near the Einstein ring and a small inner one round the core. A source at (0.06, 0.04) in the source plane — inset, with the caustics the critical curves map to — lies inside the diamond-shaped caustic and has 5 images. Their magnifications are +4.99, −3.76, +3.42, −2.63, +0.01 — a minus sign for an image turned over — and the parities sum to +1. Four sit near the outer curve and carry nearly all of the 14.8-fold total; the fifth, at the centre, is magnified by only 0.013. A source further out, at (0.55, 0.1), has 3.

The fifth image a galaxy hides

A single star bends light into two images of whatever lies behind it. A galaxy, whose mass is spread out, has to make an odd number — one, three or five — because the map from the sky to the source is smooth and has nowhere to put an even count. The famous cross of four quasar images is therefore five images, with the fifth thousands of times too faint to see, sitting at the centre of the galaxy that made it. Images are born and destroyed only in pairs, and only on the curves where the map folds.

astrophysics · Light deflection A lens that cannot focus a wave longer than itself. How much a point mass brightens a wave passing behind it — the factor on the wave's power, on a logarithmic scale — against the dimensionless frequency w = 8πGMf/c³, for a source exactly behind the lens and 0.1, 0.3, 1.0 Einstein radii off it, from the exact diffraction integral. Geometric optics puts the magnification at 10.0, 3.4 and 1.34 for the three offsets and at infinity on the axis, whatever the frequency. The wave disagrees: at w = 0.01 every curve is within 1.6 per cent of 1 — a wave a hundred times longer than the lens passes as if it were not there — and on the axis the power factor is πw/(1 − e^(−πw)): 3.28 at w = 1, 31.4 at w = 10, finite at every frequency. The off-axis curves rise to their geometric values only above w ≈ 1 and then oscillate about them, as the two images begin to interfere.

The lens a long wave walks round

Geometric optics says that a star passing exactly in front of a source magnifies it infinitely. No real wave agrees, because a wave cannot be focused to a point, and for the gravitational waves the detectors hear the disagreement is the whole story. A star, or a black hole of a few tens of suns, is smaller than one of those waves in the only measure that matters — the delay between paths round it — and the wave flows past it almost untouched. What decides whether a mass lenses a wave is not the mass's size, which is millions of kilometres, but its Schwarzschild radius set against the wavelength.

astrophysics · Light deflection

Named alongside it

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

Gravitational lensMagnificationCausticSchwarzschild radiusCritical curveDeflection angleDiffractionFermat's principleFocal lengthGeneral relativityGeometric opticsGravitational waves

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