Astronomy & Celestial Mechanics
the sky, computed
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The microlens-curve generator

A magnification, and a spike inside 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.

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.

7 essays call microlens-curve. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Every figure listed here is the same construction drawn at different numbers, so a correction to one is a correction to all of them.

A fold 0.06 Einstein times wide, and two configurations that make the same one. A binary lens of mass ratio 0.003 at a projected separation of 1.5 Einstein radii. Top left: the source plane, with the caustic — the set of source positions at which the magnification is formally infinite — and the track of a background star across it. A single lens has no such curve; it magnifies smoothly and diverges only at one point. A second mass makes the lens mapping fold, and the fold has edges: crossing one, the number of images changes from 3 to 5, because a pair is created out of nothing on the critical curve. The small closed curve near the origin is the central caustic, always there; the larger one at 0.83 Einstein radii is the planetary caustic, and its distance from the origin is s − 1/s, which is where the planet's own image lies. Top right: the central caustic drawn twice, once for s = 1.5 and once for s = 0.667. They are 0.0184 and 0.0169 Einstein radii across and they lie on top of each other. That is not a coincidence of these numbers: to the order that a central-caustic anomaly is measured, a close binary and a wide one with the reciprocal separation produce the same perturbation, so an event with only a central anomaly returns two separations and no way to choose. Below: the light curve along the track. The smooth part is what a single lens of the same total mass would do; the spikes are the two crossings, 0.06 Einstein times apart, so a few hours inside an event lasting a month. The two curves differ in one thing only — the size of the source. A point source diverges at each fold and reaches 27; a source of angular radius 0.006 Einstein radii averages over its own disc and reaches 9 — an eighth of a source radius inside the fold the two are 8 and 4, with the divergence replaced by a rounded shoulder whose width is the source's own diameter. Everywhere else in this collection the finite size of a star is a nuisance that degrades a measurement. Here it is the ruler: the fold is a straight edge of known sharpness sweeping across a disc, so the shape of that shoulder gives the source's angular radius, and dividing it by the crossing time gives the angular Einstein radius — which is the one quantity a light curve otherwise cannot supply. Exoplanets

A light curve with a fold in it

A single lens magnifies smoothly. A second mass makes the lens mapping fold, and a fold has an edge — a curve across which two images appear out of nothing and the magnification formally diverges. Crossing it turns the finite size of the source star from a nuisance into a ruler.

 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. Exoplanets

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.

 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. Galaxies

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.

 A light curve pulled out of shape by the Earth's own orbit. Magnification against time for a 90-day event at impact parameter 0.3, computed with the observer's orbital motion included at three values of the microlens parallax. The dashed curve is π_E = 0 and is exactly symmetric in time, because a straight-line track past a point lens has to be. The others are not. The Earth's displacement over the months the event lasts adds a term π_E times its projected orbital motion to the lens–source trajectory, so one wing is pushed closer to the lens and the other further away, and the curve acquires an asymmetry of 16 per cent at π_E = 0.15 and 34 per cent at π_E = 0.35. That asymmetry is the whole measurement. An ordinary event gives one dimensioned number, t_E, which mixes the lens mass with two distances and a proper motion and therefore weighs nothing; the parallax gives a second, and two constraints on the same lens are what a mass requires. It is only available on long events — the Earth has to move appreciably while the magnification is changing — which is why parallaxes are measured for the timescales above about fifty days and not for the short ones that a low-mass lens produces. Exoplanets

An asymmetry that the Earth's own orbit puts in

A microlensing event delivers one number with dimensions, and one is not enough to weigh anything. The Earth's motion over a long event distorts the light curve, and the distortion is the second constraint — after which the mass follows with no distance in it at all.

 Two signals that peak at different separations. The astrometric centroid shift and the photometric magnification against the source–lens separation in Einstein radii, on a common horizontal axis and their own vertical ones, for θ_E = 1 milliarcseconds. The shift is u/(u²+2) times θ_E: it vanishes at u = 0, because the two images are then symmetric about the lens and their centroid is the lens itself; it vanishes as u grows, because the minor image fades and the major one approaches the source; and it is largest in between, at u = √2 exactly, where it is θ_E/2√2 = 0.3536 mas. The magnification at that separation is only 1.15, which is a nineteen per cent brightening — a signal a survey would barely flag. The two observables are therefore complementary rather than redundant. The photometric event is short, bright and centred on the closest approach; the astrometric one is broad, largest on the wings, and lasts several times longer, falling only as 1/u once the source is well away. That slow decline is why astrometric microlensing needs years of monitoring and why it was a prediction for sixty years before it was a measurement. Exoplanets

A centroid that moves when the brightness does not

The two images a lens makes are never resolved, but their centre of light is displaced from where the source would be — by an amount that is largest at a separation where the magnification is only 1.34, long after the photometric event is over.

 How long an event lasts, against the mass that causes it. Einstein crossing time against lens mass, on logarithmic axes, for a lens at 6.5 kpc in front of a source at 8 moving at 6 milliarcseconds a year — a typical bulge geometry. The slope is a half, exactly, because θ_E goes as the square root of the mass and t_E is θ_E divided by a proper motion: an M dwarf gives 21 days, a Jupiter gives 22 hours, a Neptune gives 4 hours, an Earth gives 1 hours. Nothing else about the event changes with the mass. The peak magnification is set by the impact parameter alone, so a free-floating Earth passing close enough produces exactly the light curve a star would, at exactly the same height, and lasts 1.2 hours instead of a month. That is the entire difficulty of detecting them: the events are ordinary and brief, so what decides whether they are found is the survey's cadence rather than its sensitivity. The dashed line is where the source becomes larger than the Einstein ring — 1.5e-4 solar masses for a 6-microarcsecond source — and below it every event is finite-source-dominated, which flattens the peak and, in the same stroke, measures θ_E. Exoplanets

An event of a few hours and no host

The Einstein crossing time is the square root of the lens mass and nothing else changes, so a free-floating Earth produces exactly the light curve a star does, at exactly the same height, and is over in ninety minutes. What decides whether it is found is cadence rather than sensitivity.

 Two separations that draw the same caustic. The width of the central caustic against the separation, computed from the lens equation for a mass ratio of 0.003, for each separation s and for its reciprocal 1/s. The two curves lie nearly on top of each other. That is the close–wide degeneracy, and it is a theorem rather than a coincidence: expanding the binary lens equation near the primary shows that the central caustic depends on the separation only through s + 1/s to first order in the mass ratio, and that combination is invariant under s → 1/s. It is first order and not exact, which the figure shows rather than hides — the two agree to 0.7 per cent at s = 2.8 and only to 13.1 at s = 1.4, because a smaller separation is closer to the resonant regime where the central and planetary caustics have not yet separated. Reducing the mass ratio to 1.0e-3 brings the worst case to 4.8 per cent, which is the first-order statement being checked rather than quoted. What that means for a measurement is uncomfortable. An event whose planetary signal comes from the source passing near the central caustic — which is most of them, because the central caustic sits where the magnification is already high and the event is already being watched — cannot distinguish a companion at 2.8 Einstein radii from one at 0.357. For a typical lens that is the difference between a planet at four astronomical units and one at less than one. And the disagreement the figure measures is not the way out: at 2.8 Einstein radii the caustics differ in width by 0.7 per cent, which is far below what a light curve sampled through a night's seeing can separate, so the ambiguity is real in the data even where it is not exact in the mathematics. Exoplanets

Two systems that draw the same curve

A binary lens with separation s and one with separation 1/s have central caustics that agree to first order in the mass ratio. The same event is therefore a planet at four astronomical units or one at less than one, and no amount of photometric precision decides between them.

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