Astronomy & Celestial Mechanics
the sky, computed
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.

Assumes Microlensing and Parallax.

16 min read 7 figures Measuring the unreachableSmall perturbations, large consequences

The first rung of this anchor described a measurement that lasts a few hours, cannot be repeated, and does not require the lens to be visible. The second added a second mass, found the caustic, and used the source’s finite size as a ruler.

Neither rung weighed a lens, and the reason is arithmetic. A single-lens event yields exactly one number that carries dimensions — the Einstein crossing time — and one is not enough.

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.
Fig. 1 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 to the lens–source trajectory, so one wing is pushed closer to the lens and the other further away.

Why one number is not enough

The Einstein crossing time is θ_E divided by the relative proper motion, and the angular Einstein radius is

θE=κMπrel,κ=8.144 mas/M⊙\theta_E = \sqrt{\kappa M \pi_{\rm rel}}, \qquad \kappa = 8.144\ \text{mas}/M_\odot

with π_rel the relative parallax between lens and source. So

tE=κMπrelμrelt_E = \frac{\sqrt{\kappa M \pi_{\rm rel}}}{\mu_{\rm rel}}

and a measured t_E constrains one combination of three unknowns: the mass, the relative distance and the relative motion. A nearby low-mass lens moving slowly and a distant heavy one moving fast give identical events. That is not a measurement noise problem — the two light curves are the same curve.

The standard escape is a prior. Galactic models supply distributions for the distances and the proper motions of bulge and disc lenses, and a Bayesian estimate of the mass follows with a factor-of-two uncertainty. That is how most published microlensing planet masses are obtained, and it is worth being honest that it is a model output rather than a measurement.

The prior is not arbitrary and it is not innocent either. It encodes the stellar density along the line of sight, the velocity dispersions of the disc and the bulge, and a mass function — and the mass function is the thing being measured. So a mass derived this way carries a circularity that is small when the prior is broad and is not zero, and it is the reason an occurrence rate derived from these events has to be quoted with the assumed Galactic model named.

The alternative is to find more dimensioned quantities in the same data, and there are exactly three of them. The angular Einstein radius θ_E, from the source’s finite size. The microlens parallax π_E, from the observer’s motion. And the astrometric shift of the image centroid, which is θ_E again by another route. Any two of the three give a mass; none of them alone does.

What the Earth’s motion does

Over an event lasting a few days, the observer is on a straight line and the standard picture holds. Over an event lasting months, the observer is on a circle.

The Earth’s orbital displacement over the event, projected onto the sky and measured in Einstein radii, is the microlens parallax π_E = π_rel/θ_E. It adds a wobble to the lens–source relative trajectory, and the effect on the light curve is an asymmetry: the wing during which the Earth’s motion carries it toward the lens is brightened and the other is dimmed.

That asymmetry is the signature, and it is unambiguous in a way that a change of amplitude or duration would not be. A symmetric curve is what a straight track produces, exactly, so any departure from symmetry needs a cause and the observer’s own motion is the first one to test.

A light curve pulled out of shape by the Earth's own orbit. Magnification against time for a 160-day event at impact parameter 0.15, computed with the observer's orbital motion included at two 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 92 per cent at π_E = 0.5. 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.
Fig. 2 A longer and closer event at a larger parallax, where the distortion is no longer a subtlety. At π_E = 0.5 over a 160-day event the two wings differ by a factor the eye reads immediately, and the peak has shifted away from the nominal time of closest approach. Events like this are where parallaxes are actually measured, and they are rare for a reason the next rungs return to — a long event means a heavy or slow lens, and both are uncommon.

It is worth noticing that the effect is a geometric one rather than a photometric one. Nothing about the source or the lens changes; what changes is where the observer is, and therefore what the impact parameter of the trajectory is at each moment. That is why the distortion is a smooth low-order shape rather than a feature — a term in the trajectory rather than a term in the magnification — and why fitting it requires modelling the whole event rather than measuring something at one epoch.

It is also why the effect is bounded. The Earth moves two astronomical units across a diameter, so the wobble it adds is at most 2π_E Einstein radii, and π_E for a typical bulge lens is a few tenths. A parallax of one would mean the Earth’s orbit is the size of the Einstein ring projected to the observer, which happens only for very nearby lenses — and those, being nearby, are exactly the ones whose parallaxes are largest and best measured.

Two constraints and a mass

θ_E and π_E are each a combination of the mass and the relative parallax, and they are different combinations: θ_E² = κ M π_rel, and π_E² = π_rel/(κM). Divide one by the other and π_rel cancels:

M=θEκ πEM = \frac{\theta_E}{\kappa\,\pi_E}

There is no distance in that expression. Measuring both quantities weighs the lens without knowing where it is, which is the only route in this subject to a mass that is a measurement rather than a model.

Two constraints that cross, and a mass where they do. The lens mass against the lens distance, for the two dimensioned quantities a well-observed event delivers. Neither on its own is a mass. The angular Einstein radius θ_E = 0.32 mas — obtained from a finite-source effect, which needs a caustic crossing or a very close approach — fixes the combination M π_rel, so it gives the rising curve: a lens further away has a smaller relative parallax and must be heavier to make the same angle. The microlens parallax π_E = 0.35, obtained from the asymmetry the Earth's motion puts into the light curve, fixes the ratio π_rel/M and gives the falling one. The two cross at 0.112 solar masses and 4.22 kpc, and the crossing mass is θ_E/(κ π_E) with κ = 8.144 mas per solar mass — a closed form with no distance in it at all, which is why measuring both quantities weighs the lens without knowing where it is. What the crossing also supplies, for free, is the distance, and with the distance a check: a lens of 0.112 solar masses at 4.22 kpc would be a main-sequence star of a known brightness, and whether it is seen in a later high-resolution image is the one independent test the method has.
Fig. 3 The two constraints in the mass–distance plane. θ_E = 0.32 mas gives the rising curve — a lens further away has a smaller relative parallax and must be heavier to make the same angle — and π_E = 0.35 gives the falling one. They cross at 0.112 solar masses and 4.22 kpc, and the crossing mass is exactly θ_E/(κπ_E), with no distance in it. The crossing supplies the distance as a bonus, and with it a testable prediction: a lens of that mass at that distance is a main-sequence star of known brightness, and whether it is seen in a later image is the method’s one independent check.

There is a second reading of that formula which is more useful in practice than the first. Rearranged, π_E = π_rel/θ_E says that a large parallax means a small Einstein radius relative to the relative parallax — which happens for a nearby lens, a low-mass lens, or both. So a well-measured parallax is itself an indication of a nearby low-mass object, before any mass is computed, and the events with the largest parallaxes are the nearest lenses in the survey.

That has been used the other way round. The handful of microlensing events with π_E of order one and no light from the lens at all are candidate nearby stellar remnants — cold white dwarfs, neutron stars, isolated black holes — and the argument for what they are is made from the two constraints alone, with the absence of light as the third piece of evidence.

Getting θ_E is the harder half

The parallax is measurable on any sufficiently long event. The angular Einstein radius is not, and that asymmetry decides which events yield masses.

θ_E comes from a finite-source effect: the source has an angular radius θ_, known from its colour and brightness by the surface-brightness relation to a few per cent, and if the light curve shows the source’s size — because it crossed a caustic, or passed very close to the lens — then ρ = θ_/θ_E is measurable and θ_E follows. The fold crossing of the second rung is exactly that situation, which is why planetary events with caustic crossings are the ones with real masses attached.

Without a caustic, the source has to pass within about ρ Einstein radii of the lens, which for a bulge giant lensed by an M dwarf means an impact parameter below a few thousandths. That is a rare geometry, and it is the reason the two quantities are usually measured on different events.

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.
Fig. 4 The configuration that supplies θ_E: a binary lens folds the source plane, and crossing the fold resolves the source. The magnification of a point source diverges at the fold; a real source of finite size averages that divergence into a rise of finite steepness, and the steepness is a direct reading of the source’s size in Einstein radii. This figure is the second rung’s, and it belongs here as the other half of the pair: the fold gives θ_E and the Earth’s orbit gives π_E, and neither alone is a mass.

The space-based version

The Earth’s orbit is one baseline. A second observatory somewhere else is a better one, and it works on events of any duration.

Two telescopes separated by an appreciable fraction of an astronomical unit see slightly different lens–source trajectories, so they record slightly different light curves — different peak times and different impact parameters — and the difference is π_E directly, with no dependence on the event lasting long enough for the Earth to move. The Spitzer telescope, in an Earth-trailing orbit that took it to more than an AU of separation, ran exactly this programme on several hundred events.

What it brought with it is a discrete degeneracy. The two observatories measure the magnitude of the trajectory offset, and there are generally four combinations of signs consistent with the two light curves — the source can pass on either side of the lens as seen from either telescope. Some are excluded by the shape of the curves and some are not, and a published space-parallax mass therefore often comes as a set of two or four values rather than one.

A light curve pulled out of shape by the Earth's own orbit. Magnification against time for a 25-day event at impact parameter 0.25, computed with the observer's orbital motion included at two 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 7 per cent at π_E = 0.2. 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.
Fig. 5 Why the ground-based method fails on short events. A 25-day event at π_E = 0.2 is distorted by an amount that would be invisible under any realistic photometry: the Earth barely moves while the magnification is changing, so the wobble has nothing to act on. Every short event — which is every low-mass lens, since the timescale goes as the square root of the mass — is therefore inaccessible to annual parallax, and that is precisely the population a mass measurement would be most useful for.
The two images a point lens makes. A source at 0.15 Einstein radii from the lens is split into two images, at 1.08 and -0.93 Einstein radii on either side of it, one outside the ring and one inside. Neither is ever resolved — the ring is a milliarcsecond across — so all that is observed is their combined brightness, which is 6.72 times the source's own. A planet matters when it lands on top of one of them, which is why the method is sensitive at about one Einstein radius and almost nowhere else.
Fig. 6 Why the whole subject is measured against one angle. A point lens makes two images of the source — a major one outside the Einstein ring and a minor one inside — and the ring is the only length in the problem. Everything a light curve reports is a ratio to it: the impact parameter, the crossing time, the mass ratio of a companion. Turning any of those into a physical quantity requires knowing the ring’s angular size, and the whole of this rung is about the two independent ways of finding it out.

What the parallax says when it disagrees

One more use of the measurement is worth recording, because it is the reverse of everything above.

If θ_E and π_E are both measured and the mass that comes out is impossible — heavier than a main-sequence star that would have been seen, or lighter than a brown dwarf that could have formed — then one of the two measurements is wrong, and the disagreement is diagnostic rather than fatal. Several published events have been reanalysed on exactly that basis: a parallax that implied a lens too dark for its mass turned out to be an unmodelled orbital motion of the lens binary, or a source that was itself blended with a third star.

That is a modest kind of check and it is the only one the method has from within its own data. The external check is better and much slower: wait ten or twenty years for the lens and source to separate on the sky by a few tens of milliarcseconds, resolve them with adaptive optics or an interferometer, and measure the lens directly. That has now been done for a handful of events, the masses agree with the light-curve ones, and it is the single most reassuring result in the subject.

What the picture cannot show

The parallax curves here are computed with a circular Earth orbit projected as a circle, and three refinements separate that from what a fit actually does.

The lens is treated as a point and the source as a point. Both are approximations that hold everywhere except near the peak, and near the peak is where the finite-source effect this rung depends on for θ_E lives — so a real fit computes the magnification by integrating over the source disc at every epoch, which is expensive and is why microlensing modelling is a computational subject rather than an analytic one.

The projection is not circular. The Earth’s orbit seen from a field at ecliptic latitude β projects to an ellipse of axis ratio sin β, and the bulge fields sit near β = −5°, so the projection is very nearly a line rather than a circle. That makes the parallax measurable in one direction far better than the other, and published π_E values come with strongly correlated error ellipses rather than a single number.

The Earth’s orbit is not circular either, and over a hundred-day event its eccentricity matters at the level the fits work to. Real analyses use the ephemeris.

And the lens may be moving too. A binary lens whose components orbit each other during the event produces a distortion with the same character as parallax — the lens–source geometry changes for a reason that is not the observer — and the two are correlated in a fit. That degeneracy between orbital motion of the lens and the observer’s parallax is a live difficulty and is why some published parallaxes are reported with the caveat that the orbital solution is unconstrained.

What a measured mass has bought

Four kinds of object have been weighed this way, and the list is worth having because it is short and each entry is something no other method reaches.

Cold planets beyond the snow line, at a few astronomical units from M dwarfs — the population transits and radial velocities are both blind to, because one needs an alignment that grows rarer with period and the other needs a baseline longer than the orbit.

Free-floating objects with no host at all, whose masses come from θ_E and π_E because there is nothing else about them to measure.

Stellar remnants: white dwarfs, neutron stars and at least one black hole, identified by having a mass and no light. That combination is available from no other technique on an isolated object.

And the lens stars themselves, which is the least glamorous entry and the most useful — a microlensing mass for an M dwarf at four kiloparsecs is a check on the mass–luminosity relation at a metallicity and a distance where the eclipsing binaries that usually calibrate it do not exist.

The relationship to ordinary parallax

The word is the same and the geometry is the same, and the difference is worth stating because it is instructive.

Trigonometric parallax measures a star’s distance by the ellipse its apparent position traces as the Earth goes round the Sun: the semi-major axis of that ellipse in arcseconds is one over the distance in parsecs. It is the most direct distance measurement in astronomy and it works on anything bright enough to have its position measured.

Microlens parallax measures the same displacement of the observer against a different ruler. Instead of the star’s own angular position, the yardstick is the Einstein radius of the lens — so what comes out is not π_rel but π_rel/θ_E, a dimensionless number. It is a parallax measured in units of an angle that is itself unknown, which is why it constrains a combination rather than a distance.

That is the general shape of every measurement in this subject. Nothing about a microlensing event is measured against a known scale; everything is measured against the Einstein radius, and the whole art is finding a second observable measured against something else.

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.
Fig. 7 The event the whole ladder is about, in its simplest form: a magnification and a planetary spike inside it. Everything on this rung is about the two numbers this curve does not contain. Its height is the impact parameter, its width is t_E, the spike’s width is √q times t_E — and none of those is a mass, a distance or a physical size until something outside the curve supplies a scale.

What a parallax measurement costs in observing

It is worth saying what the programme looks like, because the requirement is unusual among the methods in this collection.

An annual parallax needs photometry spanning a substantial fraction of a year on an event that was not known to be interesting when it started. Microlensing surveys therefore monitor tens of millions of bulge stars every clear night for years, alert on anything that brightens, and hand the alert to follow-up networks — and the parallax comes out of the survey’s own baseline rather than out of the follow-up, because the follow-up starts when the event does.

That inverts the usual relationship between a survey and a measurement. In most of this collection the survey finds candidates and the measurement is made afterwards with a bigger telescope; here the survey is the measurement, and no amount of later observation recovers a parallax from an event whose wings were not watched. Every microlensing event is a measurement that cannot be repeated, and the parallax is the part of it most easily lost.

The habit

The structure this rung is an instance of is that a measurement of one dimensionless ratio never yields a dimensioned quantity, and that the way out is always a second ratio measured against a different scale.

A microlensing light curve measures everything in Einstein radii, so it delivers ratios and no lengths. A transit light curve measures everything in stellar radii, so it delivers a planet’s size relative to its star and no planet’s size — which is why the transit method needs an independent stellar radius before it says anything about a planet. A radial-velocity curve measures a velocity, which does have dimensions, and is therefore degenerate in a different way: it gives a mass times an unknown sine.

The pattern in each case is that the escape is not more precision but a second observable calibrated against something else, and that the second observable is usually much harder to get than the first. Microlensing needs a caustic crossing or a year of monitoring; transits need an interferometric or asteroseismic stellar radius; radial velocities need a transit or an astrometric orbit. In none of the three does the primary measurement improve its way out of the degeneracy.

Where this ladder goes next

The parallax needs a long event and the finite-source effect needs a close approach or a caustic. Between them they cover a minority of events, and the population they cover least well is the one a mass measurement would be most useful for.

The next rung is a third observable that needs neither. The two images a lens makes are unresolved, but their centroid is displaced from where the source would be — by an amount that peaks at a separation of √2 Einstein radii, where the magnification is only 1.34 and the photometric event is nearly over. Measuring that displacement gives θ_E directly, with no caustic and no finite source, and it is how the first isolated stellar-mass black hole was weighed.

Beyond it: free-floating planets, whose events last hours rather than weeks and whose abundance is a direct test of planet formation; and the close–wide degeneracy, an exact symmetry of the lens equation under s → 1/s that makes two physically different systems draw the same curve.

Where this goes next

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. 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. 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. Parallax for a star at 1.3 parsecs. The same star observed from two ends of a baseline. The two sight lines are 1.54″ apart, so the parallax — half of that, the shift seen from a one-astronomical-unit baseline — is 0.769″ for a star 1.3 parsecs away. The definition of the parsec is the distance at which it would be exactly one. The triangle that reaches the stars, and stops Parallax is the only distance measurement in astronomy that assumes nothing. It is also the only one with a hard ceiling, and everything beyond that ceiling rests on it.

About the same objects

Not linked from either essay — found by the objects both name.

The objects this essay names

Each one links to every other essay that touches it.

Angular einstein radiusDegeneracyEinstein radiusFinite-source effectLens equationLight curveMicrolens parallaxMicrolensingParallaxProper motion