Astronomy & Celestial Mechanics
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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.

Assumes Microlensing and Angular diameter.

16 min read 7 figures Measuring the unreachableLight is the only messenger

Every rung of this anchor so far has measured a brightness. The first read a planet out of a spike in a light curve; the second read a source’s size out of a fold crossing; the third read a parallax out of an asymmetry.

The lens does something else that a photometer cannot see, and it is larger for longer. The two images it makes are never resolved by an ordinary telescope, but they are not in the same place as the source, and their centre of light is not either.

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.
Fig. 1 The astrometric centroid shift and the photometric magnification against source–lens separation, on a common horizontal axis, for an Einstein radius of one milliarcsecond. The shift is u/(u²+2) times θ_E: zero at u = 0, because the two images are then symmetric about the lens; zero as u grows, because the minor image fades; and largest at u = √2 exactly, where it is θ_E/2√2 = 0.354 mas. The magnification there is 1.34 — a nineteen per cent brightening that a survey would barely flag.

Where the displacement comes from

A point lens makes two images of a point source, on opposite sides of the lens along the line joining it to the source. The major image sits at (u + √(u²+4))/2 Einstein radii from the lens and the minor at (u − √(u²+4))/2, which is negative — on the other side — and inside the ring.

The magnifications are unequal. The major image is always brighter, and the ratio grows as the source moves away, because the minor image shrinks toward the lens and fades. So the flux-weighted centroid of the pair is not at the midpoint; it is pulled toward the major image, and therefore outward from the lens relative to the source.

Working the weighted average out is a short piece of algebra and worth carrying, because the cancellation is not obvious. The two image positions are x± = (u ± √(u²+4))/2 and their magnifications are A± = (x±²/(x±² − 1) + 1)/2, so the flux-weighted position is (A₊x₊ + A₋x₋)/(A₊ + A₋). The radicals cancel completely, leaving u(u²+3)/(u²+2) Einstein radii from the lens; subtracting where the source itself would be — u — leaves

δ=uu2+2 θE\delta = \frac{u}{u^2+2}\,\theta_E

as the shift. Every term of that is geometry; nothing about the lens’s mass enters except through θ_E, which is exactly what makes the measurement useful.

The maximum is where the derivative vanishes, at u = √2, and its value is θ_E/2√2 = 0.354 θ_E. Two facts about that are worth separating. The size is a third of the Einstein radius, which is a large fraction — the effect is not a small correction to a position. And the location is far out, at a separation where the source is barely magnified at all.

The centroid traces an ellipse, exactly. The path the measured image centroid follows on the sky, relative to where the source would be without the lens, for three impact parameters and an Einstein radius of 1 milliarcseconds. Each track is a closed ellipse and that is a theorem rather than an appearance: for a straight-line source track past a point lens, the centroid offset u/(u²+2) traced in the plane is an ellipse with semi-major axis θ_E/2√(u₀²+2) and axis ratio u₀/√(u₀²+2). A close approach gives a large, thin ellipse — 0.2 gives 0.700 by 0.098 mas — and a distant one a small, round one. Two things follow. The size of the ellipse is θ_E directly, with no finite-source effect and no caustic needed, which is what makes astrometry the general route to a mass rather than the lucky one. And its shape is u₀, which the photometry also measures — so the two agree or one of them is wrong, and the redundancy is the check the method has.
Fig. 2 The path the centroid follows on the sky, relative to where the source would be without the lens. Each track is a closed ellipse, and that is a theorem rather than an appearance: for a straight-line source track past a point lens, the centroid offset traces an ellipse with semi-major axis θ_E/2√(u₀²+2) and axis ratio u₀/√(u₀²+2). A close approach gives a large thin ellipse and a distant one a small round one.

The ellipse is the measurement

The closed-form ellipse is what makes astrometric microlensing a measurement rather than a detection, and it delivers two numbers where the photometry delivers one.

The ellipse’s size is θ_E, directly. No finite-source effect is needed, no caustic crossing, no fortunate geometry — every event traces an ellipse, and measuring its major axis measures the angular Einstein radius. That is the quantity the previous rung had to work hard for and could only obtain when the source happened to pass close enough to be resolved.

The ellipse’s shape is u₀, the impact parameter, which the photometry also measures from the peak magnification. Two independent determinations of the same quantity is the redundancy the method has, and a disagreement between them is a signal that something in the model — a blend, an unseen companion, an orbital motion — is wrong.

Combine θ_E from the ellipse with π_E from a parallax and the mass follows as θ_E/(κπ_E), with no distance in it. That is the route this anchor has been working toward for two rungs.

There is a further consequence of the ellipse being closed, and it is the strongest argument for the method. A closed curve traced at a known rate is a complete description of the geometry — there is nothing about the lens–source encounter that the ellipse does not encode, and the encoding is redundant, so the fit is over-determined. That is unusual in this subject. Almost every other microlensing observable is a single number extracted from a curve whose shape is not otherwise informative; the astrometric ellipse is a two-dimensional track with five parameters and rather more than five independent features.

Why it took sixty years

Einstein wrote down the photometric effect in 1936 and dismissed it as unobservable. The astrometric effect was noticed at about the same time and dismissed for longer, and the reason is the size of the number.

For a solar-mass lens halfway to the Galactic centre, θ_E is about a milliarcsecond, so the peak centroid shift is 350 microarcseconds. For an M dwarf it is a third of that. Ground-based astrometry through the atmosphere reaches a few milliarcseconds on a good night — the seeing disc is an arcsecond and a centroid is measured to a fraction of it — and the effect is an order of magnitude below that.

What changed is space astrometry and interferometry. Hubble’s fine guidance sensors and its cameras reach a fraction of a milliarcsecond on a well-exposed field with enough reference stars; the Gaia mission measures positions to tens of microarcseconds; and near-infrared interferometry on a large baseline resolves the images themselves rather than measuring their centroid, which is a different and better measurement again — an interferometer’s resolution is set by its baseline rather than by its apertures, and a hundred metres at two microns is a milliarcsecond.

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 = 4 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 = 1.4142 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.
Fig. 3 The same functions for a lens with a four-milliarcsecond Einstein radius, which is what a ten-solar-mass black hole a few hundred parsecs away produces. The peak shift is 1.4 milliarcseconds — an order of magnitude above the previous figure and comfortably within reach of a space telescope with a good reference frame. The mass enters only through θ_E and θ_E goes as the square root of it, so the astrometric signal favours the heavy lenses that the photometry cannot distinguish from anything else.

The black hole

The measurement’s first decisive result was a mass for an object that emits nothing at all.

A long microlensing event toward the bulge in 2011 — 270 days, with a large and well-measured parallax — was followed astrometrically with Hubble over the following six years. The centroid traced its ellipse, θ_E came out around 5.2 milliarcseconds, and with the parallax the mass followed at about seven solar masses. No light was detected from the lens at any wavelength.

The interesting part of that result is not the number but the kind of statement it is. Every other stellar-mass black hole known has been found in a binary — inferred from the motion of a companion, or from an accretion disc, or from a merger waveform — so every one of them is in a system, and none of them is representative of the population as a whole. An isolated black hole detected by its gravity alone is a sample of a different distribution, and it is the only method that reaches it. Every technique in the census of compact objects requires a companion; this one requires only that something pass in front of a star.

Whether the object in question is a black hole or a neutron star was disputed for several years afterwards, on the basis of different treatments of the astrometric reference frame, and the published masses from two groups differed by a factor of two. That is a fair reflection of the state of the technique: the effect is real, it is measured, and the systematics are the reference stars rather than the physics.

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. 4 Why one measurement is not enough even here. The angular Einstein radius from astrometry gives the rising curve in the mass–distance plane, and the microlens parallax the falling one; they cross at a mass with no distance in it. Astrometry supplies the first cleanly and on every event, which is what it adds to the anchor — but a mass still requires both, and the parallax still requires a long event or a second observatory.

One more thing about that event deserves recording, because it is the sort of detail the summary version leaves out. The reason the astrometric follow-up was possible at all is that the source was bright and the field, for a bulge field, was unusually sparse. Both were luck. A survey designed to weigh isolated black holes cannot choose its lenses, and the fraction of long events with a source suitable for milliarcsecond astrometry is small — so the technique’s yield is governed by the source population rather than by the lens population it is trying to measure.

That is a selection effect of an unfamiliar kind and it is worth naming: the objects being counted are selected by a property of something else entirely. An occurrence rate corrected for detection efficiency assumes the efficiency depends on the object; here it depends on whatever happened to be behind it.

The signal that outlasts the event

The most practically important property of the astrometric effect is one the first figure states and does not emphasise: it decays far more slowly than the photometric one.

The magnification approaches one as u⁻⁴ for large separations. The centroid shift approaches zero as u⁻¹. At ten Einstein radii the magnification excess is a part in ten thousand and is unmeasurable; the centroid shift is a tenth of θ_E and is perfectly measurable.

So an event that photometrically lasted a month is astrometrically detectable for years, and the follow-up does not have to be arranged in advance. That inverts the observing problem the previous rung described: a parallax is lost if the wings were not watched, and an astrometric measurement can be started after the event is over and completed at leisure. It is the one measurement in this subject that does not have to be caught.

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. 5 The photometric event, for scale. It is over in weeks and its wings return to the baseline quickly, because the magnification excess falls as the inverse fourth power of the separation. The astrometric shift at the same epochs is still a third of its peak and falling only as the first power. Two observables from one geometry, with decay rates differing by a factor of three in the exponent, is why they are complementary rather than alternative.
The centroid traces an ellipse, exactly. The path the measured image centroid follows on the sky, relative to where the source would be without the lens, for three impact parameters and an Einstein radius of 1 milliarcseconds. Each track is a closed ellipse and that is a theorem rather than an appearance: for a straight-line source track past a point lens, the centroid offset u/(u²+2) traced in the plane is an ellipse with semi-major axis θ_E/2√(u₀²+2) and axis ratio u₀/√(u₀²+2). A close approach gives a large, thin ellipse — 0.05 gives 0.707 by 0.025 mas — and a distant one a small, round one. Two things follow. The size of the ellipse is θ_E directly, with no finite-source effect and no caustic needed, which is what makes astrometry the general route to a mass rather than the lucky one. And its shape is u₀, which the photometry also measures — so the two agree or one of them is wrong, and the redundancy is the check the method has.
Fig. 6 The same ellipses over a wider range of impact parameter, including one at u₀ = 0.05 where the photometric event is a factor of twenty in brightness. The nearly-degenerate ellipse there is a straight line traversed twice: at very small u₀ the centroid runs out along the track, reverses at the peak and comes back, with almost no transverse extent. The photometric and astrometric signals are therefore anti-correlated in what they measure well — a high-magnification event has a spectacular light curve and a nearly one-dimensional centroid track, and a distant passage has the opposite.

What the picture cannot show

The centroid formula assumes a point source, a point lens, and — the assumption that actually bites — that the only light in the aperture is the source’s.

It is never the only light. A bulge field at the resolution a space telescope reaches is crowded, and the source is blended with unrelated stars along the line of sight and often with the lens itself if the lens is luminous. Blended light dilutes both signals: the magnification is diluted because the unlensed component does not brighten, and the centroid shift is diluted because the unlensed component does not move. A blend fraction of a half halves the measured shift, and the blend fraction is fitted from the same data.

Worse, if the lens is the blend, the geometry changes rather than merely diluting. A luminous lens contributes light at its own position, which is displaced from the source, so the observed centroid is a three-way average and the ellipse is distorted. That is the situation in most events with stellar lenses, and it is why the clean measurements are the dark ones.

There is also the reference frame. A centroid shift of 350 microarcseconds is measured against other stars in the field, and those stars have their own proper motions and parallaxes. Separating the lensing signal from the source’s own proper motion, its own parallax, and the frame’s systematic drift is the whole difficulty of the analysis — and it is where two groups analysing the same black-hole event arrived at different masses.

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. 7 And the geometry the whole rung is a statement about: two images, one outside the Einstein ring and one inside, on opposite sides of the lens. Their separation is of order θ_E — a milliarcsecond, resolvable by an interferometer with a hundred-metre baseline and by nothing else — and their centroid is what a telescope that cannot resolve them measures instead. The astrometric effect is what is left of the image structure after the resolution has been thrown away.

The other thing the shift measures

A centroid track is a track in two dimensions, and the direction of its major axis is the direction of the lens–source relative proper motion on the sky.

That is one more piece of information the photometry cannot supply, and it does real work. A lens in the Galactic bulge and one in the foreground disc have different kinematics — the disc population moves in an ordered rotation and the bulge population does not — so the direction of the relative motion is evidence about which population the lens belongs to, independently of the mass. Several published events have had their disc-versus-bulge assignment settled that way.

It also feeds back into the parallax. The microlens parallax is a vector, not a scalar: its magnitude is π_rel/θ_E and its direction is the direction of the relative proper motion. Measuring the direction astrometrically and the magnitude photometrically is two halves of one quantity from two instruments, and their consistency is another check the method has and rarely uses.

The one number that has to be known independently

The centroid shift is θ_E times a function of u, and u is dimensionless — so everything the astrometry measures is in units of the Einstein radius, exactly as everything the photometry measures is.

That sounds like the same degeneracy again and it is not, and the difference is the whole reason the method is useful. What the photometry measures in units of θ_E is a time: it reports t_E, which is θ_E divided by a proper motion, so extracting θ_E requires knowing the proper motion, which nothing measures. What the astrometry measures in units of θ_E is an angle, and angles on the sky are measured against reference stars — a scale that exists independently and is calibrated by the survey’s own astrometric solution.

So the astrometric route converts an unknown angular scale into a measured one, and the photometric route cannot, because there is no independent clock for a proper motion nobody has seen. That is what it means for one observable to be dimensioned and the other not, and it is the difference between a measurement and a constraint.

Resolving the images

The final step in this line is not astrometry at all. It is interferometry, and it has been done.

Two images separated by a milliarcsecond are resolvable by an optical interferometer with a baseline of a hundred metres or more, and the GRAVITY instrument on the Very Large Telescope Interferometer has resolved the image pair of a microlensing event directly. What that measures is not a centroid but the separation itself, which is θ_E without any modelling of the centroid at all, and to a precision the astrometric route cannot reach.

The technique needs a bright source and a favourable event, so it applies to a handful of events a year rather than to a survey. What it supplies is a calibration: a directly measured θ_E on an event that also has a centroid measurement and a photometric model, which is the check the whole method rested on and did not have.

What the effect is worth as a survey

The natural question is whether a survey could be built around the astrometric signal rather than the photometric one, and the arithmetic says something specific about why not.

The photometric cross-section for a detectable event is the area on the sky within about one Einstein radius of a source — the impact parameter has to be of order one for the brightening to exceed a few per cent. The astrometric cross-section is larger, because the shift is detectable out to several Einstein radii, and it grows as the astrometric precision improves: at a precision of θ_E/100 the signal is detectable out to u of order fifty.

So astrometric events are commoner than photometric ones by a factor of a few tens, and the reason nobody surveys for them is duration. An astrometric event at u ~ 30 lasts thirty Einstein crossing times — years to decades — and detecting it means separating a slow curved drift from a proper motion and a parallax over that baseline. Gaia’s astrometry is in principle sensitive to exactly this, and the predicted yield is thousands of events over the mission; extracting them is a problem about modelling the reference frame rather than about the physics.

That inverts the usual relationship between the two observables. Photometry is easy and rare; astrometry is hard and common. Which one a survey uses is a statement about which difficulty is cheaper to solve, and the answer has been changing.

The habit

The pattern this rung is an instance of is that a signal thrown away by one instrument is often the cleanest signal available to another, and that which one is available is a fact about technology rather than about the object.

A photometer integrates over the point-spread function and reports one number, so it sees the sum of the two images and nothing else. An astrometer reports the first moment of the same light distribution, which is a different projection of the same information. An interferometer reports the second moment and beyond, which resolves them. The lens does the same thing in all three cases; what changes is how much of what it does survives the measurement.

The same three-way relationship appears elsewhere in this collection. A spectral line’s equivalent width, its width and its shape are successive moments of the same profile, and each is measurable at a different resolution and says a different thing. A transit’s depth, its duration and its ingress shape are the same progression. In each case the higher moment is harder to measure, arrives later historically, and answers a question the lower one could not pose.

Where this ladder goes next

Four rungs have taken this anchor from a spike in a light curve to a mass for an object that emits nothing, and each has been about finding one more dimensioned quantity in the same geometry.

The next rung goes the other way, to the events that supply the fewest. A free-floating planet has no host star, so there is no orbital period, no reflex velocity, no transit and no light — and the entire measurement is an event lasting a few hours whose only observable is its duration. The timescale goes as the square root of the mass, which is the whole of what can be said, and the abundance of such events is nonetheless a direct test of how planetary systems are built.

Beyond it: the close–wide degeneracy, an exact symmetry of the binary lens equation that makes two physically different systems draw the same curve; and what a space survey of the bulge changes, which is the answer to nearly every limitation on this ladder.

Where this goes next

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. 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. 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. 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 radiusAstrometric microlensingAstrometryBlack holeCentroidDegeneracyImage multiplicityMicrolens parallaxMicrolensingProper motion