Astronomy & Celestial Mechanics
the sky, computed
The observed sky

The whole sky drifting towards one point

Annual aberration is the Earth's velocity, and it closes every year. The Sun's velocity is not constant, so the same effect leaves a residue that never closes — every quasar in the sky creeping towards the Galactic centre at five microarcseconds a year, which is a direct measurement of the Sun's acceleration.

Assumes Aberration, Parallax and Celestial sphere.

17 min read 8 figures The sky as a clockMeasuring the unreachable

Aberration is a statement about the observer’s velocity, not about anything in the sky. A telescope moving at speed vv has to be tilted by v/cv/c into the direction of motion, so every star is displaced towards the point the observer is heading for, by the same angle regardless of distance. The Earth’s orbital motion does this at 20.5 arcseconds, and because the velocity comes back to itself once a year the displacement traces a closed ellipse and averages away.

The Sun’s velocity does not come back to itself. It is turning — once around the Galaxy in a quarter of a billion years — and a turning velocity means an acceleration, and an acceleration means an aberration that is different next year from this year. What that produces is not an ellipse. It is a permanent creep of the entire sky towards one point, and the point is the centre of the Galaxy.

Every quasar in the sky streaming at 5.23 µas a year towards one point. Above: the apparent proper motion of distant quasars, drawn in Galactic coordinates with the centre of the Galaxy at the origin. Quasars do not move — at their distances a real transverse velocity of a thousand kilometres a second would be a hundredth of a microarcsecond a year — so a pattern in their apparent motions is a statement about the observer. Annual aberration displaces every source by v/c and returns it a year later; the Sun's velocity is not constant, and a changing displacement does not return. The Sun is being accelerated towards the centre of the Galaxy at 2.4·10⁻¹⁰ m s⁻², so the aberration vector rotates at a/c and the whole sky streams towards the same point, at (a/c) sin θ for a source θ from it. Below: that amplitude against angle from the apex, with the fitted dipole and the measurement. A circular speed of 248 km s⁻¹ at 8.28 kiloparsecs predicts 5.23 microarcseconds a year; the measured dipole in the proper motions of 1.6 million quasars is 5.05 ± 0.35, pointing to within a few degrees of the Galactic centre. The picture cannot show what took so long: the effect is a twenty-thousandth of annual aberration, it accumulates over the whole mission rather than over a year, and it is degenerate with any real rotation of the quasar frame — so a measurement of the acceleration of the solar system is also, unavoidably, an assumption that the distant universe does not turn.
Fig. 1 The effect drawn twice. Above, the apparent proper motions of distant quasars in Galactic coordinates, with the Galactic centre at the origin: every arrow points towards it, and the length of each goes as the sine of the angle from it, which is the signature of a dipole. Below, that amplitude against angle, with the measurement. A circular speed of 248 kilometres a second at 8.28 kiloparsecs predicts an acceleration of 2.4×10−102.4\times10^{-10} m s⁻², and dividing by the speed of light gives 5.2 microarcseconds a year. The measured dipole in the proper motions of 1.6 million quasars is 5.05±0.355.05 \pm 0.35, pointing to within a few degrees of the Galactic centre.

Why an acceleration does not average away

The distinction between the two effects is worth stating precisely, because they are the same formula differentiated once.

A velocity v\mathbf{v} displaces a source’s apparent direction by v⊥/c\mathbf{v}_\perp/c — the component of the velocity perpendicular to the line of sight, divided by cc. If v\mathbf{v} is periodic the displacement is periodic, and a periodic displacement is an offset that returns: it changes where a star appears to be at each moment and changes nothing about its long-run position.

The same ellipse at 955 times the size, and a quarter of a year out of step. The aberration ellipse (solid) and the parallax ellipse (dashed) for γ Draconis at four ecliptic latitudes, drawn to one scale. Aberration is v/c towards the Earth's own direction of travel, so its ellipse has semi-major axis 20.49551″ for every star in the sky; parallax is 1/d towards the Sun, so its semi-major axis is that star's own 0.02147″ — 955 times smaller, and drawn 955 times smaller here. At the ecliptic pole both are circles; on the ecliptic both collapse to lines; in between both are ellipses of semi-minor axis sin β times the major, and the ratio is identical at every latitude because both effects project the same way. The marks are the same four dates on each: they are a quarter of a year apart between the two, because the aberration displacement follows the Earth's velocity and the parallax displacement follows its position, and velocity leads position by 90° on a circular orbit. That quarter-year is the only thing distinguishing the two phenomena on the sky, and it is why Bradley, looking for the dashed ellipse in 1728, spent months unable to interpret the solid one he had found instead.
Fig. 2 What the periodic version looks like. Every star traces an ellipse of semi-major axis 20.5 arcseconds along the ecliptic and semi-minor axis 20.5sin⁡β20.5\sin\beta, the same size for all of them because nothing about the star enters. It is a large effect — a thousand times the parallax of the nearest star — and it is entirely harmless to a catalogue, because it is computed and removed.

An acceleration a\mathbf{a} makes v\mathbf{v} change, so it makes the displacement change, at a rate a⊥/c\mathbf{a}_\perp/c. A rate of change of apparent position is a proper motion, and this one has no reason to reverse: the Sun keeps turning in the same sense for the next hundred million years, so the drift keeps going in the same direction.

The same ellipse at 955 times the size, and a quarter of a year out of step. The aberration ellipse (solid) and the parallax ellipse (dashed) for γ Draconis at four ecliptic latitudes, drawn to one scale. Aberration is v/c towards the Earth's own direction of travel, so its ellipse has semi-major axis 20.49551″ for every star in the sky; parallax is 1/d towards the Sun, so its semi-major axis is that star's own 0.02147″ — 955 times smaller, and drawn 955 times smaller here. At the ecliptic pole both are circles; on the ecliptic both collapse to lines; in between both are ellipses of semi-minor axis sin β times the major, and the ratio is identical at every latitude because both effects project the same way. The marks are the same four dates on each: they are a quarter of a year apart between the two, because the aberration displacement follows the Earth's velocity and the parallax displacement follows its position, and velocity leads position by 90° on a circular orbit. That quarter-year is the only thing distinguishing the two phenomena on the sky, and it is why Bradley, looking for the dashed ellipse in 1728, spent months unable to interpret the solid one he had found instead.
Fig. 3 The annual figure at four ecliptic latitudes including one very near the plane. At β=0\beta = 0 the ellipse degenerates to a line — the star oscillates back and forth along the ecliptic by 20.5 arcseconds and does not circulate at all — and at the pole it is a circle. Both are closed, which is the property that matters: whatever the shape, the star returns to where it started after a year, and a mean position over a year carries none of it. Only the part of the velocity that has not returned survives, and that is the acceleration.

The angular pattern follows from the geometry alone. For a source at angle θ\theta from the direction of the acceleration, the perpendicular component of a\mathbf{a} is asin⁡θa\sin\theta, directed along the great circle towards the apex. So every source streams towards the Galactic centre, fastest at ninety degrees from it and not at all at either pole of that axis. That is a dipole in the proper-motion field, and it is the lowest-order pattern a vector field on a sphere can have.

Every quasar in the sky streaming at 4.11 µas a year towards one point. Above: the apparent proper motion of distant quasars, drawn in Galactic coordinates with the centre of the Galaxy at the origin. Quasars do not move — at their distances a real transverse velocity of a thousand kilometres a second would be a hundredth of a microarcsecond a year — so a pattern in their apparent motions is a statement about the observer. Annual aberration displaces every source by v/c and returns it a year later; the Sun's velocity is not constant, and a changing displacement does not return. The Sun is being accelerated towards the centre of the Galaxy at 1.9·10⁻¹⁰ m s⁻², so the aberration vector rotates at a/c and the whole sky streams towards the same point, at (a/c) sin θ for a source θ from it. Below: that amplitude against angle from the apex, with the fitted dipole and the measurement. A circular speed of 220 km s⁻¹ at 8.28 kiloparsecs predicts 4.11 microarcseconds a year; the measured dipole in the proper motions of 1.6 million quasars is 5.05 ± 0.35, pointing to within a few degrees of the Galactic centre. The picture cannot show what took so long: the effect is a twenty-thousandth of annual aberration, it accumulates over the whole mission rather than over a year, and it is degenerate with any real rotation of the quasar frame — so a measurement of the acceleration of the solar system is also, unavoidably, an assumption that the distant universe does not turn.
Fig. 4 The same prediction at a circular speed of 220 kilometres a second, which was the standard value for most of the twentieth century. The amplitude falls to 4.1 microarcseconds a year — the acceleration goes as the square of the speed — and the pattern is unchanged, because the pattern depends only on the direction. So the measurement constrains the speed and not the geometry, and the six-degree offset between the measured direction and the Galactic centre is a separate finding that no choice of circular speed can produce.
Two ellipses, one year, 3 months apart. The displacement in ecliptic longitude over 12 months for γ Draconis at ecliptic latitude 75°: aberration at its true amplitude of 20.49551″, and parallax at 0.02147″ drawn 955× magnified so that it can be seen on the same axis at all. Both are cosines of the same period. Their maxima are 3.00 months apart, measured off the drawn curves, because one follows where the Earth is going and the other follows where the Earth is. A programme that measures a star's position through a year and does not know this recovers a parallax that is 955 times too large and three months early — which is exactly what happened, and why the aberration constant was found before any stellar distance was.
Fig. 5 The other way of seeing that a velocity effect closes. Aberration and parallax draw ellipses of the same shape for the same star and are ninety degrees out of phase, because one depends on where the Earth is and the other on which way it is going. Both repeat annually and both therefore contribute nothing to a position averaged over a year. Only the derivative of the velocity survives that average, and the derivative is what this essay is about.

The size of it

The arithmetic is short and worth doing, because the result is the smallest quantity anything in this collection measures.

The Sun’s circular speed is about 248 kilometres a second and its distance from the Galactic centre about 8.28 kiloparsecs, both now known to better than a per cent from the orbit of a single star around the central black hole and from the proper motion of the radio source at the centre. A circular orbit at those numbers has centripetal acceleration v2/Rv^2/R, which is

a  =  (2.48×105 m s−1)22.56×1020 m  =  2.4×10−10 m s−2.a \;=\; \frac{(2.48\times10^{5}\ \mathrm{m\,s^{-1}})^{2}}{2.56\times10^{20}\ \mathrm{m}} \;=\; 2.4\times10^{-10}\ \mathrm{m\,s^{-2}}.

That is one part in 4×10104\times10^{10} of the acceleration due to gravity at the Earth’s surface. Divided by the speed of light it is 8×10−198\times10^{-19} radians per second, which over a year is 2.5×10−112.5\times10^{-11} radians, which is 5.2 microarcseconds.

For scale: a microarcsecond is the angle a one-euro coin subtends on the Moon, and the effect accumulates at five of them per year. Over the whole of a mission it is of order fifty — still a hundred times smaller than the width of a stellar image on any detector ever flown.

Every quasar in the sky streaming at 5.77 µas a year towards one point. Above: the apparent proper motion of distant quasars, drawn in Galactic coordinates with the centre of the Galaxy at the origin. Quasars do not move — at their distances a real transverse velocity of a thousand kilometres a second would be a hundredth of a microarcsecond a year — so a pattern in their apparent motions is a statement about the observer. Annual aberration displaces every source by v/c and returns it a year later; the Sun's velocity is not constant, and a changing displacement does not return. The Sun is being accelerated towards the centre of the Galaxy at 2.7·10⁻¹⁰ m s⁻², so the aberration vector rotates at a/c and the whole sky streams towards the same point, at (a/c) sin θ for a source θ from it. Below: that amplitude against angle from the apex, with the fitted dipole and the measurement. A circular speed of 248 km s⁻¹ at 7.5 kiloparsecs predicts 5.77 microarcseconds a year; the measured dipole in the proper motions of 1.6 million quasars is 5.05 ± 0.35, pointing to within a few degrees of the Galactic centre. The picture cannot show what took so long: the effect is a twenty-thousandth of annual aberration, it accumulates over the whole mission rather than over a year, and it is degenerate with any real rotation of the quasar frame — so a measurement of the acceleration of the solar system is also, unavoidably, an assumption that the distant universe does not turn.
Fig. 6 And at a Galactic-centre distance of 7.5 kiloparsecs rather than 8.28, which is roughly the spread of pre-2010 determinations. The predicted amplitude rises to 5.8 microarcseconds a year, because the acceleration goes as one over the radius — so the two geometric inputs pull the prediction in opposite directions and the measurement constrains their ratio rather than either. That the measured 5.05 sits between the values these two figures bracket is the agreement the essay reports, and it is an agreement about v2/Rv^2/R rather than about vv or RR.

What was actually measured

The measurement is a fit of a dipole vector field to the proper motions of sources that are assumed to have none.

The sources are quasars, and the assumption is the important part. A quasar at redshift one is at a distance of some gigaparsecs, so a transverse velocity of a thousand kilometres a second — larger than anything the dispersion of a cluster permits — corresponds to a proper motion of about a hundredth of a microarcsecond a year. Individually they are the most nearly fixed objects available.

They are not perfectly fixed. An active nucleus is a compact radio and optical source whose photocentre moves as jets brighten and fade, and that jitter is of order tens of microarcseconds — far larger than the signal being looked for in any one object. What saves the measurement is that the jitter is uncorrelated between quasars and the signal is a coherent pattern across the whole sky: averaging 1.6 million of them beats the jitter down by more than a thousand.

Two ellipses, one year, 3 months apart. The displacement in ecliptic longitude over 24 months for γ Draconis at ecliptic latitude 75°: aberration at its true amplitude of 20.49551″, and parallax at 0.02147″ drawn 955× magnified so that it can be seen on the same axis at all. Both are cosines of the same period. Their maxima are 3.00 months apart, measured off the drawn curves, because one follows where the Earth is going and the other follows where the Earth is. A programme that measures a star's position through a year and does not know this recovers a parallax that is 955 times too large and three months early — which is exactly what happened, and why the aberration constant was found before any stellar distance was.
Fig. 7 Two years of the same star rather than one, which shows what “closed” means operationally. The aberration ellipse is retraced exactly and the parallax ellipse with it, ninety degrees out of phase; nothing accumulates between the first circuit and the second. That exact repetition is what a fit exploits — the annual terms are removed by solving for their amplitudes over many years, and what is left in the residual is anything with a different period or none. The drift this essay measures is in that residual, at a five-millionth of the ellipse’s own size.

The published value from the third Gaia data release is a dipole of amplitude 5.05±0.355.05 \pm 0.35 microarcseconds a year directed towards Galactic coordinates (l,b)=(5.0∘,−4.5∘)(l,b) = (5.0^\circ, -4.5^\circ), some six degrees from the Galactic centre. The direction was not fitted and then compared — it was fitted freely, in three components, and came out pointing where the Galaxy is.

That is the whole content of the result: the acceleration of the solar system has been measured directly, rather than inferred from a rotation curve and a mass model, and the two agree.

What had to be assumed

Three assumptions carry the measurement, and none of them can be tested by it.

That the quasars are not moving as a group. A dipole in the proper motions of distant sources is what an acceleration of the observer looks like — and it is also what a global streaming of the sources would look like. There is no way, within this data set, to distinguish the two, so a result of this kind is always a statement about the acceleration relative to the distant universe.

That the reference frame does not rotate. The other low-order pattern a proper-motion field can carry is a rigid rotation, three more numbers, and a rotation is exactly what the frame is defined to have none of. The fit therefore solves for the dipole and the rotation together and reports the dipole, and the rotation is constrained to a few microarcseconds a year by construction rather than measured. And that the instrument’s own systematics are not a dipole. This is the one that took the longest. A scanning astrometric satellite covers the sky in a pattern set by its spin axis and its precession, and any error that depends on where a source falls on the focal plane, or on how bright it is, or on what colour it is, maps onto the sky in the pattern of the scanning law. Several of those patterns have dipole components. The published error bar is dominated by them rather than by the number of quasars.

Why it was not measured earlier

The effect was predicted in the 1980s and named before anyone could look for it, and the delay between prediction and detection is instructive about what kind of measurement this is.

Nothing about the idea is hard. What is hard is that a proper motion is a position measured twice, so the precision needed is the precision of a position divided by the baseline in time — and both halves were out of reach until very recently. Ground-based astrometry reaches a few milliarcseconds on a good night and is limited by the atmosphere delivering a wavefront in patches rather than by anything about the telescope. Very-long-baseline radio interferometry does better — tens of microarcseconds on a compact source — and its three-decade archive of quasar positions gave the first hints, at around two sigma, from a few hundred objects.

The step that settled it was not a better instrument for one source but a survey that measured a million of them with the same instrument, the same calibration and the same scanning pattern, so that a coherent whole-sky pattern could be separated from an incoherent per-object one. That is the same structural argument as the pulsar timing arrays’: the quantity is not extracted from the best measurement but from the correlation across many mediocre ones, and the thing that had to be built was uniformity rather than precision.

The sources are not points

The measurement treats each quasar as a fixed marker, and at the resolution it is made with, a quasar is not a marker and is not fixed.

What a radio interferometer sees at milliarcsecond resolution is a compact core plus a one-sided jet, sometimes with several distinguishable components along it. The components move — outward from the core, at apparent speeds that frequently exceed the speed of light because the jet points nearly along the line of sight — and they brighten and fade over months to years.

The consequence is that the position assigned to the source moves too, because what is fitted is a brightness distribution and its centroid shifts as the components change. This source structure effect is at the level of tens to hundreds of microarcseconds for a typical source, which is one to two orders of magnitude larger than the signal this essay is about.

Three things make the measurement possible anyway.

The defining sources are selected for compactness and for a history of positional stability, out of thousands of candidates monitored for years. The structure effect is random in direction from source to source, while the aberration drift is a coherent dipole across the whole sky, so averaging over hundreds of sources suppresses one and not the other. And the analysis fits the dipole simultaneously with each source’s own linear motion, which absorbs a steady drift of a jet into that source’s own parameters rather than into the global pattern.

So the signal is recovered not by measuring any source well but by measuring the correlation between many, and the error budget is dominated by whether the structure effects really are uncorrelated across the sky — which is checked by splitting the sample and comparing, and which is the reason the quoted uncertainty is what it is.

Two ellipses, one year, 3 months apart. The displacement in ecliptic longitude over 12 months for a nearer star at ecliptic latitude 75°: aberration at its true amplitude of 20.49551″, and parallax at 0.3″ drawn 68× magnified so that it can be seen on the same axis at all. Both are cosines of the same period. Their maxima are 3.00 months apart, measured off the drawn curves, because one follows where the Earth is going and the other follows where the Earth is. A programme that measures a star's position through a year and does not know this recovers a parallax that is 68 times too large and three months early — which is exactly what happened, and why the aberration constant was found before any stellar distance was.
Fig. 8 The same pair for a star fourteen times nearer, where the parallax ellipse has grown to a size comparable with the drawing and the aberration ellipse has not moved at all. That contrast is the whole reason aberration was found before parallax: it is the same shape and the same period for every star, so it cannot be mistaken for a distance, and it is a thousand times larger than the parallax of anything. Bradley was looking for the second and found the first, and separating them took the realisation that they are ninety degrees apart in phase.

What the number is a measurement of

The drift is an acceleration, and an acceleration in a gravitational field is a statement about the mass distribution producing it.

The measured amplitude corresponds to a solar system acceleration of about 2.3×10−102.3\times10^{-10} metres per second squared, directed towards the Galactic centre. That is the centripetal acceleration of the Sun’s orbit around the Galaxy, and it equals v2/R0v^2/R_0 with vv the circular speed and R0R_0 the distance to the centre.

Both of those are measured independently — the distance from the orbits of stars around the central black hole, the speed from the proper motion of the central radio source and from stellar kinematics — so the acceleration is over-determined, and the three quantities can be checked against one another.

They agree. That is worth stating plainly, because the three measurements share nothing: one is the geometry of a stellar orbit at the Galactic centre, one is a proper motion, and one is a dipole in the apparent motions of objects billions of light years away.

There is a smaller term underneath, and it is the reason the measurement will keep improving. The solar system is also accelerating towards the Local Group’s centre of mass and towards the Virgo cluster, at some hundredths of the Galactic value, and separating those from the Galactic term requires an order of magnitude more precision than exists. When it exists, an interferometer will be weighing the local universe by watching the sky drift, which is about as indirect as a mass measurement gets.

The measurement is already used the other way round in practice. Because the Galactic term is known independently to better than the drift can be measured, it is applied as a correction to the reference frame rather than extracted from it — every catalogue position now carries an implied acceleration, and leaving it out would slowly deform the frame at the level future instruments will care about.

The correction is small enough that it changes nothing anybody does with a catalogue today and large enough that a frame maintained for decades without it would drift measurably against one maintained with it. That is the usual moment at which an effect graduates from a curiosity to part of the standard model of a measurement.

The same graduation happened to aberration itself, to nutation and to the relativistic light deflection by the Sun, each of which was a curiosity, then a correction applied by specialists, and finally a term in the standard reduction that nobody thinks about. This one is at the second stage.

It is worth adding what the effect is not, since the name invites the error. Nothing about the quasars is moving: their apparent displacement is entirely a property of the observer’s changing velocity, exactly as with the annual aberration, and an observer at rest with respect to the Galaxy would see none of it. The pattern is a measurement of the solar system’s motion, drawn on objects chosen precisely because they contribute nothing to it.

The distinction is not pedantic, because it decides what a future improvement measures. A better determination does not tell anybody anything new about quasars; it tells them about the gravitational field the solar system is falling through.

Where the model stops

The Sun’s acceleration is not purely the smooth galactic one. There is a contribution from the nearest few hundred parsecs of the disc, one from the spiral pattern if the Sun happens to be near an arm, and one from the Large Magellanic Cloud, which is massive enough and close enough to pull the inner Galaxy by an amount comparable with the measurement’s error bar.

That last one is the most interesting, because it is a prediction: if the Magellanic Cloud is as massive as its own dynamics suggest, the total acceleration should be displaced from the Galactic-centre direction by a few degrees, and the measured direction is displaced from the Galactic centre by about six. Whether that is the Cloud or a systematic is not yet settled.

There is also a term nobody can remove. The solar system is falling towards the Local Group’s own centre of mass and the Local Group towards the Virgo cluster, at accelerations smaller than the galactic one by two or three orders of magnitude. Those are below the current sensitivity and will not stay there.

The generalisation

The pattern is a familiar one turned inside out. A quantity too small to measure on any single object becomes measurable because it is coherent across a population while everything competing with it is not.

A one-per-cent shape distortion in a galaxy image is invisible against the galaxy’s own ellipticity and becomes a mass map when a million of them are averaged. A few metres a second of stellar wobble is invisible against a star’s own surface motion and becomes a planet when the signal is periodic and the noise is not. A gravitational-wave background is invisible in any pulsar’s residuals and becomes a detection when the correlation between pairs has a shape nothing else has.

In every case the measurement is not of an object but of a pattern, and the discriminating property is the pattern’s form rather than its size. Here the form is the simplest one available — a dipole — and the whole difficulty is that instruments have dipoles too.

Where this ladder goes next

Later rungs on this anchor: the relativistic aberration formula and the second-order terms an astrometric mission has to carry; the light-time correction, a different effect of the same order that is routinely conflated with this one; the deflection of starlight by the Sun and by the planets, which reaches milliarcseconds for a source at right angles to the Sun and which every space astrometry mission measures rather than assumes; the aberration of the microwave background dipole, which is the same physics applied to a source at redshift a thousand; and the frame itself — how a set of axes with no rotation is built out of objects that are individually unreliable.

Where this goes next

The same ellipse at 955 times the size, and a quarter of a year out of step. The aberration ellipse (solid) and the parallax ellipse (dashed) for γ Draconis at four ecliptic latitudes, drawn to one scale. Aberration is v/c towards the Earth's own direction of travel, so its ellipse has semi-major axis 20.49551″ for every star in the sky; parallax is 1/d towards the Sun, so its semi-major axis is that star's own 0.02147″ — 955 times smaller, and drawn 955 times smaller here. At the ecliptic pole both are circles; on the ecliptic both collapse to lines; in between both are ellipses of semi-minor axis sin β times the major, and the ratio is identical at every latitude because both effects project the same way. The marks are the same four dates on each: they are a quarter of a year apart between the two, because the aberration displacement follows the Earth's velocity and the parallax displacement follows its position, and velocity leads position by 90° on a circular orbit. That quarter-year is the only thing distinguishing the two phenomena on the sky, and it is why Bradley, looking for the dashed ellipse in 1728, spent months unable to interpret the solid one he had found instead. The other twenty arcseconds Every star in the sky traces a small ellipse over a year, of the same shape as its parallax ellipse and 90° out of step with it — and the same size for all of them, near or far. Bradley found it in 1728 while hunting for parallax, and it proved the Earth moves a century before anything's distance was known. One path, five numbers. Left: the apparent path of a star over 4 years, with a proper motion of 193 mas a year and a parallax of 50 mas, at ecliptic latitude 42°. It is one curve and there is nothing in the sky it can be compared against — the reference stars have paths of their own. Right: the same path with a straight line taken out of it. What is left is an ellipse of semi-major axis 50.0 mas and semi-minor axis 33.5 mas, closed and repeating once a year. The two are separated by their time signatures and by nothing else: proper motion is secular and parallax is annual, in a phase the Earth's position fixes in advance. That is why the five parameters can be told apart at all, and why an astrometric catalogue quotes five rather than two — a position without them is a position at one instant, which is not a direction to anything. Five numbers from one wiggle A star's path across a plate is a straight line with a one-year ellipse laid on it. Nothing measures either alone — one fit yields five parameters at once, and they are separable only because their time signatures differ. the Milky Way measured from inside it — the tangent-point construction. Left: the disc in plan, with the Sun at 8.2 kpc from the centre and four lines of sight at galactic longitudes 20°, 40°, 60°, 80°. Each one grazes a circle of radius R₀ sin l, and at that tangent point the whole circular velocity lies along the line of sight, so the largest velocity in the spectrum belongs to a radius the geometry fixes and no distance has to be measured. Right: the four radii and speeds that yields, on the model curve they are read against. The construction reaches only radii inside the Sun's, which is why the outer curve — the half of it that carries the argument — needs distances after all. A galaxy measured from inside it The Milky Way is the one galaxy nobody can photograph, and the only one whose rotation can be measured without knowing a single distance. A line of sight at a chosen longitude grazes a circle of known radius, and the fastest gas along it is orbiting there.

About the same objects

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

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

AberrationAstrometryDipoleGalactic accelerationGalactic rotationInertial frameProper motionQuasarReference frameSecular aberration driftSolar apexSystematic error