Astronomy & Celestial Mechanics
the sky, computed
Galaxies

One velocity and two distances

Inside the Sun's orbit a line of sight crosses each galactocentric radius twice, and the two crossings have identical radial velocities. So a cloud's velocity gives two candidate distances, near and far, differing by a factor — and nothing about the velocity says which, though the choice decides whether the object is ordinary or extraordinary.

Assumes Galactic structure and Rotation curves.

15 min read 6 figures Measuring the unreachableThe sky as a clock

The Milky Way is the one galaxy nobody can photograph, and the one whose structure has to be reconstructed from inside. There are no parallaxes for most of it and no standard candles in the obscured plane, so the standard distance indicator is kinematic: measure a cloud’s radial velocity, assume it is on a circular orbit, and read off where it must be.

The method works and it has an ambiguity built into its geometry. Inside the Sun’s own orbit, every line of sight crosses each galactocentric radius twice.

One velocity, two distances — 4.7 and 9.5 kiloparsecs. The radial velocity of gas along a line of sight at galactic longitude 30 degrees, against distance from the Sun, for a flat rotation curve. The velocity rises to a maximum at the tangent point — where the line of sight is tangent to a circle of radius R₀ sin l — and falls again beyond it, so every velocity below the maximum corresponds to two distances. A cloud observed at 84 kilometres a second is at either 4.7 or 9.5 kiloparsecs, and nothing about its velocity says which. The two possibilities differ by a factor in distance and by its square in luminosity and mass, so the ambiguity is not a refinement — it decides whether a star-forming region is an ordinary one nearby or a monster on the far side of the Galaxy.
Fig. 1 The radial velocity along a line of sight at thirty degrees of galactic longitude, against distance from the Sun. It rises to a maximum where the line of sight is tangent to a circle of radius R₀ sin l, and falls again beyond. Every velocity below that maximum corresponds to two distances, and nothing about the velocity distinguishes them.

The Milky Way is measured from inside it, and the tangent-point method that makes its inner rotation curve measurable at all is the same geometry that makes every other distance along the same line of sight ambiguous.

Why the curve is symmetric

The radial velocity of a cloud on a circular orbit is the difference between the projected orbital velocity at its radius and the projected orbital velocity of the Sun. Written out for a flat rotation curve,

vr=(V0R0R−V0)sin⁡l,v_r = \left(\frac{V_0 R_0}{R} - V_0\right)\sin l,

which depends on the distance from the Sun only through RR.

A line of sight at longitude ll inside the solar circle reaches a minimum galactocentric radius R0sin⁡lR_0\sin l at the tangent point, and every other radius along it is reached twice: once on the way in and once on the way out. Since the velocity depends only on RR, the two crossings give the same velocity exactly.

That is a purely geometric statement. It is not an approximation, it does not depend on the rotation curve’s shape, and no improvement in the velocity measurement touches it.

It is worth noticing what the symmetry does not survive. If the rotation curve were not axisymmetric — if the orbital speed depended on azimuth as well as radius — the two crossings would be at different azimuths and would have different velocities, and there would be no ambiguity. So the degeneracy is a consequence of the assumption of axisymmetry rather than of the geometry alone, and the same departures from axisymmetry that break it are the streaming motions that make the method inaccurate. The ambiguity and the systematic have the same cause and they cannot both be removed.

What the two choices imply

The two distances are not close together. At a moderate longitude they differ by a factor of two or more, which means the physical quantities inferred differ by factors that matter.

A cloud’s size is its angular size times its distance, so a factor of two in distance is a factor of two in size. Its mass, from a column density and a size, goes as the square. Its luminosity, from a flux, goes as the square. And its position in the Galaxy — which spiral arm it belongs to, whether it is in the inner disc or the outer — is completely different.

So the ambiguity is not a modest uncertainty on a catalogue entry. It is the difference between an ordinary star-forming region nearby and one of the most massive in the Galaxy on the far side, and both interpretations are consistent with the same spectrum.

One velocity, two distances — 2.1 and 6.1 kiloparsecs. The radial velocity of gas along a line of sight at galactic longitude 60 degrees, against distance from the Sun, for a flat rotation curve. The velocity rises to a maximum at the tangent point — where the line of sight is tangent to a circle of radius R₀ sin l — and falls again beyond it, so every velocity below the maximum corresponds to two distances. A cloud observed at 22 kilometres a second is at either 2.1 or 6.1 kiloparsecs, and nothing about its velocity says which. The two possibilities differ by a factor in distance and by its square in luminosity and mass, so the ambiguity is not a refinement — it decides whether a star-forming region is an ordinary one nearby or a monster on the far side of the Galaxy.
Fig. 2 The same construction at a larger longitude. The tangent point is nearer and the maximum velocity is lower, so the two solutions are closer together and the ambiguity is less costly. At ninety degrees the tangent point coincides with the Sun’s own radius and the near solution vanishes; beyond ninety, outside the solar circle, there is no ambiguity at all — the line of sight crosses each radius once. The problem is confined to the inner Galaxy, which is where most of the star formation is.

There is one more consequence and it operates on catalogues rather than on individual objects. A survey that resolves the ambiguity wrongly for a fraction of its sources produces a map with a spurious mirror population: objects placed at the near distance that belong at the far one appear as a reflection of the far structure into the near half of the line of sight. That artefact is recognisable — it produces structures that mirror across the tangent-point distance — and looking for it is a standard check on any kinematic map of the Galaxy.

Put the numbers on one case, because the abstraction hides how large the stakes are. A molecular cloud at thirty degrees of longitude with a velocity of eighty kilometres a second is at either four or nine kiloparsecs. Its angular diameter of ten arcminutes makes it either twelve or twenty-six parsecs across. Its integrated flux makes its mass either ten thousand or fifty thousand solar masses. And the first is an ordinary cloud while the second is among the more massive in the Galaxy. Two entries in a catalogue, differing by a bit that is not in the spectrum.

The precision is worst where the ambiguity is smallest

There is a second failure of the method that lives at the tangent point, and it is the exact complement of the first.

The two solutions merge there, so the discrete ambiguity vanishes — and the velocity is stationary with respect to distance, which is what “merge” means. A stationary function cannot be inverted linearly: a velocity error δv\delta v near the maximum corresponds to a distance error proportional to the square root of δv\delta v rather than to δv\delta v itself. So near the tangent point the distance is unique and badly determined, and away from it the distance is well determined and ambiguous.

The numbers are not marginal. At thirty degrees of longitude the velocity changes by only a few kilometres a second over the last kiloparsec on either side of the tangent point, and the observed line widths are themselves several kilometres a second. A cloud within a kiloparsec or two of the tangent point therefore has a formally unique distance with an uncertainty comparable to its distance from the tangent point — which is to say, no useful distance at all.

Velocity crowding is the same fact seen along the line of sight. Because the velocity is stationary there, a long stretch of the Galaxy is compressed into a narrow range of velocity, so a single spectral feature near the terminal velocity is a superposition of everything within a kiloparsec or two. Separating those contributions is not possible from the spectrum, and it is the reason the tangent-point rotation curve — which is the cleanest measurement the geometry offers — is quoted with error bars set by the width of the emission rather than by the noise. The curve that refuses to fall was measured this way, and the shape of its inner part carries this uncertainty throughout.

How it is resolved

Four methods break the ambiguity, and the first is by far the most used.

Self-absorption in the twenty-one centimetre line. A cold cloud in front of warmer emitting hydrogen absorbs at its own velocity, producing a dip in the profile. Warm hydrogen fills the Galactic plane at every distance, so a cloud at the near distance has warm gas behind it at the same velocity — from the far side of the ambiguity — and absorbs against it. A cloud at the far distance has nothing behind it at that velocity, and does not. The presence or absence of an absorption dip therefore decides the case, and it is a yes-or-no answer rather than a measurement.

Absorption against a continuum source. The same idea with a background radio source: a cloud in front of it absorbs, one behind it does not. This works where there happens to be a suitable source and is decisive when it does.

The vertical position. The Galactic disc is thin, so a source at a large distance and a given galactic latitude is at a large height above the plane. If the implied height is much larger than the disc’s scale height, the far distance is implausible.

And parallax, where it is available. Very long baseline interferometry of masers in star-forming regions measures parallaxes to a few per cent out to ten kiloparsecs and more. That resolves the ambiguity outright, and it has now been done for hundreds of regions — which is what turned kinematic distances from the primary method into a calibrated one.

NGC 3198 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.
Fig. 3 The one distance the method gives unambiguously: the tangent point, where the two solutions merge. At that point the line of sight is tangent to a circle of known radius, so the distance follows from geometry alone and the velocity is the maximum along the line of sight. That is how the inner rotation curve of the Galaxy was measured in the first place — one point per longitude, at a known radius, with no ambiguity and no assumption beyond circular motion.
One velocity, two distances — 3.3 and 8.3 kiloparsecs. The radial velocity of gas along a line of sight at galactic longitude 45 degrees, against distance from the Sun, for a flat rotation curve. The velocity rises to a maximum at the tangent point — where the line of sight is tangent to a circle of radius R₀ sin l — and falls again beyond it, so every velocity below the maximum corresponds to two distances. A cloud observed at 49 kilometres a second is at either 3.3 or 8.3 kiloparsecs, and nothing about its velocity says which. The two possibilities differ by a factor in distance and by its square in luminosity and mass, so the ambiguity is not a refinement — it decides whether a star-forming region is an ordinary one nearby or a monster on the far side of the Galaxy.
Fig. 4 A line of sight at forty-five degrees, where the tangent point is at the geometric mean of the two extremes and the two solutions are symmetrically placed about it. The figure makes the structure of the ambiguity plain: it is a reflection about the tangent distance, so the two solutions are equally far from it on either side, and their ratio grows as the longitude falls. Nothing about the shape of the rotation curve enters that statement — a rising, flat or falling curve gives the same symmetry.

There is a fifth route that has become available recently and is worth naming because it is the only one that uses the ambiguity rather than fighting it. A cloud’s dust emission is optically thin and its extinction of background starlight is not, so comparing the two gives a measure of how much of the Galaxy’s stellar column lies in front of the cloud and how much behind. With a three-dimensional map of the stellar distribution — which a survey of a billion parallaxes now supplies — that fraction is a distance. The method is independent of kinematics entirely, it works in directions where the velocity gradient is useless, and it is limited by the dust map’s own resolution.

What was actually measured

Three results stand out, and the third has changed the standing of the whole method.

The inner rotation curve, from tangent points. Measuring the maximum velocity along each line of sight gives the rotation curve inside the solar circle directly, and it was one of the first measurements to show that the Galaxy’s rotation does not fall in the Keplerian way the visible mass predicts.

The distances to hundreds of star-forming regions, by self-absorption. Large surveys of molecular clouds have applied the absorption test systematically, resolving the ambiguity for most catalogued sources and producing the standard maps of the Galaxy’s molecular gas.

And the maser parallaxes, which found systematic errors. Comparing parallax distances against kinematic ones for the same sources shows discrepancies of tens of per cent, larger than the quoted kinematic uncertainties. The cause is not the ambiguity but the assumption of circular motion: gas in a spiral arm streams, by ten to twenty kilometres a second, and a streaming velocity is interpreted by the method as a displacement in radius.

One velocity, two distances — 6.3 and 9.6 kiloparsecs. The radial velocity of gas along a line of sight at galactic longitude 15 degrees, against distance from the Sun, for a flat rotation curve. The velocity rises to a maximum at the tangent point — where the line of sight is tangent to a circle of radius R₀ sin l — and falls again beyond it, so every velocity below the maximum corresponds to two distances. A cloud observed at 124 kilometres a second is at either 6.3 or 9.6 kiloparsecs, and nothing about its velocity says which. The two possibilities differ by a factor in distance and by its square in luminosity and mass, so the ambiguity is not a refinement — it decides whether a star-forming region is an ordinary one nearby or a monster on the far side of the Galaxy.
Fig. 5 A line of sight closer to the Galactic centre, where the ambiguity is worst: the two solutions are far apart, the tangent point is distant, and a large range of the inner Galaxy is being sampled. Lines of sight like this one carry most of the Galaxy’s molecular gas and most of its star formation, and every distance along them is either resolved by absorption or is a coin toss.
Five named states of one medium, across 5 decades of density. The named states of the interstellar medium placed on the plane of density against temperature, both logarithmic, with three lines of constant pressure running across it. They are spread over 5 decades in density and five in temperature, and yet four of the five sit within a factor of 3.8 of one another in pressure — they lie nearly along one diagonal, because that is what mechanical contact between them requires. The exception is the molecular gas, which is over-pressured because it is held together by its own gravity rather than by the medium around it, and that exception is what makes star formation possible. The two numbers to read against each other are in the labels: the hot phase fills half the volume of the disc and holds one per cent of its mass, while the cold and molecular gas together hold half the mass in under four per cent of the volume. A map of the interstellar medium weighted by volume and one weighted by mass are pictures of different objects.
Fig. 6 Why the self-absorption test works at all: the interstellar medium contains both cold and warm neutral hydrogen at the same velocities, in a rough pressure equilibrium. The cold phase absorbs and the warm phase emits, and the Galaxy is full enough of the warm phase that there is almost always something behind a near cloud to absorb against. The test therefore depends on a property of the medium rather than on geometry, and it fails in the directions where the warm gas is thin.

Where the picture stops

Three of them, and the second is the one the parallaxes exposed.

The method fails near longitudes of zero and one hundred and eighty. There the line of sight is nearly radial, the projection factor sin⁡l\sin l is small, and a velocity of a few kilometres a second corresponds to an enormous range of distance. Kinematic distances in those directions are meaningless, and the region includes the Galactic centre.

Streaming motions are the dominant error. Spiral arms are density waves, and gas passing through one is compressed and deflected. The resulting non-circular motions of ten to twenty kilometres a second are a substantial fraction of the velocity difference between the near and far solutions in some directions, and they bias the distance systematically rather than randomly.

And the rotation curve itself has to be assumed. The method requires V(R)V(R), and the standard curve is measured from tangent points — which is to say from the same data. That circularity is mild, because the tangent-point measurement is ambiguity-free, and it means the method’s calibration and its application share a systematic.

There is a fourth, and it is worth stating because it is often forgotten in the outer Galaxy. Beyond the solar circle there is no ambiguity, and the method is correspondingly trusted there — but the velocity gradient with distance is much shallower outside than inside, because the rotation curve is flat and the projection changes slowly. So an outer-Galaxy kinematic distance has no ambiguity and a large uncertainty, and an inner one has a small uncertainty and a discrete ambiguity. The two halves of the Galaxy are measured with the same formula and fail in completely different ways, which is worth knowing before comparing a structure in one with a structure in the other.

A fifth limit is worth adding because it is the one that couples this essay to the mass modelling. Every distance derived here assumes a rotation curve, and the rotation curve is what a mass model is fitted to — so an error in the assumed curve moves the clouds, and the clouds’ positions are part of the evidence for the curve. The loop is not vicious, because the tangent points break it, and it does mean that the inner Galaxy’s mass distribution and its gas map are not independent measurements. Three mass models that fit the same curve shows how little a rotation curve constrains on its own; here that weakness propagates into a map.

Why a symmetric geometry is a special kind of degeneracy

The wider point is worth stating because this degeneracy has an unusual structure.

Most of the degeneracies in this collection are continuous: a parameter trades against another along a locus, and the answer is a band. This one is discrete. There are exactly two solutions, they are far apart, and the answer is one of them — so the question is not “how precisely” but “which”.

That changes what an observation has to do. A continuous degeneracy needs a second measurement with a different parameter dependence, and it narrows the band in proportion to its precision. A discrete one needs a measurement that need not be precise at all: a yes-or-no test with modest reliability is worth more than an exquisite measurement of the wrong quantity. The self-absorption test is exactly that — it delivers one bit, and one bit is what is missing.

That is worth recognising as a category. Two geometries and one bit to choose between them is the same shape in a different setting, and in both cases the productive question is which cheap observation supplies the bit rather than which expensive one improves the precision.

A parting observation about what the parallaxes have done to the method’s standing. Kinematic distances were, for fifty years, the only way to place anything in the obscured inner Galaxy, and the maps of the spiral arms were built entirely from them. The maser parallaxes now provide a few hundred direct distances, and comparing them against the kinematic ones has both validated the method in bulk and shown its errors to be larger than advertised. The result is a hybrid: the parallaxes anchor the arms and the kinematic distances fill in between, with the streaming motions calibrated from the difference. That is the same architecture as the distance ladder — a small number of hard, direct measurements calibrating a large number of easy indirect ones — arriving inside the Milky Way itself.

End on what the ambiguity has cost, since it is a rare case of a systematic whose historical price can be estimated. Maps of the Galaxy’s spiral structure built from kinematic distances in the 1970s and 1980s disagreed with each other about the number of arms, their pitch angles and their positions, and a substantial part of the disagreement was different treatments of the near–far choice. The modern picture — four arms or two, and where they are — was settled by the maser parallaxes rather than by any improvement in the kinematics. Fifty years of maps rested on a bit that was being guessed, and the guessing was systematic rather than random because the same criteria were applied by everyone. A galaxy measured from inside it is a harder measurement than a galaxy measured from outside, and this is the largest single reason.

One last comparison is worth drawing, because it says what kind of indicator this is. A kinematic distance is not a standard candle or a standard ruler: it does not compare an object against a calibrated population, and it carries no dependence on the object’s physics at all. It is a dynamical distance — it uses a model of the Galaxy’s velocity field the way a parallax uses the Earth’s orbit. That is a strength, since nothing about the cloud need be known, and it is the weakness too, since every error in the velocity field is inherited whole. A line width that is a distance fails in the opposite way: it knows nothing about the Galaxy and everything about the galaxy it measures.

Where the ladder goes next

The next rung is the streaming motions themselves: what a spiral density wave does to the gas passing through it, how large the deviations from circular motion are, and why they bias kinematic distances in a direction that correlates with the arms. The rung beyond it is the parallax programme — how a maser at ten kiloparsecs is measured to five per cent, and what a few hundred such measurements have done to the map of the Galaxy.

Where this goes next

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. Two geometries, one bit of data, and the bit decides. Rotation measure against galactic longitude for two field geometries that look identical in every image ever taken. An axisymmetric field runs the same way round the disc at every azimuth, so the component along the line of sight changes sign exactly 2 times as the longitude goes round — once where the field turns toward the observer and once where it turns away. A bisymmetric field reverses its own direction from one side of the galaxy to the other, and that doubles the count to 4. Nothing about the brightness, the arms or the colour distinguishes the two; the sign of a rotation measure does, and a sign is one bit. The measurement is made against several hundred background sources, each contributing one rotation measure through the whole disc — the sources are not the object of study, they are the illumination. The Milky Way's own answer is untidy: broadly axisymmetric with at least one reversal inside the solar circle, which is neither clean case, and is why the field's origin is still argued about. Two geometries and one bit to choose between them A galaxy's large-scale magnetic field either runs the same way round the disc everywhere or reverses from one side to the other. No image distinguishes the two. The sign of a rotation measure does — two reversals around the sky for one geometry and four for the other — and a sign is one bit of information. The rotation curve of NGC 3198, decomposed. Circular speed against radius for a three-component model of NGC 3198: a Hernquist bulge of 1.0×10⁹ M☉, an exponential disc of 2.20×10¹⁰ M☉ with a scale length of 2.6 kpc, and a pseudo-isothermal halo whose asymptotic speed is not chosen but solved for — it is whatever brings the total to the measured 150 km/s at 30 kpc, and comes out at 171 km/s. The components add in quadrature because accelerations add. The disc alone peaks at 5.7 kpc and falls away; the total does not. A rotation curve that refuses to fall Beyond the edge of a galaxy's light there is nothing left to enclose, so the orbital speed should fall away as the inverse square root of radius. It does not fall at all, and the shape of that refusal says the missing mass arrives at a constant rate for as far out as anyone can measure.

The objects this essay names

Each one links to every other essay that touches it.

DegeneracyDistance ambiguityGalactic planeKinematic distanceMolecular cloudRotation curveSelf-absorptionStreaming motionTangent pointThe 21-centimetre line