Two orbits of one pair, and a distance falls out

The rung below got masses out of a binary: two radial-velocity curves give the mass ratio exactly and the mass sum up to a factor of $\sin^3 i$, and an eclipse supplies the inclination.

It quietly assumed something. Converting a velocity amplitude into a mass requires knowing how large the orbit is in kilometres, and that came out of the velocities themselves — a velocity multiplied by a time is a length, so the spectroscopic orbit delivers a linear size with no distance in it at all.

That is a remarkable property and the rung below used it only to get masses. Used differently it gives something the whole subject is short of.

A length and an angle #

Take a double-lined spectroscopic binary with period $P$, eccentricity $e$, and velocity semi-amplitudes $K_1$ and $K_2$. The size of the relative orbit follows immediately:

$$a\sin i = \frac{(K_1+K_2)\,P\,\sqrt{1-e^2}}{2\pi}.$$

Every quantity on the right is measured from spectra, and the result is in kilometres. A spectrograph, which cannot resolve the pair and has no idea how far away it is, has nonetheless measured the size of the orbit.

Now resolve the same pair. An astrometric orbit — from long-baseline interferometry, from speckle imaging, or from decades of visual micrometry — gives the apparent relative orbit as an ellipse on the sky, from which the true angular semi-major axis $a''$ and the inclination $i$ can both be extracted, because the projection of an ellipse is an ellipse whose focus is displaced in a way that fixes the viewing angle.

Divide:

$$d\ (\text{pc}) = \frac{a\ (\text{AU})}{a''\ (\text{arcsec})}.$$

That is the orbital parallax. Nothing has been assumed about the stars’ luminosities, radii, colours or composition. Nothing has been calibrated against another distance. It is a length divided by an angle, in the most literal sense available.

Why this is not the parallax everyone means #

The trigonometric parallax uses the Earth’s orbit as a baseline: a star’s apparent position shifts by an angle $p$ over six months, and $d = 1/p$.

The orbital parallax uses the binary’s orbit as the baseline. The geometry is the same triangle with a different short side, and the crucial difference is that the short side is measured in kilometres by spectroscopy rather than being known in advance. The two methods therefore fail in different places, which is what makes the second worth having. A trigonometric parallax becomes useless beyond a few kiloparsecs because the angle vanishes. An orbital parallax becomes useless when the pair can no longer be resolved — but the angular separation of a binary falls as $1/d$, exactly as the parallax does, so the two limits scale together and the orbital version wins only where the orbit is wide.

Where it wins decisively is on objects with no parallax at all. The distance to the Galactic centre has been measured this way: the star S2 has been tracked through a complete orbit around the compact object at the centre, its position measured astrometrically to tens of microarcseconds and its radial velocity spectroscopically to a few kilometres per second, and the combination gives a distance of about 8.3 kiloparsecs to a fraction of a per cent. A distance to the centre of the Galaxy, obtained from one star’s orbit, with nothing borrowed from anything.

What the ladder looks like without a rung #

The reason this matters out of proportion to the number of systems it applies to is the structure of the distance scale. The two best-known applications are both about breaking that chain.

Eclipsing binaries in the Magellanic Clouds give a distance by a related but distinct route: the eclipses give the stellar radii in kilometres, a surface-brightness–colour relation gives the angular diameters, and the ratio is a distance. The distance to the Large Magellanic Cloud is now quoted to about one per cent by that method, and since the Cloud is where the Cepheid period–luminosity relation is calibrated, the number propagates directly into the Hubble constant.

Detached binaries in the Galaxy with both an interferometric and a spectroscopic orbit give distances to a per cent or better, and those distances are used to check Gaia’s parallaxes rather than the other way round — a valuable inversion, because a space astrometry mission’s systematic errors are otherwise very hard to test. The five numbers an astrometric solution delivers include a parallax with a zero-point offset that has to be determined from something outside the mission, and a distance owing nothing to astrometry is exactly what that requires.

There is a third application which is really the same idea with the roles exchanged. A star cluster containing an eclipsing binary can have its whole distance fixed by that one system, and every other star in the cluster then acquires an absolute magnitude — which turns a colour–magnitude diagram into a calibrated one. An age read off a bend needs exactly that calibration, and one well-measured binary supplies it for a few thousand stars at once.

What has to be true for the two to be the same orbit #

The method rests on an assumption so obvious it is easy to leave unstated: that the spectroscopic orbit and the astrometric orbit are orbits of the same thing.

They need not be. A spectroscopic orbit is of a photocentre — whichever component’s lines are being measured. An astrometric orbit from imaging is of the separation between two resolved points; one from a single-star astrometric solution is of the photocentre relative to the barycentre, which is a different ellipse again, smaller by a factor involving the luminosity ratio as well as the mass ratio. Combining a relative orbit with a photocentric one without noticing produces a distance wrong by that factor.

The consistency check that catches it is worth having in any case: the spectroscopic orbit gives $a\sin i$ and the astrometric one gives $a''$ and $i$, so the two determinations of the period, the eccentricity, the longitude of periastron and the epoch of periastron must all agree. Five redundant quantities between two datasets taken with completely different instruments is an unusually strong test, and disagreement in any of them means one of the orbits is of something else — a third body, a blend, or a photocentre mistaken for a component.

A distance from two orbits is only as good as the claim that there is one orbit, and the redundancy is what makes that claim checkable rather than assumed.

When only one spectrum is visible #

Most binaries are not double-lined. If one component is much fainter, only its brighter partner’s lines shift measurably, and the spectroscopic orbit gives only the mass function

$$f(M) = \frac{M_2^3\sin^3 i}{(M_1+M_2)^2} = \frac{P K_1^3 (1-e^2)^{3/2}}{2\pi G},$$

which is a lower bound on the companion’s mass and not a size.

The classical workaround is the dynamical parallax, and it is an iteration.

1. Guess the total mass. One solar mass each is the traditional start.
2. Kepler’s third law gives the linear semi-major axis: $a^3 = (M_1+M_2)P^2$ in solar units.
3. Divide by the measured angular semi-major axis to get a distance.
4. That distance converts the apparent magnitudes into absolute ones.
5. A mass–luminosity relation converts those into masses.
6. Return to step 2. The loop converges, quickly, and the reason is worth naming. From step 2, $a \propto M^{1/3}$, so the distance depends on the assumed mass only as its cube root. A factor of two error in the mass is a 26 per cent error in the distance, which is a 0.5 magnitude error in the absolute magnitude, which through a 3.5-power relation is a factor of 1.4 in the mass — smaller than what was started with. The iteration contracts, and three or four passes are enough from almost any starting point.

That same insensitivity is the method’s ceiling. Because the answer barely depends on the assumed masses, it barely depends on getting them right; and because it barely depends on them, it cannot be pushed to a precision better than the mass–luminosity relation’s own scatter, which is a few per cent for well-behaved main-sequence stars and much worse for anything evolved.

A number, and what it took to get it #

The clearest single application is worth walking through with figures attached, because the method sounds abstract until it has produced something.

The star S2 orbits the compact object at the centre of the Galaxy with a period of about 16 years, an eccentricity of 0.88, and an angular semi-major axis of roughly 125 milliarcseconds. Its radial velocity at periapsis passage reaches some 7,700 kilometres per second — nearly three per cent of the speed of light — which is measurable spectroscopically to a few kilometres per second.

The angular orbit is measured by near-infrared interferometry to tens of microarcseconds. The linear orbit follows from the velocities. The ratio is the distance to the Galactic centre: about 8.3 kiloparsecs, to a few tenths of a per cent.

That number used to be quoted with an uncertainty of ten per cent and was derived by half a dozen indirect routes — the distribution of globular clusters, the kinematics of the disc, the mean distance to RR Lyrae variables — each of them resting on a rung of the ladder. It is now a two-orbit measurement of one star, and it is the anchor for the Galaxy’s mass, its rotation curve and the calibration of every distance indicator that has been checked against Galactic-centre objects.

The same orbit, incidentally, has delivered two relativistic measurements as by-products: the gravitational redshift at periapsis and the precession of the orbit’s own perihelion. Forty-three arcseconds per century for Mercury becomes twelve arcminutes per orbit here, and it is measured.

What limits the good version #

The orbital parallax has no such intrinsic ceiling, so what limits it is measurement error, and the propagation is instructive.

$$\frac{\sigma_d}{d} = \sqrt{\left(\frac{\sigma_{a''}}{a''}\right)^2 + \left(\frac{\sigma_K}{K}\right)^2 + \left(\frac{\sigma_i}{\tan i}\right)^2 + \dots}$$

Every term enters linearly, so there is no rung-on-rung amplification and no place for a systematic to hide. The practical limits are three.

Resolving the pair. The orbit must be wide enough on the sky to measure and short enough in period to observe. Those pull against each other through the harmonic law: a wider orbit is a longer period, so the systems that work are a narrow band in separation and distance.

Measuring the inclination well. The $\sigma_i/\tan i$ term blows up for a face-on orbit, where the projection is nearly circular whatever the inclination. Highly inclined systems are better, and eclipsing ones are best.

Both spectra. A luminosity ratio beyond about five to one makes the fainter component’s lines unmeasurable, and the method degrades to the dynamical version.

Those three together are why the number of systems with a good orbital parallax is in the dozens rather than the thousands, and why each of them is worked over so carefully. It is also why the technique has not been made obsolete by space astrometry: the systems it reaches are not the systems a parallax reaches well, and the two disagree often enough to be worth comparing.

The same construction, on a disc of masers #

The method needs an orbit seen two ways, and there is one system where the orbits belong not to stars but to spots of gas, and the distance that comes out is the most important single number in the extragalactic ladder.

Water molecules in the dense inner disc of an active galaxy can be pumped into a population inversion and radiate as masers — extremely bright, extremely narrow radio lines from small spots of gas. Interferometry resolves the spots individually, at angular resolutions of tens of microarcseconds, and their velocities are read straight off the line frequencies.

For one nearby galaxy the spots lie in a thin, nearly edge-on, nearly Keplerian disc around the central black hole, and three quantities are measurable for them at once: the angular position of each spot, its line-of-sight velocity, and — over years of monitoring — its acceleration, from the drift of its line frequency.

That is enough to close the geometry with no assumption about the disc’s physical size. The velocity gives the orbital speed; the acceleration gives the centripetal term, so the physical radius follows as $v^2/a$; and the angular radius is measured directly. A physical length against an angle is a distance.

The result — about 7.6 megaparsecs, to better than two per cent — is a purely geometric distance to a galaxy far beyond any parallax, resting on no stellar model, no standard candle and no calibration.

Its importance is what sits on top of it. That galaxy contains Cepheid variables, so the geometric distance calibrates the period–luminosity relation absolutely, and the relation then carries out to galaxies containing type Ia supernovae. It is one of the three anchors of the local expansion measurement, and it is the only one that is a single object.

Expanding shells, measured the same way #

The pattern of the essay — an angular quantity and a physical one measured independently — has a third instance that needs no orbit at all.

An expanding shell of gas, from a nova or a supernova, has an angular radius that grows measurably over months to years, and a physical expansion speed that is read off the Doppler width of its lines. The ratio of the two is a distance, by the same logic as everything above.

The method is old and it works, with two difficulties that recur in every version. The shell has to be resolved, which restricts it to nearby objects. And the velocity measured spectroscopically is the velocity of the material along the line of sight, while the angular growth is the expansion across it — so the two are the same number only if the shell is spherical, which real ejecta rarely are.

Both difficulties are geometry rather than physics, which is the family resemblance running through this whole rung: every one of these distances is a length divided by an angle, and every one of them fails in the same way, by being wrong about the shape of the thing being measured.

Both of the additional methods here also share the property that makes the essay’s own construction valuable: they are checked by nothing else. A geometric distance has no calibrator above it and no standard object underneath it, so when one disagrees with a distance from the ladder the disagreement has to be settled on the merits of the geometry rather than by appeal to a third measurement — and that is exactly the situation the local expansion rate is in.

That is an uncomfortable position and it is the right one, since the alternative is a ladder whose bottom rungs agree because they were tied to each other.

A ladder of that kind is internally consistent and externally unconstrained, which is the failure mode a geometric rung exists to prevent.

Which is the whole reason a rung of this kind is built at all, at a cost in telescope time that no other distance measurement of comparable reach requires.

Where this ladder goes next #

This rung has established that a binary measured twice yields a distance with nothing underneath it, and what the cheaper iterative version costs.

The rung above is the astrometric-plus-spectroscopic solution done jointly rather than sequentially — a single fit to positions and velocities together, which is how Gaia’s own binary solutions are produced and which handles partial coverage of a long orbit far better than fitting each dataset alone.

Beside it lies the population question the previous rung raised: what the distribution of mass ratios and periods says about how binaries form, which is a statistical statement about the very sample these methods can reach.

And below it, the habit: a quantity measured two ways in two units is a conversion factor. A length from spectroscopy and an angle from imaging is a distance; a radius from an eclipse and an angular diameter from a colour is a distance; the pattern recurs, and each time the value of the result is that it borrows nothing.