A spread the projection does not hide

A projected spin–orbit angle measured for one planet is a weak statement about the true one. The line profile of a transited star gives the angle λ between the orbit’s normal and the star’s spin axis as both are projected on the sky; the true angle ψ needs the inclination of the star’s axis to the line of sight as well, and every method of measuring that inclination is blind in the same place — a star seen equator-on, which is exactly where an aligned system sits. A projected angle of zero, with nothing known about the star’s tilt, leaves a median true obliquity of thirty degrees.

That conclusion is about one system and it is correct. It is also the reason a different question has gone less asked than it deserves. The misaligned planets, especially those clustered near right angles, have drawn most of the attention, because they demand an explanation. The aligned majority has been treated as the uninteresting baseline: aligned within the errors, consistent with zero, the default. But “how aligned are the aligned systems?” has a definite answer that matters — a population whose true obliquities spread over a degree or two says that stars and their planet-forming discs stay locked together; one that spreads over ten or twenty says they are routinely tilted against each other before any planet migrates. And the Sun, tilted seven degrees from its planets, sits somewhere in that range without anyone knowing whether it is typical.

The question turns out to be answerable from projected angles alone, and the reason is a piece of geometry that the single-system argument does not touch.

A small tilt is a vector, and the projection keeps one component #

Take a transiting planet. Its orbit is seen almost exactly edge-on, so the orbit’s normal lies in the plane of the sky; call that direction “up”. The star’s spin axis is tipped away from up by the true obliquity ψ, in some direction around it. That direction could be towards the observer, away from the observer, or sideways along the sky.

If the tilt is small, it can be treated as a short vector in the plane perpendicular to the orbit normal, with two components: one along the line of sight, one across the sky. The projection onto the sky throws away the first and keeps the second. The projected angle λ is therefore not a degraded version of ψ — it is exactly the cross-sky component of the tilt, to first order in the angle:

$$\lambda \approx \psi\cos\Phi,$$

where $\Phi$ is the direction of the tilt measured around the orbit normal. The component along the line of sight changes the star’s inclination and is invisible to the line profile, which is why a single system’s true obliquity cannot be recovered. But nothing about it has been lost for a population, as long as the direction of the tilt is unrelated to where the observer happens to be standing — which it must be, since the observer’s position is an accident.

A vector with random direction has its squared length shared equally, on average, between its components. So for a population of small tilts,

$$\langle\lambda^2\rangle = \tfrac12\langle\psi^2\rangle,\qquad \psi_{\rm rms} = \sqrt2\,\lambda_{\rm rms}.$$

The relation holds whatever the distribution of ψ is — Gaussian, uniform, a mixture of several widths — because it uses only the second moment and the randomness of the direction. It needs no measurement of any star’s inclination, no model of rotation periods or line widths, and no seismology. The quantity that defeats the single-system analysis cancels out of the population one.

A population of Suns #

The relation is easiest to see in its starkest case, where every system has the same true obliquity.

This is the point of the argument in one picture. An astronomer measuring the Solar System from a random direction in the ecliptic would quote a projected obliquity anywhere from zero to seven degrees, and almost one time in five would call the Solar System aligned to within two. No individual measurement says the Sun is tilted by seven degrees. The collection of measurements does, through its width, exactly — and a survey that found the projected angles of its aligned systems scattered with an rms of about five degrees would be looking at a population of Suns.

The distribution in the figure has a characteristic shape, peaked at its edges, because a single fixed tilt seen from a random azimuth spends most of its time near its extremes. A real population is a superposition of such shapes for many different tilts, and the superposition fills in to something closer to the bell in the opening figure. The width is what survives; the shape is a statement about how the tilts themselves are distributed and is much harder to recover.

Two population methods that answer different questions #

The idea of using a population to escape the single-system ambiguity is not new, and it is worth being clear about how this version differs from the ones already in use.

The established population methods test whether planet hosts are seen equator-on. If orbits and spins are aligned, every transiting system is viewed from within its star’s equatorial plane, so the hosts’ projected rotation speeds should be systematically higher than those of comparable stars without transiting planets, and their spot modulations systematically larger. Both tests have been run on the stars of the Kepler field, whose planets were found by the light they removed, and both show alignment for cool hosts and something closer to randomness for hot ones.

Those tests are sensitive to large misalignments and nearly blind to small ones. A star tilted by five degrees from its planet’s orbit is still seen within five degrees of equator-on, and its projected rotation speed differs from the equator-on value by the cosine of five degrees — under half a per cent, far below what a comparison of rotation speeds between two samples of stars can resolve. The equator-on tests answer the question “what fraction of systems are badly misaligned?” and answer it well.

The projected-angle method answers the complementary question, “how widely are the well-aligned systems spread?”, and it is sensitive in exactly the regime where the other methods are blind. Where one ends the other begins: a projected angle measured to a few degrees is a direct component of a tilt of a few degrees, while a spot amplitude or a rotation speed barely notices one. And because the projected-angle method needs line-profile measurements, it works on the bright systems with large planets, while the equator-on methods reach the small planets that no line profile will ever show. The two together cover the distribution of obliquities from both ends.

Seismology, which measures a star’s tilt from which of its oscillation modes are visible, sits beside both. It gives true obliquities for individual systems, but it needs stars that oscillate cleanly and rotate fast enough to split their modes, and its precision on the inclination near equator-on is poor for the same geometric reason as every other method’s. It is the best way to calibrate the population arguments on a few systems and the wrong way to measure a spread of a few degrees.

How small is small #

The relation is exact only in the limit of small tilts, and its usefulness depends on how quickly it fails.

The failure at large angles has a clear cause. The exact projected angle is $\lambda = \arctan(\tan\psi\cos\Phi)$, not $\psi\cos\Phi$. For a tilt seen nearly along the line of sight, $\cos\Phi$ is small, but for a large ψ its tangent is large, and the projected angle can be much bigger than its small-angle component — the same effect that sends every projected angle to ninety degrees for a star seen pole-on. For populations spread by more than about twenty degrees the simple $\sqrt{2}$ overestimates the spread; for a population spread by tens of degrees the single-system ambiguity returns, and the full forward model of the earlier argument is needed.

That limitation is exactly the right one. The question the relation answers is how aligned the aligned systems are, and by construction those are the systems with small tilts. The misaligned ones, spread over all angles, are a separate population, and they will need separating — of which more below.

The orbit’s own small departure from edge-on, the second curve, hardly matters. A transiting orbit is tipped by at most a few degrees from the line of sight, and the resulting correction to the projected angle is second order in the tilt.

Errors that look like a spread #

The relation converts a population’s projected scatter into its true scatter. It does not say whether the scatter is there to be measured. Every projected angle has a measurement error, typically a few degrees for a bright hot Jupiter and more for a smaller or fainter one, and a population of perfectly aligned systems measured with five-degree errors would show a projected scatter of five degrees and, read naively, an rms true obliquity of seven — a population of Suns, entirely manufactured by the instrument.

So the quantity to measure is the excess: how much wider the projected angles are than their own errors account for. For Gaussian errors the excess variance is the observed variance minus the mean squared error, and the question is how many systems it takes to see it.

The arithmetic is simple and unforgiving. Detecting an excess variance $\sigma^2$ on top of errors $e$ at three standard deviations needs $N = 18\,(\sigma^2+e^2)^2/\sigma^4$ systems. When the spread is much larger than the errors, about twenty suffice. When it is comparable, the count is in the dozens. When it is half the error, it is in the hundreds. A spread as wide as the Sun’s is within reach of the samples that already exist — several dozen aligned systems with errors of a few degrees — while a spread of a degree or two is not, and will not be until the errors fall below it.

The arithmetic also says what the measurement depends on, and it is not only the number of systems. It is whether the quoted errors are right. An excess variance is a small difference between two large numbers, and an error underestimated by a third — easily done, when the dominant uncertainty is a model of the line profile rather than photon noise — produces a spurious spread as large as the real one being sought. A projected angle is fitted with a model of the star’s rotation, its limb darkening, its turbulent broadening and its differential rotation, and each of those carries an uncertainty that a simple error bar may not include. The population measurement of alignment is, in the end, a measurement of whether those error bars are honest.

Where the error bars come from #

The size of the error on a projected angle is set by how much of the star’s rotation the planet hides and how precisely that hidden velocity can be measured. The anomalous radial velocity during transit — a velocity measured from a shape, since the star’s centre of mass has not moved — scales as the fraction of the disc covered times the rotation speed of the part covered. A planet covering one per cent of a star rotating at five kilometres a second produces an anomaly of a few tens of metres a second; a planet covering a tenth of a per cent of the same star, a few metres. The bump the planet’s shadow makes as it crosses the rotating line profile obeys the same scaling, in a form that is easier to model when the star rotates quickly.

Both methods therefore measure projected angles most precisely for large planets around fast-rotating stars. That introduces a correlation the population argument has to respect. Fast rotators are hot stars, above the break in rotation near 6,250 kelvin, and hot stars are precisely the hosts whose planets are most often misaligned. Cool stars rotate slowly, their planets’ line-profile signals are small, and their projected angles carry larger errors — which, as the counting above showed, makes their intrinsic spread hardest to detect.

The result is an uncomfortable mismatch between where the measurement is easy and where it is wanted. The systems most likely to preserve a primordial spread, untouched by tides and around stars whose discs could have been tilted gently, are the ones whose projected angles are least precise. The systems with the best-measured angles are around hot stars, where the aligned and misaligned populations overlap most and the separation of a core from a floor is least clean. A measurement of the aligned spread will, for some years, be a measurement made on the populations that were convenient rather than the ones that were cleanest.

A minority that does not widen the core #

A real sample is not all aligned. Around hot stars, and in a smaller fraction around cool ones, there are planets whose orbits are tilted by tens of degrees or more, and some go round backwards. If they were included in a measurement of the spread, they would dominate it: a single polar orbit adds more to the variance than fifty aligned systems.

The obvious response is to remove them by hand, deciding which systems count as aligned. That is unsatisfactory, because the dividing line is arbitrary and moves the answer. A better response uses the fact that the two populations have different shapes.

Spins in random directions produce projected angles spread across the whole circle, so the misaligned minority contributes a low, flat floor. The aligned majority contributes a narrow peak. A fit of a peak plus a floor separates them statistically without any system being assigned to either, and the width of the peak is the quantity the $\sqrt{2}$ relation converts into a true spread. The fit recovers both the fraction and the width, with an uncertainty set by the number of systems in the peak.

The model in the figure is deliberately crude — a perfectly flat floor, a perfectly Gaussian core — and real samples are not. The misaligned systems are not isotropic: they cluster near polar orbits, which in projection puts more of them near ±90 degrees than a flat floor allows. A small number of systems at intermediate angles of twenty or thirty degrees could belong to either population, and those are exactly the ones that decide the width. The method is robust to the gross structure of the minority and sensitive to the boundary between the populations, which is where any method would be.

Why the answer matters for how planets form #

The spread of the aligned systems is a measurement of something that happened before the planets existed.

A star and its protoplanetary disc form from the same collapsing core, and they inherit its angular momentum. If the core’s angular momentum were perfectly ordered, the star’s spin and the disc’s orbital axis would be identical, and planets forming in the disc would orbit in the star’s equatorial plane. Several processes act against that. Infall onto the disc continues after the star has formed, and material arriving late may carry angular momentum in a different direction from what fell first, tilting the disc relative to the star. A companion star, if there is one, torques the disc as well, tipping it slowly against the star’s spin. And magnetic coupling between the star and the inner disc may realign them, or under some conditions drive them apart.

Each of those predicts a different spread. Turbulent, chaotic infall into a forming cluster suggests primordial misalignments of ten or twenty degrees. A quiet, isolated collapse suggests a few. Strong magnetic coupling during the disc phase suggests a narrow distribution regardless of the infall. The spread of the aligned planetary systems, measured around stars that have not since been realigned by tides, is the record of which of those dominated.

That last condition matters. A hot Jupiter close to a cool star raises a tide in the star’s convective envelope strong enough to realign the star over its lifetime, and a population of such systems may be aligned because tides made it so rather than because it formed that way. The informative population is the one tides cannot have touched: planets on wider orbits, multi-planet systems, and stars whose envelopes are too thin to be tidally reoriented. Those are harder to measure, because their line-profile signals are smaller, and they are the population where an honest spread would say the most.

The one system measured from inside #

The Sun’s obliquity is the only one known without any of this. It was measured from inside the system, by following sunspots across the disc: Richard Carrington, timing spots through the solar rotation in the 1850s, fixed the inclination of the Sun’s equator to the ecliptic at about seven and a quarter degrees, and the number has barely moved since. Measured against the invariable plane — the plane of the Solar System’s total angular momentum, dominated by Jupiter — the tilt is about six degrees, which is the number that compares directly with a planet’s orbital plane elsewhere.

From outside the Solar System none of that would be available. An observer would see one transiting planet at a time, measure one projected angle, and have no way to fix the Sun’s inclination to better than its line widths and spot modulation allow. The figure of a population of Suns is the statement that this observer, given enough systems like the Solar System, would nevertheless recover the six or seven degrees — as a width, not as a value for any one system.

That inversion is the general point. Some quantities cannot be measured for any member of a population and can be measured exactly for the population as a whole, because the unknown that spoils each measurement averages out in a predictable way. The inclination of a star is unknowable for an equator-on system and irrelevant to the second moment of a population of small tilts. The same structure appears in the mass of a planet known only through the wobble it causes, where each mass carries an unknown sine and the distribution of masses can still be recovered. The answer to a population question is not the average of the answers to the individual ones.

What the geometry leaves out #

Two assumptions underlie the $\sqrt{2}$, and both should be stated.

The first is that the direction of each system’s tilt is random with respect to the observer. That is guaranteed for the direction around the orbit normal, because the observer’s position is unrelated to the system’s history. It is not guaranteed for the selection of which systems are measured. A survey that preferentially measures stars with large projected rotation speeds, because their line-profile signals are larger, preferentially selects stars seen equator-on, and for a tilted system that means tilts lying across the sky rather than along the line of sight — inflating the projected spread. The effect is real and small for most samples, and it acts in the direction of a population looking more misaligned than it is.

The second is that each system is described by one tilt. A star with differential rotation has no single spin axis at the precision of the smallest spreads; a planet’s orbit precesses under the influence of others; and the tilt being measured may be changing on timescales comparable to the star’s age. None of these change the geometry, but they blur what “the obliquity” of a system means at the level of a degree.

Still open: whether the Sun is tidy or typical #

The spread of the aligned planetary systems is measurable with samples that already exist, provided their errors are understood well enough that an excess of a few degrees can be trusted. Whether the Sun’s seven-degree tilt is generous, typical or unusually tidy is therefore not a question for a future instrument; it is a question for a careful reanalysis of projected angles already published, with a single treatment of the errors and with the tidally realigned systems set aside. If the answer is a spread of a degree or two, the Sun is an outlier and something tilted it — a passing star, a late infall, a distant companion now lost. If the answer is ten degrees or more, the Sun is ordinary, and every planetary system begins life slightly askew from the star it orbits.