A precession slowed by stars lying on their sides

The orbit of two stars around each other does not stay put. In an eccentric orbit the point of closest approach — the periastron — slowly turns, and in an eclipsing binary that turning shows up in the timing of the eclipses, which drift relative to each other as the line of apsides carries the two eclipses around the orbit. The rate has three causes, each computable. General relativity advances the periastron of any orbit, the same effect that accounts for the forty-three arcseconds a century of Mercury’s. The tides each star raises on the other distort both into slightly elongated shapes, whose extra gravity advances it too. And each star’s own spin flattens it at the poles, and the equatorial bulge advances it further.

For most eclipsing binaries the sum agrees with what the eclipses record, once the stars’ internal structure is modelled well and the relativistic term is included, and the agreement is one of the cleanest tests of stellar interiors there is. DI Herculis is the exception, and the reason it was an exception took a quarter of a century to find.

A prediction four times too large #

DI Herculis is a pair of hot, young stars of about five solar masses each, a little under three solar radii across, in an orbit of 10.55 days with an eccentricity of 0.489. Every quantity the apsidal rate depends on is measured: the masses and radii from the velocities and the eclipses, the eccentricity from the eclipse timing, the rotation speeds from the widths of the spectral lines.

Relativity contributes 2.34 degrees a century. That number is computed from the masses, the period and the eccentricity alone: $6\pi G(M_1 + M_2)/[c^2 a(1 - e^2)]$ per orbit, with no freedom anywhere. The stars’ distortion contributes 1.93 degrees a century in the published stellar models, of which most — about 1.50 — is rotational flattening, because the stars spin at more than a hundred kilometres a second at their equators, and the tides supply the remaining 0.43. The prediction is 4.27 degrees a century.

The observed rate, measured from eclipse timings since the 1950s, is between about 0.6 and 1.4 degrees a century depending on the analysis — a quarter to a seventh of the prediction. And the observed rate is lower than relativity’s contribution on its own. No adjustment of the stars’ interiors can remove a discrepancy of that sign, because every interior distortion of an aligned star adds to the advance rather than subtracting from it.

That is why the system became famous. Edward Guinan and Frank Maloney established the discrepancy firmly in 1985, and over the following years it was cited as possible evidence that general relativity was wrong in the regime of strong orbits — or that an unseen third body was perturbing the pair, or that the stars’ internal structure was far more extreme than any model. None of these held up. The third body was searched for and not found, the stellar structure that would have been needed was implausible, and relativity passed every other test applied to it.

A bulge that pulls the wrong way #

The resolution was in an assumption so standard that it was rarely written down: that the stars’ spins are aligned with the orbit.

A spinning star is flattened, and its extra mass sits in a ring around its equator. If the equator lies in the orbital plane, the ring’s extra pull on the companion is concentrated in the plane of the orbit, adds to the inward force near periastron, and advances the periastron, exactly as the Earth’s equatorial bulge turns a satellite’s orbit. If the star is tipped, the ring’s mass sits partly above and partly below the orbit, and its average effect on the orbit changes.

The change follows a single function of the tilt, the second Legendre polynomial $P_2(\cos\beta) = (3\cos^2\beta - 1)/2$ — the same function that describes how any quadrupole looks from a direction at angle $\beta$ to its axis. At $\beta = 0$ it is one, and the bulge advances the periastron fully. It falls as the star tips, passes through zero at 54.7 degrees, and goes negative beyond: a star tipped further than that drags the periastron backwards. A star lying exactly on its side contributes minus a half of its aligned effect.

The angle is a familiar one. It is the angle whose cosine squared is a third, which makes it the angle at which a shared spin axis does no harm to the measured brightnesses of the stars that share it and at which a magic-angle spinning sample cancels its own dipolar couplings in a spectrometer. Here it is the angle at which a star’s equatorial bulge stops mattering to its companion’s orbit. Every one of those is the quadrupole vanishing on average, seen from three different problems.

For DI Herculis the rotational term is three quarters of the stars’ total distortion, so reversing its sign does not merely shrink the predicted rate — it subtracts from relativity. If both stars were tipped to near ninety degrees, the rotational contribution would change from plus 1.50 degrees a century to about minus 0.75, and the total would fall from 4.27 towards two.

Why the tides cannot rescue it #

The tidal term does not reverse with the stars’ tilt, and the reason is worth stating because it decides how far the prediction can move. A spin’s flattening is fixed to the star: the bulge sits around the star’s own equator, wherever that points. A tide is fixed to the companion: each star is stretched along the line joining the two, and that line always lies in the orbital plane, whatever the stars’ spins are doing. So the tidal bulge always pulls in the plane of the orbit and always advances the periastron, and its 0.43 degrees a century is untouched by the tilt.

Relativity is untouched too — it depends on the masses and the orbit and on nothing about the stars’ insides. Of the three contributions only one can change sign, and only that one depends on the spins. That is what made the explanation testable rather than adjustable: the measured tilts could have come out small, and then the discrepancy would have stood.

The eclipse that showed the stars were tipped #

Whether the stars are tipped can be measured, because one passes in front of the other.

A rotating star has one limb approaching the observer and the other receding, so the light from its two halves is Doppler-shifted in opposite directions and their average sits at the star’s own velocity. When the companion passes in front, it hides part of the disc, and the light it hides is missing from the average: hiding the approaching side makes the star appear to recede, hiding the receding side makes it appear to approach. The pattern of that apparent velocity through the eclipse — the Rossiter–McLaughlin effect, which the planets transiting their stars are measured with as routinely as binaries once were — depends on the path the eclipsing body takes across the rotating disc. If the spin is aligned, the eclipse chord runs across the lines of constant velocity, and the anomaly swings symmetrically from one sign to the other. If the spin is tipped, the chord runs more nearly along them, and the anomaly becomes lopsided.

Simon Albrecht and his colleagues measured the effect in both eclipses of DI Herculis and published the result in 2009. The projected angle between each star’s spin and the orbit’s axis was 72 degrees for the primary and 84 for the secondary. Both stars are nearly on their sides.

What the spectra measure is the angle projected onto the sky, $\lambda$. The true angle $\beta$ also depends on how far each spin axis leans towards or away from the observer, which a velocity anomaly cannot see: $\cos\beta = \sin i_\star \cos\lambda$ for an orbit seen edge on, with $i_\star$ the inclination of the spin axis to the line of sight. If the spins lie in the plane of the sky, $\beta = \lambda$. If they lean towards the observer, the true angles are larger still, and the tilt of the star hides part of the angle in exactly the way it does for transiting planets.

What the precession then measures #

With the projected angles measured, the prediction can be recomputed — and the one unknown left in it is the spins’ inclination to the line of sight.

With the spins in the plane of the sky the predicted rate falls from 4.27 to 2.13 degrees a century. That is most of the way, but still above the observed band. Tilting the spins towards the observer does two things at once. The true angle between spin and orbit grows towards ninety degrees, deepening the reversal. And the stars must be turning faster than their measured $v\sin i_\star$, because the measured speed is only the component across the line of sight, so their flattening — which goes as the square of the spin — is larger than it appeared. Both push the rate down. For spin inclinations between about 35 and 45 degrees the prediction falls inside the observed band.

That turns the problem around. The apsidal motion, once a discrepancy, becomes a measurement of how far the stars’ spin axes lean towards the Earth: a quantity spectroscopy cannot reach, delivered by eclipse timings stretching back seventy years. The value is uncertain, because the observed rate is uncertain and because the stars’ internal structure constant enters too, but its existence is the point: a three-body precession argument has measured a geometric angle that light alone does not carry.

The structure constant, and what it trades against #

One number in the prediction is not measured but modelled: the apsidal constant $k_2$, which says how centrally condensed each star is and so how much a given distortion of its surface moves its external gravity. It is the stellar version of a planet’s Love number, and for hot main-sequence stars it is small, a few thousandths, because most of their mass is packed near the centre. Every term but relativity is proportional to it.

In DI Herculis that makes the inferred inclination of the spins degenerate with the stars’ structure. A star twenty per cent more centrally condensed than the models say has a smaller $k_2$, a weaker rotational term, and needs its spin tilted further towards the observer to reach the same observed rate. The range of 35 to 45 degrees in the fourth figure is therefore conditional on the models being right to the usual ten or twenty per cent; with $k_2$ free, the precession constrains a product of structure and geometry rather than either alone. That is the same kind of degeneracy every eclipsing binary’s apsidal motion carries, and the reason the method is most powerful in systems where relativity dominates and the structure enters only as a correction.

A laboratory for relativity after all #

The systems where relativity dominates are the reason DI Herculis mattered. In an eccentric binary of massive, compact stars in a tight orbit, the relativistic advance can be more than half the total, and its agreement with observation is a test of general relativity in a regime of stronger fields than the solar system offers: the stars’ gravitational potential at the orbit is nearly twenty times the Sun’s at Mercury. A few dozen such binaries have been compared, and with the stellar structure modelled and the spins assumed aligned, most agree to within the uncertainties — which is why a single large exception drew so much attention.

With the tilt included, DI Herculis joins the agreement. Its relativistic term of 2.34 degrees a century is not optional: without it the measured spins would drive the predicted rate to about zero or below, far under the observed band. The test is weaker than the one provided by binary pulsars, whose orbits shrink exactly as relativity’s gravitational radiation requires, because the stars are extended bodies with spins and tides to model. But it is a test of a different quantity — the advance of an orbit rather than its decay — in a system with nothing exotic about it except two stars lying on their sides.

How it shows in the eclipses #

The discrepancy was found in timing, and it is large enough in timing to have been beyond dispute.

In an eccentric orbit the two eclipses are not half a period apart: the secondary comes early or late depending on where periastron lies relative to the line of sight, by an amount that depends on $e\cos\omega$. As the periastron turns, the interval changes. Over seventy years the prediction would have moved it by about two hours and the observed rate by half an hour. Eclipses of DI Herculis are timed to a minute or two, so the difference between the two is dozens of times the measurement error, and accumulating it needs only patience. The figure uses only the leading term in the eccentricity, which is crude at 0.49, but the scale is right.

Tilted stars are not rare #

DI Herculis was a surprise because the stars of close binaries were assumed to be aligned: they formed together, from one rotating cloud, and tides in a close pair should drive any misalignment away. Both assumptions turn out to be weaker than they seemed.

The tidal argument depends on the stars’ structure. In a star with a deep convective envelope, like the Sun, tides are dissipated efficiently in the turbulent envelope and spins are realigned quickly. In hot stars with radiative envelopes, like the components of DI Herculis, tidal dissipation is far weaker, and an initial misalignment can survive for the star’s whole life. The same division separates the planets that orbit hot stars on tilted orbits from those around cool stars that are aligned, and DI Herculis is its binary version.

The formation argument depends on how the pair was assembled. Stars that form by the fragmentation of one rotating core share its spin; stars that are captured, or that form in separate cores and later become bound, need not. A third star, now gone or too faint to see, can tilt the orbit of an inner pair through the slow exchange of angular momentum between an inner and an outer orbit, leaving the spins where they were and the orbit somewhere else. Each of these leaves a tilt that a radiative star cannot remove.

Surveys of the Rossiter–McLaughlin effect in eclipsing binaries since 2009 have found most systems aligned and a minority misaligned, the misaligned ones tending to be hot stars, whose tides are weak, in orbits wide or eccentric enough for a third body’s influence to matter. Several of the binaries whose apsidal motions disagreed with models have been revisited in the same way.

What the calculation leaves out #

The rotational term is computed for each star as a flattened body acting on the orbit through its quadrupole, with the tilt entering through $P_2(\cos\beta)$; a complete treatment also follows the precession of the orbital plane and of each spin axis about the total angular momentum, which changes the observable apsidal rate by a geometric factor of order one and makes the spins themselves precess over thousands of years. The internal structure constant $k_2$ is set so that the aligned distortion equals the published models’ 1.93 degrees a century, rather than computed from a stellar model here, and the stars’ rotation speeds are taken as uniform with latitude. The observed rate is drawn as a band because the published values differ; a single value with a small error would turn the inclination range in the fourth figure into a measurement with a stated uncertainty, and the measurement waits on it.

What does not depend on these is the sign. A star tipped beyond 54.7 degrees subtracts from its binary’s apsidal motion, and both stars of DI Herculis are tipped beyond it.

Still open: what tipped them #

The misalignment that explained the precession is itself unexplained. DI Herculis’s stars are young and radiative, so the tilt is likely primordial or imposed early, but the two candidate mechanisms predict different things. A pair that formed misaligned from turbulent fragmentation should have spins unrelated to each other as well as to the orbit; a pair tilted by a third body should have both spins at similar angles to the orbit and to each other. The measured projected angles, 72 and 84 degrees on opposite sides, are consistent with either. A search for the third body by its effect on the eclipse timings — the light-travel term that shows a distant companion — and a measurement of the spins’ inclinations, which the apsidal motion now partly supplies, would separate the two. Either way, the system that was once cited against relativity has become one of its better tests: once the stars’ tilt is included, the general-relativistic contribution of 2.34 degrees a century is required to make the books balance.