Two damping times, one crossing, and the slope that separates them

The cut-off period in a cluster is a clock, and it was read as one within a few years of the first measurement. What it has never settled is the mechanism, and the reason is a matter of counting parameters.

Two theories of how a star dissipates a tide are on the table. Each predicts a damping timescale rising as a power of the orbital period, and each has a normalisation nobody can compute from first principles. A cluster supplies one number — the period at which the boundary stands. One number cannot choose between two curves with one free parameter each.

The two mechanisms #

The equilibrium tide is the picture everybody draws. The companion raises a bulge on the star, the star’s own rotation and the orbit carry the bulge away from the line joining the two bodies, and the lag produces a torque. The tide is a difference of forces rather than a force, and the bulge it raises is the star’s hydrostatic response to it. Dissipation happens because the fluid displaced by the bulge has to be moved against something, and in a star with a convective envelope the something is turbulent viscosity: the convective eddies scatter the tidal flow and convert its energy into heat.

That mechanism has a frequency dependence of its own. When the tidal period is shorter than an eddy’s turnover time the eddy cannot respond over a full tidal cycle and its contribution to the viscosity is reduced, by a factor that Zahn took as the ratio of the two timescales. Carrying that through gives a damping time rising as $P^{16/3}$.

The dynamical tide is a different physical process entirely. The tidal forcing excites internal gravity waves at the boundary of the convective envelope, the waves propagate into the radiative interior, and they deposit their angular momentum where they are damped — by radiative diffusion, or by breaking near the centre. The dissipation is not local to the bulge; it is at the other end of the star.

That mechanism is the dominant one in stars with radiative envelopes, where there is no turbulence to do the equilibrium tide’s work, and its damping time rises as $P^7$.

Why one boundary cannot choose #

Both timescales have the form $\tau = \tau_0(P/P_0)^{\,n}$ with $\tau_0$ unknown, and each theory’s $\tau_0$ depends on quantities — the convective turnover time, the overshoot depth, the wave damping efficiency — that are computed from stellar models and are uncertain by factors.

A cluster gives $\tau(P_{\rm cut}) = t_{\rm age}$ at one period. That is one equation and each theory has one unknown, so each theory fits exactly, and neither is tested.

The normalisation absorbs the disagreement completely. The figure above draws the two normalised to agree at one point precisely because that is the situation the measurement is in.

What a free normalisation is worth #

It is worth being explicit about the counting, because “one measurement cannot constrain two parameters” is a statement that needs the parameters named.

Each theory has the form $\tau = A\,P^{n}$ with $n$ predicted and $A$ not. A cluster supplies a pair $(P_{\rm cut}, t)$, and requiring $\tau(P_{\rm cut}) = t$ fixes $A$ exactly. Two clusters supply two pairs and over-determine one theory, which is when a test becomes possible.

So the number of clusters needed is two, and the difficulty is not that the data are insufficient in principle but that the difference the second cluster has to resolve is small. With boundaries at 6.5 and 12.5 days at 0.125 and 4 Gyr, one theory predicts the second from the first as $6.5 \times 32^{3/16} = 13.1$ days and the other as $6.5 \times 32^{1/7} = 11.4$. A measurement good to fifteen per cent distinguishes them and nothing worse does.

That is roughly where the measurements sit, which is why the question is live rather than closed or hopeless. Adding clusters helps as the square root, and adding binaries within a cluster helps the same way, and both have been done for forty years.

How much a survey of clusters buys #

The natural response is to measure the boundary in many clusters of different ages and fit the trend rather than the position. The trend is what the two theories differ in: $t^{3/16}$ against $t^{1/7}$, which is $0.1875$ against $0.1429$.

Over a factor of eighty in age — 0.125 Gyr to 10 Gyr, which is the full range of clusters with enough binaries to measure a boundary — the two predict boundary movements of $80^{0.1875} = 2.24$ and $80^{0.1429} = 1.85$.

So the whole discriminating power of a cluster survey is the difference between a factor of 2.24 and a factor of 1.85 in boundary period, and each boundary is located to perhaps twenty per cent from a few dozen systems. The measurement is possible and is not comfortable, and it explains why a result first obtained in the 1980s became an open question again only when the samples grew from tens per cluster to hundreds.

The scatter is the reason everything above is difficult. Each cluster’s boundary is inferred from a few dozen points with a real spread in initial eccentricity, in primary mass and in evolutionary state, and the inference returns a number with an error bar of perhaps twenty per cent. Four such numbers across a factor of thirty in age are what the two theories are being separated with.

The discriminator that is not an age #

There is a second axis, and it is the one the theories name in advance rather than one imposed on them.

The equilibrium tide requires a convective envelope, because turbulent viscosity is what dissipates the flow. A star above about 1.3 solar masses has a radiative envelope and a convective core — the arrangement is inverted — and there is no turbulence at the surface to do the work.

So the equilibrium tide switches off above that mass, and the dynamical tide takes over. The theories predict a change in behaviour at a place they specify before the measurement is made, and that is a far stronger test than fitting a slope.

The prediction has a direction too. The dynamical tide is the less efficient of the two for a solar-type star, so the boundary should sit at a shorter period above the transition mass than the extrapolation from below would suggest.

Where the uncertainty in the exponent comes from #

That last point deserves its own statement, because it is the reason the two power laws are not as well separated as they look.

The equilibrium tide’s exponent depends on how the convective viscosity behaves when the tidal period is shorter than the eddy turnover time. Zahn’s treatment reduces the viscosity by the ratio of the two timescales, giving one power. An alternative, argued from the response of an eddy to a rapidly oscillating shear, reduces it by the square of the ratio, giving a different power.

The two treatments differ by a factor of two in exponent and have been argued over for forty years, without a resolution from either theory or from laboratory experiment, because the relevant regime — turbulence at Reynolds numbers of $10^{10}$, forced at a frequency above the eddy turnover — is not reachable in any laboratory and is only marginally resolvable in a simulation.

So the discrimination between the two mechanisms is being attempted with one of them carrying an exponent uncertain by more than the difference between them. That is an uncomfortable position and it is the honest one.

The third thing the boundary depends on #

Both theories predict a damping time that depends on the stellar structure, and the stellar structure changes as a star evolves — which introduces a dependence neither power law contains.

A star’s convective envelope deepens as it leaves the main sequence, so the equilibrium tide’s efficiency rises steeply with evolutionary state. A binary whose primary has become a subgiant circularises far faster than one whose primary is still on the main sequence, at the same period and the same mass.

That effect is large — orders of magnitude, not per cent — and it is visible in the data. The circularised binaries in an old cluster include systems at periods far outside the boundary read from main-sequence members, and those systems have evolved primaries.

So the boundary is not a single number even for one cluster, and the measurements that treat it as one are implicitly restricting to main-sequence primaries. That restriction is what makes the comparison across clusters meaningful, and it also throws away the subset with the strongest signal.

The evolved systems are arguably the better measurement. Their circularisation happened recently and quickly, so the accumulated history is short and the inference from a population to a rate is less fraught — which is the same argument that makes a rate measurement preferable to a population one in general.

A boundary that is fitted, and exponents that are predicted #

The cut-off periods are measured, in the sense that a boundary is fitted to a scatter of eccentricities against periods for cluster members. The fit is not a trivial one — there is no sharp edge in a sample of forty systems, and the boundary’s location is inferred from a likelihood that models the eccentricity distribution on both sides.

The cluster ages come from main-sequence turn-offs and owe nothing to the tide, which is the whole reason the stellar measurement is possible at all and the planetary one is not.

The exponents are not measured; they are predicted. The figures draw them at their theoretical values and normalise them to agree at one point, which is exactly what the measurement can constrain and no more.

What has actually been extracted from the data is a normalisation, expressed as a convective viscosity efficiency, and the value required is about a factor of fifty larger than stellar models produce. That discrepancy has been known since the 1990s and is the strongest evidence that something in the picture is wrong rather than merely uncertain.

Neither mechanism is drawn #

Neither mechanism is drawn. The figures show two power laws and the ages at which they cross given thresholds; what actually happens inside a star — an eddy scattering a tidal flow, a gravity wave breaking near the centre — is not a curve on a plot and is not visible in any observation either.

The boundary is not sharp. A circularisation timescale equal to the age means the eccentricity has fallen by a factor of $e$, not to zero, so systems near the boundary are partly circularised and the wall has a real width. The width is comparable to the precision with which the boundary is located.

And the pre-main-sequence phase is left out entirely. A binary spends its first few million years with both components larger than they will be on the main sequence, and $(R/a)^5$ — the same fifth power the planetary version of this measurement turns on — is enormous during that time — so a substantial part of the circularisation may happen before the cluster’s clock starts. That contribution is a normalisation rather than a slope, so it does not affect the discrimination, and it does affect every attempt to infer a viscosity from the normalisation.

Why the normalisation discrepancy matters more than the exponent #

Everything above concerns which of two exponents is right, and there is a larger problem sitting underneath both.

Fitting the observed boundaries with the equilibrium tide requires a convective viscosity about fifty times larger than what stellar convection models produce. That is not a refinement; it is a statement that the mechanism as written cannot deliver the observed dissipation.

Several resolutions have been proposed. The viscosity reduction at high tidal frequency may be milder than either standard treatment assumes. The pre-main-sequence phase, when both stars are much larger, may contribute most of the circularisation and remove the need for an efficient main-sequence tide entirely. Or a third mechanism may be doing the work.

The second of those is the most awkward possibility of all, because if most of the circularisation happens before the cluster’s clock starts, the boundary is not a clock at all — it is a record of a formation-epoch process, and its advance with age is a smaller residual on top.

The evidence against that reading is the advance itself. A boundary set entirely before the main sequence would not move, and it does move. But it moves by less than a factor of three across the whole observed age range, so a large pre-main-sequence contribution plus a weak main-sequence one fits about as well as a strong main-sequence tide, and separating the two is the same underdetermined problem in another guise.

What would settle it #

Three things would, and all three are being pursued.

Boundaries in many clusters, measured the same way. The trend across age is weak but real, and the samples are now large enough that the statistical error on each boundary is no longer the limiting term. What limits it now is that different clusters’ boundaries have been measured with different criteria.

The transition across the convective–radiative boundary. A single cluster split by primary mass gives two boundaries with the same age, which removes the age from the comparison entirely. The prediction is specific and the measurement is hard, because the massive binaries in a cluster are few.

A measurement of the same boundary in a population of known composition. Both mechanisms depend on the star’s internal structure, and metallicity changes the depth of a convective envelope at fixed mass. So a metal-poor cluster and a metal-rich one of the same age should show boundaries displaced by an amount the theories disagree about, and globular clusters supply the metal-poor end — at the cost of being dynamically processed, so that a close binary in one may have been made by an encounter rather than born.

And an observation of a circularising system caught in the act. A binary whose eccentricity is measurably changing would give the rate directly, and the rate is what the theories differ about. The measurement needs an eccentricity precision of a part in $10^4$ over a decade, which eclipse timing can reach for the right system.

Still open: whether either mechanism is right #

The normalisation problem is the reason to doubt both. Fitting the observed boundaries with the equilibrium tide requires a convective viscosity about fifty times larger than stellar convection models produce; fitting them with the dynamical tide requires a wave damping efficiency at the upper end of what is plausible.

Neither is impossible and neither is comfortable, and the possibility that both are incomplete has support from an unrelated direction. Inertial waves — oscillations restored by the Coriolis force, which exist only in a rotating convective region — provide a third dissipation channel that neither classical theory includes, and their efficiency depends resonantly on the ratio of the tidal frequency to the rotation frequency.

That resonant dependence would make the effective dissipation vary wildly between systems with similar parameters, which is consistent with the scatter in individual measurements and inconsistent with the clean power laws drawn above. A quality factor quoted without a period is half a number is the general form of that caution, and a resonant mechanism is the case where it bites hardest.

From here: the measurement that would stop the inferring #

The obvious continuation is the one measurement that removes the boundary entirely: a direct rate. Tidal theory predicts $de/dt$ and every argument here has been about inferring it from a population’s accumulated history instead, which is the weakest possible use of a prediction about a rate.

Two systems are now close to supplying one — an eclipsing binary whose eccentricity has been monitored for four decades, and a hot Jupiter whose orbital period is measurably shrinking. Neither measures the quantity the cluster clock is calibrated in, and both measure a rate, which is the thing a population never can.