Astronomy & Celestial Mechanics
the sky, computed
Orbits

A cut-off period that is an age

Plot eccentricity against orbital period for the binary stars of one cluster and the picture has a wall in it. Below a certain period every orbit is circular; above it, the original spread survives untouched. The wall moves outward as the cluster ages, and where it stands is a measurement of how stars dissipate a tide.

Assumes Tides, Binary stars and Orbital averages.

16 min read 8 figures The sky as a clockSmall perturbations, large consequences

A tide takes energy out of an orbit and does not take angular momentum out of it. That is why the Earth’s day lengthens while the Moon recedes rather than the pair simply losing both. That asymmetry is the whole of what follows.

An eccentric orbit and a circular one with the same angular momentum do not have the same energy: the eccentric one has more. So a process that dissipates orbital energy while conserving angular momentum drives an orbit towards a circle of the same angular momentum, and it stops when it gets there. That is circularisation, and it is not a slow drift towards a vague endpoint — it is a decay towards a well-defined final state with a rate that depends ferociously on how close the two bodies are.

A period below which every orbit is round. Orbital eccentricity against period for binaries in four clusters of 0.125, 0.625, 6, 4 billion years, with the eccentricities drawn from one seeded distribution and then damped by exp(−age/τ), where τ rises as the 5.333 power of the period. Each cluster shows the same thing: below a boundary period nothing survives eccentric, above it the original distribution is untouched, and there is almost nothing in between because the timescale is so steep. The boundary is a clock. It moves as the three-sixteenths power of the age, which the figure checks against the drawn curves, and the calibration puts it at 6.5 days at 125 million years, 8.8 at 625 million and 12.5 at four billion — against measured cut-offs near 7.2, 8.5 and 12.5 days in the Pleiades, the Hyades and M67. The boundaries are also read back off the plotted points rather than trusted, and required to move outward with age. This is the cleanest measurement of tidal dissipation in ordinary stars that exists, and its cleanliness comes from the ages: a cluster's age is read off its main-sequence turn-off and owes nothing whatever to the tide being measured.
Fig. 1 Eccentricity against orbital period for binaries in four clusters, with the eccentricities drawn from one distribution and then damped by their own periods and their cluster’s age. Below a boundary nothing survives eccentric; above it the original distribution is untouched; and there is almost nothing between, because the timescale is so steep in period. The boundary moves as the three-sixteenths power of the age, and it is read back off the plotted points rather than trusted.

Why the transition is a wall and not a slope

The steepness is what makes this useful, and it comes from three separate powers of the separation multiplying together.

A tide is a difference of gravity rather than gravity itself, and the tide raised on a star by its companion has a height going as a−3a^{-3}, since it is a difference of gravity. The torque and the dissipation involve the tide’s height times the gradient of the perturbing field, another a−3a^{-3}. And the fraction of the orbit’s energy that any given amount of dissipation represents rises as the orbit shrinks. Assembling the standard equilibrium-tide calculation for a star with a convective envelope gives

1τcirc∝(Ra)8 1P,\frac{1}{\tau_{\rm circ}} \propto \left(\frac{R}{a}\right)^{8}\,\frac{1}{P},

which, converting aa to period through Kepler’s third law, is a timescale rising as P16/3P^{16/3}.

A sixteen-thirds power is very steep. Doubling the period multiplies the circularisation time by forty. That is why a population of one age shows a wall rather than a gradient: there is a period at which the timescale equals the age, and within a factor of 1.15 either side of it the timescale changes by a factor of two and a half. Anything inside is round; anything outside is untouched; the transition region is narrower than the observational scatter.

The clock

The boundary is a function of two things: the age, and how efficiently the star turns tidal flow into heat. Fix the second and it measures the first; fix the first and it measures the second. Clusters supply the first.

A cluster’s age comes from where its stars leave the main sequence — a measurement that involves stellar structure, opacity and convection, but nothing at all about tides. That independence is what makes the comparison worth making: the age and the tidal boundary come from completely disjoint physics, and if the boundary tracks the age across a factor of thirty in age, the mechanism is right.

It does. The Pleiades at 125 million years show a boundary near 7 days; the Hyades and Praesepe at 625 million show 8.5; M67 at four billion shows about 12.5; and the ratios are what a three-sixteenths power law requires.

Everything close in is circular, and nothing else has to be. Orbital eccentricity against period for fifteen real planets, with the tidal circularisation boundary computed from τ_e = (2/21)(Q′/n)(M_p/M⋆)(a/R_p)⁵ for a Jupiter with Q′ = 1e+5 and an age of 5 billion years. It falls at 8.6 days, and the reason it is a wall rather than a slope is the fifth power: at half the period the timescale is 91 times shorter. Nothing inside it has a measurable eccentricity, and outside it eccentricities run to 0.95 — which is the number to hold on to, because a planet on a 0.93 orbit at 111 days comes within 0.030 AU of its star at periastron, closer than Mercury, and is being circularised as it is observed.
Fig. 2 The same fifteen planets with the tidal quality factor set to 10510^5 rather than 10610^6 — a factor of ten in the one number the whole calculation is uncertain in. The boundary moves from 5.0 days to 8.6: a decade in Q′Q' is a factor of 1.7 in the cut-off period, because the period enters the circularisation time as a power near five and a third and the inversion takes the root. That insensitivity is what makes the method usable and what caps what it can deliver, and it is the same cube-root-like flattening that the dynamical parallax and the Jeans mass both live on.

What actually has to be measured

A cut-off is a statement about a population, and every point in that population is a hard measurement.

An orbital period comes from radial velocities and is easy — a double-lined pair is the only kind of star whose mass is known. An eccentricity is harder: it comes from the shape of the radial velocity curve, and for small eccentricities the shape differs from a sine wave only slightly. Measuring e=0.02e = 0.02 requires velocities good to a fraction of a per cent of the orbital amplitude, over a full period, and the systematic errors that matter are the ones that vary on the orbital period itself. There is a subtler problem, which is what counts as circular. A measured eccentricity is a positive quantity even when the true one is zero, because the fit will always find some departure from a sinusoid in noisy data. So a sample of genuinely circular orbits produces a distribution of small positive eccentricities, and the boundary has to be defined as a statistical statement about that distribution rather than as the largest circular period.

The number that comes out, and why it is uncomfortable

Inverting the observed boundaries for the dissipation gives an answer several times larger than the standard theory produces.

The standard theory is Zahn’s: in a star with a convective envelope, the tidal flow is damped by the turbulent viscosity of the convection. That viscosity can be estimated from the convective velocity and the size of the largest eddies, both of which come out of a stellar model. The estimate is credible and it is too small — by a factor of somewhere between a few and a hundred, depending on whose analysis is used.

There is a well-identified reason it might be too small, and it has a nice physical statement. Turbulent viscosity works when the eddy turnover time is short compared with the tidal forcing period. When the forcing is faster than the eddies, the largest eddies cannot respond within one cycle, and their contribution to the effective viscosity is reduced — but by how much is contested. One prescription reduces it by the ratio of the two times, another by its square, and the two differ by orders of magnitude in exactly the regime that matters. The alternative mechanism is a dynamical tide rather than an equilibrium one. Instead of the star deforming quasi-statically and the deformation being dragged, the tidal forcing resonantly excites internal gravity waves in the radiative interior, which propagate, break, and deposit their angular momentum. That mechanism is far more efficient, it has a completely different dependence on period, and it operates in stars with radiative interiors where the turbulent mechanism does not exist at all — so measuring the boundary in binaries of different spectral types is the way to tell them apart.

Everything close in is circular, and nothing else has to be. Orbital eccentricity against period for fifteen real planets, with the tidal circularisation boundary computed from τ_e = (2/21)(Q′/n)(M_p/M⋆)(a/R_p)⁵ for a Jupiter with Q′ = 1e+6 and an age of 1 billion years. It falls at 3.5 days, and the reason it is a wall rather than a slope is the fifth power: at half the period the timescale is 91 times shorter. Nothing inside it has a measurable eccentricity, and outside it eccentricities run to 0.95 — which is the number to hold on to, because a planet on a 0.93 orbit at 111 days comes within 0.030 AU of its star at periastron, closer than Mercury, and is being circularised as it is observed.
Fig. 3 The same population a billion years old instead of five. Circularisation goes as a high power of the separation, so the cut-off period moves inward as the square root of the fifth root of the age — slowly, which is what makes it usable as a clock and also what makes it a demanding measurement. A factor of five in age moves the cut-off by a factor under two in period, so distinguishing a one-billion-year cluster from a five-billion-year one requires the cut-off to be located to better than twenty per cent.
Everything close in is circular, and nothing else has to be. Orbital eccentricity against period for fifteen real planets, with the tidal circularisation boundary computed from τ_e = (2/21)(Q′/n)(M_p/M⋆)(a/R_p)⁵ for a Jupiter with Q′ = 1e+6 and an age of 10 billion years. It falls at 5.9 days, and the reason it is a wall rather than a slope is the fifth power: at half the period the timescale is 91 times shorter. Nothing inside it has a measurable eccentricity, and outside it eccentricities run to 0.95 — which is the number to hold on to, because a planet on a 0.93 orbit at 111 days comes within 0.030 AU of its star at periastron, closer than Mercury, and is being circularised as it is observed.
Fig. 4 And what doubling the age does. Ten billion years rather than five moves the boundary from 5.0 days to 5.9 — eighteen per cent for a factor of two in time, which is the same steep power read the other way. A cut-off period is an extremely blunt clock and an extremely sharp discriminator: it barely distinguishes a five-billion-year system from a ten-billion-year one, and it separates circular orbits from eccentric ones almost perfectly.

Spin first, then shape

There is a second tidal timescale in the same system, and it is much shorter.

The tide that circularises the orbit also synchronises the stars’ rotation with it. Both come from the same lag, but they draw on reservoirs of very different size: the orbit’s angular momentum is enormous compared with a star’s spin angular momentum, so the same torque changes the spin quickly and the orbit slowly. The ratio of the two timescales is roughly the ratio of the two angular momenta, which for a typical binary is a factor of tens.

The observational consequence is a band of periods in which the stars are already rotating in step with the orbit while the orbit is still eccentric — which is itself a check on the picture, since it is exactly what the ordering of the two reservoirs predicts.

It also removes a possible objection. A tide only dissipates if the body it is raised on is turning at a different rate from the orbit; a perfectly synchronised, perfectly circular pair has a static bulge and nothing to dissipate. So it might be thought that synchronisation, arriving first, would shut off the circularisation before it finished. It does not, because an eccentric orbit has no single angular velocity to synchronise with: the companion’s angular speed at periastron is larger than at apastron by a factor of (1+e)2/(1−e)2(1+e)^2/(1-e)^2, so a star spinning at any fixed rate is out of step for most of the orbit. The tide therefore keeps working, and it stops only when the eccentricity that produced the mismatch is gone. Synchronisation of an eccentric orbit does not settle at the orbital mean motion. The tide is strongest at periastron, where the companion is closest and moving fastest, so the equilibrium spin is faster than the mean — pseudo-synchronous, at a rate weighted towards the periastron angular velocity. Binaries in the band between the two boundaries are observed to rotate at pseudo-synchronous rather than synchronous rates, which is a detail that only makes sense if the tide is being raised where the theory says it is.

The objection that will not go away

Everything above reads the cut-off as a clock: the boundary advances with age because circularisation has had longer to work. There is a competing reading in which the boundary is set almost entirely before the cluster reaches the main sequence, and it has never been decisively refuted.

The argument rests on the eighth power of the radius. A pre-main-sequence star of one solar mass is several times its eventual size and fully convective, so the tidal dissipation in it exceeds its main-sequence value by a factor that is not small — the radius ratio raised to the eighth is thousands even for a modest inflation, and the convective envelope is the whole star rather than an outer layer. A binary spends only a few tens of millions of years in that state, but the rate during it is so much larger that the integrated effect can exceed everything that follows.

If that is right, then a cluster arrives on the main sequence with its cut-off period already established, and the boundary should be nearly the same in a hundred-million-year-old cluster and a four-billion-year-old one. The observations do show a boundary that moves, and the movement is the reason the clock reading is preferred — but the observed sequence is 7, 8.5 and 12.5 days across a factor of thirty in age, which is a factor of 1.8. A three-sixteenths power law predicts a factor of 2.2 over that range, and a model in which most of the circularisation happened before the main sequence, with a slow drift afterwards, predicts something in between. The data do not comfortably separate them.

What would separate them is a cluster young enough that main-sequence circularisation has had essentially no time. If the pre-main-sequence account is right, such a cluster should already show a cut-off near seven days; if the clock account is right, it should show none at all. Binaries in the youngest star-forming regions do show circular orbits at periods of a few days at ages of a few million years, which is evidence in the first direction, and the samples are small enough that the counter-argument is about statistics rather than about physics.

The point is worth making because the clock reading is the one that gets quoted. A boundary that moves with age is a satisfying result and it is not, on its own, proof that the moving is what set the boundary.

Everything close in is circular, and nothing else has to be. Orbital eccentricity against period for fifteen real planets, with the tidal circularisation boundary computed from τ_e = (2/21)(Q′/n)(M_p/M⋆)(a/R_p)⁵ for a Jupiter with Q′ = 1e+7 and an age of 5 billion years. It falls at 3.0 days, and the reason it is a wall rather than a slope is the fifth power: at half the period the timescale is 91 times shorter. Nothing inside it has a measurable eccentricity, and outside it eccentricities run to 0.95 — which is the number to hold on to, because a planet on a 0.93 orbit at 111 days comes within 0.030 AU of its star at periastron, closer than Mercury, and is being circularised as it is observed.
Fig. 5 And the same age with the dissipation ten times weaker. The cut-off period depends on the tidal quality factor as well as on the age, and Q′Q' for a star is known to about an order of magnitude — so a cut-off measured perfectly still gives an age uncertain by the same factor unless Q′Q' is fixed independently. Calibrating on clusters whose ages come from their turn-offs is what fixes it, which makes this a clock read against another clock rather than a primary one.
Everything close in is circular, and nothing else has to be. Orbital eccentricity against period for fifteen real planets, with the tidal circularisation boundary computed from τ_e = (2/21)(Q′/n)(M_p/M⋆)(a/R_p)⁵ for a Jupiter with Q′ = 1e+6 and an age of 5 billion years. It falls at 5.4 days, and the reason it is a wall rather than a slope is the fifth power: at half the period the timescale is 91 times shorter. Nothing inside it has a measurable eccentricity, and outside it eccentricities run to 0.95 — which is the number to hold on to, because a planet on a 0.93 orbit at 111 days comes within 0.030 AU of its star at periastron, closer than Mercury, and is being circularised as it is observed.
Fig. 6 The same planets around a star of six-tenths of a solar mass. The boundary rises from 5.0 days to 5.4, because a lighter star gives a longer orbital period at any given separation and the same tidal damping therefore reaches further out in period. The shift is small and the direction matters: the cut-off is a property of the pair and not of the planet, so a survey that mixes host masses is reading a slightly different boundary for each system and averaging over the difference.

Circularising is not free

One more consequence deserves saying, because it turns this from a statement about orbits into a statement about stars.

The orbital energy that disappears has to go somewhere, and it goes into the stars as heat. For a close eccentric binary that is a substantial luminosity — comparable, in the extreme cases, to what the star produces by fusion — and it is deposited in the envelope rather than in the core, which is not where a star is built to receive energy.

The clearest case in the collection is not a binary at all. Io is held eccentric by a resonance and heated by the tide that would otherwise circularise it, and its heat flow is the direct measurement of a dissipation rate that in the stellar case has to be inferred from a population. The distinction is that Io has something maintaining its eccentricity while a binary does not, so the moon reaches a steady state and the binary simply arrives at a circle and stops.

For stars, the deposited energy is the reason circularisation is self-limiting in a second sense: an inflated, heated envelope has a larger radius, and the rate goes as the eighth power of the radius over the separation, so a star that is being tidally heated dissipates faster still. Whether that runaway matters depends on how quickly the envelope can radiate the extra energy away, and for main-sequence stars it can, comfortably. For a star already near the top of the main sequence it cannot, which is one reason the tightest massive binaries are unstable in ways that the tidal calculation alone does not predict. The same eighth power is what makes the cut-off a wall rather than a slope, so the heating and the sharpness of the boundary are the same fact stated twice — a pair that circularises quickly enough to be inside the wall is a pair that was heated hard while it did so.

A period below which every orbit is round. Orbital eccentricity against period for binaries in four clusters of 0.125, 0.625, 6, 4 billion years, with the eccentricities drawn from one seeded distribution and then damped by exp(−age/τ), where τ rises as the 5.333 power of the period. Each cluster shows the same thing: below a boundary period nothing survives eccentric, above it the original distribution is untouched, and there is almost nothing in between because the timescale is so steep. The boundary is a clock. It moves as the three-sixteenths power of the age, which the figure checks against the drawn curves, and the calibration puts it at 6.5 days at 125 million years, 8.8 at 625 million and 12.5 at four billion — against measured cut-offs near 7.2, 8.5 and 12.5 days in the Pleiades, the Hyades and M67. The boundaries are also read back off the plotted points rather than trusted, and required to move outward with age. This is the cleanest measurement of tidal dissipation in ordinary stars that exists, and its cleanliness comes from the ages: a cluster's age is read off its main-sequence turn-off and owes nothing whatever to the tide being measured.
Fig. 7 The same four clusters with eighty binaries drawn in each rather than thirty-four. The boundary in every cluster is where it was — it is set by the damping law and not by the sample — and what improves is how well it can be located: with thirty-four systems the shortest eccentric orbit in a cluster is a single point and its position is a draw, and with eighty it is the edge of a populated region. The measurement is of an edge in a distribution, so its precision goes as the density of points near the edge, which is why this is done in clusters rich enough to supply them.

The population has been stirred

There is a second complication in using a cluster, and it comes from the cluster rather than from the tide.

A binary in a dense stellar system does not evolve in isolation. Encounters with other members exchange energy and angular momentum with the orbit, and a wide binary’s eccentricity can be pumped back up by a passing star long after the tide has damped it. The rate at which that happens depends on the local stellar density and on the binary’s own separation, and it grows with the orbit’s size for the same reason a wide binary is easier to disrupt.

So the observed distribution in the period–eccentricity plane is not the tidal one alone. It is the tidal one with a stirring term added, and the stirring acts preferentially on exactly the systems whose eccentricities the boundary is defined against. In an open cluster the densities are low enough that the term is small; in the core of a globular cluster it is not, and circularisation cut-offs measured in globulars come out at periods that do not fit the age sequence.

That is usually stated as globular clusters being unsuitable for the measurement, which is true and slightly misses the point. The systems that survive in a dense environment have been selected as well as perturbed: soft binaries are destroyed and hard ones are hardened, so the surviving population’s period distribution is not the one the cluster formed with, and a boundary drawn across it is a boundary in a censored sample.

There is a related difficulty in how the dissipation is usually quoted. The efficiency is conventionally expressed as a tidal quality factor — a dimensionless number standing for the fraction of the tidal energy lost per cycle — which is convenient and hides everything. A single QQ presumes the dissipation is independent of the forcing frequency, and every mechanism proposed for these stars depends on frequency strongly: turbulent viscosity because the eddies must keep up with the forcing, and resonant gravity waves because a resonance is a statement about frequency and nothing else. Quoting one number per star therefore averages over the variable the competing theories disagree about, which is why two groups can fit the same cut-off periods with the same QQ and mean different physics by it.

The habit persists because it makes results comparable, and comparability is worth something even when it is comparing summaries. But a quality factor inferred from a binary at a ten-day period and one inferred from a hot Jupiter at three days are quantities evaluated at forcing frequencies a factor of three apart, in bodies with entirely different internal structure, and setting them side by side in a table invites a comparison the numbers cannot support. Where the two have been compared carefully the disagreement is several orders of magnitude, and the honest reading of that is not that one measurement is wrong but that a single frequency-independent number was never the right thing to measure.

Everything close in is circular, and nothing else has to be. Orbital eccentricity against period for fifteen real planets, with the tidal circularisation boundary computed from τ_e = (2/21)(Q′/n)(M_p/M⋆)(a/R_p)⁵ for a Jupiter with Q′ = 1e+6 and an age of 5 billion years. It falls at 6.0 days, and the reason it is a wall rather than a slope is the fifth power: at half the period the timescale is 91 times shorter. Nothing inside it has a measurable eccentricity, and outside it eccentricities run to 0.95 — which is the number to hold on to, because a planet on a 0.93 orbit at 111 days comes within 0.030 AU of its star at periastron, closer than Mercury, and is being circularised as it is observed.
Fig. 8 The same calculation for a heavier, larger primary. Both the mass and the radius enter the circularisation rate, and the radius enters as a high power — so a star half again as large circularises its companions much faster and its cut-off sits at a longer period at the same age. That is why the method is applied within a single spectral range rather than across a cluster’s whole main sequence, and why the stars it works best on are the ones most like the Sun.

Where the ladder goes

The obvious extension is downward in mass. The same measurement made on planets rather than on stellar companions probes the dissipation inside the planet rather than the star, since a planet’s tidal response is much larger, and the observed circularity of hot Jupiters is the corresponding wall. The awkwardness there is that a planetary system has no cluster to date it.

The harder direction is the mechanism. A cut-off period is one number per cluster, and the theories being separated differ in how the rate depends on period, on stellar mass and on evolutionary state. Distinguishing an equilibrium tide from a dynamical one requires the shape of the boundary rather than its position, which means measuring eccentricities for hundreds of binaries per cluster instead of tens — and that is now becoming possible, which is why a measurement first made in the 1980s is an active one again.

Where this goes next

Two radial-velocity curves, and one mass ratio. The line-of-sight velocity of each star through one orbit of AI Phoenicis. Both curves are computed from the two masses and the period; what a spectrograph delivers is the reverse. The ratio of the amplitudes is the inverse ratio of the masses — 48.2 to 50.3 kilometres a second, so the heavier star moves more slowly — and the sum of the amplitudes with the period gives the mass sum, 2.437 solar masses, once the inclination is known from the eclipses. The only stars whose masses are known A star's mass cannot be measured by looking at it. It can be measured by watching two stars pull on each other, and if the pair also eclipses, the same observations give both radii as well — with no stellar model anywhere in the chain. A few hundred such systems calibrate everything else. Io's measured heat needs k₂/Q = 0.016, and Enceladus's needs 0.011. Tidal surface heat flux against orbital eccentricity, from Ė = (21/2)(k₂/Q)GM_p²R⁵ne²/a⁶ evaluated at each satellite's own orbit, drawn at a common k₂/Q of 0.015. Every line has slope 2 because the dissipation is quadratic in e and nothing else on this axis varies. The filled marks are each body at its actual eccentricity; the two ringed ones are the bodies with a measured surface heat flux, and they are the only points here that are observations. Solving each of those for k₂/Q gives 0.016 for Io and 0.011 for Enceladus — within a factor of 1.5 of one another, for a warm silicate body and a 500-kilometre ball of ice, which ought to be a coincidence and is instead the sharpest problem in the subject: nothing about Enceladus's ice can plausibly dissipate at 0.011, and the number is what the heat requires all the same. The dashed line is the Earth's measured surface heat flux, 0.087 W/m², which Io exceeds by a factor of 28 — the most volcanically active body in the solar system is the fourth largest moon of the fifth planet, and the reason is entirely in the orbit. A moon heated by not being allowed to relax Tidal dissipation goes as the square of an eccentricity that tides themselves destroy, so a moon radiating tidal heat is spending something it cannot have saved. Io's would be gone in a hundred and forty thousand years, and the resonance that keeps putting it back is the reason there are volcanoes. Three clusters, three ages, one diagram. Isochrones for populations of 100 Myr, 1 Gyr, 12 Gyr, each drawn as the main sequence up to its own turnoff and then the post-main-sequence track of the turnoff mass. The turnoffs are at 5.52 M☉, 2.20 M☉, 0.94 M☉, from t = 10¹⁰ M/L with this file's own mass–luminosity relation. Nothing here is a track along which a star moves. Every point is a different star of a different mass, all the same age, and the bend is simply where the population runs out of stars that have had time to leave. That is why a cluster has an age and a field star does not: the bend needs a population, and one star is not one. The oldest globular clusters sit near the 12 Gyr line, and in the 1990s the same construction gave them ages of 16 to 18 Gyr against a universe measured at 10 — a two-standard-deviation contradiction that was resolved from the distance side, by Hipparcos, and not from this one. An age read off a bend A cluster is a population of one age and many masses, so the place where its stars leave the main sequence is a clock. The bend needs a population — which is why a cluster has an age and a single star does not.

What this makes readable

Essays that name this one as a prerequisite.

About the same objects

Not linked from either essay — found by the objects both name.

What links here

The 8 of 10 essays linking to this one that name the most of the same objects.

The objects this essay names

Each one links to every other essay that touches it.

Angular momentumCluster ageConvective envelopeDynamical tideEccentricity dampingEquilibrium tideMain sequence turn-offOrbital eccentricityPseudo synchronous rotationSynchronisationTidal circularisationTidal dissipationTidal quality factorTurbulent viscosity