A wall measures a ratio, and a ratio is a line

A cluster’s binaries show a wall in the period–eccentricity plane, and where the wall stands measures how a star dissipates a tide. The obvious extension is downward in mass: run the same measurement on planets rather than on stellar companions and it probes the dissipation inside the planet, whose tidal response is much larger.

The planets show the wall. The measurement does not follow, and the reason is one line of algebra.

The timescale, and where the periods enter #

The eccentricity damping time for a tide raised on the planet by its star is

$$\tau_e = \frac{2}{21}\frac{Q'}{n}\frac{M_p}{M_\star}\left(\frac{a}{R_p}\right)^5,$$

with $n$ the mean motion and $Q'$ the modified tidal quality factor — the dimensionless number that stands in for everything about how the planet’s interior turns tidal flexing into heat.

The period enters twice. Once through $n = 2\pi/P$, giving a factor $P$. And once through $a^5$, which by Kepler’s third law is $P^{10/3}$. Together,

$$\tau_e \propto Q'\,P^{13/3},$$

a very steep dependence, and it is the steepness that makes the boundary a wall rather than a slope. At half the period the timescale is $2^{13/3} = 20$ times shorter.

Setting $\tau_e$ equal to the system’s age and solving gives the boundary:

$$P_{\rm cut} \propto \left(\frac{\rm age}{Q'}\right)^{3/13}.$$

Age and $Q'$ appear only as a ratio. No measurement of where the wall stands can separate them, however precisely the wall is located, because they are not two quantities entering the observable — they are one.

Why the stellar version escaped #

The binary measurement works because the cluster supplies the age from somewhere else.

A cluster’s age comes from its main-sequence turn-off: the mass at which stars are just leaving the main sequence, read off the colour–magnitude diagram, converted to an age by stellar evolution models. That measurement owes the tide nothing at all, so the age is known independently and the boundary then gives $Q'$.

The hot-Jupiter version has no cluster. A field star’s age comes from gyrochronology — its rotation period, which slows in a known way — or from isochrone fitting, or from asteroseismology if it is bright enough. All three are good to a factor of two or three for a solar-type star, and one of them has a complication specific to this problem: a star with a close-in giant planet is tidally spun up by it, so its rotation is not a clock any more.

What the boundary looks like in practice #

The planetary wall is real and is where the argument starts. Hot Jupiters inside about five days have eccentricities consistent with zero; outside about ten days the distribution is broad and reaches 0.93.

The population outside the wall is the part that carries the formation argument. A hot Jupiter that arrived by high-eccentricity migration — scattered inward by another planet, or driven by a distant companion through the Kozai–Lidov mechanism — must have spent time on a long ellipse with a small periastron. One that migrated through the disc need never have been eccentric at all.

So the objects caught between the two states, eccentric and not yet circularised, are the direct evidence for the first channel, and they are the least populated part of the diagram.

What varying Q′ does, and what it does not #

Because the boundary depends on the ratio, changing $Q'$ at fixed age and changing age at fixed $Q'$ produce identical figures. That is the degeneracy stated as a drawing rather than as an equation.

What is not degenerate is everything else in the expression. The planet’s mass, its radius and the stellar mass each enter separately, so a population spanning a range of those does constrain more than the ratio — provided $Q'$ is the same for all of them, which is exactly the assumption in question.

The exponent, and why it differs from the stellar one #

The thirteen-thirds is not the same power the binary-star version carries, and the difference is worth following because it is the clearest statement of what the two measurements are measuring.

For a planet the tide is raised on the planet by the star, and the damping timescale carries $Q'_p(a/R_p)^5$ — the planet’s own radius in the fifth power, and its own dissipation constant. Adding the $1/n$ and converting $a$ to $P$ gives $P^{13/3}$.

For a binary in a cluster the tide is raised on the star by its companion, and the dissipation is by turbulent convection in the star’s envelope. That mechanism has its own frequency dependence: the eddies that do the dissipating turn over on a timescale comparable to the tidal period, and when the tide is faster than the eddies the effective viscosity is reduced. Carrying that through gives $P^{16/3}$ rather than $P^{13/3}$.

So the two walls are steeper and shallower for a physical reason and not by convention, and it matters: at $P^{16/3}$ a factor of two in period is a factor of 40 in the timescale, and at $P^{13/3}$ it is 20. The stellar wall is the sharper of the two.

Both are far steeper than anything else in orbital dynamics, which is why the phenomenon presents as a boundary at all rather than as a gradient.

Two tides, and which one the wall is about #

There is a second tide in the same system and it has been left out so far, which is legitimate and deserves stating.

The planet raises a tide on the star as well, and that tide also dissipates. Its contribution to the eccentricity damping is smaller, by a large factor, because the star’s radius relative to the orbit is small and the fifth power punishes it — a hot Jupiter at ten stellar radii has $(a/R_\star)^5 = 10^5$ against $(a/R_p)^5 \approx 10^{7}$ for the planet, and the mass ratio enters the other way.

So the eccentricity is damped by the planetary tide and the orbit’s semi-major axis is changed by the stellar one. The two effects are separable because they depend on different bodies’ quality factors, and a measurement of one says nothing about the other.

That separation is why the orbital decay measurement is a different measurement rather than a check on this one. It gives $Q'_\star$, the star’s number, while the wall gives $Q'_p$, the planet’s — and the two are believed to differ by several orders of magnitude, with the star’s much larger because a radiative interior dissipates poorly.

What would break the degeneracy #

Two measurements would, and one of them has been made.

An orbital decay rate. The same tide that circularises an orbit also transfers angular momentum between the planet’s orbit and the star’s spin, and for a hot Jupiter orbiting faster than its star rotates the transfer removes orbital angular momentum and the planet spirals in. The decay rate goes as $Q'^{-1}_\star$ — the star’s quality factor rather than the planet’s — with no age in it whatsoever.

That measurement exists. One planet’s transits have been observed to arrive progressively early over two decades, at a rate implying an orbital period shrinking by about 30 milliseconds a year, which gives the star’s $Q'_\star$ directly and without an age.

A tidal heat flux. Dissipation deposits energy in the planet’s interior, and for a planet still circularising the rate can be large enough to inflate its radius measurably. An inflated radius is therefore a lower bound on the heating and hence an upper bound on $Q'$ — but radius inflation has several competing explanations, so the bound is weak.

What does not break it is more planets. A larger sample locates the boundary more precisely, and a more precisely located line is still a line.

The heating that goes with the damping #

The energy removed from the orbit does not vanish, and following it gives a second reading of the same measurement.

An orbit circularising at fixed angular momentum loses energy, and the energy is dissipated inside the planet as heat. The rate is largest for the planets on the eccentric side of the wall — the ones actively circularising — and for a Jupiter at $e = 0.9$ with a few-day periastron passage it can reach a substantial fraction of the stellar irradiation.

That heat has to leave, and it leaves by inflating the planet. A planet with a large internal heat source has a larger radius at a given mass than one without, and the radius enters $\tau_e$ as the fifth power — so the heating accelerates the circularisation that produced it, which is a runaway rather than a balance.

The same mechanism operating on a moon rather than a planet is what keeps Io molten, with the eccentricity maintained by a resonance rather than decaying. The difference between the two cases is entirely whether something resupplies the eccentricity, and the physics of the dissipation is identical.

For the hot Jupiters nothing resupplies it, so the heating is transient: large while the planet is circularising and gone afterwards. A population observed at one moment therefore contains a few inflated, eccentric, actively-heated planets and many circular ones that have finished, which is what the diagram shows.

Fifteen real systems, and an eccentricity that is an upper limit #

The planets drawn are real: fifteen systems with measured periods and eccentricities, from radial velocities for the eccentric ones and from transit photometry for the circular ones.

The boundary is computed from the expression above at the stated parameters, not fitted to the points.

The eccentricities of the inner planets are upper limits rather than measurements. An eccentricity of zero is not measurable; what is measured is that $e$ is consistent with zero at whatever precision the data allow, typically a few hundredths for a well-observed hot Jupiter and worse for others. So the wall’s inner side is a statement about non-detection, and its sharpness partly reflects a detection threshold rather than the population.

That is a real caveat and it runs in the direction that helps: a transit survey’s eccentricity biases very nearly cancel, so the near-circularity of the inner planets is not a selection effect, and the precision with which each is known to be circular is what limits how sharply the wall can be drawn.

Q′ is not a physical quantity, and it is not a constant #

$Q'$ is not a physical quantity. It is a parameterisation — the inverse of the fraction of tidal energy dissipated per cycle, times a factor absorbing the planet’s rigidity — and it stands in for a mechanism nobody has identified. A planet whose dissipation happens in a thin layer and one whose dissipation is distributed can have the same $Q'$ and entirely different physics.

$Q'$ is not a constant. It depends on the tidal forcing frequency, and for a planet with a fluid envelope over a solid core the dependence can be resonant — large at some frequencies and small at others. Quoting a quality factor without a period is half a number, which is the standing caution about this entire class of measurement.

And the boundary is drawn as a line where the physics is a rate. A planet does not circularise instantly on crossing it; the boundary is where the damping time equals the age, so planets just inside are partly circularised and the wall has a real thickness the drawing does not show.

The counting argument the diagram supports #

Beyond the boundary’s position, the diagram carries a population statement that does not depend on $Q'$ at all, and it is the one most often used.

Count the planets on the eccentric side of the wall that are close enough to be circularising, and compare against the number that have finished. The ratio is a ratio of timescales — how long a planet spends visibly circularising against how long it has been circular — and if the arrival rate has been steady, the ratio gives the circularisation timescale directly.

That is an age measurement made by counting rather than by dating, and it needs no stellar ages whatever. What it needs instead is an assumption about the arrival rate, which is that hot Jupiters have been delivered to short periods at a roughly constant rate over the Galaxy’s history.

The assumption is not obviously safe. If high-eccentricity migration is triggered by a secular process with its own timescale, the arrival rate could be strongly time-dependent, and the counting argument would return the migration timescale rather than the tidal one.

So the field has two measurements that each need an assumption the other does not: the boundary’s position needs an age, and the counting argument needs an arrival history. Neither is decisive and they disagree by about a factor of ten in $Q'_p$, which is the honest state of the number.

Why the degeneracy is the ordinary case #

It is worth noticing how common this structure is, because the reflex on encountering a measurement that gives only a ratio is usually to look for a better measurement rather than to ask whether one exists.

An observable that depends on two parameters only through their ratio cannot separate them, and the situation is improved only by an observable with a different dependence. That is the same structure as a velocity amplitude and an inclination, as a transmission spectrum’s reference radius and abundance, and as a colour index against reddening and metallicity.

What distinguishes the cases is whether a second observable is available, and here it is: the orbital decay rate has a different power of $Q'$ and no age at all. So the hot-Jupiter measurement is soluble in principle and hard in practice, which is a better position than the alternatives.

The cluster binaries are the counterexample worth keeping. There the second measurement — the turn-off age — was available from the beginning, which is why a result obtained in the 1980s on a few dozen stars per cluster is still the cleanest tidal dissipation measurement there is.

What a measurement of one system would need #

It is worth setting out what would be required to measure $Q'_p$ for a single named planet, because the list is short and is the reason nobody has.

An eccentricity, measured to a precision better than the value itself — which for a partly circularised planet means a few thousandths, and needs either a long radial velocity campaign or a secondary eclipse timing, since the offset of an eclipse from mid-transit measures $e\cos\omega$ directly.

An age, to better than a factor of two, which for a solar-type star means asteroseismology and therefore a bright host.

A planetary radius and mass, both of which are routine for a transiting planet with velocities.

And an assurance that nothing is resupplying the eccentricity — no second planet exciting it secularly, which requires a long-term velocity baseline to exclude.

All five exist for perhaps a dozen systems, and the resulting $Q'_p$ values span about two orders of magnitude. Whether that spread is real or is the age errors is the question the next paragraph is about, and the sample is not yet large enough to answer it.

Still open: whether one Q′ describes anything #

The assumption underneath every use of this measurement is that $Q'$ is a property of a class of objects rather than of an individual.

The evidence is mixed. Fitting a single $Q'$ to the whole hot-Jupiter population reproduces the boundary’s position adequately, which is weak support. Fitting individual systems where the age is independently known gives values spanning two orders of magnitude, which is either real variation or the age errors propagating through.

Separating those requires ages better than a factor of two for a sample of systems spanning a range of planetary masses and radii — which asteroseismology can now supply for the brightest few dozen, and which is the reason those systems are observed far more intensively than their planets warrant.

From here, when one number will not do #

A cut-off period is one number, and the theories it is supposed to distinguish differ in how the damping rate depends on period, on stellar mass and on evolutionary state.

One number cannot choose between two power laws with a free normalisation each. What can is the shape of the boundary rather than its position — how it moves with age across many clusters, and what it does at the mass where a star’s envelope stops being convective. Computing both finds that the first discriminator is much weaker than it looks.