A year too short to feel its own eccentricity

The habitable zone is a band of flux, and a planet on an eccentric orbit does not have a flux. It has a history of fluxes, one for every point of its orbit, and the history can span a wide range. At an eccentricity of 0.4 the sunlight at periastron is 5.4 times the sunlight at apastron; at 0.6 it is sixteen times. The number used to place such a planet in or out of the zone is the time-average, which the area law fixes exactly at the circular value divided by $\sqrt{1 - e^2}$ — a mild 1.09 at $e = 0.4$ and 1.25 at $e = 0.6$.

Using the average is an assumption about the planet rather than about the orbit. It assumes the surface cannot follow the flux as it changes, so that the brief blast at periastron and the long cold stretch at apastron blur into their mean. Whether that is true depends on two times, and neither of them is in the flux.

Two times #

The first time is the orbital period. The second is how long the surface takes to respond to a change in heating, which for a layer of water of depth $h$ is its heat capacity divided by how steeply its emission rises with temperature:

$$\tau_{\rm th} = \frac{\rho\, c\, h}{4\,\varepsilon\,\sigma\,T^3}$$

The numerator is the energy needed to warm a square metre of the layer by one kelvin; the denominator is how much more the surface radiates per kelvin warmer. For an Earth-like surface at 288 K each metre of water adds about fifteen days. The mixed layer of the Earth’s ocean — the top fifty metres or so, stirred by wind on a timescale of weeks — has a thermal time of about two years. Dry land has a thermal time of days, because the heat it stores is confined to a few tens of centimetres.

The climate is a low-pass filter. A forcing that changes slowly compared with $\tau_{\rm th}$ is followed in full; one that changes quickly is attenuated in proportion to how quickly. The eccentric orbit’s forcing is a periodic signal with period $P$ and a spike near periastron that lasts a small fraction of $P$, so the ratio $\tau_{\rm th}/P$ decides everything, and the figure’s crossovers sit where that ratio is near one.

This is the same filter that delays the hottest month behind the sunniest, and the same algebra. There the forcing is the seasonal cycle, the period a year, and the answer a lag of a few weeks and an attenuation of the swing. Here the forcing is the orbit itself, and the filter’s cut-off frequency is being compared with a year that can be anything from a week to a decade.

The shape of the flux matters as much as its range. Kepler’s second law makes a planet sweep through periastron quickly, so the high flux is a narrow spike occupying a small fraction of the period, and the low flux is a broad trough. A filter with a time constant comparable with the period removes the spike far more effectively than it fills the trough, because the spike contains high frequencies and the trough does not. That is why a moderately damped eccentric planet can be colder at its minimum than its average flux suggests while barely warming at periastron.

The filter, written down #

The slab’s response can be predicted before it is integrated, and the prediction is a check on the figures. For a forcing that varies sinusoidally with period $P$, a first-order system with time constant $\tau$ passes a fraction

$$\frac{1}{\sqrt{1 + (2\pi\tau/P)^2}}$$

of the swing it would have with no capacity, and lags by $\arctan(2\pi\tau/P)$ of a cycle’s phase. An eccentric orbit’s flux is not a sinusoid, but at moderate eccentricity most of its swing is in the fundamental, and the formula applied to the fundamental alone reproduces the integrated numbers well. Ten metres of water with $\tau = 0.4$ years on a one-year orbit gives $2\pi\tau/P = 2.5$ and a fraction of 0.37: 128 K becomes 47 K, the figure’s value. Fifty metres gives 12.6 and 0.08: 10 K. One metre on the four-week orbit of a small star gives 3.4 and 0.28: 36 K against the 37 K integrated.

That agreement says the physics in the figures is nothing more than a single time constant, and it says what the periastron spike does to it. The spike is made of higher harmonics — at $e = 0.6$ a large share of the swing is at two and three times the orbital frequency — and each harmonic $n$ is attenuated by $1/\sqrt{1 + (2\pi n\tau/P)^2}$, more strongly than the fundamental. A filter that halves the fundamental removes most of the spike. What remains is a smooth oscillation at the orbital period, displaced towards the apastron side of the mean, which is why the warmest moment of a damped eccentric planet arrives well after periastron and never reaches the height the flux alone would suggest.

A Sun-like star’s year #

Round a Sun-like star a planet in the habitable zone has a year of about a year. A surface with little heat capacity follows its orbit almost exactly: one metre of water, or a continent, sees the whole 128 K swing, which at $e = 0.4$ takes it from above 80 °C just after periastron to well below freezing at apastron. That planet is not habitable in any sense a climate model would recognise, whatever its average.

An ocean changes the verdict. Fifty metres of well-mixed water has a thermal time of two years, twice the orbital period, and swings by only 10 K. A planet covered by an ocean on that orbit experiences its eccentricity as a mild annual cycle superimposed on a mean that is 1.09 times the Earth’s. Three-dimensional climate models of ocean-covered Earth-like planets on eccentric orbits agree with that picture to the extent the simple slab can: with a deep ocean, a planet’s habitability tracks its average flux up to eccentricities of 0.4 or more, and without one it does not.

So for a Sun-like star the question “is an eccentric planet in the habitable zone?” has no answer without the planet’s surface. The same orbit gives an ocean world a temperate climate and a desert world an alternation between the conditions of Venus and of Mars.

The second figure shows the limit of the argument. A deep enough ocean makes eccentricity invisible, but it cannot make the mean flux smaller; at $e = 0.6$ the mean is a quarter higher than the circular orbit’s and the planet’s problem is simply that it is too close on average. The ocean’s filter converts an eccentric orbit into a circular one with a larger flux, and the habitable-zone calculation then applies unchanged — to that larger flux.

A small star’s year #

The habitable zone of a small star lies close in, because its luminosity is low, and an orbit close to a low-mass star is short. A planet receiving the Earth’s flux from a 0.3 solar-mass star orbits at about 0.12 AU with a period of four weeks; round a 0.1 solar-mass star, one week.

The contrast between that figure and the Sun-like one is the whole argument. The flux histories are identical in shape and in range; the fifty-metre ocean has the same heat capacity in both; and the planet round the small star is thermally a circular-orbit planet with a slightly raised mean, while the planet round the Sun-like star with a shallow surface is not habitable at all. The difference is the orbital period, and the orbital period is set by the star’s mass through Kepler’s third law and by its luminosity through the location of the zone.

The first figure puts the whole sequence on one axis. At fixed eccentricity, moving from a Sun-like star to an M dwarf moves the habitable-zone planet leftward along the curves by a factor of fifty in period, from where even ten metres of water leaves a large swing to where even one metre leaves little. A flux-based zone, drawn in the plane of stellar mass and distance, has no way to show this, because it discards the period.

Where tides erase the question #

There is a twist that makes the small-star half of this partly moot. A planet close to its star is also a planet on which the star raises large tides, and tides dissipate the orbit’s eccentricity. A binary’s orbit is circularised below a period that measures its age — a boundary that measures a ratio of age to dissipation rather than either alone — and the same physics acting on a planet in the zone of a late M dwarf circularises it in a time that can be well under a billion years — shorter for a planet with a large, dissipative ocean.

So the stars whose years are short enough to average any eccentricity away are also the stars whose tides are strong enough to remove it. For a planet in the zone of a 0.1 solar-mass star the thermal argument above is largely academic: its eccentricity is small because it has had time to become small, unless another planet in the system is continually pumping it back up. That exception is common. Compact systems of several planets, such as the seven known round TRAPPIST-1, hold one another in near-resonant chains that maintain small forced eccentricities of a few thousandths to a hundredth — too small to matter for the flux, and large enough that the tides they drive are a significant internal heat source.

For Sun-like stars the tides are negligible at a year, eccentricities of known giant planets near 1 AU range up to 0.9, and the thermal question is live. A terrestrial planet on such an orbit, or a large moon of such a giant, would have its climate decided almost entirely by how much water it had.

The moon is the more interesting case, because it stacks two periods. It follows the giant round the star with the giant’s year, which may be eccentric, and it orbits the giant with a period of days, passing through the giant’s shadow once a circuit and receiving reflected and thermal light from the giant on the day side of every orbit. The second period is short enough that any surface averages it; the first is not. A moon’s climate therefore filters its own orbit away and keeps its planet’s, and on top of both it has a heat source the planet lacks: the tides raised by the giant on a moon whose orbit is kept slightly eccentric by its neighbours, which is what keeps Io volcanic and Europa’s ocean liquid at five times the Earth’s distance from the Sun. A habitable-zone argument for such a moon has three times in it, not two, and the flux is the least of them.

What is actually known about the orbits #

The eccentricities that enter these arguments are the least well measured of any exoplanet property that matters to them.

A radial-velocity orbit gives an eccentricity directly, from the shape of the velocity curve, but the shape of a small signal is poorly constrained: noise preferentially inflates a fitted eccentricity, because a circular orbit is a single point in the space of fits and every noise realisation moves the best fit away from it. Published eccentricities of low-mass planets are therefore biased high, and many that are quoted as 0.1 or 0.2 are consistent with zero.

A transiting planet’s eccentricity is harder still. The transit gives a duration, and a duration that differs from the circular prediction measures a combination of eccentricity and orientation — provided the star’s density is known independently, which for most hosts it is only to ten or twenty per cent. Applied to thousands of small planets statistically, that method finds that planets in compact multi-planet systems have typical eccentricities of a few per cent, and single transiting planets are more eccentric. And every method prefers a circle for its own reason, so the catalogue’s distribution of eccentricities is itself filtered before any climate question is asked.

What follows for the habitable zone is a matter of emphasis. For the small, rocky planets that are the targets of habitability arguments, measured eccentricities are mostly small and poorly known, and the figures above say that for most of them the value would not matter much round an M dwarf. Round a Sun-like star it would matter a great deal, and it is precisely there that no Earth-sized planet in the habitable zone yet has a measured eccentricity at all.

What the averaging gets right, and what it misses #

The flux average is the right quantity for the one thing that does not depend on the surface at all: the long-term energy balance. Over many orbits a planet in equilibrium radiates exactly what it absorbs, and what it absorbs is the time-average. Any planet’s mean temperature, to first order, is set by the mean flux.

What the average misses is anything nonlinear, and a climate is full of nonlinearities. Emission rises as the fourth power of temperature, so a planet that alternates between hot and cold radiates more on average than one held at the mean, and its mean temperature is slightly lower than the average flux implies. Water freezes at a threshold, so a planet whose swing crosses 273 K has seasonal ice whose albedo feeds back on its own absorbed flux. And the runaway greenhouse is a threshold too: a brief spike above it on a planet with a shallow surface may evaporate enough water to change the atmosphere for the rest of the orbit.

The slab drawn here includes the first of those — it radiates as a grey body, not linearly — and none of the others. It has no ice, no clouds, no atmospheric heat transport, and one well-mixed layer rather than an ocean whose deep water exchanges with the surface over centuries. Its thermal times are therefore the minimum responses of each depth; a real ocean’s deep water extends the effective memory much further, which strengthens rather than weakens the conclusion for ocean worlds.

The other two cases in the solar system #

The Earth’s eccentricity of 0.0167 gives a flux swing of seven per cent, and the Earth’s seasons are set by its obliquity instead. Kepler’s second law ensures that each hemisphere’s summer half-year receives the same energy whatever the eccentricity, which is part of why the eccentric contribution is so easily overlooked.

Mars is the planet on which it is not. Its eccentricity of 0.093 gives a flux swing of forty-five per cent, and its surface is dry dust with a thermal time of hours. Mars therefore follows its orbit almost perfectly: southern summer, at perihelion, is short and intense, northern summer long and mild, and the dust storms that can envelop the whole planet begin near perihelion in southern spring. On Mars the orbit outweighs the tilt as a clock as well as a climate. A Mars with a fifty-metre ocean would barely notice its eccentricity.

An average belongs to the forcing, and the response time belongs to the planet #

A time-average is a property of a forcing. Whether a system responds to the average is a property of the system, and specifically of the ratio between its response time and the period of the forcing. That ratio is invisible in the forcing itself: the eccentric-flux figure contains everything about the orbit and nothing about whether any of it will be felt.

The habitable zone, as a band in flux, is a statement about the forcing. Turning it into a statement about planets requires the other time — and the carbonate–silicate cycle’s half-million-year response is the same requirement at the other end of the timescale, where the forcing is a star brightening over billions of years and the response is slow enough to be treated as instantaneous. Between the week-long years of the smallest stars and the billion-year brightening of the largest, every climate question about an exoplanet is a comparison of two times, and the zone keeps only one of them.

Still open: what a small star does before it settles #

For the smallest stars the habitable zone has a further problem that no orbit and no ocean can filter. Such a star takes hundreds of millions of years to contract onto the main sequence, and until it arrives it is many times brighter than it will be. A planet in its eventual zone spends that time with its ocean in the atmosphere as steam, while ultraviolet light splits the water and hydrogen escapes. How long, and how much water that costs, is the question that decides whether the zones of the commonest stars in the galaxy were ever wet.