A planet too slow to flatten, weighed by its wobble

The single number that says most about a planet’s interior is its moment of inertia, divided by its mass and the square of its radius. A uniform sphere has 0.4; a planet whose heavy material has sunk to form a core has less, because more of its mass sits close to the axis. The Earth’s is 0.331, the Moon’s 0.393, Mars’s 0.364, Mercury’s 0.346, and each of those numbers has a definite meaning: a core of a definite fraction of the radius, given a plausible density.

For most planets the number is found from the planet’s shape. A spinning planet bulges at its equator, and if it is fluid enough to have taken up the shape its spin demands, the size of the bulge depends on how concentrated its mass is — the Darwin–Radau relation turns a measured flattening into a moment of inertia. That route requires the bulge to be the spin’s doing, and the part of a shape the spin cannot explain has to be removed before it can be used.

Venus is the case where nothing can be used. It turns once in 243 days, backwards, and a spin that slow raises an equatorial bulge too small to measure. Venus is not perfectly round — its gravity field has a flattening coefficient about the size of the Moon’s — but none of that flattening is the spin’s; it comes from the slow convection of the mantle and the topography it supports. There was, until recently, no way to weigh Venus’s interior at all: the second-closest planet, nearly the Earth’s twin in size and mass, had an unknown core.

The bulge a slow spin is entitled to #

The flattening a spin raises is set by a single dimensionless number, the ratio of the centrifugal acceleration at the equator to gravity: $q = \omega^2 R^3 / GM$. For a fluid planet the gravitational flattening coefficient $J_2$ is a fraction of $q$ — about half of it for a uniform body, less for a centrally concentrated one.

The Earth’s $q$ is a third of a per cent, and its $J_2$ sits just where a fluid planet with the Earth’s concentration of mass would put it; the flattening is almost entirely the spin’s, and the Darwin–Radau relation gives a moment of inertia close to the true one. Mars’s $J_2$ is somewhat above the fluid line, because the Tharsis volcanic province adds a large non-equilibrium bulge, and the relation has to be corrected for it. Jupiter, a fluid planet spinning fast, lies on the line for a strongly concentrated interior.

Venus is off the chart in the other direction. Its $q$ is fifty-seven thousand times smaller than the Earth’s, so the flattening its spin could support is a couple of parts in a hundred million. Its measured $J_2$, from the tracking of the Magellan and Pioneer Venus orbiters, is two hundred times that. The whole of it is the shape of the mantle’s convection frozen into the gravity field, and it says nothing about whether Venus has a core of iron the size of the Earth’s, a smaller one, or none.

A torque that does not need a bulge raised by the spin #

There is another way to weigh a planet, and Venus’s frozen flattening — useless as a measure of the interior — turns out to be exactly what it needs. Any planet with a $J_2$, whatever produced it, is torqued by the Sun when its spin axis is tilted to its orbit, because the Sun pulls harder on the near side of the bulge. The torque does not tip the axis over; it makes it precess, sweeping round a cone, as the Earth’s axis does in twenty-six thousand years. The rate is

$$\dot\Psi = \frac{3}{2}\,\frac{n^2}{\omega}\,\frac{J_2}{C/MR^2}\,\cos\varepsilon ,$$

with $n$ the orbital rate, $\omega$ the spin rate and $\varepsilon$ the obliquity. The torque is set by $J_2$, which is measured; the resistance to it is the spin angular momentum, set by the spin rate, which is measured, and by the moment of inertia $C$, which is not. A measured precession rate is therefore a measured moment of inertia. The bulge does not have to be hydrostatic for this to work. It only has to be known.

For Venus the rate is a few tens of arcseconds a year, a full turn in about thirty thousand years. The Earth’s precession, fifty arcseconds a year, is of the same order although its $J_2$ is two hundred and fifty times larger: the Earth spins 243 times faster and so has that much more angular momentum to be turned, and the Moon adds twice the Sun’s torque. Venus has no moon, and only the Sun turns it. Measuring the rate means finding the direction of Venus’s spin axis to a fraction of an arcsecond at several epochs and watching it move.

Why the slow spin helps as well as hurts #

The slowness that closed the first route opens the second wider than it would otherwise be. The precession rate is inversely proportional to the spin rate, because the torque has to turn the planet’s spin angular momentum and a slowly spinning planet has little of it. Venus’s spin is 243 times slower than the Earth’s, and that factor multiplies its precession rate. Its frozen, non-hydrostatic $J_2$ — four parts in a million, a two-hundred-and-fiftieth of the Earth’s — is turned almost as fast as the Earth’s much larger bulge, because there is so little rotation resisting it.

There is a second piece of luck in the size of the non-hydrostatic bulge. Had Venus’s mantle convection left it rounder, with a $J_2$ of a few parts in a hundred million — its spin’s entitlement — the solar torque would have been two hundred times weaker and the precession two hundred times slower, a full turn in six million years, and no radar programme of human length could have measured it. The mantle’s frozen distortion, which carries no information about the core, is what makes the core measurable.

The same reasoning applies elsewhere with different weights. Mars, which spins fast and has a large non-hydrostatic bulge, precesses at about 7,600 milliarcseconds a year, and the rate was measured by tracking landers on its surface for decades; its moment of inertia is known to a part in ten thousand. Mercury, locked into a slow rotation by a resonance with its orbit, sits in an equilibrium orientation of its axis rather than precessing freely, and its moment of inertia comes from the obliquity of that equilibrium. Venus is the only planet whose moment of inertia had to come from a free precession measured from the Earth without a lander, and the only one whose bulge, being entirely non-hydrostatic, is used as a lever and not as a measurement.

It also explains why Venus’s obliquity is so small and so stable: the slow precession keeps the axis well away from resonances with the slow, planet-driven oscillations of the orbit plane — the opposite of the situation in which a tilt is not a constant because the two rates come close. Venus’s axis has nearly stood still for as long as its spin has been slow.

Echoes from a speckled planet #

The axis was found from the Earth, by radar. A radio wave sent from a large dish to Venus scatters off its rough surface and returns as a pattern of bright and dark speckles sweeping across the Earth as Venus turns. Two antennas some thousands of kilometres apart — the Goldstone station in California transmitting and receiving, and the Green Bank telescope in West Virginia receiving — see the same speckle pattern pass over them a few seconds apart, and the delay and the moment at which the two patterns match best give the direction of Venus’s spin axis and its instantaneous spin rate.

A team using this technique observed Venus at twenty-one epochs between 2006 and 2020. The axis moved, as expected, along the cone the Sun’s torque defines, and the rate of its motion was 44.58 ± 3.3 arcseconds a year. With the Magellan $J_2$, that gives $C/MR^2 = 0.337 \pm 0.024$.

The surprising connection in the measurement is with the Earth’s own rotation. The same paper found that Venus’s day is not constant: spin periods measured at different epochs differ by about twenty minutes, sixty parts in a million. Something is changing the rotation of the solid planet by more than a part in twenty thousand, on timescales of years — and the obvious candidate is the atmosphere.

A day the atmosphere can lengthen #

Venus’s atmosphere is ninety times as massive as the Earth’s, and it super-rotates: its upper layers circle the planet in four days, sixty times faster than the surface turns. It carries angular momentum, and any change in its circulation has to be balanced by an equal and opposite change in the planet’s spin.

The atmosphere holds about one per cent as much angular momentum as the solid planet. On the Earth the figure is a hundred times smaller, and the atmosphere’s seasonal exchanges change the length of the day by a millisecond. On Venus a change of half a per cent in the atmosphere’s angular momentum — a modest variation in the strength of its super-rotation — changes the day by twenty minutes. The measured variations are of that size, and they matter for the weighing: a spin rate that wanders has to be modelled alongside the axis, and an error in the spin feeds into the precession rate.

The thick atmosphere is also why Venus spins so slowly and backwards in the first place: atmospheric tides raised by the Sun’s heating push on the planet in the opposite sense to the solid-body tides, and the balance between them holds Venus near its present slow retrograde spin. The same atmosphere that sets the spin rate keeps unsettling it.

The core the number brackets #

A moment of inertia factor of 0.337 is lower than the 0.4 of a uniform sphere by an amount that requires a dense core. How large depends on how much denser the core is than the mantle, which the moment of inertia alone cannot say.

For a density contrast like the Earth’s, the measurement allows a core of between about 2,600 and 4,300 kilometres in radius, with a central value close to the Earth’s own core. That is the first measurement of Venus’s core size that does not assume one, and it is consistent with the simplest expectation: a planet of the Earth’s size and bulk density, formed from similar material, with a core of similar size.

The uncertainty is what keeps the result from being more interesting. Venus’s factor is known hundreds of times less precisely than the Earth’s or Mars’s, and the range it allows includes both an Earth-sized core and a noticeably smaller one. Whether Venus’s core is solid or liquid — the question behind its lack of a magnetic field, since a liquid iron core that is not convecting generates none — is untouched by it. Mercury’s moment of inertia was refined from a similar precision by the four numbers of Peale’s experiment, which separate the mantle’s moment from the whole planet’s through the planet’s forced libration; Venus’s slow, irregular spin makes the equivalent libration far harder to see.

What the frozen bulge records instead #

The bulge that cannot weigh the core is not uninformative. A non-hydrostatic $J_2$ two hundred times the spin’s entitlement says that Venus’s mantle holds a large-scale distortion against its own gravity for long periods, which needs either a strong, cold lithosphere or convection vigorous enough to keep regenerating it. Venus’s gravity field correlates strongly with its topography at long wavelengths, more strongly than the Earth’s, and a mountain’s gravity says what holds it up: on Venus, the answer at the largest scales is the flow of the mantle itself, pressing up beneath highlands and drawing down beneath lowlands, with no weak layer between the crust and the convecting interior to decouple them.

That is part of why the moment of inertia is wanted. A planet without plate tectonics loses its interior heat differently, and the size and state of its core is one of the inputs that determines how. The bulge says how the mantle moves; the precession says how much of the planet is core; and the two together are the first constraints on Venus’s interior that are not borrowed from the Earth.

What patience buys #

The precession rate is the slope of a slowly changing direction, and the precision of a slope improves faster than the time spent measuring it.

If each epoch’s axis direction were measured as well as it is now, doubling the record would cut the error by nearly a factor of three. The radar observations continue. So do two planned orbiters, which would track the orientation of Venus’s surface directly from orbit by imaging the same features again and again, and measure its gravity field to higher degree. Together they would bring the moment of inertia to the percent level, enough to tell an Earth-like core from a small one and, with a measured tidal Love number, a liquid core from a solid one.

What the calculation leaves out #

The precession formula keeps only the Sun’s torque on the planet’s quadrupole, the leading term. The planets perturb Venus’s orbit plane, so that the axis precesses relative to a moving reference, and the measurement has to be expressed relative to the orbit’s own slow motion; the obliquity changes as well, through the same perturbations, over tens of thousands of years. The $J_2$ used is the orbiters’ value, itself uncertain by about a per cent, which adds a per cent to the moment of inertia.

The two-layer interior is the simplest model, with uniform core and mantle densities and no crust. Real interiors have densities rising with depth through compression, and a crust of lower density; realistic models move the inferred core radius by a few per cent at a given moment of inertia. The atmospheric angular momentum is estimated from a single super-rotation speed, where the real distribution varies with height and latitude and is measured only in part.

Still open: whether Venus’s core is liquid #

The moment of inertia says that Venus has a core and roughly how large. It does not say what state the core is in, and that is the more pressing question. The Earth’s magnetic field is generated by convection in its liquid outer core, driven partly by the slow freezing of its inner core and partly by heat flowing out into the mantle. Venus has no field. Either its core is entirely solid, or it is liquid and not convecting — perhaps because Venus, without plate tectonics, loses heat from its mantle too slowly to keep the core’s top cool enough for convection to run.

The two explanations make different predictions about Venus’s past as well as its present. A core that froze solid long ago would mean a planet that has been magnetically silent for billions of years, its atmosphere exposed to the solar wind throughout, which bears on how Venus lost its water. A core that is liquid but stagnant would mean a planet whose mantle has kept its heat in, perhaps punctuated by episodes of global resurfacing when the stored heat escaped — the kind of event that Venus’s young, uniformly cratered surface has long been taken to record. The size of the core, now bracketed, is one input to both stories; its state is the other.

Distinguishing the two needs a measurement of how much Venus flexes under the Sun’s tides, which is much larger for a planet with a liquid core. The tidal Love number from the Magellan and Pioneer Venus tracking is uncertain enough to allow either, and is the measurement the coming orbiters were designed to improve. Combined with the precession, which weighs the core, the flexing will say what the core is made of in the sense that matters — whether it is still molten — and so whether the Earth’s twin has a dynamo that stopped or one that never started.