Astronomy & Celestial Mechanics
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Exoplanets

The young Sun's spin decides what the Earth kept

The calculation that strips sub-Neptunes applies just as well to a planet with a trace of hydrogen. An Earth that captured a few hundredths of a per cent of its mass from the gas it formed in — several times the hydrogen now in its oceans — would have lost all of it, or kept most of it, depending on something nobody has measured — how fast the Sun was spinning in its first few hundred million years.

Assumes Radius valley and Magnetic braking.

14 min read 5 figures Small perturbations, large consequencesMeasuring the unreachable

The radius valley is the mark photoevaporation leaves on planets with a few per cent of their mass in hydrogen: a peak in the time needed to strip an envelope sends planets either to a bare core or to a planet roughly twice its core’s size, early in the star’s life and at a radius set by the star’s mass. Nothing in that arithmetic requires a few per cent. The same escape, driven by the same young star, acts on a planet with a hundred times less hydrogen, and the planets of the inner Solar System are candidates.

The Earth did not form in an empty system. It grew from embryos that existed while the disc of gas round the young Sun was still present, for its first few million years, and an embryo of even a tenth of the Earth’s mass sitting in that gas gathers some of it as a thin envelope of hydrogen and helium. That envelope is gone now; the Earth’s atmosphere contains no primordial hydrogen to speak of. Whether it was removed by the young Sun’s light, and how much could have been removed, depends on how much light there was — and that depends on how quickly the young Sun was spinning.

The XUV energy that reached one astronomical unit, for three young Suns. The cumulative X-ray and extreme-ultraviolet energy delivered per square metre at the Earth's distance from the Sun, against age, for three histories that differ only in how long the young Sun stayed magnetically saturated: 20 Myr for a slow rotator, 100 Myr for a medium rotator, 300 Myr for a fast rotator. After saturation each declines to a common track by 1 Gyr, as rotation histories are observed to converge. By 4.5 Gyr the totals are 2.06·10¹⁵ J/m² for the slow rotator, 3.66·10¹⁵ J/m² for the medium rotator, 6.47·10¹⁵ J/m² for the fast rotator: the fast rotator delivered 3.1 times as much as the slow rotator, nearly all of it in the first few hundred million years. The Sun's own rotation at that age is not measured; it is inferred from the spread of rotation periods in young clusters, and all three histories are consistent with a Sun that ends up rotating as it does now.
Fig. 1 The cumulative X-ray and extreme-ultraviolet energy delivered per square metre at the Earth’s distance, against age, for three young Suns that differ only in how long they stayed magnetically saturated: 20 Myr for a slow rotator, 100 Myr for a medium rotator and 300 Myr for a fast rotator. Each declines to a common track by 1 Gyr. By 4.5 Gyr the totals are 2.06·10¹⁵, 3.66·10¹⁵ and 6.47·10¹⁵ J/m²: the fast rotator delivered 3.1 times as much as the slow one, nearly all of it in the first few hundred million years.

How a planet in the making gathers gas

An embryo embedded in the disc holds gas when its gravity at its surface beats the thermal motion of the gas around it, and the envelope it binds grows steeply with its mass. A body the size of the Moon gathers almost nothing. A body of Mars’s mass gathers a thin skin. An embryo of the Earth’s mass, sitting in the gas for a few million years, gathers an envelope that estimates put anywhere from a hundredth of a per cent of its mass to around a per cent, depending on how dense the nebula was, how hot the embryo’s own surface was, and how quickly heat from colliding rocks kept the envelope puffed up.

Not all of that survives even the first moments without the gas. While the nebula is present its pressure holds the envelope down; when the nebula clears, within a few million years, that pressure disappears and a hot, inflated envelope expands and loses much of its mass rapidly. What the star’s light then has to remove is what is left after that boil-off. The thresholds that follow are for that remainder.

A spin that was not recorded

Young stars of the Sun’s mass are born with a wide range of rotation rates. In clusters a hundred million years old, stars that will become Suns spin anywhere from about once a day to about once every ten days, and the fast ones stay in the saturated, X-ray-bright state for hundreds of millions of years while the slow ones leave it within a few tens of millions. Over the following billion years the magnetised wind that carries away their angular momentum brakes the fast rotators hardest, and by a billion years the spread has closed: every star of the Sun’s mass is spinning at about the same rate, and from then on it forgets how it started.

That convergence is why the Sun’s present rotation says nothing about its early rotation, and why the three histories in the figure are all consistent with the Sun as it is. It is also why their energy totals differ by a factor of three and still come to nearly the same thing late on. Everything that separates them happened before the Sun was a billion years old, while the stripping of planets was taking place.

The Sun’s early spin has been inferred indirectly. One argument uses the Moon, whose surface rocks are depleted in easily evaporated elements like sodium and potassium in a way that depends on how strongly the young Sun’s wind and radiation drove them off its early atmosphere; it favours a Sun that was a slow or moderate rotator rather than a fast one. The inference passes through several models, and the three histories here span the range it leaves open.

The threshold an Earth-mass planet cannot keep

Consider a rocky core of one Earth mass at one astronomical unit, starting with some small fraction of its mass as a hydrogen envelope, and follow it for four and a half billion years under each young Sun.

What an Earth-mass core keeps of a hydrogen envelope, under three young Suns. The envelope an Earth-mass rocky core at one astronomical unit still has after 4.5 billion years, against the envelope it started with, for the three young-Sun histories. The interior fit is being used at the lowest mass it was made for, and the result is a statement about orders of magnitude rather than per cents. Below a threshold the envelope is lost entirely and the curve leaves the bottom of the figure: 0.0081 per cent of the planet's mass for the slow rotator, 0.015 per cent of the planet's mass for the medium rotator, 0.029 per cent of the planet's mass for the fast rotator. Above it the planet keeps most of what it had. The threshold moves by a factor of 3.5 between the gentlest and the most active history, so whether an Earth-mass planet that captured a small primordial hydrogen envelope keeps it is decided, in this model, by how fast its star was spinning in its first few hundred million years.
Fig. 2 The envelope an Earth-mass rocky core at 1 AU still has after 4.5 Gyr, against the envelope it started with, for the three young-Sun histories. Below a threshold the envelope is lost entirely: 0.0081 per cent of the planet’s mass for the slow rotator, 0.015 per cent for the medium rotator and 0.029 per cent for the fast rotator. Above it the planet keeps most of what it had. The threshold moves by a factor of 3.5 between the gentlest and the most active history.

The curves are nearly step functions, and that is the runaway again. An envelope just above the threshold survives with most of its mass, because its planet sits on the side of the loss-time peak where losing gas makes the remainder harder to lose. An envelope just below it is on the other side, where each loss makes the next one faster, and it is removed completely. There is no planet in between with a little hydrogen left; there is a line, and it moves with the young Sun.

Those thresholds are small fractions of a planet, but they are not small amounts of hydrogen. A hundredth of a per cent of the Earth’s mass is about 6×10206\times10^{20} kilograms. The hydrogen in all the Earth’s oceans, bound into water, is about 1.6×10201.6\times10^{20}. The slow rotator’s threshold, 0.0081 per cent, is three times that; the medium rotator’s six times; the fast rotator’s eleven. An Earth that captured several oceans’ worth of hydrogen from the nebula would have lost all of it under any of the three Suns — and one that captured twenty would have lost it under a fast rotator and kept most of it under a slow one.

The threshold on the back of an envelope

The medium rotator’s threshold can be checked without the full model. That young Sun delivered 3.66×10153.66\times10^{15} joules to each square metre at the Earth’s distance. The Earth intercepts it over a disc of 1.28×10141.28\times10^{14} square metres, so the total is about 4.7×10294.7\times10^{29} joules. If a tenth of that energy goes into lifting gas out of the Earth’s gravity — the efficiency the model assumes — and each kilogram needs 6.3×1076.3\times10^{7} joules to escape, the young Sun could remove about 7.5×10207.5\times10^{20} kilograms: 0.013 per cent of the Earth’s mass.

The model’s threshold for that history is 0.015 per cent, within a fifth of the rough estimate. The difference is the envelope itself: a hydrogen layer stands above the rock and catches light over a larger disc than the bare planet, and the model follows that as the envelope thins. The threshold is, to first order, simply the young Sun’s total XUV energy at the planet, times an efficiency, divided by the energy it costs to lift a kilogram away — which is why it scales with the light and falls so steeply with the planet’s mass.

A very slow and a very fast Sun

The three standard histories are not the extremes.

What an Earth-mass core keeps of a hydrogen envelope, under three young Suns. The envelope an Earth-mass rocky core at one astronomical unit still has after 4.5 billion years, against the envelope it started with, for the three young-Sun histories. The interior fit is being used at the lowest mass it was made for, and the result is a statement about orders of magnitude rather than per cents. Below a threshold the envelope is lost entirely and the curve leaves the bottom of the figure: 0.0068 per cent of the planet's mass for the very slow rotator, 0.015 per cent of the planet's mass for the medium rotator, 0.041 per cent of the planet's mass for the very fast rotator. Above it the planet keeps most of what it had. The threshold moves by a factor of 6.0 between the gentlest and the most active history, so whether an Earth-mass planet that captured a small primordial hydrogen envelope keeps it is decided, in this model, by how fast its star was spinning in its first few hundred million years.
Fig. 3 The same retained envelopes for young Suns saturated for 10, 100 and 500 Myr. The thresholds are 0.0068 per cent of the planet’s mass for the very slow rotator, 0.015 per cent for the medium rotator and 0.041 per cent for the very fast rotator — a factor of 6.0 between the extremes.

The response is lopsided. Halving the slow rotator’s saturated phase, from 20 million years to 10, moves the threshold only from 0.0081 per cent to 0.0068. Stretching the fast rotator’s, from 300 million years to 500, moves it from 0.029 to 0.041. The reason is the common decline. Every history ends on the same track once its rotation has converged, and that shared tail delivers a floor of XUV energy that no amount of early slowness removes, while a long saturated phase adds energy on top of it without limit. A gentle young Sun cannot save much hydrogen that a typical one would take; an unusually active one can take a great deal more.

Three planets under one Sun

The same threshold, computed for Venus and Mars as well as the Earth, shows how differently the same young Sun treated its three inner rocky planets.

The largest hydrogen envelope each inner planet would have lost, for three young Suns. For Venus (0.815 Earth masses at 1.91 times the Earth's insolation), the Earth (1 Earth mass at 1 times the Earth's insolation), Mars (0.107 Earth masses at 0.43 times the Earth's insolation), the largest starting hydrogen envelope, as a share of the planet's mass, that the young Sun's XUV light removes entirely within 4.5 billion years, under three rotation histories (slow rotator, saturated for 20 Myr; medium rotator, saturated for 100 Myr; fast rotator, saturated for 300 Myr). Venus: 0.023 per cent (slow rotator), 0.045 per cent (medium rotator), 0.094 per cent (fast rotator). The Earth: 0.0081 per cent (slow rotator), 0.015 per cent (medium rotator), 0.029 per cent (fast rotator). Mars: 0.32 per cent (slow rotator), more than 20 per cent (medium rotator), more than 20 per cent (fast rotator). Mars is far below the mass the interior fit was built for and its values are an extrapolation; the ordering is the point. A world loses a larger envelope the more light it receives and, far more steeply, the smaller it is — Venus, closer in, cannot keep 2.8 to 3.3 times what the Earth cannot, and Mars, further out but a tenth of the mass, cannot keep 39 times what the Earth cannot under the slow rotator, and more than the fit covers under the other two — and the more active the young Sun, the larger every threshold. So the absence of primordial hydrogen on the terrestrial planets constrains the Sun's early activity only in combination with how much hydrogen each captured, which is itself unknown.
Fig. 4 For Venus (0.815 Earth masses at 1.91 times the Earth’s insolation), the Earth, and Mars (0.107 Earth masses at 0.43 times the Earth’s insolation), the largest starting hydrogen envelope, as a share of the planet’s mass, that the young Sun removes entirely within 4.5 Gyr. Venus: 0.023, 0.045 and 0.094 per cent for the slow, medium and fast rotators. The Earth: 0.0081, 0.015 and 0.029 per cent. Mars: 0.32 per cent under the slow rotator and more than 20 per cent under the other two. Venus cannot keep 2.8 to 3.3 times what the Earth cannot, and Mars 39 times, under the slow rotator.

Venus is a little smaller than the Earth and receives nearly twice the light, and its threshold is about three times the Earth’s under every history — close to what the extra light alone would do, with a small addition from its weaker gravity. Mars is further out and receives less than half the Earth’s light, and it still cannot keep an envelope forty times larger under the gentlest Sun, and anything the model can describe under the other two. The dependence on the planet’s mass is far steeper than the dependence on its distance, for the reasons that shaped the valley: a small core’s envelope stands taller, catches more light, and is held by weaker gravity, and each of those multiplies the loss.

Mars is far below the mass the interior fit was made for, and its numbers are an extrapolation. The ordering is not in doubt. Mars is thought to have finished forming within a few million years, while the nebula was still present, which makes it the likeliest of the three to have captured gas and the least able to keep it.

Heavier planets keep what the Earth cannot

The dependence on mass runs the other way too, and it matters for the planets of other stars.

The largest hydrogen envelope each planet would have lost, for three young Suns. For an Earth-mass planet at 0.7 AU (1 Earth mass at 2.04 times the Earth's insolation), a 2 Earth-mass planet at 1 AU (2 Earth masses at 1 times the Earth's insolation), a 5 Earth-mass planet at 1 AU (5 Earth masses at 1 times the Earth's insolation), the largest starting hydrogen envelope, as a share of the planet's mass, that the young Sun's XUV light removes entirely within 4.5 billion years, under three rotation histories (slow rotator, saturated for 20 Myr; medium rotator, saturated for 100 Myr; fast rotator, saturated for 300 Myr). An Earth-mass planet at 0.7 AU: 0.018 per cent (slow rotator), 0.035 per cent (medium rotator), 0.069 per cent (fast rotator). A 2 Earth-mass planet at 1 AU: 0.0032 per cent (slow rotator), 0.0058 per cent (medium rotator), 0.010 per cent (fast rotator). A 5 Earth-mass planet at 1 AU: 0.0011 per cent (slow rotator), 0.0018 per cent (medium rotator), 0.0032 per cent (fast rotator). A world loses a larger envelope the more light it receives and, far more steeply, the smaller it is, and the more active the young Sun, the larger every threshold. What any of these planets keeps depends as much on how much hydrogen it captured as on its star's early activity, and neither threshold says anything without the other.
Fig. 5 The same thresholds for an Earth-mass planet at 0.7 AU, receiving 2.04 times the Earth’s insolation — 0.018, 0.035 and 0.069 per cent for the three histories — and for planets of 2 and 5 Earth masses at 1 AU: 0.0032, 0.0058 and 0.010 per cent, and 0.0011, 0.0018 and 0.0032 per cent.

Moving an Earth-mass planet in to 0.7 astronomical units roughly doubles its threshold, in proportion to the light. Doubling its mass instead lowers the threshold by more than half, and a planet of five Earth masses at the Earth’s distance loses only envelopes smaller than about a thousandth of a per cent under a typical young Sun — a seventh of what the Earth loses. A rocky planet a few times the Earth’s mass, forming in the habitable zone of a Sun-like star while gas is present, would keep a primordial envelope that an Earth in the same place would lose. Such a planet might carry a layer of hydrogen for billions of years, with consequences for its surface temperature that are very different from those of an atmosphere of nitrogen and carbon dioxide.

Molecular hydrogen at high pressure absorbs infrared radiation through collisions between its molecules, and a thick enough hydrogen atmosphere is a powerful greenhouse. Calculations of such atmospheres find that they can keep a planet’s surface warm enough for liquid water far outside the range a single number sets for an ordinary atmosphere. A primordial envelope that a young star failed to remove is therefore not only a question about a planet’s past; it can move where a planet is habitable now.

A young Sun seen in its twins

The Sun’s own history is not observable, but stars like it at every age are. Nearby stars of about the Sun’s mass and composition, dated by their motions, their rotation or the clusters they belong to, form a sequence from a hundred million years old to several billion, and their X-ray and ultraviolet brightness can be measured directly from space. The youngest are hundreds to thousands of times brighter in X-rays than the Sun is now, their output falls with age in roughly the way the model assumes, and at a given young age they show the same wide spread that the three histories represent. Space telescopes watching thousands of Sun-like stars have also recorded flares on young ones far more energetic than any seen on the Sun, adding bursts of XUV energy to the steady output.

The cluster ages on which that sequence rests come from the point at which a cluster’s stars leave the main sequence, and the spread of rotation at each age from counting how fast the stars in each cluster spin. The young Sun was one of those stars; which one is the unknown the thresholds depend on.

What the Earth’s own gases say

The Earth carries some evidence of its own. Neon in rocks from the deep mantle has isotopic proportions close to the Sun’s, unlike the neon in the atmosphere, which has been read as a trace of nebular gas dissolved into an early magma ocean from an envelope that was later lost. The water in the oceans has a ratio of deuterium to hydrogen close to that of primitive meteorites rather than that of the Sun, which says most of it was delivered by rocky material that formed beyond the line in the disc where water froze, not captured as nebular gas. And the heavy noble gases in the atmosphere are depleted and isotopically fractionated in a way that suggests an early escaping flow of hydrogen dragged some of them along.

None of those is a measurement of how much hydrogen the Earth captured. Together they are consistent with an Earth that had a modest nebular envelope and lost it early — which, on these figures, any of the three young Suns could have arranged for an envelope of a few oceans’ worth, and only a fast rotator for one of several tens.

Why the answer is conditional

The absence of primordial hydrogen on the terrestrial planets constrains the young Sun’s activity only in combination with how much hydrogen each planet captured, and the capture is as uncertain as the activity. It depends on how massive the embryos were while the gas lasted, how long the gas lasted — a few million years, with a spread — and how hot the envelopes were, since a hot envelope holds less gas. A small capture is consistent with any young Sun; a large one would require an active one; and the thresholds here say where the dividing lines fall.

What the model leaves out

The planet’s own heat. A newly formed rocky planet has a molten surface, and the heat escaping from it inflates and drives off an envelope with no help from the star, particularly for small planets. For Mars and small embryos that process may act faster than photoevaporation.

Impacts. The Earth’s growth ended in giant collisions, one of which formed the Moon, and a collision of that size can remove a thin atmosphere outright. The threshold calculation assumes the envelope’s only enemy is starlight.

The envelope’s composition. The model’s envelope is hydrogen and helium over a dry rock surface. An envelope in contact with a magma ocean exchanges hydrogen with it, storing some in the interior and releasing water and other gases, which changes both how much there is to lose and what the escaping flow is made of.

And the limits of the fit. The envelope thickness comes from interior models built for planets of one to twenty Earth masses with envelopes of a tenth of a per cent and more. Earth-mass planets with a hundredth of a per cent are at its edge, and Mars is beyond it; the figures here are statements about orders of magnitude and ordering, not about per cents.

The same star, seen from its planets

The radius valley in the census of other stars’ planets and the missing hydrogen of the Earth are the same calculation at two different scales, driven by the same young-star activity. For sub-Neptunes that activity decides which cores end bare. For the terrestrial planets of the Solar System it decides whether a thin primordial envelope survived, and so, indirectly, what the earliest atmospheres were made of. In both cases the star’s rotation in its first few hundred million years — a quantity no one can observe for the Sun, and one its own magnetic activity has long since forgotten — sets the outcome.

Still open: what the escaping hydrogen took with it

A planet losing a hydrogen envelope in a fast escaping flow does not lose only hydrogen. The flow drags heavier atoms with it, more efficiently the lighter they are, and leaves the ones it could not lift behind with their isotopes sorted by mass. The pattern of noble gases and their isotopes that such a flow leaves in a planet’s atmosphere is a record of how fast and for how long it ran — a measurement that exists for Venus, the Earth and Mars, and that ties the young Sun’s light to the air of each.

Where this goes next

Where photoevaporation puts the valley, round stars of different mass. The radius of the largest core stripped bare, against orbital period, round stars of 0.5 M☉, 0.75 M☉, 1 M☉, 1.25 M☉, in the same energy-limited model, with each star's luminosity taken as its mass to the fourth power and its saturated phase lengthened for smaller stars as the mass to the −1.5, 100 Myr for the Sun. At ten days the valley is at 1.22 Earth radii round the 0.5 M☉ star, 1.38 Earth radii round the 0.75 M☉ star, 1.50 Earth radii round the 1 M☉ star, 1.60 Earth radii round the 1.25 M☉ star, and every one of them tilts as the −0.176 power of the orbital period, because the tilt comes from the exponents of the escape law and the interior fit, which do not depend on the star. A lower-mass star is far fainter, so at a given period its planets receive much less light, and even its longer active phase does not make up the difference: this model puts the valley at smaller radii round smaller stars while keeping the same sign of tilt. The stellar scalings are rough, and they are the least certain part of the calculation; what is robust is that photoevaporation ties the valley to the XUV energy a planet received, which falls with the star's mass at fixed period, and a mechanism tied to something else would tie it differently. A smaller star puts the valley lower Round stars of half the Sun's mass, the gap between bare rocky cores and sub-Neptunes should sit at a smaller radius than round the Sun, because at the same orbital period their planets receive a tenth of the light. Their stars also stay young and active far longer, which pushes the other way. Photoevaporation weighs those two against each other in a definite proportion, and the answer is a valley that scales as the star's mass to about the power three tenths. When the stripping happens, in a model population. For the same 3,000 model planets, the share that have lost their whole envelope by each age, scaled to the share bare at 5.0 Gyr (35 per cent of the population), beside the share of the star's lifetime XUV energy delivered by then. Half of all the stripping in this model is finished by 72 Myr and nine tenths by 457 Myr, while the star has delivered 27 and 76 per cent of its XUV energy. That is the clock photoevaporation keeps: the valley is essentially finished within the first few hundred million years, because the planets near the boundary are the ones that run away, and they run away early. A mechanism powered instead by the slow cooling of the planets' own cores would keep moving planets across the valley for billions of years. The difference is in when, not where, and it is why the ages of the stars hosting planets on either side of the valley are the measurement that can separate the two. The stripping runs ahead of the starlight that drives it If young stars carve the radius valley with their X-ray light, the valley should be finished early — half of it before the star has delivered a third of that light, and nine tenths of it within a few hundred million years. If the planets' own cooling cores carve it instead, planets should still be crossing it billions of years later. The two accounts put the valley in the same place, and they are separated by the one thing a histogram cannot show — when. Pluto and Triton sit on the nitrogen line, and the Earth sits between helium and nitrogen. Escape speed against exospheric temperature, with a criterion line for each molecular species at v_esc = 6 v_th — the speed at which the Jeans loss time is comparable to the age of the solar system. Every line has slope one half, because the thermal speed goes as √T; a body above a line keeps that gas and a body below it does not. The horizontal axis is the temperature at the exobase, which for the Earth is near 1000 K rather than the 255 K of its equilibrium — the difference is extreme-ultraviolet heating, and using the wrong temperature puts the Earth above the hydrogen line, keeping an atmosphere it observably lost. Three readings are worth making. The Earth falls between helium and nitrogen and does exactly that: it loses helium as fast as radioactive decay supplies it, and keeps nitrogen for ever. Titan sits just above nitrogen and just above methane, which is why it has a thick nitrogen atmosphere and is slowly losing its methane. And Pluto and Triton sit on the nitrogen line, within 1 per cent — which is why both have atmospheres that are marginally bound and measurably escaping. Where it fails it fails in one direction only. Mercury, the Moon and the Galilean satellites all plot above lines for gases they do not have, because retention is necessary and not sufficient: a body also needs a source, and needs to survive the non-thermal losses this criterion says nothing about. Venus is the sharpest case — it sits above the hydrogen line and has still lost an ocean, because that hydrogen left by charge exchange with the solar wind rather than by moving fast enough. The gas a planet cannot keep Escape is a statement about the tail of a distribution rather than about its mean, so the threshold is not a speed but a dimensionless number near thirty — and past that number the loss rate falls by twelve orders of magnitude. Then, for a hot Jupiter, the whole picture fails and the atmosphere leaves as a wind.

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Energy-limited escapeNebular gasPhotoevaporationPrimordial atmosphereSaturationStellar rotationSuper-EarthTerrestrial planetsXUV fluxYoung sun