Astronomy & Celestial Mechanics
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Orbits

Ninety-nine per cent of the mass and none of the spin

The Sun holds 99.87 per cent of the solar system's mass and 0.6 per cent of its angular momentum. Jupiter holds a thousandth of the mass and three-fifths of the spin. That is not a curiosity of accounting — it is the record of the single operation that had to succeed before a star could form at all.

Assumes Angular momentum, Harmonic law and The two-body problem.

16 min read 8 figures Scales that defeat intuitionTwo bodies, and no more

Kepler’s second law is a conservation law in disguise, and the quantity it conserves is one number per orbiting body. Add those numbers up across a planetary system, add the star’s own rotation, and the sum is a constant that the system has been carrying since it formed. It is worth asking where it sits.

The answer is peculiar enough that it constrained theories of the solar system’s origin for a century before anyone could compute what to do about it.

Who holds the mass, and who holds the spin. The solar system's two ledgers on one logarithmic axis, each row a body or a group of them, with the pale bar its share of the mass and the dark bar its share of the angular momentum. Both columns are computed rather than quoted: the Sun's spin from 0.07 M R² Ω at a 25.38-day rotation, each planet's orbit from M √(GM☉ a (1 − e²)) with its own semi-major axis and eccentricity, and both sums are required to close to one part in a billion. The Sun holds 99.866 per cent of the mass and 0.61 per cent of the angular momentum. Jupiter holds 0.095 per cent of the mass and 61.1 per cent of the angular momentum, so a body a thousandth of the system by weight carries most of its rotation. The four inner planets together account for 0.0016 of it. A cloud collapsing to make this system had to move nearly all of its spin outward onto a small fraction of its mass, and the ledger is what that operation looks like when it is finished. What the figure cannot show is where the transfer happened, because everything that carried it away has either fallen in or left.
Fig. 1 The two ledgers on one logarithmic axis. Each planet’s orbital contribution is computed from its own semi-major axis and eccentricity; the Sun’s rotational contribution from its measured moment of inertia coefficient and its 25.4-day rotation. Both columns are required to sum to one. The Sun holds 99.866 per cent of the mass and 0.61 per cent of the angular momentum; Jupiter holds 0.095 per cent of the mass and 61 per cent of the angular momentum.

Where the numbers come from

Nothing in that figure is quoted. Each planet’s orbital angular momentum is

L=MGM⊙ a (1−e2),L = M\sqrt{GM_\odot\,a\,(1-e^2)},

which needs only the planet’s mass and two of its orbital elements — quantities known to many digits from tables that are fits to centuries of observation. The Sun’s rotational angular momentum is k2MR2Ωk^2 M R^2 \Omega, where k2=0.070k^2 = 0.070 is its moment of inertia coefficient, and that number is the least secure in the calculation because it comes from a solar model rather than from a measurement — though helioseismology now constrains the interior density profile well enough that the coefficient is good to a per cent or so.

The dependence on a\sqrt{a} is what does the work. Angular momentum per unit mass rises as the square root of orbital radius, so a gram at Jupiter carries five times what a gram at Earth’s orbit carries and twenty-five times what a gram at Mercury’s does. That is why the outer planets dominate a budget they do not dominate by mass.

One slope, three heights. Specific angular momentum against stellar mass, both logarithmic, for disc galaxies, for spheroids, and for the dark haloes they formed in. All three loci are drawn with slope 0.6667 — measured off the drawn line rather than assumed — because a spin parameter that does not depend on mass makes j scale as the two-thirds power of mass, and the observed relations do have that slope over three decades of mass. What separates the three lines is not their shape but their height: a disc keeps roughly 53 per cent of the specific angular momentum of its halo, and a spheroid of the same stellar mass has about 20 per cent of a disc's. That is the whole morphological sequence written as one number. A galaxy is a disc because it kept its spin and a spheroid because it lost it, and the losing happens in mergers, where the orbital angular momentum of the pair goes into the outer halo and the remnant keeps almost none of it. The figure cannot show the scatter, which is about a factor of two at fixed mass and is itself the spread in λ from the previous panel.
Fig. 2 The square root drawn on its own, with the mass taken out. Specific angular momentum — per unit mass rather than in total — is a single line of slope one half against orbital radius, and every planet sits on it. So the ledger’s lopsidedness is not a fact about the planets at all: it is that one line, sampled at eight places spread over three decades of radius, weighted by masses that happen to be largest where the line is highest. Had Jupiter formed at Mercury’s distance it would hold about a fiftieth of what it holds now, and the budget would look entirely different with the same bodies in it.

The problem it poses

A star forms from a molecular cloud core that collapses when gravity beats pressure, and a cloud core rotates. It has to: the interstellar medium is turbulent and differentially rotating with the galaxy, and a region a tenth of a parsec across picks up a velocity gradient across it. The observed rotation rates of dense cores are around 10−1410^{-14} radians a second — one turn in twenty million years, which sounds negligible.

It is not negligible, because collapse is a contraction by an enormous factor and angular momentum per unit mass is conserved through it.

A collapse that stops 1 astronomical units short. Equatorial rotation speed against radius for a collapsing cloud core of 1 solar mass, turning at 2·10⁻¹⁵ radians a second at a radius of 0.05 parsecs, compared with the Keplerian speed at the same radius. Both axes are logarithmic; both curves are computed from the same specific angular momentum, 4.8·10¹⁵ square metres a second, held fixed. The rotation speed rises as the reciprocal of the radius and the orbital speed only as its inverse square root, so they must cross, and the crossing is measured off the drawn curves at 1.71·10¹¹ metres — 1 astronomical units, or 245 solar radii. Inside that radius the material is orbiting rather than falling, and no further collapse happens along the equator at all. To arrive at a star turning once in 25.38 days the core must dispose of a factor of 3433 in specific angular momentum, and nothing in the collapse itself removes any: it has to be handed to a magnetic field, to a disc, or to a companion. The figure assumes uniform rotation and a spherical core, both of which are simplifications a real core violates in the direction that makes the problem worse.
Fig. 3 The most favourable core anybody has measured, turning five times more slowly than the one above — and the obstruction is still there. The centrifugal barrier falls at one astronomical unit rather than twenty-nine, which is a factor of thirty in radius and still four hundred times the Sun’s own. A core rotating this slowly is at the bottom of the observed distribution, so this is the best case rather than a typical one, and even the best case cannot make a star without disposing of most of its spin. The problem is not that clouds rotate quickly; it is that the contraction is by seven orders of magnitude.
A collapse that stops 29 astronomical units short. Equatorial rotation speed against radius for a collapsing cloud core of 1 solar mass, turning at 10⁻¹⁴ radians a second at a radius of 0.05 parsecs, compared with the Keplerian speed at the same radius. Both axes are logarithmic; both curves are computed from the same specific angular momentum, 2.4·10¹⁶ square metres a second, held fixed. The rotation speed rises as the reciprocal of the radius and the orbital speed only as its inverse square root, so they must cross, and the crossing is measured off the drawn curves at 4.27·10¹² metres — 29 astronomical units, or 6135 solar radii. Inside that radius the material is orbiting rather than falling, and no further collapse happens along the equator at all. To arrive at a star turning once in 25.38 days the core must dispose of a factor of 1.7·10⁴ in specific angular momentum, and nothing in the collapse itself removes any: it has to be handed to a magnetic field, to a disc, or to a companion. The figure assumes uniform rotation and a spherical core, both of which are simplifications a real core violates in the direction that makes the problem worse.
Fig. 4 The obstruction, drawn. A one-solar-mass core turning once in twenty million years at a radius of a twentieth of a parsec has a specific angular momentum of about ten to the sixteen square metres a second. Hold that fixed and the rotation speed rises as the reciprocal of the radius while the orbital speed rises only as its inverse square root, so they must cross — and the crossing is at twenty-nine astronomical units, about ten thousand solar radii. Inside that radius the material orbits rather than falls.

The core cannot become a star. It becomes a disc, twenty or thirty astronomical units across, and stops. Every gram of the material that eventually made the Sun had to lose a factor of about ten thousand in specific angular momentum before it could arrive.

A collapse that stops 2435 astronomical units short. Equatorial rotation speed against radius for a collapsing cloud core of 3 solar mass, turning at 10⁻¹⁴ radians a second at a radius of 0.2 parsecs, compared with the Keplerian speed at the same radius. Both axes are logarithmic; both curves are computed from the same specific angular momentum, 3.8·10¹⁷ square metres a second, held fixed. The rotation speed rises as the reciprocal of the radius and the orbital speed only as its inverse square root, so they must cross, and the crossing is measured off the drawn curves at 3.64·10¹⁴ metres — 2435 astronomical units, or 5.2·10⁵ solar radii. Inside that radius the material is orbiting rather than falling, and no further collapse happens along the equator at all. To arrive at a star turning once in 25.38 days the core must dispose of a factor of 2.7·10⁵ in specific angular momentum, and nothing in the collapse itself removes any: it has to be handed to a magnetic field, to a disc, or to a companion. The figure assumes uniform rotation and a spherical core, both of which are simplifications a real core violates in the direction that makes the problem worse.
Fig. 5 A larger and more massive core, at a fifth of a parsec and three solar masses, which is the regime a small cluster forms from. The barrier moves out to two and a half thousand astronomical units — the size of a small planetary system’s outer reaches, and far beyond where any star is. That is the argument for why massive cores fragment rather than collapsing: a rotating body that cannot contract past a thousand astronomical units has room inside that radius for several objects orbiting each other, and a binary’s orbit is a reservoir the spin can be put into without anybody having to remove it.

That is the operation the ledger records. The Sun’s 0.6 per cent is not what is left of the cloud’s rotation; it is what remains after the overwhelming majority was moved outwards, onto a small fraction of the mass which is now Jupiter, Saturn, Uranus and Neptune.

Three ways to move it

The mechanisms are known, and which of them mattered most is still argued about.

A disc that transports it internally. A disc cannot accrete without moving angular momentum outwards, and any process that couples adjacent annuli does so: material at smaller radius spins faster, and friction between annuli speeds up the outer one at the expense of the inner. Most of the disc’s mass moves in and a small fraction of it moves out, carrying nearly all of the angular momentum. That process is the standard account, and its difficulty is that ordinary molecular viscosity is many orders of magnitude too small — the coupling has to be turbulent, and the turbulence has to be driven by something. A wind that removes it. Material lifted off the disc’s surface and accelerated along magnetic field lines carries angular momentum away entirely, rather than merely moving it further out. The lever arm is the same one that brakes a star’s own rotation, and it is efficient for the same reason: the field forces the gas to keep turning with its launch point long after it has left. A disc wind removes a small mass and a large amount of spin.

A companion. If the excess is handed to another body, the problem is solved by construction. That is what happened here: the excess went to Jupiter. It is also why binary stars are common — a cloud core with too much angular momentum to make one star can make two orbiting each other, and the orbit is the reservoir.

Who holds the mass, and who holds the spin. The solar system's two ledgers on one logarithmic axis, each row a body or a group of them, with the pale bar its share of the mass and the dark bar its share of the angular momentum. Both columns are computed rather than quoted: the Sun's spin from 0.07 M R² Ω at a 10-day rotation, each planet's orbit from M √(GM☉ a (1 − e²)) with its own semi-major axis and eccentricity, and both sums are required to close to one part in a billion. The Sun holds 99.866 per cent of the mass and 1.54 per cent of the angular momentum. Jupiter holds 0.095 per cent of the mass and 60.5 per cent of the angular momentum, so a body a thousandth of the system by weight carries most of its rotation. The four inner planets together account for 0.0016 of it. A cloud collapsing to make this system had to move nearly all of its spin outward onto a small fraction of its mass, and the ledger is what that operation looks like when it is finished. What the figure cannot show is where the transfer happened, because everything that carried it away has either fallen in or left.
Fig. 6 The counterfactual the essay computes later, drawn now: the same eight planets against a Sun turning once in ten hours instead of twenty-five days. The Sun’s bar rises by a factor of sixty and it still does not reach Jupiter’s. That is worth having before the mechanisms are discussed, because it bounds what they have to achieve — putting all of the system’s angular momentum back into the Sun’s rotation would leave the Sun holding a bare majority of it, not an overwhelming one, since the outer planets’ contribution is that large.

What the ledger cannot say

The record is a record of the outcome and not of the process, and the reason is that everything which carried the angular momentum away is gone.

The disc dispersed within ten million years. A wind, if there was one, left the system. The material that moved outwards to absorb the spin was itself mostly lost — the outer disc is not there. What remains is four planets that between them hold 99.4 per cent of the system’s angular momentum in about a seven-hundredth of its mass, and no way to distinguish, from that alone, between an arrangement produced by viscous transport and one produced by a wind. That last point deserves emphasis. Planetary orbits have migrated, resonances have swept through the system, and the eccentricities have been reshuffled by secular interactions. None of that changes the total. Angular momentum can be traded between planets — and a resonance is precisely a channel for trading it — but the sum is conserved by the whole system, so the number in the ledger is inherited directly from the cloud.

What a bigger reservoir would have looked like

There is an instructive counterfactual buried in the arithmetic, and it takes one line to run.

Give the Sun back all of the angular momentum in the system — collapse Jupiter’s share and everyone else’s into the Sun’s rotation, holding its present structure fixed — and its rotation period falls from 25 days to about ten hours. That is a fifth of the break-up rate for a body of the Sun’s mass and radius, and it would make the Sun an oblate, gravity-darkened, rapidly rotating star of exactly the kind that the early-type main sequence is full of.

So the solar system did not merely move some spin around. It moved enough to change what kind of star sits at the middle of it, and the difference between a star with a magnetic brake and a star without one is the difference between the Sun and a body whose surface temperature depends on which way its pole happens to point.

The number in other systems

The solar system is one draw from a distribution, and the distribution turns out to be wide.

Systems with a hot Jupiter have most of their angular momentum in a planet a hundredth of an astronomical unit from the star, which by the square-root law is a tiny fraction of what the same planet would carry at five astronomical units. Such a system’s total is far smaller than the Sun’s — either because it started with less, or because the planet arrived where it is by giving angular momentum to a disc on its way in. Systems of compact inner planets — the most common architecture found so far — have their angular momentum distributed among several small bodies close in, and the totals are smaller again. What is not yet known is whether the solar system’s arrangement, with nearly all the spin in one distant giant, is typical or unusual, because a survey sensitive to distant giants is a survey that has to run for decades.

There is a second reason the comparison is harder than it looks, and it is a selection effect rather than a shortage of time. The ledger’s square-root weighting means the answer is dominated by whatever is furthest out, and every detection method in use is biased against exactly that. A transit becomes geometrically unlikely and temporally rare as the orbit widens; a radial-velocity signal falls off as the inverse square root of the separation and requires a baseline longer than the period; a direct image needs the planet to be both wide and young. So the systems whose angular momentum has been measured are the systems whose angular momentum is small, and the measured distribution is not the underlying one but a projection of it through an instrument that cannot see the term that matters most.

What can be said is that the stellar half of the ledger looks the same everywhere. Sun-like stars of the Sun’s age rotate slowly whether or not they are known to host planets, so whatever removed the star’s spin was not the planets — it was the disc, and then the wind, both of which operate on any star that forms from a rotating cloud. The planets are where the angular momentum ended up in this system, not the reason the Sun lost it.

Who holds the mass, and who holds the spin. The solar system's two ledgers on one logarithmic axis, each row a body or a group of them, with the pale bar its share of the mass and the dark bar its share of the angular momentum. Both columns are computed rather than quoted: the Sun's spin from 0.12 M R² Ω at a 25.38-day rotation, each planet's orbit from M √(GM☉ a (1 − e²)) with its own semi-major axis and eccentricity, and both sums are required to close to one part in a billion. The Sun holds 99.866 per cent of the mass and 1.05 per cent of the angular momentum. Jupiter holds 0.095 per cent of the mass and 60.8 per cent of the angular momentum, so a body a thousandth of the system by weight carries most of its rotation. The four inner planets together account for 0.0016 of it. A cloud collapsing to make this system had to move nearly all of its spin outward onto a small fraction of its mass, and the ledger is what that operation looks like when it is finished. What the figure cannot show is where the transfer happened, because everything that carried it away has either fallen in or left.
Fig. 7 The one number in the ledger that is not measured, moved to the edge of what a solar model allows. The moment of inertia coefficient is 0.070 from helioseismology and is quoted here at 0.12 — a more centrally concentrated Sun than any model gives. The Sun’s share of the angular momentum rises accordingly and remains under one per cent, which is the robustness check the ledger needs: the headline conclusion survives being wrong about the least secure input by nearly a factor of two, because the shortfall it describes is three orders of magnitude and the uncertainty is one part in two.
A torque that stops when the patch lets go. Left, the mechanism: a protogalactic patch drawn as an ellipsoid of axis ratios 1:0.72:0.5, with the principal axes of the surrounding tidal field drawn across it at 30 degrees to its own. The torque is proportional to the difference of the patch's principal moments times the sine of twice that angle, so it is exactly zero when the two sets of axes agree — checked at both alignments — and largest at forty-five degrees. A spherical patch takes no torque whatever the field around it does, which is why the spin of every galaxy begins as a statement about its shape. Right, the angular momentum against time in units of the turnaround time: in linear theory the torque acts on a patch still expanding with the universe and the angular momentum grows as the first power of time, measured off the drawn curve as t^1.000. At turnaround the patch detaches from the expansion, its quadrupole shrinks, and the torque switches off — so a galaxy's spin is fixed before it has collapsed at all, by neighbours it will never interact with again. What the figure cannot show is the sign: the same mechanism gives no preferred direction, and the observed near-absence of alignment between neighbouring galaxies' spins is the check on that.
Fig. 8 And the mechanism that supplies spin rather than removing it, one scale up. A protogalaxy acquires its rotation from the tidal torque of its neighbours acting on its own quadrupole while it is still expanding — which requires the body to be genuinely triaxial, since a patch with two equal principal moments in the plane takes no torque at all and the generator refuses to draw one. Galaxies are spun up from outside; a collapsing stellar core has no neighbours close enough for that, and has to be braked from inside instead. The two problems have opposite signs and the same conserved quantity.

The same problem, inside the Sun

The ledger above treats the Sun as a single rotating body with one period, and that is an approximation which hides a second angular-momentum problem of the same kind.

The Sun’s outer third is convective and rotates differentially: the equator turns once in about twenty-five days and the poles in about thirty-five. Below the convection zone the interior rotates as a solid body, at a rate near the average of the surface’s, and the transition between the two is a thin shear layer.

That solid-body interior is not what a straightforward account predicts. The Sun has been losing angular momentum from its surface for four and a half billion years through its magnetised wind, and the loss acts on the outer layers. If the interior were not coupled to them it would have kept its original rotation, and since the Sun spun perhaps ten times faster when it arrived on the main sequence, the core should now be turning far faster than the surface.

It is not. Helioseismology measures the interior rotation directly, and it is flat to within a few per cent down to the limit the measurement reaches. So something transports angular momentum outward through the radiative interior, on a timescale short compared with the Sun’s life, and the radiative interior has no convection to do it.

Two mechanisms are proposed. A weak internal magnetic field would couple the layers, and a field of a few gauss suffices — but a primordial field of that kind, left over from the star’s formation, ought to have imposed its own geometry on the rotation profile and does not appear to have. Internal gravity waves generated at the base of the convection zone can also carry angular momentum, and can deposit it where they are absorbed; the transport is plausible and the calculation is delicate.

A star has the same disposal problem as the cloud that formed it, one level down, and the evidence that it was solved is a rotation profile with no gradient in it.

The disc that has to be reconstructed

The mechanisms in this essay all operate on a disc that is no longer present, so the disc has to be reconstructed from what it left behind, and there is a standard way of doing it whose weaknesses are instructive.

Take each planet, restore its heavy elements to solar composition by adding back the hydrogen and helium a body of that composition would have had, and spread the result over an annulus centred on its present orbit. Sum the annuli and the result is a surface density profile — the minimum-mass solar nebula, so called because every step of the construction underestimates.

The profile that comes out falls roughly as the inverse three-halves power of radius and totals about a hundredth of a solar mass. That is the number every model of planet formation was calibrated against for thirty years.

Three assumptions in the construction are known to be wrong and all push the same way. The planets did not necessarily form where they now are, so smearing each over its present annulus misplaces the material. Not all of the solids in an annulus ended up in a planet — a substantial fraction was ejected or accreted by the star. And the gas was largely lost rather than incorporated, so the reconstruction accounts for the mass that stayed and not for the mass that passed through.

The angular momentum version of the same reconstruction is the one that matters here, and it is more robust, because angular momentum is dominated by the outermost material and the outermost material is the least likely to have moved inward. Spreading the giant planets’ angular momentum back into a disc gives a reservoir consistent with the cloud core the previous section computed — which is a genuine consistency check between two calculations sharing no inputs.

A fossil reconstruction is only as good as its assumption about what has been lost, and the reason the angular momentum ledger is trusted more than the mass one is that the quantity is carried by the part of the system that moved least.

An old argument that the ledger settled the wrong way round

The ledger is worth a paragraph of history, because it was for a long time the strongest argument against the theory that turned out to be right.

The nebular hypothesis — that the Sun and planets condensed from one rotating cloud — was proposed in the eighteenth century and had an obvious problem. A contracting cloud spins up. If the Sun formed by contraction from a rotating nebula, the Sun should be the fastest-rotating thing in the system, and it is nearly the slowest. That objection was taken seriously enough that alternatives were constructed to avoid it: the planets were proposed to have been torn out of the Sun by a passing star, precisely because a tidal filament pulled from a slowly rotating Sun would carry the orbital angular momentum of the encounter rather than the Sun’s own.

The encounter theories failed on other grounds — material torn from the Sun would be far too hot to condense, and the required close passages are far too rare. What rescued the nebular hypothesis was not a defence of the angular momentum argument but the discovery of a mechanism the argument had not considered: magnetic fields, which couple a star to material at a distance and let it hand its spin outwards without handing over any appreciable mass.

The lesson is a general one about conservation arguments. A conserved quantity that appears to be in the wrong place is never evidence that the history is wrong; it is evidence that a transport mechanism has been left out. The objection was sound in every step except its unstated premise, which was that a cloud’s angular momentum has nowhere to go but into the rotation of whatever the cloud becomes. Nothing forces that, and once a field line is allowed to reach from the star into the disc it stops being true by a factor of a thousand. The same shape of argument runs through the rest of this collection — a rotation curve that refuses to fall, a core turning too slowly, a binary that should not have merged — and in each case the missing ingredient is a way of moving something rather than a way of destroying it.

Where the ladder goes

The direct continuation is the obstruction itself: what a collapsing core actually does when it hits its own rotation barrier, which is where the mechanisms above are tested against an object still in the middle of the process rather than against a fossil.

The other direction is to notice that the same ledger, drawn for a galaxy, has the same shape. A disc galaxy’s stars carry a specific angular momentum set by the halo they formed in, and the halo — like the Sun — holds nearly all the mass and very little of the coherent rotation. In both cases the visible, rotating, structured part of the system is a small minority that inherited the spin, and the majority of the mass is something that got rid of it.

Where this goes next

Equal areas in equal times, at eccentricity 0.65. Positions of an orbiting body at equal intervals of time, obtained by solving Kepler's equation. The two shaded sectors span the same interval and enclose the same area — a long thin one at the far end, a short fat one at the close approach. Equal areas in equal times, which is angular momentum in disguise Kepler's second law is a statement about the area a radius line sweeps. It looks like an odd thing to have noticed, and it turns out to be a conservation law arriving eighty years early. The temperature of a disc around a stellar black hole. Effective temperature against radius, in units of the inner edge, for a stellar black hole of 10 solar masses accreting 10⁻⁸ solar masses a year. Two features are structural. The profile turns over rather than rising all the way in: the factor (1 − √(r_in/r)) is the statement that no torque acts across the inner edge, so nothing is dissipated there and the peak sits at 49/36 of it, measured here at 1.361. And outside a few inner radii the run is exactly r^−3/4, drawn as the dashed line, which is what makes a disc's spectrum broad: every decade of radius contributes at a temperature a factor of 5.6 lower. The peak is 3.46·10⁶ K here, so the disc radiates in X-rays, and integrating the whole profile gives 4.72·10³⁰ W — which is GMṀ/2r_in to a per cent, half the binding energy released and no more, because the other half is still going round. The disc that has to throw angular momentum away Matter cannot simply fall onto a compact object. At the energy it arrives with, it has far too much angular momentum, and the only way in is for some of it to be carried outwards — which is what a disc is for. A collapse that stops 29 astronomical units short. Equatorial rotation speed against radius for a collapsing cloud core of 1 solar mass, turning at 10⁻¹⁴ radians a second at a radius of 0.05 parsecs, compared with the Keplerian speed at the same radius. Both axes are logarithmic; both curves are computed from the same specific angular momentum, 2.4·10¹⁶ square metres a second, held fixed. The rotation speed rises as the reciprocal of the radius and the orbital speed only as its inverse square root, so they must cross, and the crossing is measured off the drawn curves at 4.27·10¹² metres — 29 astronomical units, or 6135 solar radii. Inside that radius the material is orbiting rather than falling, and no further collapse happens along the equator at all. To arrive at a star turning once in 25.38 days the core must dispose of a factor of 1.7·10⁴ in specific angular momentum, and nothing in the collapse itself removes any: it has to be handed to a magnetic field, to a disc, or to a companion. The figure assumes uniform rotation and a spherical core, both of which are simplifications a real core violates in the direction that makes the problem worse. A cloud that cannot become a star A molecular cloud core turning once every twenty million years sounds like a body at rest. Conserve its angular momentum through a collapse by a factor of ten thousand and it stops at the orbit of Neptune, spinning, having failed to make anything at all.

About the same objects

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

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

Accretion discAngular momentumAngular momentum transportBarycentreConservation lawDisc windKepler's third lawMagnetic brakingMoment of inertiaPlanet migrationProtoplanetary discThe snow lineSolar nebulaSpecific angular momentum