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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.

Assumes Star formation, Molecular clouds and Angular momentum.

16 min read 8 figures Scales that defeat intuitionMeasuring the unreachable

A cold cloud collapses when gravity beats pressure, and the condition for that gets easier as the collapse proceeds, so once it starts it runs away. That is the standard account of how a star begins, and it is complete only if the cloud is not turning.

Clouds turn. Not fast — the measured velocity gradients across dense cores correspond to angular velocities of around 10−1410^{-14} radians a second, one revolution in twenty million years — but the collapse is a contraction by an enormous factor, and specific angular momentum is conserved through it.

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. 1 The obstruction. A one-solar-mass core of radius a twentieth of a parsec, turning once in twenty million years, has a specific angular momentum of about ten to the sixteen square metres a second. Rotation speed rises as the reciprocal of the radius during collapse; the orbital speed needed to hold material up rises only as the inverse square root. The two curves must cross, and the crossing — measured off the drawn curves rather than asserted — is at twenty-nine astronomical units.

The barrier, and how far short it falls

The arithmetic is one line. Material with specific angular momentum jj cannot fall inside the radius at which j/rj/r equals the circular speed GM/r\sqrt{GM/r}, which is

rbarrier=j2GM.r_{\rm barrier} = \frac{j^2}{GM}.

For the numbers above that is 4×10124 \times 10^{12} metres — about six thousand solar radii, or the orbit of Neptune. The collapse stops there, in the equatorial plane, and what exists is a rotationally supported disc rather than a star.

The square in that expression deserves attention, because it is what makes the barrier’s position so much less certain than the barrier’s existence. The radius goes as the square of the specific angular momentum, so a factor of two in the measured velocity gradient — well within what the projection ambiguities of the previous section allow — is a factor of four in where the collapse stops. Estimates for the same core from different tracers, or from the same tracer modelled two ways, routinely differ by that much.

What survives the uncertainty is the comparison that matters. The Sun’s own specific angular momentum is about 101510^{15} square centimetres a second, and a core’s is four orders of magnitude above it; a factor of four in the barrier radius is a factor of two in jj, which changes four orders of magnitude into three and a half. The obstruction is stated in decades and measured in factors of two, which is why the qualitative conclusion has been secure since the 1960s while the quantitative one is still being argued about.

To arrive at a Sun turning once in twenty-five days, the material has to shed a factor of about ten thousand in specific angular momentum. Nothing in the collapse itself removes any: gravity is a central force and exerts no torque about the centre of mass. The excess has to be handed to something.

This is the same statement that the solar system’s ledger records after the fact. The difference is that a ledger describes an outcome and this describes an obstruction, and the obstruction is observable in objects that are still in the middle of it.

A core with a flat middle, and the singular solution it never reaches. The density profile of the critical isothermal sphere, in units of its own central density and of the radius at which it is truncated, on logarithmic axes. Two features are the whole of it. The centre is flat: inside about a fifth of the outer radius the density hardly changes, because gravity there has almost no enclosed mass to pull with, and this is what distinguishes a real core from the singular isothermal sphere. The outer part falls a little steeper than the −2 of that singular solution — measured off the drawing at −2.11 — because a truncated sphere crosses the singular profile rather than settling onto it, and the dashed line drawn for comparison has infinite central density and no stability question to ask about it at all. Observed cores in nearby dark clouds are fitted with exactly this shape, and the fitted density contrast is how a core is said to be near its critical state or far from it — a contrast well under fourteen means the thing is stable and will still be there in a million years.
Fig. 2 What a core looks like before the barrier matters. The density profile is flat in the middle and falls as an inverse square outside, approaching but never reaching the singular isothermal solution. That inner flat region is where the collapse begins, and it is also the region whose rotation is measurable — the velocity gradient across it is the observable that sets the number in the opening figure.
A core with a flat middle, and the singular solution it never reaches. The density profile of the critical isothermal sphere, in units of its own central density and of the radius at which it is truncated, on logarithmic axes. Two features are the whole of it. The centre is flat: inside about a fifth of the outer radius the density hardly changes, because gravity there has almost no enclosed mass to pull with, and this is what distinguishes a real core from the singular isothermal sphere. The outer part falls a little steeper than the −2 of that singular solution — measured off the drawing at −2.11 — because a truncated sphere crosses the singular profile rather than settling onto it, and the dashed line drawn for comparison has infinite central density and no stability question to ask about it at all. Observed cores in nearby dark clouds are fitted with exactly this shape, and the fitted density contrast is how a core is said to be near its critical state or far from it — a contrast well under fourteen means the thing is stable and will still be there in a million years.
Fig. 3 The same critical isothermal sphere at twice the temperature. The profile is identical — it is a solution of the dimensionless Lane–Emden equation and carries no temperature in its shape — and what the temperature sets is the scale: the radius the flat centre extends to and the mass the whole sphere holds both move, while the picture does not. That separation is why a core’s density profile is a weak test of anything. Two cores of different temperature and different mass have the same shape, so a fitted profile constrains the scale and says nothing about the physics the next sections are about.

What is actually measured about a core’s rotation

The measurement is a velocity gradient, and turning it into an angular momentum requires assumptions worth stating.

A dense core is mapped in a molecular line — ammonia, or a nitrogen-bearing species that survives at high density — and the line centroid is measured across the map. A systematic gradient of a fraction of a kilometre per second per parsec is common. Interpreting that gradient as solid-body rotation gives an angular velocity, and multiplying by the core radius squared gives a specific angular momentum.

The assumptions are that the gradient is rotation rather than infall or outflow projected onto the sky, that the rotation is about a single axis, and that it is solid-body rather than differential. None is safe. A core threaded by a magnetic field and undergoing collapse has systematic motions along the field lines too, and a projected gradient does not distinguish them. Despite all that, the numbers are consistent across many cores and the conclusion is robust: the specific angular momentum of a core exceeds a star’s by three to four orders of magnitude, and the barrier is real.

There is one further check, and it is a good one. The specific angular momentum measured for cores and the specific angular momentum measured for wide binary orbits are the same number to within the scatter. That is what a disposal by fragmentation would produce, and it is not what would be expected if the core measurements were dominated by projection effects — a systematic error in the maps would not know about the orbits of stars formed from them.

Three disposals

The mechanisms available are the same three the solar system’s ledger points to, and here they can be watched working.

Magnetic braking. A collapsing core is threaded by the interstellar magnetic field, and the field is anchored in the much larger, much slower-rotating envelope around it. As the core spins up, the field lines are twisted, and a twisted field carries a torque — transmitting angular momentum outwards along the field lines at the Alfvén speed. The mechanism is efficient. In fact it is embarrassingly efficient: an ideal calculation, with the field frozen into the gas, brakes the core so completely that no rotationally supported disc forms at all.

That is the magnetic braking catastrophe, and it is a real problem rather than a rhetorical one. Discs are observed around the youngest protostars; ideal calculations say they should not exist. The resolutions on offer all involve breaking the field’s grip on the gas — misalignment between the field and the rotation axis, turbulence that tangles the field, or non-ideal effects in which the neutral gas slips through the ionised component. Fragmentation. A core with too much angular momentum for one star can make two, with the excess in their mutual orbit. This is not a marginal channel: roughly half of solar-type stars are in binaries, and the binary fraction rises with mass. The orbit is a reservoir that will hold any amount of angular momentum, and a wide binary holds a great deal of it.

Fragmentation stops at 0.005 solar masses, and the reason is opacity. The Jeans mass along a collapse rather than across a family of clouds. To the left of the knee the gas is isothermal at 10 K, because molecular material at these densities radiates away the heat of compression as fast as compression supplies it; the critical mass therefore falls, and the cloud fragments into pieces, and each piece fragments again. At 10¹⁰ particles per cubic centimetre the gas becomes opaque to its own cooling lines, the collapse turns adiabatic with an index of 1.67, and the temperature starts to rise as density to the 0.67. The Jeans mass turns with it. The minimum, read off the drawn curve at 0.0054 solar masses, is the smallest object this cascade can produce, and it is close to the hundredth of a solar mass usually quoted as the opacity limit. Nothing in the argument mentions hydrogen burning: the lower end of the stellar mass function is set by when a gas stops being able to get rid of heat, not by whether the result can ignite.
Fig. 4 The competition fragmentation is part of. As a collapsing region gets denser it can support smaller and smaller unstable masses, so it fragments — until the gas becomes opaque to its own cooling radiation and the temperature starts to rise, which ends the cascade. Rotation adds to this: a rotating collapsing region flattens into a disc, and a disc is more unstable to fragmentation than a sphere, so the same angular momentum that obstructs the collapse also promotes the splitting that solves it.

A disc that transports it. If a disc forms and survives, it can move angular momentum outwards internally, letting most of its mass spiral in while a small fraction takes the spin away. That is what an accretion disc is for, and it requires a coupling between adjacent annuli that ordinary viscosity cannot supply. In a protostellar disc the leading candidates are magnetic turbulence in the ionised regions and gravitational instability in the massive early phase.

Fragmentation stops at 0.002 solar masses, and the reason is opacity. The Jeans mass along a collapse rather than across a family of clouds. To the left of the knee the gas is isothermal at 10 K, because molecular material at these densities radiates away the heat of compression as fast as compression supplies it; the critical mass therefore falls, and the cloud fragments into pieces, and each piece fragments again. At 10¹¹ particles per cubic centimetre the gas becomes opaque to its own cooling lines, the collapse turns adiabatic with an index of 1.67, and the temperature starts to rise as density to the 0.67. The Jeans mass turns with it. The minimum, read off the drawn curve at 0.0017 solar masses, is the smallest object this cascade can produce, and it is close to the hundredth of a solar mass usually quoted as the opacity limit. Nothing in the argument mentions hydrogen burning: the lower end of the stellar mass function is set by when a gas stops being able to get rid of heat, not by whether the result can ignite.
Fig. 5 The same collapse with the gas made adiabatic a decade later in density — at 10¹¹ particles per cubic centimetre rather than 10¹⁰. The smallest fragment falls from a hundredth of a solar mass to two thousandths, because the minimum of the Jeans mass sits exactly at the knee and the knee has moved. The lower end of the stellar mass function is set by where the gas stops being able to radiate, which is a statement about dust opacity at densities no observation reaches, and the factor of five between these two drawings is the honest width of that uncertainty.

The other quantity with the same problem

Angular momentum is not the only conserved thing a collapsing core carries too much of, and the second one is instructive because its resolution is known.

Interstellar gas is threaded by a magnetic field, and in a perfectly conducting fluid the field is frozen into the material: compress the gas and the field is compressed with it, in proportion. A core collapsing by a factor of ten thousand in radius would carry its field along, and the field it arrived with is far more than a star can hold. The Sun’s surface field is a few gauss; the field a frozen-in collapse would deliver is many orders of magnitude larger, and the magnetic pressure would have halted the collapse long before stellar density.

That is the flux problem, and it has the same shape as the angular momentum problem: a conserved quantity, a large contraction, an impossible result. Its resolution is that the freezing is not perfect. The field is tied to the ionised fraction of the gas, which in a dense molecular core is a part in ten million, and the neutral majority drifts slowly through the ions — ambipolar diffusion — leaving the field behind.

The two problems are coupled, and the coupling runs both ways. The field is what brakes the rotation, so a core that loses its field early keeps its angular momentum; a core that holds its field is braked hard and may end up with no disc at all. The observed outcome — most young stars have discs, and none has an impossible magnetic field — requires the two disposals to have run at rates that are within an order of magnitude of each other, and there is no reason built into the physics for them to be. Simulations that carry both quantities through the collapse reproduce that coincidence only when the resistivity is treated in detail, and the discs they produce are small — tens of astronomical units rather than hundreds — which is a standing tension with what is observed. The usual escape is turbulence, which misaligns the field from the rotation axis and makes the braking substantially less efficient than an aligned calculation predicts.

The barrier, seen

For most of the history of this problem the centrifugal barrier was a calculated radius rather than an observed one. That changed, and the way it changed is worth recording, because the barrier turned out to announce itself chemically rather than kinematically.

Material falling in along the envelope arrives at the barrier and stops falling. It cannot continue inward, so it piles up and shocks — and a shock at those densities is enough to heat the dust grains past the point at which their icy mantles evaporate. Species locked into grain surfaces for the whole life of the cloud are returned to the gas phase in a thin annulus, and nowhere else.

Sulphur monoxide is the clearest of them. In several of the best-studied protostars — L1527 and IRAS 16293−2422 among them — its emission forms a ring rather than a centrally peaked blob, at a radius of tens of astronomical units, with the infalling envelope traced by one set of molecules outside it and a rotationally supported disc traced by another inside. The transition between an infall-dominated velocity field and a Keplerian one happens at the same radius, measured independently from the position–velocity diagram.

So the barrier is now a place on a map. Its radius is a direct measurement of the specific angular momentum that actually arrived, rather than of the value inferred from a gradient across a core half a parsec wide — which means the two ends of the argument in this essay can at last be compared against each other rather than each against a model.

What that comparison shows is that the arriving angular momentum is smaller than an unbraked collapse would deliver, by a factor of a few. Not by the four orders of magnitude the star ultimately needs, and not by nothing: braking is happening in the envelope, at the level the non-ideal calculations predict rather than at the level ideal magnetohydrodynamics demands, and the disc that forms is small but is not absent.

The critical mass falls as the cloud contracts, with a slope of −0.50. The Jeans mass — the least mass of molecular gas that its own pressure cannot support — against number density, for gas at 8, 30, 100 K, both axes logarithmic. The curves are straight because the mass goes as T^(3/2) ρ^(−1/2), and the slope measured off the drawn 8 K curve is −0.500 against the −0.5 that exponent requires. The consequence is the one that matters and it is a matter of sign: a cloud collapsing at constant temperature moves to the right along one of these lines, so the mass it takes to be unstable keeps falling, and sub-regions that were individually stable when the collapse began become individually unstable during it. A giant molecular cloud at 100 particles per cubic centimetre has a Jeans mass of 99 solar masses; a dense core at 10⁵ has one of 1.70. That is why a cloud of ten thousand solar masses makes a cluster rather than a star, and why the question about star formation is not what makes gas collapse but what stops the fragmenting.
Fig. 6 The Jeans mass across a wider range of temperature than a molecular cloud offers: 8 kelvin, which is about as cold as dust-shielded gas gets, up to 100, which is warm cloud material near a young star. The mass goes as T3/2T^{3/2}, so an order of magnitude in temperature is a factor of thirty in the critical mass, and the drawn slope against density is −0.500 on every curve. Temperature is the variable that decides what a cloud can make, and density only decides when.

What the observations settle and what they do not

Three things are firmly established.

Discs exist around the youngest protostars — objects identified by light that nothing young emits directly — at ages of a hundred thousand years, with radii of tens to hundreds of astronomical units — which is where the barrier argument puts them. Discs disperse within a few million years, which sets the time available for planet formation and for angular momentum transport. And the specific angular momentum of a disc is comparable to that of the core it formed from, which means that the disc is where the excess went rather than a place it passed through.

What is not settled is which disposal dominates, and the difficulty is that the observation which would decide it is the one hardest to make. The braking happens in the envelope, at scales of thousands of astronomical units, in gas that is cold, diffuse and mixed along the line of sight with everything in front of and behind it. Resolving the rotation profile of a collapsing envelope from a hundred parsecs away is at the edge of what interferometry can do, and the profile is exactly what distinguishes an efficiently braked collapse from an unbraked one.

The shape of that profile is worth stating, because it is the whole of the test. Unbraked collapse conserves each shell’s angular momentum, so the rotation speed of infalling material rises inversely with radius and the specific angular momentum is flat. Braked collapse removes angular momentum from the shell as it falls, so the specific angular momentum falls inwards and the disc that eventually forms is smaller than the barrier argument predicts. The two give measurably different velocity fields — a difference of a few tenths of a kilometre a second across an arcsecond — and both are consistent with the position–velocity diagrams that were available until recently, which is why the question survived so long on data that looked adequate.

There is also a systematic that is easy to overlook. A rotating envelope and an infalling one produce velocity gradients in the same direction on the sky, and separating rotation from infall requires modelling both at once rather than reading a gradient as a rotation curve. Several early measurements of core rotation, and therefore several early estimates of how far short the barrier falls, were gradients of collapse read as gradients of spin.

A pressure-bounded cloud has a heaviest stable version, and it is 1.16 M☉. The mass of an isothermal sphere held together by gravity and held in by an external pressure, in the dimensionless form M P^(1/2) G^(3/2)/c_s⁴, plotted against how much denser its centre is than its edge. Every point is a genuine hydrostatic solution — the isothermal Lane–Emden equation integrated from the centre outwards and cut off at a radius — so the curve is a family of clouds that could exist, not a stability argument imposed on them. It has a maximum, at a density contrast of 14.0 and a dimensionless mass of 1.182, and past the maximum the same mass is served by two solutions of which the more centrally concentrated one is unstable. A cloud on the rising side that is squeezed harder settles at a smaller radius and stays; one past the peak that is squeezed has no solution to settle into and collapses. For molecular gas at 10 K squeezed by a pressure of 10⁵ K cm⁻³ the peak is at 1.16 solar masses, which is why the cores seen in nearby clouds are the mass of stars rather than the mass of clouds.
Fig. 7 The other stability question that has to be settled at the same time. A pressure-bounded isothermal cloud has a heaviest stable version, and above it there is no equilibrium at all. Rotation raises that limit, because centrifugal support adds to thermal support — so a core that would be unstable at rest can be stable when turning, and the mass at which collapse begins is not a property of the gas alone. The two effects have to be solved together, and the usual treatment solves them separately.
A pressure-bounded cloud has a heaviest stable version, and it is 0.37 M☉. The mass of an isothermal sphere held together by gravity and held in by an external pressure, in the dimensionless form M P^(1/2) G^(3/2)/c_s⁴, plotted against how much denser its centre is than its edge. Every point is a genuine hydrostatic solution — the isothermal Lane–Emden equation integrated from the centre outwards and cut off at a radius — so the curve is a family of clouds that could exist, not a stability argument imposed on them. It has a maximum, at a density contrast of 14.0 and a dimensionless mass of 1.182, and past the maximum the same mass is served by two solutions of which the more centrally concentrated one is unstable. A cloud on the rising side that is squeezed harder settles at a smaller radius and stays; one past the peak that is squeezed has no solution to settle into and collapses. For molecular gas at 10 K squeezed by a pressure of 10⁶ K cm⁻³ the peak is at 0.37 solar masses, which is why the cores seen in nearby clouds are the mass of stars rather than the mass of clouds.
Fig. 8 The same stability curve at ten times the external pressure. The heaviest stable sphere falls to 0.37 solar masses, because the critical mass goes as P−1/2P^{-1/2} — squeeze a core harder and the largest one it can hold together without collapsing is smaller. The shape of the curve does not change at all, and the peak stays at the same central-to-edge density contrast of about fourteen. A core in a high-pressure environment is closer to collapse at every mass, which is the mechanism behind triggered star formation and the reason the same cloud makes different stars in different places.

The problem does not end at the disc

Getting the material onto a disc disposes of the barrier and not of the excess. A disc that transports angular momentum outwards delivers material to the star’s surface, and the material arriving from the inner edge of a Keplerian disc carries the specific angular momentum of that edge — which for a young star is close to the breakup value.

The arithmetic is unforgiving. A protostar accreting at 10−810^{-8} solar masses a year for a million years takes on a per cent of its mass from an inner disc edge at a few stellar radii, and that alone would spin it to a substantial fraction of breakup. Add the whole accretion history and the star should be rotating at its limit by the time the disc clears.

It is not. The observed rotation periods of young stars still surrounded by discs cluster around a week — a tenth or so of breakup — and, more tellingly, stars with discs rotate more slowly than stars of the same age and mass without them. Something is removing angular momentum at the same rate accretion is supplying it, and it stops when the disc goes.

The mechanism is the same lever the earlier figure draws, operating at the inner edge. The star’s magnetic field threads the disc, is twisted by the differential rotation between them, and transmits a torque; the material that is not accreted is launched along the field lines as a wind, carrying away the angular momentum of the radius it was launched from rather than of the radius it started at. The star is held near a rotation rate set by where its own magnetosphere truncates the disc, and the accretion supplies mass without supplying spin.

So the disposal happens three times over, at three scales. The envelope is braked magnetically against the cloud, the disc transports outwards against itself, and the star is braked magnetically against the disc — and after all of that the Sun still ends up with a spin it has spent four billion years shedding into its own wind. The four orders of magnitude are not removed by one mechanism at one moment; they are paid off in instalments, and each instalment has its own unsolved problem attached.

Why it matters beyond one star

The problem generalises upwards without changing shape.

Where the ladder goes

The nearest rung is the disc itself, and specifically the mechanism by which it moves angular momentum. That has been an open problem for half a century in a form that is quantitative rather than vague: the required transport can be parameterised by one number, that number is measured from disc lifetimes and accretion rates, and no first-principles mechanism reliably produces it in the cold, weakly ionised conditions of a protoplanetary disc.

The other direction is the leftover. Whatever the disc did not accrete became planets, and the resulting ledger puts three-fifths of the system’s angular momentum in Jupiter. The same problem run at galactic scale gives a disc whose size is set by the spin its halo happened to be born with, and in both cases the visible, structured object is a minority component that inherited the angular momentum the majority had to be rid of.

Where this goes next

The critical mass falls as the cloud contracts, with a slope of −0.50. The Jeans mass — the least mass of molecular gas that its own pressure cannot support — against number density, for gas at 10, 20, 50 K, both axes logarithmic. The curves are straight because the mass goes as T^(3/2) ρ^(−1/2), and the slope measured off the drawn 10 K curve is −0.500 against the −0.5 that exponent requires. The consequence is the one that matters and it is a matter of sign: a cloud collapsing at constant temperature moves to the right along one of these lines, so the mass it takes to be unstable keeps falling, and sub-regions that were individually stable when the collapse began become individually unstable during it. A giant molecular cloud at 100 particles per cubic centimetre has a Jeans mass of 99 solar masses; a dense core at 10⁵ has one of 1.70. That is why a cloud of ten thousand solar masses makes a cluster rather than a star, and why the question about star formation is not what makes gas collapse but what stops the fragmenting. The cloud that cannot hold itself up A cold cloud collapses when gravity beats pressure, and the mass at which that happens falls as the square root of the density — so the collapse makes the condition for collapse easier, over and over, until the gas can no longer get rid of the heat. That is why a cloud of ten thousand solar masses makes a cluster and not a star. 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. 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. 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.

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.

Ambipolar diffusionAngular momentum problemAngular momentum transportBinary fractionCentrifugal barrierDisc windFlux problemFragmentationFree-fall timeMagnetic brakingMass-to-flux ratioMolecular cloud coreProtostellar discSpecific angular momentum