Astronomy & Celestial Mechanics
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Galaxies

A disc the size its halo was born with

A galaxy's mass says how much light it makes. It does not say how big it is. What sets a disc's size is a single dimensionless number describing how fast the dark halo around it happens to be turning — a number the disc had no part in choosing, distributed the same way for every halo mass in the universe.

Assumes Dark matter, Rotation curves and Virial theorem.

16 min read 8 figures Scales that defeat intuitionMeasuring the unreachable

Two galaxies with the same stellar mass can differ in size by a factor of five. One is a compact disc a couple of kiloparsecs across; the other is a diffuse sheet ten times the area at a tenth the surface brightness. They have the same amount of material, the same rotation speed to within a factor approaching one, and the same place on the relations that link mass to luminosity. What they do not have in common is how far out that material sits, and what decides it is the mass that is not the light.

There is a single quantity that predicts the difference, and it is not a property of the visible galaxy at all.

A halo is born with a spin of a few hundredths. The distribution of the dimensionless spin parameter λ = J|E|^½ ÷ (G M⁵ᐟ²) across dark matter haloes, drawn as a lognormal of median 0.035 and logarithmic width 0.5, with the disc scale length each λ implies printed along the lower axis. The distribution is required to integrate to one and to peak at 0.0272, which is the median times e raised to minus sigma squared, and is the signature of a lognormal rather than of a bell curve drawn to look like one. λ is small because a halo is supported by random motion rather than by rotation: a value of 0.035 means the halo turns at about three and a half per cent of the rate it would need to hold itself up centrifugally. It is also nearly independent of halo mass, which is what points at a common origin. Mapping it to a disc through R_d = λ R₂₀₀/√2 with all of the specific angular momentum retained, a 10¹²-solar-mass halo of radius 206 kiloparsecs gives 5.1 kiloparsecs at the median, against the 2.6 kiloparsecs the Milky Way's disc actually has. The gap is not a failure of the estimate; it is the measurement that the baryons arrived with less spin per unit mass than the halo they arrived in.
Fig. 1 The distribution of the halo spin parameter λ, drawn as a lognormal of median 0.035 and logarithmic width 0.5, with the disc scale length each value implies along the lower axis. The distribution is required to integrate to one and to peak at the median times e to the minus sigma squared, which is what makes it a lognormal rather than a bell curve drawn to look like one. A median-spin halo of a million million solar masses gives a five-kiloparsec disc; the Milky Way’s is 2.6, and the gap is the essay’s second half.

A dimensionless measure of turning

The quantity is λ, and its definition looks awkward until it is read as a ratio.

λ=J ∣E∣1/2G M5/2.\lambda = \frac{J\,|E|^{1/2}}{G\,M^{5/2}}.

The combination on the right has no units. What it compares is the halo’s actual angular momentum against the angular momentum a system of that mass and that binding energy would have if rotation alone held it up. A halo with λ=1\lambda = 1 would be rotationally supported, a flat disc of dark matter. A halo with λ=0\lambda = 0 would not be turning at all.

Real haloes have λ≈0.035\lambda \approx 0.035.

That is the first surprising fact and it is worth stopping on. A dark matter halo is not a rotating structure. It is held up by the random motions of its particles, exactly as an elliptical galaxy is held up by disorder, and the coherent rotation on top of that disorder is a three-and-a-half per cent effect. Everything visible in a spiral galaxy — a thin disc, an ordered rotation curve, spiral arms — is built out of the small fraction of the system that is rotating, and the object that fraction sits inside is not. It is worth saying why the distribution in the figure is a lognormal rather than a Gaussian, because the reason is structural rather than empirical.

A halo’s angular momentum is the accumulated result of many torques and many mergers, each of which multiplies the existing value by some factor rather than adding a fixed amount to it. A quantity built by multiplying many independent random factors has a logarithm built by adding many independent random terms, and a sum of many independent terms is Gaussian. So the logarithm of λ is normal, which is to say λ is lognormal, and the width in the logarithm is the meaningful parameter rather than any width in λ itself.

That is worth more than a note on the plotting convention. A lognormal has no negative tail, which a spin parameter cannot have; it is skewed, so the mean and the median differ and quoting the wrong one shifts every derived size; and its width does not shrink as more contributions are added, unlike a sum of additive terms. The observed width — about a factor of e in the logarithm — is therefore not a measurement error and does not average down. It is the intrinsic spread of a multiplicative process, and it is what makes galaxies of one mass differ in size by a factor of several.

From a spin to a size

The mapping from λ to a disc size is one line of algebra and one assumption.

Suppose the baryons that make the disc started out mixed through the halo, sharing its specific angular momentum, and then cooled and settled without gaining or losing any. Angular momentum conservation then fixes where they can end up: material with specific angular momentum jj settles on a circular orbit of radius j/vcj/v_c. Integrating over the halo’s distribution of jj and requiring the result to be an exponential disc gives

Rd≃λ2 R200,R_d \simeq \frac{\lambda}{\sqrt 2}\,R_{200},

with R200R_{200} the radius inside which the halo’s mean density is two hundred times the critical density of the universe.

For a halo of a million million solar masses, R200R_{200} is a little over two hundred kiloparsecs. At the median spin that gives a disc scale length of about five kiloparsecs.

The Milky Way’s is about 2.6.

Why the estimate is too large, and what that measures

A factor of two is not a failure of the argument. It is the argument’s most useful output, because the assumption it violates is a specific and checkable one.

The estimate assumed that the baryons retained all of their specific angular momentum. Several things remove some of it.

Dynamical friction on clumps. Gas that arrives in lumps rather than smoothly is dragged by the halo it moves through, losing orbital angular momentum to the dark matter on the way in. The lumps sink further than smooth gas would, and the disc they build is smaller.

Selective loss. Feedback from young stars and supernovae drives gas out of a galaxy, and it does not drive out a random sample: the gas most easily expelled is the gas nearest the centre, where the star formation is, and that is the low-angular-momentum material. Expelling low-angular-momentum gas raises the average of what remains, which works in the opposite direction — and this is one of the few processes in galaxy formation that can make a disc bigger rather than smaller.

Contraction of the halo. As the baryons settle into the middle they deepen the potential there, and the dark matter responds by contracting inwards. That raises the circular speed at every radius, and since a given jj settles at j/vcj/v_c, a higher vcv_c means a smaller radius.

The measured discrepancy — a disc about half the size the naive estimate gives — is therefore a measurement of the net of these, and it is one of the few numerical constraints on galaxy formation that comes from a conservation law rather than from a simulation. There is an awkwardness in the definition that anyone reading two papers on this will hit, and it is worth naming because the two conventions differ by tens of per cent.

The expression above contains the halo’s total energy EE, which is not an observable and is not straightforward even in a simulation: it requires summing the potential energy over every pair of particles, and it depends on where the halo is declared to end. A halo has no edge, the virial radius is a convention, and the energy inside it is sensitive to that convention in a way the mass and the angular momentum are not.

The standard repair replaces the energy with a circular speed, defining

λ′=J2 M V200R200,\lambda' = \frac{J}{\sqrt2\,M\,V_{200}R_{200}},

which needs no potential-energy sum and is exactly equal to the original for a singular isothermal sphere in equilibrium. For a real halo the two differ, typically by ten to twenty per cent, and not by a fixed factor — the ratio depends on the halo’s concentration and on how far it is from equilibrium.

That matters here because the median quoted for one definition is used with a mapping derived for the other more often than it should be. It also matters for the mass-independence claimed below: concentration does depend on halo mass, so a quantity defined through the energy and a quantity defined through the circular speed do not have the same weak trend with mass, and part of the residual trend reported in some analyses is a statement about which λ was measured.

The part that does not depend on mass

The most informative property of λ is not its value but its lack of a trend.

Across four decades of halo mass — from dwarf galaxies to clusters — the median spin parameter is the same to within the measurement, and the width of its distribution is the same too. Nothing else in galaxy formation behaves this way. Star formation efficiency depends strongly on mass; gas fraction depends on mass; metallicity depends on mass. Spin does not.

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 consequence, in the plane where it is measured. Specific angular momentum against stellar mass, for discs, for spheroids and for the haloes they formed in, all three with slope two thirds — which is what a mass-independent λ requires, since j scales as λ M to the two thirds at fixed density. The three lines differ only in height, and the differences are retained fractions: a disc keeps about half its halo’s specific angular momentum and a spheroid about a fifth of a disc’s. The whole morphological sequence is one number.

A quantity whose distribution does not know about mass is a quantity produced by a scale-free process. That points directly at where the spin came from: gravitational torques exerted on a growing perturbation by its neighbours, in a universe whose density fluctuations have no preferred scale. It is one of the cleaner inferences available about the early universe, and it is made from the sizes of nearby galaxies.

A halo is born with a spin of a few hundredths. The distribution of the dimensionless spin parameter λ = J|E|^½ ÷ (G M⁵ᐟ²) across dark matter haloes, drawn as a lognormal of median 0.045 and logarithmic width 0.6, with the disc scale length each λ implies printed along the lower axis. The distribution is required to integrate to one and to peak at 0.0314, which is the median times e raised to minus sigma squared, and is the signature of a lognormal rather than of a bell curve drawn to look like one. λ is small because a halo is supported by random motion rather than by rotation: a value of 0.045 means the halo turns at about three and a half per cent of the rate it would need to hold itself up centrifugally. It is also nearly independent of halo mass, which is what points at a common origin. Mapping it to a disc through R_d = λ R₂₀₀/√2 with all of the specific angular momentum retained, a 10¹²-solar-mass halo of radius 206 kiloparsecs gives 6.6 kiloparsecs at the median, against the 2.6 kiloparsecs the Milky Way's disc actually has. The gap is not a failure of the estimate; it is the measurement that the baryons arrived with less spin per unit mass than the halo they arrived in.
Fig. 3 The same distribution shifted to a higher median spin and widened. The predicted disc size scales directly with the spin parameter, so a median of 0.045 rather than 0.035 makes every disc thirty per cent larger — and the observed distribution of disc sizes is narrower than either prediction. That mismatch is the finding: the spread in halo spin is a robust output of simulations and the spread in disc size is not as wide, so something is regulating the outcome that the angular momentum alone does not.

The discs that came out too small

The mapping in the previous section has a history worth recording, because for about fifteen years the simulations that were supposed to confirm it produced the wrong answer by a factor of ten rather than two.

Early hydrodynamic simulations of galaxy formation followed the gas as it cooled and settled, and the discs that emerged had scale lengths several times smaller than real galaxies and rotation curves that rose far too steeply in the middle. The cause was identified fairly quickly and was the first of the three mechanisms listed above, running much harder than expected: the gas cooled early, fragmented into dense clumps, and those clumps sank to the centre by dynamical friction against the halo, delivering their angular momentum to the dark matter on the way.

That is the angular momentum catastrophe, and what made it a catastrophe rather than a discrepancy was that it was insensitive to almost everything. Better resolution made it worse, because it allowed the gas to cool into smaller and denser clumps. Different numerical schemes changed the size of it and not the direction. For a while it was read as evidence against the whole cold-dark-matter picture of galaxy formation.

The resolution was feedback, and specifically feedback that acts early — energy returned by young stars and supernovae before the gas has had time to condense and sink. That does two things at once: it prevents the clumps from forming, so the friction never happens, and it preferentially removes the low-angular-momentum gas that had already reached the centre. Simulations with sufficiently vigorous early feedback produce discs of about the right size, and the strength required is at the upper end of what the stellar physics comfortably supplies.

So the factor of two this essay is about is what remains after a factor of ten was fixed by adjusting a process the argument does not contain. That is worth remembering before reading the residual as a measurement of anything: it is a measurement of the net angular-momentum loss, and it is quoted against simulations whose feedback prescription was calibrated to reproduce disc sizes.

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 50 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. 4 The same ladder with spheroids at half the discs’ specific angular momentum rather than a fifth. Where a galaxy sits between those two lines is the single number that separates a disc from an elliptical at fixed mass — and the slope of two thirds is what a collapse conserving angular momentum predicts, independent of the mass. So the ordering of galaxy types is an ordering in one conserved quantity, and the disagreement above is about how much of it survives the collapse.

Where a spin can be seen directly

None of this would be worth much if λ were only a parameter in a simulation. There are two places it is measured.

The first is in the sizes themselves, run backwards. Given a rotation curve, a halo mass follows; given a surface brightness profile, a scale length follows; and the ratio is λ. Doing this across a large survey returns a distribution — approximately lognormal, median a few hundredths, width about a factor of e — which is the distribution the simulations predict. That is not an independent measurement in the strict sense, since it uses the mapping being tested, but the agreement of the shape is meaningful, since nothing in the mapping would force it to be lognormal. The second is more direct and applies to the halo rather than the disc. Weak gravitational lensing measures a halo’s mass distribution from the distortion it imprints on background galaxies, and a rotating halo is not quite symmetric; the effect is far too small to detect for a single galaxy, and stacking many is what makes it approachable.

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. 5 And the same conservation law where it fails by twenty orders of magnitude. A cloud core rotating at its observed rate, collapsing to stellar size with its angular momentum intact, would be turning far above break-up — so almost all of it has to be removed, by fragmentation, by magnetic braking, or into a disc. The galactic version of this essay’s argument assumes the opposite: that the disc keeps what the halo had. Both cannot be right, and the difference is which scales have a mechanism for losing spin.

What the number cannot say

Two limits are worth stating plainly.

λ is a single number describing an entire halo, and a halo’s angular momentum is not a single number — it is a distribution over radius, and it need not point the same way at every radius. Simulated haloes routinely have their inner and outer angular momentum vectors misaligned by tens of degrees, and a disc built out of the inner material can be tilted relative to the outer halo. The whole apparatus above uses one number where a profile is required, and the disagreement between the predicted and observed disc sizes may be partly that.

And λ is measured at one instant, whereas a galaxy is assembled over ten billion years. A halo’s spin changes with every merger. The value quoted for a present-day halo is not the value its disc was built from, and the connection between the two runs through the whole assembly history.

The way that second limit bites is worth being concrete about, because it is not simply an added scatter. A merger delivers angular momentum in a direction set by the orbit of the incoming object, which is uncorrelated with what the halo already had, so λ performs a random walk with steps of order its own size. The distribution of λ over a population can therefore be stationary — every simulated snapshot returns the same lognormal — while no individual halo holds its value for longer than the interval between its mergers. A quantity that is stable as a population statistic and unstable object by object is exactly the kind that supports an argument about why discs are the sizes they are and refuses one about why any particular disc is the size it is.

That is also the answer to an objection the argument invites. If λ wanders, why is the predicted size distribution as narrow as it is? Because the size depends on λ linearly and on the halo mass to the one-third power, and the halo mass grows monotonically while λ does not. Over the interval in which most of a disc’s stars form, the mass term dominates the drift and the spin term supplies the spread — which is the division of labour the two observed relations of the previous section are reading separately.

The collapse the whole estimate rests on can be run from two other starting conditions, and it is worth seeing how far the answer moves.

A collapse that stops 257 astronomical units short. Equatorial rotation speed against radius for a collapsing cloud core of 1 solar mass, turning at 3·10⁻¹⁴ 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, 7.1·10¹⁶ 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 3.84·10¹³ metres — 257 astronomical units, or 5.5·10⁴ 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 5.1·10⁴ 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. 6 The same collapse from a core rotating three times faster. Angular momentum is conserved, so the radius at which the collapse stops moves out in proportion to the square of the rotation rate — from a few hundred astronomical units to a few thousand, which is a protostellar disc and not a star.
A collapse that stops 7304 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 0.2 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 1.09·10¹⁵ metres — 7304 astronomical units, or 1.6·10⁶ 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. 7 And from a core four times larger at the same rotation rate. The stopping radius moves out again, and by a larger factor, because the starting angular momentum goes as the square of the radius. No plausible starting condition lets a rotating core collapse to stellar dimensions, which is the angular-momentum problem stated in one picture.

What is fixed and what is free

It helps to be precise about which of a galaxy’s properties this argument claims to set and which it leaves alone.

The rotation speed is not set by λ. It is set by the halo’s mass and concentration, and a disc’s circular speed at a few scale lengths is essentially the halo’s circular speed there. That is why the relation between rotation speed and luminosity is tight while the relation between size and luminosity is not: the first is asking about mass, which is well determined, and the second is asking about spin, which has a factor-of-three spread at every mass. The surface brightness, on the other hand, is set almost entirely by λ. The same mass of stars spread over a disc twice as large is four times fainter per unit area, which is a surface brightness rather than a luminosity, so a high-spin halo makes a low-surface-brightness galaxy. That is the standard explanation for the diffuse discs that dominate by number and contribute almost nothing to the light: they are not failed galaxies, they are ordinary galaxies whose haloes happened to be turning faster than average, sitting in the upper tail of the same lognormal every other disc is drawn from.

It is a satisfying account and it is not fully secured. The most extreme diffuse galaxies require spins in the top few per cent of the distribution, and whether the tail is populated as densely as the model requires is exactly what a survey of such objects is trying to establish — the difficulty being that a galaxy is hardest to find precisely when its light is most spread out.

And the observed relation at a shallower slope, which is the alternative the data have not yet excluded.

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 0.5000 — 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. 8 The specific angular momentum against stellar mass at a slope of a half rather than two thirds. The three tracks keep their vertical separation — that separation is the morphology, and it is the robust part — while the slope is the part the samples disagree about, because it is measured over a mass range where the selection changes.

Where the ladder goes

The direct continuation is backwards in time, to the mechanism that put the spin there — a torque exerted before the halo had collapsed at all, growing linearly with time and switching off at turnaround.

The other direction is forwards, into what happens to a disc afterwards. A disc that keeps its angular momentum stays a disc; one that loses it becomes a spheroid, and the losing happens in mergers. The two-thirds slope in the figure above is the same for both, so morphology is not about how much a galaxy has but about how much it kept — and the mechanism that moves angular momentum around inside an intact disc, without any merger at all, is the spiral pattern itself.

Where this goes next

The rotation curve of NGC 3198, decomposed. Circular speed against radius for a three-component model of NGC 3198: a Hernquist bulge of 1.0×10⁹ M☉, an exponential disc of 2.20×10¹⁰ M☉ with a scale length of 2.6 kpc, and a pseudo-isothermal halo whose asymptotic speed is not chosen but solved for — it is whatever brings the total to the measured 150 km/s at 30 kpc, and comes out at 171 km/s. The components add in quadrature because accelerations add. The disc alone peaks at 5.7 kpc and falls away; the total does not. A rotation curve that refuses to fall Beyond the edge of a galaxy's light there is nothing left to enclose, so the orbital speed should fall away as the inverse square root of radius. It does not fall at all, and the shape of that refusal says the missing mass arrives at a constant rate for as far out as anyone can measure. An average that arrives, and a snapshot that never does. 2T/|U| against time for Burrau's problem — three masses released from rest, and never repeating, in two forms: the instantaneous ratio, and the ratio of the running time-averages. The instantaneous one runs between 0.03 and 1.97 — the system is a long way from equilibrium at almost every moment, because the bodies are alternately falling together and flying apart. The averaged one settles: after 50 time units it is 1.0142, against the exact 1 the virial theorem requires of any bound system. This system is not periodic and is not even permanently bound — Burrau's problem ejects its lightest body — so the average is drifting rather than converged, and that is the theorem's own condition made visible: it is a statement about bound systems and says nothing about anything that is leaving. What the figure cannot show is the error: a cluster observed once gives 2T/|U| with a scatter of this size, and what makes its mass believable is not the measurement but the relaxation. An average that weighs what cannot be watched A cluster's mass can be had from its speeds alone — no orbit followed, no period observed, no distance to any single star. The theorem that allows it is an average over time, which is exactly what a photograph is not. 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. Spin acquired before there was anything to spin Every galaxy turns, and nothing in a smooth expanding universe turns. The rotation was applied while the material was still a mildly overdense patch spread across megaparsecs — torqued by the tidal field of its neighbours, growing steadily with time, and switching off the moment the patch stopped expanding.

What this makes readable

Essays that name this one as a prerequisite.

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.

Adiabatic contractionAngular momentumDark matter haloDisc scale lengthDynamical frictionExponential discGalaxy formationLognormal distributionSpecific angular momentumSpin parameterSurface brightnessTidal torque theoryVirial radiusVirial theorem