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A hole that says mass times time

A gap in a stellar stream is the strongest evidence available that dark subhaloes exist, and its depth depends on the perturber's mass, the impact parameter and the elapsed time only through one combination. Two of those three are unobservable, so a gap is a measurement of a product.

Assumes Stellar streams, Dark matter and Impulse approximation.

16 min read 8 figures Small perturbations, large consequencesMeasuring the unreachable

Cold dark matter predicts, unavoidably, that a galaxy’s halo is lumpy. Structure forms from the bottom up, small haloes merge into large ones, and the survivors orbit inside as subhaloes. Above about a billion solar masses those subhaloes hold stars and are visible as satellite galaxies, each with a dispersion that says how much mass is not light in it. Below it they hold nothing that emits, and there are supposed to be thousands of them.

How many is the question the whole exercise exists to answer, and the disagreement is old. Simulations of a Milky-Way-sized halo produce hundreds of subhaloes above ten million solar masses inside thirty kiloparsecs; the count of satellite galaxies is a few dozen. That gap has been closed from the visible end — deeper surveys keep finding fainter satellites, and the physics of how star formation shuts off in a small halo keeps improving — but closing it from the visible end can never rule out the alternative, which is that the small haloes are not there because dark matter is not cold. A warm particle erases structure below a mass set by how far it streamed before it slowed down, and the erasure is sharp. Distinguishing the two needs a count of objects that contain no stars at all.

A thin stellar stream is the only instrument anyone has found for detecting an object that emits nothing, is not massive enough to lens, and is not dense enough to be found by its own gravity. The stream is dynamically cold — its stars share an orbit to within a few kilometres a second — so a perturbation of a few kilometres a second is not a small correction to it. It is everything.

A gap's depth is one number: 2GmT ÷ v b². The central density of the gap against impact parameter, 4 billion years after the encounter, for subhaloes of 10⁶, 10⁷, 10⁸ solar masses. Points are read off the drawn density profiles; the curves are 1/(1 + 2GmT/v b²), the stretch the map applies at the encounter point, and the two agree to 3.0 per cent wherever the histogram has enough stars left in the gap to measure a depth at all. Everything about the encounter enters the depth through that one combination. The consequence is the horizontal reading: a gap of a given depth is produced by every point along a locus on which the mass rises as the square of the impact parameter — half depth at 0.42, 1.33, 4.19 kiloparsecs for the three masses drawn, an exponent of 0.500 against the half the algebra requires. What breaks that particular degeneracy is the gap's width, which scales as the impact parameter itself while the depth does not: rescale s by b and the map is identical, so the profile is one shape stretched. Depth and width together give the impact parameter and the product mT. They do not give the mass.
Fig. 1 The depth of the hole a passing subhalo leaves, four billion years afterwards, against how close it passed, for three subhalo masses. The points are read off simulated density profiles and the curves are 1/(1 + 2GmT/v b²) — the factor by which the encounter stretches the stream at the closest point, which is what a dilution is. Everything about the encounter reaches the depth through that one combination. A gap of half the undisturbed density is produced at 0.42, 1.33 and 4.19 kiloparsecs for the three masses, an impact parameter rising exactly as the square root of the mass.

The rest of this essay is about how much of that one combination can be taken apart, and the answer is: about half of it.

The mechanism, and why nothing collides

The first rung of this anchor established the stream as an object: stars leaking through the two Lagrange saddles of a dissolving cluster, each carrying a small energy offset, spreading along a track that is not the progenitor’s orbit. What follows here happens to that track.

A subhalo passing at a few kiloparsecs never touches a star. What it does is deliver an impulse — the deflection obtained by integrating the force along the path the star would have followed anyway — and the component of that impulse along the stream is what matters, because the stream is a one-dimensional object and motion across it is quickly forgotten.

A gap that opens where nothing hit anything, and widens with age. The density along a thin stream 0.5, 1.5, 4 billion years after a 5·10⁶ solar-mass subhalo passed it at 0.4 kiloparsecs, in units of the undisturbed density. Nothing collided. The subhalo gave each star a small velocity kick, larger for the stars it passed closest to, and directed forward on one side and backward on the other; a change of velocity in a stream is a change of orbital energy and therefore of the rate at which a star moves along it. The two reversals in that sentence matter and nearly cancel: the kick pulls stars toward the encounter point, but a star given energy moves outward onto a slower orbit, so the stars ahead end up running faster and those behind slower, and the stream is pulled apart at the point of closest approach rather than piled up there. The earliest profile shows the beginning of it. By the latest, the centre is emptied to 0.16 of its former density and the material has gone into two shoulders either side. The gap widens from 1.7 to 3.1 kiloparsecs between 1.5 and 4 billion years, and it goes on widening, which is what makes the width a measurement of how long ago the encounter happened. Counting gaps along real streams is therefore a way of counting dark subhaloes that emit nothing and have never been seen any other way.
Fig. 2 The density along a stream half a billion, one and a half billion and four billion years after a five-million solar-mass subhalo passed within 0.4 kiloparsecs. Nothing collided. The kick pulls stars toward the encounter point, but a star given energy moves outward onto a slower orbit, so the two reversals leave the stars ahead running faster and those behind running slower — and the stream is pulled apart at the encounter point rather than piled up there. The hole deepens and widens with time, which is what makes the picture a clock as well as a scale.

The magnitudes involved are worth stating because they are so small. A ten-million solar-mass subhalo passing at one kiloparsec with a relative speed of two hundred kilometres a second delivers a kick of about 0.4 kilometres a second — a tenth of the escape speed from a globular cluster, a thousandth of the orbital speed. Over four billion years that translates into an along-stream drift of about 1.7 kiloparsecs, because a stream is a marker moving at a fixed rate and four billion years is a long lever. The detectability of the encounter has almost nothing to do with the size of the kick and almost everything to do with the coldness of what it is applied to and the length of time it has had.

The sign is worth dwelling on because it is the one part of the mechanism that is genuinely counter-intuitive, and getting it backwards draws a caustic where the gap belongs. Two reversals nearly cancel: the impulse points toward the perturber’s track, and the response of an orbit to added energy is to slow down. The survivor of that near-cancellation is a stretching, not a compression.

Why the depth is a Jacobian

The closed form in the first figure is not fitted and it is not an approximation to a simulation. It is a change of variables.

Stars are conserved, so the density after the map is the density before, divided by the map’s own stretching factor. At the encounter point the stretching factor is

dds(s+T δv(s))∣s=0  =  1+2GmTvrel b2\frac{\mathrm{d}}{\mathrm{d}s}\Big(s + T\,\delta v(s)\Big)\bigg|_{s=0} \;=\; 1 + \frac{2GmT}{v_{\rm rel}\,b^{2}}

because the derivative of s/(s2+b2)s/(s^2+b^2) at the origin is 1/b21/b^2. The depth is the reciprocal of that. It is exact for as long as the map remains one-to-one, which is until the stream folds — and a folded stream produces caustics rather than a gap, which is a different figure and a different observational signature.

That derivation is why the depth carries m/b2m/b^2 and the width carries bb: the first is a derivative at a point and the second is the scale over which the derivative changes.

One combination, and what it costs

The map that opens the gap is

s  ⟼  s+T δv(s),δv(s)=2Gm svrel (s2+b2)s \;\longmapsto\; s + T\,\delta v(s), \qquad \delta v(s) = \frac{2Gm\,s}{v_{\rm rel}\,(s^2 + b^2)}

and reading it is the whole of the difficulty. The mass and the elapsed time appear only as their product. There is no operation on the density that separates them.

4 times the mass, one profile. Density along a stream after encounters with 3 subhaloes whose masses differ by a factor of 4 — 10⁷ M☉ 1 Gyr ago, 5·10⁶ M☉ 2 Gyr ago, 2.5·10⁶ M☉ 4 Gyr ago — all at an impact parameter of 0.6 kiloparsecs. The curves are the same curve. The velocity kick is proportional to the mass and the drift it produces is proportional to the time, so the map that opens the gap contains the two only as their product; a light subhalo long ago and a heavy one recently are indistinguishable in the density, not approximately but identically. The depth all three reach, 0.45 of the undisturbed density, is 1/(1 + 2GmT/v b²) — the stretch of the map at the encounter point, which is what a dilution is — and the drawn profiles agree with that closed form to 0.2 per cent. Nothing observes when the encounter happened. The stream's own age is a few billion years and the encounter could have been at any point in it, so a gap's depth converts into a subhalo mass only after a time has been assumed.
Fig. 3 Three encounters with the same impact parameter, drawn from the same map: a ten-million solar-mass subhalo one billion years ago, five million two billion years ago, and two and a half million four billion years ago. The three profiles are one profile. This is not a near-coincidence to be broken with better data — the product mT is the only way either quantity enters, so any measurement of the density is a measurement of that product and of nothing else.

The elapsed time is not observable. A stream is a few billion years old, the encounter happened at some point during that, and nothing about the gap says when. So the honest statement of what a gap measures is a product with a factor of several in it before any other uncertainty is considered.

12 times the mass, one profile. Density along a stream after encounters with 3 subhaloes whose masses differ by a factor of 12 — 3·10⁷ M☉ 0.5 Gyr ago, 7.5·10⁶ M☉ 2 Gyr ago, 2.5·10⁶ M☉ 6 Gyr ago — all at an impact parameter of 1 kiloparsecs. The curves are the same curve. The velocity kick is proportional to the mass and the drift it produces is proportional to the time, so the map that opens the gap contains the two only as their product; a light subhalo long ago and a heavy one recently are indistinguishable in the density, not approximately but identically. The depth all three reach, 0.60 of the undisturbed density, is 1/(1 + 2GmT/v b²) — the stretch of the map at the encounter point, which is what a dilution is — and the drawn profiles agree with that closed form to 0.4 per cent. Nothing observes when the encounter happened. The stream's own age is a few billion years and the encounter could have been at any point in it, so a gap's depth converts into a subhalo mass only after a time has been assumed.
Fig. 4 The same construction over a wider range of epochs — thirty million solar masses half a billion years ago against two and a half million six billion years ago, a factor of twelve, at an impact parameter of one kiloparsec. The depth is 0.60 in all three cases. Twelve is roughly the span of subhalo masses the whole method is trying to distinguish, and it is the span a single unmeasured epoch covers.

What the width does break

Depth is not the only observable. The gap has a width too, and the width scales differently.

Rescale the along-stream coordinate by the impact parameter and the map above becomes independent of bb altogether: the profile is one shape, stretched horizontally in proportion to how far away the perturber passed. So the width gives the impact parameter directly, once the depth has fixed the combination.

A gap's depth is one number: 2GmT ÷ v b². The central density of the gap against impact parameter, 2 billion years after the encounter, for subhaloes of 3·10⁵, 3·10⁶, 3·10⁷ solar masses. Points are read off the drawn density profiles; the curves are 1/(1 + 2GmT/v b²), the stretch the map applies at the encounter point, and the two agree to 6.6 per cent wherever the histogram has enough stars left in the gap to measure a depth at all. Everything about the encounter enters the depth through that one combination. The consequence is the horizontal reading: a gap of a given depth is produced by every point along a locus on which the mass rises as the square of the impact parameter — half depth at 0.16, 0.51, 1.62 kiloparsecs for the three masses drawn, an exponent of 0.500 against the half the algebra requires. What breaks that particular degeneracy is the gap's width, which scales as the impact parameter itself while the depth does not: rescale s by b and the map is identical, so the profile is one shape stretched. Depth and width together give the impact parameter and the product mT. They do not give the mass.
Fig. 5 The same depth relation at two billion years instead of four, for masses of three hundred thousand, three million and thirty million. Half-depth now occurs at 0.16, 0.51 and 1.62 kiloparsecs — every impact parameter smaller by the square root of two, because halving the time halves the product and the locus moves with it. The exponent along the locus is 2.00 in both figures: it is a property of the map, not of the numbers put into it.

So two observables give two combinations: the impact parameter, cleanly, and the product of mass and time. Three unknowns, two equations, and the missing one is the epoch.

That is not the end of the matter, because a real stream is perturbed more than once, and the encounters have different epochs. The distribution of gap widths and depths along a long stream, taken together, does constrain the joint distribution of masses and times — but as a population, statistically, and under an assumed history for the stream. No individual gap yields a mass.

A gap that opens where nothing hit anything, and widens with age. The density along a thin stream 0.5, 1.5, 4 billion years after a 10⁸ solar-mass subhalo passed it at 2 kiloparsecs, in units of the undisturbed density. Nothing collided. The subhalo gave each star a small velocity kick, larger for the stars it passed closest to, and directed forward on one side and backward on the other; a change of velocity in a stream is a change of orbital energy and therefore of the rate at which a star moves along it. The two reversals in that sentence matter and nearly cancel: the kick pulls stars toward the encounter point, but a star given energy moves outward onto a slower orbit, so the stars ahead end up running faster and those behind slower, and the stream is pulled apart at the point of closest approach rather than piled up there. The earliest profile shows the beginning of it. By the latest, the centre is emptied to 0.18 of its former density and the material has gone into two shoulders either side. The gap widens from 7.3 to 12.9 kiloparsecs between 1.5 and 4 billion years, and it goes on widening, which is what makes the width a measurement of how long ago the encounter happened. Counting gaps along real streams is therefore a way of counting dark subhaloes that emit nothing and have never been seen any other way.
Fig. 6 A hundred-million solar-mass subhalo at two kiloparsecs, which is the regime where the method is unambiguous and where nothing dark is expected to be. A subhalo this massive would ordinarily hold stars and be a visible satellite; the interest is entirely in the decade or two below. The figure is drawn to show what the easy case looks like, and how much easier it is than anything the method is actually asked to do.

The other things that make gaps

Even granted a perfectly measured gap, attributing it to a dark subhalo requires that nothing else in the galaxy makes one, and several things do.

Giant molecular clouds. A cloud of a few million solar masses is exactly the mass range in question, is entirely capable of passing a stream, and is not dark. For a stream on an orbit that crosses the disc — most of them do — the encounter rate with clouds is comparable to or larger than the rate with subhaloes at the same mass, because although the clouds are confined to a thin layer they are far more numerous within it. The two are distinguishable only statistically, by where along the orbit the encounters happened, and the statistics need more streams than are known.

The bar. A rotating bar drives resonances that heat and structure a stream on scales of kiloparsecs, and the effect grows for streams on orbits that reach into the inner galaxy. It produces density variations that are periodic in a way an encounter is not, which is a discriminator, but only for a stream long enough to show the periodicity. Palomar 5’s stream is about twenty kiloparsecs long and its orbit reaches within eight kiloparsecs of the centre, so it is exactly the case where the bar matters and the periodicity is barely resolvable.

Spiral arms. Transient and recurrent, of the right amplitude, and the least well characterised of the three.

The progenitor itself. Stars leave a cluster in bursts at each pericentre passage, and the epicyclic motion of the escaping stars produces alternating overdensities near the release point. These are periodic, they are strongest near the progenitor, and they are the reason the first claimed gap detections concentrated on the parts of streams furthest from where the cluster used to be. Their spacing is set by the ratio of the epicyclic frequency to the orbital one, so it is a property of the host potential rather than of the stream — which makes it predictable, and therefore removable, in a way the other three are not.

The halo itself, if it is not smooth. A halo assembled from mergers retains large-scale structure — shells, sheets, the debris of earlier accretions — on scales far larger than a subhalo and with no particular symmetry. That contribution is not a gap-maker so much as a slowly varying background, and it is the hardest of all to characterise, because a galaxy measured from inside it gives no external view of its own halo’s lumpiness.

A stream that is not the orbit it came from. 556 stars released in pairs from the two saddles of a 5·10⁴ solar-mass cluster over 4.9 billion years, integrated in a halo whose circular speed is 200 kilometres a second, drawn with the progenitor's own orbit. The cluster runs between 12 and 30 kiloparsecs and the orbit is the thin closed-looking curve; the stars are everything else. The point of the figure is the discrepancy. Stars leaving through the inner saddle are on slightly smaller orbits, turning round at a median of 29.78 kiloparsecs rather than the progenitor's 30.00, and therefore running ahead; stars leaving through the outer saddle reach 30.21 and fall behind. The whole spread is 1.4 per cent of the apocentre, which is the number worth carrying: an offset far too small to see in this drawing builds the entire stream, because it acts for four billion years. The two arms are therefore not merely displaced along the orbit, they are on different orbits, and the track a survey measures is a family of them rather than any single one. Fitting a Galactic potential by demanding that a stream lie along an orbit is wrong by exactly this much, and the size of the error grows with the mass of the progenitor, because the mass is what sets the distance between the two doors.
Fig. 7 A stream from a fifty-thousand solar-mass progenitor on a moderately eccentric orbit over five billion years, with the progenitor’s own orbit drawn beside it. The stream is not on that orbit and the departure grows with distance from the release point. Every gap analysis needs the unperturbed density along the track as its baseline, and the unperturbed density is not uniform: it varies with the release rate, which varies with the pericentre passages, which are what set the shape drawn here.

The list is not a set of caveats appended to a result. It is the reason the field has moved from counting gaps to modelling the whole density field of a stream with every known perturber included, and comparing the residual — which is a much weaker constraint and a much more honest one.

What is actually measured

The observation is a count of stars. A stream is identified as an overdensity in position, proper motion and colour–magnitude space; its members are selected by a cut in all three; and the density along it is a histogram of those members.

Every step of that leaks into the answer. A selection cut that varies along the stream — because the survey’s depth varies, or because a foreground cloud of dust sits across part of it, or because the stream crosses a region of higher stellar density — produces density variations that are indistinguishable from gaps in the counts. The Palomar 5 stream, the best-studied case, has had gaps claimed, disputed, attributed to the bar, and re-attributed to the selection function, in that order.

Two streams carry most of the weight and they are worth naming, because two is a small number for a statistical argument. Palomar 5 is the long tail of a dissolving globular cluster on an orbit that reaches into the inner galaxy — bright, well populated, and exposed to every confuser in the list above. GD-1 is longer, thinner, and further out, so it is much less exposed to the disc, and its progenitor is gone entirely, which removes the epicyclic problem and removes the baseline with it. GD-1 shows a pronounced underdensity and an accompanying spur of stars offset from the track, which is the signature an encounter should produce and which no confuser predicts as naturally. It is the single most-cited piece of evidence for a dark subhalo, and it is one feature in one stream.

The proper-motion data that made both of these measurable are recent. Selecting stream members by their motion rather than only by their position and colour cuts the foreground contamination by more than an order of magnitude, and it is the reason the field went from arguing about whether streams have structure to arguing about what causes it.

Two doors, 1.14 tidal radii apart on either side, and neither at the same height. The effective potential along the line joining a 2·10⁵ solar-mass cluster to the centre of its host galaxy, in the frame that turns with the cluster's orbit at 16.0 kiloparsecs. The deep well in the middle is the cluster; the two maxima either side of it are the saddles of the combined field, and they sit at 1.14 tidal radii, which is what the tidal radius is defined to approximate. A star that wanders above the level of a saddle can leave, and it can only leave through one of the two: inward, toward the galactic centre, or outward, away from it. That is the first fact about a stream and it is geometric rather than statistical. The second is the energy. A star released at the inner saddle is on a slightly smaller orbit than the cluster and one released at the outer saddle on a slightly larger one, differing by about 781 squared kilometres a second either way, and in a galaxy where the period grows with radius the inner star runs ahead while the outer one falls behind. Two arms, one leading and one trailing, from one door each.
Fig. 8 And the baseline problem at its source: the two saddle points through which stars leave a two-hundred-thousand solar-mass progenitor, 1.14 tidal radii either side, at slightly different heights in the effective potential. The asymmetry means the two tails are not released at the same rate or with the same energy spread, so the leading and trailing arms of a real stream have different intrinsic density profiles before anything perturbs them.

There is a second measurement problem underneath the first, and it is about what a stream is made of. A stream’s intrinsic density profile depends on how its progenitor dissolved — a cluster boiling itself away loses stars at a rate that depends on its own internal relaxation as much as on the tide, and the rate changes as the cluster shrinks. So the baseline against which a gap is measured is the output of a model of a cluster that no longer exists, whose initial mass and concentration are inferred from the stream it left.

The honest summary is that the method is sound, the instrument is exquisite, and the systematics are of the same order as the signal. What has been established is that stream densities are not uniform at a level consistent with a lumpy halo. What has not been established is a mass function of dark subhaloes, and the reason is contained in this essay’s title.

The generalisation

The structure worth extracting is that a perturbation seen long afterwards records an impulse, and an impulse is a product.

The kick a passing body delivers is proportional to its mass over its speed and impact parameter; the displacement that kick produces is proportional to the time since. Nothing about the observed displacement remembers which factor was large. That is a general property of anything measured after the fact rather than during, and it is why so much of this collection’s dynamics is about finding a second observable with a different time dependence.

The contrast with the rest of this field is instructive. The mass that is not the light is measured from a rotation curve, which is an instantaneous balance and carries no history at all; three mass models fitting the same curve is a degeneracy of a different kind, between profiles rather than between a product’s factors, and it is broken by measuring at more radii. A gap cannot be broken that way, because there is only one gap.

A collisional family is dated by a scatter plot precisely because it has one — the drift rate depends on a body’s size, so the family’s V-shape in size against semi-major axis separates the age from the strength of the drift. A stream gap has no such handle, because every star in it responds to the impulse identically.

A second shape worth naming is what the width does here, since it is the only degeneracy in this essay that gets broken. It is broken because the two observables depend on the unknowns through different functional forms — one on a derivative at a point, the other on a scale — rather than because one of them is more precise. Precision never breaks a degeneracy; a different dependence does. The mass and the epoch enter the map through one product and there is no second form for them to enter through, which is why no improvement in the data can help and why the next rung goes to a statistical argument instead.

The corollary is a rule for judging a new technique. Count the unknowns before counting the significance. A detection at five sigma of a quantity that is a product of three things, two of them unmeasured, is a five-sigma detection of the product.

What would settle it

It is worth saying what a decisive measurement would look like, because the shape of it explains why nobody has one.

Two things are needed. The first is many streams — dozens, spanning a range of orbits, so that the encounter rate can be compared between streams that cross the disc and streams that do not, which separates the clouds from the subhaloes by their spatial distribution rather than by their individual signatures. The second is velocities across each gap, since an encounter leaves a characteristic pattern in the line-of-sight velocity as well as in the density, and the velocity signature is not degenerate in the same way — it retains information about the direction the perturber was moving.

Neither is a matter of a better idea. Both are catalogues, of a depth and a spectroscopic completeness that do not yet exist, and both are among the stated purposes of the surveys now being built.

Where the ladder goes next

The next rung leaves the individual gap and takes the stream’s whole density as a random field. Many weak encounters superpose, and the resulting power spectrum of density fluctuations along the stream has an amplitude and a slope that depend on the subhalo mass function — statistically, over the population, with no epoch assumed for any single event. That is where the method’s constraints actually come from, and it is a considerably less photogenic argument than a hole.

Further rungs on this anchor: the velocity structure across a gap, which is a second observable and is beginning to be measurable; streams from globular clusters against streams from dwarf galaxies, which are thicker and much less sensitive; the use of a stream’s track to measure the shape of the halo potential, which is a completely different question asked of the same data; and the gaps that would be produced by a primordial black-hole population, which is the same arithmetic with a different mass function.

Where this goes next

A stream that is not the orbit it came from. 456 stars released in pairs from the two saddles of a 10⁵ solar-mass cluster over 4.0 billion years, integrated in a halo whose circular speed is 220 kilometres a second, drawn with the progenitor's own orbit. The cluster runs between 10 and 25 kiloparsecs and the orbit is the thin closed-looking curve; the stars are everything else. The point of the figure is the discrepancy. Stars leaving through the inner saddle are on slightly smaller orbits, turning round at a median of 24.77 kiloparsecs rather than the progenitor's 25.00, and therefore running ahead; stars leaving through the outer saddle reach 25.22 and fall behind. The whole spread is 1.8 per cent of the apocentre, which is the number worth carrying: an offset far too small to see in this drawing builds the entire stream, because it acts for four billion years. The two arms are therefore not merely displaced along the orbit, they are on different orbits, and the track a survey measures is a family of them rather than any single one. Fitting a Galactic potential by demanding that a stream lie along an orbit is wrong by exactly this much, and the size of the error grows with the mass of the progenitor, because the mass is what sets the distance between the two doors. A stream is not the orbit it came from A cluster torn apart by a galaxy leaves a thin trail of stars across the sky, and the obvious thing to do with it is fit an orbit. That is wrong by a knowable amount, because stars leave through two doors with a small energy offset and end up on a family of orbits rather than on one. The whole kick, delivered in about two encounter times. The transverse force a body feels while a mass sweeps past it on a straight line, and the velocity that force has delivered so far, both against time in units of the impact parameter divided by the relative speed. The force is the component of the inverse-square attraction perpendicular to the path, which is the impact parameter over the cube of the distance, and it is drawn at its peak value of one at closest approach. The rising curve is its running integral, scaled by twice the gravitational constant times the mass over the impact parameter and the speed. Two things are visible and both are the point. The integral of the force over all time is exactly two in these units, so the kick is exactly 2GM/bv with no free constant anywhere — an answer to a three-body-shaped question obtained without solving anything. And it arrives quickly: 71 per cent of it within a single encounter time of closest approach and 98.6 per cent within the 6 drawn, which is what licenses calling the whole thing an impulse. On the scale of anything slower, the velocity changes discontinuously. An answer obtained along a path that was not taken Integrate the force of a passing mass along the straight line the body would have followed if the encounter had not happened, and out comes an exact deflection with no free constant in it. The approximation is circular, it is wrong in a known direction by a known amount, and it is the reason stellar dynamics has closed forms at all. The mass-to-light ratio of NGC 3198, radius by radius. The dynamical mass inside each radius divided by the light inside it, in solar units. Inside two disc scale lengths it is nearly flat at about 2.2 — a galaxy made of stars, weighing what stars weigh. Outside the disc the light stops and the mass does not, so the ratio climbs to 10 by 30 kpc with no sign of turning over. The curve is not fitted: it is the rotation curve's enclosed mass divided by the light profile's enclosed light, both drawn elsewhere in this collection. The mass that is not the light A galaxy's mass-to-light ratio is not a number, it is a curve, and it rises without turning over. What stellar populations can plausibly weigh sets a ceiling; the dynamics sit far above it, and the gap is a shape rather than a discrepancy.

The objects this essay names

Each one links to every other essay that touches it.

Cold dark matterDark subhaloEpicyclic overdensityGalactic barGiant molecular cloudImpulse approximationStellar streamStream gapTidal radiusVelocity kick