Astronomy & Celestial Mechanics
the sky, computed
Gravitation

The part of a kick that heats nothing

Most of the impulse a passing mass delivers to a star cluster is delivered equally to every star in it, so the cluster moves and nothing inside it changes. What heats it is the difference across it — one power of the impact parameter smaller, squared in the energy — and that one distinction decides which encounters matter and which cannot.

Assumes Impulse approximation, Two-body relaxation and Virial theorem.

17 min read 8 figures Small perturbations, large consequencesTwo bodies, and no more

The first rung of this anchor established the device and its limits: integrate the force along the path the body would have taken had nothing happened, and get an exact deflection with no free constant in it, valid when the deflection is small and circular when it is not.

Apply that device to a star cluster rather than to a star and something happens which the first rung had no occasion to notice. The kick is delivered to every star in the cluster, and to a very good approximation it is the same kick. A cluster ten parsecs across, passed at a hundred parsecs, sees an impulse that varies by a few per cent from one side of it to the other. Ninety-odd per cent of the impulse simply moves the cluster.

A cluster that has been moved is not a cluster that has been changed. Its internal energy is untouched; its stars are exactly where they were relative to one another, moving exactly as they were. The heating is entirely in the residue after the common part is subtracted, and that residue is smaller by one power of the impact parameter.

The kick falls as b⁻² and the heating as b⁻⁴. Two quantities delivered by the same distant encounter, against impact parameter in units of the target's half-mass radius, both logarithmic and both scaled to cross near one. The upper line is the impulse itself, 2Gm/vb, which every star in a bound system receives almost equally — so it moves the system and changes nothing inside it. The lower line is what is left after the common part is subtracted: the difference of the impulse across the system, which is its gradient multiplied by the system's own size, one power of b smaller and squared in the energy. The slopes are −2 and −4 exactly, and they are measured off the drawn curves rather than quoted. What follows is the point. Encounters at impact parameter b arrive at a rate proportional to b db, so summing the impulse over all of them gives ∫b⁻¹ db, which diverges logarithmically and is the origin of the Coulomb logarithm that appears in every treatment of relaxation. Summing the heating gives ∫b⁻³ db, which converges: extending the population from ten half-mass radii out to 300 multiplies the summed impulse by 2.48 and the summed heating by 1.010. Distant encounters diffuse velocities and heat nothing, and a tidal-heating calculation therefore needs no cutoff at large impact parameter, where a relaxation calculation cannot proceed without one.
Fig. 1 Two quantities delivered by the same distant encounter, against impact parameter in units of the target’s half-mass radius. The upper line is the impulse itself, which falls as b⁻² and does no heating. The lower is the difference of that impulse across the system — its gradient times the system’s own size — which falls as b⁻⁴. The consequence is in the labels: encounters arrive at a rate proportional to b db, so summing the impulse over ever more distant encounters gives an integral that keeps growing, ×2.48 on extending the population from ten half-mass radii to two hundred, while the same extension multiplies the summed heating by 1.010. Distant encounters diffuse velocities and heat nothing.

Two powers, and what each one buys

The arithmetic is worth setting out because the whole of the essay is one subtraction.

A star at position r\mathbf{r} relative to the cluster’s centre receives an impulse Δv(r)\Delta \mathbf{v}(\mathbf{r}). Expanding about the centre,

Δv(r)  =  Δv(0)  +  (r⋅∇) Δv  +  …\Delta\mathbf{v}(\mathbf{r}) \;=\; \Delta\mathbf{v}(0) \;+\; (\mathbf{r}\cdot\nabla)\,\Delta\mathbf{v} \;+\;\ldots

The first term is the same for every star and is a translation. The second is the tide — it is what a tide always is, a difference of a force rather than a force — and it carries an extra factor of r/br/b because differentiating 1/b1/b brings down a power. The energy is quadratic in the velocity change, so the heating carries (r/b)2(r/b)^2 relative to the bulk kinetic energy, and the whole thing goes as

ΔE  =  43 G2m2v2b4 M ⟨r2⟩\Delta E \;=\; \frac{4}{3}\,\frac{G^2 m^2}{v^2 b^4}\,M\,\langle r^2\rangle

for an isotropic system. The mean square radius appears because a bigger cluster catches more of the gradient. The fourth power of bb is the point of it.

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 99.2 per cent within the 8 drawn, which is what licenses calling the whole thing an impulse. On the scale of anything slower, the velocity changes discontinuously.
Fig. 2 Where the kick comes from, drawn over eight encounter times either side of closest approach. The transverse force rises, peaks and falls, and the cumulative deflection reaches its final value within about two encounter times of the moment of closest approach — which is what justifies calling it an impulse and what makes the encounter time the relevant clock. Everything in this essay is about the difference between two of these curves evaluated at slightly different places.

The factor of 4/34/3 and the ⟨r2⟩\langle r^2\rangle both deserve a sentence. The four-thirds comes from averaging the tidal tensor over directions: the tide stretches along one axis and compresses along the other two, and squaring before averaging leaves that particular number for an isotropic system. It is not a fudge and it is not adjustable. The mean square radius is the cluster’s own moment of inertia per unit mass, which means the heating is dominated by the stars furthest out — a cluster with a long tenuous envelope is shocked far harder, per unit mass, than a compact one of the same mass.

Both facts point the same way, and they are why an already-loose cluster loses more and becomes looser. The alternative reading, that a shock deposits energy uniformly and a cluster expands as a whole, is what the impulse approximation would say if the translation were not subtracted, and it is wrong in the same direction everywhere.

The divergence that does not happen

The difference between b−2b^{-2} and b−4b^{-4} is not a detail of two exponents. It decides whether a calculation needs to be told where to stop.

Encounters at impact parameter between bb and b+dbb + \mathrm{d}b arrive at a rate proportional to b dbb\,\mathrm{d}b, because that is the area of the annulus. So summing the velocity diffusion over all impact parameters gives ∫b−2⋅b db=∫db/b\int b^{-2}\cdot b\,\mathrm{d}b = \int \mathrm{d}b/b, which is logarithmic and diverges at both ends. That integral is the Coulomb logarithm. It is finite only because somebody chooses a maximum impact parameter — the size of the system, usually — and the whole theory of two-body relaxation carries a factor whose value depends on that choice, which is the difficulty a drag computed with a logarithm nobody can pin down is entirely about.

Equal energy from every decade of impact parameter, totalling 6.12. How much energy a passing population delivers, split by impact parameter on a logarithmic axis running from a tenth of the ninety-degree radius to 300 times it. The flat curve is the contribution from each decade: distant encounters are weak, close ones are strong, and there are exactly enough more of the former to cancel the difference, so every decade contributes the same. That cancellation is the reason a Coulomb logarithm appears in every relaxation calculation in stellar dynamics, and it is the reason the answer depends on the ratio of the largest useful impact parameter to the smallest rather than on either of them. The rising curve, on the right-hand scale, is the running total; it reaches 6.125, against a logarithm of 5.704 for the ratio drawn. The left-hand end is the part usually skipped. Written with the impulsive kick the contribution stays flat all the way down and the integral diverges, and the textbook cuts it off at the ninety-degree radius by hand. Written with the exact deflection it falls as the fourth power of the impact parameter instead, so it converges on its own and the hand-placed cutoff is a convenience rather than a necessity.
Fig. 3 The relaxation integral, drawn as equal contributions from every decade of impact parameter — which is what a logarithm looks like. The total here is 6.12 for a ratio of three hundred between the largest and smallest impact parameters, and it would be 6.9 for a ratio of a thousand: the answer depends on the cutoff and depends on it weakly, which is the least satisfactory kind of dependence, because the weakness is what allows the choice to go unexamined.

Summing the heating gives ∫b−4⋅b db=∫db/b3\int b^{-4}\cdot b\,\mathrm{d}b = \int \mathrm{d}b/b^3, which converges at large bb. There is no cutoff to choose, because the answer is dominated by the closest encounters and the distant ones contribute a negligible tail.

The kick falls as b⁻² and the heating as b⁻⁴. Two quantities delivered by the same distant encounter, against impact parameter in units of the target's half-mass radius, both logarithmic and both scaled to cross near one. The upper line is the impulse itself, 2Gm/vb, which every star in a bound system receives almost equally — so it moves the system and changes nothing inside it. The lower line is what is left after the common part is subtracted: the difference of the impulse across the system, which is its gradient multiplied by the system's own size, one power of b smaller and squared in the energy. The slopes are −2 and −4 exactly, and they are measured off the drawn curves rather than quoted. What follows is the point. Encounters at impact parameter b arrive at a rate proportional to b db, so summing the impulse over all of them gives ∫b⁻¹ db, which diverges logarithmically and is the origin of the Coulomb logarithm that appears in every treatment of relaxation. Summing the heating gives ∫b⁻³ db, which converges: extending the population from ten half-mass radii out to 1000 multiplies the summed impulse by 3.00 and the summed heating by 1.012. Distant encounters diffuse velocities and heat nothing, and a tidal-heating calculation therefore needs no cutoff at large impact parameter, where a relaxation calculation cannot proceed without one.
Fig. 4 The same comparison extended to a thousand half-mass radii. Extending the population that far multiplies the summed impulse by 3.00 and the summed heating by 1.012. The two curves are separated by four decades in energy at the outer edge, and the separation grows as the square of the impact parameter — so any question of the form “how far out do the encounters matter?” has a definite answer for heating and no answer at all for relaxation.

There is a piece of physics hiding in the asymmetry and it is worth naming, because it explains why the two integrals behave differently rather than merely recording that they do. Velocity diffusion is a random walk: each distant encounter contributes a small random kick, the kicks add in quadrature, and the number of encounters grows as b2b^2 while the effect of each falls as b−2b^{-2} — so every decade of impact parameter contributes equally and the total is a logarithm. Tidal heating is not a random walk. Each encounter’s contribution falls as b−4b^{-4} and the number grows as b2b^2, so a decade of impact parameter contributes a hundredth of what the decade inside it did, and the sum is a geometric series that has already converged.

The difference is one power of bb, and it is the difference between a quantity that has a value and a quantity that has a value only after somebody chooses a boundary.

The other subtraction: what the system gives back

The tidal term is what is left after the translation is removed. There is a second removal, and it is subtler because it does not come from geometry.

A star inside the cluster is not free. It is on an orbit, with a period of its own, and whether it keeps the energy it is given depends on how that period compares with the duration of the encounter.

What a bound system keeps, against how slow the encounter was. The energy a harmonic oscillator retains from a passing mass, divided by what it would have taken if it were unbound, against the product of its own angular frequency and the encounter time. The vertical axis is logarithmic and spans five decades. At the left the encounter is over before the oscillator has moved and it takes the full impulsive kick; at the right it follows the perturbation quasi-statically, gives the energy back on the way out, and keeps a fraction that falls by a factor of ten for every 1.34 added to the product. That is adiabatic invariance, drawn: the action of a slowly perturbed oscillator is conserved, so a system tightly bound compared with the duration of the disturbance is protected from it. The curve is the squared Fourier amplitude of the forcing at the oscillator's frequency; the dots are obtained a completely different way, by starting the oscillator at rest and stepping the equation of motion through the whole encounter, and the two agree to 0.01 per cent. The consequence for a real system is that an encounter does not heat it uniformly. The outer parts, whose orbital periods are long, are struck impulsively and lose stars; the core, whose periods are short, barely notices.
Fig. 5 The energy a bound oscillator keeps from an encounter, against how slow that encounter was in units of the oscillator’s own period. Struck quickly, it keeps everything the impulse approximation predicts. Perturbed slowly, it follows the perturbation quasi-statically, returns at the end what it took at the beginning, and keeps a residue that falls faster than any power. The suppression is computed here twice — once as the squared Fourier amplitude of the forcing at the oscillator’s frequency, once by integrating the equation of motion from rest — and the two agree, which they must by Parseval’s theorem and by nothing else.

That suppression is not uniform across a cluster, because the orbital period is not. Deep in the core the orbital frequency is high and stars respond adiabatically; in the outskirts the frequency is low and stars are struck.

A shock that strips the outside and leaves the core. The energy a passing tide deposits per unit mass inside a cluster, against radius in units of the half-mass radius, both axes logarithmic. The dashed line is the impulse answer: the tidal force grows linearly with distance from the centre, so the energy grows as its square, and on that reckoning the outer parts absorb everything. The solid curves are the same shock after the adiabatic correction, for shocks lasting 0.15, 0.4, 1 times the half-mass crossing time. A star deep in the core orbits many times while the shock is happening; it follows the perturbation quasi-statically, gives back at the end what it took at the beginning, and keeps almost nothing. A star in the outskirts orbits once in far longer than the shock lasts, is struck before it can respond, and keeps all of it. The suppression factor is (1 + (ωτ)²)^(−3/2), and the radius inside which the shock delivers less than half of the impulse answer goes from impulsive to half at 1.29 r_h as the shock is slowed from 0.15 to 1 crossing times. The two effects therefore multiply in the same direction: the heating is largest outside and the protection largest inside, and a shocked cluster is peeled rather than dissolved.
Fig. 6 The energy deposited per unit mass inside a Plummer cluster, against radius. The dashed line is the impulse answer, growing as r² because the tidal force grows linearly with distance from the centre. The solid curves are the same shock after the adiabatic correction, for three shock durations. A shock lasting 0.15 of the half-mass crossing time is impulsive everywhere in the cluster; one lasting a full crossing time has taken half the heating away inside 1.29 half-mass radii. The heating is largest outside and the protection largest inside, and the two effects multiply in the same direction.

Spitzer’s form for the suppression, (1+(ωτ)2)−3/2(1 + (\omega\tau)^2)^{-3/2}, is the one drawn. It is not the last word: an oscillator perturbed slowly returns its energy exponentially well, while a star on an orbit — which is not a harmonic oscillator, and which visits a range of frequencies — leaks a little at every pericentre, and the correct suppression for a real orbit falls as a power rather than exponentially. The corrected exponent is nearer −5/2-5/2 than −3-3 and the difference matters at the factor-of-two level for the core. What does not change is the direction: the core is protected, the envelope is not, and the protection strengthens inwards.

The consequence is that a shocked cluster is peeled rather than dissolved. The outer envelope absorbs the energy, expands, and is lost to the galactic tide; the core is barely touched, and afterwards it is a smaller, denser, more tightly bound object than it was. That last clause is the counter-intuitive one and it is the same effect as a system that gets hotter as it loses energy: a self-gravitating system responds to energy input by expanding and to mass loss by contracting, and a shock does both.

A shock that strips the outside and leaves the core. The energy a passing tide deposits per unit mass inside a cluster, against radius in units of the half-mass radius, both axes logarithmic. The dashed line is the impulse answer: the tidal force grows linearly with distance from the centre, so the energy grows as its square, and on that reckoning the outer parts absorb everything. The solid curves are the same shock after the adiabatic correction, for shocks lasting 0.05, 0.2, 0.8, 2 times the half-mass crossing time. A star deep in the core orbits many times while the shock is happening; it follows the perturbation quasi-statically, gives back at the end what it took at the beginning, and keeps almost nothing. A star in the outskirts orbits once in far longer than the shock lasts, is struck before it can respond, and keeps all of it. The suppression factor is (1 + (ωτ)²)^(−3/2), and the radius inside which the shock delivers less than half of the impulse answer goes from impulsive to half at 2.26 r_h as the shock is slowed from 0.05 to 2 crossing times. The two effects therefore multiply in the same direction: the heating is largest outside and the protection largest inside, and a shocked cluster is peeled rather than dissolved.
Fig. 7 The same picture over a wider range of shock durations and radii. The two fastest shocks are impulsive throughout — a cluster crossing a thin galactic disc at two hundred kilometres a second experiences a shock of a few million years against an internal crossing time of ten, so this is the realistic regime for the disc-crossing of a globular cluster. The two slowest are the regime of a passage by a giant molecular cloud, where the encounter is longer and only the outskirts are affected. The same cluster is therefore differently vulnerable to two perturbers of similar mass, according to how fast each goes past.

What this decides about globular clusters

The abstraction has a concrete consequence and it is a large one.

A globular cluster on an orbit through the Galactic disc crosses the disc twice per orbit, for as long as it exists. Each crossing is a shock of the kind above. Integrated over a Hubble time, disc shocking removes more mass from the cluster population than internal evaporation does for clusters on disc-crossing orbits — and it removes it preferentially from the outside, which lowers the cluster’s tidal radius, which makes the next crossing more effective.

The numbers are worth having. A globular cluster on a typical halo orbit crosses the disc every hundred million years or so, at a speed of two hundred kilometres a second through a layer a few hundred parsecs thick — so the shock lasts a few million years. A cluster of a hundred thousand solar masses and a half-mass radius of three parsecs has a half-mass crossing time of about ten million years. The ratio is a fraction, which by the figure above puts the whole cluster in the impulsive regime, and it is why disc shocking is efficient in a way that a slow passage by an equally massive cloud is not.

That is a runaway, and it is one of the two clocks that decide which clusters survive. The other is internal: a cluster boils itself away through two-body relaxation on a timescale set by its own density, with no external help at all. Which clock dominates depends on the orbit, and the observed cluster population is the survivors of both.

The distinction the essay opened with matters here in a specific way. Because the heating falls as b−4b^{-4}, the shocking is dominated by the closest passages — the pericentres, the disc crossings, the rare close encounter with a molecular cloud — rather than by the steady background of distant perturbers. A cluster’s history is a small number of large events, not an accumulation of small ones. That is the opposite of how relaxation works inside it.

The comparison with a molecular cloud is instructive because the masses are similar and the outcomes are not. A cloud of a million solar masses passing at ten parsecs delivers a comparable tidal impulse to a disc crossing — but the cluster’s relative speed through a population of clouds is tens of kilometres a second rather than two hundred, so the encounter lasts ten times longer, and by the adiabatic curve the core keeps almost nothing of it. Two perturbers of similar mass, one of which peels the cluster and the other of which shakes it and is forgotten.

What is actually measured

None of the above is observed happening. What is observed is a population, and the inference runs through an average that weighs what cannot be watched at every step.

The Galaxy’s globular clusters have a mass function that peaks near two hundred thousand solar masses, with very few below about ten thousand. Young massive clusters in other galaxies have a mass function that falls monotonically, a power law with no peak. The standard reading is that the peak is carved: the low-mass clusters were destroyed, and the mechanisms above are what destroyed them. The evidence for that reading is the shape of the surviving distribution and its dependence on galactocentric radius — clusters near the centre are on shorter, more disc-crossing orbits and are preferentially the bigger survivors.

There is one direct observation that supports the mechanism rather than the outcome, and it is worth having. Clusters that have been shocked should show extra-tidal stars — material pushed beyond the tidal radius and not yet dispersed — and many of them do, as an excess above a fitted density profile at the outside and as tidal tails where the excess has had time to shear out. The amount of that material is a direct measure of recent mass loss, and for the clusters where both are known it agrees with the shocking calculation to a factor of about two. A factor of two is what agreement means here.

That is an inference from a distribution to a history, and the parts of it that are computed rather than fitted are exactly the parts in this essay: the b−4b^{-4} scaling, the adiabatic correction, and the crossing time. The parts that are fitted are the initial mass function of clusters, the orbit distribution and the Galaxy’s mass model at the epochs concerned — none of which is known independently.

The impulse overestimates, by one over one plus the square of a single ratio. The exact transverse velocity change of a hyperbolic encounter divided by the impulsive estimate, against the ratio of the ninety-degree impact parameter to the actual one on a logarithmic axis. That ratio, the gravitational constant times the mass over the impact parameter and the square of the speed, is the only dimensionless number an encounter has, and the exact answer is the impulsive one divided by one plus its square. Two readings matter. The approximation is not merely asymptotically right: it is right to one per cent for any encounter whose deflection is under about six degrees, which is almost every encounter a star in a cluster or a galaxy ever has. And where it fails, it fails in one direction — the impulse always overestimates, because it integrates the force along a path that the deflection has already carried the body away from. At a ninety-degree encounter the estimate is exactly twice the truth, which is the definition of that impact parameter rather than a result, and it is the natural place to put the inner cutoff that the next figure turns out not to need.
Fig. 8 And the ceiling on all of it. The exact deflection for a hyperbolic encounter against the impulse approximation’s answer, as a ratio that is one over one plus the square of a single dimensionless ratio. The approximation overestimates, always, and the error is second order in the deflection angle — so the very close encounters that dominate the b⁻⁴ integral are the ones for which the approximation is least reliable. A calculation whose answer is dominated by its own worst-behaved regime is a calculation that needs a numerical check, and modern treatments of cluster shocking are numerical for that reason.

A last observation about what the essay’s opening subtraction costs in practice. Because the translation is removed, the heating depends on the cluster’s size through the mean square radius — and a cluster’s size is the least well determined of its properties, since the outer parts are faint and the profile has no edge. Two published half-mass radii for the same cluster differing by thirty per cent, which is ordinary, is a factor of 1.7 in the shock heating. That is larger than the uncertainty in the encounter itself, and it is a consequence of the subtraction rather than of the observation: a quantity that depends on a difference across a system inherits the uncertainty in how big the system is, where the quantity it was subtracted from did not.

The generalisation

The structure worth extracting is that a perturbation’s effect on an internal degree of freedom is always at least one order smaller than its effect on the system as a whole, because the leading term is a translation and a translation changes nothing internal.

That is why tides go as the cube of distance where gravity goes as the square; why the Roche limit exists; why an aeroplane in free fall is weightless inside while being accelerated; and why the heating in this essay carries b−4b^{-4}. In every case the observable is a difference, the difference is a derivative, and a derivative costs a power of the separation.

There is a second reading about the two integrals, and it generalises past dynamics. A quantity whose contributions per decade are constant has no natural scale and needs a boundary supplied from outside; a quantity whose contributions per decade fall steeply has a natural scale, which is wherever the falling starts. Which of those a calculation is doing is decided by an exponent, and the exponent is usually visible before any of the constants are known. The time a system takes to forget is the first kind; tidal shocking is the second; and every argument in this collection about whether a sum needs a cutoff comes down to the same comparison.

The corollary is a warning about orders of magnitude. Estimating an internal effect from the size of the external force overestimates it by the ratio of the system’s size to the distance, which for the cases in this collection is between a factor of ten and a factor of a million. A great many order-of-magnitude arguments in dynamics fail in exactly that way, and they fail by being too generous.

The one number a shock cannot be reduced to

It is tempting to compress all of this into a single “shocking timescale” and compare it with a relaxation timescale, and the temptation should be resisted for a reason the figures make plain.

A relaxation timescale is meaningful because relaxation is a random walk with a well-defined rate: every star in the system is being nudged continuously, and the time for the velocities to be reorganised is a property of the system. A shocking timescale is not that. The heating is concentrated at the outside, it arrives in discrete events separated by a hundred million years, and its effect depends on what the cluster does between events — whether it has had time to re-virialise, whether the stars pushed past the tidal radius have had time to leave.

So the honest form of the calculation is a simulation with the orbit in it, and the honest form of the result is a mass-loss history rather than a timescale. Quoting one number for how long a cluster survives disc shocking is quoting the answer to a question about an average cluster on an average orbit, and the distribution around that average is wider than the number.

Where the ladder goes next

The next rung takes the shock and asks what the cluster does afterwards. The energy is deposited over a crossing time and the cluster is then out of equilibrium; it re-virialises over a few crossing times, expands, loses the stars whose new energies put them outside the tidal radius, and settles denser than before. That readjustment is not a small correction to the shock — it roughly doubles the mass lost — and it is the step that turns an energy input into a number of stars.

Further rungs on this anchor: the second-order term in the expansion, which is the first one that depends on the encounter’s orientation rather than only its strength; the sum over an entire population of perturbers, which is where the shocking timescale comes from; the same arithmetic applied to a binary star rather than to a cluster, where the outcome is ionisation rather than heating; and encounters that are neither impulsive nor adiabatic, which is most of them and where nothing has a closed form.

Where this goes next

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. 7 decades of relaxation time, and a Hubble time between them. Crossing time and two-body relaxation time for 5 self-gravitating systems, against the number of bodies in each. The lower marks are the crossing time R/σ, which spans 3 decades; the upper ones are t_relax ≈ 0.1N/ln N times it, which spans 8. The horizontal rule is a Hubble time, and the same expression puts these systems on opposite sides of it. Two of them relax — an open cluster and a globular cluster — so their stars have exchanged energy, their heavy stars have sunk, and their present structure is not the one they were born with. The rest do not: a dwarf spheroidal, an elliptical galaxy, a cluster of galaxies, the elliptical missing it by a factor of 10⁶, which is why a galaxy is modelled as a smooth potential with no stars in it at all. The dependence is on N and hardly at all on anything else, because the Coulomb logarithm makes every decade of impact parameter contribute equally. A cluster that relaxes also evaporates: about a stellar mass in a hundred and forty leaves per relaxation time, so the globular cluster empties in 6.1·10¹¹ years. A cluster that boils itself away A star cluster has no thermostat. Encounters between its members push a few of them above escape speed, the cluster loses them, and losing them makes it contract — which makes it hotter, which makes more of them escape. A self-gravitating system heats up as it loses energy, and the process ends by destroying the system. The tidal field is a difference. The pull of a distant body at each point of a sphere, minus its pull at the sphere's centre. What remains stretches along the line to the source and squeezes across it — two bulges, not one. The tide is a difference, which is why there are two of them The Moon pulls the ocean toward it. That explains one bulge. The second one, on the far side, is the whole of the physics — and it comes from subtracting.

About the same objects

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

The objects this essay names

Each one links to every other essay that touches it.

Adiabatic invariantCoulomb logarithmCrossing timeDisc shockingGlobular clusterImpulse approximationPlummer sphereTidal forceTidal shockTwo-body relaxation