Galactic Potential Models

A galaxy is a swarm of a hundred billion stars, yet no astronomer integrates $10^{11}$ pairwise forces to follow a single orbit — and none needs to. Because the number of stars is enormous and the two-body relaxation time vastly exceeds the age of the Universe, a star does not feel its neighbours individually; it falls freely in the smooth, slowly-varying mean field of the whole system. A galactic potential model is an analytic function $\Phi(\mathbf{r})$ that captures that mean field, paired with the density $\rho(\mathbf{r})$ that generates it through Poisson's equation $\nabla^2\Phi=4\pi G\rho$ . Replace $10^{11}$ point masses with a handful of parameters and stellar dynamics becomes a science one can do with pencil, paper, and a laptop. The craft lies in choosing functional forms that are at once physically reasonable, mathematically closed, and flexible enough to fit data. A model earns its keep when $\Phi$ , the acceleration $-\nabla\Phi$ , the density $\rho$ , the enclosed mass $M(r)$ , and the observable circular speed $v_c(R)$ are all elementary functions, so orbits integrate cheaply and escape speeds read off directly. The point mass, Plummer sphere, Hernquist model, isochrone, logarithmic halo, and Miyamoto–Nagai disk are the standard toolkit — each a small jewel of exact potential theory. The decisive structural fact is that Poisson's equation is linear. A real galaxy — bulge plus disk plus halo — has $\Phi=\Phi_{\rm bulge}+\Phi_{\rm disk}+\Phi_{\rm halo}$ , each term an independently valid model, the densities adding likewise. Every modern Milky Way model is exactly such a superposition, tuned to the rotation curve, the local density, and the halo mass. And once $\Phi$ is fixed, everything downstream follows: orbits, action–angle variables, the epicyclic approximation, the Jeans equations. The choice of $\Phi$ is the foundational decision on which all of galactic dynamics rests.

Key takeaways

A galactic potential model is an analytic solution of Poisson's equation $\nabla^2\Phi=4\pi G\rho$ chosen so that the potential, force, density, enclosed mass, and circular speed are all closed-form — making orbit integration and dynamical modelling tractable.
The point-mass (Kepler) potential $\Phi=-GM/r$ and its harmonic core bracket the two exactly solvable limits; the isochrone $\Phi=-GM/(b+\sqrt{b^2+r^2})$ is the only spherical model besides these whose orbital actions are elementary.
Real galaxies are assembled by superposition — a Hernquist or Plummer bulge, a Miyamoto–Nagai disk, a logarithmic or NFW halo — each adding to $\Phi$ because Poisson's equation is linear.
The logarithmic halo $\Phi=\tfrac12 v_0^2\ln(R_c^2+R^2+z^2/q^2)$ gives a flat rotation curve $v_c\to v_0$ , matching the observed constancy of galactic rotation speeds and demanding $M(R)\propto R$ — dark matter.
These models feed directly into orbit theory, the Jeans equations, action–angle variables, and the epicyclic approximation: they are the scaffolding of stellar dynamics.
Key idea: a galactic potential model is a self-consistent pair $(\rho,\Phi)$ satisfying $\nabla^2\Phi=4\pi G\rho$ , engineered so that potential, force, density, and enclosed mass are all closed-form; because Poisson's equation is linear, realistic galaxies are built by adding such models component by component.

Where this lives today

Analytic potential models are the beating heart of modern Galactic archaeology. Orbit-integration packages such as galpy, gala, and AGAMA ship with libraries of exactly these components — Hernquist bulges, Miyamoto–Nagai disks, logarithmic and NFW haloes — which astronomers superpose and tune to the Milky Way's rotation curve, local density, and total mass. The reason the field is booming is data: the Gaia mission has measured positions and velocities for over a billion stars, and reconstructing where those stars have been means integrating each one backward through a Galactic potential for billions of years. Stellar streams — the shredded remains of disrupted clusters and dwarf galaxies, like the GD-1 and Sagittarius streams — are the sharpest probes yet: a stream traces a single orbit, so its shape on the sky measures $\Phi$ directly, and gaps in streams may even record encounters with dark subhaloes. The same closed-form potentials seed $N$ -body initial conditions, supply distribution functions by Eddington inversion, and feed the Jeans equations that weigh dwarf spheroidals. The open frontier is honesty about the model's limits: the real Galaxy is barred, warped, and still ringing from the Sagittarius impact, so the next generation trades rigid analytic forms for flexible, time-dependent, data-driven potentials — while still leaning on the analytic toolkit as the scaffolding underneath.

The physical picture

Think of the potential as a landscape of hills and valleys in which stars roll. The point mass is a single infinitely-deep funnel; the isochrone rounds the bottom of that funnel into a finite-density core; the logarithmic halo replaces the funnel with a broad bowl whose walls climb only as the logarithm of radius, so a star far out feels a nearly constant speed rather than a falling one — that is a flat rotation curve. The shape of the well is not decoration: it fixes the one number observers measure most cleanly, the circular speed $v_c^2(R)=R\,\partial\Phi/\partial R$ , which rises like $R$ in a harmonic core, falls like $R^{-1/2}$ in a Keplerian outskirt, and levels off in a logarithmic halo.

Figure

Three potential wells a star can roll in: the point-mass funnel dives without bound and gives a Keplerian falloff v_c ∝ 1/√R, the isochrone rounds the bottom into a finite-density harmonic core, and the broad logarithmic bowl has walls that climb only as ln R so the circular speed levels off at a constant v₀ — a flat rotation curve

Because gravity superposes, you assemble a galaxy the way you stack transparencies: a compact deep well for the bulge, a flattened trough for the disk, a wide gentle basin for the halo. Each layer is an exact, tractable model; their sum need not be tractable, but it is trivially computable, and that is enough to integrate any orbit. The payoff is the figure a galactic astronomer actually reaches for — the decomposed rotation curve, where each component announces itself by the radius at which it dominates, and only the sum reproduces the stubborn flatness that betrayed dark matter.

Figure

A model Milky Way rotation curve built by superposition: a Hernquist bulge in amber, a Miyamoto–Nagai disk peaking near 5 kpc in green, and a dark logarithmic halo in red add in quadrature to v_c squared so the total in blue stays flat at v₀ ≈ 220 km per second out past the Sun at R₀ ≈ 8 kpc — the observed flatness is halo-dominated and implies M(R) ∝ R

The core equations

Everything begins with the master relation of potential theory: the field equation that ties any density to the potential it produces. It is a second-order linear elliptic equation, and its linearity is the single most consequential property in this article.

\nabla^2\Phi=4\pi G\rho \qquad (1)

where: $\Phi$ — gravitational potential (energy per unit mass), units $\mathrm{m^2\,s^{-2}}$ ; $\rho$ — mass density generating the field; $G$ — Newton's gravitational constant; $\nabla^2$ — Laplacian operator. Reading it: fix $\rho$ and solve for $\Phi$ to find the field of a given galaxy; or fix a convenient $\Phi$ and differentiate to read off the density it implies — the trick behind every closed-form model below. The simplest solution is the field outside a point (or any spherical) mass — the Kepler potential — together with the circular speed a test star needs to orbit it. This is the exactly solvable limit every other model must reduce to at large radius.

\Phi(r)=-\frac{GM}{r},\qquad v_c^2(r)=\frac{GM}{r} \qquad (2)

where: $M$ — total mass; $r$ — spherical radius; $v_c$ — circular speed of a test particle. Reading it: the speed falls as $r^{-1/2}$ , the Keplerian signature; a rotation curve that keeps rising or stays flat well beyond the visible mass is telling you the enclosed mass is still growing. The Plummer sphere is the simplest model with a finite-density core rather than a central singularity — historically Plummer's fit to globular clusters. Its potential mimics the point mass at large $r$ but softens to a constant at the centre.

\Phi_P(r)=-\frac{GM}{\sqrt{r^2+a^2}},\qquad \rho_P(r)=\frac{3M}{4\pi a^3}\left(1+\frac{r^2}{a^2}\right)^{-5/2} \qquad (3)

where: $a$ — Plummer scale radius (core size); $M$ — total mass, finite because $\rho\propto r^{-5}$ far out. Reading it: the density is a cored profile, flat inside $a$ and steeply truncated outside — realistic for star clusters but too shallow ( $\rho\propto r^{-5}$ ) for the extended envelopes of elliptical galaxies. The Hernquist model repairs exactly that shortcoming: it adds a central cusp and an $r^{-4}$ envelope, so its projection reproduces the de Vaucouleurs $R^{1/4}$ surface-brightness law of bulges and ellipticals. Its enclosed mass is a clean rational function.

\Phi_H(r)=-\frac{GM}{r+a},\qquad \rho_H(r)=\frac{M}{2\pi}\frac{a}{r\,(r+a)^3},\qquad M(r)=M\,\frac{r^2}{(r+a)^2} \qquad (4)

where: $a$ — Hernquist scale radius; $\rho_H$ — cuspy density, $\propto r^{-1}$ at the centre and $\propto r^{-4}$ far out; $M(r)$ — mass within $r$ . Reading it: the $r^{-1}$ inner cusp and finite total mass make Hernquist the workhorse bulge and the standard analytic proxy for the NFW dark-halo profile. The isochrone is the aristocrat of the family: the only spherical potential besides the Kepler and harmonic limits it contains whose radial action, and hence orbital frequencies, are elementary functions of the integrals of motion.

\Phi_{\rm iso}(r)=-\frac{GM}{b+\sqrt{b^2+r^2}} \qquad (5)

where: $b$ — isochrone scale length setting the core size; $M$ — total mass. Its name comes from the radial action and period, which for the isochrone collapse to Kepler-like closed forms:

J_r=\frac{GM}{\sqrt{-2E}}-\frac12\Big(L+\sqrt{L^2+4GMb}\Big),\qquad T_r=2\pi\frac{\partial J_r}{\partial E}=\frac{2\pi GM}{(-2E)^{3/2}} \qquad (5')

where: $E$ — orbital energy ( $E<0$ for bound orbits); $L$ — angular momentum; $J_r$ — radial action; $T_r$ — radial period. Reading it: $T_r$ depends only on $E$ , not on $L$ — the "isochronous" property — which is why the isochrone is the ideal analytic testbed for action–angle methods. The logarithmic halo is engineered backwards — from the observation, not the density. Its potential is chosen so that the circular speed saturates at a constant $v_0$ , and a flattening parameter $q$ lets the potential be oblate.

\Phi_L(R,z)=\frac12 v_0^2\ln\!\left(R_c^2+R^2+\frac{z^2}{q^2}\right) \qquad (6)

where: $v_0$ — asymptotic (flat) circular speed; $R_c$ — core radius inside which the curve rises; $q$ — potential flattening ( $q=1$ spherical, $q<1$ oblate); $R,z$ — cylindrical radius and height. Reading it: a single parameter $v_0$ sets the halo mass scale; but as the density derivation shows, flattening the potential too hard makes the density go negative. The Miyamoto–Nagai disk supplies the flattened component. It is a razor-thin Kuzmin disk given finite thickness by softening $|z|$ to $\sqrt{z^2+b^2}$ — the softening that keeps $\Phi$ closed-form while spreading the mass into a physical disk.

\Phi_{\rm MN}(R,z)=-\frac{GM}{\sqrt{R^2+\left(a+\sqrt{z^2+b^2}\right)^2}} \qquad (7)

where: $a$ — radial scale length; $b$ — vertical scale height ( $a\gg b$ gives a thin disk); $M$ — disk mass. Reading it: two length scales let one function span the whole range from a point mass ( $a,b\to0$ ) through a thin disk ( $a\gg b$ ) to a round bulge ( $a\to0$ ), which is why it is the default disk term in Galactic models. Finally, the two operational relations that turn any $\Phi$ into an observable and let components combine: the circular speed in the disk plane, and the additivity of potentials.

v_c^2(R)=R\left.\frac{\partial\Phi}{\partial R}\right|_{z=0},\qquad \Phi_{\rm tot}=\sum_i\Phi_i,\qquad v_{c,\rm tot}^2=\sum_i v_{c,i}^2 \qquad (8)

where: $v_c(R)$ — observable rotation curve; $\Phi_i$ — potential of component $i$ . Reading it: because $\Phi$ and $\partial_R\Phi$ add, circular speeds add in quadrature — the arithmetic behind the decomposed rotation curve in the figure above.

Derivation ARecovering the Plummer density from its potential

The goal is to demonstrate the central move of closed-form modelling — that a well-chosen potential can be differentiated back through Poisson's equation to yield an everywhere-positive, physically sensible density — using the Plummer sphere as the cleanest example. We start from $\Phi_P=-GM/\sqrt{r^2+a^2}$ , apply the spherical Laplacian, and watch the singular $1/r^2$ prefactor cancel to leave a finite, positive $\rho$ , confirming that the model is self-consistent rather than merely convenient.

Step 1. Differentiate the potential once: $\mathrm{d}\Phi_P/\mathrm{d}r = GM\,r\,(r^2+a^2)^{-3/2}$ , the radial gravitational field. Why: the force per unit mass is what a star actually feels, and it is the first ingredient the spherical Laplacian needs.

Step 2. Assemble the spherical Laplacian $\nabla^2\Phi=\dfrac1{r^2}\dfrac{\mathrm d}{\mathrm dr}\!\left(r^2\dfrac{\mathrm d\Phi}{\mathrm dr}\right)$ with $r^2\,\mathrm d\Phi/\mathrm dr = GM\,r^3(r^2+a^2)^{-3/2}$ ; differentiating the bracket gives $3r^2(r^2+a^2)^{-3/2}-3r^4(r^2+a^2)^{-5/2}=3r^2a^2(r^2+a^2)^{-5/2}$ . Why: the two terms share a factor and combine so that $(r^2+a^2)-r^2=a^2$ survives — the algebra that kills the singularity.

Step 3. Divide by $r^2$ and then by $4\pi G$ :

\rho_P=\frac{1}{4\pi G}\cdot\frac{3GM\,a^2}{(r^2+a^2)^{5/2}}=\frac{3M}{4\pi a^3}\left(1+\frac{r^2}{a^2}\right)^{-5/2}.

Why: the result is finite at $r=0$ and strictly positive everywhere, so the pair $(\rho_P,\Phi_P)$ is a legitimate self-gravitating model — not every smooth $\Phi$ passes this test, as the flattened halo will show.

Go deeper · Derivation B · 1 minWhy the logarithmic halo gives a flat rotation curve

Derivation BWhy the logarithmic halo gives a flat rotation curve

The goal is to show that the logarithmic potential is not an arbitrary fitting function but the minimal analytic form whose circular speed rises solid-body in the core and then saturates at a constant $v_0$ , thereby reproducing the single most important empirical fact about spiral galaxies and forcing the conclusion that mass keeps accumulating beyond the visible light. We work in the disk plane $z=0$ , take the radial derivative, form $v_c^2$ , and examine both limits.

Step 1. In the plane $\Phi_L=\tfrac12 v_0^2\ln(R_c^2+R^2)$ , so $\partial\Phi_L/\partial R=\tfrac12 v_0^2\cdot\dfrac{2R}{R_c^2+R^2}=\dfrac{v_0^2 R}{R_c^2+R^2}$ . Why: the circular speed is built entirely from this in-plane radial force, so this is the only derivative we need.

Step 2. Multiply by $R$ to form the circular speed:

v_c^2(R)=R\,\frac{\partial\Phi_L}{\partial R}=\frac{v_0^2\,R^2}{R_c^2+R^2}.

Why: this single closed form contains both regimes, so the physics is read off by taking limits rather than solving anything further.

Step 3. Inner limit $R\ll R_c$ : $v_c\approx (v_0/R_c)\,R$ , rising linearly — solid-body rotation, the harmonic core. Outer limit $R\gg R_c$ : $v_c\to v_0$ , dead flat. Why: a constant $v_c$ means $M(R)=v_c^2 R/G\propto R$ , so mass grows without bound where the light has vanished — the dynamical fingerprint of a dark halo.

Go deeper · Derivation C · 1 minThe rotation curve of the Miyamoto–Nagai disk

Derivation CThe rotation curve of the Miyamoto–Nagai disk

The goal is to derive the in-plane rotation curve of the standard finite-thickness disk and to show that, unlike the logarithmic halo, a self-gravitating disk produces a curve that rises, peaks, and then declines Keplerian-fashion — establishing quantitatively why a disk alone cannot explain the observed outer flatness and why an extended halo is mandatory. We set $z=0$ , where the vertical softening collapses the two disk lengths into a single combination $a+b$ , then differentiate.

Step 1. In the plane $\sqrt{z^2+b^2}=b$ , so $\Phi_{\rm MN}(R,0)=-GM/\sqrt{R^2+(a+b)^2}$ — formally identical to a Plummer sphere of scale length $a+b$ . Why: the disk parameters merge in the plane, which is exactly why the plane rotation curve looks Plummer-like even though the mass is flattened.

Step 2. Differentiate and multiply by $R$ :

v_c^2(R)=R\,\partial_R\Phi_{\rm MN}(R,0)=\frac{GM\,R^2}{\left[R^2+(a+b)^2\right]^{3/2}}.

Why: this is the observable the disk contributes to the decomposition figure, and its shape is entirely fixed by the one length $a+b$ .

Step 3. Small $R$ : $v_c^2\approx GM R^2/(a+b)^3$ , rising linearly; large $R$ : $v_c^2\approx GM/R$ , the Keplerian decline; setting $\mathrm{d}v_c/\mathrm{d}R=0$ puts the peak at $R=\sqrt2\,(a+b)$ . Why: the curve must turn over, so the flat outer rotation observed in real spirals cannot come from the disk — it is halo-dominated.

Limiting cases

Each model lives between two exactly solvable regimes; checking these limits is the fastest validation of any formula.

Case	Result	Meaning
Isochrone, small $b$	$\Phi_{\rm iso}\xrightarrow{b\to0}-\dfrac{GM}{r}$	Recovers the point-mass (Kepler) potential; orbits become closed ellipses about the focus.
Isochrone, small $r$	$\Phi_{\rm iso}\approx-\dfrac{GM}{2b}+\dfrac{GM}{8b^3}\,r^2$	Harmonic core: constant plus $\tfrac12\omega^2 r^2$ with $\omega^2=GM/(4b^3)$ ; orbits are centred ellipses.
Logarithmic halo, large $R$	$v_c^2=\dfrac{v_0^2 R^2}{R_c^2+R^2}\xrightarrow{R\gg R_c}v_0^2$	Flat rotation curve at $v_0$ — the defining signature of a dark-matter halo.
Miyamoto–Nagai, $b\to0$	$\Phi_{\rm MN}\to-\dfrac{GM}{\sqrt{R^2+(a+\lvert z\rvert)^2}}$	The razor-thin Kuzmin disk; with $a\to0$ as well it collapses to a point mass.
Hernquist, $r\gg a$	$\Phi_H\to-\dfrac{GM}{r},\ \rho_H\propto r^{-4}$	Point-mass field with a steep, finite-mass envelope matching elliptical-galaxy light.

Common trapA smooth, sensible-looking potential is guaranteed to have a positive density

It is not. Poisson's equation runs both ways, but only one way is safe: a positive density always yields a sensible potential, yet a plausible, everywhere-smooth potential can imply a negative density in some region — which is unphysical, because mass density cannot be less than zero. The textbook trap is the flattened logarithmic halo: choose the potential flattening $q$ too small (too oblate) and $\nabla^2\Phi_L$ turns negative along the symmetry axis at large $\lvert z\rvert$ , so no real matter could produce it. Concretely the density stays non-negative everywhere only for $q\gtrsim1/\sqrt2\approx0.7$ . The lesson generalises: never trust a hand-picked $\Phi$ until you have differentiated it through Poisson's equation and checked $\rho\ge0$ everywhere. Flattening a potential by eye is not the same as flattening a mass distribution.

Worked examples

1 · Dynamical mass of the Milky Way inside the Sun's orbit (fully worked)

The Galactic rotation curve is flat at $v_0\approx220\ \mathrm{km\,s^{-1}}$ near the Sun's radius $R_0\approx8\ \mathrm{kpc}$ . Treating the halo as logarithmic in the flat regime ( $R\gg R_c$ ), estimate the mass enclosed within $R_0$ .

Step 1. In the flat regime $v_c^2=v_0^2=GM(R)/R$ , so the enclosed mass is $M(R_0)=v_0^2 R_0/G$ . This is the operational content of "flat rotation curve implies $M\propto R$ ."

Step 2. Insert SI values: $v_0=2.20\times10^5\ \mathrm{m\,s^{-1}}$ , $R_0=8\times3.086\times10^{19}=2.47\times10^{20}\ \mathrm{m}$ , $G=6.674\times10^{-11}$ . Then

M(R_0)=\frac{(2.20\times10^5)^2(2.47\times10^{20})}{6.674\times10^{-11}}=1.79\times10^{41}\ \mathrm{kg}.

Step 3. Convert to solar masses ( $M_\odot=1.989\times10^{30}\ \mathrm{kg}$ ): $M(R_0)\approx\mathbf{9.0\times10^{10}\ M_\odot}$ . About $9\times10^{10}\,M_\odot$ lies within 8 kpc — and it keeps growing linearly outward, the mass no starlight accounts for.

2 · Where a Miyamoto–Nagai disk peaks (fully worked)

Model the Galactic disk as a single Miyamoto–Nagai component with $M=10^{11}\,M_\odot$ , radial length $a=3\ \mathrm{kpc}$ , scale height $b=0.3\ \mathrm{kpc}$ . Find the radius and value of its peak circular speed.

Step 1. From Derivation C the peak is at $R_{\rm peak}=\sqrt2\,(a+b)=\sqrt2\times3.3=\mathbf{4.7\ \mathrm{kpc}}$ . The single in-plane length $a+b=3.3$ kpc controls everything.

Step 2. Evaluate $v_c^2$ there. Using $R_{\rm peak}^2=2(a+b)^2$ the expression simplifies to $v_{c,\rm max}^2=\dfrac{2GM}{3\sqrt3\,(a+b)}$ . With $GM=1.33\times10^{31}\ \mathrm{m^3\,s^{-2}}$ and $a+b=1.018\times10^{20}\ \mathrm{m}$ :

v_{c,\rm max}=\sqrt{\frac{2(1.33\times10^{31})}{3\sqrt3\,(1.018\times10^{20})}}=2.24\times10^5\ \mathrm{m\,s^{-1}}.

Step 3. So the disk alone peaks at $\mathbf{v_{c,\rm max}\approx224\ \mathrm{km\,s^{-1}}}$ near 4.7 kpc and then declines as $R^{-1/2}$ . The value is close to the observed $v_0$ , but the decline is the point: beyond a few scale lengths the disk contribution falls and the flat outer curve must be carried by the halo.

3 · The positivity bound on a flattened halo (analytic)

Show why a flattened logarithmic potential cannot be made arbitrarily oblate.

Step 1. Applying the cylindrical Laplacian to $\Phi_L$ and dividing by $4\pi G$ gives the density

\rho_L(R,z)=\frac{v_0^2}{4\pi G q^2}\;\frac{(2q^2+1)R_c^2+R^2+(2-q^{-2})z^2}{\left(R_c^2+R^2+z^2/q^2\right)^2}.

Step 2. The only term that can turn the numerator negative is $(2-q^{-2})z^2$ , which is worst on the symmetry axis ( $R=0$ ) at large $\lvert z\rvert$ . Requiring the coefficient $2-q^{-2}\ge0$ gives $q^2\ge1/2$ , i.e. $\mathbf{q\gtrsim0.707}$ .

Step 3. For $q<0.7$ the model predicts negative density along the poles — unphysical. The flattening of a physical logarithmic halo is bounded below at $q\approx0.7$ ; to represent a rounder mass distribution one must flatten the density directly (or switch to a flattened NFW halo) rather than squashing the potential.

Physics of the portrait

The models in this toolkit carry the names of the people who found them — Plummer, Kuzmin, Hernquist, Miyamoto and Nagai — but one figure sits closest to the article's most elegant object, the isochrone, and to the numerical revolution that made these potentials usable.

Origins · 1 minMichel Hénon and how this was discovered

Michel Hénon

1931–2013 · numerical stellar dynamics and the isochrone potential

Michel Hénon was a French astronomer and mathematician who spent his career at the Nice Observatory turning stellar dynamics into a computational science. In 1959, as a young researcher, he asked which spherical potential would make a star cluster's orbits especially simple, and discovered the isochrone: the unique mass model (bracketing the Kepler and harmonic limits) for which the radial period depends only on energy and the actions are elementary — the potential recalled by name across all of action–angle theory. He then helped invent modern numerical stellar dynamics, using some of the first digital computers to follow the relaxation and evaporation of globular clusters when analytic methods gave out. His curiosity then carried him into dynamical chaos: the Hénon–Heiles system of 1964, a toy model of a star orbiting in a galactic potential, became a founding example of chaotic motion, and the Hénon map of 1976 is a canonical strange attractor. That arc — from an exactly solvable galactic potential to the discovery that most galactic orbits are anything but solvable — is the shape of this whole subject: we build closed-form models precisely because the real dynamics they seed are too rich to solve any other way. Hénon worked productively into his eighties and died in 2013.

History

1911

— Plummer introduces his sphere to fit the light of globular clusters

1956

— Kuzmin constructs the razor-thin axisymmetric disk potential

1959

— Hénon derives the isochrone, the potential with elementary actions

1975

— Miyamoto and Nagai soften the disk to finite thickness

1990

— Hernquist publishes his cuspy model for bulges and ellipticals

Going further

Texts. Binney and Tremaine, Galactic Dynamics, 2nd ed., Ch. 2 (The Potential Theory of Galaxies) and §3.5 (the isochrone) · Mo, van den Bosch and White, Galaxy Formation and Evolution, Ch. 11 · Carroll and Ostlie, An Introduction to Modern Astrophysics, 2nd ed., Ch. 24–25 · Dodelson, Modern Cosmology, for the halo profiles in cosmological context · Ryden, Introduction to Cosmology, 2nd ed., Ch. 8. Each density can be recovered by applying the spherical or cylindrical Laplacian to the stated $\Phi$ ; the log-halo and Miyamoto–Nagai rotation curves follow by differentiation. Key self-consistency checks include the isochrone harmonic-core coefficient $GM/(8b^3)$ with $\omega^2=GM/(4b^3)$ , the $q\gtrsim0.7$ positivity bound on the flattened halo, and the worked numbers $M(R_0)\approx9\times10^{10}\,M_\odot$ and $v_{c,\rm max}\approx224\ \mathrm{km\,s^{-1}}$ . The Milky Way anchors used throughout — $v_0\approx220\ \mathrm{km\,s^{-1}}$ , $R_0\approx8\ \mathrm{kpc}$ , $v_{\rm esc}\approx530\ \mathrm{km\,s^{-1}}$ — are standard observational values.

Prerequisites. Gravitational Two-Body Problem · Gravitational N-Body Problem · Two-Body Relaxation Time. Leads to. Orbits in Spherical Potentials · Orbits in Axisymmetric and Triaxial Potentials · Epicyclic Approximation · Jeans Equations · Action–Angle Variables.

Ask the tutor: Ask me to superpose a bulge, disk, and halo into a Milky Way rotation curve, or to derive the density behind any potential and check whether it stays positive.

Why can I ignore individual stars and use a smooth potential at all?

Because a galaxy is collisionless on the timescales that matter. The two-body relaxation time — how long it takes star-by-star gravitational encounters to noticeably reshuffle orbits — scales roughly as $N/\ln N$ times the crossing time, and for $N\sim10^{11}$ stars it comes out vastly longer than the age of the Universe. So over any interval you care about, a star does not "see" its neighbours as individual masses; it feels only their collective, smoothly-varying field. That is precisely the regime in which replacing $10^{11}$ point masses with a single analytic $\Phi(\mathbf r)$ is not an approximation of convenience but an excellent physical description. Globular clusters, with far smaller $N$ , are the exception where relaxation does matter.

What is the difference between a potential model and a density model?

They are two faces of the same object, tied by Poisson's equation $\nabla^2\Phi=4\pi G\rho$ , but you specify one and derive the other. Sometimes it is natural to write down a density — a cored or cuspy profile that looks like the observed light — and then integrate to find $\Phi$ . Other times, especially when you want closed-form orbits, it is easier to write down a convenient $\Phi$ and differentiate to see what density it implies. The logarithmic halo is the classic "potential-first" model: nobody would guess its density, but its flat rotation curve is exactly what you want. The catch, covered in the warning, is that a nice potential can imply a negative density, so the derived $\rho$ must always be checked for positivity.

Why does the logarithmic halo, specifically, give a flat rotation curve?

Because its circular speed is $v_c^2=v_0^2R^2/(R_c^2+R^2)$ , which climbs through the core and then saturates at the constant $v_0$ for $R\gg R_c$ . The deeper reason is structural: a flat $v_c$ requires $M(R)\propto R$ , hence a density falling as $\rho\propto r^{-2}$ , and the potential of an $r^{-2}$ density is logarithmic in $r$ — the $\ln$ is the only function whose gradient gives a $1/R$ force that keeps $v_c$ constant. So "logarithmic potential," " $\rho\propto r^{-2}$ ," " $M\propto R$ ," and "flat rotation curve" are four descriptions of one physical situation, and that situation is what the rotation curves of real spirals show far beyond their visible edges.

Can any single analytic model describe a whole galaxy?

No, and it is not meant to. Each model captures one component well — Hernquist for a cuspy bulge, Miyamoto–Nagai for a flattened disk, logarithmic or NFW for a dark halo — but a real galaxy is all three at once, with different shapes dominating at different radii. The power of the approach is superposition: because Poisson's equation is linear, you add the components, $\Phi_{\rm tot}=\sum_i\Phi_i$ , and their circular speeds combine in quadrature. The sum is no longer a tidy elementary function, but you never needed it to be — you only need to evaluate $\Phi$ and its gradient numerically to integrate orbits, and the analytic pieces make each evaluation instant.

What is the isochrone actually good for if real galaxies are not isochrones?

It is the hydrogen atom of stellar dynamics: a fully solvable case you calibrate everything else against. Because its actions, angles, and frequencies are elementary, the isochrone is where you first learn action–angle variables and validate numerical schemes for computing actions — if a method cannot reproduce the isochrone's known answers, it is wrong. It also interpolates smoothly between the Kepler point mass and the harmonic core, so it captures the qualitative behaviour of a real cored galaxy while staying exactly tractable. You use it not because a galaxy is an isochrone but because it is the sharpest tool for building intuition and testing machinery.

How do modern surveys like Gaia actually use these potentials?

Gaia measures a star's present position and velocity, but what astronomers want is its orbit — where it has been over billions of years — because orbital properties reveal how the Galaxy formed. To get there you integrate the star's motion through an assumed Galactic potential, one of these superposed analytic models tuned to the rotation curve. Do this for a billion stars and patterns emerge: co-moving groups, accreted debris, and stellar streams that trace single orbits and so measure $\Phi$ directly. The potential is thus both an input (assumed to compute orbits) and an output (streams and kinematics constrain it), and the field iterates toward an ever more accurate map of the Milky Way's mass, most of it dark.

Why do dark-matter haloes use NFW or logarithmic profiles instead of Plummer or Hernquist?

Because cosmological $N$ -body simulations of structure formation consistently produce haloes with a specific shape — the Navarro–Frenk–White profile, $\rho\propto r^{-1}(r+r_s)^{-2}$ — cuspy inside and falling as $r^{-3}$ outside, giving a rotation curve that rises and stays high over a large range. The logarithmic halo is the simpler analytic stand-in that reproduces the crucial flat- $v_c$ behaviour with one parameter, ideal for quick orbit work. Plummer and Hernquist, by contrast, were built to match luminous components and fall off too fast to hold a galaxy's outer rotation curve flat. Choosing a halo model means choosing which fact you most need to reproduce: flatness (logarithmic), cosmological shape (NFW), or a finite-mass luminous profile (Hernquist).