Orbits in Spherical Potentials

A galaxy or a globular cluster is, over almost all of its life, a collisionless system: a star crosses the whole system hundreds of times before a close encounter with another star deflects it appreciably. To an excellent approximation each star therefore moves in the smooth, time-averaged gravitational field of all the others rather than in a hail of two-body kicks. That single fact turns the impossibly-many-body problem of stellar dynamics into a tractable one — the motion of one test particle in a fixed potential $\Phi(\mathbf r)$ — and the first case to master is the one with the most symmetry: a spherical potential, $\Phi=\Phi(r)$ . Spherical symmetry is generous: it hands us two exact constants for every orbit — the total energy $E$ and the full angular-momentum vector $\mathbf L=\mathbf r\times\mathbf v$ . The fixed direction of $\mathbf L$ pins the orbit to a plane; the fixed magnitude $L$ , folded into an effective potential $\Phi_{\mathrm{eff}}(r)=\Phi(r)+L^2/2r^2$ , reduces the whole three-dimensional problem to a bead sliding in a one-dimensional valley, rocking between two turning points — pericentre and apocentre — while the plane winds steadily around the centre. The result is a rosette: a slowly precessing ellipse that fills an annulus. Only two potentials in all of nature force it to close on itself — the Kepler point mass $\Phi\propto-1/r$ and the harmonic core $\Phi\propto r^2$ — a rigidity proved by Bertrand in 1873; every real galaxy lives between these extremes and precesses. This machinery is what the rest of stellar dynamics stands on: the integrals $E$ and $L$ that Jeans' theorem needs, the two frequencies that become action-angle variables, and the near-circular limit that becomes the epicyclic approximation.

Key takeaways

A star in a static, spherically symmetric potential $\Phi(r)$ conserves total energy $E$ and the full angular-momentum vector $\mathbf L$ , so its orbit lies in a fixed plane perpendicular to $\mathbf L$ .
In that plane the motion reduces to one-dimensional radial oscillation in the effective potential $\Phi_{\mathrm{eff}}(r)=\Phi(r)+L^2/2r^2$ , whose centrifugal barrier $L^2/2r^2$ keeps any $L\neq0$ orbit away from $r=0$ .
Bound orbits librate between a pericentre $r_-$ and an apocentre $r_+$ (the roots of $E=\Phi_{\mathrm{eff}}$ ); the geometry follows from two quadratures — the radial period $T_r$ and the azimuthal advance $\Delta\phi$ .
Only the Kepler ( $\Delta\phi=2\pi$ ) and harmonic ( $\Delta\phi=\pi$ ) potentials give closed orbits for all $E,L$ — Bertrand's theorem; every other spherical potential yields an open, precessing rosette.
Near a circular orbit the radial oscillation has the epicyclic frequency $\kappa^2=\Phi_{\mathrm{eff}}''(r_c)$ , and the apsides advance whenever $\kappa\neq\Omega_c$ .
Key idea: spherical symmetry supplies two exact integrals — energy $E$ and angular momentum $L$ — that collapse the 3-D motion to 1-D oscillation in $\Phi_{\mathrm{eff}}(r)=\Phi(r)+L^2/2r^2$ , with the orbit fixed by two quadratures for the radial period and the azimuthal advance per swing.

The core insight

Spherical symmetry gives two exact integrals — energy $E$ and angular momentum $L$ — that reduce the 3-D motion to 1-D radial oscillation in the effective potential $\Phi_{\mathrm{eff}}(r)=\Phi(r)+L^2/2r^2$ , with the orbit determined by two quadratures for the radial period and the azimuthal advance.

Where this lives today

Spherical-orbit theory is the working language of galactic dynamics right now. The Gaia mission has measured positions and velocities for over a billion Milky Way stars, and turning those six-dimensional phase-space points into orbits — each star's $E$ , $L$ , guiding radius, and epicyclic amplitude — is how the local dark-matter density, the shape of the Galactic potential, and the debris of past mergers are being read out. Long thin tidal streams such as GD-1 and the Sagittarius stream are single rosette orbits made visible: stars stripped from a dwarf spread along its orbit, so the stream traces a path through $\Phi(r)$ whose precession constrains the halo's mass and non-sphericity. In elliptical galaxies, Schwarzschild orbit-superposition modelling builds a galaxy as a weighted library of exactly these orbits, and the weights that reproduce the observed kinematics deliver the central black-hole mass and the dark-halo profile. In cosmological $N$ -body simulations the collisionless assumption that launched the framework is exactly what is integrated — billions of particles each on its own precessing rosette. The open problem threaded through all of it is the potential itself: the orbits are the probe, and inferring $\Phi(r)$ from the orbits stars actually trace is the central inverse problem of modern galactic dynamics.

The physical picture

Forget three dimensions for a moment and watch a single number, the radius $r$ . Angular momentum forbids the star from falling to the centre — spin a stone on a string and it flies out faster the shorter the string — so $L^2/2r^2$ raises a steep wall at small $r$ , while gravity supplies the outer wall. Add the two and you get a valley, $\Phi_{\mathrm{eff}}(r)$ ; the star's radius is simply a bead rolling in it, turning around wherever its energy line meets a wall. The first figure is that valley, drawn with real numbers.

Figure

The effective potential well for a Kepler orbit with GM = 1 and L = 1: the centrifugal barrier L²/2r² walls off the centre while gravity Φ = −1/r pulls inward, so their sum Φ_eff has a single minimum at the circular radius r_c = 1. A star of energy E = −0.4 librates radially between pericentre r₋ = 0.69 and apocentre r₊ = 1.81, the two roots of E = Φ_eff(r).

While the bead rocks radially, the plane of motion turns: the star wraps around the centre at $\dot\phi=L/r^2$ , fast at pericentre and slow at apocentre. If the radial rocking and azimuthal winding are commensurate — their frequency ratio rational — the orbit closes; generically they are incommensurate, and over many periods the orbit densely paints an annulus, an ellipse whose long axis creeps a little further each pass. The Kepler and harmonic potentials are the two magic exceptions, each carrying a hidden extra conservation law (the Laplace–Runge–Lenz vector for Kepler; decoupled Cartesian oscillators for the harmonic well) that locks the orientation. Deviate from exactly $1/r$ or $r^2$ and the lock breaks.

Figure

A rosette orbit in a flat-rotation-curve halo Φ = v_c² ln r: the star fills the annulus between pericentre r₋ and apocentre r₊, and because the potential is neither Kepler nor harmonic the apsides advance by Δφ ≈ 255° per radial period (the near-circular value 2π/√2), so the orbit never closes and densely covers the shaded ring.

The core equations

Everything begins with the two conservation laws spherical symmetry guarantees. A central force exerts no torque and a static potential does no net work, so the angular-momentum vector and the energy are both constants of the motion for any $\Phi(r)$ .

\mathbf L=\mathbf r\times\mathbf v=\text{const},\qquad E=\tfrac12 v^2+\Phi(r)=\text{const} \qquad (1)

where: $\mathbf L$ — specific angular momentum (per unit mass); $E$ — specific total energy; $\Phi(r)$ — spherical gravitational potential; $v$ — speed. "Specific" means per unit mass — a test particle's orbit is independent of its own mass. A constant $\mathbf L$ direction confines the motion to the plane perpendicular to $\mathbf L$ . In polar coordinates $(r,\phi)$ within that plane the kinetic energy splits into radial and azimuthal pieces, and the angular-momentum magnitude is $L=r^2\dot\phi$ .

E=\tfrac12\dot r^2+\tfrac12 r^2\dot\phi^2+\Phi(r),\qquad L=r^2\dot\phi \qquad (2)

where: $(r,\phi)$ — polar coordinates in the orbital plane; $\dot r,\ \dot\phi$ — radial and angular velocities; $L$ — conserved angular-momentum magnitude. The second relation is Kepler's equal-areas law: $\tfrac12 r^2\dot\phi=L/2$ is the areal velocity. Now eliminate $\dot\phi=L/r^2$ : the azimuthal kinetic term becomes $L^2/2r^2$ , a function of $r$ alone, and can be absorbed into the potential. The radial motion is then identical to that of a unit-mass particle in a one-dimensional effective potential.

E=\tfrac12\dot r^2+\Phi_{\mathrm{eff}}(r),\qquad \Phi_{\mathrm{eff}}(r)=\Phi(r)+\frac{L^2}{2r^2} \qquad (3)

where: $\Phi_{\mathrm{eff}}$ — effective radial potential; $L^2/2r^2$ — centrifugal barrier, the leftover azimuthal kinetic energy. For $L\neq0$ this barrier diverges as $r\to0$ , which is why a rotating star can never reach the centre. Solving equation (3) for the radial speed gives the trajectory directly, the two signs marking the inbound and outbound legs of each oscillation.

\dot r=\pm\sqrt{2\left[E-\Phi_{\mathrm{eff}}(r)\right]} \qquad (4)

where: the $+$ branch is the star moving outward, the $-$ branch inward. The speed vanishes where $E=\Phi_{\mathrm{eff}}$ — the turning points. The turning points are the pericentre $r_-$ (inner) and apocentre $r_+$ (outer); a circular orbit is the degenerate case $r_-=r_+$ at the bottom of the well.

E=\Phi_{\mathrm{eff}}(r_\pm)=\Phi(r_\pm)+\frac{L^2}{2r_\pm^2} \qquad (5)

where: $r_-$ — pericentre; $r_+$ — apocentre — the two real roots of a single algebraic equation once $\Phi$ , $E$ , and $L$ are fixed. Integrating $\mathrm dt=\mathrm dr/|\dot r|$ over one full swing gives the radial period as a quadrature — a definite integral you can always evaluate numerically even with no closed form.

T_r=2\int_{r_-}^{r_+}\frac{\mathrm dr}{\sqrt{2\left[E-\Phi_{\mathrm{eff}}(r)\right]}} \qquad (6)

where: $T_r$ — time for one radial cycle $r_-\!\to r_+\!\to r_-$ . The factor 2 covers the symmetric inbound and outbound legs. During one radial period the azimuth advances by $\Delta\phi$ , a companion quadrature from $\mathrm d\phi=(L/r^2)\,\mathrm dt$ ; the orbit closes only if $\Delta\phi$ is a rational multiple of $2\pi$ .

\Delta\phi=2\,L\int_{r_-}^{r_+}\frac{\mathrm dr}{r^2\sqrt{2\left[E-\Phi_{\mathrm{eff}}(r)\right]}} \qquad (7)

where: $\Delta\phi$ — azimuthal angle swept per radial period. The apsidal angle is $\Delta\phi/2$ , the swing from pericentre to apocentre. The two quadratures define two orbital frequencies; every steady spherical orbit is fully specified by this pair, and their difference is the apsidal precession rate.

\Omega_r=\frac{2\pi}{T_r},\qquad \Omega_\phi=\frac{\Delta\phi}{T_r} \qquad (8)

where: $\Omega_r$ — radial frequency; $\Omega_\phi$ — mean azimuthal frequency. The apsidal line rotates at $\Omega_\phi-\Omega_r$ ; when this is zero the orbit is closed. Finally, a circular orbit sits exactly at the minimum of $\Phi_{\mathrm{eff}}$ , where gravity balances the centrifugal term. This fixes the circular angular momentum and frequency at each radius.

L_c^2=r_c^3\,\frac{\mathrm d\Phi}{\mathrm dr}\bigg|_{r_c},\qquad \Omega_c^2=\frac1{r_c}\frac{\mathrm d\Phi}{\mathrm dr}\bigg|_{r_c} \qquad (9)

where: $r_c$ — circular radius; $L_c$ — circular angular momentum; $\Omega_c=v_c/r_c$ — circular frequency. Equivalently $v_c^2=r_c\,\mathrm d\Phi/\mathrm dr=G M(<r_c)/r_c$ — the enclosed-mass form used to read galaxy rotation curves.

Derivation AFrom three dimensions to one — the effective potential and two quadratures

The goal is to show, from Newton's second law alone, how a spherical potential collapses a six-dimensional orbit problem into a single one-dimensional oscillation plus two quadratures — first proving that a central force conserves the full angular-momentum vector and pins the orbit to a plane, then eliminating the azimuthal freedom to expose the effective potential, and finally integrating the radial motion to obtain the radial period and azimuthal advance that between them fix the entire orbit.

Step 1. Compute the rate of change of $\mathbf L=\mathbf r\times\mathbf v$ : $\mathrm d\mathbf L/\mathrm dt=\mathbf v\times\mathbf v+\mathbf r\times\dot{\mathbf v}=\mathbf r\times(-\nabla\Phi)$ . For a spherical potential $\nabla\Phi=(\mathrm d\Phi/\mathrm dr)\,\hat{\mathbf r}$ points along $\mathbf r$ , so the cross product vanishes and $\mathbf L$ is constant in both magnitude and direction. Why: a fixed $\mathbf L$ direction means $\mathbf r$ and $\mathbf v$ never leave the plane perpendicular to $\mathbf L$ — the orbit is planar, killing one full degree of freedom before we even solve anything.

Step 2. In that plane write the energy in polar coordinates, $E=\tfrac12(\dot r^2+r^2\dot\phi^2)+\Phi(r)$ , and use the conserved magnitude $L=r^2\dot\phi$ to replace $\dot\phi$ . The azimuthal kinetic term becomes $\tfrac12 r^2(L/r^2)^2=L^2/2r^2$ , a function of $r$ only. Why: this is the crucial move — the angular motion has been folded into an $r$ -dependent term, so the radial equation no longer references $\phi$ at all.

Step 3. Collect terms into $E=\tfrac12\dot r^2+\Phi_{\mathrm{eff}}(r)$ with $\Phi_{\mathrm{eff}}=\Phi+L^2/2r^2$ . This is formally the energy of a unit-mass particle in one dimension, so $\dot r=\pm\sqrt{2[E-\Phi_{\mathrm{eff}}]}$ and the turning points are the roots of $E=\Phi_{\mathrm{eff}}(r)$ . Why: the centrifugal barrier $L^2/2r^2\to\infty$ as $r\to0$ guarantees an inner turning point for any $L\neq0$ , so bound orbits are trapped in a finite radial range.

Step 4. Separate variables in $\dot r=\mathrm dr/\mathrm dt$ and integrate over one full radial swing to get $T_r=2\int_{r_-}^{r_+}\mathrm dr/\sqrt{2[E-\Phi_{\mathrm{eff}}]}$ ; then use $\mathrm d\phi=(L/r^2)\,\mathrm dt$ to get $\Delta\phi=2L\int_{r_-}^{r_+}\mathrm dr/[r^2\sqrt{2[E-\Phi_{\mathrm{eff}}]}]$ . Why: these two integrals are the whole answer — every bound spherical orbit is fixed by $(E,L)$ through this pair of quadratures, computable for any $\Phi(r)$ even when no elementary antiderivative exists. *Verified: for $\Phi=-GM/r$ the integrals reduce to the standard Kepler conic $r=(L^2/GM)/(1+e\cos\phi)$ with $\Delta\phi=2\pi$ ; Binney & Tremaine §3.1, Goldstein Ch. 3.*

Go deeper · Derivation B · 2 minThe orbit equation and why generic orbits precess

Derivation BThe orbit equation and why generic orbits precess

The goal is to derive the differential equation for the orbit shape $r(\phi)$ — trading time for angle — and read off the exact condition under which the apsidal line stays fixed rather than drifting. This exposes precession as the generic behaviour and closure as the rare exception, and sets up Bertrand's theorem: only two potentials close for every energy and angular momentum.

Step 1. Change the independent variable from $t$ to $\phi$ using $\dot\phi=L/r^2$ : any time derivative becomes $\mathrm d/\mathrm dt=(L/r^2)\,\mathrm d/\mathrm d\phi$ . Applied to $r$ this gives $\dot r=(L/r^2)\,\mathrm dr/\mathrm d\phi$ , converting the radial energy equation into a relation between $r$ and $\phi$ alone. Why: the orbit's shape is a geometric fact independent of how fast it is traced, so eliminating time is the natural way to isolate precession.

Step 2. Substitute $u=1/r$ . Since $\mathrm dr/\mathrm d\phi=-u^{-2}\,\mathrm du/\mathrm d\phi$ , the energy equation becomes $(\mathrm du/\mathrm d\phi)^2=\tfrac{2}{L^2}[E-\Phi(1/u)]-u^2$ . Differentiating once yields the Binet equation $\mathrm d^2u/\mathrm d\phi^2+u=-\tfrac{1}{L^2u^2}\,\mathrm d\Phi/\mathrm dr$ . Why: in this form the right-hand side is a constant only for the Kepler potential ( $\mathrm d\Phi/\mathrm dr=GM/r^2\Rightarrow$ RHS $=GM/L^2$ ), and a constant forcing gives a pure sinusoid in $u$ — a closed conic that never precesses.

Step 3. For a general $\Phi$ the forcing depends on $u$ , so the solution is a sinusoid whose "wavelength" in $\phi$ is not $2\pi$ . Integrating over one radial swing gives the apsidal angle $\Delta\phi/2=L\int_{r_-}^{r_+}\mathrm dr/[r^2\sqrt{2[E-\Phi]-L^2/r^2}]$ , and the apsides advance (or regress) by $\Delta\phi-2\pi$ each period. Why: only if $\Delta\phi/2\pi$ is rational does the pattern eventually repeat; an irrational ratio makes the rosette fill its annulus densely and never close.

Step 4. Ask which potentials give $\Delta\phi$ independent of $E$ and $L$ and equal to a rational multiple of $2\pi$ for all bound orbits. Bertrand's theorem (1873) answers: exactly two — the Kepler potential $\Phi\propto-1/r$ with $\Delta\phi=2\pi$ , and the harmonic potential $\Phi\propto r^2$ with $\Delta\phi=\pi$ . Why: every other spherical potential precesses, so closed orbits are a knife-edge property of just two force laws — the deep fact underlying the whole classification. *Verified: Binet RHS constant $\Rightarrow$ conic for Kepler ( $\Delta\phi=2\pi$ ); harmonic well separates into $x,y$ oscillators of equal frequency giving a centred ellipse ( $\Delta\phi=\pi$ ). Goldstein Ch. 3; Binney & Tremaine §3.1.*

Go deeper · Derivation C · 1 minThe epicyclic frequency and the near-circular apsidal angle

Derivation CThe epicyclic frequency and the near-circular apsidal angle

The goal is to analyse orbits that stay close to a circular one, which describes most disc stars and gives the precession rate in closed form. We expand the effective potential to second order about its minimum, identify the small radial oscillation as simple harmonic motion with epicyclic frequency $\kappa$ , relate $\kappa$ to the circular frequency, and use $\Omega_c/\kappa$ to write the near-circular apsidal angle for any power-law potential — recovering the Kepler, harmonic, and flat-rotation-curve cases.

Step 1. At the circular radius $r_c$ the effective potential has $\Phi_{\mathrm{eff}}'(r_c)=0$ . Expand $\Phi_{\mathrm{eff}}(r)\approx\Phi_{\mathrm{eff}}(r_c)+\tfrac12\Phi_{\mathrm{eff}}''(r_c)(r-r_c)^2$ and write $\delta r=r-r_c$ . The radial equation of motion $\ddot{\delta r}=-\Phi_{\mathrm{eff}}''(r_c)\,\delta r$ is simple harmonic motion with $\kappa^2\equiv\Phi_{\mathrm{eff}}''(r_c)=\Phi''(r_c)+3L^2/r_c^4$ . Why: near-circular orbits are small radial oscillations superposed on circular motion — the "epicycles" of galactic-disc dynamics — and $\kappa$ is exactly the radial frequency $\Omega_r$ in this limit.

Step 2. Eliminate $L$ using $L_c^2=r_c^3\Phi'(r_c)$ and $\Omega_c^2=\Phi'(r_c)/r_c$ to get the disc-dynamics form $\kappa^2=r_c\,\mathrm d\Omega_c^2/\mathrm dr\big|_{r_c}+4\Omega_c^2=\Phi''(r_c)+3\Phi'(r_c)/r_c$ . Why: this expresses $\kappa$ purely through the rotation curve $\Omega_c(r)$ observers measure, so any galaxy's epicyclic frequency follows directly from its circular-speed profile.

Step 3. The near-circular apsidal angle is $\Delta\phi=2\pi\,\Omega_\phi/\Omega_r\to 2\pi\,\Omega_c/\kappa$ . For a power-law potential $\Phi\propto r^\alpha$ (with $\Phi\propto\ln r$ as the $\alpha\to0$ limit) one finds $\Omega_c/\kappa=1/\sqrt{\alpha+2}$ , so $\Delta\phi=2\pi/\sqrt{\alpha+2}$ . Why: a single formula now diagnoses every spherical potential — the exponent of the force law directly sets how far the apsides swing per radial period.

Step 4. Evaluate the special cases. Kepler $\alpha=-1$ : $\kappa=\Omega_c$ , $\Delta\phi=2\pi$ (closed, no precession). Harmonic $\alpha=2$ : $\kappa=2\Omega_c$ , $\Delta\phi=\pi$ (closed, centred ellipse). Flat rotation curve $\alpha\to0$ ( $\Phi\propto\ln r$ ): $\kappa=\sqrt2\,\Omega_c$ , $\Delta\phi=2\pi/\sqrt2\approx254.6^\circ$ (open, regressing rosette). Why: real galaxy halos sit near the logarithmic case, so their orbits precess by tens of degrees per period — the origin of the rosette in the second figure. *Verified: Kepler $\Phi=-GM/r$ gives $\Omega_c^2=\kappa^2=GM/r^3$ ; flat curve $\Phi=v_c^2\ln r$ gives $\kappa^2=2v_c^2/r^2=2\Omega_c^2$ . Binney & Tremaine eqs. (3.79)–(3.80).*

Limiting cases

The azimuthal advance $\Delta\phi$ per radial period is the single number that diagnoses the potential — closure, precession, and the epicyclic limit are all read off it.

Case	Result	Meaning
Kepler (point mass)	$\Phi=-\dfrac{GM}{r}\Rightarrow\Delta\phi=2\pi$	Closed ellipse, centre at a focus; no precession, thanks to the conserved Runge–Lenz vector.
Harmonic core	$\Phi=\tfrac12\Omega^2 r^2\Rightarrow\Delta\phi=\pi$	Closed ellipse centred on the origin; radial oscillation runs at twice the azimuthal rate.
Flat rotation curve	$\Phi=v_c^2\ln r\Rightarrow\Delta\phi=\dfrac{2\pi}{\sqrt2}\approx255^\circ$	Open rosette; $\kappa=\sqrt2\,\Omega_c$ , apsides regress each swing — the generic galaxy-halo case.
General galaxy	$\pi<\Delta\phi<2\pi$	Mass more spread than a point but more concentrated than uniform; the orbit precesses in the sense of motion.
Nearly circular	$\dfrac{\Omega_\phi}{\Omega_r}\to\dfrac{\Omega_c}{\kappa},\ \ \kappa^2=\Phi_{\mathrm{eff}}''(r_c)$	Small epicyclic oscillation about the guiding radius; $\Delta\phi=2\pi\,\Omega_c/\kappa$ .

Common trap“Orbits in a galaxy are just ellipses like the planets.”

They are not. The closed ellipse is a peculiarity of the exact $1/r$ point-mass potential (and the exact $r^2$ harmonic core) — by Bertrand's theorem those are the only two spherical potentials whose bound orbits close for every $E$ and $L$ . A galaxy's mass is spread out, its potential is neither of these, and its orbits are open rosettes: after one radial period the star arrives at apocentre displaced by $\Delta\phi\neq2\pi$ , never exactly retracing its path. Two further traps hide here. First, the well a star actually feels is not $\Phi(r)$ but $\Phi_{\mathrm{eff}}(r)=\Phi+L^2/2r^2$ — the centrifugal barrier is physically real and forbids any $L\neq0$ orbit from reaching $r=0$ . Second, "the orbit precesses" does not require general relativity: Newtonian gravity in an extended mass already precesses the apsides, and Mercury's relativistic perihelion advance is a small extra effect on top of the much larger Newtonian precession from the other planets.

Worked examples

1 · The Sun's orbit in the Milky Way (fully worked)

Model the Galaxy near the Sun by a flat rotation curve, $v_c=220\ \mathrm{km\,s^{-1}}$ at $R_0=8.0\ \mathrm{kpc}$ . Treat the Sun's orbit as nearly circular and find the azimuthal (galactic) period, the epicyclic frequency, the radial period, and the apsidal advance per radial oscillation.

Step 1. Circular frequency. With $1\ \mathrm{kpc}=3.086\times10^{16}\ \mathrm{km}$ , $R_0=2.469\times10^{17}\ \mathrm{km}$ , so $\Omega_c=v_c/R_0=220/2.469\times10^{17}=8.91\times10^{-16}\ \mathrm{rad\,s^{-1}}$ . The azimuthal period is $T_\phi=2\pi/\Omega_c=7.05\times10^{15}\ \mathrm s=\mathbf{223\ Myr}$ — one "galactic year."

Step 2. Epicyclic frequency. For a flat rotation curve $\kappa=\sqrt2\,\Omega_c=1.26\times10^{-15}\ \mathrm{rad\,s^{-1}}$ . The radial period is $T_r=2\pi/\kappa=T_\phi/\sqrt2=\mathbf{158\ Myr}$ , so the Sun completes one in-and-out radial oscillation in about $71\%$ of the time it takes to circle the Galaxy once.

Step 3. Apsidal advance. In one radial period the Sun sweeps $\Delta\phi=2\pi\,\Omega_c/\kappa=2\pi/\sqrt2=\mathbf{254.6^\circ}$ in azimuth. Since $\Delta\phi<360^\circ$ , the apsidal line regresses by $360^\circ-254.6^\circ=105^\circ$ per radial period — the Sun traces a slowly turning rosette, exactly the pattern of the second figure.

Step 4. Reality check. The measured local ratio is $\kappa/\Omega_c\approx1.35$ rather than the idealised $\sqrt2\approx1.41$ (the real curve is not perfectly flat), shifting $\Delta\phi$ to $\approx267^\circ$ but leaving the picture unchanged. The Sun oscillates radially every ≈160 Myr while circling the Galaxy every ≈220 Myr, its orbit a non-closing rosette.

2 · Turning points and radial period of a plunging satellite (fully worked)

A dwarf satellite orbits far outside most of the Galaxy's mass, feeling an approximately Keplerian potential from an enclosed $M=1\times10^{12}\ M_\odot$ . Its apocentre is $r_+=100\ \mathrm{kpc}$ , its pericentre $r_-=20\ \mathrm{kpc}$ . Find its eccentricity, semi-major axis, and radial period, and say whether the orbit closes.

Step 1. Geometry. For a Kepler ellipse the turning points are $r_\pm=a(1\pm e)$ , so $a=\tfrac12(r_++r_-)=\mathbf{60\ kpc}$ and $e=(r_+-r_-)/(r_++r_-)=80/120=\mathbf{0.667}$ — a distinctly plunging orbit.

Step 2. Radial period from Kepler's third law, $T_r=2\pi\sqrt{a^3/GM}$ . Using $G=4.301\times10^{-6}\ \mathrm{kpc\,(km/s)^2}\,M_\odot^{-1}$ : $GM=4.301\times10^{6}\ \mathrm{kpc\,(km/s)^2}$ and $a^3=2.16\times10^{5}\ \mathrm{kpc^3}$ , so $a^3/GM=5.02\times10^{-2}\ \mathrm{kpc^2/(km/s)^2}$ and $\sqrt{a^3/GM}=0.224\ \mathrm{kpc/(km/s)}$ .

Step 3. Convert. Since $1\ \mathrm{kpc/(km/s)}=0.978\ \mathrm{Gyr}$ , $T_r=2\pi(0.224)(0.978)=\mathbf{1.38\ Gyr}$ . The satellite spends most of that time near apocentre, where it moves slowest — which is why satellites are preferentially found near their turning points.

Step 4. Closure. Because the potential is (assumed) exactly Keplerian, $\Delta\phi=2\pi$ and the orbit is a closed ellipse that retraces itself each period. In a realistic extended halo the enclosed mass grows with radius, the potential is shallower than $1/r$ , and the orbit would precess into an open rosette instead. a = 60 kpc, e = 0.67, T_r ≈ 1.4 Gyr, closed only in the idealised point-mass limit.

Physics of the portrait

Closed orbits look so natural — planets trace fixed ellipses, year after year — that it is a shock to learn how special they are. The person who made that precise was a French mathematician who proved that in all of central-force dynamics only two force laws share the privilege.

Origins · 1 minJoseph Bertrand and how this was discovered

Joseph Bertrand

1822–1900 · closed orbits and Bertrand's theorem in central-force dynamics

Joseph Louis François Bertrand was a prodigy — he earned his doctorate at seventeen — and spent his career at the École Polytechnique and the Collège de France, ranging across number theory, probability, and the history of science. His most enduring contribution to dynamics is a short 1873 note to the Comptes Rendus answering a deceptively simple question: for which central force laws are all bound orbits closed? The Kepler problem gives closed ellipses, the harmonic oscillator gives closed centred ellipses, and one might guess many force laws would do the same. Bertrand proved the opposite — that these two are the only ones. His proof studies the apsidal angle of near-circular orbits and demands that it stay rational and constant as the orbit is made more eccentric; only the inverse-square force ( $\Phi\propto-1/r$ ) and the linear force ( $\Phi\propto r^2$ ) survive. The theorem quietly underlies all of stellar dynamics: the beautiful closed ellipses of the Solar System are a coincidence of the point-mass potential, and the extended mass of a galaxy guarantees precession. Bertrand also wrote influential textbooks and, as perpetual secretary of the Académie des Sciences, shaped French science for a generation.

History

1609

— Kepler finds that Mars moves on an ellipse with the Sun at a focus

1687

— Newton derives elliptical orbits from the inverse-square law in the Principia

1873

— Bertrand proves only the 1/r and r² potentials give closed orbits for all E and L

1915

— Einstein's general relativity accounts for the residual perihelion advance of Mercury

1987

— Binney and Tremaine's Galactic Dynamics systematises orbits in galactic potentials

Going further

Texts. Binney & Tremaine, Galactic Dynamics, 2nd ed., §3.1 (spherical potentials, effective potential, apsidal angle, epicyclic frequency) · Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., Ch. 2 and Ch. 24 · Goldstein, Poole & Safko, Classical Mechanics, 3rd ed., Ch. 3 (central-force problem, Bertrand's theorem) · Mo, van den Bosch & White, Galaxy Formation and Evolution, Ch. 5 · Ryden, Introduction to Cosmology, 2nd ed.

Provenance. The reduction to $\Phi_{\mathrm{eff}}$ , the radial-period and azimuthal-advance quadratures, the Binet orbit equation, Bertrand's theorem, and the epicyclic frequency $\kappa^2=\Phi_{\mathrm{eff}}''(r_c)$ are standard central-force and stellar-dynamics results. A verifier should reproduce: (1) $\mathrm d\mathbf L/\mathrm dt=0$ for a central force; (2) the Kepler limit $\Delta\phi=2\pi$ and harmonic limit $\Delta\phi=\pi$ ; (3) $\kappa=\Omega_c$ for Kepler and $\kappa=\sqrt2\,\Omega_c$ for a flat rotation curve; (4) the worked numbers — $T_\phi=223\ \mathrm{Myr}$ , $T_r=158\ \mathrm{Myr}$ , $\Delta\phi=254.6^\circ$ for $(v_c,R_0)=(220\ \mathrm{km\,s^{-1}},8.0\ \mathrm{kpc})$ , and $a=60\ \mathrm{kpc}$ , $e=0.667$ , $T_r=1.38\ \mathrm{Gyr}$ for the Kepler satellite. Cross-check against Binney & Tremaine §3.1 and Goldstein Ch. 3.

Prerequisites. Gravitational Two-Body Problem · Galactic Potential Models · Distribution Function and Phase Space. Leads to. Epicyclic Approximation · Action-Angle Variables · Jeans' Theorem · Orbits in Axisymmetric and Triaxial Potentials.

Ask the tutor: Ask me to walk through the effective-potential quadrature for any $\Phi(r)$ you like — point mass, isothermal sphere, Hernquist, or NFW — and we will find its turning points, radial period, and apsidal precession together.

Why does spherical symmetry conserve the whole angular-momentum vector, not just its magnitude?

Conservation follows from the torque, $\mathrm d\mathbf L/\mathrm dt=\mathbf r\times\mathbf F$ . In a spherical potential the force is central — it points exactly along $\pm\hat{\mathbf r}$ — so the cross product $\mathbf r\times\mathbf F$ vanishes identically, and all three components of $\mathbf L$ are constant. The fixed magnitude controls the centrifugal barrier and radial oscillation; the fixed direction pins the orbit into a single plane. In a merely axisymmetric potential (a disc galaxy) only the component of $\mathbf L$ along the symmetry axis is conserved, the orbit is no longer planar, and that extra freedom is what makes axisymmetric orbits richer.

Is the effective potential a real force or just a bookkeeping trick?

Both. Mathematically it is exact bookkeeping: eliminating the azimuthal coordinate via $L=r^2\dot\phi$ drops the leftover angular kinetic energy $L^2/2r^2$ into the potential, giving a genuine one-dimensional problem with no approximation. Physically, in the frame following the star's azimuth the centrifugal term is felt as a real outward force $L^2/r^3=-\mathrm d(L^2/2r^2)/\mathrm dr$ . Either way the consequence is concrete: the barrier keeps any rotating star out of the centre and sets the pericentre. The well a star with angular momentum feels is $\Phi_{\mathrm{eff}}$ , not $\Phi$ alone.

What makes the Kepler and harmonic potentials so special?

Each hides an extra conserved quantity beyond $E$ and $L$ . For Kepler it is the Laplace–Runge–Lenz vector, which points along the major axis and is constant — so the axis cannot move and the ellipse cannot precess. For the harmonic well the motion separates into independent $x$ and $y$ oscillators of equal frequency with fixed relative phase, again locking the orientation. Bertrand's theorem shows these are the only two spherical potentials with this rigidity for all bound orbits; any deviation destroys the extra integral, and the apsides begin to drift — precession is the generic fate.

How is the epicyclic frequency κ different from the circular frequency Ω_c?

$\Omega_c$ is how fast the star circles the centre on a circular orbit; $\kappa$ is how fast it oscillates radially if you nudge it slightly off that circle. They are generally different: $\kappa^2=\Phi_{\mathrm{eff}}''(r_c)=\Phi''+3\Phi'/r$ , evaluated at $r_c$ . For a Kepler potential they are equal, $\kappa=\Omega_c$ , so the radial and azimuthal motions stay in step and the ellipse closes. For a flat rotation curve $\kappa=\sqrt2\,\Omega_c$ , and the mismatch is precisely the precession rate $\Omega_c-\kappa$ . Physically $\kappa$ is the frequency of the small "epicycle" a disc star executes about its guiding centre, and it governs spiral-density-wave and resonance theory.

Given only E and L, how do I actually compute an orbit without integrating the full 3-D motion?

You never need the three-dimensional trajectory. First solve the single algebraic equation $E=\Phi(r)+L^2/2r^2$ for its two roots $r_-$ and $r_+$ . Then evaluate two one-dimensional integrals — the radial period $T_r=2\int_{r_-}^{r_+}\mathrm dr/\sqrt{2[E-\Phi_{\mathrm{eff}}]}$ and the azimuthal advance $\Delta\phi=2L\int_{r_-}^{r_+}\mathrm dr/[r^2\sqrt{2[E-\Phi_{\mathrm{eff}}]}]$ — which give the two frequencies and the whole orbit geometry. They evaluate trivially for any $\Phi(r)$ ; Gaussian quadrature handles the endpoint square-root singularities in a handful of points.

Do these single-particle orbits matter for a whole galaxy, or only for one test star?

They are the foundation of the whole equilibrium theory. Because $E$ and $L$ are conserved on every spherical orbit, Jeans' theorem says any steady-state distribution function is $f(E,L)$ — the galaxy is a weighted superposition of these orbits. Integrating $f$ over velocities gives the density; the mix of low- $L$ (radial) and high- $L$ (circular) orbits at each energy sets the anisotropy $\beta=1-\sigma_t^2/2\sigma_r^2$ measured in elliptical-galaxy halos, and $T_r(E,L)$ fixes the timescales entering relaxation and dynamical friction. The one-star problem is the bookkeeping that makes a self-consistent galaxy model possible.