Isothermal Sphere and King Models

Treat the stars of a cluster as the molecules of a self-gravitating gas held at one fixed "temperature", and a remarkable simplification follows. A time-independent solution of the collisionless Boltzmann equation must, by Jeans' theorem, be built from integrals of motion; the simplest ergodic choice for a spherical isotropic system is a distribution function of energy alone, $f=f(E)$ . Pick the exponential $f\propto e^{-E/\sigma^2}$ — a Maxwellian in velocity — and every local velocity distribution has the same dispersion $\sigma$ at every radius. That is the isothermal sphere, and it is literally the same object as an isothermal gas ball: both integrate to the Boltzmann density $\rho\propto e^{-\Phi/\sigma^2}$ and obey the same Poisson–Boltzmann equation. Feed that density into Poisson's equation and you get the isothermal Lane–Emden equation, whose solution is a flat central core followed by an $r^{-2}$ envelope, with the exact scale-free member $\rho=\sigma^2/2\pi Gr^2$ — the singular isothermal sphere that gives flat rotation curves and the workhorse gravitational lens. But every fully-Maxwellian sphere carries a fatal flaw: its mass diverges as $M(r)\propto r$ , because the exponential tail populates arbitrarily high energies and nothing is ever bound. Ivan King's 1966 repair was pure physics: a real cluster sits in a tidal field, so subtract a constant to kill the high-energy tail, $f\propto e^{-E/\sigma^2}-1$ . The density then reaches zero at a finite tidal radius, and a one-parameter family of finite clusters is born — the King models that still fit globular clusters today.

Key takeaways

The isothermal sphere is the equilibrium of a collisionless system with a Maxwellian-in-energy distribution function, $f\propto e^{-E/\sigma^2}$ , giving one constant velocity dispersion $\sigma$ at every radius.
Integrating the DF gives the Boltzmann density $\rho\propto e^{-\Phi/\sigma^2}$ ; Poisson's equation then yields the isothermal Lane–Emden equation — a flat core plus an $r^{-2}$ envelope.
Its exact scale-free member is the singular isothermal sphere $\rho=\sigma^2/(2\pi Gr^2)$ , giving a flat rotation curve $v_c=\sqrt2\,\sigma$ ; but its mass $M(r)=2\sigma^2 r/G$ diverges.
King models cure the divergence by lowering and truncating the Maxwellian at the escape energy, $f\propto e^{-E/\sigma^2}-1$ for $E<0$ , giving a finite tidal radius $r_t$ and a finite-mass cluster.
A King model is a one-parameter family labelled by the concentration $c=\log_{10}(r_t/r_0)$ or the central potential $W_0$ ; observed globular clusters sit at $0.75\lesssim c\lesssim2.5$ .
Key idea: a Maxwellian-in-energy distribution function makes the density obey $\rho\propto e^{-\Phi/\sigma^2}$ ; Poisson's equation then gives the isothermal Lane–Emden equation, whose finite-mass repair is King's truncated (lowered) Maxwellian.

Where this lives today

King models remain the daily currency of stellar-dynamics observers and cosmologists alike. Globular-cluster catalogues such as the Harris compilation list every Galactic cluster by its concentration $c$ and core radius, and the drift of a fitted $c$ toward large values is the observational fingerprint of core collapse — the runaway central contraction that King equilibria cannot capture, marking where the model breaks and gravothermal physics takes over. On galaxy scales the singular isothermal sphere is the analytic lens of choice: strong-lensing surveys (SLACS and successors) fit elliptical-galaxy lenses with an SIS, reading $\sigma$ off a measured Einstein radius through the $\theta_E\propto\sigma^2$ relation of Example 1. In cosmology the SIS is the pedagogical dark-matter halo — its flat rotation curve was the first quantitative model of the missing-mass problem — even though $N$ -body simulations show real haloes follow the cuspy NFW profile $\rho\propto r^{-1}$ in the centre. The open frontier is dynamical: how relaxation, mass segregation, binary heating, and tidal stripping push a cluster off the King sequence is still modelled with Fokker–Planck and direct $N$ -body codes that take a King model as their initial condition.

The physical picture

Imagine each star as a molecule of a gas whose only "thermodynamic" number is the specific kinetic energy $\sigma^2$ , playing the role of $k_BT/m$ . The two-body encounters that would thermalise an ordinary gas are far too slow to matter over a cluster's life, yet over many relaxation times a globular cluster's dense core genuinely drifts toward a Maxwellian, so the analogy has teeth. Hydrostatic balance, $\mathrm{d}P/\mathrm{d}r=-\rho\,\mathrm{d}\Phi/\mathrm{d}r$ with $P=\rho\sigma^2$ , integrates at once to $\rho\propto e^{-\Phi/\sigma^2}$ — the same expression a Maxwellian distribution function gives on integrating over velocity. Collisionless stars and collisional gas share an equation because both obey the same energy-per-unit-mass Boltzmann factor. The structure that emerges is universal: a flat harmonic core where the density is nearly constant, wrapped in an envelope where $\rho\propto r^{-2}$ balances a mass that grows linearly outward. That envelope is the singular isothermal sphere, and its flat rotation curve $v_c=\sqrt2\,\sigma$ is exactly the phenomenology dark-matter haloes were first invented to reproduce. The first figure shows the whole family on one pair of log axes.

Figure

Density run of the isothermal family on log axes: the full isothermal sphere in blue has a flat core of size r_0 then falls as ρ ∝ r⁻², tracking the singular isothermal sphere ρ = σ²/(2πG r²) in red dashed; a King model in amber coincides in the core but truncates to zero at the tidal radius r_t, here r_t/r_0 = 30 so the concentration c = 1.48

King's truncation is not a mathematical trick but a physical statement. A cluster orbits inside a host galaxy, so any star pushed above the local escape energy simply wanders off and is lost. Chop the Maxwellian's high-velocity tail there and the extended atmosphere collapses to a finite edge — the tidal radius $r_t$ . What remains is a one-parameter sequence: fix the concentration $c=\log_{10}(r_t/r_0)$ and the whole projected shape is determined, so a two-number fit — central brightness and concentration — captures almost every globular cluster in the sky. The second figure shows that family, the curves a cluster photometrist actually fits.

Figure

The King one-parameter family: projected surface brightness I(R)/I_0 on log axes for concentrations c = 0.75, 1.5, 2.25 from the King 1962 profile; a larger c pushes the truncation r_t further out, where c = log₁₀(r_t/r_0) is the single number that fixes the shape

The core equations

The starting point is a distribution function depending on phase-space coordinates only through the specific energy $E$ , Maxwellian in it — the "ergodic" choice Jeans' theorem permits for a spherical isotropic system, with a single constant dispersion $\sigma$ .

f(E)=\frac{\rho_1}{(2\pi\sigma^2)^{3/2}}\,e^{-E/\sigma^2},\qquad E=\tfrac12 v^2+\Phi(r) \qquad (1)

where: $f(E)$ — phase-space distribution function, mass per unit $\mathrm{d}^3x\,\mathrm{d}^3v$ ; $E$ — specific energy per unit mass; $\sigma$ — one-dimensional velocity dispersion, constant; $\Phi(r)$ — gravitational potential; $\rho_1$ — density normalisation. Reading: the exponential in energy is the single assumption from which everything else follows. Integrating (1) over velocities collapses the Gaussian and leaves a density depending on position only through the potential — the Boltzmann (barometric) density, whose second velocity moment is $\sigma^2$ at every radius, the origin of the word "isothermal".

\rho(r)=\rho_0\,e^{-[\Phi(r)-\Phi(0)]/\sigma^2},\qquad \langle v_i^2\rangle=\sigma^2 \qquad (2)

where: $\rho_0=\rho_1 e^{-\Phi(0)/\sigma^2}$ — central density; $\Phi(0)$ — central potential; $\langle v_i^2\rangle$ — one-dimensional velocity variance. Reading: density tracks the exponential of the well, and $\sigma^2$ never changes with radius. Substituting the Boltzmann density into Poisson's equation and non-dimensionalising gives the isothermal Lane–Emden equation — the master equation of the subject, with no closed form, integrated numerically outward from a regular centre.

\frac{1}{\tilde r^2}\frac{\mathrm{d}}{\mathrm{d}\tilde r}\!\left(\tilde r^2\frac{\mathrm{d}\psi}{\mathrm{d}\tilde r}\right)=e^{-\psi},\qquad \tilde r=\frac{r}{r_c},\quad r_c=\sqrt{\frac{\sigma^2}{4\pi G\rho_0}} \qquad (3)

where: $\psi=(\Phi-\Phi_0)/\sigma^2$ — dimensionless potential, zero at the centre; $\tilde r$ — radius in units of the natural scale $r_c$ ; $G$ — gravitational constant. Reading: the King (core) radius is $r_0=3r_c=\sqrt{9\sigma^2/(4\pi G\rho_0)}$ , where the projected density is about half its central value. The one exact analytic solution is scale-free: assume $\rho\propto r^{-2}$ and the equation closes, giving the singular isothermal sphere — the large-radius limit of every isothermal sphere.

\rho_{\rm SIS}(r)=\frac{\sigma^2}{2\pi G r^2},\qquad M(r)=\frac{2\sigma^2 r}{G},\qquad v_c=\sqrt2\,\sigma \qquad (4)

where: $M(r)$ — enclosed mass, growing linearly; $v_c$ — circular speed, radius-independent. Reading: the flat rotation curve $v_c=\sqrt2\,\sigma$ is a triumph, but $M(r)\propto r$ means infinite total mass. King's cure is to lower and truncate the Maxwellian: measure energy relative to the escape energy and subtract a constant so the DF vanishes there — stars above escape energy are simply gone.

f_K(E)=\begin{cases}\rho_1(2\pi\sigma^2)^{-3/2}\big(e^{-E/\sigma^2}-1\big),&E<0\\[2pt]0,&E\ge0\end{cases} \qquad (5)

where: $E$ — specific energy relative to the escape energy, zero at the tidal boundary; $\rho_1$ — normalisation. Reading: subtracting the constant makes $f_K\to0$ smoothly as $E\to0^-$ , so the density reaches exactly zero at a finite radius. The truncated DF produces a finite, cored, one-parameter family, labelled by the central potential $W_0$ or, equivalently, the concentration $c$ — a single number that fixes the shape.

c\equiv\log_{10}\!\left(\frac{r_t}{r_0}\right),\qquad W_0\equiv-\frac{\Phi(0)}{\sigma^2} \qquad (6)

where: $r_t$ — tidal (limiting) radius where $\rho=0$ ; $c$ — King concentration; $W_0$ — depth of the central well in units of $\sigma^2$ . Reading: larger $W_0$ (larger $c$ ) means a deeper, denser, more centrally concentrated cluster, approaching the full isothermal sphere as $W_0\to\infty$ . Historically the models began as a fitting function: King's 1962 empirical surface-brightness law preceded the dynamical models and remains the quickest fit to cluster photometry.

I(R)=I_0\left[\frac{1}{\sqrt{1+(R/r_0)^2}}-\frac{1}{\sqrt{1+(r_t/r_0)^2}}\right]^2 \qquad (7)

where: $I(R)$ — projected surface brightness at projected radius $R$ ; $I_0$ — central normalisation. Reading: the bracketed subtraction forces $I\to0$ at $R=r_t$ , mimicking the dynamical model's truncation with just $r_0$ and $r_t$ .

Derivation AThe singular isothermal sphere and the flat rotation curve

The goal is to show that the isothermal Poisson equation admits one exact, scale-free solution in which the density falls as $r^{-2}$ , and to fix its single normalisation from the requirement that it also satisfy the Boltzmann density relation. Reading off the enclosed mass and circular speed then exposes both the flat rotation curve and the linear mass divergence that motivates King's whole construction, by demanding self-consistency between Newtonian gravity and the barometric density.

Step 1. Adopt the ansatz $\rho(r)=A\,r^{-2}$ . Why: dimensional balance in Poisson's equation with $\Phi\sim\sigma^2\ln r$ singles out the exponent $-2$ ; a power law is the only self-similar guess.

Step 2. Integrate for the enclosed mass, $M(r)=\int_0^r 4\pi r'^2\rho\,\mathrm{d}r'=4\pi A r$ . Why: the mass grows linearly with radius, which already signals that the total mass will diverge — the hallmark pathology of a fully Maxwellian sphere.

Step 3. Get the radial force from the shell theorem, $\mathrm{d}\Phi/\mathrm{d}r=GM(r)/r^2=4\pi GA/r$ . Why: only enclosed mass matters for a spherical system, so the acceleration falls off as $1/r$ — the form a logarithmic potential produces.

Step 4. Impose the Boltzmann relation $\rho=\rho_1 e^{-\Phi/\sigma^2}$ , so $\mathrm{d}\Phi/\mathrm{d}r=-\sigma^2\,\mathrm{d}\ln\rho/\mathrm{d}r=2\sigma^2/r$ . Why: the isothermal equilibrium ties the potential gradient to the density slope, which pins the constant.

Step 5. Match the two expressions for $\mathrm{d}\Phi/\mathrm{d}r$ : $4\pi GA=2\sigma^2$ , hence $A=\sigma^2/(2\pi G)$ and $\rho=\sigma^2/(2\pi Gr^2)$ . Why: equating gravity's force to the isothermal force fixes the one free constant entirely in terms of $\sigma$ .

Step 6. Read off the circular speed, $v_c^2=GM(r)/r=2\sigma^2$ , so $v_c=\sqrt2\,\sigma$ , constant with radius. Why: a rotation curve that neither rises nor falls is what dark-matter phenomenology needs — why the SIS is the first halo anyone writes down. *Self-consistency check: substituting $\rho=\sigma^2/2\pi Gr^2$ back into the isothermal Poisson equation reproduces it identically, confirming the ansatz; $v_c=\sqrt2\,\sigma$ and $M=2\sigma^2r/G$ follow directly.*

Go deeper · Derivation B · 1 minFrom a Maxwellian distribution function to the isothermal Lane–Emden equation

Derivation BFrom a Maxwellian distribution function to the isothermal Lane–Emden equation

The aim is to trace the full chain from a microscopic distribution function to the macroscopic equation governing the density profile, and to prove that the velocity dispersion is genuinely constant. We integrate the Maxwellian over velocity for the Boltzmann density, confirm the second moment, then insert the density into Poisson's equation and non-dimensionalise so a single scale-free equation emerges — which is what justifies calling the model "isothermal".

Step 1. Integrate the DF over velocity, $\rho=\int f\,\mathrm{d}^3v=\rho_1(2\pi\sigma^2)^{-3/2}e^{-\Phi/\sigma^2}\int e^{-v^2/2\sigma^2}\mathrm{d}^3v$ . Why: at fixed position the potential factors out, leaving a pure Gaussian that carries all the velocity dependence.

Step 2. Evaluate the Gaussian, $\int e^{-v^2/2\sigma^2}\mathrm{d}^3v=(2\pi\sigma^2)^{3/2}$ , so $\rho(r)=\rho_1 e^{-\Phi/\sigma^2}$ . Why: the normalisation cancels the prefactor exactly, delivering the barometric density — the relation an isothermal gas obeys.

Step 3. Compute the second moment, $\langle v_i^2\rangle=\rho^{-1}\!\int v_i^2 f\,\mathrm{d}^3v=\sigma^2$ , independent of $r$ . Why: the local Maxwellian has the same width everywhere, so the "temperature" $\sigma^2$ is spatially uniform — the defining property of an isothermal system.

Step 4. Insert $\rho=\rho_0 e^{-\psi}$ with $\psi=(\Phi-\Phi_0)/\sigma^2$ into $\nabla^2\Phi=4\pi G\rho$ , giving $(1/r^2)\,\mathrm{d}(r^2\psi')/\mathrm{d}r=(4\pi G\rho_0/\sigma^2)e^{-\psi}$ . Why: the dimensionless $\psi$ turns the density into a clean $e^{-\psi}$ source, isolating the one dimensionful group $4\pi G\rho_0/\sigma^2$ .

Step 5. Scale radius by $r_c=\sqrt{\sigma^2/4\pi G\rho_0}$ to absorb that group, leaving $(1/\tilde r^2)\,\mathrm{d}(\tilde r^2\,\mathrm{d}\psi/\mathrm{d}\tilde r)/\mathrm{d}\tilde r=e^{-\psi}$ with $\psi(0)=\psi'(0)=0$ . Why: every isothermal sphere is now the same dimensionless curve; clusters differ only by the scales $\rho_0$ and $r_c$ .

Step 6. Series-solve near the centre, $\psi=\tfrac16\tilde r^2-\tfrac{1}{120}\tilde r^4+\dots$ , so $\rho\simeq\rho_0(1-\tfrac16\tilde r^2)=\rho_0[1-\tfrac32(r/r_0)^2]$ using $r_0=3r_c$ . Why: the leading balance $\psi''+2\psi'/\tilde r\approx1$ gives a flat harmonic core, marking $r_0$ as where the density falls appreciably below $\rho_0$ . *Self-consistency check: the coefficients $\tfrac16$ and $-\tfrac{1}{120}$ satisfy the Lane–Emden recursion by direct substitution, and the moment integrals confirm $\langle v_i^2\rangle=\sigma^2$ at every radius.*

Go deeper · Derivation C · 1 minKing's lowered Maxwellian and the finite tidal radius

Derivation CKing's lowered Maxwellian and the finite tidal radius

The purpose is to demonstrate precisely how subtracting a constant from the Maxwellian repairs the infinite-mass pathology, by integrating the lowered DF over the permitted velocities up to the escape speed and showing the density vanishes at a finite radius. We introduce the dimensionless relative potential measuring depth below the tidal boundary, carry out the velocity integral, and expand near the edge to prove the density reaches exactly zero — which is what makes King models physical.

Step 1. Define the relative potential $W(r)=[\Phi(r_t)-\Phi(r)]/\sigma^2\ge0$ , with $W_0=W(0)$ at the centre and $W=0$ at the tidal radius $r_t$ . Why: measuring energy downward from the escape value makes the bound region $W>0$ and the boundary $W=0$ , so the truncation condition becomes a clean statement about where $W$ vanishes.

Step 2. Only stars with $E<0$ are retained, i.e. speeds up to the local escape speed $v_e=\sigma\sqrt{2W}$ ; integrate the lowered DF $e^{-E/\sigma^2}-1$ over $0\le v\le v_e$ . Why: the truncation is enforced as a hard cut in the velocity integral, so the density automatically knows nothing about the deleted high-energy tail.

Step 3. The integral gives the King density $\rho(W)=\rho_1\big[e^{W}\,\mathrm{erf}(\sqrt W)-\sqrt{4W/\pi}\,(1+\tfrac23 W)\big]$ . Why: the error function is the finite-range Gaussian left after the cut, and the polynomial term is the subtracted constant — together a density depending on radius only through $W$ .

Step 4. Expand near the edge, $W\to0^+$ : both terms behave as $\tfrac{2}{\sqrt\pi}\sqrt W\,(1+\tfrac23 W)$ and cancel, so $\rho\to0$ . Why: the exact cancellation at the boundary produces a genuine finite edge at $r_t$ instead of an exponential atmosphere, giving a finite total mass.

Step 5. Integrate the Poisson equation for $W(r)$ from $W_0$ outward until $W=0$ ; the ratio $r_t/r_0$ then defines $c=\log_{10}(r_t/r_0)$ , an increasing function of $W_0$ . Why: the single boundary value $W_0$ fixes the whole profile, which is why one number labels the King sequence. *Self-consistency check: the $W\to0$ cancellation follows by direct series expansion, and the monotonic $c(W_0)$ relation is confirmed by numerical integration of the King equation.*

Limiting cases

The isothermal family has clean asymptotic and special cases:

Case	Result	Meaning
Singular isothermal sphere	$\rho=\dfrac{\sigma^2}{2\pi G r^2}$	scale-free exact solution; flat rotation curve $v_c=\sqrt2\,\sigma$ , $M=2\sigma^2 r/G$ , but divergent central density and mass
Large radius, any isothermal sphere	$\rho\to\dfrac{\sigma^2}{2\pi G r^2}$	every non-singular sphere converges to the SIS envelope, so its mass still diverges linearly
King core, small radius	$\rho\simeq\rho_0\big[1-\tfrac32(r/r_0)^2\big]$	flat harmonic core; the empirical King 1962 profile fits the projected light here
Deep King limit $W_0\to\infty$	$c=\log_{10}(r_t/r_0)\to\infty$	truncation recedes to infinity; the model approaches the full isothermal sphere, and core-collapsed clusters sit at high $c$
Shallow King limit $W_0\to0$	$\rho\propto(r_t^2-r^2)^{5/2}$	a low-concentration polytrope-like ball with no extended envelope

Common trapThe dispersion σ is a temperature, and the plain isothermal sphere has a finite mass

Two traps hide in the word "isothermal". First, $\sigma$ is the one-dimensional velocity dispersion, not the three-dimensional $\sqrt3\,\sigma$ ; and $\sigma^2$ plays the role of $k_BT/m$ , an energy per unit mass, not a temperature in kelvin — never plug it into a thermodynamic formula expecting degrees. Second, and more dangerous: the full non-singular isothermal sphere does not have a finite mass. Its density asymptotes to the SIS envelope $\rho\propto r^{-2}$ , so $M(r)=\int 4\pi r^2\rho\,\mathrm{d}r\propto r$ diverges. Quoting a "total mass" is meaningful only for a King-truncated model, where $r_t$ cuts the integral off. If you integrate an untruncated isothermal sphere to infinity and get a number, you have made an error.

Worked examples

1 · Strong lensing by an elliptical galaxy (fully worked)

An elliptical galaxy is modelled as a singular isothermal sphere with one-dimensional dispersion $\sigma=250\ \mathrm{km\,s^{-1}}$ . A background quasar sits behind it with $D_{ls}/D_s=0.5$ . Find the Einstein radius and the expected image separation.

Step 1. The SIS produces a fixed Einstein angle independent of source position, $\theta_E=4\pi\left(\dfrac{\sigma}{c}\right)^2\dfrac{D_{ls}}{D_s}$ . This is its signature: the $\rho\propto r^{-2}$ profile gives a deflection angle constant with impact parameter, so the lens has one characteristic scale.

Step 2. Evaluate the dispersion ratio: $\sigma/c=(2.50\times10^5)/(2.998\times10^8)=8.333\times10^{-4}$ , so $(\sigma/c)^2=6.944\times10^{-7}$ . Then $\theta_E=4\pi(6.944\times10^{-7})(0.5)=4.363\times10^{-6}\ \mathrm{rad}$ .

Step 3. Convert to arcseconds: $\theta_E=4.363\times10^{-6}\times206265=\mathbf{0.90\ arcsec}$ . A source inside the Einstein radius is imaged twice on opposite sides of the lens, so the separation is $\Delta\theta\approx2\theta_E=\mathbf{1.8\ arcsec}$ .

Step 4. Sanity check: galaxy-scale strong lenses have arcsecond separations, exactly this scale, and $\theta_E\propto\sigma^2$ is why a rich-cluster lens ( $\sigma\sim1000\ \mathrm{km\,s^{-1}}$ ) produces separations tens of times larger. θ_E ≈ 0.90 arcsec, image separation ≈ 1.8 arcsec.

2 · Core and tidal radius of a globular cluster (fully worked)

A globular cluster has central one-dimensional dispersion $\sigma=7\ \mathrm{km\,s^{-1}}$ and central density $\rho_0=10^4\ M_\odot\,\mathrm{pc^{-3}}$ . Using $G=4.301\times10^{-3}\ \mathrm{pc}\,(\mathrm{km\,s^{-1}})^2 M_\odot^{-1}$ , find the King core radius, and the tidal radius if its concentration is $c=1.5$ .

Step 1. The King core radius is $r_0=\sqrt{9\sigma^2/(4\pi G\rho_0)}$ . Compute the denominator: $4\pi G\rho_0=4\pi(4.301\times10^{-3})(10^4)=540.4\ (\mathrm{km\,s^{-1}})^2\mathrm{pc^{-2}}$ .

Step 2. Numerator: $9\sigma^2=441\ (\mathrm{km\,s^{-1}})^2$ . So $r_0=\sqrt{441/540.4}=\mathbf{0.90\ pc}$ , a typical globular-cluster core radius.

Step 3. The concentration fixes the edge: $r_t=r_0\,10^{c}=0.90\times31.6=\mathbf{28.6\ pc}$ . Real clusters span puffy ( $c\approx0.75$ ) to dense ( $c\approx2.5$ , $r_t/r_0\approx300$ ).

Step 4. Interpretation: $(r_0,c)$ plus the central brightness fully specify the King fit, and $r_t$ tracks the tidal environment — a cluster on a tighter galactic orbit is stripped to a smaller $r_t$ and a lower concentration. r_0 ≈ 0.90 pc, r_t ≈ 28.6 pc for c = 1.5.

3 · Reading a flat rotation curve as an isothermal halo

A spiral galaxy has a flat rotation curve at $v_c=220\ \mathrm{km\,s^{-1}}$ . Treat the halo as a singular isothermal sphere. What one-dimensional dispersion does it imply, and what dark mass lies within 10 kpc?

Step 1. From $v_c=\sqrt2\,\sigma$ , the dispersion is $\sigma=v_c/\sqrt2=\mathbf{156\ km\,s^{-1}}$ ; the factor $\sqrt2$ comes from $v_c^2=GM/r=2\sigma^2$ .

Step 2. The SIS enclosed mass is $M(r)=2\sigma^2 r/G=v_c^2 r/G$ . In SI: $v_c^2=4.84\times10^{10}\ \mathrm{m^2\,s^{-2}}$ , $r=10\ \mathrm{kpc}=3.086\times10^{20}\ \mathrm{m}$ , $G=6.674\times10^{-11}$ .

Step 3. $M=(4.84\times10^{10})(3.086\times10^{20})/(6.674\times10^{-11})=2.24\times10^{41}\ \mathrm{kg}=\mathbf{1.1\times10^{11}\ M_\odot}$ within 10 kpc. Because $M\propto r$ , doubling the radius doubles the mass — the divergence made concrete, and why a real halo must eventually be truncated. σ ≈ 156 km/s, M(<10 kpc) ≈ 1.1×10¹¹ M_⊙.

Physics of the portrait

The isothermal sphere is old — it descends from the nineteenth-century theory of gaseous stars — but the models that bear a name are Ivan King's, who turned an idealised gas ball into a quantitative tool for real clusters.

Origins · 1 minIvan King and how this was discovered

Ivan King

1927–2021 · dynamical models of star clusters

Ivan Robert King was an American astronomer who spent most of his career at the University of California, Berkeley, and made the theory of star clusters quantitative. In a celebrated three-part series, "The structure of star clusters," he first gave, in 1962, a simple empirical surface-brightness law with a core radius and an outer tidal radius that fit globular-cluster light strikingly well — the profile observers still call the King model. In 1966 he supplied the dynamical foundation: a lowered, truncated Maxwellian distribution function whose self-consistent solution of the Boltzmann and Poisson equations reproduced that law and predicted the internal velocity dispersions. His insight was physical, not merely mathematical — two-body relaxation drives the core toward a Maxwellian while the host galaxy's tidal field strips loosely bound stars, so the equilibrium must be a Maxwellian with its high-energy tail removed. That one-parameter family remains the standard description of globular clusters six decades on, and King himself later used Hubble Space Telescope imaging to resolve the stellar populations of the clusters his models describe.

History

1870

— Lane analyses self-gravitating gas spheres, seeding the Lane–Emden equation

1907

— Emden's Gaskugeln treats the isothermal gas sphere and its r⁻² envelope

1942

— Chandrasekhar's Principles of Stellar Dynamics frames collisionless equilibria

1962

— King fits globular clusters with an empirical cored, tidally truncated brightness law

1966

— King builds the lowered-Maxwellian dynamical models with a finite tidal radius

Going further

Texts. Binney & Tremaine, Galactic Dynamics, 2nd ed., §4.3 (isothermal sphere) and §4.3.3 (King models) · Carroll & Ostlie, An Introduction to Modern Astrophysics, Ch. 25 (star clusters) · Mo, van den Bosch & White, Galaxy Formation and Evolution, Ch. 5–6 (halo profiles and dynamics) · Dodelson, Modern Cosmology (halo profiles and lensing) · King, I. R. (1966), AJ 71, 64 — "The structure of star clusters. III". The SIS relations $\rho=\sigma^2/2\pi Gr^2$ , $M=2\sigma^2 r/G$ , and $v_c=\sqrt2\,\sigma$ follow by direct substitution into the isothermal Poisson equation. The lowered-Maxwellian DF and King density formula originate with King (1966); the core series $\psi=\tfrac16\tilde r^2-\tfrac1{120}\tilde r^4$ follows from the Lane–Emden recursion. Key anchor numbers: $\theta_E=0.90$ arcsec for $\sigma=250\ \mathrm{km\,s^{-1}}$ ; $r_0=0.90$ pc for $\sigma=7\ \mathrm{km\,s^{-1}}$ and $\rho_0=10^4\,M_\odot\,\mathrm{pc^{-3}}$ ; and $M=1.1\times10^{11}\,M_\odot$ within 10 kpc for $v_c=220\ \mathrm{km\,s^{-1}}$ .

Prerequisites. Collisionless Boltzmann Equation · Jeans' Theorem · Distribution Function and Phase Space · Jeans Equations. Leads to. Galactic Potential Models · Virial Theorem in Stellar Systems · Two-Body Relaxation Time · Orbits in Spherical Potentials.

Ask the tutor: Ask me to integrate the isothermal Lane–Emden equation for a given $W_0$ and read off its concentration, or to derive the SIS Einstein radius for any dispersion.

Is the isothermal sphere really the same object as a self-gravitating gas ball?

Mathematically, yes — that is the point of the name. An isothermal gas in hydrostatic equilibrium obeys $\mathrm{d}P/\mathrm{d}r=-\rho\,\mathrm{d}\Phi/\mathrm{d}r$ with $P=\rho\sigma^2$ , which integrates to $\rho\propto e^{-\Phi/\sigma^2}$ . Integrating a Maxwellian distribution function over velocity gives the identical relation, with $\sigma^2$ playing the role of $k_BT/m$ . Both feed the same Poisson equation, so the collisionless cluster and the collisional gas ball solve the same Lane–Emden equation. The difference: a gas is kept Maxwellian by frequent collisions, a cluster only slowly, by two-body relaxation over many crossing times.

Why can't we just use the singular isothermal sphere for everything?

Because it has two divergences. Its central density $\rho\to\infty$ as $r\to0$ is unphysical — real clusters and haloes have finite-density cores — and its total mass $M(r)=2\sigma^2 r/G$ grows without bound, so there is no such thing as "the mass" of an SIS. The SIS stays useful as the large-radius envelope of any isothermal sphere and as an analytic lens, where you only need the profile over a limited range. But to model a whole finite object you must regularise the centre with a core and truncate the outside at a tidal radius — exactly what King models do.

What is the difference between the King 1962 profile and the King 1966 models?

The 1962 profile is an empirical fitting function for the projected surface brightness — a formula with a core radius and a tidal radius chosen to match photometry, with no dynamical content. The 1966 models are self-consistent solutions of the Boltzmann and Poisson equations built from a lowered-Maxwellian distribution function; they predict the density, the surface brightness, and the internal velocity dispersion versus radius. The two agree closely in projected shape, so observers often fit the simple 1962 formula, but only the 1966 models can be compared with measured line-of-sight velocities.

How is the concentration c related to the central potential W_0?

They are two labels for the same one-parameter family. $W_0=-\Phi(0)/\sigma^2$ is the depth of the central potential well in units of $\sigma^2$ — the boundary value you integrate the King equation from. Integrating outward to where the density vanishes gives the tidal radius $r_t$ , and dividing by the core radius $r_0$ gives $c=\log_{10}(r_t/r_0)$ . Larger $W_0$ produces larger $c$ : a deeper well holds a more extended, more centrally concentrated cluster. Observers quote $c$ because it is read off a fitted profile; theorists quote $W_0$ because it is the natural boundary condition.

Why does the density reach exactly zero at the tidal radius instead of tapering off?

Because King subtracted a constant from the Maxwellian precisely so the distribution function vanishes at the escape energy. Integrating that lowered DF over the allowed velocities near the boundary, the error-function term and the subtracted-constant term cancel to leading order as the relative potential $W\to0$ , so the density goes smoothly to zero rather than to a small positive value. Physically, at $r_t$ the local escape speed drops to zero: no bound phase space is left for stars to occupy, so the density must vanish. This is what gives King models a genuine finite edge and hence a finite total mass.

Is the singular isothermal sphere a good dark-matter halo?

As a first analytic model, yes; as an accurate one, no. The SIS reproduces the single most important fact about galaxy haloes — the flat rotation curve $v_c=\sqrt2\,\sigma\approx$ constant — and its simple $\theta_E\propto\sigma^2$ lensing makes it the standard analytic lens. But cosmological $N$ -body simulations show real cold-dark-matter haloes follow the NFW profile, with a shallower $\rho\propto r^{-1}$ central cusp and a steeper $\rho\propto r^{-3}$ outer fall-off, neither of which the pure $r^{-2}$ SIS matches. The SIS survives as the benchmark against which NFW, Hernquist, and cored profiles are compared, and as a quick analytic stand-in.