Singular Isothermal Sphere

The singular isothermal sphere (SIS)—together with its cored non-singular counterpart—is a foundational self-gravitating spherical mass model in astrophysics and galactic dynamics in which a collisionless stellar or dark matter system (or a hydrostatic ideal gas sphere) maintains a spatially uniform one-dimensional velocity dispersion σ (or uniform temperature T) at all radii. Investigated in nineteenth-century studies of self-gravitating gas spheres by Jonathan Homer Lane, August Ritter, and Robert Emden and applied to star clusters and galaxies by James Jeans and Subrahmanyan Chandrasekhar, the singular isothermal sphere satisfies the spherically symmetric Poisson and Jeans equations via an inverse-square space density profile, ρ(r) = σ²/(2π*Gr*²). Because its enclosed mass grows linearly with radius (M(r) ∝ r), the singular isothermal sphere generates the flat circular rotation curves (v_c = 2^1/2σ = const) observed across spiral galaxies and yields a constant light-deflection angle in gravitational lensing, while its divergences at r → 0 and r → ∞ motivate cored and tidally truncated generalizations such as the King profile, Jaffe profile, and Burkert profile.

A self-gravitating spherical system is termed isothermal when its velocity distribution at every position follows a Maxwell–Boltzmann Gaussian with a spatially constant one-dimensional velocity dispersion σ² = k_BT/(μm_p) (for a collisional gas of mean molecular weight μ) or a uniform radial and tangential velocity variance σ_r² = σ_θ² = σ_ϕ² = σ² (for a collisionless system of stars or dark-matter particles). In six-dimensional phase space (x, v), the ergodic distribution function f(E) solving the steady-state collisionless Boltzmann equation is a Maxwellian exponential of the specific orbital energy E = ½v² + Φ(r):

f(E) = [ρ₁ / (2πσ²)^3/2] exp(−E/σ²) = [ρ₁ / (2πσ²)^3/2] exp{−[Φ(r) + ½v²]/σ²}

Integrating f(E) over all three velocity components (v ∈ ℝ³) relates the three-dimensional mass density ρ(r) to the Newtonian gravitational potential Φ(r) as ρ(r) = ρ₁ exp[−Φ(r)/σ²]. Taking the natural logarithm Φ(r) = −σ² ln[ρ(r)/ρ₁] (equivalent to integrating the isotropic spherical Jeans equations, σ² dρ/dr = −ρ dΦ/dr) and substituting Φ(r) into the spherically symmetric Poisson equation, (1/r²) d/dr(r² dΦ/dr) = 4πGρ(r), yields the second-order nonlinear isothermal Lane–Emden differential equation:

(1/r²) d/dr [(r²/ρ)(dρ/dr)] = −(4πG/σ²) ρ(r)

Inspecting the isothermal Lane–Emden equation for a pure power-law solution ρ(r) = Ar^−b shows that matching powers of r on both sides requires b = 2 and fixes the normalization constant A = σ²/(2πG). This particular solution is the singular isothermal sphere (SIS):

ρ_SIS(r) = σ² / (2π*Gr*²)

Integrating 4πr′²ρ_SIS(r′) from the origin to radius r gives the cumulative enclosed mass M(r) and the circular orbital speed v_c(r) = [GM(r)/r]^1/2:

M(r) = (2σ²/G)r and v_c(r) = [GM(r)/r]^1/2 = 2^1/2σ = constant

The corresponding gravitational potential increases logarithmically with radius relative to a reference radius r₀:

Φ_SIS(r) = 2σ² ln(r/r₀) = v_c² ln(r/r₀)

In disk galaxies such as the Milky Way (v_c ≈ 220–235 km s⁻¹, corresponding to an equivalent halo velocity dispersion σ = v_c/2^1/2 ≈ 155–166 km s⁻¹), H I 21-cm rotation curves remain flat out to tens of kiloparsecs beyond the optical stellar disk, demonstrating that the enclosed dark-matter mass scales as M(r) ∝ r over intermediate halo radii where the local logarithmic slope of an Einasto profile or NFW halo passes through d ln ρ / d ln r ≈ −2. However, the SIS cannot hold across all radii because ρ_SIS(r) ∝ r⁻² diverges at r → 0 and M(r) ∝ r diverges at r → ∞.

To eliminate the central singularity at r = 0 while preserving the isothermal equation of state P = ρσ², one integrates the isothermal Lane–Emden equation outward from the origin subject to regular central boundary conditions: a finite central density ρ(0) = ρ₀ and zero central density gradient (dρ/dr)_r=0 = 0. Introducing the King radius (or isothermal core radius) r₀ ≡ (9σ²/4πGρ₀)^1/2 and dimensionless potential ψ = [Φ(r) − Φ(0)]/σ² transforms the ODE into (1/ξ²) d/dξ(ξ² dψ/dξ) = 9 exp(−ψ), where ξ = r/r₀. Within two King radii (r ≤ 2r₀), the numerical solution is closely approximated by the modified Hubble law, ρ(r) ≈ ρ₀[1 + (r/r₀)²]^−3/2 and projected surface density Σ(R) ≈ 2ρ₀r₀[1 + (R/r₀)²]⁻¹, while at r ≫ r₀ it converges via damped oscillations to ρ_SIS(r) = (2/9)ρ₀(r₀/r)². When bounded by an external pressure or enclosure of radius R_max, isothermal spheres exceeding density contrasts of ρ₀/ρ(R_max) > 14.04 (the Bonnor–Ebert Jeans instability limit) or > 708.81 (the microcanonical Antonov limit) undergo core collapse via the gravothermal catastrophe.

In extragalactic gravitational lensing, projecting ρ_SIS(r) = σ²/(2π*Gr*²) along the line of sight z gives the two-dimensional surface mass density Σ(R) = σ²/(2*GR*) at impact parameter R, and enclosed cylinder mass M_proj(R) = (πσ²/G)R. In general relativity, the gravitational bending angle α̂(R) = 4*GM*_proj(R)/(c²R) is independent of impact parameter R:

α̂_SIS = 4π (σ/c)² ≈ 1.15″ (σ / 200 km s⁻¹)²

For a background source at distance D_S lensed by an SIS galaxy at distance D_L (with lens-source distance D_LS), the angular Einstein radius is θ_E = 4π(σ/c)²(D_LS/D_S), producing two collinear images at θ_± = β ± θ_E separated by Δθ = 2θ_E whenever the source lies inside the Einstein ring (β < θ_E).

How is the circular rotation velocity related to the velocity dispersion in an isothermal sphere?

In a singular isothermal sphere, the enclosed mass is M(r) = 2σ²r/G, so the circular orbital velocity v_c = [GM(r)/r]^1/2 equals 2^1/2 ≈ 1.414 times the one-dimensional velocity dispersion σ at all radii (v_c = 2^1/2σ).

Why is the deflection angle of a singular isothermal sphere lens constant?

In gravitational lensing, the deflection angle α̂(R) is proportional to the enclosed projected mass M_proj(R) divided by impact parameter R. Because the projected surface density of a singular isothermal sphere scales as Σ(R) ∝ R⁻¹, the mass inside cylinder R grows linearly as M_proj(R) ∝ R, causing R to cancel out and leaving α̂ = 4π(σ/c)².

What is the difference between a cored isothermal sphere and a pseudo-isothermal sphere?

The non-singular cored isothermal sphere is the numerical solution of the Poisson–Boltzmann equation ∇²Φ = 4πGρ₀ exp(−Φ/σ²) derived via the Eddington inversion formula with uniform σ everywhere. The pseudo-isothermal sphere ρ(r) = ρ₀/(1 + r²/r_c²) is an algebraic approximation that allows closed-form integration of rotation curves.

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