Jaffe Profile

The Jaffe profile (or Jaffe model) is an analytical, spherically symmetric gravitational potential–density pair proposed by astronomer Walter Jaffe in 1983 to model the three-dimensional mass distribution, gravitational field, and phase-space dynamics of pressure-supported elliptical galaxies and spherical stellar bulges. Prior to Jaffe's formulation, dynamical modeling of early-type galaxies relied either on numerical deprojections of Gerard de Vaucouleurs's empirical R^1/4 surface-brightness law or on cored models such as the Plummer model and King profile that failed to reproduce the central surface-brightness peaks and extended r⁻⁴ outer envelopes of giant ellipticals. By constructing a double power-law space density ρ(r) that transitions smoothly from an inner singular isothermal sphere cusp (ρ ∝ r⁻²) to an outer Keplerian halo envelope (ρ ∝ r⁻⁴), the Jaffe profile yields closed-form expressions for the enclosed mass M(r), logarithmic gravitational potential Φ(r), sky-projected surface density Σ(R), Jeans equations velocity dispersions, and the ergodic phase-space distribution function f(ℰ).

A spherical Jaffe galaxy of total mass M and characteristic Jaffe scale radius r_J (often denoted a) is defined by the three-dimensional mass density distribution:

ρ(r) = (Mr_J) / [4πr²(r + r_J)²]

Within the unified Dehnen profile family (γ-models, where ρ ∝ r^−γ(r + a)^γ−4), the Jaffe model corresponds to the γ = 2 case, whereas the Hernquist profile corresponds to γ = 1. Its radial structure bridges two asymptotic regimes:

• Inner Isothermal Cusp (r ≪ r_J): At radii much smaller than the scale radius, (r + r_J)² ≈ r_J² and the density scales as ρ(r) ≈ (M/4πr_J)r⁻². Unlike the γ = 1 Hernquist cusp where the isotropic velocity dispersion drops to zero at the origin, the r⁻² inner slope of the Jaffe profile behaves locally like a singular isothermal sphere with a finite, non-zero central velocity dispersion σ_r(0) = (GM/2r_J)^1/2 and a flat inner circular rotation speed v_c(0) = (GM/r_J)^1/2.
• Outer Violent-Relaxation Halo (r ≫ r_J): At large galactocentric distances, (r + r_J)² ≈ r² and the density declines as ρ(r) ≈ (Mr_J/4π)r⁻⁴. As Jaffe (1987) proved, this r⁻⁴ outer falloff is the universal signature of incomplete violent relaxation whenever the energy distribution N(ℰ) of marginally bound stars remains continuous across the escape threshold ℰ = 0.

Integrating 4πr′²ρ(r′) from the center to radius r yields a simple rational expression for the cumulative enclosed mass M(r):

M(r) = ∫₀^r 4πr′² ρ(r′) dr′ = Mr / (r + r_J)

Setting M(r_h) = ½M reveals the geometric meaning of the scale parameter r_J: in three dimensions, the Jaffe scale radius r_J equals the spherical half-mass radius (r_h = r_J). Integrating the radial gravitational field g(r) = −GM(r)/r² = −GM/[r(r + r_J)] from r to infinity yields the Newtonian gravitational potential Φ(r):

Φ(r) = (GM / r_J) ln[r / (r + r_J)] = −(GM / r_J) ln(1 + r_J/r)

When a spherical Jaffe galaxy is observed on the sky at projected cylindrical radius R = sr_J, integrating ρ(r) along the line of sight yields an analytical expression for the projected surface mass density Σ(R):

Σ(R) = (M/r_J²) { 1/(4s) − [(2s² − 1)X(s) − 1] / [2π(s² − 1)] }

where X(s) = (1 − s²)^−1/2 arccosh(1/s) for s < 1 and X(s) = (s² − 1)^−1/2 arccos(1/s) for s > 1. Near the center (R ≪ r_J), the projected surface density diverges inversely with radius as Σ(R) ≈ M/(4r_JR) ∝ R⁻¹, whereas at large projected radii (R ≫ r_J) it declines as Σ(R) ≈ (Mr_J/8)R⁻³. Half of the total projected mass is enclosed within the projected effective (half-light) radius R_e ≈ 0.7632*r*_J (r_J ≈ 1.3103*R*_e).

Inverting the relative gravitational potential Ψ(r) = −Φ(r) = (GM/r_J) ln(1 + r_J/r) gives r(Ψ) = r_J/[exp(ψ) − 1], where ψ = Ψr_J/(GM), so the density is ρ(ψ) = (M/4πr_J³) exp(−2ψ)[exp(ψ) − 1]⁴. Substituting ρ(ψ) into the Eddington inversion formula expresses the isotropic phase-space distribution function f(ℰ) solving the collisionless Boltzmann equation in closed form via Dawson's integral F₊(x) = exp(−x²) ∫₀^x exp(t²) dt and F₋(x) = exp(x²) ∫₀^x exp(−t²) dt:

f(ℰ) = [M / (2π^5/2r_J³(GM/r_J)^3/2)] [F₋((2ε)^1/2) − 2^1/2F₋(ε^1/2) − 2^1/2F₊(ε^1/2) + F₊((2ε)^1/2)]

where ε = ℰr_J/(GM) > 0. Integrating the isotropic spherical Jeans equations yields the radial velocity dispersion profile:

σ_r²(r) = (GM/r_J) { −(r/r_J)²(r/r_J + 1)² ln(1 + r_J/r) + (r/r_J)³ + (3/2)(r/r_J)² + (1/2)(r/r_J) }

which starts at the isothermal value σ_r²(0) = GM/(2r_J) at the center and declines as GM/(5r) at large radii, while total self-gravitational energy is W = −GM²/(2r_J).

Because the Jaffe profile combines an inner isothermal r⁻² mass distribution with a finite total mass M, it is widely applied as a pseudo-Jaffe or truncated isothermal lens model in strong and weak gravitational lensing studies of galaxy clusters and giant early-type galaxies (such as Messier 87). Inside R ≲ R_e, the Jaffe profile produces the nearly constant gravitational deflection angle α̂ ≈ 4π(σ₀/c)² characteristic of a singular isothermal sphere, while outside r_J the transition to ρ ∝ r⁻⁴ assigns each cluster subhalo a finite total mass M without an artificial step-function cutoff.

How does the Jaffe profile differ from the Hernquist profile?

Both belong to the Dehnen profile family and share an outer ρ(r) ∝ r⁻⁴ halo slope. However, the Hernquist profile (γ = 1) has a shallow ρ ∝ r⁻¹ inner cusp with a finite central potential and vanishing central velocity dispersion σ_r(0) = 0, whereas the Jaffe profile (γ = 2) has a steeper isothermal ρ ∝ r⁻² cusp with a logarithmic central potential and a non-zero central velocity dispersion σ_r(0) = (GM/2r_J)^1/2.

Why is the Jaffe scale radius equal to the 3D half-mass radius?

The cumulative enclosed mass of a Jaffe sphere is M(r) = Mr/(r + r_J). Setting r = r_J yields M(r_J) = Mr_J/(2r_J) = ½M, proving that half of the three-dimensional mass lies inside r = r_J.

What is a pseudo-Jaffe profile in gravitational lensing?

A pseudo-Jaffe profile modifies the Jaffe density to ρ(r) = ρ₀r_core²r_cut²/[(r² + r_core²)(r² + r_cut²)], smoothing the central r⁻² cusp inside a core radius r_core while retaining the isothermal r⁻² zone and r⁻⁴ outer falloff beyond r_cut.

1. Jaffe, W. (1983). "A simple model for the distribution of light in spherical galaxies." Monthly Notices of the Royal Astronomical Society, 202(4), 995–999. https://doi.org/10.1093/mnras/202.4.995
2. Jaffe, W. (1987). "Note on violent relaxation." In P. T. de Zeeuw (Ed.), Structure and Dynamics of Elliptical Galaxies (IAU Symposium No. 127, pp. 511–512). D. Reidel Publishing. https://doi.org/10.1007/978-94-009-3971-4_69
3. Dehnen, W. (1993). "A family of potential-density pairs for spherical galaxies and bulges." Monthly Notices of the Royal Astronomical Society, 265(1), 250–256. https://doi.org/10.1093/mnras/265.1.250
4. Merritt, D. (1985). "Spherical stellar systems with spheroidal velocity distributions." The Astronomical Journal, 90, 1027–1037. https://doi.org/10.1086/113810
5. Binney, J., & Tremaine, S. (2008). Galactic Dynamics (2nd ed.). Princeton University Press. https://doi.org/10.1515/9781400828722