Cosmology · The evidence motivating dark matter

Cluster dynamics, gas and lensing mass

Connect random tracer motions, a gas pressure gradient and light deflection to one supplied spherical mass, keeping projected and spherical masses distinct.

Preparation: ratios, powers of ten and ordinary arithmetic. Physical quantities and notation are introduced as they are needed.

Back to the chapter

After reading, you can

Describe the physical response behind three cluster mass methods.
Compare conditional dynamical recoveries with a cylinder-projected mass.
Recognize shared assumptions and the limits of built-in agreement.

Mass affects motions, pressure balance and light

Random galaxy motions can support a cluster against gravity. Relating those motions to mass requires a tracer distribution, a velocity model and an equilibrium assumption.

Gas supplies another response: a pressure that decreases outward can balance inward gravity. Inferring mass this way needs a gas density and pressure gradient, together with a model of which stresses provide support.

Gravitational lensing uses the deflection of light rather than the same gas or tracer equilibrium. It depends on projected mass and the geometry between observer, lens and source.

These are physically different responses, but their errors need not be statistically independent. Shared distances, calibration, selection and mass-model assumptions must be considered in a joint analysis.

Build a conditional comparison with known profiles

Choose positive total mass M and length L. Supply Φ(r) = −GM/√(r²+L²) and total density ρ_tot = 3M/(4πL³)(1+x²)⁻⁵ᐟ², with x = r/L and GM/(Lc²) ≪ 1.

Let thermal gas have density ρ_g = fρ_tot for a constant 0 < f < 1. The remaining mass is collisionless. Passive tracers have negligible additional mass and number density ν = ν₀(1+x²)⁻⁵ᐟ²; their density is not a second contribution to M.

Supply zero mean streaming, isotropic tracer motions and radial-component variance σ_r² = (GM/L)/[6√(1+x²)]. This is one component’s variance, not the total three-dimensional mean squared speed or an observed line-of-sight dispersion.

Define ρ_g0 = f3M/(4πL³) and thermal pressure P = ρ_g0 GM/(6L)(1+x²)⁻³. For an ideal gas of fixed composition, P/ρ_g = k_BT/(μm_p) = (GM/L)/[6√(1+x²)]. The temperature therefore varies with radius; no actual cluster temperature or gas fraction is assigned.

The two force-balance formulae recover the supplied mass

Under the stated spherical stationary tracer assumptions, the Jeans equation gives M_J = −rσ_r²/G [d ln ν/d ln r + d ln σ_r²/d ln r + 2β]. Here β measures velocity anisotropy and is set to zero.

The supplied logarithmic slopes are −5x²/(1+x²) and −x²/(1+x²). Inserting them recovers M_J/M = x³/(1+x²)³ᐟ².

Thermal hydrostatic balance gives M_H = −r²/(Gρ_g) dP/dr. Differentiating the supplied pressure recovers the same spherical enclosed mass.

This exact agreement was built into the forward profiles. It illustrates consistency of two mechanisms under their assumptions; it is not independent observational confirmation. Real anisotropy, rotation, acceleration or nonthermal support can change the inference.

A cylinder contains a different mass than a sphere

Project the same density along the full line of sight. With X = R/L for projected radius R, its surface density is Σ = [M/(πL²)](1+X²)⁻² and cylinder-enclosed mass is M_2D(<R) = MX²/(1+X²).

For one supplied source geometry, define Σ_crit = c²D_s/(4πGD_lD_ls) using positive angular-diameter distances and a source behind the lens. Choose M/(πL²Σ_crit) = 1/50 as a mathematical normalization.

The convergence is κ = Σ/Σ_crit. In this axisymmetric example, tangential shear is γ_t = (mean Σ−Σ)/Σ_crit and reduced shear is g_t = γ_t/(1−κ). The weak-field thin-lens model has small convergence and no critical curve.

The table compares r = xL for spherical quantities with R = xL for projected quantities. At x = 1 the cylinder mass is 0.5M, while the sphere contains about 0.353553M. The difference is the projection integral, not a disagreement between forces.

One supplied Plummer system; dynamical masses refer to spheres at r=xL, projected mass to cylinders at R=xL
x	Spherical M(<r)/M	Jeans recovery/M	Hydrostatic recovery/M	Projected M_2D(<R)/M	Reduced shear g_t
0.5	0.0894427	0.0894427	0.0894427	0.2	0.00324149
1	0.353553	0.353553	0.353553	0.5	0.00502513
2	0.715542	0.715542	0.715542	0.8	0.00320256
4	0.913075	0.913075	0.913075	0.941176	0.00110734

Use one hypothetical positive Plummer total density ρ_tot = 3M/[4πL³](1+x²)^(−5/2) and potential −GM/√(r²+L²), with x = r/L, M > 0, L > 0 and GM/(Lc²) ≪ 1. Thermal gas is fρ_tot for a supplied constant 0 < f < 1; the remaining density is collisionless. Passive tracers have negligible additional mass, zero streaming and isotropic radial-component variance σ_r² = (GM/L)/[6√(1+x²)]. Their number density is proportional to (1+x²)^(−5/2), and gas pressure to (1+x²)^(−3). These chosen stationary Jeans and hydrostatic equations recover exactly the same spherical mass. Panel A draws that one shared sphere/Jeans/hydrostatic curve as solid blue circles, without fake offsets or three separate measurements. Its dimensionless mass is x³/(1+x²)^(3/2). The dashed orange squares are instead the cylinder-enclosed mass X²/(1+X²), obtained by integrating the same density over the full line of sight, where X = R/L is projected radius. Both mass definitions share the numerical horizontal scale from 0.5 to 4, but r is spherical radius and R is cylinder radius. The ordinate is zero-based mass divided by total M, from zero to one. At x = X = 1, the sphere mass is about 0.353553 M and the cylinder mass is 0.5 M, with ratio √2 because the cylinder also contains mass outside the sphere. Panel B has a separate zero-based linear reduced-shear axis from zero to 0.006. A solid green curve and circles show g_t = γ_t/(1−κ) for the chosen isolated weak-field thin lens and one positive source geometry. Set M/(πL²Σ_crit) = 1/50, where Σ_crit = c²D_s/(4πGD_lD_ls) uses angular-diameter distances and the source is behind the lens. κ = 0.02/(1+X²)², mean κ = 0.02/(1+X²), and γ_t = mean κ−κ. At X = 0.5, 1, 2, 4 the reduced-shear samples are about 0.00324149, 0.00502513, 0.00320256 and 0.00110734. All samples are model evaluations, not measured shapes. κ never exceeds 0.02 and this prescribed lens has no critical curve. Built-in equilibrium agreement does not constitute independent observational confirmation. The gas is already within the total mass, the radial dispersion is not observed line-of-sight or total three-dimensional dispersion, and temperature is not constant. Projection, anisotropy, nonthermal support, disturbance, calibration, selection and covariance remain real-method limitations. No actual gas fraction, mass, temperature, distance, unique three-dimensional reconstruction or particle identity is supplied. Original Ark teaching model.

Carry each method’s response and uncertainty into the comparison

Lensing does not require the same hydrostatic gas assumption, but it still requires gravitational light propagation, source geometry, shape interpretation and a mass-model convention. A shear measurement alone is not a unique three-dimensional mass reconstruction.

Real tracer observations project velocity components and may not determine anisotropy uniquely. Real gas measurements also project density and temperature information, while unmodeled stresses or a disturbed state can affect pressure balance.

A joint comparison should therefore predict the actual primitive observations through a common physical model and each method’s response. It should compare matched apertures and definitions, with selection and covariance retained.

Agreement can test that common explanation; disagreement can identify missing physics, calibration or modeling assumptions. Neither the agreement built into this mathematical system nor an inferred gravitating mass names its microscopic constituent.

Worked reasoning

Compare the masses at the same scale L

Why does the projected mass at R = L exceed the spherical mass at r = L in the supplied system?

Both conditional force-balance recoveries give M(<L) = M/(2√2) ≈ 0.353553M.
The full line-of-sight cylinder projection gives M_2D(<L) = M/2.
Their ratio is √2 ≈ 1.414214 because the cylinder also includes material outside the sphere along the line of sight.

They are different integrals of the same positive density, so equality is not expected.

Try the reasoning

What spherical mass do both supplied force-balance methods recover at r = 2L?
Show the answer and reasoning
About 0.715542M.
The profiles were constructed to satisfy the two equations in the common potential.
Does lensing require the same thermal hydrostatic assumption as the gas method?
Show the answer and reasoning
No.
It uses light propagation and projected mass with source geometry, while retaining its own assumptions.
Does the exact agreement here constitute three independent observed mass measurements?
Show the answer and reasoning
No.
The example supplies consistent mathematical profiles rather than observations, and projected mass is separately defined.

What to carry forward

Random motions, gas pressure gradients and lensing probe mass through different mechanisms.
Spherical and cylinder-projected mass must be compared through a common model.
Joint inference retains method assumptions, calibration, selection and covariance.

References

Sources supporting the explanations, physical distinctions and examples in this lesson.