Cosmology · Clusters and gravitational lensing

Hot gas, X-rays and the thermal Sunyaev–Zel’dovich signal

Compare two unresolved gas segments with the same electron column and temperature but different squared-density emission measures.

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

Distinguish a thermal pressure-column signal from squared-density emission.
Calculate the effect of supplied unresolved density variation on a smooth-gas inference.
State the temperature, composition, geometry and scattering assumptions behind the comparison.

Hot intracluster gas offers several measurements

Gas in galaxy clusters can emit X-rays and can change the spectrum of background radiation by scattering. These signals weight gas properties differently, so agreement or disagreement between them can reveal modelling assumptions.

For a fixed ionized composition and a specified thermal emissivity function, an optically thin X-ray emission measure can be represented by EM = ∫n_e² dl, up to the common composition factor relating electron and ion densities. The integral follows physical path length l.

For nonrelativistic thermal hot electrons, the thermal Compton parameter is y = [σ_T/(m_ec²)]∫n_ek_BT_e dl in the selected approximation. It weights electron pressure along that path and is dimensionless.

Use small scattering optical depth, k_BT_e ≪ m_ec² and electron temperature much greater than the background radiation temperature. The example isolates thermal scattering; it does not model peculiar-velocity scattering or a measured frequency-channel signal.

The same mean density can hide a different mean square

Choose a total physical path length L divided into two equal segments L/2. Supply electron densities n₀(1−δ) and n₀(1+δ), with positive n₀ and 0 ≤ δ ≤ 0.75.

Both segments share a fixed temperature T₀ and composition, so their emissivity function is the same. The model is a fully ionized optically thin snapshot with unresolved density variation, not an isobaric atmosphere or a solved gas flow.

Their mean electron density is n₀ and their electron column is n₀L for every δ. The average of squared density is n₀²(1+δ²), which grows even though the mean density does not.

Let the uniform reference have density n₀ over the same path and the same T₀. Its y_ref is positive under the stated conditions, allowing a meaningful ratio y/y_ref.

Pressure column stays fixed while emission measure rises

The two-segment emission measure is (L/2)n₀²[(1−δ)²+(1+δ)²] = n₀²L(1+δ²). The normalized emission measure is therefore 1+δ².

The pressure integral is k_BT₀(L/2)n₀[(1−δ)+(1+δ)] = n₀k_BT₀L. Consequently y/y_ref = 1 for every displayed δ.

Define the path-averaged clumping factor C = ⟨n_e²⟩/⟨n_e⟩². In this finite snapshot C = 1+δ². This definition names the averaging domain; it is not a universal measured cluster clumping factor.

At δ = 0.5, the densities are 0.5n₀ and 1.5n₀. The normalized electron column and y remain one, while the emission measure is 1.25 times the uniform value.

Two equal physical path segments with the same supplied T₀ and composition; separate snapshots, no observed cluster values
δ	Segment densities / n₀	Column / (n₀L)	EM / (n₀²L)	y/y_ref	Uniform-inferred density / n₀
0	1 / 1	1	1	1	1
0.25	0.75 / 1.25	1	1.0625	1	1.030776
0.5	0.5 / 1.5	1	1.25	1	1.118034
0.75	0.25 / 1.75	1	1.5625	1	1.25

A smooth-gas assumption can bias an inferred density

Suppose an analysis knows L and the common emissivity but incorrectly imposes uniform density n_inf. Equating its n_inf²L to the actual emission measure gives n_inf/n₀ = √(1+δ²).

For δ = 0.75, this inferred density is 1.25n₀, although the true path mean remains n₀. A larger squared-density signal is being attributed to a larger uniform mean instead of unresolved variation.

The effect is conditional on fixed path length, composition and temperature. Temperature variation, changes in atomic emissivity, absorption or detector response can alter an actual inference and do not cancel merely because two observables are compared.

The fixed-temperature segments have different thermal pressures when their densities differ. No pressure-balance or stability conclusion follows from this snapshot, and no heating or transport law maintains it in time.

Choose a physical path L split into two equal lengths L/2. With positive density scale n₀, give the segments electron densities n₀(1−δ) and n₀(1+δ), for 0 ≤ δ ≤ 0.75. Both have the same fixed temperature T₀ and ionized composition, so their chosen thermal emissivity is common. The top two equal-width blocks illustrate δ = 0.5, with labels 0.5n₀ and 1.5n₀. Their colors identify the two segments; equal widths represent equal physical path lengths. The blocks do not encode an equilibrium, brightness image or pressure balance. The mean density is n₀ and electron column n₀L for every supplied snapshot. The fixed-emissivity optically thin emission measure EM = ∫n_e² dl is n₀²L(1+δ²), whereas thermal y = [σ_T/(m_ec²)]∫n_e k_BT_e dl stays at its positive uniform reference. The linear graph spans δ from zero to 0.75 and normalized ratios from one to 1.6; its vertical baseline is one, not zero. Solid blue circles show EM/(n₀²L) = 1+δ², dashed orange squares y/y_ref = 1, and dotted green diamonds the uniform-density inference n_inf/n₀ = √(1+δ²), which would incorrectly attribute density variation to a larger uniform mean. All three initial markers coincide at one; that is an exact identity, not missing data. At δ = 0, 0.25, 0.5 and 0.75, the EM ratios are 1, 1.0625, 1.25 and 1.5625. The y and column ratios are always one. Inferred density ratios are 1, 1.030776, 1.118034 and 1.25. Segment density ratios are respectively (1,1), (0.75,1.25), (0.5,1.5), (0.25,1.75). EM has units m⁻⁵, electron column m⁻² and y is dimensionless. The thermal scattering approximation requires small optical depth, nonrelativistic thermal electrons k_BT₀ much smaller than m_ec² and T₀ much greater than the background radiation temperature. Common fixed geometry, temperature and composition are necessary for the stated cancellations. Equal temperature and unequal densities mean unequal pressures; no stability, heating, time evolution or isobaric atmosphere is solved. Neither EM nor y is total gravitating mass. This is not a measured cluster clumping factor, frequency-channel observation or model-independent three-dimensional reconstruction. Original Ark teaching model.

Neither signal alone is total gravitating mass

The emission measure has units of inverse fifth power of length: n_e² contributes m⁻⁶ and dl contributes metres. The electron column has units m⁻²; multiplying it by σ_T and k_BT₀/(m_ec²) gives dimensionless y.

These units describe gas observables. An X-ray signal is not total matter mass, and a thermal y map is not the same quantity as an emission-measure map. Their angular distributions also require a geometry and distance model for interpretation.

Combining X-ray and thermal scattering information can constrain complementary gas properties, with assumptions about line-of-sight structure, temperature and composition. It does not automatically yield a unique three-dimensional density or cluster mass.

Mass estimation can add hydrostatic or gravitational-lensing information, each with its own physical and calibration assumptions. Keeping those assumptions explicit prevents a clean two-segment illustration from becoming an unjustified correction for every observed cluster.

Worked reasoning

Compare a column with a squared-density measure

For δ = 0.5, what changes relative to the uniform reference?

The densities are 0.5n₀ and 1.5n₀ in equal path lengths, so their mean is n₀.
The mean of their squared density ratios is (0.5²+1.5²)/2 = 1.25.
At common T₀ the pressure column and y are unchanged, but a uniform emission-measure fit gives n_inf = √1.25 n₀ ≈ 1.118034n₀.

The column and y stay fixed; emission measure rises by a factor 1.25 and the wrongly imposed uniform density is about 11.8 percent above the path mean.

Try the reasoning

What are the units of the thermal Compton parameter y in the supplied expression?
Show the answer and reasoning
It is dimensionless.
Cross-sectional area times electron column is dimensionless, as is the thermal-to-rest-energy ratio.
Are the two unequal-density segments in thermal pressure balance simply because T₀ is the same?
Show the answer and reasoning
No.
At fixed composition and temperature, thermal pressure is proportional to density; the snapshot supplies no equilibrium solution.
Does a larger emission measure directly prove a larger total gravitating mass?
Show the answer and reasoning
No.
Squared gas density, temperature, composition, geometry and other modelling steps separate the signal from total mass.

What to carry forward

Thermal scattering weights electron pressure, while a fixed-emissivity X-ray measure weights squared density.
Unresolved variation can change mean-square density without changing the mean column.
Gas geometry and thermodynamics must remain explicit when signals are turned into density or mass.

References

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