Table 1
Gravitational scaling relations.
| Property | Proportionality | Scaling for var. Δ | r-dependencea | Scaling for fixed Δ | Deviation |
|
|
|||||
| radius, RΔ | ∝ M1/3 F−2/3 | ∝ T1/2 F-1 | ∝ r | T1/2 E-1 | |
| density profilea | ρ ∝ MR-3 | ∝ F(z)2 | ∝ r-2 | ∝ E(z)2 | − |
| luminosityb | LX ∝ ρ2 R3 | ∝ T1.5 F | ∝ r1/2 | ∝  |
39% |
| bolom. lum. | Lbol ∝ ρ2 R3 T1/2 | ∝ T2 F | ∝ r1/2 | ∝  |
39% |
| entropy, K | ∝ T/ρ2/3 | ∝ T F−4/3 | ∝ r | ∝ TXE−4/3Δ(z)−1/6 | 7.5% |
| gas mass | ∝ ρR3 | ∝ T3/2 F-1 | ∝ r |
|
− |
| YX | ∝ TρR3 | ∝ T5/2 F-1 | ∝ r |
|
− |
| pressure | ∝ Tρ | ∝ T F2 | ∝ r−7/3 | ∝ TXE2Δ(z)−1/6 | 7.5% |
| surf. bright.a | SX ∝ ρ2 R | ∝ T1/2 F3 | ∝ r-3 | ∝  |
− |
|
|
|||||
| temperature | TX | ∝ M2/3 F2/3 | (M(r) ∝ r) | ∝  |
− |
| luminosityb | L X | ∝ M F2 | ∝ M ⋆ E2 Δ(z)3/4 | 39% | |
| bolom. lum. | Lbol | ∝ M4/3 F7/3 | ∝  |
39% | |
| entropy | K | ∝ M2/3 F−2/3 | ∝  |
7.5% | |
| gas mass | Mgas | ∝ M | ∝ M⋆ | − | |
| YX | YX | ∝ M5/3 F2/3 | ∝  |
− | |
| pressure | P | ∝ M2/3 F8/3 | ∝  |
7.5% | |
| surf. bright.a | SX | ∝ M1/3 F10/3 | ∝  |
− | |
Notes. Columns 1 and 2 give the property and its definition, and Col. 3 provides the scaling relation where the parameter F(z) can be read in two ways: (i) for recent formation approximation F(z) ≡ E(z) × Δ(z)1/2 and for the fixed overdensity model F = E(z). Column 4 gives the assumed radial dependence of the property of Col. 1, and Col. 5 the scaling relation for the case that the property of Col. 1 is taken at a fixed overdensity radius. The temperature TX is in all cases defined as a mean measured temperature. Entropy, K, pressure, surface brightness are local values at the fiducial radius. Luminosity, gas mass, and YX are integral parameters. M ⋆ is the mass at fixed overdensity. Column 6 lists the deviation of the scaling relation if the Δ(z) term is neglected for a comparison of clusters at z = 0 and z = 1.
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