Table 2
Hydrodynamic scaling relations.
| Property | Proportionality | Scaling no evoluion of fgas | Scaling with empirical evolution |
|
|
|||
| density profile | ∝ ρ fgas | ∝  |
∝ T0.45E(z)1.385 |
| luminositya |
|
∝  |
∝  |
| bolom. lum. |
|
∝  |
∝  |
| entropy, K | ∝  |
∝  |
∝  |
| gas mass | ∝ ρ R3 fgas | ∝  |
∝  |
| YX | ∝ TX ρ R3 fgas | ∝  |
∝  |
| pressure, P(r) | ∝ TX ρ fgas | ∝  |
∝  |
| surf. bright., SX(r) | ∝  |
∝  |
∝  |
|
|
|||
| temperature | TX | ∝ M2/3E2/3 | |
| gas mass fr. | fgas | ∝ M0.3E0.3 | ∝ M0.3E-0.32 |
| density profile | ρ | ∝ M0.3E(z)2.3 | ∝ M0.3E1.69 |
| luminositya | LX | ∝ M1.6 E2.6 | ∝ M1.6 E1.37 |
| bolom. lum. | Lbol | ∝ M1.93 E2.93 | ∝ M1.93 E2.01 |
| entropy | K(r) | ∝ M0.47 E-0.87 | ∝ M0.47 E0.98 |
| gas mass | Mgas | ∝ M1.3 E0.3 | ∝ M1.3 E-0.32 |
| YX | TX Mgas | ∝ M1.97 E0.97 | ∝ M1.97 E0.35 |
| pressure | P(r) | ∝ M0.97 E2.97 | ∝ M0.97 E2.35 |
| surf. bright. | SX(r) | ∝ M0.93 E3.93 | ∝ M0.93 E2.70 |
Notes. Columns 1 and 2 give the property and its definition (see also Table 1); Col. 3 the scaling relations modified for hydrodynamical effects as described in Sect. 6, based on a redshift-independent fgasTX relation, and Col. 4, redshift dependence implied from the results of Reichert et al. (2011) as given in Eq. (33).
For temperature independent emissivity (see Table 1 and text).
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