T

This sum equals zero if the fluid's velocity is constant. satisfies the non-linear differential equation, With perfect X-ray and distance data, we could calculate the baryon density at each point in the cluster and thus the dark matter density.

B are the temperature and density of the baryonic matter, and is dependent on the redshift {\displaystyle r^{2}/\rho _{B}(r)}

ρ θ According to the definition of planet adopted by the International Astronomical Union in 2006, one defining characteristic of planets and dwarf planets is that they are objects that have sufficient gravity to overcome their own rigidity and assume hydrostatic equilibrium. ( ρ

It can also be a satisfactory approximation when flow speeds are low enough that acceleration is negligible. B

Hydrostatic equilibrium is the distinguishing criterion between dwarf planets and small Solar System bodies, and has other roles in astrophysics and planetary geology. [3] For instance, the pressure-gradient force prevents gravity from collapsing Earth's atmosphere into a thin, dense shell, whereas gravity prevents the pressure gradient force from diffusing the atmosphere into space. − {\displaystyle {\mathcal {L}}_{X}=\Lambda (T_{B})\rho _{B}^{2}}

The smallest object that appears to have an equilibrium shape is the icy moon Mimas at 396 km, whereas the largest object known to have an obviously non-equilibrium shape is the rocky asteroid Vesta at 525 km (573 × 557 × 446 km). {\displaystyle k} Hydrostatic equilibrium is also important for the intracluster medium, where it restricts the amount of fluid that can be present in the core of a cluster of galaxies. is the angular width of the cluster and {\displaystyle s} For example, the massive base of the tallest mountain on Earth, Mauna Kea, has deformed and depressed the level of the surrounding crust, so that the overall distribution of mass approaches equilibrium.

The baryonic density satisfies the above equation − Under planetary conditions, "solid" rock and ice are actually fluid, and will lapse into hydrostatic equilibrium if the body is massive or warm enough.

r

) θ

, the index i runs for the coordinates r and Also, the purported dwarf planet Haumea is scalene due to its rapid rotation, though it may not currently be in equilibrium. This occurs when external forces such as gravity are balanced by a pressure-gradient force. d / r {\displaystyle z}

2 Superficial deviations from perfect equilibrium, such as peaks, valleys, craters, and other irregularities of the crust which are small in scale compared to the body as a whole, are not considered significant for these definitions.

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