A spherical gravitating object of radius R and mass M...

A spherical gravitating object of radius $R$ and mass $M$ consists of a monatomic ideal gas in hydrostatic equilibrium that has average density $\langle\rho\rangle$ and average temperature $\langle T\rangle$. Show that $\langle T\rangle=$ constant $\times\langle\rho\rangle^{1 / 3} M^{2 / 3}$, so that for two such objects of the same composition at the same average temperature the more massive object must have lower average density. Hint: Use the virial theorem.

A spherical gravitating object of radius $R$ and mass $M$ consists of a monatomic ideal gas in hydrostatic equilibrium that has average
density $\langle\rho\rangle$ and average temperature $\langle T\rangle$. Show that $\langle T\rangle=$ constant $\times\langle\rho\rangle^{1 / 3} M^{2 / 3}$, so that for two such objects of the same composition at the same average temperature the more massive object must have lower average density. Hint: Use the virial theorem.

Key Concepts

Scaling Relationships in Astrophysics

Scaling relationships involve using dimensional analysis to relate different physical quantities through power-law expressions. In the context of gravitating objects, such scaling laws show how average temperature, density, and mass interdepend, such as deducing that the average temperature scales as the one-third power of the density and the two-thirds power of the mass. These arguments are crucial for comparing astrophysical objects and understanding how variations in mass affect their internal states.

Ideal Gas Law in Astrophysical Context

The ideal gas law, relating pressure, temperature, and density, is essential in understanding the thermodynamic behavior of a monatomic gas in astrophysical settings. It allows us to express the thermal pressure in terms of temperature and density, thereby linking the internal energy of the gas to macroscopic properties like mass and average density when combined with the virial theorem.

Hydrostatic Equilibrium

Hydrostatic equilibrium refers to the balance between the inward gravitational force and the outward pressure gradient within a star or any self-gravitating object. This balance ensures the stability of such objects over time and provides a framework to relate the mass and density distribution with the internal pressure, which is often connected to the thermal state of the gas.

Virial Theorem

The virial theorem provides a vital relation between the kinetic energy and gravitational potential energy for a system in equilibrium. In astrophysics, it is used to connect the thermal energy of a self-gravitating gas with its binding energy, typically expressed as 2K + U = 0. This relation allows one to derive scaling laws between physical properties such as mass, radius, temperature, and density of gravitating objects in hydrostatic equilibrium.

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