Under normal conditions we can assume that , then we can see that in the virial theorem the kinetic energy is related to the potential energy not by the coefficient , but rather by the coefficient close to 0.6. The difference from the classical case arises due to considering the pressure field and the field of particles’ acceleration inside the system, while the derivative of the scalar is not equal to zero and should be considered as the material derivative. An analysis of the integral theorem of generalized virial makes it possible to find, on the basis of field theory, a formula for the root-mean-square speed of typical particles of a system without using the notion of temperature: where is the speed of light, is the acceleration field constant, is the mass density of particles, is the current radius. Unlike the virial theorem for particles, for the electromagnetic field the virial theorem is written as follows: where the energy considered as the kinetic field energy associated with four-current , and sets the potential field energy found through the components of the electromagnetic tensor. In astrophysics The virial theorem is frequently applied in astrophysics, especially relating the gravitational potential energy of a system to its kinetic or thermal energy. Some common virial relations are for a mass , radius , velocity , and temperature . The constants are Newton's constant , the Boltzmann constant , and proton mass . Note that these relations are only approximate, and often the leading numerical factors (e.g. or ) are neglected entirely. Galaxies and cosmology (virial mass and radius) In astronomy, the mass and size of a galaxy (or general overdensity) is often defined in terms of the "virial mass" and "virial radius" respectively. Because galaxies and overdensities in continuous fluids can be highly extended (even to infinity in some models, such as an isothermal sphere), it can be hard to define specific, finite measures of their mass and size. The virial theorem, and related concepts, provide an often convenient means by which to quantify these properties. In galaxy dynamics, the mass of a galaxy is often inferred by measuring the rotation velocity of its gas and stars, assuming circular Keplerian orbits. Using the virial theorem, the velocity dispersion can be used in a similar way. Taking the kinetic energy (per particle) of the system as , and the potential energy (per particle) as we