Virial theorem
The virial theorem (Latin vis, force ') is a temporal relationship between the arithmetic mean value of the kinetic energy and the time average of the potential energy of a closed stationary physical system. He was executed in 1870 by Rudolf Clausius installed in the factory Over a mechanical theorem applicable to heat.
Scalar formulation
The virial V of a system of N particles is the sum of the inner products of the time derivatives of the pulses and the locations of these particles, ie
If the virial limited, then:
It is the resultant of the forces acting on the i-th particle forces, which are exerted by the other particles in the system. As a closed system is considered, there are no external forces.
If the force is conservative and has a potential U, which is homogeneous of degree k, ie for α > 0, then the above simplifies to the form
A many-particle system is in equilibrium, the system can be regarded as ergodic, ie the time average is equal to the ensemble average for all observables. Because this particular applies to the kinetic and the potential energy and the ensemble average of the energies from the sum of the individual energies, divided by the number N of objects is formed, the ensemble average can be expressed by the total energy. We therefore obtain for equilibrium systems:
Without averaging over time, since the values are constant in time.
Sign
The virial in the general form of a homogeneous potential function of degree s
Are clear information about the sign of the energy quantities involved.
For the best-known case (gravitation, Coulomb's force ) results, for example:
The signs are as follows ( see Figure ):
- Is positive definite.
- For and for
Discussion
A stationary system of point particles must for s = -1 according to the virial theorem to surrender half of the potential energy when it is converted into a spatially narrower. On the other hand has only half the lifting work ( which corresponds to the total energy ) can be supplied when it is converted into a broader space and there will be stationary again. Of course, other ratios, as one can easily recalculate apply to other s.
For -2 > s> 0 coulombähnliche conditions apply, s> -2 results in metastable systems with only specific stable solutions.
For 0
S = 0 has no associated potential function. With s = -2, the energy difference and the energy disappear itself, of this system can be removed arbitrarily far apart. 37f
Example: Mass determination of astronomical pile
Applies the virial theorem, for example, in astrophysics and celestial mechanics. There one uses the Newtonian gravitational potential, which is homogeneous of degree -1. Then we have
The virial theorem makes it possible to find reasonably good estimates of the total masses dynamically bound systems such as star clusters or clusters of galaxies. The total mass of such a cluster may then be fully expressed through observables such as radial velocities, angular distances and apparent magnitudes of the individual objects. The only requirement for the application of the virial theorem is the knowledge of the distance of the cluster. We want to outline the procedure for a mass determination of such a cluster here:
The total kinetic energy of a star or galaxy cluster is given by
Neither the individual masses mi nor amounts velocity | vi |, however, are observables.
The introduction of the total mass allows the transformation
Now you meet two assumptions:
A) The individual masses mi are proportional to the individual luminosities li and is therefore considered
Where the last term refers to the weighted average over the luminance speeds.
B ) The system is spherically symmetric and is in equilibrium ( one also says it is virialisiert ). Therefore, the speeds are about the spatial directions uniformly distributed ( equipartition theorem ). Then we have
Where vR denotes the radial peculiar velocities, ie the deviations of the radial velocity from the cluster mean.
Thus we obtain:
Other hand, for the total potential energy under the condition of the spherical symmetry
With
- The gravitational constant G
- The total radius R of the system
- The morphological factor α, which is dependent on the radial distribution function, which is the geometry of the cluster; for a (however unrealistic ) uniform distribution within the radius R is, for example, α = 5 /3 System. In general, the factor from the observed angular distances of the individual systems is to determine the cluster center.
By applying the virial for gravitation we get the total mass of the cluster as
Generalization to tensors
In the framework of continuum mechanics of the tensor virial theorem from the collisionless Boltzmann equation and its derivation Jeans criterion is proved. When the interaction in turn gravity is adopted, the rate takes the form:
With
- The inertia tensor
- The tensor of kinetic energy
- The stress tensor and
- The tensor of potential energy.
In the static case, the time derivative falls on the left side of the equation away, and since the stress tensor is traceless, the trace of the equation again yields the scalar virial theorem.
For the quantum mechanics of the virial theorem remains valid, as was shown by Fock.