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Topic: Virial theorem

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	covalent bond
		Slater's method of analyzing the covalent bond using the virial theorem is presented for the hydrogen molecule.
		It is not widely appreciated that John C. Slater used the virial theorem to come to a similar conclusion about the importance of electron kinetic energy in chemical bond formation in the early days of quantum mechanics.
		It is ironic that Slater correctly used the virial theorem to show that electron kinetic energy is essential to understanding the chemical bond in the 1930s, while today many use it incorrectly to show that electron kinetic energy is irrelevant.
	www.users.csbsju.edu /~frioux/h2-virial/virial-h2.htm (2466 words)

	Gravitational Contraction and Star Formation
		Knowledge of the mass and distribution of the gas cloud permits some fairly detailed modeling, because half of the energy from gravitational collapse goes into kinetic energy according to the virial theorem.
		A general theorem from the mathematics of physics becomes a useful part of the picture of gravitational collapse.
		One application of this theorem would be to a known mass of hydrogen gas in a proto-star.
	hyperphysics.phy-astr.gsu.edu /hbase/astro/gravc.html (433 words)

	Re: What is the virial theorem?
		The statement and proof of the theorem are on pp.
		The theorem applies if the motion is periodic (in which case the average is over a period), or if all positions and velocities are bounded, in which case it applies in the limit as the time over which the average is taken tends to infinity.
		In that case, the virial theorem could be taken to apply, at least approximately, without the time-averages.
	www.lns.cornell.edu /spr/2000-08/msg0027486.html (565 words)

	The Virial Theorem
		This general proposition is known as Clausius’s virial theorem, and it has important applications in a variety of fields ranging from statistical mechanics to astrophysics.
		Kepler’s third law is essentially a special case of the virial theorem for gravitationally bound systems consisting of a small particle in circular orbit around a large massive body.
		Needless to say, Clausius’s virial theorem is just one of infinitely many propositions that follow from the fact that the average of the derivative of a bounded function is zero.
	www.mathpages.com /home/kmath572/kmath572.htm (1253 words)

	Virial theorem - Wikipedia, the free encyclopedia
		The expression "virial" derives from vis, the Latin word for "force" or "energy" and was coined by Clausius in 1870.
		The significance of the virial theorem is that it allows the average total kinetic energy to be calculated even for very complicated systems that defy an exact solution, such as those considered in statistical mechanics.
		For example, the virial theorem may be used to derive the equipartition theorem or to compute the Chandrasekhar limit for the stability of white dwarf stars.
	en.wikipedia.org /wiki/Virial_theorem (822 words)

	The Virial Theorem (Site not responding. Last check: 2007-11-03)
		The virial theorem states that, for a stable, self-gravitating, spherical distribution of equal mass objects (stars, galaxies, etc), the total kinetic energy of the objects is equal to minus 1/2 times the total gravitational potential energy.
		We usually assume that all of the orbits travel on similar orbits that are isotropic, that is, are not flattened in any way and have no preferential direction; we say these are random orbits.
		The virial theorem then requires that the kinetic energy equals one half the potential energy, that is:
	astrosun2.astro.cornell.edu /academics/courses/astro201/vt.htm (299 words)

	[No title]
		Virial Theorem and Heat Theorem This section of the chapter discussed connecting the microscopic pressure tensor and heat flux vector to the particle coordinates and momenta.
		To do this, the virial theorem and heat theorem are used.
		For equilibrium systems, the virial theorem can be written as: EMBED Equation.3 For nonequilibrium systems, this version is not applicable.
	www.ee.duke.edu /~skb/docs/MD_lectures_II.doc (812 words)

	[No title]
		Using the virial theorem: the mass of a cluster of galaxies
		The virial theorem postulates a simple relationship between the average kinetic energy and average gravitational potential energy of bodies in a gravitationally bound system:
		use the virial theorem to estimate the mass of the Coma Cluster.
	spiff.rit.edu /classes/phys440/lectures/gal_clus/gal_clus.html (1288 words)

	Gravitational collapse of particles
		The Virial Theorem states that, for a system of bound particles in a gravitational potential, the time average of the potential energy is equal to twice the total energy (or equivalently, the time average of the kinetic energy is equal to
		The accuracy of the simulation is demonstrated by Figure 9(b), which shows the percentage change in the total energy from its initial value.
		Therefore, the moment of collapse of the outermost shell is a short while later than the moment of maximum kinetic energy: this is a flaw in the assumptions and cannot be corrected by improvements to the numerical methods.
	compsoc.dur.ac.uk /~tw/nbody/report/node3.html (2479 words)

	[No title]
		I have included this derivation of the Virial Theorem in order to illustrate how it can be derived in the most general terms.
		You must, however, be extremely familiar with the equations at the end of this section that describe the Virial Theorem.
		Be sure you understand, and are familiar with, the three cases that describe the balance of the gravitational and kinetic terms in the Virial equation.
	bison.ph.bham.ac.uk /~wjc/Teaching/SIUstud4.html (514 words)

	Virial Relations (Site not responding. Last check: 2007-11-03)
		Abstract: A virial relation is an integral identity involving the solution of a differential equation.
		A well-known example of a viral relation relates the time-average kinetic and potential energies of an n-particle system under the influence of central forces (usually referred to as the virial theorem).
		In mathematics virial relations have been used extensively, for example, in deriving necessary conditions for the existence of solutions of differential equations.
	www.sfu.ca /~rpyke/research/virial.html (324 words)

	Virial Theorem (Site not responding. Last check: 2007-11-03)
		There is an important theorem in statistical physics, according to which all the different types of energy in a physical system must be related.
		For a star in ``hydrostatic equilibrium'', the Virial theorem states that the total kinetic energy must be equal to one half the total potential energy by magnitude.
		In the case of stationary equilibrium, the kinetic energy is all in the random thermal motion of the gas particles and photons.
	www.ipp.mpg.de /~bds/astro/virial.html (331 words)

	[No title]
		This is known as the virial theorem, and it implies that a satellite rocketed into low Earth orbit is given just one half the energy that would be required to blast it completely away into interplanetary space.
		Still, the total kinetic energy remains one half the total gravitational energy since there is no energy to or from outside the star.
		The main implication of the virial theorem is that, when the star decreases in size, releasing gravitational energy, half of this must be radiated into space.
	www.gpc.edu /~fbuls/other/virial/virial2.html (652 words)

	The partition function and pressure estimators
		The primitive pressure estimator can be manipulated to produce another pressure estimator in the same manner as the path integral virial kinetic energy estimator is obtained from the primitive kinetic energy estimator [13, 6].
		This pressure estimator, the virial pressure estimator, also must be made consistent with periodic boundary conditions before it can be applied.
		Finally, the virial expression can also be derived by assuming that only the particle centroid is scaled by the volume or the limits of integration of the other normal modes are not bound by the volume (see below).
	homepages.nyu.edu /~mt33/PINPT/node7.html (496 words)

	Determining Masses in Galactic Centers (Site not responding. Last check: 2007-11-03)
		One thing that makes us rather certain that massive fl holes can exist in the center of active galaxies is that there is strong experimental evidence for enormous amounts of unseen mass in the centers of some galaxies (whether active or not).
		There are some simple ideas concerning gravitational energy in astronomy constituting what is called the Virial Theorem that allow us to estimate the amount of mass contained in a region.
		The basic idea is that a large concentration of mass (whether we can see it or not) causes a very large gravitational field, and that by observing the velocity of objects in that region we can infer the magnitude of the gravitational field and therefore the mass.
	csep10.phys.utk.edu /astr162/lect/active/virial.html (312 words)

	Pubs.GISS: Kalman et al. 1976. 'Self-consistency condition and high-density virial theorem in relativistic ... (Site not responding. Last check: 2007-11-03)
		Self-consistency condition and high-density virial theorem in relativistic many-particle systems.
		In order for the thermodynamic and kinetic definitions of the chemical potential and the pressure to lead to identical results, a nontrivial self-consistency criterion has to be satisfied.
		This, in turn, leads to a virial-like theorem in the high-density limit.
	pubs.giss.nasa.gov /abstracts/1976/Kalman_etal.html (68 words)

	virial
		If the system is roughly in equilibrium so the time averages of kinetic and potential energy are close to their current values, the virial theorem implies that T = -(1/2) V. we know that = -/2.
		Before I sketch the proof of the virial theorem, let's consider the simplest possible case: a single light particle in circular orbit around a heavy one.
		This is especially important in the theory of gases, where we measure the deviation from being an ideal gas using "virial coefficients".
	math.ucr.edu /home/baez/virial.html (945 words)

	Density Determination
		In words, the virial theorem states that twice the kinetic energy is equal to the magnitude of the potential energy.
		If we use the virial theorem on galaxies instead of clusters of galaxies then we get a mass-to-luminosity ratio that is about 30.
		Thus the mass-to-luminosity ratio appears to vary with the size of the region measured, from 3 in the solar neighborhood to 30 in galaxies to 300 in clusters of galaxies.
	www.astro.ucla.edu /~wright/density.html (1243 words)

	13. Rotation & Velocity Anisotropy
		Integrating the collisionless Boltzmann equation over all velocities and positions yields a simple relationship, the tensor virial theorem, which links a galaxy's shape to its kinematics.
		To derive the tensor virial equation, multiply the CBE by
		We will come back to this term shortly, but it is worth noting that for a system in equilibrium the integral is constant, and thus the LHS is zero.
	www.ifa.hawaii.edu /faculty/barnes/ast626_97/rva.html (723 words)

	Virial Theorem
		In fact this documents states: "it (The Virial Theorem) is really the reason we think that dark matter exists." which is the real reason behind the first three questions.
		Spiral Galaxiesdon't consists of stars which move randomly, which is one precondition to use the Virial Theorem, but to a large extend in circles.
		Nor can the Virial Theorem be used to calculate the amount of dark matter in Galaxies which belong to a Cluster.
	users.pandora.be /nicvroom/virial.htm (913 words)

	Gravitational collapse, star precipitation and black hole formation.
		Is it possible that the classical virial theorem is at the heart of the pervasive presence of fl holes at the centres of galaxies?
		Therefore the virial equilibrium of a star-filled collapsing structure is one in which the stars themselves can be treated as particles to be virialised.
		Individual assembly histories are discussed, the degree of virialization is demonstrated and various commonly used formation times are measured and inter-compared.
	www.physicsforums.com /showthread.php?t=121865 (3932 words)

	Equipartition of Energy
		The theorem of equipartition of energy states that molecules in thermal equilibrium have the same average energy associated with each independent degree of freedom of their motion and that the energy is
		Each vibrational mode will get kT/2 for kinetic energy and kT/2 for potential energy - equality of kinetic and potential energy is addressed in the virial theorem.
		Equipartition of energy also has implication for electromagnetic radiation when it is in equilibrium with matter, each mode of radiation having kT of energy in the Rayleigh-Jeans law.
	hyperphysics.phy-astr.gsu.edu /hbase/kinetic/eqpar.html (475 words)

	Re: What is the virial theorem?
		In article <39A07D89.1E24AFDF@phys.canterbury.ac.nz>, Hamish Rose wrote: >> If you want to generalize to, say, galaxy clusters, in which things >> don't move in nice circular orbits, Kepler's law doesn't apply, and >> you need to go straight to the virial theorem.
		So you don't have enough information from the dynamics of any one object work out the mass of the cluster.
		You can, though, measure the velocity dispersion of a large number of galaxies in the cluster, use that to estimate the average kinetic energy of a galaxy, and apply the virial theorem to figure things out about the total mass of the cluster.
	www.lns.cornell.edu /spr/2000-08/msg0027549.html (338 words)

	Completing Knowledge in Mechanics (virial)
		I would like to know: 1)what is the so-called VIRIAL in physics, 2) what is the VIRIAL THEOREM and 3) comments on its importance in physics.
		It is worth noting that even if there are non conservative forces present, say frictive forces proportional to the velocity of the particles, that the virial theorem still holds.
		In the case of a gas with N atoms within a container of volume V, the equipartition theorem holds and the average kinetic energy is given by
	www.physicsforums.com /showthread.php?t=98954 (662 words)

	(FT/P1-13) A Tokamak with Nearly Uniform Coil Stress Based on Virial Theorem
		A novel tokamak concept with a new type of toroidal field (TF) coils and a central solenoid (CS) whose stress is much reduced to a theoretical limit determined by the virial theorem has been devised.
		Recently, we had developed a tokamak with force-balanced coils (FBCs) which are multi-pole helical hybrid coils combining TF coils and a CS coil.
		In this work, we have extended the FBC concept using the virial theorem.
	www.iaea.org /programmes/ripc/physics/fec2002/html/node355.htm (223 words)

	virial theorem concept from the Astronomy knowledge base
		virial theorem concept from the Astronomy knowledge base
		abstraction > hypothesis > theory > law > thermodynamics law > virial theorem
		Next thermodynamics law: zeroth law of thermodynamics Up: thermodynamics law Previous thermodynamics law: third law of thermodynamics
	www.site.uottawa.ca:4321 /astronomy/virialtheorem.html (71 words)

	Orbital Mechanics II
		(This is a key relationship for a larger problem in orbital mechanics known as the virial theorem.)
		Circular orbits arise whenever the gravitational force on a satellite equals the centripetal force needed to move it with uniform circular motion.
		For the second and third parts of this problem, use the impulse-momentum and work-energy theorems, respectively.
	hypertextbook.com /physics/mechanics/orbital-mechanics-2 (1030 words)

	[18.04] Precise Calibration of the Virial Theorem from Hubble Volume Cluster Catalogs (Site not responding. Last check: 2007-11-03)
		Subdividing the population into two classes --- `parents' and `children' --- we identify the minority child component as the source of the skewness and infer that the children are merger debris that has not yet been fully incorporated into the parent population.
		For the parents alone, the probability distribution function of the velocity dispersion residuals is very well modeled by a Gaussian distribution, suggesting a central limit theorem interpretation.
		The accuracy of the calibration will be addressed by examining Virgo simulations with higher mass resolution and smaller volumes.
	www.aas.org /publications/baas/v32n3/head2000/443.htm (397 words)

	Virial Theorem (Site not responding. Last check: 2007-11-03)
		It allows us to prove an extremely general result, the Virial theorem, regarding the averages of the various terms in the energies for systems in pure energy states.
		In its most general form, the Virial theorem is true even for systems containing of many particles such as macroscopic objects.
		In fact, any system composed of electrons and nuclei, such as yourself, satisfies the conditions of the virial theorem.
	people.ccmr.cornell.edu /~muchomas/8.04/Lecs/lec_TISE/node8.html (257 words)

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