6.1 Virial theorem

The virial theorem is a key concept in astrophysics, linking kinetic and potential energy in equilibrium systems. It's crucial for understanding galaxy dynamics, estimating masses, and revealing dark matter's presence. This powerful tool helps astronomers study various celestial objects.

Applying the virial theorem to galaxies and clusters has led to significant discoveries. It's used to calculate masses, investigate dark matter distribution, and explore the properties of intracluster gas. These applications have shaped our understanding of the universe's largest structures.

Definition of virial theorem

• The virial theorem is a fundamental concept in astrophysics that relates the average kinetic energy and potential energy of a system in equilibrium
• It provides a powerful tool for understanding the dynamics and structure of various astrophysical objects, including galaxies, galaxy clusters, and star clusters
• The theorem is based on the assumption that the system has reached a steady state where the forces acting on it are balanced

Relation between kinetic and potential energy

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• The virial theorem states that for a system in equilibrium, the time-averaged total kinetic energy $\langle T \rangle$ is related to the time-averaged total potential energy $\langle U \rangle$ by the equation: $2\langle T \rangle + \langle U \rangle = 0$
• This implies that the total kinetic energy is equal to half the absolute value of the total potential energy
• The negative sign in the equation indicates that the potential energy is negative for bound systems, as the gravitational force is attractive

Applications in astrophysics

• The virial theorem has wide-ranging applications in astrophysics, particularly in the study of gravitationally bound systems
• It allows astronomers to estimate the masses of galaxies and galaxy clusters based on the observed velocities of their constituent stars or galaxies
• The theorem also helps explain the presence of dark matter in galaxies, as the observed velocities often exceed what can be accounted for by visible matter alone

Mathematical formulation

• The virial theorem can be derived using various mathematical approaches, each highlighting different aspects of the concept
• These formulations provide a rigorous foundation for understanding the theorem and its implications

Clausius's original formulation

• The virial theorem was first formulated by Rudolf Clausius in 1870 in the context of kinetic theory of gases
• Clausius's formulation relates the average kinetic energy of gas molecules to the virial, which is defined as the scalar product of the force acting on a particle and its position vector
• For a system of $N$ particles with masses $m_i$, positions $\vec{r}_i$, and velocities $\vec{v}_i$, the virial $G$ is given by: $G = \sum_{i=1}^{N} \vec{r}_i \cdot \vec{F}_i$, where $\vec{F}_i$ is the force acting on the $i$-th particle

Lagrange's identity

• Joseph-Louis Lagrange's identity is a mathematical relation that forms the basis for the virial theorem in classical mechanics
• It states that for a system of particles with kinetic energy $T$ and potential energy $U$, the time derivative of the moment of inertia tensor $I$ is given by: $\frac{1}{2}\frac{d^2I}{dt^2} = 2T - \vec{r} \cdot \nabla U$
• This identity relates the second time derivative of the moment of inertia tensor to the kinetic and potential energies of the system

Time averaging

• The virial theorem involves time averaging of the quantities in Lagrange's identity
• For a system in a steady state, the time average of the second time derivative of the moment of inertia tensor vanishes: $\langle \frac{d^2I}{dt^2} \rangle = 0$
• Applying this condition to Lagrange's identity and rearranging the terms leads to the virial theorem: $2\langle T \rangle + \langle \vec{r} \cdot \nabla U \rangle = 0$

Assumptions and limitations

• The virial theorem is based on several assumptions and has certain limitations that must be considered when applying it to real astrophysical systems
• Understanding these assumptions and limitations is crucial for the proper interpretation of the results obtained using the theorem

Steady-state systems

• The virial theorem assumes that the system under consideration is in a steady state or equilibrium
• This means that the system's statistical properties, such as the average kinetic and potential energies, do not change over time
• In reality, many astrophysical systems may not be in perfect equilibrium, and the theorem's applicability may be limited in such cases

Isolated systems

• The virial theorem is strictly valid for isolated systems that do not exchange energy or mass with their surroundings
• In practice, most astrophysical systems are not completely isolated and may be subject to external forces or influences
• The presence of external forces can modify the virial theorem, leading to the concept of the modified virial theorem

Non-relativistic velocities

• The classical virial theorem is derived under the assumption of non-relativistic velocities
• It may not be directly applicable to systems where relativistic effects are significant, such as in the vicinity of black holes or in the early universe
• Relativistic formulations of the virial theorem have been developed to address this limitation

Applications to galaxies

• The virial theorem has been extensively applied to the study of galaxies, providing valuable insights into their structure, dynamics, and the presence of dark matter
• It allows astronomers to estimate galaxy masses and investigate the distribution of matter within galaxies

Estimating galaxy masses

• By measuring the velocities of stars or gas clouds within a galaxy and applying the virial theorem, astronomers can estimate the total mass of the galaxy
• The virial mass estimate is based on the assumption that the galaxy is in equilibrium and that the observed velocities reflect the gravitational potential of the system
• This method is particularly useful for galaxies that are not well-resolved or for which direct mass measurements are difficult

Dark matter in galaxies

• The application of the virial theorem to galaxies has provided strong evidence for the presence of dark matter
• The observed velocities of stars and gas in the outer regions of galaxies are often much higher than what can be explained by the visible matter alone
• This discrepancy suggests the presence of an additional, invisible mass component, which is attributed to dark matter halos surrounding galaxies

Velocity dispersion vs mass

• The virial theorem predicts a relation between the velocity dispersion (a measure of the random motions of stars or gas) and the total mass of a galaxy
• This relation, known as the Faber-Jackson relation for elliptical galaxies and the Tully-Fisher relation for spiral galaxies, has been observed empirically
• These relations provide a powerful tool for estimating galaxy masses based on their observable properties, such as velocity dispersion or rotation velocity

Applications to galaxy clusters

• Galaxy clusters are the largest gravitationally bound structures in the universe, consisting of hundreds to thousands of galaxies, hot intracluster gas, and dark matter
• The virial theorem has been successfully applied to the study of galaxy clusters, providing insights into their masses, structure, and the properties of the intracluster medium

Estimating cluster masses

• By measuring the velocities of galaxies within a cluster and applying the virial theorem, astronomers can estimate the total mass of the cluster
• This method assumes that the cluster is in equilibrium and that the galaxy velocities trace the cluster's gravitational potential
• Virial mass estimates of galaxy clusters have revealed that they are dominated by dark matter, with a dark matter to baryonic matter ratio of about 5:1

Intracluster gas

• Galaxy clusters contain a significant amount of hot, X-ray emitting gas known as the intracluster medium (ICM)
• The ICM is heated to temperatures of 10^7 to 10^8 K by the gravitational potential of the cluster and the energy released from galaxy interactions and mergers
• The virial theorem can be used to relate the temperature of the ICM to the total mass of the cluster, providing an independent method for estimating cluster masses

X-ray emissions from clusters

• The hot ICM emits X-rays through thermal bremsstrahlung and line emission processes
• The X-ray luminosity of a cluster is related to its total mass through the virial theorem and the self-similar model of cluster formation
• By measuring the X-ray luminosity and temperature of the ICM, astronomers can estimate the total mass and gas mass fraction of the cluster, which provides insights into the baryon content and the role of dark matter in clusters

Modified virial theorem

• The classical virial theorem assumes that the system is isolated and in equilibrium, but many astrophysical systems are subject to external influences or boundary conditions
• The modified virial theorem takes into account the effects of external pressure and surface terms, extending the applicability of the theorem to a wider range of astrophysical scenarios

Including external pressure

• In the presence of an external pressure $P_{ext}$, the virial theorem is modified to include a surface integral term: $2\langle T \rangle + \langle U \rangle + 3P_{ext}V = 0$, where $V$ is the volume of the system
• This modification is particularly relevant for systems embedded in a larger medium, such as molecular clouds within the interstellar medium or galaxies within galaxy clusters
• The external pressure term can have significant effects on the dynamics and structure of the system, altering the balance between kinetic and potential energies

Bounded vs unbounded systems

• The modified virial theorem distinguishes between bounded and unbounded systems
• In bounded systems, the surface pressure term is non-zero and contributes to the virial balance, while in unbounded systems, the surface term vanishes
• The distinction between bounded and unbounded systems is important for understanding the stability and evolution of astrophysical structures

Implications for galaxy formation

• The modified virial theorem has implications for the formation and evolution of galaxies within the context of hierarchical structure formation
• The external pressure from the intergalactic medium can influence the collapse and equilibrium of dark matter halos, affecting the properties of the galaxies that form within them
• The interplay between the gravitational potential, kinetic energy, and external pressure shapes the observed scaling relations and the diversity of galaxy properties

Observational tests

• The virial theorem makes specific predictions about the relations between the observable properties of astrophysical systems
• Various observational tests have been conducted to verify these predictions and to probe the validity of the assumptions underlying the theorem

Measuring velocity dispersions

• Velocity dispersion measurements are crucial for testing the virial theorem in galaxies and galaxy clusters
• The velocity dispersion can be determined from spectroscopic observations of the Doppler shifts of spectral lines, such as the 21 cm hydrogen line or optical absorption lines
• By comparing the observed velocity dispersions with the predictions from the virial theorem, astronomers can test the assumption of dynamical equilibrium and the presence of dark matter

Gravitational lensing

• Gravitational lensing provides an independent method for measuring the total mass of galaxies and galaxy clusters
• The gravitational potential of a massive object can deflect the light from background sources, creating distorted or multiple images
• By analyzing the gravitational lensing effects, astronomers can map the mass distribution of the lensing object and compare it with the virial mass estimates

Tully-Fisher relation

• The Tully-Fisher relation is an empirical scaling relation between the luminosity and rotation velocity of spiral galaxies
• It arises as a consequence of the virial theorem, connecting the observable rotation velocity to the total mass of the galaxy
• Observational studies of the Tully-Fisher relation have provided support for the virial theorem and have been used to constrain the mass-to-light ratios of galaxies and the nature of dark matter

Theoretical extensions

• The classical virial theorem has been extended and generalized to account for various physical effects and to explore its implications in different astrophysical contexts
• These theoretical extensions expand the scope and applicability of the virial theorem, providing a more comprehensive framework for understanding the structure and evolution of astrophysical systems

Tensor virial theorem

• The tensor virial theorem is a generalization of the scalar virial theorem that takes into account the anisotropy and non-spherical nature of astrophysical systems
• It relates the components of the kinetic energy tensor to the components of the potential energy tensor and the moment of inertia tensor
• The tensor virial theorem is particularly useful for studying the dynamics and shape of elliptical galaxies, which often exhibit significant anisotropy in their velocity distributions

Relativistic formulations

• The classical virial theorem is based on Newtonian mechanics and is not directly applicable to systems where relativistic effects are important
• Relativistic formulations of the virial theorem have been developed to account for the effects of special and general relativity
• These formulations are relevant for studying the dynamics of compact objects, such as neutron stars and black holes, and for understanding the behavior of matter in the strong gravitational fields near these objects

Virial theorem in cosmology

• The virial theorem has been extended to the cosmological context, where it is used to study the formation and evolution of large-scale structures in the universe
• In cosmology, the virial theorem is often applied to dark matter halos, which are the building blocks of cosmic structure
• The cosmological virial theorem takes into account the expansion of the universe and the effects of dark energy, providing insights into the growth of structure and the nature of dark matter and dark energy