The Jeans Equations and the Tensor Virial Theorem

Before theoretical ideas in this subject can be compared with observational data, it is necessary to consider the properties of galaxies that are likely to be relics of their formation. Most astronomers would agree that the list of important parameters should be headed by the total mass M, energy E and angular momentum J. Next on the list should probably be the relative contributions to these quantities from the disc and bulge components of galaxies and denoted D/B for the mass ratio. They can be estimated from the median (i.e. half-mass) radius R, velocity dispersion σ and rotation velocity v of each component, either through the virial theorem or through the luminosity L and an assumed value of M/L. As a first approximation, it is reasonable to suppose that galaxies of a given disc-to-bulge ratio or morphological type form a sequence with mass as the fundamental parameter. The comparison of theory with data is further simplified by considering the extreme cases of ellipticals, with D/B << 1, and late-type spirals, with D/B >> 1. The approach outlined below is to explore the consequences of relaxing in succession the constraints that E, J and M be conserved during the collapse of proto-galaxies. In this article I concentrate on theories that are based on some form of hierarchical clustering because the pancake and related theories are not yet refined enough for a detailed confrontation with observations.

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Towards B-Spline Atomic Structure Calculations

Atoms ◽

10.3390/atoms9030050 ◽

2021 ◽

Vol 9 (3) ◽

pp. 50

Author(s):

Charlotte Froese Fischer

Keyword(s):

Atomic Structure ◽

Bound States ◽

Virial Theorem ◽

B Spline ◽

B Splines ◽

Self Consistent ◽

History Of ◽

Consistent Method ◽

Atomic Structure Calculations ◽

Structure Calculations

The paper reviews the history of B-spline methods for atomic structure calculations for bound states. It highlights various aspects of the variational method, particularly with regard to the orthogonality requirements, the iterative self-consistent method, the eigenvalue problem, and the related sphf, dbsr-hf, and spmchf programs. B-splines facilitate the mapping of solutions from one grid to another. The following paper describes a two-stage approach where the goal of the first stage is to determine parameters of the problem, such as the range and approximate values of the orbitals, after which the level of accuracy is raised. Once convergence has been achieved the Virial Theorem, which is evaluated as a check for accuracy. For exact solutions, the V/T ratio for a non-relativistic calculation is −2.

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Dwarf galaxies without dark matter: constraints on modified gravity

Monthly Notices of the Royal Astronomical Society ◽

10.1093/mnras/staa3653 ◽

2020 ◽

Vol 501 (1) ◽

pp. 254-260

Author(s):

Ali Rida Khalifeh ◽

Raul Jimenez

Keyword(s):

Dark Matter ◽

Modified Gravity ◽

Virial Theorem ◽

Dwarf Galaxies ◽

Strong Support ◽

Gravity Models ◽

Birkhoff Theorem ◽

Astronomical Surveys ◽

Stringent Test ◽

Modified Gravity Models

ABSTRACT The discovery of 19 dwarf galaxies without dark matter (DM) provides, counterintuitively, strong support for the ΛCDM standard model of cosmology. Their presence is well accommodated in a scenario where the DM is in the form of cold dark particles. However, it is interesting to explore quantitatively what is needed from modified gravity models to accommodate the presence of these galaxies and what extra degree of freedom is needed in these models. To this end, we derive the dynamics at galaxy scales (Virial theorem) for a general class of modified gravity models. We distinguish between theories that satisfy the Jebsen–Birkhoff theorem, and those that do not. Our aim is to develop tests that can distinguish whether DM is part of the theory of gravity or a particle. The 19 dwarf galaxies discovered provide us with a stringent test for models of modified gravity. Our main finding is that there will always be an extra contribution to the Virial theorem coming from the modification of gravity, even if a certain galaxy shows very small, if not negligible, trace of DM, as has been reported recently. Thus, if these and more galaxies are confirmed as devoid (or negligible) of DM, while other similar galaxies have abundant DM, it seems interesting to find modifications of gravity to describe DM. Our result can be used by future astronomical surveys to put constraints on the parameters of modified gravity models at astrophysical scales where DM is described as such.

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