Certain particular solutions of the clairaut equation

A study has been conducted to ascertain conditions on the density distribution within a fluid, rotating planet in order that the deformation of its outer shell be expressible in terms of the Bessel functions of the first kind and the Gaussian hypergeometric series.It has been established that (1) the density must be a solution to an ordinary differential equation of the Riccati type, and (2) two already considered density distributions are the only closed-form solutions of physical relevance.     

The dark matter distribution in disc galaxies

We use high-quality optical rotation curves of nine low-luminosity disc galaxies to obtain the velocity profiles of the surrounding dark matter haloes. We find that they increase linearly with radius at least out to the edge of the stellar disc, implying that, over the entire stellar region, the density of the dark halo is about constant.
The properties of the mass structure of these haloes are similar to those found for a number of dwarf and low surface brightness galaxies, but provide a more substantial evidence of the discrepancy between the halo mass distribution predicted in the cold dark matter scenario and those actually detected around galaxies. We find that the density law proposed by Burkert reproduces the halo rotation curves, with halo central densities ( ρ ₀∼1–4×10⁻²⁴ g cm⁻³) and core radii ( r ₀∼5–15 kpc) scaling as ρ ₀∝ r ₀^−2/3.     

The equilibrium of a rotating body of arbitrary density

This paper extends Clairaut's theory of rotational equilibrium to third order terms in a small parameter and is meant to be a sequel to a 1962 publication by the author bearing on the same topic. It has been feasible to obtain the Clairaut equation, which governs the deformation of the equipotential surfaces within a rapidly rotating mass in hydrostatic equilibrium, as an ordinary differential equation. This has been achieved by eliminating the two integral terms which appeared in the original formulation. It is expected that the numerical integration of this newly obtained equation will contribute toward a more precise solution of certain geophysical problems — e.g., the determination of the geoid to an accuracy of ±1 m, and the correction to the travel-time of seismic waves; it should also assist in some planetary questions like the determination of the exterior shape for the rapidly rotating planets Jupiter and Saturn.     

The extended rotation curve and the dark matter halo of M33

We present the 21-cm rotation curve of the nearby galaxy M33 out to a galactocentric distance of 16 kpc (13 disc scalelengths). The rotation curve keeps rising out to the last measured point and implies a dark halo mass ≳5×10¹⁰ M_⊙. The stellar and gaseous discs provide virtually equal contributions to the galaxy gravitational potential at large galactocentric radii, but no obvious correlation is found between the radial distribution of dark matter and the distribution of stars or gas.
Results of the best fit to the mass distribution in M33 picture a dark halo which controls the gravitational potential from 3 kpc outward, with a matter density which decreases radially as R ^−1.3. The density profile is consistent with the theoretical predictions for structure formation in hierarchical clustering cold dark matter (CDM) models, and favours lower mass concentrations than those expected in the standard cosmogony.     

Evolutionary and structural properties of mirror star MACHOs

There can exist a hidden sector of the Universe in the form of parallel “mirror” world which has the same particle physics as the observable world and interacts with the latter only gravitationally. Big Bang nucleosynthesis bounds demand that the mirror sector should have a smaller temperature than the ordinary one. This implies that the mirror matter could play a role of dark matter, and in addition its chemical content should be dominated by helium. Here we study the evolutionary and structural properties of the mirror stars which essentially are similar to that of the ordinary stars but with higher helium contents. Being invisible in terms of photons, they could be observed only as MACHOs in the microlensing experiments. Using a numerical code, we compute evolution of stars with large helium abundances (Y = 0.30–0.80) and a wide range of masses, from 0.5 to 10 M. We found that helium dominated mirror star should have much faster evolutionary time (up to a factor 30) than the ordinary star with the same mass. In addition, we show the diagrams of luminosities, effective temperatures, central temperatures and densities, and compute the masses of the He-core at ignition and the minimum mass for carbon ignition, for different chemical compositions. The general conclusion is that mirror stars evolve faster as compared to ordinary ones, and explode earlier as type II supernovae, thus enriching the galactic halo of processed mirror gas with higher metallicity, with implications for MACHO observations and galaxy evolution.