Tidal Evolution of Close Binary Asteroid Systems

We provide a generalized discussion of tidal evolution to arbitrary order in the expansion of the gravitational potential between two spherical bodies of any mass ratio. To accurately reproduce the tidal evolution of a system at separations less than five times the radius of the larger primary component, the tidal potential due to the presence of a smaller secondary component is expanded in terms of Legendre polynomials to arbitrary order rather than truncated at leading order as is typically done in studies of well-separated system like the Earth and Moon. The equations of tidal evolution including tidal torques, the changes in spin rates of the components, and the change in semimajor axis (orbital separation) are then derived for binary asteroid systems with circular and equatorial mutual orbits. Accounting for higher-order terms in the tidal potential serves to speed up the tidal evolution of the system leading to underestimates in the time rates of change of the spin rates, semimajor axis, and mean motion in the mutual orbit if such corrections are ignored. Special attention is given to the effect of close orbits on the calculation of material properties of the components, in terms of the rigidity and tidal dissipation function, based on the tidal evolution of the system. It is found that accurate determinations of the physical parameters of the system, e.g., densities, sizes, and current separation, are typically more important than accounting for higher-order terms in the potential when calculating material properties. In the scope of the long-term tidal evolution of the semimajor axis and the component spin rates, correcting for close orbits is a small effect, but for an instantaneous rate of change in spin rate, semimajor axis, or mean motion, the close-orbit correction can be on the order of tens of percent.Comment: 40 pages, 2 tables, 8 figure

Lipolytic enzymes and hydrolytic rancidity

Lipolysis, the enzymic hydrolysis of milk lipids to free fatty acids and partial glycerides, is a constant concern to the dairy industry because of the detrimental effcts it can have on the flvor and other properties of milk and milk products. However, free fatty acids also contribute to the desirable flavor of milk and milk products when present at low concentrations and, in some cheeses, when present at high concentrations. The enzymes responsible for the detrimental effects of lipolysis are of two main types: those indigenous to milk, and those of microbial origin. The major indigenous milk enzyme is lipoprotein lipase. It is active on the fat in natural milk fat globules only after their disruption by physical treatments or if certain blood serum lipoproteins are present. The major microbial lipases are produced by psychrotrophic bacteria. Many of these enzymes are heat stable and are particularly significant in stored products. Human milk differs from cows' milk in that it contains two lipases, a lipoprotein lipase and a bile salt-stimulated lipase. The ability of the latter to cause considerable hydrolysis of ingested milk lipids has important nutritional implications