Choreographic Three Bodies on the Lemniscate

We show that choreographic three bodies {x(t), x(t+T/3), x(t-T/3)} of period T on the lemniscate, x(t) = (x-hat+y-hat cn(t))sn(t)/(1+cn^2(t)) parameterized by the Jacobi's elliptic functions sn and cn with modulus k^2 = (2+sqrt{3})/4, conserve the center of mass and the angular momentum, where x-hat and y-hat are the orthogonal unit vectors defining the plane of the motion. They also conserve the moment of inertia, the kinetic energy, the sum of square of the curvature, the product of distance and the sum of square of distance between bodies. We find that they satisfy the equation of motion under the potential energy sum_{i<j}(1/2 ln r_{ij} -sqrt{3}/24 r_{ij}^2) or sum_{i<j}1/2 ln r_{ij} -sum_{i}sqrt{3}/8 r_{i}^2, where r_{ij} the distance between the body i and j, and r_{i} the distance from the origin. The first term of the potential energies is the Newton's gravity in two dimensions but the second term is the mutual repulsive force or a repulsive force from the origin, respectively. Then, geometric construction methods for the positions of the choreographic three bodies are given

Studying marine microorganisms from space

Microorganisms are but a few micrometers in diameter and are not visible to the naked eye. Yet, the large numbers of microorganisms present in the oceans and the global impact of their activities make it possible to observe them from space. Here a few examples of how microorganisms can be studied from satellites are presented. The first case is the best known: the main pigment used in photosynthesis (chlorophyll a) can be determined from satellites. These kinds of studies have contributed a tremendous amount of understanding about the distribution and dynamics of primary production in the oceans. Two other examples will concern analysis of heterotrophic prokaryotic production and estimates of dimethyl sulfide (DMS) concentration and flux to the atmosphere. These three processes are of fundamental importance for the functioning of the biosphere. Marine microbes carry out about half of the total primary production in the planet. A substantial fraction of the respiration in the oceans is due to the activity of heterotrophic prokaryotes. Finally, the flux of DMS to the atmosphere is believed to constitute one of the mechanisms by which the biota can regulate climate. The global implications of microbial processes in the oceans can only be addressed with the help of satellites