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Applications of Systems of Equations
Published in Roberts Charles, Elementary Differential Equations, 2018
For our purposes we can think of the infinitesimally small mass as a satellite, space-station, or spaceship and the two large bodies as the earth and moon or as the Sun and Jupiter. To be specific, let us consider an earth-moon-spaceship system. Let E denote the mass of the earth and M denote the mass of the moon. The unit of mass is chosen to be the sum of the masses of the earth and moon—hence, E+M=1 $ E + M = 1 $ . It is customary to let μ $ \mu $ represent the mass of the smaller of the two large bodies, so μ=M $ \mu = M $ . The distance between the two large masses—E and M, in this instance—is selected as the unit of length. The unit of time is chosen so that the gravitational constant is 1. This choice means the two large bodies complete one circular revolution in 2π $ 2\pi $ units of time. We will present the equations of motion of the infinitesimally small body (the spaceship) in a special two-dimensional rectangular coordinate system—the barycentric coordinate system. The origin of this system is at the center of mass of the earth and moon. The x-axis passes through the gravitational centers of the earth and moon. In this coordinate system, the earth is located at (-μ,0) $ (-\mu , 0) $ and the moon is located at (1-μ,0) $ (1-\mu , 0) $ . The location of the spaceship is (x(t), y(t)). See Figure 10.27.
Synthesis of Protocols with Indexed Samples: Single-Cell Analysis
Published in Mohamed Ibrahim, Krishnendu Chakrabarty, Optimization of Trustworthy Biomolecular Quantitative Analysis Using Cyber-Physical Microfluidic Platforms, 2020
Mohamed Ibrahim, Krishnendu Chakrabarty
Consider a continuous space Q⊂[0,1)rg constructed using input variables RRq={RR(q,1),RR(q,2),...,RR(q,rg)}∈Q. A point q in the space Q is defined as q={q1,q2,...,qtm}⊂Q, where qi = {q(i,1), q(i,2),..., q(i,rg)}| q(i,j) ∈ RR(q,j), and it must satisfy the following condition ∑j=1rgq(i,j)=1;∀i∈{1,...,tm}. The space Q can be graphically represented using a simplex that is formed using a barycentric coordinate system [222]. Figure 7.4(a) depicts the shapes of 1-simplex (2-D space) and 2-simplex (3-D space).
The angular characteristics of Moon-based Earth observations
Published in International Journal of Digital Earth, 2020
Huadong Guo, Yuanzhen Ren, Guang Liu, Hanlin Ye
The International Celestial Reference System (ICRS) is the idealized barycentric coordinate system to which celestial positions are referred. It is non-rotating with respect to an ensemble of distant extragalactic objects and has no intrinsic orientation; however, the system is aligned close to the mean equator and dynamical equinox of J2000.0 for continuity with previous fundamental reference systems. The ICRS orientation is independent of epoch, ecliptic, or equator and is realized from a list of adopted coordinates of extragalactic sources (Gérard and Luzum 2010). The J2000 epoch is defined in the framework of General Relativity by IAU 1994 Resolution C7 as occurring on January 1, 2000 Terrestrial Time (TT; Julian Date 2451545.0 TT).