Semimajor axis

To understand this effect more quantitatively, consider an idealized measurement of Doppler shifts from one member of a binary, call it star 1. Assume that both objects are effectively point masses. One can measure the period P of the orbit and the velocity semiamplitude Vi of star 1 in the direction of the line of sight. If the two stars have masses mi and m2 and are orbiting in a circle with a semimajor axis a and an inclination i (such that i = 90° means an orbit edge-on to our line of sight), then... 

The shapes of liquid drops falling through air can be conveniently represented by two oblate semispheroids with a common semimajor axis a and minor semiaxes bi and 2 (B8, FI). Several workers have reported measurements of the aspect ratio, (hj + b2)/2a, and these are shown as a function of Eo in Fig. 7.10. The data can be represented by the relationships... 

In addition to its handedness, a vibration ellipse is characterized by its ellipticity, the ratio of the length of its semiminor axis to that of its semimajor axis, and its azimuth, the angle between the semimajor axis and an arbitrary reference direction (Fig. 2.13). Handedness, ellipticity, and azimuth, together with irradiance, are the ellipsometric parameters of a plane wave. 

Figure 11.15 shows Asano s calculations of extinction by nonabsorbing spheroids for an incident beam parallel to the symmetry axis, which is the major axis for prolate and the minor axis for oblate spheroids. Because of axial symmetry extinction in this instance is independent of polarization. Calculations of the scattering efficiency Qsca, defined as the scattering cross section divided by the particle s cross-sectional area projected onto a plane normal to the incident beam, are shown for various degrees of elongation specified by the ratio of the major to minor axes (a/b) the size parameter x = 2ira/ is determined by the semimajor axis a. 

Yarkovsky effect a thrust produced when small, rotating bodies absorb sunlight, heat up, and then re-radiate the energy after a short delay produced by thermal inertia. The Yarkovsky effect produces a slow but steady drift in the semimajor axis of an object s orbit. 

X distance along semimajor axis to point of interest... 

Semimajor axis, in AU, where 1 AU = Earth-Sun mean distance — 1.496 X 10 cm. Inclination of orbit relative to an invariable plane. 

Figure 13.16. Optimum field enhancement / (vq) for silver (optimized with respect to wavelength), and the optimum wavelength for various ellipticities and particle sizes. Labels (e.g., 3 1) indicate the ratio of the semimajor to the semiminor axis, bja. The large dots indicate the enhancement and optimum wavelength for a 30 nm-diameter sphere, while the squares indicate those for a 3 1 ellipsoid with a 35 nm semimajor axis. (Adapted from Reference 6 with permission.)...

Consider an elliptical hole characterized by a semimajor axis of length la and semiminor axis of length lb. The result of a detailed elastic analysis of a plane strain geometry with remote loading parallel to the semiminor axis (call this direction y) is an enhancement of the local stresses well above the remote value of (To- The key point that emerges from an analysis of the stresses associated with an elliptical hole are summarized in schematic form in fig. 2.12. In particular, the maximum value of the stress Oyy is given by... 

NUMBER NAME DIAMETER (KM) MASS (1015 KG) PERIOD OF ROTATION (HRS) ORBITAL PERIOD (YRS) SEMIMAJOR AXIS (AU)... 

All invariant curves on the X Y plane are circles with constant radii y/2Ji, where J = /Gma — J20 It is clear that the radius of the invariant curve depends on a fixed value of the semimajor axis a. 

As J increases, the semimajor axis increases. Consequently the ratio n/n varies, and passes through resonant values. 

We remark that all the fixed points on a resonant invariant curve of the Poincare map correspond to elliptic motion of the small body, with the same semimajor axis a, such that n/n is rational, and the same eccentricity e. They differ only in the orientation, which means that all these orbits have different values of w, as shown in Figure (15). 

A small libration island close to the periodic orbit can be interpreted, through linearization, by an ellipse of ratio 7 = a/b between the semimajor axis a and the semi-minor axis b. Since there is no differential rotation in the case of periodic orbits, we consider a model without... 

When 7 >> 1, as it is in our study of the Fibonacci sequence, then the FLI, i.e. the supremum of the norm of v behaves like 7 (0), i.e. like a since vy(0) b. We remark that if vx(0) = 0 then vy(0) = b. A question remains about the transitory regime in which the FLI grows linearly with time. Actually, in order to reach its maximum value, the vector v(t) has to visit the ellipse from the the semi-minor to the semimajor axis, i.e. has to rotate at least an angle of 90 degrees. We were able to reproduce very well the evolution of the FLI with time of the seven Fibonacci periodic orbits (Lega and Froeschle 2000), confirming the validity of the simple model for explaining the behavior of the FLI for periodic orbits. 

The total surface area related to the semimajor axis is a function of the two aspect ratios (3 and y. The special functions F(ty, k) and E(ty, k) are incomplete elliptical integrals of the first and second kind, respectively. They depend on the amplitude angle <[> and the modulus k. These special functions can be computed quickly and accurately by means of computer algebra systems such as Mathematica [153]. Their properties are given in Abramowitz and Stegun [1], The relationship between the square root of the total surface area and the semimajor axis is [150] ... 

In June 2008 the International Astronomical Union decided on the name plutoid for the category of transneptunian dwarf planets. Plutoids are celestial bodies in orbit around the sun at a semimajor axis greater than that of Neptune and sufficiently massive to adopt a near-spherical shape. See . 

In the case of elliptical orbits, Newton s mechanics relates the two parameters that specify an ellipse to the dynamic parameters that define orbital motion. The size of an ellipse depends on its semimajor axis, its shape depends on the eccentricity, and these respective parameters are proportional to the energy and angular momentum of the rotational motion. 

The unit circle is the special case when x + y = 1. A circle viewed from an angle has the apparent shape of an ellipse. An ellipse centered at (xo, yo) with semimajor axis a and semiminor axis b, as shown in Fig. 5.5, is described by the equation... 

The hyperbola can likewise be characterized by a semimajor axis a and a semiminor axis b, which, in this case, define a rectangle centered about... 

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