Orbital period

The band-structure code, called BAND, also uses STO basis sets with STO fit functions or numerical atomic orbitals. Periodicity can be included in one, two, or three dimensions. No geometry optimization is available for band-structure calculations. The wave function can be decomposed into Mulliken, DOS, PDOS, and COOP plots. Form factors and charge analysis may also be generated. 

Short-period comets these display a strong tendency for their farthest point from the sun (aphelia) to coincide with a giant planet s orbital radius, so that we can distinguish so-called comet families . The Jupiter family of comets is the largest and numbers around 70 comets. The shortest orbital period known is that of the short-period comet Encke—about 3.3 years. 

Using the numbers quoted above and the derived mass of the Earth gives pc = 5.52 gem-3, which, by comparison with the density of other materials measured in the laboratory, means that the Earth must be made of rock, and heavy rock at that. The mass of the other planets can be determined from their orbital periods and their radii can be measured, for example, from rates of transit in front of the Sun, and so the density of the other planets within the solar system can then be determined (Table 7.1). 

The density estimates in Table 7.1 show a distinction between the structures of the planets, with Mercury, Venus, Earth and Mars all having mean densities consistent with a rocky internal structure. The Earth-like nature of their composition, orbital periods and distance from the Sun enable these to be classified as the terrestrial planets. Jupiter, Saturn and Uranus have very low densities and are simple gas giants, perhaps with a very small rocky core. Neptune and Pluto clearly contain more dense materials, perhaps a mixture of gas, rock and ice. 

Current extrasolar planets are all much larger than the Earth. The total count at present (9 September 2005) is 168 found in 144 planetary systems, of which 18 contain multiple planets. The first to be discovered was 51-Pegasi in the constellation of Pegasus by the radial velocity method. It is about 0.45 Mjupiter and has an orbital period around the star of about 4.5 days. Of the 168 planets found so far only nine are present within a habitable zone around their star. The survey of the star catalogue for planets has only just started and we have found a large number of planets very quickly - solar systems, at least, are not special. 

Orbital period of the Saturnian system around the Sun 29.5 years... 

Halley The comet whose passage around the Sun has been observed 30 times from 239 bc to 1986. The orbital period is 75, with aphelion outside Neptune s orbit and perihelion at 0.59 AU. 

Even stronger constraints are potentially available from the high-mass X-ray binary Vela X-l. This source contains a 20 M0 star, and radial velocity variations from the star have been measured as well as periodic timing variations from X-ray pulses. The orbital period is 8.96 days and the eccentricity of... 

Table 1. Here are summarized the used models. The letter (A-E) indicates the mass and number part indicates the 1-hadronic and 2-hybrid Equations of State (EsoS). Mao denotes the gravitational mass of one NS in isolation, C is the compactness, P the orbital period, few,o the corresponding GW frequency, do the initial orbital separation and Roo the radius of one single isolated NS measured in Schwarzschild coordinates.

Comets are generally considered to be weakly consolidated, and active comets are commonly observed to split into fragments. This is sometimes due to the tidal forces of a close planetary encounter, such as affected comet Shoemaker-Levy when it passed close to Jupiter in 1992 and broke into 21 pieces. More commonly, a comet spontaneously fragments multiple times over its orbit period, without any obvious cause. Disintegrating comets leave trails of small particles in their wakes. These trails are known as meteor streams, and when the Earth passes through such a meteor stream, as it does several times a year, a meteor shower occurs. Meter-sized rocks are known to occur within cometary meteor streams. 

As we discussed in Section II in relation to (2.41), a survival amplitude has a semiclassical behavior that is directly related to the periodic orbits by the Gutzwiller or the Berry-Tabor trace formulas, in contrast to the quasi-classical quantities (2.42) or (3.3). Therefore, we may expect the function (3.7) to present peaks on the intermediate time scale that are related to the classical periodic orbits. For such peaks to be located at the periodic orbits periods, we have to assume that die level density is well approximated as a sum over periodic orbits whose periods Tp = 3eSp and amplitudes vary slowly over the energy window [ - e, E + e]. A further assumption is that the energy window contains a sufficient number of energy levels. At short times, the semiclassical theory allows us to obtain... 

Figure 1. Orbital period of an electron moving in a Coulomb field, the time scales of some internal and external perturbations [3a-3d], and the observed (shorter, see below) lifetime for the polyatomic molecule known as BBC [4]. Note that at the highermost values of n the decay lifetime begins to shorten cf. Fig. 4.

Even at high n s one needs to follow the system for many orbital periods if one is to mimic the experimental results. The difficulty is compounded if one measures the time in units of periods of the core motion. This suggests that the time evolution be characterized using the stationary states of the Hamiltonian rather than propagating the initial state. We have done so, but our experience is that in the presence of DC fields of experimental magnitude (which means that Stark manifolds of adjacent n values overlap), and certainly so in the presence of other ions that break the cylindrical symmetry and hence mix the m/ values, the size of the basis required for convergence is near the limit of current computers. In our experience, truncating the quan-... 

Classical nova (CN) and dwarf nova (DN) systems have the same binary components (a low-mass main sequence star and a white dwarf) and the same orbital periods. An important question that therefore arises is are these systems really different (and if so, what is the fundamental difference ) or, are these the same systems, metamorphosing from one class to the other ... 

White dwarf masses were taken from Ritter s Catalogue of Cataclysmic Binaries (1987, and references therein). When the mass was not known, we took the average of known masses above or below the period gap, depending on the system s orbital period. 

Accretion rates were taken from Patterson (1984) and Verbunt and Wade (1984). When the accretion rate was not known, we used Patterson s accretion rate-orbital period relation. 

The most fashionable theories of cataclysmic variables evolution assume that mass transfer above the period gap is driven by magnetic braking (e.g. Lamb and Melia 1987, Hameury et al. 1987). Available magnetic braking laws give accretion rates that are dependent on the orbital period but are quite insensitive to other parameters of the binary system (Verbunt and Zwaan 1981, Mestel and Spruit 1987). In particular, the spread in the values of A observed at a given orbital period (Warner 1987), is significantly wider than predicted by the theory. 

In the context of planetary motion, this relationship between the square of the orbital period (inversely proportional to ft)) and the cube of the orbital radius is known as Kepler s Third Law. 

These equations predict that the spheroid will repeatedly rotate through the same orbit, the particle will not migrate across the streamline, and that the orbit period is independent of the initial orientation. 

---