Differences In Moments Of Inertia Research Articles

Neutron star crust can support a large ellipticity

ABSTRACT Non-axisymmetrical deformations of the crust on rapidly rotating neutron stars are one of the main targets of searches for continuous gravitational waves. The maximum ellipticity, or fractional difference in moments of inertia, which can be supported by deformations of the crust (known as ‘mountains’), provides an important upper limit on the strength of these continuous gravitational wave sources. We use the formalism developed by Gittins and Andersson, along with a deforming force that acts mainly in the transverse direction, to obtain a maximum ellipticity of 7.4 × 10−6. This is larger than the original results that Gittins and Andersson obtained but consistent with earlier calculations by Ushomirsky, Cutler, and Bildsten. This suggests that rotating neutron stars could be strong sources of continuous gravitational waves.

Increasing the regulating ability of a wind turbine in a local power system using magnetic continuous variable transmission

The paper is devoted to investigations of dynamic processes in a local power system consisting of wind turbines with a magnetic continuously variable transmission. Due to low inertia of wind turbine generator rotors, there is a problem of ensuring dynamic stability at sharp load changes or at short circuits in an autonomous power system. To increase dynamic stability of the system, two algorithms for controlling a magnetic continuously variable transmission are presented. The first algorithm stabilizes a rotation speed of the high-speed rotor of a magnetic continuously variable transmission from the generator side in a local power system consisting of wind turbines with uniform synchronous generators with permanent magnets having equal moments of inertia. Undoubtedly, local power systems having only the wind turbines with equal mechanical inertia time constants are not widely used due to stochastic nature of wind energy. Therefore, wind power systems are combined with a diesel generator or a gas-turbine unit. Investigations show that the use of the only speed stabilization algorithm is not enough for such power systems, because there is a possibility for occurrence of asynchronous operation under specific power changes due to the difference in moments of inertia of generator rotors. Thus, the second algorithm uses the phase shift compensation in accordance with a primary generator in an autonomous power system consisting of non-uniform generators having different mechanical inertia time constants. As a primary generator, a diesel generator or a gas-turbine unit having a primary speed controller may be used. It should be noted that algorithms of stabilization for speed and phase angle are extended by an inertial circuit of aerodynamic compensation for torque of rotation from the wind turbine side to reduce loading on an energy storage unit of the magnetic continuously variable transmission at disturbances from the generator side and the turbine side.

Gravitational waves from low mass neutron stars

Low mass neutron stars may be uniquely strong sources of gravitational waves. The neutron star crust can support large deformations for low mass stars. This is because of the star's weaker gravity. We find maximum ellipticities $ϵ$ (fractional difference in moments of inertia) that are 1000 times larger, and maximum quadrupole moments ${Q}_{22}$ over 100 times larger, for low mass stars than for $1.4{M}_{\ensuremath{\bigodot}}$ neutron stars. Indeed, we calculate that the crust can support an $ϵ$ as large as 0.005 for a minimum mass neutron star. A $0.12{M}_{\ensuremath{\bigodot}}$ star, that is maximally strained and rotating at 100 Hz, will produce a characteristic gravitational wave strain of ${h}_{0}=2.1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}24}$ at a distance of 1 kpc. The gravitational wave detector Advanced LIGO should be sensitive to such objects through out the Milky Way Galaxy. A low mass neutron star could be uniquely identified from a large observed spin down rate and its discovery would have important implications for general relativity, supernova mechanisms, and possibly nucleosynthesis.

Microscopic Mechanism of Large Fluctuation in Odd-Even Differences in Momentsof Inertia for Actinide Nuclei

The experimental large fluctuation in odd-even differences in moments ofinertia of deformed actinide nuclei is investigated using the particle-numberconserving (PNC) method for treating the cranked shell model with monopole andquadrupole pairing interactions. PNC calculations show that the large odd-evendifference in moments of inertia mainly comes from the interferencecontributions j(μν) from particles in high j intruder orbitals μand ν quite near the Fermi surface, which have no counterpart in the BCSformalism. The effective monopole and quadrupole pairing interaction strengthsare determined to fit the experimental odd-even differences in bindingenergies and bandhead moments of inertia. The experimental results for thevariation of moments of inertia with rotational frequency ω arereproduced well by the PNC calculation. The nearly identical experimentalmoments of inertia between 236U(gsb) and 238U(gsb) at lowfrequencies ℏω⩽0.20 MeV are also reproduced quite well.

Configuration dependence of moments of inertia in the odd-Anucleus from BCS theory

The moments of inertia (MoI) of odd-$A$ and even-even rare-earth nuclei are calculated using the Nilsson + BCS method with and without the blocking effect taken into account. The calculated MoI's with blocking considered is smaller than that without blocking considered. The MoI of an odd-$A$ nucleus depends strongly on which orbital is occupied by the odd nucleon. Although the calculated MoI's are systematically smaller than the data, the large fluctuations in the experimental odd-even differences in MoI's can be reproduced qualitatively. This is contrary to the common assertion that the MoI of an odd-$A$ nucleus is some 15% larger than those of its neighboring even-even nuclei with the BCS theory and is helpful to understand the appearance of normally deformed identical bands in an odd-$A$ nucleus and its even-even neighbors under the mean-field (BCS) approximation.

Microscopic Mechanism of the Odd-Even Difference in Moments of Inertia of Superdeformed Bands

The large odd-even difference in moments of inertia between the SD bands 195Tl(1,2) and 196Pb(1) is investigated by the particle-number conserving method, in which the blocking effect on pairing is taken into account strictly. The ω variation of J(2) for the three SD bands are reproduced quite well and the underlying microscopic mechanism is demonstrated clearly. Calculations show that the blocking of the proton [642]5/2 intruder orbital and the excitation of [651]1/2 intruder orbital play crucial roles for the odd-even difference in J(2)'s.

Odd-even differences in moments of inertia

Blocking effect and the effect of the Coriolis interaction are studied as the causes of the odd-even difference in the moments of inertia. The cranking formula is used to calculate the moments of inertia and the pairing Hamiltonian is diagonalized to obtain the state vectors and their energies that are necessary to calculate the moments of inertia. The moment of inertia is divided into two parts. One part ${I}_{\ensuremath{\Delta}v=2}$ comes from a part of the angular momentum operator that changes the seniority by 2. This part is the contribution from the even core in the odd-particle system and the blocking effect emerges in this part. The other part ${I}_{\ensuremath{\Delta}v=0}$ comes from that which does not change the seniority. This part corresponds to the contribution of the Coriolis interaction to the moment of inertia. Calculations are made for systems with $N=95--101.$ Three typical examples are analyzed. The first example shows the importance of ${I}_{\ensuremath{\Delta}v=0},$ the second example shows that the moment of inertia becomes an increasing function of $G,$ the strength of the pairing interaction, because of the predominance of ${I}_{\ensuremath{\Delta}v=0}$, and the third example shows the situation where ${I}_{\ensuremath{\Delta}v=0}$ is negligibly small. It is also shown that the blocking of a special level by the last odd particle brings about the smaller value for ${I}_{\ensuremath{\Delta}v=2}$ relative to the moment of inertia for the neighboring even-particle system.

Difference in moments of inertia and paring interaction strength in superdeformed nuclei

The difference in the bandhead moments of inertia between the superdeformed (SD) band194Hg(1) and the two-quasiparticle SD bands194Hg(2, 3) was investigated using the particle-number-conserving treatment for the cranked shell model Hamiltonian. and the pairing interaction strength in SD nuclei is estimated to be much weaker than that in normally deformed nuclei.

A comparison of the known SD bands in the mass A≈80 region

The mass A≈80 region is now firmly established as the newest region of superdeformation, with up to 11 nuclei exhibiting as many as 21 superdeformed bands. The dynamic moments of inertia of these bands reveal a variety of interesting features, such as superdeformed band crossings and the first pair of identical superdeformed bands in the mass A≈80 region. Cranked shell model calculations have been performed and the bands are interpreted in terms of the number of high-N intruder orbitals occupied. The differences in dynamic moments of inertia and the proposed single particle configurations are discussed.

Microwave spectroscopic detection of BFHOH and BH(OH) 2

Transient molecules, BFHOH and BH(OH) 2, have been detected by microwave spectroscopy in the hydrolyses of diborane and BF 3. Their conformations have been determined from the inertia defects and the differences in moments of inertia between normal and deuterated species.

The far infrared absorption spectrum and the assignment of the lattice modes of sodium formate

The far i.r. spectra of room temperature and liquid nitrogen cooled polycrystalline samples of sodium formate and sodium formate- d 1 and the i.r. spectrum of sodium formate- d 1 are reported. Factor group analysis predicts three rotatory and three translatory modes, of which the first three could be observed separately and the latter three coincide in one broad band. Based on the difference in moment of inertia of the formate and the formate- d 1 ion the three rotatory mode absorptions could be assigned. Not all lattice modes, deduced by some authors from combination bands in the i.r. region, are observed in the far i.r. region; arguments have been found to consider these bands to be Raman active.

MOMENS OF INERTIA OF EXCITED BANDS AND THE EVEN-ODD DIFFERENCE

The formulae for calculating nuclear moments of inertia with the particle-numbes-conserving method for treating pairing forces are presented. Taking 170 Yb and 169 Yb as typical examples, the even-odd difference of the moments of inertia and the dif-ference in moment of inertia between excited bands and the ground band are treated with this method, which automatically takes the blocking effect into account. The calculated results agree relatively well with the experiments.

Shape isomerism in mercury isotopes

We study the energies of the mercury isotopes in their oblate, prolate, and bubble shapes within the framework of the single-particle K-matrix theory. A change from an oblate ground state to a prolate ground state is observed when the mass is decreased from 184 to 182. Self-consistent stable bubble configurations were also obtained for these isotopes. They lie a few MeV from the ground deformed state, coming down lower for the mass region around A ≈ 200 and 182. For the isotopes of 188Hg, 186Hg, and 182Hg, the oblate and prolate minima are nearly degenerate; a strong mixing of the prolate and oblate configuration is expected. This mixing is occasionally enhanced for states with non-zero angular momentum when their energies become accidentally near-degenerate due to their differences in moments of inertia. We study such hybridization of shapes in terms of a phenomenological two-center collective model. Good qualitative agreement with experimental energy spectra and branching is observed, indicating a decrease of the prolate-oblate energy difference as the mass number decreases from 188 to 184.

The Lunar Laser Ranging Experiment

The lunar ranging measurements now being made at the McDonald Observatory have an accuracy of 1 nsec in round-trip travel time. This corresponds to 15 cm in the one-way distance. The use of lasers with pulse-lengths of less than 1 nsec is expected to give an accuracy of 2 to 3 cm in the next few years. A new station is under construction in Hawaii, and additional stations in other countries are either in operation or under development. It is hoped that these stations will form the basis for a worldwide network to determine polar motion and earth rotation on a regular basis, and will assist in providing information about movement of the tectonic plates making up the earth's surface. Several mobile lunar ranging stations with telescopes having diameters of 1.0 m or less could, in the future, greatly extend the information obtainable about motions within and between the tectonic plates. The data obtained so far by the McDonald Observatory have been used to generate a new lunar ephemeris based on direct numerical integration of the equations of motion for the moon and planets. With this ephemeris, the range to the three Apollo retro-reflectors can be fit to an accuracy of 5 m by adjusting the differences in moments of inertia of the moon about its principal axes, the selenocentric coordinates of the reflectors, and the McDonald longitude. The accuracy of fitting the results is limited currently by errors of the order of an arc second in the angular orientation of the moon, as derived from the best available theory of how the moon rotates in response to the torques acting on it. Both a new calculation of the moon's orientation as a function of time based on direct numerical integration of the torque equations and a new analytic theory of the moon's orientation are expected to be available soon, and to improve considerably the accuracy of fitting the data. The accuracy already achieved routinely in lunar laser ranging represents a hundredfold improvement over any previously available knowledge of the distance to points on the lunar surface. Already, extremely complex structure has been observed in the lunar rotation and significant improvement has been achieved in our knowledge of lunar orbit. The selenocentric coordinates of the retroreflectors give improved reference points for use in lunar mapping, and new information on the lunar mass distribution has been obtained. Beyond the applications discussed in this article, however, the history of science shows many cases of previously unknown, phenomena discovered as a consequence of major improvements in the accuracy of measurements. It will be interesting to see whether this once again proves the case as we acquire an extended series of lunar distance observations with decimetric and then centimetric accuracy.

Shape and internal structure of the Moon

Studies of the libration of points on the Moon's surface and the occultation of stars on the limb reveal a 10-km range in heights on the Moon's surface. These data and the 1-km separation of the center of mass from the center of the Moon's disk indicate that the continental areas are approximately 3 km higher than the maria. The values of the differences in the Moon's principal moments of inertia require that there be a concentration of material in the equatorial regions and around the axis pointing toward the Earth. If continental areas were so concentrated, the density distribution within the Moon could be spherically symmetrical. Utilizing charts prepared from Russian photographs of a portion of the Moon's far side and maps of the near side, it is shown that the actual continental-maria distribution is opposite to that required to explain the moment-of-inertia data. This discrepancy between the differences in moments of inertia and the gross shape, as reflected in the distribution of continents, indicates that density variations exist within the Moon. The density variations and internal structures which would lead to the observed differences in moments for a rigid Moon with a random lumpiness, as envisioned by Urey, Elsasser, and Rochester (1959), and a Moon in isostatic equilibrium and thermal convection are calculated. It is found that either model of internal structure is consistent with our knowledge of the Moon.

Possible effects of convection on lunar moments of inertia

The differences in moments of inertia of a homogeneous globe of viscous liquid arising from steady convection in its interior are evaluated on the basis of a linearized theory of such motion as developed by Chandrasekhar in 1952. It is shown that if the physical characteristics of such a globe are made to approximate the well-known properties of the Moon, with the same difference C - A of momenta about its principal axes of inertia, Chandrasekhar's theory requires the velocity of convective motion to be of the order of 10 −8 cm/sec (too small for the establishment of steady flow for a globe of lunar dimensions in 10 9 years); and the observed Rayleigh number is several hundred times as large as that required theoretically for the stability of the respective flow. As the physical basis of Chandrasekhar's theory should be closely applicable to the lunar problem, it is concluded that internal convection conforming to it—if it exists—does not lend itself for the actual explanation of the difference in momenta C - A of the lunar globe as deduced from its librations.

Close equatorial satellites of the moon

The paper deals with the orbits of near satellites of the moon moving in its equatorial plane. The figure of the moon is a triaxial ellipsoid, and the effects of the perturbation caused by the second and fourth harmonic terms in the gravitational potential of the moon are considered. It is shown that the areal velocity of the satellite is not a constant, and its deviation from the constant value is, in the first order, proportional to the difference in the principal moments of inertia about the two axes lying in the equatorial plane. Expressions for the rotation of the major axis of the orbit are given. The equations of motion are solved forh (areal velocity) andu (reciprocal of the radial distance) correct to second-order terms in J1 and J2.