The advanced molybdenum-based rare process experiment (AMoRE) aims to search for neutrinoless double beta decay (
0
ν
β
β
) of
100
Mo with
∼
100
kg
of
100
Mo-enriched molybdenum embedded in cryogenic ...detectors with a dual heat and light readout. At the current, pilot stage of the AMoRE project we employ six calcium molybdate crystals with a total mass of 1.9 kg, produced from
48
Ca-depleted calcium and
100
Mo-enriched molybdenum (
48
depl
Ca
100
MoO
4
). The simultaneous detection of heat (phonon) and scintillation (photon) signals is realized with high resolution metallic magnetic calorimeter sensors that operate at milli-Kelvin temperatures. This stage of the project is carried out in the Yangyang underground laboratory at a depth of 700 m. We report first results from the AMoRE-Pilot
0
ν
β
β
search with a 111 kg day live exposure of
48
depl
Ca
100
MoO
4
crystals. No evidence for
0
ν
β
β
decay of
100
Mo is found, and a upper limit is set for the half-life of
0
ν
β
β
of
100
Mo of
T
1
/
2
0
ν
>
9.5
×
10
22
years
at 90% C.L. This limit corresponds to an effective Majorana neutrino mass limit in the range
⟨
m
β
β
⟩
≤
(
1.2
-
2.1
)
eV
.
An extended approach to the model of circular Gaussian rings is developed to study the secular evolution of the orbits of two planets under the influence of mutual gravitational perturbation. The ...orbits of the planets have a small mutual inclination angle and are represented by circular rings, on which the masses, semimajor axes and inclination angles of the orbits, as well as the orbital angular momenta of the planets are transferred. The role of the perturbation function in the problem is played by the mutual gravitational energy of the rings, which is obtained in integral form and as a series in powers of slope angles. The coefficients of the series are expressed in terms of elliptic integrals. The method for the first time considers the discrepancy between the nodes of the planets’ orbits and has been developed in two versions: (i) with a large and (ii) small angles between the nodes of the orbits. For each of these options, systems of 4 differential equations describing the secular evolution of orbits are compiled and solved in the final analytical form. It is proved that the angle of mutual inclination of the orbits remains constant in the process of evolution in both cases. Option (i) is tested on the example of the Sun–Jupiter–Saturn; option (ii) is tested on the evolution of orbits of exoplanets Kepler-10b and Kepler-10c. For both systems, the precession parameters were calculated and the diagrams were plotted.
An analytical method is developed for finding the coefficients of zonal spherical harmonics of the azimuthally averaged potential of a rotating stratified inhomogeneous triaxial ellipsoid. Two models ...are considered: i) an ellipsoid of discrete layers of finite thickness, including the two-component Core – Shell model; ii) an inhomogeneous ellipsoid consisting of infinitely thin equidensity layers with arbitrary elliptic profiles from the center to the periphery. For both types of models, equations were obtained that allow calculating the coefficients of zonal spherical harmonics of any degree according to a single scheme. Along with the general stratification, special cases of ellipsoids consisting of homeoids and focaloids were also considered. It is proved that for confocal stratification the coefficients of zonal spherical harmonics of homogeneous and congruent inhomogeneous ellipsoids coincide. The method is applied to the construction of a near-equilibrium model of the dwarf planet Haumea. It has been established that the nodal precession period of the Haumea’s ring is
T
prec
=
12.9
±
0.7
days
.
The new analytical R-toroid method is applied to study the apsidal and nodal precession of test orbits in the circumbinary exo-systems Kepler-16, Kepler-35, Kepler-38, Kepler-413, Kepler-453, ...Kepler-1661, Kepler-1647, and TOI-1338. For each system from the sample, we (1) created superposition of three R‑toroids, (2) estimated angular momenta of the stellar pair and planet relative to the Laplace plane, (3) found coefficients of the second and fourth zonal harmonics, and (4) deduced and solved equations for the frequencies of both types of the precession at test orbits. In the R-toroid gravitational field, a ratio of the apsidal and nodal precession periods at the zero inclination Gaussian ring was found to equal (–2). Research methods for the circumbinary system known from the literature proved to be the subcases of the approach developed herewith; out method additionally considers the eccentricities and orbital inclinations of the bodies to the Laplace plane, as well as the gravitational perturbation from the third body (the planet).
Mutual Energy of Gaussian Rings Kondratyev, B. P.; Kornoukhov, V. S.
Technical physics,
10/2019, Letnik:
64, Številka:
10
Journal Article
Recenzirano
A problem of mutual potential energy of two elliptical gravitating (or electrostatically charged) Gaussian rings is formulated and solved. The rings are coplanar, and their apsid lines generally have ...an angle of inclination to each other. The mutual energy of the rings is found in quadratic approximation with respect to ring eccentricities
e
1
and
e
2
. At the first stage, the potential of the Gaussian ring is represented as a series in terms of eccentricity and determined at the points of another elliptical ring (note theoretical importance of such a result). Linear (with respect to quantities
e
1
and
e
2
) terms are absent in the expression for the mutual energy of the rings, and the coefficients of the second-order terms (
and
) are equal to each other. Only one coefficient of mixed term (
e
1
e
2
) is determined by the tilt angle of the apse lines. Such a result can be used to easily determine the moment of force between the rings that is needed for the study of small mutual oscillations of the Gaussian rings.
Two methods have been developed to study secular (apsidal and nodal) orbital precession in circumbinary systems consisting of a binary star and an exoplanet. The first method is based on a model of ...three R-toroids and is intended to study the precession of test orbits. For exosystems Kepler-413 and Kepler-453, the mutual orientation of the angular momenta of the stellar pair
and the planet
was found relative to the Laplace plane, the ratio
and zonal harmonics of the potential of R-toroids were calculated. Equations for the frequencies of both types of precession were obtained and solved, and the dominant influence of the toroids of the stellar pair was established. The second method is based on the model of interacting Gaussian rings and is intended to study the secular evolution of the orbits of the stars and the planet of the circumbinary system itself. This approach made it possible to accurately calculate the periods of nodal precession for the stars and the planet; for example, in the Kepler-413 system, these periods are, respectively,
years,
years,
years. A subtle effect of the planet’s influence on the disruption of the 1 : 1 resonance for the periods of nodal precession of the stars was revealed.
A new approach to the study of long-period and secular perturbations in celestial mechanics is developed. In contrast to the traditional use of the apparatus of the perturbing Lagrange function, we ...rely on the mutual potential energy of elliptic Gaussian rings. This approach is important due to the fact that instead of averaging the expression for the perturbing Lagrange function obtained in a very complicated way, it is methodologically simpler to immediately calculate the mutual energy of Gaussian rings. In this paper, we consider the problem for two Gaussian rings with one common focus, with small eccentricities, a small angle of mutual inclination, and an arbitrary angle between the lines of apsides. An expression for the mutual energy of such a system of rings is obtained in the form of a series up to the terms of the 4th order of smallness inclusively. This expression is used to derive and solve a system of differential equations describing the evolution of rings in an ecliptic reference frame. The method is used for a detailed study of the two-planetary Sun–Jupiter–Saturn problem. The results complement and refine the results of other authors. The new expression of the perturbing function can be applied not only to the planetary problem, in which all inclinations should be small, but also to the problem with nonplanetary rings found around small celestial bodies.
A scheme of the modified method of round Gaussian rings, designed to study the secular evolution of orbits in systems consisting of a central star and two planets, is presented. The reason for the ...secular evolution of the nodes and inclinations of the orbits of the planets is their mutual gravitational attraction. The orbits of the planets are modeled by homogeneous round Gaussian rings, to which the masses, sizes and angles of inclination of the orbits, as well as orbital angular momenta of the planets, are transferred. The method takes into account the fact that, in general, the ascending nodes of the orbits may not coincide. The mutual gravitational energy of the rings
is represented as a series in the quadratic approximation in powers of small inclination angles. Using this function
, a closed system of four differential equations describing the secular evolution of the planets’ orbits is composed. The solution to the equations is obtained in finite analytic form, which simplifies the interpretation of the investigated planetary motions. The method was tested on the example of the Sun–Jupiter–Saturn system; for it, in particular, the difference in the longitudes of the nodes of the orbits of Jupiter and Saturn was calculated as a function of time. New approach is also used to study the precession of nodes in the exoplanetary system K2-36; graphs of all unknown quantities are obtained. It has been established that in the course of evolution the mutual inclination angle of the orbits remains constant, and the librations of the orbits in the inclination angle and in the motion of the nodes occur synchronously.
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