Determination of the body of the dwarf planet Haumea from observations of a stellar occultation and photometry data

Secular Evolution of Rings around Rotating Triaxial Gravitating Bodies

“…This method considers all information on the planetoid available in 2018, which made it possible to calculate for each value of the photometric parameter not only the shape and density of the Haumea model, but also the orientation of its body relative to the ring and orbits of the satellites. The most probable characteristics of Haumea were obtained: In [10], it was also established that the Haumea's ring has a slight inclination relative to the equatorial plane, (5) Comparing ( 4) with the results of Ortiz et al [4], we see that our Haumea's figure is not so elongated and is more dense. These details are important in the analysis of the ring's dynamics.…”

“…Therefore, formula (38) yields (39) But it was found in (34) that the precession period of the ring's node is days; therefore, the ratio of the relaxation time to the precession period is approximately (40) This means that the influence of perturbations from the indicated non-sharp orbital resonance can be neglected during the precession of the ring plane. [10]; therefore, studying the secular evolution and precession of this ring structure is a topical problem.…”

“…Due to the problem's condition that the central ellipsoid rotates rapidly around the minor axis, it is necessary to average potential (10) over longitude in order to study the secular effects in the motion of the ring's particles. As a result of this averaging, the harmonics disappear and potential (10) in the cylindrical coordinates takes the form ( 11)…”

“…where is the internal pressure and the total potential is (23) Equation ( 22) requires that the inner and outer surfaces of the ice shell be equipotential, i.e., This imposes restrictions on the shell's shape: it must be a focaloid in which the squares of the semiaxes of the two boundary surfaces are related as (24) where is the largest root of the cubic equation (25) As is known [3], the external potential of a twolayer ellipsoid can be represented as a potential combination of uniform ellipsoids: (26) Therefore, for the coefficients of zonal harmonics, we find the expression (27) Using the generalized Maclaurin-Laplace theorem [16], expression (27) can be simplified to (28) For further calculations, we use the refined Haumea's parameters [10]. Some of these parameters were given above in formulas ( 4) and ( 5); in addition, it is also necessary to know the semiaxes of the planetoid's rocky core:…”

“…(3) A more complete approach to studying the problem of rotating ellipsoidal bodies is given in [10]. To determine the Haumea's space form, a method based on a system of eight equations was developed.…”