A closed-form expression is obtained for the density of a simple layer, equipotential to an oblate level ellipsoid of revolution in an outer space. The potential of any level spheroid of positive ...mass with the inward direction of the attracting force on its surface can be represented in this way. A family of density functions defined within the whole volume of a level ellipsoid of revolution is found. Several density examples are considered.
ABSTRACT
Statistical analysis of samples of the orbits of celestial bodies is complicated by the fact that the Keplerian orbit is a multidimensional object, the coordinate representation of which ...non-linearly depends on the choice of orbital elements. In this work, using the construction of the Fréchet mean, concepts of mean orbit and dispersion of the orbit family are introduced, consistent with the distance function on the orbit set. The introduced statistical characteristics serve as analogues of sample mean and variance of a one-dimensional random variable. Exact formulas for calculating the elements of mean orbits and dispersion quantities with respect to two metrics on the orbit space are derived. For a large sample of meteoroid orbits from the Geminid stream, numerical simulations of orbit evolution over 20 000 yr in the past were conducted. By analysing the dependency of statistical characteristics on time, estimates for the age of the stream and the gas outflow velocity are obtained under the assumption of the birth of the Geminids due to the rapid destruction of the cometary nucleus. The estimate of the age of the stream lies in the interval from 1200 to 2400 yr, and the speed of gas outflow at perihelion should have been more than 1.2 km s−1.
Geometric properties of spaces of Keplerian orbits are of interest for celestial mechanics problems related to the search for groups of celestial bodies with close orbits. Those groups include ...asteroid families and meteor streams. Studying these groups provides important information on the evolution of the Solar System, as well as on the characteristics of objects within a family and their parent bodies. The local properties of a distance function between orbits are of primary importance for the problems of the search for families of related celestial bodies, because orbits of family members cluster together in a small region of the orbit space. Several metrics on the set of Keplerian orbits
and its quotient sets are considered in this paper. For each of these metrics we solve the question: is there a normed vector space that is locally isometric to the orbit metric space? In two of the considered cases, the answer turns out to be positive: the quotient space of
by the equivalence relation neglecting the magnitude of the pericenter argument of the orbit can be isometrically embedded into
. The embedding into
also exists for the quotient space by the pair of elements: the longitude of ascending node and the pericenter argument. It is shown in this paper that for other metrics, the answer to the stated question is negative. The possibility of an isometric embedding of the orbit space or its part into Euclidean space is useful in application to the aforementioned problems of celestial mechanics. The isometric map helps to define the mean of the orbit family in a natural way: the arithmetic mean of images corresponds to the orbit with the minimum square deviation of distances from the orbits of the family.
The distance functions on the set of Keplerian orbits play an important role in solving the problems of searching for the parent bodies of meteoroid streams. A special kind of function is the ...distances in the quotient spaces of orbits. Three metrics of this type were developed earlier. These metrics allow disregarding the longitude of the ascending node or the argument of pericenter, or both. Here, we introduce one more quotient space, where two orbits are considered to be identical if they differ only in their longitudes of nodes and arguments of pericenters, but have the same sum of these elements (the longitude of pericenter). Function
is defined to calculate the distance between two equivalence classes of orbits. The algorithm for calculation of
from given elements of the orbits is provided along with a reference to the corresponding program written in the C++ programming language. Unfortunately, function
is not a full-fledged metric. We have proven that it satisfies the first two axioms of a metric space; however, it does not satisfy the third axiom: the triangle inequality does not hold, at least in the case of large eccentricities. However, there are two important cases when the triangle axiom is satisfied: one of three orbits is circular and the longitudes of pericenters of all three orbits coincide. Perhaps, the inequality holds for all elliptic orbits, but this is a matter of further research.
Food borne transmission of antibiotic resistant strains of Escherichia coli from contaminated food has been recognised as an important hazard for human health in the past few decades and pathogenic ...strains of E. coli have long been considered as serious zoonotic hazards. Poultry meat is attractive for consumers worldwide, so the measures to preserve the safety of poultry meat are important issues. Tetracyclines are one of the most frequently used antibiotics in poultry farming and are still used in human medicine. The widespread use of tetracyclines in poultry farming may have resulted in the development and transmission of resistance strains from poultry to humans via the food chain. The relationship between the use of tetracyclines and the presence of resistant strains of E. coli in poultry meat, mainly due to the irregular use of antibiotics, is reviewed in this paper. The need for prudent use of antibiotics, particularly those which are used in human medicine is highlighted. Continuous monitoring and reporting on incidents in the future should improve the current regulations.
The importance of gulls as bioindicators, reservoirs and vectors of Escherichia coli strains resistant to the older generation of antibiotics, broad spectrum cephalosporins and fluoroquinolones is ...reviewed in the following paper. The aim is to highlight the fact that they could be a hot spot for the development of new resistance types. Even though gulls do not naturally come into contact with antibiotics, they are omnivorous and they often eat food in agricultural, rural and urban areas so they can be infected with resistant strains from livestock or human sources which they can spread again into the environment. They may then come into contact with poultry kept under free range conditions. More intensive investigations of this subject are required, as well as the need to find accurate and reliable preventive measures, are demonstrated in the present paper.
The outer gravitational potential
V
of the level ellipsoid of revolution
T
is uniquely determined by two quantities: the eccentricity
ε
of the ellipsoid and Clairaut parameter
q
, proportional to the ...angular velocity of rotation squared and inversely proportional to the mean density of the ellipsoid. Quantities
ε
and
q
are independent, though they lie in a rather strict two-dimensional domain. It follows that Stokes coefficients
I
n
of Laplace series representing the outer potential of
T
are uniquely determined by
ε
and
q
. In this paper, we have found explicit expressions for Stokes coefficients via
ε
and
q
, as well as their asymptotics when
n
→
∞
. If
T
does not coincide with a Maclaurin ellipsoid, then
|
I
n
|
∼
B
ε
n
/
n
with a certain constant
B
. Let us compare this asymptotics with one of
I
n
for ellipsoids constrained by the only condition of increasing (even nonstrict) of oblateness from the centre to the periphery:
|
I
n
|
∼
B
¯
ε
n
/
(
n
2
)
. Hence,
level ellipsoids with ellipsoidal equidensites do not exist
. The only exception represents Maclaurin ellipsoids. It should be recalled that we confine ourselves by ellipsoids of revolution.
In this article, we prove the relaxed triangle inequality for Southworth and Hawkins, Drummond and Jopek orbital similarity criteria on the set of non-rectilinear Keplerian orbits with the ...eccentricity bounded above. We give estimates of the minimal coefficients in the inequality for each criterion and show that one of the calculated coefficients is exactly minimal. The obtained inequalities can be used for the acceleration of algorithms involving pairwise distances calculations between orbits. We present an algorithm for calculation of all distances not exceeding a fixed number in a quasi-metric space and demonstrate that the algorithm is faster than the complete calculation on the set of meteors orbits. Finally, we estimate the correlation dimensions of the set of main belt asteroids orbits and meteors orbits with respect to various orbital metrics and quasi-metrics.
In order to solve the problem of deflecting a dangerous asteroid from a collision orbit with the Earth, using a low-thrust engine directed tangentially to the trajectory is considered. The engine can ...be mounted on the asteroid or on a “gravity tractor.” The purpose of this study is to establish the fundamental possibility of steering away an asteroid to a safe distance over times of approximately a month and a year. This is acceptable since an asteroid with about a 100-m diameter is unlikely to strike immediately after its discovery. We limited ourselves to a model statement of the problem: the engine provides constant tangential acceleration. Previously, we transformed the respective Euler equations using the averaging method. Here, we solve them by the method of series in powers of “slow time” and demonstrate the adequacy of the solution on the time intervals of decades. It turns out that asteroids up to 55 m in diameter can be deflected in a year with an engine thrust of 1 N. With a thrust of 20 N, asteroids up to 50 m in diameter can be deflected in a month, and asteroids with a diameter of up to 150 m, in a year. Diverting larger asteroids requires more time or more powerful engines. The results are compared with the previously obtained similar data for the case of the transversal perturbing acceleration. The tangential traction leads to better results in all cases; however, both variants nearly coincide for orbits with eccentricities up to 0.4. The difference becomes significant at
.
Abstract
Yogurt is one of the most popular fermented dairy products with a worldwide acceptance. There are many types of yogurt differing in flavor, physical and chemical properties. Yogurt is ...produced by adding bacterial culture of
Lactobacillus delbrueckii
subsp.
bulgaricus
and
Streptococcus thermophilus
to milk and cream products. During the period from January 2017 to December 2020, a total of 202 yogurts from different small and medium sized dairy plants were analyzed as part of HACCP self-control programs. The determination of
Enterobacteriaceae
was performed as an alternative indicator of good hygiene practice. The results showed that 21.29% of analyzed yogurts contained more than 10 CFU/g
Enterobacteriaceae
, which is the evidence of poor hygiene or inadequate processing, process failure and post-process contamination. Generally, dairy products are potential vehicles for microorganisms from the
Enterobacteriaceae
family. Good manufacturing practices and good hygiene practices must be followed throughout the production line thoroughly. The absence of classic foodborne pathogens does not indicate that the yogurt is fit for consumption, since other potentially pathogenic bacteria of the
Enterobacteriaceae
family could be present. Thus, rather than pathogen testing, using
Enterobacteriaceae
to monitor the effectiveness of implemented preventive prerequisite measures could offer a better view of the quality, sanitary conditions, and safety of yogurt products.
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