On the elimination of short-period terms in second-order general planetary theory investigated by Hori's method

1. The short-period terms of a second-order general planetary theory are removed through the Hori's method based on a development of the HamiltonianF in a Lie series which involves a determining functionS not depending upon mixed canonical variables as in the Von Zeipel's method but upon all the canonical variables resulting from the elimination of the short period terms ofF. Canonical variables adopted are the slow Delaunay variables. Eccentricitiese _j and sines γ_j of the semi inclinations are respectively replaced by the Jacques Henrard variablesE _j ,J _j which lead to formulas remarkably simple.F is reduced to the sumF ₀+F ₁ of its terms of degrees 0,1 in small parameter ε of the order of the masses. Only one disturbing planet is considered.F ₁ is not calculated beyond its terms of degree 3 inE _j ,E _j ,J _j , the determining functionS ₂ of degree 2 in ε not being therefore calculated beyond its terms of degree 2 inE′ _j ,E _j ,J _j and the expressions of slow Delaunay canonical variables of the disturbed planetP ₁ and the disturbing planetP ₂ in terms of the new slow Delaunay canonical variables ofP ₁ andP ₂ which result from the elimination of the short period terms ofF ₁ being therefore reduced to their terms of degree <1 in theE′ _j ,E′ _j ,J′ _j . Calculation of the principal partF _1m ofF ₁ is carried out through Laplace coefficients and operatorD=α(d/dα) applied to Laplace coefficients, α ratio of the semi major axis ofP ₁ andP ₂. Eccentricitye ₂ of the disturbed planetP ₂ is assumed to be zero, such an assumption not restricting our aim which is to investigate the mechanism of the elimination of short period terms in a second order general planetary theory carried out through the Hori's method, not to perform the elimination of those terms for a complete second order general planetary theory. Expressions of the slow Delaunay canonical variables in terms of the new ones resulting from the elimination of the short period terms ofF ₁ are written down only for the disturbed planetP ₁.
2. Small divisors in 1/E′ ₁ and 1/E′ ₁ ² appear in the longitude ?₁ of perihelia ofP ₁. No small divisors appear in the other five slow Delaunay variables ofP ₁. The only Jacques Henrard variables which appear in the longitude Ω₁ of the ascending node ofP ₁ are the J _j′ j=1, 2 and no Jacques Henrard variables appear in the slow Delaunay canonical variablesX ₁,Y ₁,Z ₁, λ₁. The solving of the ten canonical equations ofP ₁ andP ₂ in the slow Delaunay canonical variablesX′ _j ,Y′ ₁,Z′ _j ,λ′ _j ,ω′ _j ,Ω′ _j resulting from the elimination of the short period terms ofF ₁ reduces to that of four canonical equations inZ′ _j ,©′ _j and to six quadratures three of them expressing theX′ _j ,Y′ ₁ are constants and the three others expressingλ′ _j ,?′ _j as functions of timet. Solving of the four canonical equations inZ′ _j ,Ω′ _j reduces to that of a first order non linear differential equation and to two quadratures. Sinceγ′ ₁ is then constant, so is the Jacques Henrard variableE′ ₁. If the eccentricitye ₂ ofP ₂ is no more assumed to be zero, additive small divisors inE′ ₂/E′ ² ₁ appear in longitude ?′₁ of perihelia ofP ₁ and the solving of the twelve canonical equations ofP ₁ andP ₂ inX′ _j ,Y′ _j ,Z′ _j ,λ′ _j ,?′ _j ,Ω′ _j is reduced to that of eight canonical equations inY′ _j ,?′ _j ,Z′ _j ,Ω′ _j and to four quadratures expressingX′ _j are constants andλ′ _j as functions oft. Those eight canonical equations split into two systems of four canonical equations, one of them inY′ _j ,?′ _j and the other one inZ′ _j ,Ω′ _j . Each of those two systems is identical to the system inZ′ _j ,Ω′ _j corresponding toe ₂=0 and its solving reduces to that of a first order non linear differential equation and to two quadratures identical to those of the casee ₂=0.
3. Expressions ofX ₁,Y ₁,Z ₁,λ ₁,? ₁,Ω ₁ as functions ofX′ _j ,Y′ ₁,Z′ _j ,λ′ _j ,?′ ₁,Ω′ _j ;j=1, 2 are sums of sines and cosines of the multiples ofλ′ _j ,?′ ₁,Ω′ _j for the terms arising from the indirect partF _1j ofF ₁, Fourier series in those sines and cosines or products of two such Fourier series for the terms arising from the principal partF _1m ofF ₁, coefficients of those sums and Fourier series having one of the eight forms: $$A,{\text{ }}\frac{B}{{E'}},{\text{ }}\frac{C}{{E'^2 }},{\text{ }}D\frac{{j'^{2_1 } }}{{E'^{2_1 } }},{\text{ }}E\frac{{j'^{2_2 } }}{{E'^{2_1 } }},{\text{ }}F\frac{{j'^{_1 } j'^2 }}{{E'^{2_1 } }},{\text{ }}G\frac{{j'^2 }}{{j'^{_1 } }},{\text{ }}H\frac{{j'^{22} }}{{j'^{2_1 } }}{\text{.}}$$ A,..., H being constants which depend upon ratio α. Numerical calculation of the constantsA,..., H arising from the terms ofF _1j is easily carried out; that of theA,..., H arising from the terms ofF _1m require more manipulations, Fourier series in sines and cosines of the multiples ofλ′ _j ,?′ _j ,Ω _ij and products of two such Fourier series having then to be reduced to sums of a finite number of terms and treated through the methods of harmonic analysis. Divisors inp+qα^3/2;p, q relative integers, or products of such divisors appear inA,..., H.
4. the method extends to the case whenF ₁ is calculated beyond its terms of degree 3 in the Jacques Henrard variables.F ₁ being calculated up to its terms of degree 8 in the Jacques Henrard variables which is the precision required to eliminate the short period terms of a complete second order general planetary theory,S ₂ has to be calculated up to its terms of degree 7 and the expression of the slow Delaunay canonical variables ofP ₁ andP ₂ in terms of the slow Delaunay canonical variables ofP ₁ andP ₂ resulting from the elimination of the short period terms ofF ₁ have, therefore, to be calculated up to their terms of degree 5 in the Jacques Henrard variables.

     

Trigonometric series representations of the orbital inclination function

In this paper, new trigonometric series representations of the orbital inclination functionF _imp (i) in multiples of cosines or sines will be established for all possible values ofl, m, andp. For such representations, the literal analytical expressions and the recurrence formulae satisfied by their coefficients will be established. Moreover, an economic algorithm for the table formulation of these coefficients for the possible values ofl, m, andp is given. Finally, numerical examples of the representations forl=2(1)4;m=0(1)l;p=0(1)l are also included.     

Faint galaxies: merging model of E/SO galaxies

We study a model of mergers affecting only the progenitors of present E/SO. We adopt the standard scenarios of star formation as prescribed by Guiderdoni & Rocca-Volmerange. The merging process is parametrized under the assumptions of(1) self-similarity of the Schechter MF and(2) mass conservation. Nine models are discussed. The predictions are compared with counts ofB _J ,U ⁺,F ⁺,N ⁺ bands. E/SO mergers account for the excess of the faintest blue galaxies without causing excess in redder bands. However, as we no longer have enough mergers at brighter magnitudes, a plain E/SO merging model fits less tightly for the redshift and the colour distributions. Detection effect, a steeper slope of LF may be ways to improve. Our models predict acceptable merger frequencies atz = 0.5 although some models predict more interacting galaxies than observation atz = 0.     

Fourier analysis of the light curves of eclipsing variables,XXII

The theoretical values of the momentsA _2m for any type of eclipses, expressed in terms of the elementsL ₁,a andc ₀, have been derived in the simple forms of rapidly convergent expansions to the series of Chebyshev polynomials, Jacobi polynomials and KopalJ-integrals (Kopal, 1977c) and hold good for any real (not necessarily integral) value ofm0.The aim of the present paper has been to establish explicit expressions for the Jacobian and its fast enough computation in the light changes of close eclipsing systems, arising from the partial derivative of different pairs ofg-functions (Kopal and Demircan, 1978, Paper XIV) with respect toa andc ₀ ² , for any type of eclipses (be these occultations or transit, partial, total or annular) and for any arbitrary degreel of the adopted law of limb-darkening. The functional behaviour of this Jacobian would determine the reasonable light curve in connection with geometrical determinacy of the parametersa andc ₀. In the expansion of Jacobian, the terms consist of two polynomials which satisfy certain three-term recursion relations having the eclipse parametersa andc ₀, as their arguments.Closed form expressions forf-functions, as well as of the Jacobian (e.g.,m=1, 2, 3), obtaining in the case of total eclipses, are given for a comparative discussion with the theoretical values of Jacobian derived from partial derivative of different pairs ofg-functions.The numerical magnitude of Jacobian would determine the best combination of the momentsA _2m in the different pairs ofg-functions and definite results would follow in the subsequent paper of this series (Edalati, 1978c, Paper XXIV).     

Third-order Jupiter — Saturn planetary theory

The construction of a third order J-S theory is presented. The Hori theory of planetary perturbations is employed. No Critical J-S terms due to the 2:5 commensurabilities and its multiples exist, when we take into account the periodic terms of order 0, 1, 2 with respect to the eccentricity- inclination. In this case the Lie series transformation degenerates and is meaningless. The J-S equations of motion for secular perturbations are solved when we neglect in our treatment, the Poisson terms of degree > 2 in the Poincaré canonical variables H _u , K _u , P _u Q _u (u = 1, 2). The Jacobi-Radau referential is adopted, and the theory is expressed in terms of the canonical variables of H. Poincaré.Now at the Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California, U.S.A.     

General time elements for the two body problem

When integrating a perturbed two-body problem, very often the propagation of the numerical error is reduced by using a new time variables defined by dt/ds=|q| ⁿ , (|q| is the radial distance,t the time). This paper introduces a time element for such transformations, i.e., a new variablet _n is defined so that dt _n/ds=1+ (perturbing terms) andt=F _n(t_n), whereF _n is a known function. The time element equation should be useful in reducing the error in the determination of the timet.F _n is given explicitly forn=1, 3/2, 2, 5/2 and 3, and a general expression is given for other values.The work was performed while the author was an NRC Senior Research Associate, Goddard Space Flight Center, Greenbelt, Md., U.S.A.     

Fourier analysis of the light curves of eclipsing variables,XXIV

The aim of the present paper will be to evaluate numerically Jacobian and other functions which have been discussed in more detail in a previous paper of this series (Edalati, 1978b, Paper XXII), and also choose the most convenient moments to obtain a good determination for the unknown eclipse parametersa andc ₀. More than 12 different pairs ofg-functions for real values ofm have been investigated numerically and diagrammatically. The behaviour ofg-functions depends but very little on different combination of the moments, and related diagrams are approximately the same asg ₂ andg ₄ (Kopal and Demircan, 1978, Paper XIV).The behaviour of the vanishing Jacobian, arising from different pairs ofg-functions for real values ofm1 has been shown diagrammatically in terms ofa andc ₀. Accordingly, we obtain the optimum combination of the moments (i.e.,A ₆,A ₇,A ₈ andA ₉) ing-functionsg ₇ andg ₈. It has been noted that the behaviour of theg-functions which depend on the combinations of the higher order moments (i.e.,m5) have been ruled out, because the proportional error of the momentsA _2m increases with increasing values of realm. The automated method has been tested successfully on the light curve of RT Per (Mancusoet al., 1977; Edalati, 1978a). Finally, a comparison is given of the elements of RT Per arising from two different pairs ofg-functions, i.e.,g ₂,g ₄ (Edalati, 1978a) andg ₇,g ₈ for the light curves analysis.     

Spectra of the ammonium radical: the schüler bands

The main bands of the Schüler system of ND₄ and NH₄ have been observed at high resolution. On the basis of these spectra, Watson, in a separate paper, has analysed the ND₄ main band showing that it represents a²F₂ → ² A₁ transition of a tetrahedral molecule. The observed wavenumber data for both ND₄ and NH₄ are presented; the latter have not yet been analysed. Isotopic bands for¹⁵ND₄,¹⁴ND₃H,¹⁴ND₂H₂,¹⁴NDH₃ have also been obtained and as previously pointed out confirm the assumed carrier of the spectrum. The much weaker bands accompanying the main Schüler band on the short and long wavelength sides are photographed at medium resolution. The interpretation of these bands in terms of the vibrational levels of upper and lower states is briefly discussed.     

THE MILLS,NEW MEXICO,CHONDRITE — A NEW FIND

A mineralogical and chemical analysis has been performed on a slice of the Mills, New Mexico, chondrite, which was found in August 1970. The mineralogical composition is olivine Fa_19.5, bronzite, Fs_17.4, plagioclase, nickel-iron, troilite and ilmenite. The chemical analysis confirms that Mills is a typical bronzite, H5 chondrite, with considerable degree of weathering, as indicated by the presence of large amounts of Fe₂O₃. A comparison with other stones found in the same New Mexico region by Nininger may show a possible relation between Mills and previous finds.     

Chromospheric downflow velocities as a diagnostic in solar flares

We explore the dynamics of chromospheric condensations driven by evaporation during the impulsive phase of solar flares. Specifically, we find that the maximum chromospheric downflow speed obeys the approximate relation υ_d= 0.4 (F/ϱ_ch)^1/3, where F is that part of the flare energy flux driving chromospheric evaporation, and ϱ_ch is the mass density in the preflare chromosphere just below the preflare transition region. This implies that chromospheric downflows as measured by Hα asymmetries may be a powerful probe of flare energetics.

     

The curious case of the greenwich faculae

We analyze the record of facular areas compiled by the Royal Greenwich Observatory (RGO) from daily white-light observations between 1874 and 1976. Curiously, the relative amplitudes of the three largest sunspot cycles 17, 18, and 19 in this record are reversed when they are ranked by facular area. We show that this negative correlation arises from a general decrease of the ratioA _F/A _S, of facular to sunspot area, with increasingA _S. Within a given cycle,A _F/A _Sdecreases in active regions of largeA _S, butA _F/A _Sis also lower at allA _S, in cycles of higher peak amplitude inA _S. This decrease ofA _F/A _Sin large spot groups is consistent with its decrease in younger, more active solar-mass stars, and it may explain why stars only slightly more magnetically active than the Sun tend to exhibit much greater variability in broad-band photometry. We suggest that the physical explanation is an increased spatial filling factor of magnetic flux, favoring formation of sunspots over faculae. We also explain why the decrease inA _F/A_Sis not seen in the disc-integrated Ca K plage areas, nor in theF10.7 microwave index, both of which exhibit remarkable linearity when plotted against smoothed sunspot area. This explanation suggests how complementary data on faculae and plages from RGO and Mt. Wilson could be used to improve empirical models of total irradiance variation, extending back to 1874.     

Fourier techniques for an analysis of eclipsing binary light curves. VIa

The aim of the present paper is to deduce relations between the integral transformsA _2m, B_2m,andF _1,2 of the light curves of eclipsing binary systems. The integral transformsA _2m, B_2m,andF _1,2 have been related to one another by means of finite or rapidly converging infinite summations obtained by integrations of the series expansions of trigonometric functions.     

A second order Jupiter-Saturn planetary theory

We present a second order secular Jupiter-Saturn planetary theory through Poincaré canonical variables, von Zeipel's method and Jacobi-Radau referential. We neglect in our expansions terms of power higher than the fourth with respect to eccentricities and sines of inclinations. We assume that the disturbing function is composed of secular and critical terms only. We shall deriveF _2si and writeF _2s in terms of Poincaré canonical variables in Part II of this problem.     

Time-dependent chemistry in expanding circumstellar envelopes

The rate equations of a restricted set of gas-phase chemical reactions occuring in an expanding circumstellar envelope are integrated numerically on the assumption that no chemical evolution has occurred in the stellar atmosphere. Abundances of all species are found to peak at a time on the order ofr ₀/u ₀, wherer ₀ is the initial radius andu ₀ the expansion velocity. After this time geometrical dilution dominates. For an initial density of 10⁸ cm^–3, on the order of 1% of hydrogen is converted to H₂, and CH and CO have comparable densities of 10^–6 relative to H, OH and O₂ remain very low in abundance. For higher initial densities, H and H₂ are more nearly comparable, and nearly all carbon is in CO, CH, OH and O₂ remain low in abundance. The relevance of these results to M giants and other objects is discussed.     

OnH-functions of radiative transfer

Chandrasekhar'sH-functionH(z) corresponding to the dispersion functionT(z)=| _rs –frs(z)|, where [f _rs (z)] is of rank 1, is obtained in terms of a Cauchy integral whose density functionQ(x, ₁, ₂,...) can be approximated by approximating polynomials (uniformly converging toQ(x)) having their coefficients expressed as known functions of the parameters _r 's. A closed form approximation ofH(z) to a sufficiently high degree of accuracy is then readily available by term by term integration.     

The elimination of the critical terms of a first order Uranus-Neptune theory by Hori's method

In this part we determine the value ofS ₁, and in terms of the canonical variables of H. Poincaré. A complete solution of the auxiliary system of equations generated by the Hamiltonian is presented.     

Evolutionary considerations involving the internal density concentration parameter of binary stars

The evolutionary changes that occur in the internal density concentration parameterk ₂ (called the apsidal constant for brevity) for a star of given mass and initial composition are examined in detail. The purpose is to ascertain whether or not such an approach leads to a reduction in the differences now noted between the theoretically derived values ofk ₂ and the observed values derived from the secular advance of the periastron in close eclipsing binary systems.A series of stellar models of mass 2.0, 5.0, 10.0 and 20.0M were employed, with an initial compositional mixture ofX=0.739,Y=0.24 andZ=0.021. These models cover an evolutionary range from a point in time where the star has just completed the Hayashi phase of its pre-Main Sequence contraction through its entire Main Sequence phase to the point where hydrogen depletion in the core is complete.For each model, a value ofk ₂ is determined by numerically integrating Radau's equation and using the values of , the ratio of the star's density at pointa to its mean density, as taken from the models. The result is the time history ofk ₂ for each stellar mass over the evolutionary range of interest. The results are then summarized in the (logk ₂, logT _e) plane which, for the first time, quantitatively indicates the variation ink ₂ as a function of the evolutionary state of the star.A comparison between these theoretically derived values ofk ₂ and a selected set of observationally determined ones immediately indicates that the secular variation ink ₂ plays an extremely important part in any comparison between theory and observation. For most of the cases studied, the difference between the theoretically and observationally determined values ofk ₂ can be reconciled in terms of the evolutionary history of the binary system.While tentatively providing a satisfactory explanation for the previously noted differences in the determination ofk ₂, there now exists the problem of accurately pinpointing the evolutionary state of the observed binary system.     

The influence of satellite flexibility on orbital motion

The orbital perturbations induced by the librational motion and flexural oscillations are studied for satellites having large flexible appendages. Using a Lagrangian procedure, the equations for coupled motion are derived for a satellite having an arbitrary number of appendages in the nominal orbital plane and two flexible members normal to it. The formulation enables one to study the influence of flexibility on both the orbital and attitude motions. The orbital coordinates are expanded as perturbation series in =(l/a ₀)²,l anda ₀ being a characteristic length of the satellite and unperturbed semi-major axis of the orbit, respectively. The first order perturbation equations are solved in terms of elastic deformations and librational angles using the WKBJ method in conjunction with the variation of parameter technique. Existence of secular perturbations is noted for certain librational flexural motions. Three specific examples, Alouette II, Radio Astronomy Explorer and Tethered Orbiting Interferometer, are considered subsequently and their possible secular drifts estimated.List of Symbols A _ij, B_ij coefficients in the eigenfunction expansion ofv _i andw _i respectively, Equation (10) - C _k, D_k constants, Equation (21) - EI _i flexural rigidity of theith appendage - E(u₀) ²(1+e ₀ cosu ₀)² h ₀ ³ - F(u₀) perturbation function, Equation (17b) - F ,F ,F functions of librational angles and flexural displacements, Equation (11i) - F ,F ,F F ,F ,F with change of independent variable fromt tou ₀ - I _xx, I_yy, I_zz principal moments of inertia of the undeformed satellite - [J ⁱ] inertia dyadic of the deformedith appendage - [J _d] inertia dyadic of the deformed satellite - M mass of the satellite - P _R, P_u functions of librational angles and flexural displacements, Equation (15d) and (15e), respectively - R _c magnitude ofR _c - R _c0, R₁ unperturbed value and first order perturbation ofR _c, respectively - R _c ,R ₀ position vectors of the c.m. of the deformed and undeformed satellite, respectively - T kinetic energy of the satellite - U potential energy of the satellite - U _e, U_g elastic and gravitational potential energy, respectively - X, Y, Z orbital co-ordinate axes, located at the c.m. of the deformed satellite - Y ₁(u₀), Y₂(u₀) functions ofu ₀, Equation (18b) and (18c), respectively - a semi-major axis - a ₀ unperturbed value ofa - e eccentricity - e ₀ unperturbed value ofe - h ₀ unperturbed angular momentum per unit mass of the satellite - i inclination of the orbital plane to the ecliptic - i, j, k unit vectors alongx (or ),y (or ) andz (or ) axes, respectively - l characteristic length of the satellite - l _i length of theith appendage - [l ⁱ] matrix of direction cosines ofx _i, v_i andw _i - l ,l ,l direction cosines ofR _c - m ₀, m_i mass of the main body andith appendage, respectively - p _i ² - q _m, Q_m generalized co-ordinate and force, respectively - r ₁ R ₁/R_c0 - r position vector of an element of the body referred toxyz axes - r _u position vector of an element after deformation, referred to axes - r _c x _c i+y _c j+z _c k, position vector of the c.m. of the deformed body referred toxyz axes - s x _i/l_i - t time - u true anomaly - u ₀, u₁ unperturbed value and the first order perturbation ofu, respectively - u elastic displacement vector - u _c u–r _c - velocity of an element relative to axes - v _i, w_i flexural deformations - x, y, z body co-ordinate axes with origin at the c.m. of the undeformed satellite - x _i distance of an element of theith appendage from the root - _j jth eigenfunction (normalized) of a cantilever - angle between the line of nodes and vernal equinox - , , components of nondimensionalized angular velocity of the satellite - , , pitch (spin), yaw and roll, respectively - _i nominal inclination of theith appendage in the orbital plane - - small parameter, (l/a ₀)² - _j jth eigenvalue of a cantilever - gravitational constant - _jk constant, Equation (11j) - , , body co-ordinate axes with origin at the c.m. of the deformed satellite - ( i + j + k), angular velocity of the satellite     

Ultraviolet observations of cataclysmic variables

The preliminary results of the analysis of more than 1000 spectra of cataclysmic variables in the archive of the International Ultraviolet Explorer were presented at the meeting. To characterize the slope of the spectra I useF=log(f _1460Å/f _2880Å). For most spectraF lies between 0.2 and 0.7. No correlation of F with orbital period, inclination, system type or (for dwarf novae) length of the interoutburst interval are found, apart from somewhat lower values ofF for DQ Her type systems. Out of 16 dwarf novae for which spectra both at outburst maximum and minimum are available 11 show no large difference inF between maximum and minimum, and in 5F declines with the flux level. Out of 6 dwarf novae 5 show very red spectra during the rise to maximum, and 1 shows slopes during rise similar to those during decline.In the ultraviolet resonance lines, due to a wind from the disc, no correlation is found between inclination and terminal velocity.Paper presented at the IAU Colloquium No. 93 on Cataclysmic Variables. Recent Multi-Frequency Observations and Theoretical Developments, held at Dr. Remeis-Sternwarte Bamberg, F.R.G., 16–19 June, 1986.     

Fourier analysis of the light curves of eclipsing variables,XV

A new general expression for the theoretical momentsA _2m of the light curves of eclipsing systems has been presented in the form of infinite series expansion. In this expansion, the terms have been given as the product of two different polynomials which satisfy certain three-term recursion formulae, and the coefficients diminish rapidly with increasing number of terms. Thus, the numerical values of the theoretical momentsA _2m can be generated recursively up to four significant figures for any given set of eclipse elements. This can be utilized to solve the eclipse elements in two ways: (i) with an indirect method (for the procedures see Paper XIV, Kopal and Demircan, 1978), (ii) with a direct method as minimization to the observational momentsA _2m (area fitting). The procedures given in Paper XIV for obtaining the elements of any eclipsing system consisting of spherical stars have been automated by making use of the new expression for the momentsA _2m of the light curves. The theoretical functionsf ₀,f ₂,f ₄,f ₆,g ₂ andg ₄ which are the functions ofa andc ₀, have been used to solve the eclipse elements from the observed photometric data. The closed-form expressions for the functionsf ₂,f ₄ andf ₆ have also been derived (Section 3) in terms of Kopal'sI-integrals.The automated methods for obtaining the eclipse elements from one minimum alone have been tested on the light curves of YZ (21) Cassiopeiae under the spherical model assumptions. The results of these applications will be given in Section 5 which follows a brief introduction to the procedure we followed.