On the convergence of an algorithm constructing the normal form for elliptic lower dimensional tori in planetary systems

We give a constructive proof of the existence of elliptic lower dimensional tori in nearly integrable Hamiltonian systems. In particular we adapt the classical Kolmogorov normalization algorithm to the case of planetary systems, for which elliptic tori may be used as replacements of elliptic Keplerian orbits in Lagrange-Laplace theory. With this paper we support with rigorous convergence estimates the semi-analytic work in our previous article (Sansottera et al., Celest Mech Dyn Astron 111:337–361, 2011), where an explicit calculation of an invariant torus for a planar model of the Sun-Jupiter-Saturn-Uranus system has been made. With respect to previous works on the same subject we exploit the characteristic of Lie series giving a precise control of all terms generated by our algorithm. This allows us to slightly relax the non-resonance conditions on the frequencies.     

Spectroscopic studies of yellow supergiants in the Cepheid instability strip

High-resolution spectra of nine yellow nonvariable supergiants (NVSs) located within the canonical Cepheid instability strip from Sandage and Tammann (1969) (α Aqr, ? Leo, μ Per, ω Gem, BD+60 2532, HD 172365, HD 187299, HD 190113, and HD 200102) were taken with the 1-m Zeiss and 6-m BTA telescopes at the Special Astrophysical Observatory of the Russian Academy of Sciences in the 1990s. These have been used to determine the atmospheric parameters, chemical composition, radial velocities, reddenings, luminosities, distances, and radii. The spectroscopic estimates of T _eff and the luminosities determined from the Hipparcos parallaxes have shown eight of the nine program NVSs on the T _eff?log(L/L _⊙) diagram to be outside the canonical Cepheid instability strip. When the edges of the Cepheid instability strip from Bono et al. (2000) are used, out of the NVSs from the list on the diagram one is within the Cepheid instability strip but closer to the red edge, two are at the red edge, three are beyond the red edge, two are at the blue edge, and one is beyond the blue edge. The evolutionary masses of the objects have been estimated. The abundances of α-elements, r- and s-process elements for all program objects have turned out to be nearly solar. The СNO, Na, Mg, and Al abundance estimates have shown that eight of the nine NVSs from the list have already passed the first dredge-up. Judging by the abundances of the key elements and its position on the T _eff?log(L/L _⊙) diagram, the lithium-rich supergiant HD 172365 is at the post-main-sequence evolutionary stage of gravitational helium core contraction and moves toward the first crossing of the Cepheid instability strip. The star ? Leo should be assigned to bright supergiants, while HD 187299 and HD 190113 may have already passed the second dredge-up and move to the asymptotic branch.     

The <Emphasis Type="Italic">z</Emphasis> distribution of hydrogen clouds and masers with kinematic distances

Data on HII regions, molecular clouds, and methanol masers have been used to estimate the Sun’s distance from the symmetry plane z _⊙ and the vertical disk scale height h. Kinematic distance estimates are available for all objects in these samples. The Local-arm (Orion-arm) objects are shown to affect noticeably the pattern of the z distribution. The deviations from the distribution symmetry are particularly pronounced for the sample of masers with measured trigonometric parallaxes, where the fraction of Local-arm masers is large. The situation with the sample of HII regions in the solar neighborhood is similar. We have concluded that it is better to exclude the Local arm from consideration. Based on the model of a self-gravitating isothermal disk, we have obtained the following estimates from objects located in the inner region of the Galaxy (R ≤ R ₀): z _⊙ = ?5.7 ± 0.5 pc and h ₂ = 24.1 ± 0.9 pc from the sample of 639 methanol masers, z _⊙ = ?7.6±0.4 pc and h ₂ = 28.6±0.5 pc from 878HII regions, z _⊙ = ?10.1 ± 0.5 pc and h ₂ = 28.2 ± 0.6 pc from 538 giant molecular clouds.     

Correlation function of quasars from SDSS DR3

We calculate the parameters of the two-point correlation function of quasars w(r) = (r _c /r) ^γ on the basis of the SDSS DR3 data. The correlation functions are first determined from projected distances with the use of a special technique for compiling randomized catalogs. Next the parameters of the spatial correlation function are obtained with the assumption of local isotropy. For the quasars with redshifts z = 0.8–2.1, we obtained the estimates γ = 1.76 ± 0.14, r _c = 6.60 ± 0.85 h ^?1 Mpc in the comoving distance range 2–30 Mpc and γ = 1.90 ± 0.11, r _c = 6.95±0.57 h ^?1 Mpc in the range 2–50 Mpc. These estimates agree, within the limits of errors, with the estimates obtained for the redshifts 0.4 < z < 2.1. The original catalog shows some deficit of pairs with separations less than 1 Mpc.     

An analysis of the convergence of Newton iterations for solving elliptic Kepler’s equation

In this note a study of the convergence properties of some starters \( E_0 = E_0(e,M)\) in the eccentricity–mean anomaly variables for solving the elliptic Kepler’s equation (KE) by Newton’s method is presented. By using a Wang Xinghua’s theorem (Xinghua in Math Comput 68(225):169–186, 1999) on best possible error bounds in the solution of nonlinear equations by Newton’s method, we obtain for each starter \( E_0(e,M)\) a set of values \( (e,M) \in [0, 1) \times [0, \pi ]\) that lead to the q-convergence in the sense that Newton’s sequence \( (E_n)_{n \ge 0}\) generated from \( E_0 = E_0(e,M)\) is well defined, converges to the exact solution \(E^* = E^*(e,M)\) of KE and further \( \vert E_n - E^* \vert \le q^{2^n -1}\; \vert E_0 - E^* \vert \) holds for all \( n \ge 0\). This study completes in some sense the results derived by Avendaño et al. (Celest Mech Dyn Astron 119:27–44, 2014) by using Smale’s \(\alpha \)-test with \(q=1/2\). Also since in KE the convergence rate of Newton’s method tends to zero as \( e \rightarrow 0\), we show that the error estimates given in the Wang Xinghua’s theorem for KE can also be used to determine sets of q-convergence with \( q = e^k \; \widetilde{q} \) for all \( e \in [0,1)\) and a fixed \( \widetilde{q} \le 1\). Some remarks on the use of this theorem to derive a priori estimates of the error \( \vert E_n - E^* \vert \) after n Kepler’s iterations are given. Finally, a posteriori bounds of this error that can be used to a dynamical estimation of the error are also obtained.     

Systematic error of the Gaia DR1 TGAS parallaxes from data for the red giant clump

Based on the Gaia DR1 TGAS parallaxes and photometry from the Tycho-2, Gaia, 2MASS, andWISE catalogues, we have produced a sample of ~100 000 clump red giants within ~800 pc of the Sun. The systematic variations of the mode of their absolute magnitude as a function of the distance, magnitude, and other parameters have been analyzed. We show that these variations reach 0.7 mag and cannot be explained by variations in the interstellar extinction or intrinsic properties of stars and by selection. The only explanation seems to be a systematic error of the Gaia DR1 TGAS parallax dependent on the square of the observed distance in kpc: 0.18R ² mas. Allowance for this error reduces significantly the systematic dependences of the absolute magnitude mode on all parameters. This error reaches 0.1 mas within 800 pc of the Sun and allows an upper limit for the accuracy of the TGAS parallaxes to be estimated as 0.2 mas. A careful allowance for such errors is needed to use clump red giants as “standard candles.” This eliminates all discrepancies between the theoretical and empirical estimates of the characteristics of these stars and allows us to obtain the first estimates of the modes of their absolute magnitudes from the Gaia parallaxes: mode(M _H ) = ?1.49 ^m ± 0.04 ^m , mode(M _Ks ) = ?1.63 ^m ± 0.03 ^m , mode(M _W1) = ?1.67 ^m ± 0.05 ^m mode(M _W2) = ?1.67 ^m ± 0.05 ^m , mode(M _W3) = ?1.66 ^m ± 0.02 ^m , mode(M _W4) = ?1.73 ^m ± 0.03 ^m , as well as the corresponding estimates of their de-reddened colors.     

A Check on the Validity of Magnetic Field Reconstructions

We investigate a method to test whether a numerically computed model coronal magnetic field \({\boldsymbol {B}}\) departs from the divergence-free condition (also known as the solenoidality condition). The test requires a potential field \({\boldsymbol {B}}_{0}\) to be calculated, subject to Neumann boundary conditions, given by the normal components of the model field \({\boldsymbol {B}}\) at the boundaries. The free energy of the model field may be calculated using \(\frac{1}{2\mu _{0}}\int ({\boldsymbol {B}}-{\boldsymbol {B}}_{0})^{2}\mathrm{d}V\), where the integral is over the computational volume of the model field. A second estimate of the free energy is provided by calculating \(\frac{1}{2\mu _{0}}\int {\boldsymbol {B}}^{2}\,\mathrm{d}V-\frac{1}{2\mu _{0}}\int {\boldsymbol {B}}_{0}^{2}\,\mathrm{d}V\). If \({\boldsymbol {B}}\) is divergence free, the two estimates of the free energy should be the same. A difference between the two estimates indicates a departure from \(\nabla \cdot {\boldsymbol {B}}=0\) in the volume. The test is an implementation of a procedure proposed by Moraitis et al. (Solar Phys.289, 4453, 2014) and is a simpler version of the Helmholtz decomposition procedure presented by Valori et al. (Astron. Astrophys.553, A38, 2013). We demonstrate the test in application to previously published nonlinear force-free model fields, and also investigate the influence on the results of the test of a departure from flux balance over the boundaries of the model field. Our results underline the fact that, to make meaningful statements about magnetic free energy in the corona, it is necessary to have model magnetic fields that satisfy the divergence-free condition to a good approximation.     

Is the Sun a Magnet?

It has been argued (Gough and McIntyre in Nature 394, 755, 1998) that the only way for the radiative interior of the Sun to be rotating uniformly in the face of the differentially rotating convection zone is for it to be pervaded by a large-scale magnetic field, a field which is responsible also for the thinness of the tachocline. It is most likely that this field is the predominantly dipolar residual component of a tangled primordial field that was present in the interstellar medium from which the Sun condensed (Braithwaite and Spruit in Nature 431, 819, 2004), and that advection by the meridional flow in the tachocline has caused the dipole axis to be inclined from the axis of rotation by about \(60^{\circ}\) (Gough in Geophys. Astrophys. Fluid Dyn., 106, 429, 2012). It is suggested here that, notwithstanding its turbulent passage through the convection zone, a vestige of that field is transmitted by the solar wind to Earth, where it modulates the geomagnetic field in a periodic way. The field variation reflects the inner rotation of the Sun, and, unlike turbulent-dynamo-generated fields, must maintain phase. I report here a new look at an earlier analysis of the geomagnetic field by Svalgaard and Wilcox (Solar Phys. 41, 461, 1975), which reveals evidence for appropriate phase coherence, thereby adding support to the tachocline theory.     

Determination of the combination of cosmological parameters Ω<Subscript><Emphasis Type="Italic">m</Emphasis></Subscript><Superscript><Emphasis Type="Italic">α</Emphasis></Superscript><Emphasis Type="Italic">σ</Emphasis><Subscript>8</Subscript>

We have obtained new constraints on the cosmological parameters Ω _m and σ ₈ from the peculiar velocities of flat edge-on spiral galaxies from the RFGC catalog. Based on these results presented graphically, we have found the quantitative condition (Ω _m /0.3)^0.37 σ ₈ = 0.92 ± 0.05. The estimates of Ω _m and σ ₈, along with their combinations Ω _m ^α σ ₈ for various α, are compared with the estimates by other authors.     

Line-of-Sight Observables Algorithms for the <Emphasis Type="Italic">Helioseismic and Magnetic Imager</Emphasis> (HMI) Instrument Tested with <Emphasis Type="Italic">Interferometric Bidimensional Spectrometer</Emphasis> (IBIS) Observations

The Helioseismic and Magnetic Imager (HMI) instrument onboard the Solar Dynamics Observatory produces line-of-sight (LOS) observables (Doppler velocity, magnetic-field strength, Fe i line width, line depth, and continuum intensity) as well as vector magnetic-field maps at the solar surface. The accuracy of LOS observables is dependent on the algorithm used to translate a sequence of HMI filtergrams into the corresponding observables. Using one hour of high-cadence imaging spectropolarimetric observations of a sunspot in the Fe i line at 6173 Å through the Interferometric Bidimensional Spectrometer installed at the Dunn Solar Telescope, and the Milne–Eddington inversion of the corresponding Stokes vectors, we test the accuracy of the observables algorithm currently implemented in the HMI data-analysis pipeline: the MDI-like algorithm. In an attempt to improve the accuracy of HMI observables, we also compare this algorithm to others that may be implemented in the future: a least-squares fit with a Gaussian profile, a least-squares fit with a Voigt profile, and the use of second Fourier coefficients in the MDI-like algorithm.     

Fine-resolution analysis of exoplanetary distributions by wavelets: hints of an overshooting iceline accumulation

We investigate 1D exoplanetary distributions using a novel analysis algorithm based on the continuous wavelet transform. The analysis pipeline includes an estimation of the wavelet transform of the probability density function (p.d.f.) without pre-binning, use of optimized wavelets, a rigorous significance testing of the patterns revealed in the p.d.f., and an optimized minimum-noise reconstruction of the p.d.f. via matching pursuit iterations.In the distribution of orbital periods, \(P\), our analysis revealed a narrow subfamily of exoplanets within the broad family of “warm Jupiters”, or massive giants with \(P\gtrsim 300~\mbox{d}\), which are often deemed to be related with the iceline accumulation in a protoplanetary disk. We detected a p.d.f. pattern that represents an upturn followed by an overshooting peak spanning \(P\sim 300\mbox{--}600~\mbox{d}\), right beyond the “period valley”. It is separated from the other planets by p.d.f. concavities from both sides. It has at least 2-sigma significance.In the distribution of planet radii, \(R\), and using the California Kepler Survey sample properly cleaned, we confirm the hints of a bimodality with two peaks about \(R=1.3R_{\oplus }\) and \(R=2.4R_{ \oplus }\), and the “evaporation valley” between them. However, we obtain just a modest significance for this pattern, 2-sigma only at the best. Besides, our follow-up application of the Hartigan and Hartigan dip test for unimodality returns 3 per cent false alarm probability (merely 2.2-sigma significance), contrary to 0.14 per cent (or 3.2-sigma), as claimed by Fulton et al. (2017).     

Shannon Entropy-Based Prediction of Solar Cycle 25

A new model is proposed to forecast the peak sunspot activity of the upcoming solar cycle (SC) using Shannon entropy estimates related to the declining phase of the preceding SC. Daily and monthly smoothed international sunspot numbers are used in the present study. The Shannon entropy is the measure of inherent randomness in the SC and is found to vary with the phase of an SC as it progresses. In this model each SC with length \(T_{\mathrm{cy}}\) is divided into five equal parts of duration \(T_{\mathrm{cy}}/5\). Each part is considered as one phase, and they are sequentially termed P1, P2, P3, P4, and P5. The Shannon entropy estimates for each of these five phases are obtained for the \(n\)th SC starting from \(n=10\,\mbox{--}\,23\). We find that the Shannon entropy during the ending phase (P5) of the \(n\)th SC can be efficiently used to predict the peak smoothed sunspot number of the \((n+1)\)th SC, i.e. \(S_{\mathrm{max}}^{n+1}\). The prediction equation derived in this study has a good correlation coefficient of 0.94. A noticeable decrease in entropy from 4.66 to 3.89 is encountered during P5 of SCs 22 to 23. The entropy value for P5 of the present SC 24 is not available as it has not yet ceased. However, if we assume that the fall in entropy continues for SC 24 at the same rate as that for SC 23, then we predict the peak smoothed sunspot number of 63±11.3 for SC 25. It is suggested that the upcoming SC 25 will be significantly weaker and comparable to the solar activity observed during the Dalton minimum in the past.     

Kinematics of B-F Stars as a Function of Their Dereddened Color from Gaia and PCRV Data

Parallaxes with an accuracy better than 10% and proper motions from the Gaia DR1 TGAS catalogue, radial velocities from the Pulkovo Compilation of Radial Velocities (PCRV), accurate Tycho-2 photometry, theoretical PARSEC, MIST, YaPSI, BaSTI isochrones, and the most accurate reddening and interstellar extinction estimates have been used to analyze the kinematics of 9543 thin-disk B-F stars as a function of their dereddened color. The stars under consideration are located on the Hertzsprung–Russell diagram relative to the isochrones with an accuracy of a few hundredths of a magnitude, i.e., at the level of uncertainty in the parallax, photometry, reddening, extinction, and the isochrones themselves. This has allowed us to choose the most plausible reddening and extinction estimates and to conclude that the reddening and extinction were significantly underestimated in some kinematic studies of other authors. Owing to the higher accuracy of TGAS parallaxes than that of Hipparcos ones, the median accuracy of the velocity components U, V, W in this study has improved to 1.7 km s^?1, although outside the range ?0.1 ^m < (B _T ? V _T )₀ < 0.5 ^m the kinematic characteristics are noticeably biased due to the incompleteness of the sample. We have confirmed the variations in the mean velocity of stars relative to the Sun and the stellar velocity dispersion as a function of their dereddened color known from the Hipparcos data. Given the age estimates for the stars under consideration from the TRILEGAL model and the Geneva–Copenhagen survey, these variations may be considered as variations as a function of the stellar age. A comparison of our results with the results of other studies of the stellar kinematics near the Sun has shown that selection and reddening underestimation explain almost completely the discrepancies between the results. The dispersions and mean velocities from the results of reliable studies fit into a ±2 km s^?1 corridor, while the ratios σ _V /σ _U and σ _W /σ _U fit into ±0.05. Based on all reliable studies in the range ?0.1 ^m < (B _T ? V _T )₀ < 0.5^m, i.e., for an age from 0.23 to 2.4 Gyr, we have found: W_⊙ = 7.15 km s^?1, \({\sigma _U} = 16.0{e^{1.29({B_T} - {V_T})o}}\), \({\sigma _V} = 10.9{e^{1.11({B_T} - {V_T})o}}\), \({\sigma _W} = 6.8{e^{1.46({B_T} - {V_T})o}}\), the stellar velocity dispersions in km s^?1 are proportional to the age in Gyr raised to the power β _U = 0.33, β _V = 0.285, and β _W = 0.37.     

On the connection between the Nekhoroshev theorem and Arnold diffusion

The analytical techniques of the Nekhoroshev theorem are used to provide estimates on the coefficient of Arnold diffusion along a particular resonance in the Hamiltonian model of Froeschlé et al. (Science 289:2108–2110, 2000). A resonant normal form is constructed by a computer program and the size of its remainder ||R _opt || at the optimal order of normalization is calculated as a function of the small parameter ${\epsilon}$ . We find that the diffusion coefficient scales as ${D \propto ||R_{opt}||^3}$ , while the size of the optimal remainder scales as ${||R_{opt}|| \propto {\rm exp}(1/\epsilon^{0.21})}$ in the range ${10^{-4} \leq \epsilon \leq 10^{-2}}$ . A comparison is made with the numerical results of Lega et al. (Physica D 182:179–187, 2003) in the same model.     

Space distribution of evolving stars in M13

Information about space distribution is collected for selected classes of evolving stars in the globular cluster M13. After a rigorous elimination of field stars, three samples are examined, corresponding to the red giant stage (G), the blue (B) and the yellow (YG) parts of the horizontal branch. It is shown that results are easy to understand in terms of:

1. A substantial mass loss in the H-shell burning stage;
2. Evolution along the horizontal branch from the blue side to the red one;
3. A mixing in the observed giant branch of two populations with sensible differences in masses.

Theoretical interpretations are briefly discussed with reference to the reliability and usefulness of this type of investigation.     

3D stellar reddening map from 2MASS photometry: An improved version

An improved version of the 3D stellar reddening map in a space with a radius of 1200 pc around the Sun and within 600 pc of the Galactic midplane is presented. As in the previous 2010 and 2012 versions of the map, photometry with an accuracy better than 0.05 ^m in the J and Ks bands for more than 70 million stars from the 2MASS catalogue is used in the new version. However, the data reduction technique is considerably more complicated. As before, an analysis of the distribution of stars near the main-sequence turnoff on the (J ? Ks)?Ks diagram, where they form a distribution maximum, provides a basis for the method. The shift of this maximum, i.e., the mode (J ? Ks), along (J ? Ks) and Ks, given the spatial variations of the mean dereddened color (J ? Ks)₀ of these stars, is interpreted as a growth of the reddening with increasing distance. The main distinction of the new method is that instead of the fixed mean absolute magnitude, dereddened color, distance, and reddening for each cell, the individual values of these quantities are calculated for each star by iterations when solving the system of equations relating them. This has allowed one to increase the random accuracy of the map to 0.01 ^m and its spatial resolution to 20 pc in coordinates and distance and to 1° in longitude and latitude. Comparison with other reddening estimates for the same spatial cells and Gaia DR1 TGAS stars shows that the constructed map is one of the best maps for the space under consideration. Its systematic errors have been estimated to be σ(E(J ? Ks)) = 0.025 ^m , or σ(E(B ? V)) = 0.04 ^m . The main purpose of the map is to analyze the characteristics of Galactic structures, clouds, and cloud complexes. For this purpose, the reddening map within each spatial cell has also been computed by analyzing the reddening along each line of sight.     

High order normal form stability estimates for co-orbital motion

We present estimates of the size of the analytic domain of stability for co-orbital motions obtained by a high order normal form in the framework of the elliptic restricted three body problem. As a demonstration example, we consider the motion of a Trojan body in an extrasolar planetary system with a giant planet of mass parameter $\mu =0.005$ μ = 0.005 and eccentricity $e^{\prime }=0.1$ e ′ = 0.1 . The analysis contains three basic steps: (i) derivation of an accurate expansion of the Hamiltonian, (ii) computation of the normal form up to an optimal order (in the Nekhoroshev sense), and (iii) computation of the optimal size of the remainder at various values of the action integrals (proper elements) of motion. We explain our choice of variables as well as the method used to expand the Hamiltonian so as to ensure a precise model. We then compute the normal form up to the normalisation order $r=50$ r = 50 by use of a computer-algebraic program. We finally estimate the size $||R||$ | | R | | of the remainder as a function of the normalization order, and compute the optimal normalization order at which the remainder becomes minimum. It is found that the optimal value $\log (||R_{opt}||)$ log ( | | R o p t | | ) can serve in order to construct a stability map for the domain of co-orbital motion using only series. This is compared to the stability map found by a purely numerical approach based on chaotic indicators.     

An on Orbit Determination of Point Spread Functions for the <Emphasis Type="Italic">Interface Region Imaging Spectrograph</Emphasis>

Using the 2016 Mercury transit of the Sun, we characterize on orbit spatial point spread functions (PSFs) for the Near- (NUV) and Far- (FUV) Ultra-Violet spectrograph channels of NASA’s Interface Region Imaging Spectrograph (IRIS). A semi-blind Richardson–Lucy deconvolution method is used to estimate PSFs for each channel. Corresponding estimates of Modulation Transfer Functions (MTFs) indicate resolution of 2.47 cycles/arcsec in the NUV channel near 2796 Å and 2.55 cycles/arcsec near 2814 Å. In the short (\({\approx}\,1336~\mathring{\mathrm{A}}\)) and long (\({\approx}\,1394~\mathring{\mathrm{A}}\)) wavelength FUV channels, our MTFs show pixel-limited resolution (3.0 cycles/arcsec). The PSF estimates perform well under deconvolution, removing or significantly reducing instrument artifacts in the Mercury transit spectra. The usefulness of the PSFs is demonstrated in a case study of an isolated explosive event. PSF estimates and deconvolution routines are provided through a SolarSoft module.     

Stellar velocity dispersion and mass estimation for galactic disks

Available velocity dispersion estimates for the old stellar population of galactic disks at galactocentric distances r?2L (where L is the photometric radial scale length of the disk) are used to determine the threshold local surface density of disks that are stable against gravitational perturbations. The mass of the disk M_d calculated under the assumption of its marginal stability is compared with the total mass M_t and luminosity L _B of the galaxy within r=4L. We corroborate the conclusion that a substantial fraction of the mass in galaxies is probably located in their dark halos. The ratio of the radial velocity dispersion to the circular velocity increases along the sequence of galactic color indices and decreases from the early to late morphological types. For most of the galaxies with large color indices (B–V)₀>0.75, which mainly belong to the S0 type, the velocity dispersion exceeds significantly the threshold value required for the disk to be stable. The reverse situation is true for spiral galaxies: the ratios M_d/L_B for these agree well with those expected for evolving stellar systems with the observed color indices. This suggests that the disks of spiral galaxies underwent no significant dynamical heating after they reached a quasi-equilibrium stable state.     

A Model of the Mira–Type Star T UMi

Stellar evolution calculations were carried out from the main sequence to the final stage of the asymptotic giant branch for stars with initial masses 1 M_⊙ ≤ M_ZAMS ≤ 2 M_⊙ and metallicity Z = 0.01. Selected models of evolutionary sequences were used as initial conditions for solution of the equations of radiation hydrodynamics and time–dependent convection describing radial stellar pulsations. The study was aimed to construct the hydrodynamic models of Mira–type stars that show the secular decrease in the pulsation period Π commenced in 1970th at Π = 315 day. We show that such a condition for the period change is satisfied with evolutionary sequences 1 M_⊙ ≤ M_ZAMS ≤ 1.2 M_⊙ and the best agreement with observations is obtained for M_ZAMS = 1.2 M_⊙. The pulsation period reduction is due to both the stellar radius decrease during the thermal pulse of the helium burning shell and mode switch from the fundamental mode to the first overtone. Theoretical estimates of the fundament parameters of the star at the onset of pulsation period reduction are as follows: the mass is M = 0.93 M_⊙, the luminosity is L = 4080 L_⊙, and the radius is R = 220 R_⊙. The mode switch occurs 35 years after the onset of period reduction.