The cosmic age crisis and the Hubble constant in a non-expanding universe

The present paper outlines a cosmological paradigm based upon Dirac’s large number hypothesis and continual creation of matter in a closed static (nonexpanding) universe. The cosmological redshift is caused by the tired-light phenomenon originally proposed by Zwicky. It is shown that the tired-light cosmology together with continual matter creation has a universal Hubble constant H ₀=(512π ²/3)^1/6(GC ₀)^1/3 fixed by the universal rate C ₀ of matter creation, where G is Newton’s gravitational constant. It is also shown that a closed static universe has a finite age τ ₀=(243π ⁵/8GC ₀)^1/3 also fixed by the universal rate of matter creation. The invariant relationship H ₀ τ ₀=3π ²6^1/2 shows that a closed static universe is much older (≈one trillion years) than any expanding universe model based upon Big-Bang cosmology. It is this property of a static universe that resolves any cosmic age crisis provided that galaxy formation in the universe is a continual recurring process. Application of Dirac’s large number hypothesis gives a matter creation rate C ₀=4.6×10^?48 gm?cm^?3?s^?1 depending only on the fundamental constants of nature. Hence, the model shows that a closed static universe has a Hubble constant H ₀=70 km?s^?1?Mpc^?1 in good agreement with recent astronomical determinations of H ₀. By using the above numerical value for H ₀ together with observational data for elongated cellular-wall structures containing superclusters of galaxies, it is shown that the elongated cellular-wall configurations observed in the real universe are at least one hundred billion years old. Application of the microscopic laws of physics to the large-scale macroscopic universe leads to a static eternal cosmos endowed with a matter-antimatter symmetry. It is proposed that the matter-antimatter asymmetry is continuously created by particle-antiparticle pair annihilation occurring in episodic cosmological gamma-ray bursts observed in the real universe.     

Cosmological evolution of agegraphic dark energy with the sign-changeable interaction

In this paper, we study a cosmological model with the sign-changeable interaction between agegraphic dark energy (ADE) and dark matter. For the accelerated expansion of the universe, the model parameters n and β should satisfy the condition n>1 and $-\frac{2}{3}<\beta<0$ . We also investigate the effect of the parameters n and β on the evolutive behavior of our universe. Furthermore, by analysis it is shown that the equation of state of ADE with the sign-changeable interaction can cross the phantom divide from w _d >?1 to w _d <?1 for the appropriate n and β. This is different from that of ADE with usual interaction, whose equation of state changes from w _d <?1 to w _d >?1.     

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.

     

Periodicity in the distribution of quasar redshifts and density perturbation in the early universe

We made a power spectrum analysis on the quasar emission redshift distribution, and further confirmed the existence of periodicity in respect of the quantity x = F_(z,qo) defined at (8).The existence of this periodicity does not mean that the quasar redshift is non-cosmological, for it can be interpreted as a remnant of density (acoustic) perturbations in the early big-bang universe. For this model, we made a number of tests. We found: 1) the ratio of periodic to non-periodic components falls as the sample size increases; 2) the periodicity should be more marked for quasars in one region of the sky than for all quasars, and 3) the Jeans wavelength before the recombination epoch determines the length of the period. Using this model we also found that qo > 0.5, lending further support to the conclusion reached by other means that the universe may be closed.     

Effect of non-zero rest-mass of neutrinos on clustering in the early universe

This paper discusses the effect of neutrinos with non-zero rest-mass on the clustering process in the early universe. The early universe is regarded as a two-component fluid, one component being the de-coupled neutrinos, and the other being matter and radiation, between the two there is only the gravitational coupling. The main conclusions are: (1) such neutrinos will cause clustering of matter before the epoch of re-combination; (2) the mass so clustered will be in the range of the mass of clusters of galaxies; (3) there exists a preferential clustering scale, corresponding to the earliest onset of Jeans instability; (4) if the rest-mass is below a certain value, then there will be no effect.     

Jovian chromophore characteristics from multispectral HST images

The chromophores responsible for coloring the jovian atmosphere are embedded within Jupiter’s vertical aerosol structure. Sunlight propagates through this vertical distribution of aerosol particles, whose colors are defined by ?₀(λ), and we remotely observe the culmination of the radiative transfer as I/F(λ). In this study, we employed a radiative transfer code to retrieve ?₀(λ) for particles in Jupiter’s tropospheric haze at seven wavelengths in the near-UV and visible regimes. The data consisted of images of the 2008 passage of Oval BA to the south of the Great Red Spot obtained by the Wide Field Planetary Camera 2 on-board the Hubble Space Telescope. We present derived particle colors for locations that were selected from 14 weather regions, which spanned a large range of observed colors. All ?₀(λ) curves were absorbing in the blue, and ?₀(λ) increased monotonically to approximately unity as wavelength increased. We found accurate fits to all ?₀(λ) curves using an empirically derived functional form: ?₀(λ) = 1 − A exp(−Bλ). The best-fit parameters for the mean ?₀(λ) curve were A = 25.4 and B = 0.0149 for λ in units of nm. We performed a principal component analysis (PCA) on our ?₀(λ) results and found that one or two independent chromophores were sufficient to produce the variations in ?₀(λ). A PCA of I/F(λ) for the same jovian locations resulted in principal components (PCs) with roughly the same variances as the ?₀(λ) PCA, but they did not result in a one-to-one mapping of PC amplitudes between the ?₀(λ) PCA and I/F(λ) PCA. We suggest that statistical analyses performed on I/F(λ) image cubes have limited applicability to the characterization of chromophores in the jovian atmosphere due to the sensitivity of I/F(λ) to horizontal variations in the vertical aerosol distribution.     

Cosmic Background Bose Condensation (CBBC)

Degeneracy effects for bosons are more important for smaller particle mass, smaller temperature and higher number density. Bose condensation requires that particles be in the same lowest energy quantum state. We propose a cosmic background Bose condensation, present everywhere, with its particles having the lowest quantum energy state, ?c/λ, with λ about the size of the visible universe, and therefore unlocalized. This we identify with the quantum of the self gravitational potential energy of any particle, and with the bit of information of minimum energy. The entropy of the universe (～10¹²² bits) has the highest number density (～10³⁶ bits/cm³) of particles inside the visible universe, the smallest mass, ～10^?66 g, and the smallest temperature, ～10^?29 K. Therefore it is the best candidate for a Cosmic Background Bose Condensation (CBBC), a completely calmed fluid, with no viscosity, in a superfluidity state, and possibly responsible for the expansion of the universe.     

Dark Matter from the inflaton field

We present a model where inflation and Dark Matter takes place via a single scalar field ?. Without introducing any new parameters we are able unify inflation and Dark Matter using a scalar field ? that accounts for inflation at an early epoch while it gives a Dark Matter WIMP particle at low energies. After inflation our universe must be reheated and we must have a long period of radiation dominated before the epoch of Dark Matter. Typically the inflaton decays while it oscillates around the minimum of its potential. If the inflaton decay is not complete or sufficient then the remaining energy density of the inflaton after reheating must be fine tuned to give the correct amount of Dark Matter. An essential feature here, is that Dark Matter-Inflaton particle is produced at low energies without fine tuning or new parameters. This process uses the same coupling g as for the inflaton decay. Once the field ? becomes non-relativistic it will decouple as any WIMP particle, since n_? is exponentially suppressed. The correct amount of Dark Matter determines the cross section and we have a constraint between the coupling g and the mass m_o of ?. The unification scheme we present here has four free parameters, two for the scalar potential V(?) given by the inflation parameter λ of the quartic term and the mass m_o. The other two parameters are the coupling g between the inflaton ? and a scalar filed φ and the coupling h between φ with standard model particles ψ or χ. These four parameters are already present in models of inflation and reheating process, without considering Dark Matter. Therefore, our unification scheme does not increase the number of parameters and it accomplishes the desired unification between the inflaton and Dark Matter for free.     

The future evolution of white dwarf stars through baryon decay and time varying gravitational constant

Motivated by the possibility that the fundamental “constants” of nature could vary with time, this paper considers the long term evolution of white dwarf stars under the combined action of proton decay and variations in the gravitational constant. White dwarfs are thus used as a theoretical laboratory to study the effects of possible time variations, especially their implications for the future history of the universe. More specifically, we consider the gravitational constant G to vary according to the parametric relation G=G ₀(1+t/t _? )^?p , where the time scale t _? is the same order as the proton lifetime t _P . We then study the long term fate and evolution of white dwarf stars. This treatment begins when proton decay dominates the stellar luminosity, and ends when the star becomes optically thin to its internal radiation.     

Bianchi type-V universe with wet dark fluid

The Bianchi type-V universe filled with dark energy from a wet dark fluid has been considered. A new equation of state for the dark energy component of the universe has been used. It is modeled on the equation of state p=γ(ρ?ρ _? ) which can describe a liquid, for example water. The exact solutions to the corresponding field equations are obtained in quadrature form. The solution for constant deceleration parameter have been studied in detail for power-law and exponential forms both. The case $\gamma =\frac{1}{3}$ has been also analysed.     

Density perturbations in a two-fluid component Universe

In a two-fluid component universe consisting of visible matter and neutrinos, the developed features of perturbations in the two components are quite different. If the densities ₁ and ₂, and the Jeans lengths _1J and _2J of the two components satisfy the relations ₁₂, _1J2J, the developed inhomogeneities in the non-dominant component 1 are larger than those in the dominant component 2. Moreover, the increase of perturbations is in some situations not monotonous but oscillatory, and such oscillations in the two components are contrary.     

Measurements of the Matter Density Perturbation Amplitude from Cosmological Data

The various measurements of the linear matter density perturbation amplitude obtained from the observations of the cosmic microwave background (CMB) anisotropy, weak gravitational lensing, galaxy cluster mass function, matter power spectrum, and redshift space distortions are compared. The Planck data on the CMB temperature anisotropy spectrum at high multipoles, ? > 1000 (where the effect of gravitational lensing is most significant), are shown to give a measurement of the matter density perturbation amplitude that contradicts all other measurements of this quantity from both Planck CMB anisotropy data and other data at a significance level of about 3.7σ. Thus, at present these data should not be combined together for the calculations of constraints on cosmological parameters. Except for the Planck data on the CMB temperature anisotropy spectrum at high multipoles, all the remaining measurements of the density perturbation amplitude agree well between themselves and give the following constraints: σ₈ = 0.792± 0.006 on the linear matter density perturbation amplitude, Ω_m = 0.287± 0.007 on the matter density parameter, and H₀ = 69.4 ± 0.6 km s^?1 Mpc^?1 on the Hubble constant. Various constraints on the sum of neutrino masses and the number of neutrino flavors can be obtained by additionally taking into account the data on baryon acoustic oscillations and (or) direct Hubble constant measurements in the local Universe.     

On the spin distributions of ΛCDM haloes

We used merger trees realizations, predicted by the extended Press-Schechter theory, in order to study the growth of angular momentum of dark matter haloes. Our results showed that:

1. The spin parameter λ′ resulting from the above method, is an increasing function of the present day mass of the halo. The mean value of λ′ varies from 0.0343 to 0.0484 for haloes with present day masses in the range of 10⁹h^?1 M _⊙ to 10¹⁴h^?1 M _⊙.
2. The distribution of λ′ is close to a log-normal, but, as it is already found in the results of N-body simulations, the match is not satisfactory at the tails of the distribution. A new analytical formula that approximates the results much more satisfactorily is presented.
3. The distribution of the values of λ′ depends only weakly on the redshift.
4. The spin parameter of an halo depends on the number of recent major mergers. Specifically the spin parameter is an increasing function of this number.

     

A new type of interaction between generalized Chaplygin gas and dark matter

In this paper, we investigate the model with a new type of interaction between generalized Chaplygin gas (GCG) and dark matter. It is shown that there exists a stable scaling attractor, which provides the possibility to alleviate the coincidence problem. The equation of state (EoS) of GCG approaches the attractor phase from either w _g >?1 or w _g w _g w _g w _g >?1), and next cross again the phantom divide (the transition from w _g >?1 to w _g Q can change its sign from Q<0 to Q>0 as the universe expands, which is different from the usual interaction. Moreover, we investigate the model from statefinder viewpoint. The statefinder diagnostic can not only discriminate the model with different coupling constant but also distinguish the model from other dark energy models.     

Galactic kinematics from data on open star clusters from the MWSC catalogue

Open star clusters from the MWSC (Milky Way Star Clusters) catalogue have been used to determine the Galactic rotation parameters. The circular rotation velocity of the solar neighborhood around the Galactic center has been found from data on more than 2000 clusters of various ages to be V ₀ = 236 ± 6 km s^?1 for the adopted Galactocentric distance of the Sun R ₀ = 8.3 ± 0.2 kpc. The derived angular velocity parameters are Ω ₀ = 28.48 ± 0.36 km s^?1 kpc^?1, Ω’₀ = ?3.50 ± 0.08 km s^?1 kpc_?2, and Ω″₀ = 0.331 ± 0.037 km s^?1 kpc^?3. The influence of the spiral density wave has been detected only in the sample of clusters younger than 50 Myr. For these clusters the amplitudes of the tangential and radial velocity perturbations are f ^θ = 5.6 ± 1.6 km s^?1 and f _R = 7.7 ± 1.4 km s^?1, respectively; the perturbation wavelengths are λ _θ = 2.6 ± 0.5 kpc (i _θ = ^?11? ± 2?) and λ _R = 2.1 ± 0.5 kpc (i _R = ?9? ± 2?) for the adopted four-armed model (m = 4). The Sun’s phase in the spiral density wave is (χ_⊙)_θ = ?62? ± 9? and (χ_⊙)_R = ?85? ± 10? from the residual tangential and radial velocities, respectively.     

Diagnostics of the early explosion phase of a classical nova using its X-ray emission: A model for the X-ray outburst of CI Camelopardalis in 1998

We have computed a spherically symmetric model for the interaction of matter ejected during the outburst of a classical nova with the stellar wind from its optical component. This model is used to describe the intense X-ray outburst (the peak 3–20 keV flux was ～2 Crab) of the binary system CI Camelopardalis in 1998. According to our model, the stellar wind from the optical component heated by a strong shock wave produced when matter is ejected from the white dwarf as the result of a thermonuclear explosion on its surface is the emission source in the standard X-ray band. Comparison of the calculated and observed time dependences of the mean radiation temperature and luminosity of the binary system during its outburst has yielded very important characteristics of the explosion. We have been able to measure the velocity of the ejected matter immediately after the onset of the explosion for the first time: it follows from our model that the ejected matter had a velocity of ～2700 km s^?1 even on 0.1–0.5 day after the outburst onset and it flew with such a velocity for the first 1–1.5 day under an external force, possibly, the radiation pressure from the white dwarf. Subsequently, the matter probably became transparent and began to decelerate. The time dependence of the mean radiation temperature at late expansion phases has allowed us to estimate the mass of the ejected matter, ～10^?7–10^?6 M _⊙. The mass loss rate in the stellar wind required to explain the observed peak luminosity of the binary system during its outburst has been estimated to be \(\dot M\) ～ (1 ? 2) × 10^?6 M _⊙ yr^?1.     

Titan: Far-infrared and microwave remote sensing of methane clouds and organic haze

It is shown that Titan's surface and plausible atmospheric thermal opacity sources—gaseous N₂, CH₄, and H₂, CH₄ cloud, and organic haze—are sufficient to match available Earth-based and Voyager observations of Titan's thermal emission spectrum. Dominant sources of thermal emission are the surface for wavelenghts λ ? 1 cm, atmospheric N₂ for 1 cm ? λ ? 200 μm,, condensed and gaseous CH₄ for 200 μm ? λ ? 20 μm, and molecular bands and organic haze for λ ? 20 μm. Matching computed spectra to the observed Voyager IRIS spectra at 7.3 and 52.7° emission angles yields the following abundances and locations of opacity sources: CH₄ clouds: 0.1 g cm^? at a planetocentric radius of 2610–2625 km, 0.3 g cm^?2 at 2590–2610 km, total 0.4 ± 0.1 g cm^–2 above 2590 km; organic haze: $\text{4 ± 2 × 10}{}^{\mathrm{?6}}\text{,}\text{g cm}\text{,}{}^{\mathrm{?2}}$ above 2750 km; tropospheric H₂: 0.3 ± 0.1 mol%. This is the first quantitative estimate of the column density of condensed methane (or CH₄/C₂H₆) on Titan. Maximum transparency in the middle to far IR occurs at 19 μm where the atmospheric vertical absorption optical depth is $\text{?0.6}$ A particle radius $\text{r}\text{? 2 μ}\text{m}$ in the upper portion of the CH₄ cloud is indicated by the apparent absence of scattering effects.     

Kinematics of the Galaxy from OB Stars with Data from the Gaia DR2 Catalogue

We have selected and analyzed a sample of OB stars with known line-of-sight velocities determined through ground-based observations and with trigonometric parallaxes and propermotions from the Gaia DR2 catalogue. Some of the stars in our sample have distance estimates made from calcium lines. A direct comparison with the trigonometric distance scale has shown that the calcium distance scale should be reduced by 13%. The following parameters of the Galactic rotation curve have been determined from 495 OB stars with relative parallax errors less than 30%: (U, V,W)_⊙ = (8.16, 11.19, 8.55)± (0.48, 0.56, 0.48) km s^?1, Ω₀ = 28.92 ± 0.39 km s^?1 kpc^?1, Ω'₀ = ?4.087 ± 0.083 km s^?1 kpc^?2, and Ω″ ₀ = 0.703 ± 0.067 km s?1 kpc^?3, where the circular velocity of the local standard of rest is V0 = 231 ± 5 km s^?1 (for the adopted R₀ = 8.0 ± 0.15 kpc). The parameters of the Galactic spiral density wave have been found from the series of radial, V_R, residual tangential, ΔV_circ, and vertical, W, velocities of OB stars by applying a periodogram analysis. The amplitudes of the radial, tangential, and vertical velocity perturbations are f_R = 7.1± 0.3 km s^?1, f_θ = 6.5 ± 0.4 km s^?1, and f_W = 4.8± 0.8 km s^?1, respectively; the perturbation wavelengths are λ_R = 3.3 ± 0.1 kpc, λ_θ = 2.3 ± 0.2 kpc, and λ_W = 2.6 ± 0.5 kpc; and the Sun’s radial phase in the spiral density wave is (χ_⊙)_R = ?135? ± 5?, (χ_⊙)θ = ?123? ± 8?, and (χ_⊙)_W = ?132? ± 21? for the adopted four-armed spiral pattern.     

Galactic rotation curve and spiral density wave parameters from 73 masers

Based on kinematic data on masers with known trigonometric parallaxes and measurements of the velocities of HI clouds at tangential points in the inner Galaxy, we have refined the parameters of the Allen-Santillan model Galactic potential and constructed the Galactic rotation curve in a wide range of Galactocentric distances, from 0 to 20 kpc. The circular rotation velocity of the Sun for the adopted Galactocentric distance R ₀ = 8 kpc is V ₀ = 239 ± 16 km s^?1. We have obtained the series of residual tangential, ΔV _θ , and radial, V _R , velocities for 73 masers. Based on these series, we have determined the parameters of the Galactic spiral density wave satisfying the linear Lin-Shu model using the method of periodogram analysis that we proposed previously. The tangential and radial perturbation amplitudes are f _θ = 7.0±1.2 km s^?1 and f _R = 7.8±0.7 km s^?1, respectively, the perturbation wave length is λ = 2.3±0.4 kpc, and the pitch angle of the spiral pattern in a two-armed model is i = ?5.2° ±0.7°. The phase of the Sun ζ _⊙ in the spiral density wave is ?50° ± 15° and ?160° ± 15° from the residual tangential and radial velocities, respectively.