Satellite de-orbiting via controlled solar radiation pressure

We analyze the families of central configurations of the spatial 5-body problem with four masses equal to 1 when the fifth mass m varies from 0 to \(+\infty \). In particular we continue numerically, taking m as a parameter, the central configurations (which all are symmetric) of the restricted spatial (\(4+1\))-body problem with four equal masses and \(m=0\) to the spatial 5-body problem with equal masses (i.e. \(m=1\)), and viceversa we continue the symmetric central configurations of the spatial 5-body problem with five equal masses to the restricted (\(4+1\))-body problem with four equal masses. Additionally we continue numerically the symmetric central configurations of the spatial 5-body problem with four equal masses starting with \(m=1\) and ending in \(m=+\infty \), improving the results of Alvarez-Ramírez et al. (Discrete Contin Dyn Syst Ser S 1: 505–518, 2008). We find four bifurcation values of m where the number of central configuration changes. We note that the central configurations of all continued families varying m from 0 to \(+\infty \) are symmetric.     

Secular tidal changes in lunar orbit and Earth rotation

Small tidal forces in the Earth–Moon system cause detectable changes in the orbit. Tidal energy dissipation causes secular rates in the lunar mean motion n, semimajor axis a, and eccentricity e. Terrestrial dissipation causes most of the tidal change in n and a, but lunar dissipation decreases eccentricity rate. Terrestrial tidal dissipation also slows the rotation of the Earth and increases obliquity. A tidal acceleration model is used for integration of the lunar orbit. Analysis of lunar laser ranging (LLR) data provides two or three terrestrial and two lunar dissipation parameters. Additional parameters come from geophysical knowledge of terrestrial tides. When those parameters are converted to secular rates for orbit elements, one obtains dn/dt = \(-25.97\pm 0.05 ''/\)cent\(^{2}\), da/dt = 38.30 ± 0.08 mm/year, and di/dt = ?0.5 ± 0.1 \(\upmu \)as/year. Solving for two terrestrial time delays and an extra de/dt from unspecified causes gives \(\sim \) \(3\times 10^{-12}\)/year for the latter; solving for three LLR tidal time delays without the extra de/dt gives a larger phase lag of the N2 tide so that total de/dt = \((1.50 \pm 0.10)\times 10^{-11}\)/year. For total dn/dt, there is \(\le \)1 % difference between geophysical models of average tidal dissipation in oceans and solid Earth and LLR results, and most of that difference comes from diurnal tides. The geophysical model predicts that tidal deceleration of Earth rotation is \(-1316 ''\)/cent\(^{2}\) or 87.5 s/cent\(^{2}\) for UT1-AT, a 2.395 ms/cent increase in the length of day, and an obliquity rate of 9 \(\upmu \)as/year. For evolution during past times of slow recession, the eccentricity rate can be negative.     

Constraints on the location of a putative distant massive body in the Solar System from recent planetary data

We analytically work out the long-term variations caused on the motion of a planet orbiting a star by a very distant, pointlike massive object X. Apart from the semi-major axis a, all the other Keplerian osculating orbital elements experience long-term variations which are complicated functions of the orbital configurations of both the planet itself and of X. We infer constraints on the minimum distance d _X at which X may exist by comparing our prediction of the long-term variation of the longitude of the perihelion \({\varpi}\) to the latest empirical determinations of the corrections \({\Delta\dot\varpi}\) to the standard Newtonian/Einsteinian secular precessions of several solar system planets recently estimated by independent teams of astronomers. We obtain the following approximate lower bounds on d _X for the assumed masses of X quoted in brackets: 150–200 au (Mars), 250–450 au \(({0.7 m_{\oplus}})\), 3500–4500 au (4 m _Jup).     

A Possible Cause of the Diminished Solar Wind During the Solar Cycle 23 – 24 Minimum

Interplanetary magnetic field and solar wind plasma density observed at 1 AU during Solar Cycle 23?–?24 (SC-23/24) minimum were significantly smaller than those during its previous solar cycle (SC-22/23) minimum. Because the Earth’s orbit is embedded in the slow wind during solar minimum, changes in the geometry and/or content of the slow wind region (SWR) can have a direct influence on the solar wind parameters near the Earth. In this study, we analyze solar wind plasma and magnetic field data of hourly values acquired by Ulysses. It is found that the solar wind, when averaging over the first (1995.6?–?1995.8) and third (2006.9?–?2008.2) Ulysses’ perihelion (\({\sim}\,1.4~\mbox{AU}\)) crossings, was about the same speed, but significantly less dense (\({\sim}\,34~\%\)) and cooler (\({\sim}\,20~\%\)), and the total magnetic field was \({\sim}\,30~\%\) weaker during the third compared to the first crossing. It is also found that the SWR was \({\sim}\,50~\%\) wider in the third (\({\sim}\,68.5^{\circ}\) in heliographic latitude) than in the first (\({\sim}\,44.8^{\circ}\)) solar orbit. The observed latitudinal increase in the SWR is sufficient to explain the excessive decline in the near-Earth solar wind density during the recent solar minimum without speculating that the total solar output may have been decreasing. The observed SWR inflation is also consistent with a cooler solar wind in the SC-23/24 than in the SC-22/23 minimum. Furthermore, the ratio of the high-to-low latitude photospheric magnetic field (or equatorward magnetic pressure force), as observed by the Mountain Wilson Observatory, is smaller during the third than the first Ulysses’ perihelion orbit. These findings suggest that the smaller equatorward magnetic pressure at the Sun may have led to the latitudinally-wider SRW observed by Ulysses in SC-23/24 minimum.     

Secular resonances between bodies on close orbits: a case study of the Himalia prograde group of jovian irregular satellites

The gravitational interaction between two objects on similar orbits can effect noticeable changes in the orbital evolution even if the ratio of their masses to that of the central body is vanishingly small. Christou (Icarus 174:215–229, 2005) observed an occasional resonant lock in the differential node \(\varDelta \varOmega \) between two members in the Himalia irregular satellite group of Jupiter in the N-body simulations (corresponding mass ratio \(\sim 10^{-9}\)). Using a semianalytical approach, we have reproduced this phenomenon. We also demonstrate the existence of two additional types of resonance, involving angle differences \(\varDelta \omega \) and \(\varDelta (\varOmega +\varpi )\) between two group members. These resonances cause secular oscillations in eccentricity and/or inclination on timescales \(\sim \)1 Myr. We locate these resonances in (a, e, i) space and analyse their topological structure. In subsequent N-body simulations, we confirm these three resonances and find a fourth one involving \(\varDelta \varpi \). In addition, we study the occurrence rates and the stability of the four resonances from a statistical perspective by integrating 1000 test particles for 100 Myr. We find \(\sim \)10 to 30 librators for each of the resonances. Particularly, the nodal resonance found by Christou is the most stable: 2 particles are observed to stay in libration for the entire integration.     

Contractivity-preserving explicit multistep Hermite–Obrechkoff series differential equation solvers

New optimal, contractivity-preserving (CP), explicit, d-derivative, k-step Hermite–Obrechkoff series methods of order p up to \(p=20\), denoted by CP HO(d, k, p), with nonnegative coefficients are constructed. These methods are used to solve nonstiff first-order initial value problems \(y'=f(t,y)\), \(y(t_0)=y_0\). The upper bound \(p_u\) of order p of HO(d, k, p) can reach, approximately, as high as 2.4 times the number of derivatives d. The stability regions of HO(d, k, p) have generally a good shape and grow with decreasing \(p-d\). We, first, note that three selected CP HO methods: 4-derivative 7-step HO of order 13, denoted by HO(4, 7, 13), 5-derivative 6-step HO of order 13, denoted by HO(5, 6, 13), and 9-derivative 2-step HO of order 13, denoted by CMDAHO(13) compare favorably with Adams–Cowell of order 13, denoted by AC(13), in solving standard N-body problems over an interval of 1000 periods on the basis of the relative error of energy as a function of the CPU time. Next, the three HO methods compare positively with AC(13) in solving standard N-body problems on the basis of the growth of relative positional error and relative energy error over 10, 000 periods of integration. Finally, these three methods compare also well with P-stable methods of Cash and Franco et al. on some quasi periodic, second-order linear and nonlinear problems. The coefficients of selected HO methods are listed in the appendix.     

Inverse problem of central configurations and singular curve in the collinear 4-body problem

In this paper, we consider the inverse problem of central configurations of n-body problem. For a given \({q=(q_1, q_2, \ldots, q_n)\in ({\bf R}^d)^n}\), let S(q) be the admissible set of masses denoted \({ S(q)=\{ m=(m_1,m_2, \ldots, m_n)| m_i \in {\bf R}^+, q}\) is a central configuration for m}. For a given \({m\in S(q)}\), let S _m (q) be the permutational admissible set about m = (m ₁, m ₂, . . . , m _n ) denoted

$S_m(q)=\{m^\prime | m^\prime\in S(q),m^\prime \not=m \, {\rm and} \, m^\prime\,{\rm is\, a\, permutation\, of }\, m \}.$

The main discovery in this paper is the existence of a singular curve \({\bar{\Gamma}_{31}}\) on which S _m (q) is a nonempty set for some m in the collinear four-body problem. \({\bar{\Gamma}_{31}}\) is explicitly constructed by a polynomial in two variables. We proved:

1. (1)
   If \({m\in S(q)}\), then either ^# S _m (q) = 0 or ^# S _m (q) = 1.
    
2. (2)
   ^#S _m (q) = 1 only in the following cases:

   1. (i)
      If s = t, then S _m (q) = {(m ₄, m ₃, m ₂, m ₁)}.
       
   2. (ii)
      If \({(s,t)\in \bar{\Gamma}_{31}\setminus \{(\bar{s},\bar{s})\}}\), then either S _m (q) = {(m ₂, m ₄, m ₁, m ₃)} or S _m (q) = {(m ₃, m ₁, m ₄, m ₂)}.
       
    

     

Predictions on the core mass of Jupiter and of giant planets in general

More than 80 giant planets are known by mass and radius. Their interior structure in terms of core mass, number of layers, and composition however is still poorly known. An overview is presented about the core mass M _core and envelope mass of metals M _Z in Jupiter as predicted by various equations of state. It is argued that the uncertainty about the true H/He EOS in a pressure regime where the gravitational moments J ₂ and J ₄ are most sensitive, i.e. between 0.5 and 4 Mbar, is in part responsible for the broad range \(M_{\mathit{core}}=0{-}18\:M_{\oplus }\), \(M_{Z}=0{-}38\:M_{\oplus }\), and \(M_{\mathit{core}}+M_{Z}=14{-}38\:M_{\oplus }\) currently offered for Jupiter. We then compare the Jupiter models obtained when we only match J ₂ with the range of solutions for the exoplanet \(\mathrm{GJ}\:436\mathrm{b}\), when we match an assumed tidal Love number k ₂ value.     

Correlations Between CME Parameters and Sunspot Activity

Smoothed monthly mean coronal mass ejection (CME) parameters (speed, acceleration, central position angle, angular width, mass, and kinetic energy) for Cycle 23 are cross-analyzed, showing that there is a high correlation between most of them. The CME acceleration (a) is highly correlated with the reciprocal of its mass (M), with a correlation coefficient r=0.899. The force (Ma) to drive a CME is found to be well anti-correlated with the sunspot number (R _z), r=?0.750. The relationships between CME parameters and R _z can be well described by an integral response model with a decay time scale of about 11 months. The correlation coefficients of CME parameters with the reconstructed series based on this model (\(\overline{r}_{\mathrm{f1}}=0.886\)) are higher than the linear correlation coefficients of the parameters with R _z (\(\overline{r}_{\mathrm{0}}=0.830\)). If a double decay integral response model is used (with two decay time scales of about 6 and 60 months), the correlations between CME parameters and R _z improve (\(\overline{r}_{\mathrm{f2}}=0.906\)). The time delays between CME parameters with respect to R _z are also well predicted by this model (19/22=86%); the average time delays are 19 months for the reconstructed and 22 months for the original time series. The model implies that CMEs are related to the accumulation of solar magnetic energy. These relationships can help in understanding the mechanisms at work during the solar cycle.     

Variability of hot supergiant IRAS 19336-0400 in the early phase of planetary nebula ionization

We present photoelectric and spectral observations of a hot candidate proto-planetary nebula—early B-type supergiant with emission lines in spectrum—IRAS 19336-0400. The light and color curves display fast irregular brightness variations with maximum amplitudes \(\Delta V = 0_ \cdot ^m 30\), \(\Delta B = 0_ \cdot ^m 35\), \(\Delta U = 0_ \cdot ^m 40\) and color-brightness correlations. By the variability characteristics IRAS 19336-0400 appears similar to other hot proto-planetary nebulae. Based on low-resolution spectra in the range λ4000–7500 Å we have derived absolute intensities of the emission lines Hα, Hβ, Hγ, [S II], [N II], physical conditions in gaseous nebula: n _e = 10⁴ cm^?3, T _e = 7000 ± 1000 K. The emission line Hα, Hβ equivalent widths are found to be considerably variable and related to light changes. By UBV-photometry and spectroscopy the color excess has been estimated: E _B-V = 0.50–0.54. Joint photometric and spectral data analysis allows us to assume that the star variability is caused by stellar wind variations.     

Sounding Rocket Observations of Active Region Soft X-Ray Spectra Between 0.5 and 2.5 nm Using a Modified SDO/EVE Instrument

Spectrally resolved measurements of individual solar active regions (ARs) in the soft X-ray (SXR) range are important for studying dynamic processes in the solar corona and their associated effects on the Earth’s upper atmosphere. They are also a means of evaluating atomic data and elemental abundances used in physics-based solar spectral models. However, very few such measurements are available. We present spectral measurements of two individual ARs in the 0.5 to 2.5 nm range obtained on the NASA 36.290 sounding rocket flight of 21 October 2013 (at about 18:30 UT) using the Solar Aspect Monitor (SAM), a channel of the Extreme Ultaviolet Variability Experiment (EVE) payload designed for underflight calibrations of the orbital EVE on the Solar Dynamics Observatory (SDO). The EVE rocket instrument is a duplicate of the EVE on SDO, except the SAM channel on the rocket version was modified in 2012 to include a freestanding transmission grating to provide spectrally resolved images of the solar disk with the best signal to noise ratio for the brightest features, such as ARs. Calibrations of the EVE sounding rocket instrument at the National Institute of Standards and Technology Synchrotron Ultraviolet Radiation Facility (NIST/SURF) have provided a measurement of the SAM absolute spectral response function and a mapping of wavelength separation in the grating diffraction pattern. We discuss techniques (incorporating the NIST/SURF data) for determining SXR spectra from the dispersed AR images as well as the resulting spectra for NOAA ARs 11877 and 11875 observed on the 2013 rocket flight. In comparisons with physics-based spectral models using the CHIANTI v8 atomic database we find that both AR spectra are in good agreement with isothermal spectra (4 MK), as well as spectra based on an AR differential emission measure (DEM) included with the CHIANTI distribution, with the exception of the relative intensities of strong Fe?xvii lines associated with \(2p^{6}\)–\(2p^{5}3{s}\) and \(2p^{6}\)–\(2p^{5}3{d}\) transitions at about 1.7 nm and 1.5 nm, respectively. The ratio of the Fe?xvii lines suggests that the AR 11877 is hotter than the AR 11875. This result is confirmed with analysis of the active regions imaged by X-ray Telescope (XRT) onboard Hinode.     

Improved Determination of the Location of the Temperature Maximum in the Corona

The most used method to calculate the coronal electron temperature [\(T_{\mathrm{e}} (r)\)] from a coronal density distribution [\(n_{\mathrm{e}} (r)\)] is the scale-height method (SHM). We introduce a novel method that is a generalization of a method introduced by Alfvén (Ark. Mat. Astron. Fys. 27, 1, 1941) to calculate \(T_{\mathrm{e}}(r)\) for a corona in hydrostatic equilibrium: the “HST” method. All of the methods discussed here require given electron-density distributions [\(n_{\mathrm{e}} (r)\)] which can be derived from white-light (WL) eclipse observations. The new “DYN” method determines the unique solution of \(T_{\mathrm{e}}(r)\) for which \(T_{\mathrm{e}}(r \rightarrow \infty) \rightarrow 0\) when the solar corona expands radially as realized in hydrodynamical solar-wind models. The applications of the SHM method and DYN method give comparable distributions for \(T_{\mathrm{e}}(r)\). Both have a maximum [\(T_{\max}\)] whose value ranges between 1?–?3 MK. However, the peak of temperature is located at a different altitude in both cases. Close to the Sun where the expansion velocity is subsonic (\(r < 1.3\,\mathrm{R}_{\odot}\)) the DYN method gives the same results as the HST method. The effects of the other free parameters on the DYN temperature distribution are presented in the last part of this study. Our DYN method is a new tool to evaluate the range of altitudes where the heating rate is maximum in the solar corona when the electron-density distribution is obtained from WL coronal observations.     

Quantification of tidal parameters from Solar System data

Tidal dissipation is the main driver of orbital evolution of natural satellites and a key point to understand the exoplanetary system configurations. Despite its importance, its quantification from observations still remains difficult for most objects of our own Solar System. In this work, we overview the method that has been used to determine, directly from observations, the tidal parameters, with emphasis on the Love number \(k_2\) and the tidal quality factor Q. Up-to-date values of these tidal parameters are summarized. Last, an assessment on the possible determination of the tidal ratio \(k_2/Q\) of Uranus and Neptune is done. This may be particularly relevant for coming astrometric campaigns and future space missions focused on these systems.     

Orbit and physical characteristics of the components of the massive Algol V622 Per,a member of the open star cluster χ Per

Analysis of the radial velocities based on spectra of high (near the H _α line) and moderate (4420–4960 Å) resolutions supplemented by the published radial velocities has revealed the binarity of a bright member of the young open star cluster χ Per, the star V622 Per. The derived orbital elements of the binary show that the lines of both components are seen in its spectrum, the orbital period is 5.2 days, and the binary is in the phase of active mass exchange. The photometric variability of the star is caused by the ellipsoidal shape of its components. Analysis of the spectroscopic and photometric variabilities has allowed the absolute parameters of the binary’s orbit and its components to be found. V622 Per is shown to be a classical Algol with moderate mass exchange in the binary. Mass transfer occurs from the less massive (\({M_1} = 9.1 \pm 2.7{M_ \odot }\)) but brighter (\(\log {L_1} = 4.52 \pm 0.10{L_ \odot }\)) component onto the more massive (\({M_2} = 13.0 \pm 3.5{M_ \odot }\)) and less bright (\(\log {L_2} = 3.96 \pm 0.10{L_ \odot }\)) component. Analysis of the spectra has confirmed an appreciable overabundance of CNO-cycle products in the atmosphere of the primary component. Comparison of the positions of the binary’s components on the T _eff–log g diagram with the age of the cluster χ Per points to a possible delay in the evolution of the primary component due to mass loss by no more than 1–2Myr.     

Phase transition in the Reissner–Nordström black hole

The thermodynamics of the Reissner–Nordström black hole at the charge-to-mass ratio of \(\sqrt{3}/2\) is studied. We show that there exists infinite discontinuity not only in the heat capacity but also in the electromagnetic potential and hence in the Gibbs free energy. Despite the existence of an infinite discontinuity in the heat capacity, it is shown that there exists no phase transition in the Reissner–Nordström black hole. The discontinuity may be merely an artefact of the change in the slope of the |Q| vs. M curve when the charge-to-mass ratio crosses \(\sqrt{3}/2\). This may signify that the horizon temperature is not a well-behaved thermodynamic parameter in the black hole thermodynamics.     

Photometry and spectroscopy of the luminous red nova PSNJ14021678+5426205 in the galaxy M101

We present the results of the study of a red nova from the observations carried out with the Russian 6-m telescope (BTA) along with other telescopes of SAO RAS and SAI MSU. To investigate the nova progenitor,we used the data from the Digital Sky Survey and amateur photos available on the Internet. In the period between April 1993 and July 2014, the brightness of the progenitor gradually increased by \(2_ \cdot ^m 2\) in the V-band. At the peak of the first outburst in mid-November 2014, the star reached an absolute visual magnitude of \(- 12_ \cdot ^m 75\) but was discovered later, in February 2015, in a repeated outburst at the magnitude of \(- 11_ \cdot ^m 65\). The amplitude of the outburst was minimum among the red novae, only \(5_ \cdot ^m 6\) in V-band. The Hα emission line and the background of a cool supergiant continuum with gradually decreasing surface temperature were observed in the spectra. Such process is typical for red novae, although the object under study showed extreme parameters: maximum luminosity, maximum outburst duration, minimum outburst amplitude, unusual shape of the light curve. This event is interpreted as a massive OB star system components’merging accompanied by formation of a common envelope and then the expansion of this envelope with minimal energy losses.     

Prospects of Measuring the Angular Power Spectrum of the Diffuse Galactic Synchrotron Emission with SKA1 Low

The Diffuse Galactic Syncrotron Emission (DGSE) is the most important diffuse foreground component for future cosmological 21-cm observations. The DGSE is also an important probe of the cosmic ray electron and magnetic field distributions in the turbulent interstellar medium (ISM) of our galaxy. In this paper we briefly review the Tapered Gridded Estimator (TGE) which can be used to quantify the angular power spectrum C _? of the sky signal directly from the visibilities measured in radio-interferometric observations. The salient features of the TGE are: (1) it deals with the gridded data which makes it computationally very fast, (2) it avoids a positive noise bias which normally arises from the system noise inherent to the visibility data, and (3) it allows us to taper the sky response and thereby suppresses the contribution from unsubtracted point sources in the outer parts and the side lobes of the antenna beam pattern. We also summarize earlier work where the TGE was used to measure the C _? of the DGSE using 150 MHz GMRT data. Earlier measurements of C _? are restricted to \(\ell \le \ell _{\max } \sim 10^{3}\) for the DGSE, the signal at the larger ? values is dominated by the residual point sources after source subtraction. The higher sensitivity of the upcoming SKA1 Low will allow the point sources to be subtracted to a fainter level than possible with existing telescopes. We predict that it will be possible to measure the C _? of the DGSE to larger values of \(\ell _{\max }\) with SKA1 Low. Our results show that it should be possible to achieve \(\ell _{\max }\sim 10^{4}\) and ～10⁵ with 2 minutes and 10 hours of observations respectively.     

New Hamiltonian expansions adapted to the Trojan problem

A number of studies, referring to the observed Trojan asteroids of various planets in our Solar System, or to hypothetical Trojan bodies in extrasolar planetary systems, have emphasized the importance of so-called secondary resonances in the problem of the long term stability of Trojan motions. Such resonances describe commensurabilities between the fast, synodic, and secular frequency of the Trojan body, and, possibly, additional slow frequencies produced by more than one perturbing bodies. The presence of secondary resonances sculpts the dynamical structure of the phase space. Hence, identifying their location is a relevant task for theoretical studies. In the present paper we combine the methods introduced in two recent papers (Páez and Efthymiopoulos in Celest Mech Dyn Astron 121(2):139, 2015; Páez and Locatelli in MNRAS 453(2):2177, 2015) in order to analytically predict the location of secondary resonances in the Trojan problem. In Páez and Efthymiopoulos (2015), the motion of a Trojan body was studied in the context of the planar Elliptic Restricted Three Body or the planar Restricted Multi-Planet Problem. It was shown that the Hamiltonian admits a generic decomposition \(H=H_b+H_{sec}\). The term \(H_b\), called the basic Hamiltonian, is a model of two degrees of freedom characterizing the short-period and synodic motions of a Trojan body. Also, it yields a constant ‘proper eccentricity’ allowing to define a third secular frequency connected to the body’s perihelion precession. \(H_{sec}\) contains all remaining secular perturbations due to the primary or to additional perturbing bodies. Here, we first investigate up to what extent the decomposition \(H=H_b+H_{sec}\) provides a meaningful model. To this end, we produce numerical examples of surfaces of section under \(H_b\) and compare with those of the full model. We also discuss how secular perturbations alter the dynamics under \(H_b\). Secondly, we explore the normal form approach introduced in Páez and Locatelli (2015) in order to find an ‘averaged over the fast angle’ model derived from \(H_b\), circumventing the problem of the series’ limited convergence due to the collision singularity at the 1:1 MMR. Finally, using this averaged model, we compute semi-analytically the position of the most important secondary resonances and compare the results with those found by numerical stability maps in specific examples. We find a very good agreement between semi-analytical and numerical results in a domain whose border coincides with the transition to large-scale chaotic Trojan motions.     

Instability of radial modes at the Wolf-Rayet evolutionary stage

The evolution of a Population-I star with an initial mass M _ZAMS = 60 M _⊙ has been calculated. At the stage when a red giant turns into an early-type helium star, the vast bulk of the stellar mass is concentrated in a compact core surrounded by an extended envelope that is unstable with respect to radial oscillations. The range of effective temperatures within which the instability arises extends to T _eff ? 10⁵ K. For the models corresponding to the Wolf-Rayet evolutionary stage (5 × 10⁴ K ≤ T _eff ≤ 1.05 × 10⁵ K), hydrodynamic calculations of self-exciting radial stellar pulsations have been performed. The pulsational instability develops in a time interval comparable to the dynamic timescale. Once the amplitude has ceased to grow, the pulsational motions are nonlinear traveling waves propagating from the core boundary to the stellar surface. The velocity amplitude of the outer layers is 500 km s^?1 < ΔU < 10³ km s^?1, depending on the effective temperature. During the evolution of a helium star, the mean ratio of the maximum expansion velocity of the outer layers to the local escape velocity decreases and lies within the range 0.25 < U _max/v _esc < 0.6 for the models considered. The nonlinearity of the stellar pulsations is responsible for the increase in the mean radius \(\bar r\) of the Lagrangian layers compared to the equilibrium radius r _eq. The effect of the increase in mean radius decreases with rising effective temperature from\(\bar r\)/r ～ 10 at T _eff = 7 × 10⁴ K to \(\bar r\)/r ≈ 2 at T _eff = 10⁵ K. The radial pulsation periods for the models considered lie within the range 0.1 day ≤ Π ≤ 1.6 day and the amplitude of the bolometric magnitude variations does not exceed 0 _. ^m 2.