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)}.
       
    

     

The matter in the Big-Bang model is dust and not any arbitrary perfect fluid!

We consider a spherically symmetric general relativistic perfect fluid in its comoving frame. It is found that, by integrating the local energy momentum conservation equation, a general form of g ₀₀ can be obtained. During this study, we get a cue that an adiabatically evolving uniform density isolated sphere having ρ(r,t)=ρ ₀(t), should comprise “dust” having p ₀(t)=0; as recently suggested by Durgapal and Fuloria (J. Mod. Phys. 1:143, 2010) In fact, we offer here an independent proof to this effect. But much more importantly, we find that for the homogeneous and isotropic Friedmann-Robertson-Walker (FRW) metric having p(r,t)=p ₀(t) and ρ(r,t)=ρ ₀(t), \(g_{00} = e^{-2p_{0}/(p_{0} +\rho_{0})}\). But in general relativity (GR), one can choose an arbitrary t→t _?=f(t) without any loss of generality, and thus set g ₀₀(t _?)=1. And since pressure is a scalar, this implies that p ₀(t _?)=p ₀(t)=0 in the Big-Bang model based on the FRW metric. This result gets confirmed by the fact the homogeneous dust metric having p(r,t)=p ₀(t)=0 and ρ(r,t)=ρ ₀(t) and the FRW metric are exactly identical. In other words, both the cases correspond to the same Einstein tensor \(G^{a}_{b}\) because they intrinsically have the same energy momentum tensor \(T^{a}_{b}=\operatorname {diag}[\rho_{0}(t), 0,0, 0]\).     

On the possibility of determining the imaginary part of the complex refractive index of aerosol particles in an individual altitudinal cloud layer of Jupiter’s atmosphere

We suggest the method for determining the imaginary part n _i of the complex refractive index of aerosol particles forming a cloud layer at a specified altitude in the atmosphere of a giant planet. From the data of spectral measurements of the geometric albedo of Jupiter (carried out in 1993), the value of n _i was calculated for the whole atmospheric column and the pressure range of 0.52 to 0.78 bar in the cloud layer presumably composed of ammon _i um hydrosulfides. The values of n _i obtained for the cloud layer and the whole atmospheric column substantially differ and amount to 0.00098 and 0.00012, respectively.     

Spectroscopic and photometric characteristics of stars in the field of the cluster vdB 130

The observational data for 24 stars toward the young cluster vdB 130 are analyzed. The spectroscopic and photometric BV RIJHK observations have been carried out with the following telescopes: 6-m at the Special Astrophysical Observatory of the Russian Academy of Sciences, 60-cm at the Southern Station of the Moscow State University, and 2.5-m at the Caucasus Observatory of the Sternberg Astronomical Institute of theMoscow State University. Nine stars previously selected as cluster members have been found to belong to different subtypes of type B. A minimum color excess toward the cluster, E(B ? V) = 0.9 mag, has been revealed for the vdB 130 stars lying outside the molecular cloud. Maximum color excesses, E(B ? V) = 1.3?1.4 mag, have been found in the spectra of cluster stars 1r and 5r observed in dust blobs. Inside the cluster R _v is shown to differ from the standard one. The overwhelming majority of the remaining investigated stars belong to late types and have minor color excesses (≤0.3) typical of close distances.     

Structure of the current sheets in the near-Mars magnetotail. Maven observations

During the last 15 years, the Current Sheets (CSs) have been intensively studied in the tail of the terrestrial magnetosphere, where protons are the dominated ion component. On the contrary, in the Martian magnetotail heavy ions (O⁺ and⁺ ₀) usually dominate while the abundance of protons can be negligible. Hence it is interesting to study the spatial structure and plasma characteristics of such “oxygen” CSs. MAVEN spacecraft (s/c) currently operating on the Martian orbit with a unique set of scientific instruments allows observation of the magnetic field and three-dimensional distribution functions of various ion components and electrons with a high time resolution. In this paper, we analyse nine intervals of the CSs observed by MAVEN in the near-Mars tail at the distances from the planet ~1.5–1R _M , where R _M is the radius of Mars. We analyse the spatial structure of the CSs and estimate their thickness for different magnetic configurations and relative abundance of the heavy and light ions in the sheets. It is shown that, similarly to the CSs in the Earth’s magnetotail, the thickness and complexity of the spatial structure of the Maritan CSs (i.e. the presence of embedded and / or peripheral current structures) depend on the magnetic configuration of the sheets, which, in turn, affects the fraction of the quasi-adiabatic particles in the CSs.     

Rational approximation formula for Chandrasekhar’s <Emphasis Type="Italic">H</Emphasis>-function for isotropic scattering

In this work, we first establish a simple procedure to obtain with 11-figure accuracy the values of Chandrasekhar’s H-function for isotropic scattering using a closed-form integral representation and the Gauss-Legendre quadrature. Based on the numerical values of the function produced by this method for various combinations of ? ₀, the single scattering albedo, and μ, the cosine of the zenith angle θ of the direction of radiation emergent from or incident upon a semi-infinite scattering-absorbing medium, we propose a rational approximation formula with μ ^1/4 and \(\sqrt{1-\varpi_{0}}\) as the independent variables. This allows us to reproduce the correct values of H(? ₀,μ) within a relative error of 2.1×10^?5 without recourse to any iterative procedure or root-finding process.     

Interstellar extinction toward the ultradeep Galactic field of the Chandra observatory from 2MASS data

We have measured the interstellar extinction in the region of ultradeep Galactic-field observations by the Chandra telescope (l ^II, b ^II) ≈ 0.1–1.42 using photometric data from the 2MASS infrared allsky survey. The angular resolution of our interstellar extinction map is 1′.8. We show that the interstellar extinction has a minimum, A _V ～ 3.4, near the center of the Chandra field of view and increases to A _V ～ 5.8–6 at the edge of the field of view. In addition, we show that the bulk of the extinction is gained in the Galactic disk and is approximately the same for all bulge stars. Our results will be subsequently used to process the Chandra data and to estimate the properties of the stellar population in this region.     

On a family of well behaved perfect fluid balls as astrophysical objects in general relativity

A family of well behaved perfect fluid balls has been derived starting with the metric potential g ₄₄=B(1+Cr ²) ⁿ for all positive integral values of n. For n≥4, the members of this family are seen to satisfy the various physical conditions e.g. c ² ρ≥p≥0,dp/dr<0,dρ/dr<0, along with the velocity of sound \((\sqrt{dp/c^{2}d\rho} )< 1\) and the adiabatic index ((p+c ² ρ)/p)(dp/(c ² dρ))>1. Also the pressure, energy density, velocity of sound and ratio of pressure and energy density are of monotonically decreasing towards the pressure free interface (r=a). The fluid balls join smoothly with the Schwarzschild exterior model at r=a. The well behaved perfect fluid balls so obtained are utilised to construct the superdense star models with their surface density 2×10¹⁴  gm/cm³. We have found that the maximum mass of the fluid balls corresponding to various values of n are decreasing with the increasing values of n. Over all maximum mass for the whole family turns out to be 4.1848M _Θ and the corresponding radius as 19.4144 km while the red shift at the centre and red shift at surface as Z ₀=1.6459 and Z _a =0.6538 respectively this all happens for n=4. It is interesting to note that for higher values of n viz n≥170, the physical data start merging with that of Kuchowicz superdense star models and hence the family of fluid models tends to the Kuchowicz fluid models as n→∞. Consequently the maximum mass of the family of solution can not be less than 1.6096 M _Θ which is the maximum mass occupied by the Kuchowicz superdense ball. Hence each member of the family for n≥4 provides the astrophysical objects like White dwarfs, Quark star, typical neutron star.     

A model of possible variations of the galactic cosmic ray intensity over the recent billion years

Based on the analysis of published data on exposure ages of iron meteorites determined with the ⁴⁰K/K method (T _K) and ages calculated using short-lived cosmogenic radionuclides (with the half-life T _1/2 < 1 Myr) in combination with stable cosmogenic isotopes of noble gases (T_RS), the following results have been obtained. (1) The distribution of T _RS ages (106 values) has an exponential shape, similar to that for ordinary chondrites, but different from the distribution of T _K ages (80 values). The difference is most likely due to small amounts of data for meteorites with low T _K ages (less than ~200–300 Myr). The latter can be ascribed to the difficulty of measurement of small concentrations of cosmogenic potassium isotopes. This circumstance makes the selection of meteorites with ⁴⁰K/K ages nonrepresentative and casts doubt on the correctness of conclusions about the variations of the intensity of galactic cosmic rays (GCR) based on the analysis of distribution of these ages. (2) The magnitude of the known effect (systematic overestimation of T _K ages in comparison with T _RS ages) has been refined. The value k = T _K/T _RS = 1.51 ± 0.03 is acquired for the whole population of data. We have shown the inefficiency of the explanation of this effect on account of an exponential change in the GCR intensity (I _T ) with time (T) according to the relation I _T = I ₀exp(–γT) over the whole range of ages of iron meteorites. (3) In order to explain the overestimation of T _K ages in comparison with T _RS ages, a model has been proposed, according to which the GCR intensity has exponentially increased in the interval of 0–1500 Myr governed by the relation: I _T = I _{T = 1500} (1 + αexp(–βT)). For one of the variants of this model, the GCR intensity has exponentially increased by a factor of two only over the recent ~300 Myr, remaining approximately constant for the rest of the time. The data acquired with the use of this model indicate that the measured T _K ages are close to the actual time that the meteorites existed in space; the data are in agreement with the observed exponential distribution of T _RS ages.     

Investigation of heliogeoactivity impact on the dynamics of orbital parameters of Earth’s artificial satellites. I

The influence of active processes on the Sun and their response on the dynamics of Earth’s artificial satellites has been investigated. The relationship between the characteristics of solar activity and variations of the periods P of the orbital motion of Earth’s artificial satellites has been found. These variations mainly indicate the variations in the Earth’s atmosphere density caused by solar activity (index F_10.7) and geomagnetic activity (ΣK_p index). High values of the correlation coefficients between P and ^F_10.7 (–0.77…–0.91) and between P and ΣK_p (–0.67…–0.89) exhibit significant effect of solar and geomagnetic activity on the orbital periods of satellites.     

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.     

Evolutionary studies of the young star clusters: NGC 1960, NGC 2453 and NGC 2384

We report the analysis of the young star clusters NGC 1960, NGC 2453 and NGC 2384 observed in the J (1.12 μm), H (1.65 μm) and K′ (2.2 μm) bands. Estimates of reddening, distance and age as E(B?V)=0.25, d=1380 pc and t=31.6 to 125 Myr for NGC 1960, E(B?V)=0.47, d=3311 pc and t=40 to 200 Myr for NGC 2453 and E(B?V)=0.25, d=3162 pc and t=55 to 125 Myr for NGC 2384 have been obtained. Also, we have extended the color–magnitude diagrams of these clusters to the fainter end and thus extended the luminosity functions to fainter magnitudes. The evolution of the main sequence and luminosity functions of these clusters have been compared with themselves as well as Lyngå 2 and NGC 1582.     

Fundamental parameters and intrinsic colors of F,G, and K supergiants and classical cepheids

We used high-resolution echelle spectra with high signal-to-noise ratio to determine with a high degree of accuracy some atmospheric parameters (T _eff, log g and [Fe/H]) for 68 non-variable supergiants of types F, G, and K and 26 classical Cepheids in 302 pulsation phases. Very accurate effective temperatures, with errors of only 10–30 K, were determined by the line-depth ratio method. We found that the observed intrinsic color indices (B ? V)₀ can be related to these parameters: (B ? V)₀ = 57.984? 10.3587(log T _eff)² + 1.67572(log T _eff)³ ? 3.356 log g+ 0.321 V _t + 0.2615[Fe/H] + 0.8833log g(log T _eff). With this empirical relation, the intrinsic colors of individual supergiants and classical Cepheids of spectral types F0-K0 and of luminosity classes I and II can be estimated with an accuracy as high as 0.05 ^m , which is comparable to the accuracy of the most elaborate photometric procedures. In view of large distances to supergiants, the method we propose here allows a large-scale mapping of interstellar extinction with an accuracy of 0.1–0.2 ^m in a quite large region of the Galaxy.     

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.     

Rotation curve and mass distribution in the Galaxy from the velocities of objects at distances up to 200 kpc

Three three-component (bulge, disk, halo) model Galactic gravitational potentials differing by the expression for the dark matter halo are considered. The central (bulge) and disk components are described by the Miyamoto–Nagai expressions. The Allen–Santillán (I), Wilkinson–Evans (II), and Navarro–Frenk–White (III) models are used to describe the halo. A set of present-day observational data in the range of Galactocentric distances R from 0 to 200 kpc is used to refine the parameters of thesemodels. For the Allen–Santillán model, a dimensionless coefficient γ has been included as a sought-for parameter for the first time. In the traditional and modified versions, γ = 2.0 and 6.3, respectively. Both versions are considered in this paper. The model rotation curves have been fitted to the observed velocities by taking into account the constraints on the local matter density ρ _⊙ = 0.1 M _⊙ pc^?3 and the force K _z =1.1/2πG = 77 M _⊙ pc^?2 acting perpendicularly to the Galactic plane. The Galactic mass within a sphere of radius 50 kpc, M _G (R ≤ 50 kpc) ≈ (0.41 ± 0.12) × 10¹² M _⊙, is shown to satisfy all three models. The differences between the models become increasingly significant with increasing radius R. In model I, the Galactic mass within a sphere of radius 200 kpc at γ = 2.0 turns out to be greatest among the models considered, M _G (R ≤ 200 kpc) = (1.45 ±0.30)× 10¹² M _⊙, M _G (R ≤ 200 kpc) = (1.29± 0.14)× 10¹² M _⊙ at γ = 6.3, and the smallest value has been found in model II, M _G (R ≤ 200 kpc) = (0.61 ± 0.12) × 10¹² M _⊙. In our view, model III is the best one among those considered, because it ensures the smallest residual between the data and the constructed model rotation curve provided that the constraints on the local parameters hold with a high accuracy. Here, the Galactic mass is M _G (R ≤ 200 kpc) = (0.75 ± 0.19) × 10¹² M _⊙. A comparative analysis with the models by Irrgang et al. (2013), including those using the integration of orbits for the two globular clusters NGC 104 and NGC 1851 as an example, has been performed. The third model is shown to have subjected to a significant improvement.     

Analysis of the Z distribution of young objects in the Galactic thin disk

We have obtained new estimates of the Sun’s distance from the symmetry plane Z _⊙ and the vertical disk scale height h using currently available data on stellar OB associations, Wolf–Rayet stars, HII regions, and Cepheids. Based on individual determinations, we have calculated the mean Z _⊙ = ?16 ± 2 pc. Based on the model of a self-gravitating isothermal disk for the density distribution, we have found the following vertical disk scale heights: h = 40.2 ± 2.1 pc from OB associations, h = 47.8 ± 3.9 pc from Wolf–Rayet stars, h = 48.4 ± 2.5 pc from HII regions, and h = 66.2 ± 1.6 pc from Cepheids. We have estimated the surface, Σ = 6 kpc^?2, and volume, D(Z _⊙) = 50.6 kpc^?3, densities from a sample of OB associations. We have found that there could be ～5000 OB associations in the Galaxy.     

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.     

Evolution of Intergalactic Gas in the Neighborhood of Dwarf Galaxies and Its Manifestations in the HI 21 cm Line

Low-mass galaxies are known to have played the crucial role in the hydrogen reionization in the Universe. In this paper we investigate the contribution of soft x-ray radiation (E ~ 0.1–1 keV) from dwarf galaxies to hydrogen ionization during the initial reionization stages. The only possible sources of this radiation in the process of star formation in dwarf galaxies during the epochs preceding the hydrogen reionization epoch are hot intermediate-mass stars (M ~ 5–8 M_⊙) that entered the asymptotic giant branch (AGB) stage and massive x-ray binaries. We analyze the evolution of the intergalactic gas in the neighborhood of a dwarf galaxy with a total mass of 6 × 10⁸M_⊙ formed at the redshift of z ~ 15 and having constant star-formation rate of 0.01–0.1 M_⊙ yr^?1 over a starburst with a duration of up to 100 Myr. We show that the radiation from AGB stars heats intergalactic gas to above 100 K and ensures its ionization x_e ? 0.03 within about 4–10 kpc from the galaxy in the case of a star-formation rate of star formation 0.03–0.1 M_⊙ yr^?1, and that after the end of the starburst this region remains quasi-stationary over the following 200–300 Myr, i.e., until z ~ 7.5. Formation of x-ray binaries form in dwarf galaxies at z ~ 15 results in a 2–3 and 5–6 times greater size of the ionized and heated region compared to the case where ionization is produced by AGB stars exclusively, if computed with the “x-ray luminosity–star-formation rate” dependence (L_X ~ f_XSFR) factor f_X = 0.1 and f_X ~ 1, respectively. For f_X ? 0.03 the effect of x-ray binaries is smaller that that of AGB star population. Lyα emission, heating, and ionization of the intergalactic gas in the neighborhood of dwarf galaxies result in the excitation of the 21 cm HI line. We found that during the period of the starburst end at z ~11.5–12.5 the brightness temperature in the neighborhood of galaxies is 15–25 mK and the region where the brightness temperature remains close to its maximum has a size of about 12–30 kpc. Hence the epoch of the starburst end is most favorable for 21 cm HI line observations of dwarf galaxies, because at that time the size of the region of maximum brightness temperature is the greatest over the entire evolution of the dwarf galaxy. In the case of the sizes corresponding to almost 0.’1 for z ~ 12 regions with maximum emission can be detected with the Square Kilometre Array, which is currently under construction.     

Global structure of the local universe according to 2MRS survey

We report the results of a statistical analysis of the space distribution of galaxies of the 2MRS catalog, which contains redshifts of 43533 galaxies of the 2MASS all-sky IR survey. Because of the unique features of the 2MRS survey, such as its 90% sky coverage, galaxy selection in the IR, the complete incorporation of the old stellar population of galaxies, weakness of the dust extinction effects, and the smallness of the k- and e-corrections allowed us to determine the statistical properties of the global distribution of galaxies in the Local Universe. We took into account the main methodological factors that distort the theoretically expected relations compared to those actually observed. We construct the radial galaxy number counts N(R), SL(R, r) statistics, and the complete correlation function (conditional density) Γ(r) for volume-limited (VL) galaxy samples. The observed conditional density Γ(r) in the redshift space is independent of the luminosity of galaxies and has the form of a power-law function with exponent γ ≈ 1.0 over a large range scale-length spanning from 0.1 to 100 Mpc. We compare the statistical properties of the space distribution of galaxies of the 2MRS catalog with the corresponding properties of simulated catalogs: stochastic fractal distributions and galaxies of the Millennium catalog.     

Effect of Coulomb collisions on the particle acceleration in collapsing magnetic traps

The problem of particle acceleration in collapsing magnetic traps in the solar corona has been solved by taking into account the particle scattering and braking in the high-temperature plasma of solar flares. The Coulomb collisions are shown to be weak in traps with lifetimes t _l < 10 s and strong for t _l > 100 s. In the approximation of strong collisions, collapsing magnetic traps are capable of confining up to 20% of the injected particles in the corona for a long time. In the collisionless approximation, this value exceeds 90%. The question about the observational manifestations of collisions is examined. For collision times comparable to t _l , the electron spectrumat energies above 10 keV is shown to be a double-power-law one. Such spectra were found by the RHESSI satellite in flares.