Chromospheric Height and Density Measurements in a Solar Flare Observed with RHESSI – II. Data Analysis

We present an analysis of hard X-ray imaging observations from one of the first solar flares observed with the Reuven Ramaty High-Energy Solar Spectroscopic Imager (RHESSI) spacecraft, launched on 5 February 2002. The data were obtained from the 22 February 2002, 11:06 UT flare, which occurred close to the northwest limb. Thanks to the high energy resolution of the germanium-cooled hard X-ray detectors on RHESSI we can measure the flare source positions with a high accuracy as a function of energy. Using a forward-fitting algorithm for image reconstruction, we find a systematic decrease in the altitudes of the source centroids z(ε) as a function of increasing hard X-ray energy ε, as expected in the thick-target bremsstrahlung model of Brown. The altitude of hard X-ray emission as a function of photon energy ε can be characterized by a power-law function in the ε=15–50 keV energy range, viz., z(ε)≈2.3(ε/20 keV)^−1.3 Mm. Based on a purely collisional 1-D thick-target model, this height dependence can be inverted into a chromospheric density model n(z), as derived in Paper I, which follows the power-law function n _e(z)=1.25×10¹³(z/1 Mm)^−2.5 cm⁻³. This density is comparable with models based on optical/UV spectrometry in the chromospheric height range of h≲1000 km, suggesting that the collisional thick-target model is a reasonable first approximation to hard X-ray footpoint sources. At h≈1000–2500 km, the hard X-ray based density model, however, is more consistent with the `spicular extended-chromosphere model' inferred from radio sub-mm observations, than with standard models based on hydrostatic equilibrium. At coronal heights, h≈2.5–12.4 Mm, the average flare loop density inferred from RHESSI is comparable with values from hydrodynamic simulations of flare chromospheric evaporation, soft X-ray, and radio-based measurements, but below the upper limits set by filling-factor insensitive iron line pairs.     

Effect of fluctuations in Doppler width on the center of strong absorption lines

Stochastic temperatures and turbulence are characterized by average velocities u _th and < u _turb  > ≡ u ₀ and fluctuations u￠_th {u'_{th}} and u′ (<u′ > = 0). Thus, the Doppler width of a line also has a fluctuating component Dl￠_D \Delta {\lambda '_D} . Observed spectra correspond to the radiative flux averaged over time and over a star’s surface, <H_λ>. Usually, only the average velocities u _th and u ₀ are taken into account in photospheric models and these yield the Doppler width Dl_D⁽⁰⁾ \Delta \lambda_D^{(0)} of a line in the customary way. The fluctuations Dl￠_D \Delta {\lambda '_D} mean that near a line center the average absorption coefficient < α_λ > is larger than the usual α_λ, which depends only on the average velocities u _th and u ₀. This enhances the absorption line near the center and is not explained by the photospheric models. This new statistical effect depends on the wavelength of the line. A comparison of observed lines with model profiles yields an estimate for the average level of fluctuations in the Doppler width, h = á | Dl￠_D | ñ

The PL calibration for Milky Way Cepheids and its implications for the distance scale

The rationale behind recent calibrations of the Cepheid PL relation using the Wesenheit formulation is reviewed and reanalyzed, and it is shown that recent conclusions regarding a possible change in slope of the PL relation for short-period and long-period Cepheids are tied to a pathological distribution of HST calibrators within the instability strip. A recalibration of the period-luminosity relation is obtained using Galactic Cepheids in open clusters and groups, the resulting relationship, described by log L/L _⊙=2.415(±0.035)+1.148(±0.044)log P, exhibiting only the moderate scatter expected from color spread within the instability strip. The relationship is confirmed by Cepheids with HST parallaxes, although without the need for Lutz-Kelker corrections, and in general by Cepheids with revised Hipparcos parallaxes, albeit with concerns about the cited precisions of the latter. A Wesenheit formulation of W _V =−2.259(±0.083)−4.185(±0.103)log P for Galactic Cepheids is tested successfully using Cepheids in the inner regions of the galaxy NGC 4258, confirming the independent geometrical distance established for the galaxy from OH masers. Differences between the extinction properties of interstellar and extragalactic dust may yet play an important role in the further calibration of the Cepheid PL relation and its application to the extragalactic distance scale.     

Intensity Distribution and Contrast of the Solar EUV Network

Intensity distributions of the EUV network and the cell interior in the solar atmosphere have been obtained in fourteen emission lines from Solar and Heliospheric Observatory (SOHO)/Coronal Diagnostic Spectrometer (CDS) observations. The formation temperature of the observed lines is in the range log T=4.90 – 6.06 (T in Kelvin), and hence they represent increasing heights in the solar atmosphere from the upper chromosphere and the transition region to the low corona. Intensity distributions of the cell interior have been found to be different in the quiet Sun and the coronal hole even at the lower transition region, which is at variance with some earlier results. The intensity contrast of the network with respect to the cell interior has been obtained for each line, and differences in the quiet Sun and the coronal hole have been examined. The network contrast, in general, is lower for the coronal hole as compared to the quiet Sun, but becomes equal to it in the upper transition region. The maximum contrast for both the regions is at about log T=5.3. Also obtained are the relative contributions of the network and the cell interior to the total intensity. The implications of the results for models of the transition region are briefly mentioned.     

Is the Entropy S q Extensive or Nonextensive?

The cornerstones of Boltzmann-Gibbs and nonextensive statistical mechanics respectively are the entropies S _BG ≡ −k ∑ _{i = 1} ^W p _i ln p _i and S _q ≡ k (1−∑ _{i = 1} ^W p _i ^q )/(q−1) (q∊ℜ S ₁ = S _BG ). Through them we revisit the concept of additivity, and illustrate the (not always clearly perceived) fact that (thermodynamical) extensivity has a well defined sense only if we specify the composition law that is being assumed for the subsystems (say A and B). If the composition law is not explicitly indicated, it is tacitly assumed that A and B are statistically independent. In this case, it immediately follows that S _BG (A+B) = S _BG (A)+S _BG (B), hence extensive, whereas S _q (A+B)/k = [S _q (A)/k]+[S _q (B)/k]+(1−q)[S _q (A)/k][S _q (B)/k], hence nonextensive for q ≠ 1. In the present paper we illustrate the remarkable changes that occur when A and B are specially correlated. Indeed, we show that, in such case, S _q (A+B) = S _q (A)+S _q (B) for the appropriate value of q (hence extensive), whereas S _BG (A+B) ≠ S _BG (A)+S _BG (B) (hence nonextensive). We believe that these facts substantially improve the understanding of the mathematical need and physical origin of nonextensive statistical mechanics, and its interpretation in terms of effective occupation of the W a priori available microstates of the full phase space. In particular, we can appreciate the origin of the following important fact. In order to have entropic extensivity (i.e., lim _N→∞ S(N)/N < ∞, where N≡ numberof elements of the system), we must use (i) S _BG , if the number W ^eff of effectively occupied microstates increases with N like W ^{{eff}}∼ W ∼ μ ^N (μ ≥ 1); (ii) S _q with q = 1−1/ρ, if W ^{{eff}}∼ N^ρ < W (ρ ≥ 0). We had previously conjectured the existence of these two markedly different classes. The contribution of the present paper is to illustrate, for the first time as far as we can tell, the derivation of these facts directly from the set of probabilities of the W microstates.     

Intensity and polarization line profiles in a semi-infinite Rayleigh-scattering planetary atmosphere. II. Variations of equivalent width over the disk

The study of the variation of equivalent width in a Rayleighscattering planetary atmosphere along the intensity equator and along the mirror meridian on whichμ =μ ₀ shows that the equivalent widths decrease monotonically towards the poles, the limb and the terminator with the following characteristics: (i) the weakest lines exhibit the maximum change; (ii) theI ^e _r component shows more change than theI ^e _r component; (iii) the decrease towards the limb or the terminator is not as sharp as that towards the poles; (iv)I ^e _r component shows more decrease towards the limb whileI ^e _r component shows more decrease towards the terminator; and (v) the relationW (μ, φ;μ ₀,φ ₀)= W (μ ₀,φ ₀;μ, φ) holds for the total intensity. These results are qualitatively in agreement with the observations of absorption bands in the spectra of Venus, Jupiter and Saturn.     

Some Peculiarities of Intensity Oscillations in Ca <Emphasis Type="SmallCaps">ii</Emphasis> Lines under Coronal Holes

Observations of the central intensity of the Ca ii K and 849.8 nm lines are used to derive the ratios of the oscillation power in the frequency ranges of the “five-minute” (W ₅) and “three-minute” (W ₃) oscillations. It is shown that at high significance level ratios, (W ₅/W ₃) >1 at coronal hole bases, and W ₅/W ₃ ≈1 in quiet chromospheric areas far from holes.     

Energy-Dependent Timing of Thermal Emission in Solar Flares

We report solar flare plasma to be multi-thermal in nature based on the theoretical model and study of the energy-dependent timing of thermal emission in ten M-class flares. We employ high-resolution X-ray spectra observed by the Si detector of the “Solar X-ray Spectrometer” (SOXS). The SOXS onboard the Indian GSAT-2 spacecraft was launched by the GSLV-D2 rocket on 8 May 2003. Firstly we model the spectral evolution of the X-ray line and continuum emission flux F(ε) from the flare by integrating a series of isothermal plasma flux. We find that the multi-temperature integrated flux F(ε) is a power-law function of ε with a spectral index (γ)≈−4.65. Next, based on spectral-temporal evolution of the flares we find that the emission in the energy range E=4 – 15 keV is dominated by temperatures of T=12 – 50 MK, while the multi-thermal power-law DEM index (δ) varies in the range of −4.4 and −5.7. The temporal evolution of the X-ray flux F(ε,t) assuming a multi-temperature plasma governed by thermal conduction cooling reveals that the temperature-dependent cooling time varies between 296 and 4640 s and the electron density (n _e) varies in the range of n _e=(1.77 – 29.3)×10¹⁰ cm⁻³. Employing temporal evolution technique in the current study as an alternative method for separating thermal from nonthermal components in the energy spectra, we measure the break-energy point, ranging between 14 and 21±1.0 keV.     

A dynamical model of type I supernova atmosphere with the velocity gradient

A compact structure of a low-mass Type I presupernovae is assumed to be an essential feature of the hydrodynamical problem dealing with the supernova Type I (SNI) envelope outbursts. This structure is characterized by a degenerate carbon-oxygen core, which suffers a thermonuclear explosion of carbon fuel (M ₀≃1.40M _⊙), and by a compact lowmass envelope (M _e ≲0.1M _⊙) with external radiusR _e≃10⁹ cm. The parameters, of this hydrostatic envelope are specified and then, for a relatively small explosion energy, ofW ₀≃(2–10)×10⁴⁹ erg, hydrodynamic problem of the envelope ejection is solved numerically. This energy comes from neutrino-induced detonative carbon burning. The resulting structure of the SNI atmosphere expanding with the velocity gradient can be employed for an interpretation of the observed SNI spectra. In accordance with our previous papers, the SNI light curves are considered to occur due to an additional slow (with time-scale 10⁶–10⁷ s) release of the bulk of the SNI energy,W≃10⁵¹, erg. The slow energy release does not, however, affect the structure of the outermost expanding layers of the envelope which are responsible for the SNI spectra. A short (Δt≃10⁻² s) burst of soft (2–10 keV) X-rays with total radiated energy of about 10⁴⁰ erg is found to appear 10–20 days before the SNI optical maximum.     

A Search for Warm Neutral Gas Associated with the Giant HII Region NGC 604

The characteristics of the halo gas and its interaction with the galactic disk in spiral galaxies are poorly known; this is particularly true for the warm neutral gas associated with HVCs and galactic chimneys. The detection of absorption features such as the NaI D or the CaII (H,K)lines is instrumental to study its detailed physical properties but requires very long integration times. In this work very deep optical spectra of NGC 604, the brightest giant HII region among the nearby spirals, are presented. The detection of two absorption components at LSR velocities -255 km/s and −20 km/s respectively, is reported; the first component is associated with the HII region. The inferred line width after deconvolution is 155 km/s; this large width is produced by the blending of the multiple absorption components produced by the diverse sources of internal motion (expanding shells and general turbulence). The radial velocity of the CaII absorption is slightly larger than the measured in the HII emission lines suggesting a possible flow of gas into the halo above the young star cluster. The large ratio x = W _λ (NaID_2)/W _λ (CaIIK) = 0.7indicates the probable presence of shocks which release Ca from the dust grains into the gas phase. The lower velocity component most likely trace galactic gas. This revised version was published online in July 2006 with corrections to the Cover Date.     

Spectroscopic study of the variable planetary nebula IC 4997

Based on our spectroscopic observations of the variable planetary nebula IC 4997 in 2003–2009, we have obtained the relative fluxes in optical emission lines. The interstellar extinction c = 0.35 has been found from the Balmer decrement by taking into account the effect of self-absorption in hydrogen lines in dense nebular regions. The variations in the Balmer decrement point to variability of the self-absorption and circumstellar extinction. We have investigated the variations in the relative intensities of some spectral lines and their ratios with time. The drop in the ratios F(λ4363)/F(Hγ) and F(λ363)/F(λ4959) that began back in 1990–1995 has continued, suggesting a decrease in the electron density and temperature in the central nebular region. The ratio F(λ6731)/F(λ6717) has remained constant. It gives an estimate for the electron density in the outer regions of IC 4997, N _e ∼ 10⁴ cm⁻³.     

Do EUV Nanoflares Account for Coronal Heating?

Recent observations with EUV imaging instruments such as SOHO/EIT and TRACE have shown evidence for flare-like processes at the bottom end of the energy scale, in the range of E _th≈10²⁴–10²⁷ erg. Here we compare these EUV nanoflares with soft X-ray microflares and hard X-ray flares across the entire energy range. From the observations we establish empirical scaling laws for the flare loop length, L(T)∼T, the electron density, n _e(T)∼T ², from which we derive scaling laws for the loop pressure, p(T)∼T ³, and the thermal energy, E _th∼T ⁶. Extrapolating these scaling laws into the picoflare regime we find that the pressure conditions in the chromosphere constrain a height level for flare loop footpoints, which scales with h _eq(T)∼T ^−0.5. Based on this chromospheric pressure limit we predict a lower cutoff of flare loop sizes at L _∖min≲5 Mm and flare energies E _∖min≲10²⁴ erg. We show evidence for such a rollover in the flare energy size distribution from recent TRACE EUV data. Based on this energy cutoff imposed by the chromospheric boundary condition we find that the energy content of the heated plasma observed in EUV, SXR, and HXR flares is insufficient (by 2–3 orders of magnitude) to account for coronal heating.     

Calculation of effective optical depth of absorption line formation in homogeneous semi-infinite planetary atmosphere during anisotropic scattering

The algorithm for determining effective optical thickness of absorption line formation in a plane-parallel homogeneous planetary atmosphere is presented. The case of anisotropic scattering is considered. The results of numerical calculations of τ _e (μ₀) at the scattering angle γ = π for some values of the single scattering albedo λ and the parameter of the Heyney-Greenstein scattering indicatrix g are given. The refined equation for the function T ^m (−μ, μ₀) is presented.     

Determination of mass-loss rates from the first order momentW 1 of unsaturated P Cygni line profiles

The escape probability method introduced by Sobolev to treat the transfer of line photons is used in order to derive the expressions of thenth order momentW _n (E()/E _c –1)(–₁₂) ⁿ ·d of a P Cygni profile formed in rapidly expanding envelopes around a central point-like source under various physical and geometrical conditions.With the only assumption that there is mass-conservation of the species in the flow, we state for the case of optically thin lines that the relation between the first order momentW ₁ and the quantityM _n (level), first established by Castoret al. (1981) under more restrictive conditions, is in fact independent of the type of velocity fieldv(r) and a fortiori of the distribution adopted for the radial opacity ₁₂ ^r (X). These results also remain unchanged when including collisions (0) and/or an additional rotational velocity fieldv (r) in the expanding atmosphere. We investigate the presence of an underlying photospheric absorption line and conclude that for realistic cases, neglecting this boundary condition to the radiative transfer leads to an underestimate of the mass-loss rate by a factor of about 20%. By means of a three-level atom model, we demonstrate that all results derived for a single line transition equally apply for an unresolved doublet profile providedW ₁ andM _n (level) are calculated with the weighted wavelength _D and total oscillator strengthf _D of the doublet.Considering the occultation and inclination effects caused by the finite size of the central core, we refine the value of the multiplicative constant fixing the ratio ofW ₁ toM _n (level). We show that this relation allows a determination of the mass-loss rate with an uncertainly less than 30%, irrespective of the sizeL _max of the atmosphere and of the limb-darkening affecting the stellar core. Reviewing all possible sources of error, we finally conclude that this method of deriving a mass-loss rate from the analysis of an unsaturated P Cygni profile is very powerful. The total uncertainty affecting the determination ofM _n (level) from the measurement ofW ₁ should be smaller than 60%.     

Hamiltonian Stability of Spin–Orbit Resonances in Celestial Mechanics

The behaviour of ‘resonances’ in the spin-orbit coupling in celestial mechanics is investigated in a conservative setting. We consider a Hamiltonian nearly-integrable model describing an approximation of the spin-orbit interaction. The continuous system is reduced to a mapping by integrating the equations of motion through a symplectic algorithm. We study numerically the stability of periodic orbits associated to the above mapping by looking at the eigenvalues of the matrix of the linearized map over the full cycle of the periodic orbit. In particular, the value of the trace of the matrix is related to the stability character of the periodic orbit. We denote by ε_* (p/q) the value of the perturbing parameter at which a given elliptic periodic orbit with frequency p/q becomes unstable. A plot of the critical function ε_* (p/q) versus the frequency at different orbital eccentricities shows significant peaks at the synchronous resonance (for low eccentricities) and at the synchronous and 3:2 resonances (at higher eccentricities) in good agreement with astronomical observations. This revised version was published online in July 2006 with corrections to the Cover Date.     

Minimal Energy and Gravitation

Quantum theory in Robertson – Walker spacetime suggests the existence of a minimal energy ε of the order of 10⁻⁴⁵ erg. Reasonable forms for ε give the expansion factor R=R(t)(t= the cosmic time) with no need of gravitational field equations.Einstein's theory should be modified in gravitational fields of strength less than ε c/ħ ∼ 10⁻⁸ cm/s² where c is the speed of light and ħ is the reduced Planck constant. The cosmological term λ is expected to decrease as the universe expands.In the Appendix, ε is derived from a big bang – big crunch Newtonian cosmology.     

Dust Activity in Comet 67P/Churyumov–Gerasimenko from February 20 to April 20, 2003

Broadband imaging of Comet 67P/Churyumov–Gerasimenko has provided more data on the characterisation of the target of the ESA Rosetta Mission. The comet monitoring between r _h=2.37 and r _h=2.78 AU postperihelion shows a prominent dust coma which extends up to ≈ 25,000 km from the nucleus, and a long dust structure in approximately anti-tail direction, reaching at least 230,000 km, identified as a neck-line structure. The non-isotropic dust emission is detected from the structures in the inner coma, and it is reflected on the slope of linear fits of surface brightness profiles vs. cometocentric projected distance in log–log representation as m ≈ 0.83−0.941. Besides the long dust spike at position angle of 295°, the morphological study of the dust coma confirms the presence of two structures at position angles of 95 and 195° where the overabundance of dust can be as high as 50% at ρ ≤ 30,000 km. The A f ρ parameter derived from our R broadband data is 26.0 and 29.8 cm at r _h=2.37 and 2.48 AU postperihelion. The dust reflectivity S′(λ), a measurement of the dust colour, is 0.061±0.019, a rather neutral colour.     

Investigation of the Neupert Effect in the Various Intervals of Solar Flares

The Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI) gives us a chance to investigate the theoretical Neupert effect using the correlation between the thermal-energy derivative and the nonthermal energy, or the thermal energy and the integral nonthermal energy. Based on this concept, we analyze four M-class RHESSI flares on 13 November 2003, 4 November 2004, 3 and 25 August 2005. According to the evolution of the temperature [T], emission measure [EM], and thermal energy [E _th], each event is divided into three phases during the nonthermal-energy input [ \frac dE_nthdt\frac {\mathrm{d}E_{\mathrm{nth}}}{\mathrm{d}t} in the units of erg s⁻¹]. Phase 1 is identified as the interval before the temperature maximum, while after the thermal-energy maximum is phase 3, between them is phase 2. We find that these four flares show the Neupert effect in phase 1, but not in phase 3. The Neupert effect still works well in the second phase, although the cooling becomes slightly important. We define the parameter μ in the relation of \fracdE_thdt=m\fracdE_nth(t)dt\frac{\mathrm {d}E_{\mathrm{th}}}{\mathrm{d}t}=\mu\frac{\mathrm{d}E_{\mathrm {nth}}(t)}{\mathrm{d}t} or E_th(t₀)=mò₀^t₀\fracdE_nth(t)dt dtE_{\mathrm{th}}(t_{0})=\mu\int_{0}^{t_{0}}\frac{\mathrm{d}E_{\mathrm{nth}}(t)}{\mathrm{d}t}\,\mathrm{d}t when the cooling is ignored in phase 1. Considering the uncertainties in estimating the energy from the observations, it is not possible to precisely determine the fraction of the known energy in the nonthermal electrons transformed into the thermal energy of the hottest plasma observed by RHESSI. After a rough estimate of the flare volume and the assumption of the filling factor, we investigate the parameter μ in these four events. Its value ranges from 0.02 to 0.20, indicating that a small fraction (2% – 20%) of the nonthermal energy can be efficiently transformed into thermal energy, which is traced by the soft X-ray emission, and the bulk of the energy is lost possibly due to cooling.     

Equivalence principle (EP) and solar system constraints on $R(1\pm\epsilon\ln({R \over R_{c}}))$ model of gravity

Experiments on the violation of equivalence principle (EP) and solar system give a number of constraints in which any modified gravity model must satisfy them. We study these constraints on a kind of f(R) gravity as f(R) = R(1±eln([(R)/(R_c)]))f(R) = R(1\pm\epsilon\ln({R \over R_{c}})). For this investigation we use of chameleon mechanism and show that a spherically body has thin-shell in this model. So that we obtain an effective coupling of the fifth force which is suppressed through a chameleon mechanism. Also, we obtain γ _PPN =1±1.13×10⁻⁵ which is agreement with experiment results. At last, we show that for R _c ≈ρ _c this model is consistent with EP, thin shell condition and fifth force of chameleon mechanism for ε⋍10⁻¹⁴.     

CCD BV and 2MASS photometric study of the open cluster NGC 1513

We present CCD BV and JHK _s 2MASS photometric data for the open cluster NGC 1513. We observed 609 stars in the direction of the cluster up to a limiting magnitude of V∼19 mag. The star-count method showed that the centre of the cluster lies at α ₂₀₀₀=04 ^h 09 ^m 36 ^s , δ ₂₀₀₀=49°28^′43^″ and its angular size is r=10 arcmin. The optical and near-infrared two-colour diagrams revealed the colour excesses in the direction of the cluster as E(B−V)=0.68±0.06, E(J−H)=0.21±0.02 and E(J−K _s )=0.33±0.04 mag. These results are consistent with normal interstellar extinction values. Optical and near-infrared Zero Age Main-Sequences (ZAMS) provided an average distance modulus of (m−M)₀=10.80±0.13 mag, which can be translated into a distance of 1440±80 pc. Finally, using Padova isochrones we determined the metallicity and age of the cluster as Z=0.015±0.004 ([M/H]=−0.10±0.10 dex) and log (t/yr)=8.40±0.04, respectively.