Refinement of the geopotential scale factorR 0 on

Altimetric measurements of the GEOSAT satellite were used for the determination of geopotential scale factorR ₀. The geopotential valueW ₀ on the geoid surface was then computed (W ₀ =GM/R ₀).The GEOSAT Geophysical Data Records (GDR's) covering an initial period of the Exact Repeat Mission (ERM) were filtered and processed. The necessary corrections were made in order to allow a precise detection of the sea surface. Gravitational geopotential, rotation and permanent tides were taken into account and the equipotential surface which is the best approximation of the sea surface was found.The determination of the potential valueW ₀ on the mean geoid surface in this way is very promising. An associated value withW ₀ - the geopotential scale factorR ₀ - seems to be a very good Earth dimension defining quantity. Moreover, there are many possible applications ofW ₀ (R ₀) in modern geophysics.The incorporation of one of these parameters - we now recommendR ₀ - into the set of the Primary Geodetic Parameters (PGP) is discussed and suggested.     

Testing geopotential models

The actual accuracy of the geopotential value on the geoid computed from satellite altimetry recently asW ₀ = (62 636 857.5 ± 1.0) m² s^–2 makes it possible to adopt this quantity as geopotential models testing (GMT) value. However, GMT network should be established consisting of points situated near the gauge stations and of other points at small sea level heights, globally distributed. As numerical example illustrating the GMT method suggested, the recent Satellite Laser Ranging network points have been used.     

Dimension of the Earth's General Ellipsoid

The problem of specifying the Earth's mean (general)ellipsoid is discussed. This problem has been greatly simplified in the era of satellite altimetry, especially thanks to the adopted geoidal geopotential value, W₀ = (62 636 856.0 ± 0.5) m² s^-2.Consequently, the semimajor axis a of the Earth's mean ellipsoid can be easily derived. However, an a priori condition must be posed first. Two such a priori conditions have been examined, namely an ellipsoid with the corresponding geopotential that fits best W₀ in the least squares sense and an ellipsoid that has the global geopotential average equal to W₀. It has been demonstrated that both a priori conditions yield ellipsoids of the same dimension, with a–values that are practically identical to the value corresponding to the Pizzetti theory of the level ellipsoid: a = (6 378 136.68 ± 0.06) m.     

Long-Term Stability Of Geoidal Geopotential FromTopex/Poseidon Satellite Altimetry 1993–1999

The TOPEX/POSEIDON (T/P) satellite altimeter data from January 1, 1993to October 24, 1999 (cycles 11–261) was used for investigating thelong-term variations in the geoidal geopotential W₀ and/orin the geopotential scale factor R₀ = GM/W₀ (GM is theadopted geocentric gravitational constant). The mean valuesdetermined for the whole period covered are: W₀ =(62 636 856.161 ± 0.002) m² s^-2, R₀ =(6 363 672.5448 ± 0.0002) m. The actual accuracy is limited bythe altimeter calibration error (2–3 cm) and it isestimated to be about ± 0.5 m² s^-2 (± 5 cm).The yearly variations of the above mean values are at the formalerror level. No long-term trend in W₀, representing the oceanvolume change, was found for the seven years period 1993–9 on thebasis of T/P altimeter (AVISO) data. No sea surface topography modelwas used in the solution. This revised version was published online in July 2006 with corrections to the Cover Date.     

Geoidal potential free of zero-frequency tidal distortion

It has been proved that the geoidal valueW ₀ is not dependent on the system used for defining the geoid surface. It is the same for the zero-frequency tidal system, mean system and tide-free system. It has been suggested,W ₀ be adopted as primary constant defining the length dimensions of celestial bodies.     

Interstellar extinction and diffuse absorption features

The equivalent width of the 2175 Å band,W ₂₁₇₅, well known as the big bump in the interstellar extinction curves, has been found to be closely correlated with the colour excessE _B-V as well as with the extinction differencesE _8–6 andE _9–7 defined to characterize quantitatively the steep slopes of the extinction curves in the far ultraviolet.The equivalent widths of the 5780 and 5797 Å diffuse lines show good correlation withE _B-V. The correlations ofW ₅₇₈₀ andW ₅₇₉₇ withE _8–6 resp.E _9–7 are, however, rather weak. Correlations betweenW ₂₁₇₅ andW ₅₇₈₀ and betweenW ₂₁₇₅ andW ₅₇₉₇ are indicated.The results have been qualitatively interpreted in favour of the dust model consisting of a mixture of small silicate grains and larger silicate grains coated by molecular mantles.Paper presented at the Symposium on Solid State Astrophysics, held at the University College, Cardiff, Wales, between 9–12 July 1974.     

The significance of observed rotational magnetic hysteresis in lunar samples

Rotational Magnetic hysteresis (W _R ) curves for lunar soils 10084, 12070, 14259, and rock 14053, have been published. There is no adequate explanation to date for the observed largeW _R at high fields. Lunar rock magnetism researchers consider fine particle iron to be the primary source of stable magnetic remanence in lunar samples. Iron has cubic anisotropy with added shape anisotropy for extreme particle shapes. The observed high fieldW _R must have its source in uniaxial or unidirectional anisotropy. This implies the existence of minerals with uniaxial anisotropy or exchange coupled spin states. Therefore, the source of this observed high fieldW _R must be identified and understood before serious paleointensity studies are made. It is probable that the exchange coupled spin states and/or the source of uniaxial anisotropy responsible for the high fieldW _R might be influenced by the lunar surface diurnal temperature cycling. The possible sources of high fieldW _R in lunar samples are presented and considered.     

GEOS-II and 13th order terms of the geopotential

The resonance of GEOS-II (1968-002A) with 13th-order terms of the geopotential is analyzed. The odd-degree geopotential coefficients (13, 13), (15, 13), and (17, 13) given by Yionoulis most accurately model the resonance effects on GEOS-II of any of the published sets of 13th-order coefficients. However, this set is not adequate for precision orbit determination; additional even-degree coefficients are required.Values ofC _14,13(=0.57×10^–21) andS _14,13(=6.5×10^–21) to be used with the odd-degree set of Yionoulis were obtained from an analysis of the observed along-track position variation of GEOS-II. These coefficients, when used with those of Yionoulis, yield greatly improved fits to the data and orbital prediction capability. However, further refinement is possible because the small effects of the remaining even-degree resonant terms were not modeled.The composite coefficientsC _13,13(=1.7×10^–20) andS _13,13(=+2.7×10^–20) were obtained under the assumption that the (13, 13) spherical harmonic of the geopotential is responsible for all of the observed along-track variation of GEOS-II due to resonance. The good agreement of these deliberately composite values with some published values ofC _13,13 andS _13,13 suggests that some of the published values may also be composite to some extent.These coefficients are hereinafter referred to as the APL coefficients.     

Quality analysis of Moore,Minnaert, and Houtgast's equivalent widths

A quality analysis of the equivalent widths (W) for solar lines tabulated in the Atlas of Moore et al. (1966) is undertaken by comparison with other modern sources. No significant correlation with is detected, and only a weak systematic dependence with W turned out which may be neglected for most practical purposes. An approximate trend for the internal dispersion as a function of W is also derived.Fellowship holder of CONICET, Buenos Aires, Argentina.     

A New Reference Equipotential Surface, and Reference Ellipsoid for the Planet Mars

Using the shape model of Mars GTM090AA in terms of spherical harmonics complete to degree and order 90 and gravitational field model of Mars GGM2BC80 in terms of spherical harmonics complete to degree and order 80, both from Mars Global Surveyor (MGS) mission, the geometry (shape) and gravity potential value of reference equipotential surface of Mars (Areoid) are computed based on a constrained optimization problem. In this paper, the Areoid is defined as a reference equipotential surface, which best fits to the shape of Mars in least squares sense. The estimated gravity potential value of the Areoid from this study, i.e. W ₀ = (12,654,875 ± 69) (m²/s²), is used as one of the four fundamental gravity parameters of Mars namely, {W ₀, GM, ω, J ₂₀}, i.e. {Areoid’s gravity potential, gravitational constant of Mars, angular velocity of Mars, second zonal spherical harmonic of gravitational field expansion of Mars}, to compute a bi-axial reference ellipsoid of Somigliana-Pizzetti type as the hydrostatic approximate figure of Mars. The estimated values of semi-major and semi-minor axis of the computed reference ellipsoid of Mars are (3,395,428 ± 19) (m), and (3,377,678 ± 19) (m), respectively. Finally the computed Areoid is presented with respect to the computed reference ellipsoid.     

Solar flares,microflares, nanoflares,and coronal heating

Solar flare occurrence follows a power-law distribution against total flare energy W: dN/dW W ^– with an index 1.8 as determined by several studies. This implies (a) that microflares must have a different, steeper distribution if they are energetically significant, and (b) there must be a high-energy cutoff of the observed distribution. We identify the distinct soft distribution needed for coronal heating, if such a distribution exists, with Parker's nanoflares.This paper considers a microflare to be a member of the normal X-ray burst population, with comparable physical parameters except for a smaller total energy.     

Hydromagnetic stability of some non-dissipative flows subject to non-axisymmetric disturbances

We study the linear stability of nondissipative flow of an electrically conducting fluid subject to non-axisymmetric disturbances in the following cases: (i) the radial flow of an incompressible fluid between two concentric porous circular cylinders in the presence of a radial magnetic field and (ii) axial flow of a compressible fluid between two concentric circular cylinders permeated by a helical magnetic field (0,B ₀(r),B _0z) in a cylindrical coordinate system. It is shown that in case (i), the flow is stable if the Alfvén velocity based on the undisturbed radial magnetic field exceeds the radial velocity due to suction or injection at the cylinder surfaces. In case (ii), it is found that under certain conditions the complex wave speed for an unstable mode lies within a circle of diameterW _max-W _min, whereW _max andW _min are the maximum and minimum values of the axial velocity in the flow region. In the presence of a purely axial magnetic field, however, the complex wave speed for an unstable mode always lies within the above circle.     

Estimation of low degree geopotential coefficients using SLR data

Geodetic satellites have been providing the low frequency part of the geopotential models used for precise orbit determination purposes (e.g. JGM3, EGM96, …). Nevertheless they can be used to estimate the temporal variation of selected coefficients, helping to clarify the complex interrelations in the earth-ocean-atmosphere system. In this paper we present the two years long analysis of SLR data from the seven available geodetic satellites (Lageos I–II, Stella, Starlette, Ajisai, Etalon I–II) to recover monthly estimates of low degree geopotential coefficients; the results are obtained analysing the satellites separately and in proper combination. An accurate modelling of the satellite orbits is required in order to separate the geopotential coefficients: we assume as a priori geopotential the JGM3 model together with its associated tides and we take care of non-gravitational effects on the satellites by means of proper empirical estimated accelerations. The time series of the estimated coefficients (J₂, J₃, J₄, J₅) are inspected to detect the sub-annual perturbations related to seasonal variation of mass distribution. Huge residual seasonal signals in the orbit of Stella indicate a strong model deficiency related to the Sun's influence on the environment. The remaining six satellites are homogeneously modelled and build up a three cycles per year oscillation on J₂ and a seasonal oscillation (1 year and six month periods) revealed on the J₄. The origin and possible causes of these signals are further discussed in the text. We also present a preliminary estimate, using twelve years of Lageos-I and Lageos-II observations, that is compared with previous obtained values.     

Equilibrium Structures of Differentially Rotating and Tidally Distorted White Dwarf Models of Stars

In this paper we present a method for computing the equilibrium structures and various physical parameters of a primary component of the binary system assuming that the primary is more massive than the secondary and is rotating differentially according to the law of the w² = b₀ + b₁ × s² + b₂ × s⁴, w being the angular velocity of rotation of a fluid element distant s from the axis of rotation and b₀, b₁, b₂ suitably chosen numerical constants. This method utilizes the averaging approach of Kippenhahn and Thomas (1997) and the concept of Roche equipotentials in a manner earlier used by Mohan et al. (1997) to incorporate the effects of rotation and tidal distortions on the equilibrium structures of certain rotationally and tidally distorted stellar models. The use of the method has been illustrated by applying it to obtain the structures and some observable parameters of certain differentially rotating and tidally distorted binary systems whose primary component is assumed to be a white dwarf star.     

The equivalent width of the Hβ emission line and the evolution of the Hii regions

The synthetic equivalent width (W _H) of the line H in emission is obtained for Hii regions opaque to the Lyman photon flux, with embedded OB associations with different initial chemical compositions and initial mass functions. The variation ofW _H as a function of the evolution of the ionizing stars is analysed. The observations ofW _H for M33, M101, and M51 by Searle (1971) are discussed.     

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%.     

The interpretation of the Wilson-Bappu effect

The problem of the physical interpretation of the Wilson-Bappu effect is examined. A postulate is forwarded according to which this effect expresses the dependence of the complete kinetic energyE _t of turbulent motions in the medium generating magnesium emission in the doublet 2800 Mgii on the velocityW ₀ of turbulent motion. One of the consequences of the application of this postulate is the appearance of the relationships (8) and (11) which give the magnitude of the kinetic energyE _t of turbulent motions in the first case (Figure 2) and the magnitude of the mass of emitting mediumM ₀ in the second (Figure 3) depending on the turbulent velocityW ₀ (k). This postulate unit in a common conception all variety of the magnesium emission escaped from a single star as well as from binary systems on the one hand and the magnesium emission generated in chromospheres as well as in intercomponent clouds or streams on the other.     

The motion of a satellite in an axi-symmetric gravitational field

The equations for the variation of the osculating elements of a satellite moving in an axi-symmetric gravitational field are integrated to yield the complete first-order perturbations for the elements of the orbit. The expressions obtained include the effects produced by the second to eighth spherical harmonics. The orbital elements are presented in the most general form of summations by means of Hansen coefficients. Due to their general forms it is a simple matter to estimate the perturbations of any higher harmonic by simply increasing the index of summation. Finally, this paper gives the respective general expressions for the secular perturbations of the orbital elements. The formulae presented should be useful for the reductions of Earth-satellite observations and geopotential studies based on them.List of Symbols semi-major axis - C _jk ⁿ (, ) cosine functions of and - e eccentricity of the orbit - f acceleration vector of perturbing force - f sin²t - i inclination of the orbit - J _n coefficients in the potential expansion - M mean anomaly - n mean motion - p semi-latus rectum of the orbit - R, S, andW components of the perturbing acceleration - r radius-vector of satellite - r magnitude ofr - S _jk ⁿ (, ) sine functions of and - T time of perigee passage - u argument of latitude - U gravitational potential - true anomaly - V perturbing potential - G(M₊+m) (gravitational constant times the sum of the masses of Earth and satellite) - _n,k coefficients ofR component of disturbing acceleration (funtions off) - _n,k coefficients ofS andW components of disturbing acceleration (functions off) - mean anomaly at timet=0 - X ₀ ^n,m zero-order Hansen coefficients - argument of perigee - right ascension of the ascending node     

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.