One-armed oscillations of a non-selfgravitating polytropic disk

One-armedglobal oscillations in a non-selfgravitating polytropic disk rotating around a star are investigated. The unperturbed disk is axisymmetric, geometrically thin, and extends infinitely in the radial direction keeping its thickness constant. Perturbations considered are inviscid and adiabatic. It is found that there are one-armed retrograde wave modes which are trapped in an inner region of the disk. The eignefrequency of the lowest order mode is given by _K(r _s)(z ₀/r _s)₂, wherer _s is the radius of the central star,z ₀ is the half-thickness of the disk, and _K(r _s) is the Keplerian angular frequency at the surface of the star.Paper presented at the IAU Third Asian-Pacific Regional Meeting, held in Kyoto, Japan, between 30 September–6 October, 1984.     

Green's theorem and Green's functions for the steady-state cosmic-ray equation of transport

Green's Theorem is developed for the spherically-symmetric steady-state cosmic-ray equation of transport in interplanetary space. By means of it the momentum distribution functionF _o(r,p), (r=heliocentric distance,p=momentum) can be determined in a regionr _arr_bwhen a source is specified throughout the region and the momentum spectrum is specified on the boundaries atr _a andr _b . Evaluation requires a knowledge of the Green's function which corresponds to the solution for monoenergetic particles released at heliocentric radiusr _o , Examples of Green's functions are given for the caser _a =0,r _b = and derived for the cases of finiter _a andr _b . The diffusion coefficient is assumed of the form = _o(p)r ^b . The treatment systematizes the development of all analytic solutions for steady-state solar and galactic cosmic-ray propagation and previous solutions form a subset of the present solutions.     

High accuracy precession measurement with an autometric gyro

A simple reticle with parallel equally spaced lines to transmit light in an autometric gyro is shown to give a signal which can be used for very accurate angular precession measurements. Frequency analysis of the signal is chosen over time domain analysis and/or over a specially designed reticle.Gyro precession will induce a change in the interference structure of the spectrum in addition to dilating the overall frequency distribution. The expected output spectrum has been analytically determined, exhibiting the dependence on spot and reticle line cross sections and line spacing so that they can be precisely determined from the output data.The spectrum structure is extremely sensitive to precession. Computer simulations indicate that accuracy will be limited by reticle line spacing variations to ±4×10^–9 rad. This would allow a Lense-Thirring precession measurement accurate to ±20% in 30 days (accuracy being limited by physical effects common to all such experiments) or a ±1% de Sitter precession measurement in four days.Nomenclature a _n() Fourier coefficients ofI() - a _n ^r () rapidly varying part ofa _n() - a _m ^x ,b _m ^x Fourier coefficients ofI(x) - a (),b () Fourier sine and cosine transforms of _x(x) - angle between star and gyro spin axis - ₀ initial alignment angle - f focal length of optical system - I transmitted light intensity, signal function - l reticle line width - r distance from light spot center - R light-circle radius - = reduced light-circle radius - ₀ reduced reticle center-spin center offset - s reticle line space - T(x) reticle transmission function - angle of gyro rotation - w _g gyro angular velocity This work was supported by the M.I.T. Center for Space Research under NASA Grant NGL-22-009-019.     

Compatible potentials and orbits in synodic systems

For a given family of orbits f(x,y) = c ^* which can be traced by a material point of unit in an inertial frame it is known that all potentials V(x,y) giving rise to this family satisfy a homogeneous, linear in V(x,y), second order partial differential equation (Bozis,1984). The present paper offers an analogous equation in a synodic system Oxy, rotating with angular velocity . The new equation, which relates the synodic potential function (x,y), = –V(x, y) + % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSqaaSqaai% aaigdaaeaacaaIYaaaaaaa!3780!\[\tfrac{1}{2}\]²(x ² + y ²) to the given family f(x,y) = c ^*, is again of the second order in (x,y) but nonlinear.As an application, some simple compatible pairs of functions (x,y) and f(x, y) are found, for appropriate values of , by adequately determining coefficients both in and f.     

Rigorous estimates for the series expansions of Hamiltonian perturbation theory

In the present paper we prove a theorem giving rigorous estimates in the problem of bringing to normal form a nearly integrable Hamiltonian system, using methods of classical perturbation theory, i.e. series expansions in the small parameter . For any order of normalization, we give a lower bound _* ^r for the convergence radius of the normalized Hamiltonian, and a greater bound for the remainder, i.e. the non normalized part of the Hamiltonian. As an application, we consider the case of weakly coupled harmonic oscillators with highly nonresonant frequencies and show how, by optimizing, for fixed , the orderr of normalization, one gets for the remainder a greater bound of the formAe ^–(*₁/) ^a , with positive constantsA,a and ₁ ^* exponential estimate of Nekhoroshev's type.     

Size Distribution of Cometary Dust Particles

We considered two types of size distributions of cometary dust particles, namely, a: f(a) a ^{–( + 1)}(for comet Encke) and (for comet Halley). Using these distributions, expressions for the loss of the comet's absolute brightness per revolution, H ₁₀, are obtained; this quantity depends on the relative volume content of dust in the nucleus, f _v; the granulometric composition of dust ( or A _d, B _d, a _min, anda _max); the nucleus size and density; and the orbital elements P, q, and e. Under reasonable assumptions on the values of the above-mentioned physical parameters, a nucleus radius can be estimated from the value of H ₁₀obtained from observations.     

Propagation of spiral density waves in a magneto-active disk

The propagation of spiral density waves in a differentially rotating, self-gravitating, magnetoactive and highly flattened disk is investigated by using the asymptotic theory for tightly wound spirals developed by Lin and his collaborators. We adopt the continuum fluid model as the primary basis, and our treatment will be largely analytical. The disk plasma is studied in the frozen field approximation and inhomogenceous magnetic fields in the plane of the disk are considered in detail.In a differentially rotating disk with strong magnetic fields, the field lines will be considerably distorted and the mutual influence of magnetic fields and differential rotation is by no means obvious.In this paper we present a new asymptotic dispersion relation for tightly wound spiral density waves with magnetic fields along the spiral armsB (r). The effects of the magnetic fields lead to such terms likek ²(a ² +V _A ² ), wherek is the wave number,a represents the speed of sound,V _A = (B ²/4)^1/2 is the Alfvén speed,B denotes the field strength, and is the plasma density. These terms depict the well-known magnetoacoustic waves and could have been anticipated without a detailed computation. However the interaction of magnetic fields and differential rotation may give rise to other significant terms which are not so easy to foresee.We also present a more exact local dispersion relation by using the WKB approximation and study the effects of magnetic fields on the growth rate through the parametersQ andJ defined in the literature.Although the effects of the magnetic fields are rather insignificant for applications to Galactic dynamics, the effects of the magnetic fields are important for applications to star formation and problems related to the solar nebula.     

Evidence for enhanced star formation in iras-detected markarian galaxies

The IR emission of 640 Markarian galaxies (MrkG), included in the IRAS Survey, is considered as an evidence for enhanced star formation rate (SFR) in these objects. About 73% of the MrkG have high far-infrared luminosities (ca. 10E + 44 erg s^–1) in 1–500 mcm IR spectral band. The distribution of log(f ₆₀/f ₁₀₀), peaked at about 45 K, shows that IRAS MrkGs have a tendency to extend the relationf ₆₀/f ₁₀₀ vsL _ir/L _bifor normal S glaxies. They emit up to hundred times more IR energy in 40–120 mcm band than in optics. The mean ratio log L _ir/L _b for 621 IRAS MrkG with known redshifts is 2.2.It is suggested that there are two IR emitting components in the IRAS MrkG - a warm one connected with the UV-fluxes of the newborn massive stars, re-radiated by dust, and a cool one, originated from the dust in galactic disks and heated by the general interstellar radiation field. The warm IR luminosities and warm IR fractions are determined on the basis of IR colour-colour diagrams(25/12),(60/25), and(100/60). The mean warm IR fraction for all Mrk IRAS detected galaxies with well-defined IR fluxes is 0.83 when the grain mass absorption coefficient model withn = 0.0 is used. The dust mass responsible for the IR flux at 60 mcm is derived to be about 10E + 5M , assuming the dust clouds are optically thin, and using the dust temperatureT _d 46 K (deduced from thef(60)/f(100) ratio). There is a relation betweenL _irandL _blwhich points out that the most IRAS MrkG have rather enhanced SFR.     

Solution of Lambert's problem for short arcs

Approximation formulas are found for and , wherex(t) satisfies ,x(0)=x ₀,x(1)=x ₁. The results are applied to an example of two-body motion.     

Theorie de Weyl et principe de moindre action: Application a l'etude des mouvements

Sommaire Les lois du mouvement dans une variété riemannienneV ₄ peuvent être déduites d'un principe de moindre action. Nous établissons dans cet article l'équivalence des relations ds=0 et dL=-L _k d^k, où ds ²=L ² est une métrique riemannienne et d _k /dt une fonction homogène de degré 1 des variables dx ⁱ/dt qui définit un espace de Weyl. Ce théorème permet de ramener une théorie de jaugen à un principe de moindre action. Il peut être utilisé dans la théorie de la double métrique de Dirac, obtenue en choissant la théorie des grands nombres comme condition de jauge. Une fibration de l'espace physiqueV ₃ basée sur le théorème de Huyghens permet de déduire les propriétés dynamiques des particules des propriétés des photons dansV ₃, et constitue en ce sens une unification des propriétés dynamiques des particules.

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The laws of motion in a RiemannianV ₄ manifold can be deduced from the principle of least action. We state in this work the equivalence between the equations ds=0 and dL=-L _k d^k, where ds ² =L ² is the Riemannian metric and d _k /dt the homogeneous functions of first degree of the dx ⁱ/dt which define a Weylian space. This theorem can then reduce a gauge theory to a principle of least action. It can be used in the double metric theory of Dirac, obtained by means of the Large Number Hypothesis as a gauging condition. A fibration of the physical spaceV ₃ based on Huyghens' theorem allows the deduction of the dynamical properties of particles by means of the properties of photons inV ₃, and constitutes in this way an unification of the dynamical properties of particles.

     

On Geovenelli's approximate form of j(τ)

The accuracy of the approximate replacement of the frequency meanj() of the total intensityJ () byJ _c+a{J ₀()–J _c} by Geovenelli, whereJ _c is the total intensity at the continuum,J ₀() is that at the line center anda is independent of depth, is studied. Numerical calculations for strong solar lines show very high values of the residual intensities. A slight change in the approximation is then suggested to get a result quite consistent with observational and existing calculated data.     

The influence of satellite flexibility on orbital motion

The orbital perturbations induced by the librational motion and flexural oscillations are studied for satellites having large flexible appendages. Using a Lagrangian procedure, the equations for coupled motion are derived for a satellite having an arbitrary number of appendages in the nominal orbital plane and two flexible members normal to it. The formulation enables one to study the influence of flexibility on both the orbital and attitude motions. The orbital coordinates are expanded as perturbation series in =(l/a ₀)²,l anda ₀ being a characteristic length of the satellite and unperturbed semi-major axis of the orbit, respectively. The first order perturbation equations are solved in terms of elastic deformations and librational angles using the WKBJ method in conjunction with the variation of parameter technique. Existence of secular perturbations is noted for certain librational flexural motions. Three specific examples, Alouette II, Radio Astronomy Explorer and Tethered Orbiting Interferometer, are considered subsequently and their possible secular drifts estimated.List of Symbols A _ij, B_ij coefficients in the eigenfunction expansion ofv _i andw _i respectively, Equation (10) - C _k, D_k constants, Equation (21) - EI _i flexural rigidity of theith appendage - E(u₀) ²(1+e ₀ cosu ₀)² h ₀ ³ - F(u₀) perturbation function, Equation (17b) - F ,F ,F functions of librational angles and flexural displacements, Equation (11i) - F ,F ,F F ,F ,F with change of independent variable fromt tou ₀ - I _xx, I_yy, I_zz principal moments of inertia of the undeformed satellite - [J ⁱ] inertia dyadic of the deformedith appendage - [J _d] inertia dyadic of the deformed satellite - M mass of the satellite - P _R, P_u functions of librational angles and flexural displacements, Equation (15d) and (15e), respectively - R _c magnitude ofR _c - R _c0, R₁ unperturbed value and first order perturbation ofR _c, respectively - R _c ,R ₀ position vectors of the c.m. of the deformed and undeformed satellite, respectively - T kinetic energy of the satellite - U potential energy of the satellite - U _e, U_g elastic and gravitational potential energy, respectively - X, Y, Z orbital co-ordinate axes, located at the c.m. of the deformed satellite - Y ₁(u₀), Y₂(u₀) functions ofu ₀, Equation (18b) and (18c), respectively - a semi-major axis - a ₀ unperturbed value ofa - e eccentricity - e ₀ unperturbed value ofe - h ₀ unperturbed angular momentum per unit mass of the satellite - i inclination of the orbital plane to the ecliptic - i, j, k unit vectors alongx (or ),y (or ) andz (or ) axes, respectively - l characteristic length of the satellite - l _i length of theith appendage - [l ⁱ] matrix of direction cosines ofx _i, v_i andw _i - l ,l ,l direction cosines ofR _c - m ₀, m_i mass of the main body andith appendage, respectively - p _i ² - q _m, Q_m generalized co-ordinate and force, respectively - r ₁ R ₁/R_c0 - r position vector of an element of the body referred toxyz axes - r _u position vector of an element after deformation, referred to axes - r _c x _c i+y _c j+z _c k, position vector of the c.m. of the deformed body referred toxyz axes - s x _i/l_i - t time - u true anomaly - u ₀, u₁ unperturbed value and the first order perturbation ofu, respectively - u elastic displacement vector - u _c u–r _c - velocity of an element relative to axes - v _i, w_i flexural deformations - x, y, z body co-ordinate axes with origin at the c.m. of the undeformed satellite - x _i distance of an element of theith appendage from the root - _j jth eigenfunction (normalized) of a cantilever - angle between the line of nodes and vernal equinox - , , components of nondimensionalized angular velocity of the satellite - , , pitch (spin), yaw and roll, respectively - _i nominal inclination of theith appendage in the orbital plane - - small parameter, (l/a ₀)² - _j jth eigenvalue of a cantilever - gravitational constant - _jk constant, Equation (11j) - , , body co-ordinate axes with origin at the c.m. of the deformed satellite - ( i + j + k), angular velocity of the satellite     

Some requirements of a colliding comet source of gamma ray bursts

Colliding comets in the Solar System may be an important source of gamma ray bursts. The spherical gamma ray comet cloud required by the results of the Venera Satellites (Mazets and Golenetskii, 1987) and the BATSE detector on the Compton Satellite (Meeganet al., 1992a, b) is neither the Oort Cloud nor the Kuiper Belt. To satisfy observations ofN(>P _max) vsP _max for the maximum gamma ray fluxes,P _max > 10^–5 erg cm^–2 s^–1 (about 30 bursts yr^–1), the comet density,n, should increase asn a ¹ from about 40 to 100 AU wherea is the comet heliocentric distance. The turnover above 100 AU requiresn a ^–1/2 to 200 AU to fit the Venera results andn a ^1/4 to 400 AU to fit the BATSE data. Then the masses of comets in the 3 regions are from: 40–100 AU, about 9 earth masses,m _E; 100–200 AU about 25m _E; and 100–400 AU, about 900m _E. The flux of 10^–5 erg cm^–2 s^–1 corresponds to a luminosity at 100 AU of 3 × 10²⁶ erg s^–1. Two colliding spherical comets at a distance of 100 AU, each with nucleus of radiusR of 5 km, density of 0.5 g cm^–3 and Keplerian velocity 3 km s^–1 have a combined kinetic energy of 3 × 10²⁸ erg, a factor of about 100 greater than required by the burst maximum fluxes that last for one second. Betatron acceleration in the compressed magnetic fields between the colliding comets could accelerate electrons to energies sufficient to produce the observed high energy gamma rays. Many of the additional observed features of gamma ray bursts can be explained by the solar comet collision source.     

Gas,dust and molecules in the Galaxy

The relative abundances of cool neutral hydrogen, carbon monoxide and formaldehyde are studied using all the available observational data in the literature. The obtained mean valuesN _{H 1}/ ,N _{H 1}/N _CO,N _CO/ are approximately constant in the dark clouds of the solar neighbourhood and in the distant molecular clouds.The observed correlationsN _CO,A _v and ,A _v show that formaldehyde can also be used as an indicator of molecular hydrogen. The ratioN H₁/A _v depends on densities and decays considerably in the ranges of visual absorptions in which the molecules become detectable (A _v 2 mg); an average of /N _{H 1}10 is calculated for the dense dark clouds.Indications of systematic temperature gradiens T/A _v are found for formaldehyde and neutral hydrogen inside the dark clouds, and qualitative comparisons are made with theoretical quantum mechanics calculations.The observed carbon monoxide and formaldehyde abundances, the free electron layer in the Galaxy, the distribution of neutral hydrogen in different states are only compatible if an ionization rate of 10^–16 is accepted, provided presumably by 2 MeV protons of cosmic radiation.Three main states for neutral hydrogen and dust are identified from different kinds of observational data (21 cm line in emission, absorption in galactic radio sources and self-absorption in the hot gas background): (1) a homogeneous intercloud stratum of tenuous gas and dust with a galactic halfwidth of 350 pc and mean parametersn _H=0.2 atom cm^–3, spin temperatureT _s 10000 K andn _d 0.3 mg kpc^–1; (2) cool gas and dust concentrated in spiral features with a galactic half-width of less than 100 pc, probably forming clouds with diffuse and indefinite limits, with mean parametersn _H2 atom cm^–3,T _s <1100 K (probable average,T _s =135 K) andn _d 3 mg kpc^–1; (3) dense gas and dust clouds with a mean diameter of 7 pc and mean parametersn _H700 atom cm^–3 (90% in a molecular state),T _s 63 K andn _d 1 mg pc^–1 on which molecules as CO and H₂CO are formed.The application of the Jeans criteria for gravitational instability shows that the dense clouds are gravitationally bound while the gas in the intermediate state (2) can be protected against collapse by the total internal energy in the medium increasing due to cosmic rays and the magnetic field in the Galaxy.The observed velocity halfwidths and galacticZ-halfwidths in states (1) and (2) are compatible with a total mass density in the galactic layer of 90M pc^–2 (gas plus stars) according to the barometric equation.The relative abundancesN _{H 1}/N _CO, calculated from C¹²O and C¹³O data and comparisons with studies in the 21 cm emission line, show that the antenna temperatureT _A ⁺ in the 2.6 mm line of C¹²O is a good indicator of the cool gas densities in the Galaxy. The possible application of this for studies in galactic structure is discussed and hypothetical distributions of carbon monoxide in the zones outside the galactic planeB=0° are presented.From a synthesis based on the results obtained, a cycle is postulated for the neutral hydrogen in the Galaxy: condensation and cooling of gas molecular formation gravitational collapse and star formation gas dissipation and heating by cosmic rays and UV radiation.     

Towards Calibration of the Mean Magnetic Field of the sun

The 1968–2000 data on the mean magnetic field (MMF, longitudinal component) of the Sun are analysed to study long-time trends of the Sun's magnetic field and to check MMF calibration. It is found that, within the error limits, the mean intensity of photospheric magnetic field (the MMF strength, |H|), did not change over the last 33 years. It clearly shows, however, the presence of an 11-year periodicity caused by the solar activity cycle. Time variations of |H| correlate well with those of the radial component, |B _r|, of the interplanetary magnetic field (IMF). This correlation (r=0.69) appears to be significantly higher than that between |B _r| and the results of a potential source-surface extrapolation, to the Earth's orbit, of synoptic magnetic charts of the photosphere (using the so-called `saturation' factor <sup>–1</sup> for magnetograph measurements performed in the line Fei 525.0 nm; Wang and Sheeley, 1995). It seems therefore that the true source surface of IMF is the `quiet' photosphere – background fields and coronal holes, like those for MMF. The average `effective' magnetic strength of the photospheric field is determined to be about 1.9 G. It is also shown that there is an approximate linear relation between |B <sub>r</sub>| and MMF intensity |H| (in gauss)|B <sub>r</sub>|(H <sub>0</sub>)<sub>min</sub>×(1+C|H|)where =1.5×10<sup>–5</sup> normalizes the photospheric field strength to 1 AU distance from the Sun, (H <sub>0</sub>)<sub>min</sub>=1.2 G is some minimal `effective' intensity of photospheric background fields and C=1.3 G^–1 an empirical constant. It is noted that good correlation between time variations of |H| and |B _r| makes suspicious a correction of the photospheric magnetic fields with the use of saturation factor ^–1.     

Ar XIII line ratios in solar flares

Theoretical ArXIII electron-density-sensitive emission line ratios, derived using electron impact excitation rates interpolated from accurateR-matrix calculations, are presented forR ₁ =I(242.22 )/I(236.27 ),R ₂ =I(210.46 )/I(236.27 ), andR ₃ =I(248.68 )/I(236.27 ). Electron densities deduced from the observed values ofR ₁,R ₂, andR ₃ for solar flares obtained with the NRL S082A slitless spectrograph on boardSkylab are in excellent agreement, and furthermore compare favorably with those determined from line ratios in CaXV, which is formed at a similar electron temperature to that of ArXIII. These results provide experimental support for the accuracy of the atomic data adopted in the analysis, as well as for the techniques used to calculate the line ratios.     

Kinetic theory of MHD instabilities in a nonuniform plasma

Most of the MHD instabilities originating from the nonuniformity of a plasma excite MHD surface wave. When the excited wave has a frequency _s which corresponds to the local shear Alfvén wave resonance (_s = k v _a (x), where v _a is the Alfvén speed and k is the wave number in the direction of the magnetic field), the surface wave resonantly mode converts to the kinetic Alfvén wave, the Alfvén wave having a perpendicular wavelength comparable to the ion gyroradius and being able to propagate across the magnetic field. We discuss various linear and nonlinear effects of this kinetic Alfvén wave on the plasma including particle acceleration and heating. A specific example for the case of a MHD Kelvin-Helmholtz instability is given.     

Time transformations and Cowell's method

The precise numerical integration of Cowell's equations of satellite motion is frequently performed with an independent variables defined by an equation of the form dt=cr ⁿ ds, wheret represents time,r the radial distance from the center of attraction,c is a constant, andn is a parameter. This has been primarily motivated by the uniformizing effects of such a transformation resulting in desirable analytic stepsize control for elliptical orbits. This report discusses the proper choice of the parametern defining the independent variables for various types of orbits and perturbation models, and develops a criterion for its selection.     

Spatial dependence of the pitch-angle and associated spatial diffusion coefficients for cosmic rays in interplanetary space

The spatial dependence of the pitch-angle and associated spatial diffusion coefficients for cosmic ray particles in interplanetary space is calculated in the WKB approximation. The model considers only Alfven waves of solar origin to be responsible for scattering of moderate energy particles. After developing the general theory results are presented for the asymptotic case corresponding to radial distancesr greater than about 1 to 2 AU. The radial diffusion coefficient _r increases with energyE like _r E , wherev2/3. The radial mean free path turns out to increase proportional tor ³ at medium and low heliographic latitudes. This behaviour is consistent with a very small radial cosmic ray gradient and the existence of a free boundary for particle diffusion. At equal radial distances the high latitude mean free path is not only much smaller than the one calculated at the lower latitudes but in addition increases only weakly with distance. Some conceivable dynamical implications for the outer solar system are indicated.     

Hall effects on natural convective flow in slip flow regime

Hall effects on the flow of electrically conducting rarefied gas due to combined buoyant effects of thermal and mass diffusion past an infinite porous plate with constant suction in the presence of strong transverse magnetic field have been investigated. The equations governing the flow poblem have been solved for primary, secondary velocities and temperature. The effects of Hall current, magnetic field and the effect of rarefication have been discussed graphically followed by a discussion.Nomenclature x,y coordinate system - u velocity inx direction - v ₀ suction velocity - w velocity inz direction - E Eckert number - G, G^* Grashof numbers - h ₁ velocity slip coefficient - h ₂ temperature jump coefficient - h ₃ concentration jump coefficient - M, m magnetic field parameter, Hall parameter - Pr Prandtl number - Sc Schmidt number - T, T _w, T temperature in flow regime, plate temperature, temperature outside the boundary layer very away from the plate - C, C _w, C concentration of the gas in flow, concentration at the plate, concentration far away from the plate - thermal conductivity - D coefficient of chemical molecular diffusion - coefficient of kinematic viscosity - coefficient of viscosity - electrical conductivity - C _p specific heat of gas at constant pressure density