Experimental studies of magnetite formation in the solar nebula

Abstract— Oxidation of Fe metal and Gibeon meteorite metal to magnetite via the net reaction 3 Fe (metal) + 4 H₂O (gas) = Fe₃O₄ (magnetite) + 4 H₂ (gas) was experimentally studied at ambient atmospheric pressure at 91–442 °C in H₂ and H₂-He gas mixtures with H₂/H₂O molar ratios of ～4–41. The magnetite produced was identified by x-ray diffraction. Electron microprobe analyses showed 3.3 wt% NiO and 0.24 wt% CoO (presumably as NiFe₂O₄ and CoFe₂O₄) in magnetite formed from Gibeon metal. The NiO and CoO concentrations are higher than expected from equilibrium between metal and oxide under the experimental conditions. Elevated NiO contents in magnetite were also observed by metallurgists during initial stages of oxidation of Fe-Ni alloys. The rate constants for magnetite formation were calculated from the weight gain data using a constant surface area model and the Jander, Ginstling-Brounshtein, and Valensi-Carter models for powder reactions. Magnetite formation followed parabolic (i.e., diffusion-controlled) kinetics. The rate constants and apparent activation energies for Fe metal and Gibeon metal are: These rate constants are significantly smaller than the parabolic rate constants for FeS growth on Fe metal in H₂S-H₂ gas mixtures containing 1000 or 10 000 ppmv H₂S (Lauretta et al., 1996a). The experimental data for Fe and Gibeon metal are used to model the reaction time of Fe alloy grains in the solar nebula as a function of grain size and temperature. The reaction times for 0.1–1 μm radius metal grains are generally within estimated lifetimes of the solar nebula (0.1–10 Ma). However, the calculated reaction times are probably lower limits, and further study of magnetite formation at larger H₂/H₂O ratios, at lower temperatures and pressures, and as a function of metal alloy composition is needed for further modeling of nebular magnetite formation.     

Spin-dependent limits from the DRIFT-IId directional dark matter detector

Data are presented from the DRIFT-IId detector operated in the Boulby Underground Science Facility in England. A 0.8 m³ fiducial volume, containing partial pressures of 30 Torr CS₂ and 10 Torr CF₄, was exposed for a duration of 47.4 live-time days with sufficient passive shielding to provide a neutron free environment within the detector. The nuclear recoil events seen are consistent with a remaining low-level background from the decay of radon daughters attached to the central cathode of the detector. However, charge from such events must drift across the entire width of the detector, and thus display large diffusion upon reaching the readout planes of the device. Exploiting this feature, it is shown to be possible to reject energy depositions from these Radon Progeny Recoil events while still retaining sensitivity to fiducial-volume nuclear recoil events. The response of the detector is then interpreted, using the F nuclei content of the gas, in terms of sensitivity to proton spin-dependent WIMP-nucleon interactions, displaying a minimum in sensitivity cross section at 1.8 pb for a WIMP mass of 100 GeV/c². This sensitivity was achieved without compromising the direction sensitivity of DRIFT.     

Supersonic meteorology of Io: Sublimation-driven flow of SO2

The horizontal flow of SO₂ gas from day side to night side of Io is calculated. The surface is assumed to be covered by a frost whose vapor pressure at the subsolar point is orders of magnitude larger than that on the night side. Temperature of the frost is controlled by radiation. The flow is hydrostatic and turbulent, with velocity and entropy per particle independent of height. The vertically integrated conservation equations for mass, momentum, and energy are solved for atmospheric pressure, temperature, and horizontal velocity as functions of solar zenith angle. Formulas from boundary layer theory govern the interaction between atmosphere and surface. The flow becomes supersonic as it expands away from the subsolar point, as in the theory of rocket nozzles and the solar wind. Within 35° of the subsolar point atmospheric pressureis less than the frost vapor pressure, and the frost sublimes. Elsewhere, atmospheric pressure is greater than the frost vapor pressure, and the frost condenses. The two pressures seldom differ by more than a factor of 2. The sublimation rate at the subsolar point is proportional to the frost vapor pressure, which is a sensitive function of temperature. For a subsolar temperature of 130°K, the sublimation rate is 10¹⁵ molecules/cm²/sec. Diurnally averaged sublimation rates at the equator are comparable to the 0.1 cm/year resurfacing rate required for burial of impact craters. At the poles where both the vapor pressures and atmospheric pressures are low, the condensation rates are 100 times smaller. Surface pressures near the terminator are generally too low to account for the ionosphere discovered by Pioneer 10. The possibility of a noncondensable gas in addition to SO₂ must be seriously considered.     

Low-mass red giants and masses of NPNs

The evolutionary track of low-mass red giant stars (0.7–0.9M ) is computed with the aim to demonstrate the conditions under which low-mass white dwarfs (WDs) can form through the evolution of single stars. Also, the influence of the mixing length to the scale height ratio on the radius of the star is calculated and the coupling between the mixing-length and the mass-loss rate parameters is investigated. Our conclusions are that the uncertainties in mass-loss and mixing-length to scale-height ratio leave enough parameter space to allow the formation of low-mass WD via single star evolution. We also conclude that the gap between proto-WD stars without any nebula and stars with well-defined nebulae is bridged by stars which have a dilute gas cloud around them which cannot be observed as a nebula.     

On inhomogeneous scattering models of Titan's atmosphere

The spectrum of Titan from 4800 to 11 000 Å has many CH₄ absorption bands which cover a range of intensities of several orders of magnitude. Yet even the strongest of these bands in Titan's spectrum has considerable residual central intensity. Some investigators have concluded that these strong CH₄ bands must be highly saturated, but recent laboratory measurements of the bands made at room temperature show that curve-of-growth saturation is very small. At the presumed low pressures and temperatures in Titan's atmosphere, we show that saturation is very dependent on the band model parameters. However, in either a simple reflecting layer model or in a homogeneous scattering model saturation cannot be the principal cause of the filling in of these strong CH₄ bands if our best estimates of the band model parameters are correct. We find that an inhomogeneous scattering model atmosphere with fine “Axel dust” above most ot the CH₄ gas is needed to fill in the band centers. The calculated spectrum of one particular model of this class is compared to observations of Titan. Our essential conclusion is that Titan does have most of its scattering particles above most of the CH₄ gas which has an abundance of at least 2 km-am. This large abundance of CH₄ is necessary to produce the 6420-Å feature recently discovered in Titan's spectrum.     

Solar wind heating beyond 1 AU

The effect of an interplanetary atomic hydrogen gas on solar wind proton, electron and α-particle temperatures beyond 1 AU is considered. It is shown that the proton temperature (and probably also the α-particle temperature) reaches a minimum between 2 AU and 4 AU, depending on values chosen for solar wind and interstellar gas parameters. Heating of the electron gas depends primarily on the thermal coupling of the protons and electrons. For strong coupling (whenT _p ≳T _e ), the electron temperature reaches a minimum between 4 AU and 8 AU, but for weak coupling (Coulomb collisions only), the electron temperature continues to decrease throughout the inner solar system. A spacecraft travelling to Jupiter should be able to observe the heating effect of the solar wind-interplanetary hydrogen interaction, and from such observations it may be possible of infer some properties of the interstellar neutral gas. Currently a National Research Council Resident Research Associate.     

The Spectrum of Torsional Oscillations of the Moon

The diagnostic potentialities of the torsional oscillations for probing the structure of the interiors of the Moon are investigated. Models with no core, a liquid core, and a solid core are considered. The profiles of compressional and shear wave velocities V _P and V _S for the lunar interior estimated by Bills and Ferrari (1977), Goins et al. (1981), and Nakamura (1983) from the Apollo lunar seismic network are used. For all these models, the periods of torsional oscillations for n = 2–100 and four overtones have been calculated. The derivatives of the dimensionless eigenfrequency with respect to the dimensionless shear modulus and density are calculated and tabulated for use. These data can be used to determine corrections to the model density and shear modulus distributions due to their small change. The damping of torsional oscillations is studied. Several trial radial distributions of the dissipative function Q are considered.     

Two Regularities in the Coronal Green-Line Brightness – Magnetic Field Coupling and the Heating of the Corona

To study the quantitative relationship between the brightness of the coronal green line 530.5 nm Fe xiv and the strength of the magnetic field in the corona, we have calculated the cross-correlation of the corresponding synoptic maps for the period 1977 – 2001. The maps of distribution of the green-line brightness I were plotted using every-day monitoring data. The maps of the magnetic field strength B and the tangential B _t and radial B _r field components at the distance 1.1 R _⊙ were calculated under potential approximation from the Wilcox Solar Observatory (WSO) photospheric data. It is shown that the correlation I with the field and its components calculated separately for the sunspot formation zone ±30° and the zone 40 – 70° has a cyclic character, the corresponding correlation coefficients in these zones changing in anti-phase. In the sunspot formation zone, all three coefficients are positive and have the greatest values near the cycle minimum decreasing significantly by the maximum. Above 40°, the coefficients are alternating in sign and reach the greatest positive values at the maximum and the greatest negative values, at the minimum of the cycle. It is inferred that the green-line emission in the zone ±30° is mainly controlled by B _t, probably due to the existence of low arch systems. In the high-latitude zone, particularly at the minimum of the cycle, an essential influence is exerted by B _r, which may be a manifestation of the dominant role of large-scale magnetic fields. Near the activity minimum, when the magnetic field organization is relatively simple, the relation between I and B for the two latitudinal zones under consideration can be represented as a power-law function of the type I ∝ B ^q. In the sunspot formation zone, the power index q is positive and varies from 0.75 to 1.00. In the zone 40 – 70°, it is negative and varies from −0.6 to −0.8. It is found that there is a short time interval approximately at the middle of the ascending branch of the cycle, when the relationship between I and B vanishes. The results obtained are considered in relation to various mechanisms of the corona heating.     

Sensitivity plots for WIMP direct detection using the annual modulation signature

Annual modulation due to the Earth's motion around the Sun is a well-known signature of the expected weakly interacting massive particle (WIMP) signal induced in a solid state underground detector. In the present paper, we discuss the prospects of this technique on statistical grounds, introducing annual-modulation sensitivity plots for the WIMP–nucleon scalar cross-section for different materials and experimental conditions. The highest sensitivity to modulation is found in the WIMP mass interval 10m_W130 GeV, the actual upper limit depends on the choice of the astrophysical parameters, while the lowest values of the explorable WIMP–nucleon elastic cross-sections fall in most cases within one order of magnitude of the sensitivities of the present direct detection WIMP searches.     

An experimental study on Fischer‐Tropsch catalysis: Implications for impact phenomena and nebular chemistry

Abstract— Fischer‐Tropsch catalysis, by which CO and H₂ are converted to CH₄ on the surface of transition metals, has been considered to be one of the most important chemical reactions in many planetary processes, such as the formation of the solar and circumplanetary nebulae, the expansion of vapor clouds induced by cometary impacts, and the atmospheric re‐entry of vapor condensate due to asteroidal impacts. However, few quantitative experimental studies have been conducted for the catalytic reaction under conditions relevant to these planetary processes. In this study, we conduct Fischer‐Tropsch catalytic experiments at low pressures (1.3 times 10^?4 bar ≤ P ≤ 5.3 times 10^?1 bar) over a wide range of H₂/CO ratios (0.25–1000) using pure iron, pure nickel, and iron‐nickel alloys. We analyze what gas species are produced and measure the CH₄ formation rate. Our results indicate that the CH₄ formation rate for iron catalysts strongly depends on both pressure and the H₂/CO ratio, and that nickel is a more efficient catalyst at lower pressures and lower H₂/CO ratios. This difference in catalytic properties between iron and nickel may come from the reaction steps concerning disproportionation of CO, hydrogenation of surface carbon, and the poisoning of the catalyst. These results suggest that nickel is important in the atmospheric re‐entry of impact condensate, while iron is efficient in circumplanetary subnebulae. Our results also indicate that previous numerical models of iron catalysis based on experimental data at 1 bar considerably overestimate CH₄ formation efficiency at lower pressures, such as the solar nebula and the atmospheric re‐entry of impact condensate.     

Intergalactic medium: X-ray background and compton parameter

A model of intergalactic medium heated by QSOs and cooled by the expansion of the universe and Compton cooling is studied in the framework of a Friedmann-Robertson-Walker universe. Cosmological evolution functions of the comoving density of QSO's as well as the case of no evolution are considered. The theoretical X-ray background spectrum (through thermal bremsstrahlung) and Comptony parameter are calculated including relativistic corrections in the electron-electron, electron-proton and electron-photon interactions. The observed X-ray background and the upper limit of the Compton parametery cobe given by the COBE satellite are used to adjust, for each value of reheating redshiftsz _c ranging from 0.1 to 5.0, the present values of the temperatureT ₀ and densityn ₀ of the intergalactic gas. Forz _c > 0.25, when the theoretical X-ray spectrum fits the observed one, the adjusted values ofT ₀ andn ₀ imply iny >y cobe. On the other hand, whenT ₀ andn ₀ are consistent withy cobe, the calculated X-ray spectrum is lower than the observed one. Unless 100% of the observed X-ray background is due to discrete sources and if the intergalactic medium contributes more than 2.5% to such background we come to the interesting result that the medium must have been heated atz _c < 1. In this case we shall have to explain the high energy rates necessary to heat the intergalactic medium. Forz _c 0.25, it is possible to find values ofT ₀ andn ₀ such that both the calculated X-ray background and the y parameter simultaneously reproduce the corresponding observed values. However, in this case, unless it could be shown to be otherwise by future observations or theoretical studies, it seems that the model of hot intergalactic medium is not plausible because of the high energies required to heat the intergalactic gas.     

Intrinsic neutron background of nuclear emulsions for directional Dark Matter searches

Recent developments of the nuclear emulsion technology led to the production of films with nanometric silver halide grains suitable to track low energy nuclear recoils with submicrometric length. This improvement opens the way to a directional Dark Matter detection, thus providing an innovative and complementary approach to the on-going WIMP searches. An important background source for these searches is represented by neutron-induced nuclear recoils that can mimic the WIMP signal. In this paper we provide an estimation of the contribution to this background from the intrinsic radioactive contamination of nuclear emulsions. We also report the neutron-induced background as a function of the read-out threshold, by using a GEANT4 simulation of the nuclear emulsion, showing that it amounts to about 0.06 per year per kilogram, fully compatible with the design of a 10 kg × year exposure.     

The G‐dwarf problem in the solar neighbourhood: II. Halo subdwarfs

The oxygen abundance distribution in solar neighbourhood halo subdwarfs is deduced, using two alternative, known empirical relationships, involving the presence or the absence of [O/Fe] plateau for low [Fe/H] values, from a sample of 372 kinematically selected halo stars, for which the iron abundance distribution has been determined by Ryan & Norris (1991). The data are interpreted by a simple, either homogeneous or inhomogeneous model of chemical evolution, using an updated value of the solar oxygen abundance. The effect of changing the solar oxygen abundance, the power‐law exponent in the initial mass function, and the rate of oxygen nucleosyntesis, keeping the remaining input parameters unchanged, is investigated, and a theorem is stated. In all cases, part of the gas must necessarily be inhibited from forming stars, and no disk contamination has to be advocated for fitting the empirical oxygen abundance distribution in halo subdwarfs of the solar neighbourhood (EGD). Then a theorem is stated, which allows a one‐to‐one correspondence between simple, homogeneous models with and without inhibited gas, related to same independent parameters of chemical evolution, except lower stellar mass limit, real yield, and inhibition parameter. The mutual correlations between the latter parameters are also specified. In addition the starting point, and the point related to the first step, of the theoretical distribution of oxygen abundance (TGD) predicted by simple, inhomogeneous models, is calculated analytically. The mean oxygen abundance of the total and only inhibited gas, respectively, are also determined. Following the idea of a universal, initial mass function (IMF), a power‐law with both an exponent p = 2.9, which is acceptably close to Scalo IMF for m ≳ m_⊙, and an exponent p = 2.35, i.e. Salpeter IMF, have been considered. In general, both the age‐metallicity relationship and the empirical distribution of oxygen abundance in G dwarfs of the disk solar neighbourhood, are fitted by power‐law IMF exponents in the range 2.35 ≤ p ≤ 2.9. Acceptable models predict about 15% of the total mass in form of long‐lived stars and remnants, at the end of halo evolution, with a mean gas oxygen abundance which is substantially lower than the mean bulge and initial disk oxygen abundance. To avoid this discrepancy, either the existence of a still undetected, baryonic dark halo with about 15% of the total mass, or an equal amount of gas loss during bulge and disk formation, is necessary. The latter alternative implies a lower stellar mass limit close to 0.2 m_⊙, which is related to a power‐law IMF exponent close to 2.77. Acceptable models also imply a rapid halo formation, mainly during the first step, Δt = 0.5 Gyr, followed by a period (three steps) where small changes occur. Accordingly, statistical fluctuations are found to produce only minor effects on the evolution.     

The influence of radiation pressure on equipotential surfaces in high-temperature binary systems

In this paper we considered the effect of the radiation pressure from both components of the high-temperature binary systems in terms of a modified Roche potential. Usually, the parameters of radiation pressures ₁, ₂ can be written as a function of radiiR, effective temperatureT _eff , and massM of components. The simulation calculation shows that the positions of saddle pointsL ₁ ,L ₂ ,L ₃ and shape of critical surface are changed with radiation pressure factors ₁, ₂. Consequently, use of the conventional critical Roche model to estimate value of radii and mass ratios may be completely unwarranted if effective temperature of both componentsT _eff 30 000 K.These conclusions are applied to Wolf-Rayet binary systems V444 Cygni,v ₂ Velorum, and O-type binary system Y Cygni. The modified critical potentials are calculated and discussed.     

Reclassification and thermal history of Trenzano chondrite

Abstract— We present a new single‐crystal X‐ray diffraction (XRD) study performed on a suite of six orthopyroxene grains from the low‐shocked H6 Trenzano meteorite. The quenched intracrystalline Fe²⁺‐Mg ordering state in orthopyroxene preserves the memory of the cooling rate near closure temperature T_c, thus yielding useful constraints on the last thermal event undergone by the host rock. The orthopyroxene T_c of 522 ± 13 °C, calculated using a new calibration equation obtained by Stimpfl (2005b), is higher than in previously published H chondrite data. The orthopyroxene cooling rate at this T_c is about 100 °C/kyr. This fast rate is inconsistent with the much slower cooling rate expected for H6 in the onion shell structural and thermal model of chondrite parent bodies. A petrographic study carried out at the same time indicated that the Trenzano meteorite is an H5 chondrite and not an H6 chondrite, as it is officially classified. Furthermore, the two‐pyroxene equilibrium temperature of Trenzano (824 ± 24 °C), calculated with QUILF95, is similar to the two‐pyroxene temperature of 750–840 °C obtained for the Carcote (H5) chondrite (Kleinschrot and Okrusch 1999).     

Radial migration of planetesimals

Planetesimals orbiting a protostar in a circumstellar disk are affected by gravitational interaction among themselves and by gas drag force due to disk gas. Within the Kyoto model of planetesimal accretion, the migration rate is interpreted as the inverse of the planetary formation time scale. Here, we study time scales of gravitational interaction and gas drag force and their influence on planetesimal migration in detail. Evaluating observations of 86 T Tauri stars (Beckwithet al., 1990), we find the mean radial temperature profile of circumstellar disks. The disk mass is taken to be 0.01M in accordance with minimum mass models and observed T Tauri disks. The time scale of gravitational interaction between planetesimals is studied analogously to Chandrasekhar's stellar dynamics. Hence, Chandrasekhar's coefficient , defined as the fraction between the mean separation of planetesimals and the impact parameter, plays an important role in determining the migration rate. We find ln to lie between 5 and 10 within the protosolar disk. Our result is that, at the stage of disk evolution considered here, gas drag force affects the radial migration of planetesimals by a few orders of magnitude more than gravitational interaction.Paper presented at the Conference on Planetary Systems: Formation, Evolution, and Detection held 7–10 December, 1992 at CalTech, Pasadena, California, U.S.A.     

Structure of the region of solar wind—Interstellar medium interaction and its influence on H atoms penetrating the solar wind

The structure of the region of interaction between the solar wind and the interstellar medium in the two-shocks model (TSM), first suggested by Baranovet al. (1970), is numerically calculated.For this problem our model is true only for charged particles of the interstellar medium interacting with the solar wind, since the free paths of neutral particles are very long and any hydrodynamical approximation would be incorrect.The shapes of the outer and inner shocks, the shape of the contact surface and the distribution of the parameters inside the interaction region are calculated, and are universal and correct for other astrophysical applications such as interstellar bubbles (Weaveret al., 1977), the stellar wind flow around a globule (Dyson, 1975), the interaction of stellar winds in binaries (Prilutzky and Usov, 1976), and so on.The problem of the effect of the charge exchange of H atoms with interstellar gas protons decelerated by an outer shock on H atoms penetrating the solar system is considered using the calculated results (Wallis, 1975). This effect is shown to influence essentially the estimate of H-atom concentration in the interstellar medium based on theL -scattering data.     

Effects of redistribution with dipole scattering on line source functions

The partial frequency redistribution function for zero natural line width with dipole scattering (R_I) has been considered in obtaining the simultaneous solution of the statistical equilibrium and line transfer equations in the comoving frame of the expanding gas. We have considered a non-LTE two level atom in an expanding spherical medium whose outer radii are 3, 10 and 20 times the stellar radius with a total optical depthT ≃ 2 × 10³. In all the cases, we have calculated the population ratio of the two levels N₂/N₁ and compared these results with those obtained by using different expansion velocities and geometrical extensions. Initially, the upper level population (N₂) is set equal to zero. The converged simultaneous solution shows that the upper level population is enhanced considerably from the initial value. Variation in velocity gradients seem to have little effect on the ratio N₂/N₁ when the geometrical thickness of the medium is 3 or 10 times the stellar radius. However, when the thickness is increased to 20 times the central radius, the velocity gradients change the ratio N₂/N₁ considerably in the region where log T ≤ 2. The effect of variation of geometrical thickness is to reduce the N₂/N₁ ratio atτ = 0.     

The evolution of an impact-generated atmosphere

Shock wave and thermodynamic data for rock-forming and volatile-bearing minerals are used to determine minimum impact velocities (v_cr) and minimum impact pressures (p_cr) required to form a primary H₂O atmosphere during planetary accretion from chondritelike planetesimals. The escape of initially released water from an accreting planet is controlled by the dehydration efficiency. Since different planetary surface porosities will result from formation of a regolith, v_cr and p_cr can vary from 1.5 to 5.8 km/sec and from 90 to 600 kbar, respectively, for target porosities between 0 and ～45%. On the basis of experimental data, hydration rates for forsterite and enstatite are derived. For a global regolith layer on the Earth's surface, the maximum hydration rate equals 6 × 10¹⁰ g H₂O sec^?1 during accretion of the Earth. Attenuation of impact-induced shock pressure is modeled to the extent that the amount of released water as a function of projectile radius, impact velocity, weight fraction of water in the target, target porosity, and dehydration efficiency can be estimated. The two primary processes considered are the impact release of water bound in hydrous minerals (e.g., serpentine) and the subsequent reincorporation of free water by hydration of forsterite and enstatite. These processes are described in terms of model calculations for the accretion of the Earth. Parameters which lead to a primary atmosphere/hydrosphere are: an accretion time of ? 1.6 × 10⁸years, the use of an accretion model defined by Weidenschilling (1974, 1976), a mean planetesimal radius of 0.5 km, a hydration rate of 6 × 10¹⁰ g H₂O sec^?1 inferred from a mean porosity of ～ 10% for the upper 1 km of the accreting Earth, and values for the dehydration efficiency, DE, of 0.55 and 0.07 for the maximum and minimum pressure decay model, respectively. Conditions which prohibit the formation of a primary atmosphere include an accretion time much longer than 1.6 × 10⁸ years, a hydration rate for forsterite and enstatite well in excess of 6 × 10¹⁰ g H₂O sec^?1, and a dehydration efficiency DE < 0.07. We conclude that the concept of dehydration efficiency is of dominant importance in determining the degree to which an accreting planet acquires an atmosphere during its formation.     

Radiative cooling flows of self-gravitating filamentary clouds

We study the dynamics of a self-gravitating cooling filamentary cloud using a simplified model. We concentrate on the radial distribution and restrict ourselves to quasi-hydrostatic, cylindrically symmetric cooling flows. For a power-law dependence of cooling function on the temperature, self-similar solutions which describe quasi-hydrostatic cooling flows are derived. We consider obtically thin filaments with a constant mass per unit length and the solutions are parameterized by their line masses. There is no polytropic relation between the density and the pressure. The filament experiences radiative condensation, irrespective of the γ,the gas specific heat ratio. So, the filament becomes denser due to the quasi-hydrostatic flows and the density at the center (ρ_c) increases in proportion to (t ₀-t)^-1, where t denotes the time. The term,t ₀, denotes an epoch at which the central density increases infinitely. We also found that the radius of the filament (r _c) decreases in proportion to (t ₀-t)^1/2. This revised version was published online in July 2006 with corrections to the Cover Date.