Mean Solar Day Research Articles

Free decay and excitation of the chandler wobble: self-consistent estimates of the period and quality factor

The period TCW and quality factor QCW of the Chandler wobble (CW) as well as polar motion (PM) transfer functions are all determined by the Earth’s layered structure, mass distribution, elasticity, rheology and energy dissipation, via the Earth’s dynamic figure parameters and complex degree-2 Love numbers. However, most previous studies used geophysical excitations derived from real-valued PM transfer functions to invert for TCW and QCW, thus leading to results that are not self-consistent. By separating the observed PM into the freely decaying CW and the excited PM, a traverse-based method is proposed to search values of TCW and QCW that can fit both sides simultaneously, yielding the self-consistent estimates of TCW = 430.4 mean solar days and QCW = 130. This implies the degree-2 tidal Love number k = 0.35011 − 0.00226i and load Love number k' = − 0.36090 + 0.00233i, and the PM transfer functions TNL = 1.80001 − 0.00692i (non-loading) and TL = 1.15040 − 0.00023i (loading) valid at the Chandler period.

Chandler period estimated from frequency domain expression solving the Liouville equation for polar motion

SUMMARY Accurate determination of the Chandler wobble (CW) period (TCW) and quality factor (QCW) is of great significance to our understanding of the Earth's dynamic figure parameters, elasticity, rheology and energy dissipation. TCW and QCW were typically determined in the time domain using the digital filter designed by Wilson; however, we developed an alternative method to estimate TCW in the frequency domain. We adopted the frequency domain expression solving the Liouville equation for polar motion (eq. 3 in the following) rather than the time domain to separate the free-damping CW and excited parts. Next, we substituted various excitation functions derived from the outputs of several general circulation models and selected monthly gravity models into the above frequency domain expression; hence we estimate TCW. The preferred TCW value using this method and the least difference combination mgm90 model is 430.4 ± 2.0 mean solar days. Comparing with previous studies within the error range, our results provide an independent way of estimating TCW.

Core–mantle topographic coupling: a parametric approach and implications for the formulation of a triaxial three-layered Earth rotation

SUMMARYWe propose a parametric approach to the topographic (TOP) coupling between the mantle and outer core for refinement of the latest triaxial three-layered Earth rotation theory. Based on three models of the core–mantle boundary (CMB) topography, we obtain the axial components of the TOP torque as −2.08 × 1019, −2.72 × 1018 and −1.97 × 1017 N m, respectively. Under the frame of the triaxial three-layered Earth rotation theory, we solve the corresponding periods of free core nutation as −(329.83 ± 28.12), −(457.54 ± ∼) and −(428.23 ± 1.09) mean solar days (d), respectively. The other three normal modes, namely, Chandler wobble, inner core wobble and free inner core nutation, are almost not affected by the TOP coupling of the CMB, their period values being 433.24, 2718.69 and 934.02 d, respectively. Calculations show that the TOP torque is highly sensitive to the adopted model of the topography, which is known to be robust. Taking into account the normal modes of the triaxial three-layered Earth rotation, the results of the CMB topography obtained by seismic tomography can be constrained in the future to a certain extent. In this study, considering the TOP coupling with the appropriate topography model, the estimates for the dynamic ellipticity ef of the fluid core lie between 0.0026340 and 0.0026430, values that are 3.56 % higher than the hydrostatic equilibrium value.

New estimation of triaxial three-layered Earth's inertia tensor and solutions of Earth rotation normal modes

Until now, the calculation of the principal inertia moment of the triaxial three-layered Earth mainly adopts the scaling method. This method assumes that the corresponding principal inertia axes of the layers coincide each other, but this is not the case. In this paper, a rigorous tensor transformation rule is adopted to calculate the principal inertia moments (PIMs) of different layers. Appling the new estimated PIMs to the triaxial three-layered Earth rotation theory with considering various couplings, the numerical calculations show that the periods of the Chandler Wobble (CW), Free Core Nutation (FCN), Free Inner Core Nutation (FICN) and Inner Core Wobble (ICW) are respectively 433.0, 430.8, 943.9 and 2735.9 mean solar days, which are well comparable with the corresponding values accepted at present in geoscience community. Better estimates of the PIMs of different layers may provide better constrains on relevant physical parameters of the Earth's interior.

Formulation of a Triaxial Three‐Layered Earth Rotation: Theory and Rotational Normal Mode Solutions

AbstractIn this study, we formulated a triaxial three‐layered anelastic Earth rotation theory considering various core‐mantle couplings, including the pressure and gravitational couplings acting on the elastic inner core by the outer core and mantle, and the viscoelectromagnetic couplings between the outer core and mantle and between the outer and inner cores. With this formulation, using the eigenvalue method, we solved four rotational normal modes, including the Chandler Wobble (CW), Free Core Nutation (FCN), Free Inner Core Nutation (FICN), and the Inner Core Wobble (ICW). The triaxiality of the actual Earth has notable effects on rotational normal modes, that is, 0.005, 0.0, 0.071, and 1.174 d (mean solar days), respectively, for the CW, FCN, FICN, and ICW. In the case that the triaxiality is bigger, for instance, if the Earth's equatorial dynamical ellipticity were , which is 1 order of magnitude larger than the presently determined ellipticity ( ), the CW and ICW periods will be lengthened by 3.579 and 94.912 d, and the FICN period will be decreased by 0.591 d. Based on our present knowledge, the numerical solutions of the four normal modes under the new rotation frame suggest that the periods of CW, FCN, FICN, and ICW are, respectively, 432.2, 429.9, 934.0, and 2,718.7 d. Our theory for Earth can be extended to other planets in our solar system.

Numerical solutions of rotational normal modes of a triaxial two-layered anelastic Earth

The Earth's rotational normal modes depend on Earth model used, including the layer structures, principal inertia moments of different layers and the compliances. This study focuses on providing numerical solution of the rotational normal modes of a triaxial two-layered anelastic Earth model without external forces but with considering the complex forms of compliances and the electromagnetic coupling between the core and mantle. Based on the present knowledge of the Chandler wobble (CW) and Free Core Nutation (FCN), we provide a set of complete compliances which could be used for reference in further investigations. There are eight rotational normal mode solutions, four of which might exist in nature. However, in reality only two of these four solutions correspond to the present motion status of the prograde CW and the retrograde FCN. On one hand, our numerical calculations show that the periods and quality factors (Qs) of CW and FCN are respectively 434.90 and 429.86 mean solar days (d) and 76.56 and 23988.47 under frequency-dependent assumption, and the triaxiality prolongs CW about 0.01 d and has hardly effect on FCN. On the other hand, we analyze the sensibility of compliances and electromagnetic coupling parameter on the periods and Qs of CW and FCN and find the sensitive parameters with respect to them.

New method for determining free core nutation parameters, considering geophysical effects

Context. In addition to the torques exerted by the Moon, Sun, and planets, changes of precession-nutation are known to be caused also by geophysical excitations. Recently studies suggest that geomagnetic jerks (GMJ) might be associated with sudden changes of phase and amplitude of free core nutation. We showed that using atmospheric and oceanic excitations with those by GMJ improves substantially the agreement with observed celestial pole offsets.Aims. Traditionally, the period T f and quality factor Q f of the free core nutation (FCN) are derived from VLBI-based celestial pole offsets (CPO). Either direct analysis of the observed CPO, or indirect method using resonant effects of nutation terms with frequencies close to FCN, are used. The latter method is usually preferred, since it yields more accurate results. Our aim is to combine both approaches to better derive FCN parameters.Methods. We numerically integrated the part of CPO that is due to geophysical excitations for different combinations of T f , Q f , using Brzezinski’s broadband Liouville equations (, Manuscripta geodaetica, 19, 157), and compared the results with the observed values of CPO. The values yielding the best fit were then estimated. The observed CPO, however, must be corrected for the change of nutation that is caused by the T f , Q f values different from those used to calculate IAU 2000 model of nutation. To this end, we have used the Mathews-Herring-Buffet transfer function and applied it to the five most affected terms of nutation (with periods 365.26, 182.62, 121.75, 27.55 and 13.66 days).Results. The results, based on the CPO data in the interval 1986.0—2016.0 and excitations with three different models, are presented. We demonstrate that better results are obtained if the influence of additional excitations at GMJ epochs is added to excitations by the atmosphere and oceans. Our preferred values are T f = 430.28 ± 0.04 mean solar days and Q f = 19 500 ± 200.

Measurement of the Earth's rotation: 720 BC to AD 2015.

New compilations of records of ancient and medieval eclipses in the period 720 BC to AD 1600, and of lunar occultations of stars in AD 1600–2015, are analysed to investigate variations in the Earth’s rate of rotation. It is found that the rate of rotation departs from uniformity, such that the change in the length of the mean solar day (lod) increases at an average rate of +1.8 ms per century. This is significantly less than the rate predicted on the basis of tidal friction, which is +2.3 ms per century. Besides this linear change in the lod, there are fluctuations about this trend on time scales of decades to centuries. A power spectral density analysis of fluctuations in the range 2–30 years follows a power law with exponent −1.3, and there is evidence of increased power at a period of 6 years. There is some indication of an oscillation in the lod with a period of roughly 1500 years. Our measurements of the Earth’s rotation for the period 720 BC to AD 2015 set firm boundaries for future work on post-glacial rebound and core–mantle coupling which are invoked to explain the departures from tidal friction.

Estimation of the free core nutation period by the sliding-window complex least-squares fit method

Estimation of the free core nutation (FCN) period is a challenging prospect. Mostly, two methods, one direct and one indirect, have been applied in the past to address the problem by analyzing the Earth orientation parameters observed by the very long baseline interferometry. The indirect method estimates the FCN period from resonance effects of the FCN on forced nutation terms, whereas the direct method estimates the FCN period using the Fourier Transform (FT) approach. However, the FCN period estimated by the direct FT technique suffers from the non-stationary characteristics of celestial pole offsets (CPO). In this study, the FCN period is estimated by another direct method, i.e., the sliding-window complex least-squares fit method (SCLF). The estimated values of the FCN period for the full set of 1984.0–2014.0 and four subsets (1984.0–2000.0, 2000.0–2014.0, 1984.0–1990.0, 1990.0–2014.0) range from −428.8 to −434.3 mean solar days. From the FT to the SCLF method, the estimate uncertainty of the FCN period falls from several tens of days to several days. Thus, the SCLF method may serve as an independent direct way to estimate the FCN period, complementing and validating the indirect resonance method that has been frequently used before.

Discrete sources of cryovolcanism on the nucleus of Comet 29P/Schwassmann–Wachmann and their origin

Evidence for long-lived sources of cryovolcanism on the nucleus of the Comet 29P/Schwassmann–Wachmann has been found from a study of its times of outburst (t0) and the morphological development of inner coma structures. Analysis of data from the Minor Planet Center observations archive spanning 2002–2014 and other observations have yielded 64 outburst times of mainly well-observed events with a median timing uncertainty of 0.40d. Outbursts comprise largely (i) isolated explosive events; or (ii) multiple outbursts occurring typically within 5–15d of each other. On rare occasions, a form of continuous or gradually increasing activity is manifest, appearing to be the result of a series of mini-outbursts. Quasi-periodicity in t0 is manifested as an excess of outbursts every 52–60d, along with a paucity of events every ∼30d and ∼90d. Seasonal changes in activity are evident from the temporal analysis of the outburst data. An unambiguous periodicity of 57.6±0.4d has been found in the times of 26 outbursts during 2010–2014, with all active sources at that time localised within a longitude span of ∼135–150°. Cluster analysis of t0 data for 2002–2010 and 2010–2014, and HST imaging from 1996 confirm and refine the apparent periodicity, indicating that outbursts appear to be grouped in longitude centred on at least 6 circumferential locations. Sources of activity generally persist for at least 10–20yr, and some appear discrete in nature, able to re-outburst after a single day–night cycle. Given that outbursts are triggered by solar heating, the analysis yields a value for the mean solar day of 57.71±0.06d, equivalent to a sidereal rotation period of 57.09±0.06d, assuming the more probable prograde direction of spin. A novel outburst mechanism is outlined in which some cometary ices, principally solid CH4, confined under pressure (>12kPa) beneath a stabilisation crust, begin to melt and absorb supervolatile gases, mainly CO and N2. These gases liberate considerable heat (5–7kJmol−1) via their enthalpy of solution inducing further melting deep within the nucleus where direct insolation heating is absent. This gas-solute process is most active near the solid–liquid interface, where the solvent temperature is lowest and gas solubility is highest. An outburst occurs when insolation heating of the crust above a gas-laden subsurface reservoir softens paraffinic hydrocarbons and causes a crustal plate to dislodge under the accumulated gas pressure, the sudden release of which provokes the explosive ex-solution of dissolved gases, principally CO, propelling entrained dust and debris into space. Fissures reseal as the plate sinks back under the gravitational influence of the large nucleus and the adhesive, waxy hydrocarbon fraction solidifies, permitting a new outburst cycle to begin. A detailed account of the gas ex-solution mechanism is the subject of a partner paper (Miles, R. [2015]. Icarus).

Temporal behavior of a ventilated claystone at the Tournemire URL: Cross-spectral analyses focused on daily harmonics

The main topic of this communication is the presentation of study, auscultation and supervision procedures of deep geological radioactive waste storage repositories using natural harmonic forcings. In this paper, the effects of natural ventilation on the macroscopic behavior of a clayrock are investigated by means of time series recorded underground over a period of two years in the eastern part of Gallery 1996 at the Tournemire Underground Research Laboratory (URL). This study is based on time series acquired in the atmosphere, at the gallery wall surface, and inside the rock mass. It includes measured signals from 6 thermo-hygrometers, 5 crack-meters (measuring the displacement of 2 shrinkage cracks and 3 tectonic fractures), and a 1 meter-FDR (Frequency Domain Reflectometry) profile probe equipped with 6 sensors for measuring the volumetric pore-water content into the rock mass. Auto-spectral and cross-spectral analyses using the concept of Singular Spectrum Harmonics (SSHs and cross-SSHs) are developed. Our analyses and interpretations focus here mainly on the solar diurnal atmospheric tide (denoted S1). This tide corresponds to the insolation cycle of the Earth atmosphere during a mean Solar Day (24h 00min). This component is tracked throughout the various measured signals (“Spectral Tracking” of tide fluctuations across signals). This is equivalent, in a way, to analyzing the temporal behavior of the URL during a “Mean Solar Day on Earth”. Results indicate that the daily natural forcing caused mainly by a combination of barometric and temperature related fluctuations, is the most important effect overall on our various signals. The daily harmonic induces the fluctuations of gallery air temperature, relative and absolute air humidity, and it leads to desaturation of the claystone, which in turn leads to the claystone deformation and damage. The effects of the annual harmonic SA may also be significant (it was fully analyzed in the more complete version of this work); however the SA results are not presented here because the spectral resolution on the annual harmonic is insufficient at this stage with only 2years and 2months of clean recorded data. Focusing the present study on daily fluctuations, the cross-spectral time shifts were obtained from the phase spectrum, for the daily component S1, for various pairs of pore-water content sensors located at different distances from the gallery wall. These time shifts were then used to quantify a “spectral velocity” which is found to be on the same order than the hydraulic conductivities deduced previously from pneumatic tests performed in the Excavation Damaged Zone of Gallery 1996.

Shortwave Cloud and Aerosol Radiative Forcings and Their Effects on the Vertical Local Heating/Cooling Rates

An analysis of atmospheric SW-radiative forcing and local heating/cooling rate is made using a one year temporal and vertical profiles of aerosol and cloud over Yaounde (11.51°E, 3.83°N). It appears that the direct influence of aerosols on the surface compared to the TOA can be 3 times larger. Annual mean value obtained at 559 mb altitude is +27.74 W/m2 with range from 0 to +43 W/m2. At 904 mb, we obtained an annual mean of ﹣46.22 W/m2 with range from ﹣65 to ﹣9 W/m2. Frequency distribution indicates that more than 95% of ARF are between +10 and +70 W/m2 at 559 mb (upper limit of UL), and more than 85% of ARF are between ﹣70 and ﹣10 W/m2 at 904 mb (upper limit of PBL). This sign change is explained by the fact that the backscattering peaks at the upper limit of the aerosol PBL layer. The maximum CRF is noted at TOA where it reaches ﹣600 W/m2 based on the time interval and the structure of clouds. The highest values occur between 11.50 and 13.50 LST. Clouds lead to a general heating of the entire atmospheric column with a much greater effect near the surface. Aerosols effect on the heating rate profile show strong cooling during the day for the lower atmosphere, with slight heating at the upper atmosphere. This cooling contribution generally increases from the surface and peacks at the upper boundary of aerosol layer where reflectivity is the most important. Depending on the moment of the day, average heating effect of clouds peacks at surface or within the middle troposphere due to the absorption by clouds particles. Vertical profiles deeply evolve exhibiting differences that exceed ﹣3 K/day according to altitude from one hour to another during a given mean solar day.

The evolution of time measurement, Part 4: the atomic second [Recalibration

Before the invention of atomic clocks, the second was defined by dividing the period of an astronomical event into a shorter time interval. For example, the second was once defined by dividing the average period of one revolution of the Earth on its axis. The mean solar second was equal to 1/86,400 of the mean solar day. To create a more stable unit of time interval, the second was redefined in 1956 as 1/31,556,925.9747 of the tropical year 1900. The ephemeris second was indeed more stable than the mean solar second but was nearly impossible to use as a time reference and of little use to metrologists or engineers. In retrospect, it seems almost ridiculous that another astronomical definition of the second was accepted during a period when atomic clocks were already being built [1], [2]. A clean transition from the mean solar second to the atomic second would have made more sense. Doomed from the start, the ephemeris second would be easy to forget about except for one thing - it became the comparison reference for the atomic second.

The new SI based on fundamental constants

The papers in this Discussion Meeting Issue of Philosophical Transactions A are based on the presentations given at a Discussion Meeting held at the Royal Society in London on 24 and 25 January 2011 entitled ‘The new SI: units of measurement based on fundamental constants’. The meeting was

New estimates of the inertia tensor and rotation of the triaxial nonrigid Earth

The Earth's rotation is perturbed by mass redistributions and relative motions within the Earth system, as well as by the torques from both the internal Earth and celestial bodies. The present study aims to establish a theory to incorporate all these factors perturbing the rotation state of the triaxial Earth, just like the traditional rotation theory of the axial‐symmetric Earth. First of all, we reestimate the Earth's inertia tensor on the basis of two new gravity models, EIGEN‐GL05C and EGM2008. Then we formulate the dynamic equations and obtain their normal modes for an Earth model with a triaxial anelastic mantle, a triaxial fluid core, and dissipative oceans. The periods of the Chandler wobble and the free core nutation are successfully recovered, being ∼433 and ∼430 mean solar days, respectively. Further, the Liouville equations and their general solutions for that triaxial nonrigid Earth are deduced. The Liouville equations are characterized by the complex frequency‐dependent transfer functions, which incorporate the effects of triaxialities and deformations of both the mantle and the core, as well as the effects of the mantle anelasticity, the equilibrium, and dissipative ocean tides. Complex transfer functions just reflect the fact that decays and phase lags exist in the Earth's response to the periodic forcing. Our theory reduces to the traditional rotation theory of the axial‐symmetric Earth when assuming rotational symmetry of the inertia tensor. Finally, the present theory is applied to the case of atmospheric‐oceanic excitation. The effective atmospheric‐oceanic angular momentum function (AMF)χeff=χeff1+iχeff2for the present theory is compared with the AMFχeffsym=χeff1sym+iχeff2symfor the traditional theory and the observed AMFχobs=χ1obs+iχ2obs; we find that the difference betweenχeffandχeffsymis of a few milliseconds of arc (mas) and can sometimes exceed 10 mas. In addition, spectrum analyses indicate thatχeffis in good agreement withχeffsymand, further, show an increase of coherency withχobsespecially in the low‐frequency band. The obvious advantage ofχeffin the low‐frequency band with respect toχeffsymis the critical support of the present theory. However, still better performance of our theory can be expected if the models of the mantle anelasticity and oceanic dynamics were improved. Thus we conclude that the traditional Earth rotation theory should be revised and upgraded to include the effects of the Earth's triaxiality, the mantle anelasticity, and oceanic dynamics. The theory presented in this study might be more appropriate to describe the rotation of the triaxial Earth (or other triaxial celestial bodies such as Mars), though further studies are needed to incorporate the effects of the solid inner core and other possible influences.

Inversion of polar motion data: Chandler wobble, phase jumps, and geomagnetic jerks

We reconsider the analysis of the polar motion with a method totally different from the wavelet analysis used in previous papers. The total polar motion is represented as the sum of oscillating annual and Chandler terms whose amplitude and phase perturbations are inverted with a nonlinear simulated annealing method. The phase variations found in previous papers are confirmed with the huge phase change (≈3π/2) occurring in the 1926–1942 period and another less important one (≈π/3) in the 1970–1980 epoch. The best Chandler period, Tc, is found to be 434 ± 0.5 mean solar days. The epochs of significant phase changes found in the Chandler wobble are also those where important geomagnetic jerks occur in the secular variation of the geomagnetic field. Turbulent viscous friction produced by small‐scale topography at the core‐mantle boundary might be responsible for the observed phase jumps.

Combined tidal ice drift and ice-induced changes in the dynamics and energy of the combined tide on the Siberian continental shelf

The results of a simulation of the combined tidal ice drift corresponding to a linear superposition of the M2, S2, K1, and O1 harmonics of the tidal generating force are discussed. Also, ice-induced maximal (during the tropical month, i.e., over 27.322 mean solar days) values of the dynamic and energy characteristics of combined motions are estimated in the marginal seas of the Siberian continental shelf. Special attention was paid to the revealing of zones of compression-rarefaction and zones of ice floe ridging.

Chandler Wobble Period andQDerived by Wavelet Transform

We apply complex Morlet wavelet transform to three polar motion data series, and derive quasi-instantaneous periods of the Chandler and annual wobble by differencing the wavelet transform results versus the scale factor, and then find their zero points. The results show that the mean periods of the Chandler (annual) wobble are 430.71 ± 1.07 (365.24 ± 0.11) and 432.71 ± 0.42 (365.23 ± 0.18) mean solar days for the data sets of 1900–2001 and 1940–2001, respectively. The maximum relative variation of the quasi-instantaneous period to the mean of the Chandler wobble is less than 1.5 % during 1900–2001 (3%–5% during 1920–1940), and that of the annual wobble is less than 1.6 % during 1900–2001. Quasi-instantaneous and mean values of Q are also derived by using the energy density—period profile of the Chandler wobble. An asymptotic value of Q = 36.7 is obtained by fitting polynomial of exponential of σ-2 to the relationship between Q and σ during 1940–2001.

Modeling of nutation and precession: New nutation series for nonrigid Earth and insights into the Earth's interior

The analytical formulation of the theories of nutation and wobble reveals the combinations of basic Earth parameters that govern the nutation‐wobble response of the Earth to gravitational (tidal) forcing by heavenly bodies and makes it possible to estimate several of them through a least squares fit of the theoretical expressions to the high‐precision data now available. This paper presents the essentials of the theoretical framework, the procedure that we used for least squares estimation of basic Earth parameters through a fit of theory to nutation‐precession data derived from an up‐to‐date very long baseline interferometry data set, the results of the estimation and their geophysical interpretation, and the nutation series constructed using the estimated values of the parameters. The theoretical formulation used here differs from earlier ones in the incorporation of anelasticity and ocean tide effects into the basic structure of the dynamical equations of the theory and in the inclusion of electromagnetic couplings of the mantle and the solid inner core to the fluid outer core, though this generalization comes at the cost of making some of the system parameters complex and frequency dependent; it is also more complete, as it takes account of nonlinear terms in these equations, including effects of the time‐dependent deformations produced by zonal and sectorial tides, which had been traditionally neglected in nonrigid Earth theories. Among the geophysical results obtained from our fit are estimates for the dynamic ellipticity e of the Earth (e = 0.0032845479 with an uncertainty of 12 in the last digit), for the dynamical ellipticity ef of the fluid core (3.8% higher than its hydrostatic equilibrium value, rather than ∼5% as hitherto), and for the two complex electromagnetic coupling constants. Our best estimates for the RMS radial magnetic fields at the core mantle boundary and at the inner core boundary, based on the estimates for these coupling constants, are ~6.9 and 72 gauss, respectively, when the magnetic field configurations are restricted to certain simple classes. The field strength needed at the inner core boundary could be lower if the density of the core fluid at this boundary or the ellipticity of the solid inner core were lower than that for the Preliminary Reference Earth Model. Our estimate for the resonance frequency of the prograde free core nutation mode, with an uncertainty of ∼10%, constitutes the first firm detection of the resonance associated with this mode; the period found is ∼1025 days, double that with electromagnetic couplings ignored. (Throughout this work, “days,” referring to periods, stands for “mean solar days.”) A new nutation series (MHB2000) is constructed by direct solution of the linearized dynamical equations (with our best fit values adopted for all the estimated Earth parameters) for each forcing frequency, and adding on the contributions from the nonlinear terms and other effects not included in the linearized equations. This series gives a considerably better fit to the nutation data than any of the earlier series based on geophysical theory. In particular, the residuals in the out of phase amplitudes of the retrograde 18.6 year and annual nutations, which had long remained at ∼0.5 milliseconds of arc (mas), are now reduced to the level of the uncertainties in the observational estimates, thanks mainly to the role played by the electromagnetic couplings. The largest remaining discrepancy is that in the out of phase prograde 18.6 year nutation, of ∼72 micorseconds of arc (μas). The frequency dependence of the nutation amplitudes cannot be exactly represented through a resonance formula, nor may the resonance frequencies themselves be interpreted as the eigenfrequencies of free modes because of the presence of complex and frequency‐dependent system parameters. Nevertheless, we have constructed a new resonance formula which reproduces our nutation series accurately for almost all nutation frequencies; for the few remaining frequencies, a listing is given of the corrections to be applied in order to reproduce the exact results of the direct solution.

A post-Pathfinder evaluation of areocentric solar coordinates with improved timing recipes for Mars seasonal/diurnal climate studies

The accurate determination of the Mars pole vector derived from Pathfinder and Viking Lander radio data ( Folkner et al., 1997. Science 278, 1749–1752), together with the VSOP87 representation of planetary orbits (Bretagnon and Francou, 1988. Astron. Astrophys. 202, 209–315), have been applied to a new evaluation of the right ascension of the “Fictitious Mean Sun” (FMS) at Mars. With Δ t J2000 the elapsed time in days from the J2000 epoch (JD 2451545.0 TT), α FMS=270°.3863+0.5240384(°/d)·Δ t J2000−4×10 −13(°/d 2)·Δ t 2 J2000 represents a best least-squares quadratic fit of the FMS, including aberration, to each instance of the four equinox and solstice passages for each of 134 Mars orbits spanning the calendar years 1874–2126. The implied tropical orbit period for Mars, 686.9726 d, closely agrees with the recent evaluations by Surán (1997. Planet. Space Sci. 45, 705–708) and Allison (1997. Geophys. Rev. Lett. 24, 1967–1970). Together with the Pathfinder radio determination of the Mars sidereal rotation, the derived FMS rate corresponds to a mean solar day (or “sol”) of 1.02749125 d. The new FMS determination would serve to define the Mean Solar Time at Mars to the nearest tenth-second, according to historical conventions originally established for terrestrial time-keeping, once the Mars prime meridian defined by the crater Airy-0 is determined within inertial space to the same accuracy. For convenient reference to current epochs, 2000 January 06 00:00:00 UTC (=MJD 51549.000 UTC) corresponds to a coincidence of α FMS and the rotation angle of the crater Airy-0 measured with respect to the Mars equinox (i.e. “mean solar midnight” on the planet’s prime meridian), to within the current uncertainty in the locational definition of the planet’s cartographic grid. As a further result of the analysis, the consistently derived Mars obliquity of date is ϵ=25°.192+3.45×10 −7(°/d)·Δ t J2000. An improved analytic recipe for the calculation of the solar areocentric longitude ( L S) of Mars to an accuracy better than 0°.01 is also provided, accounting for the primary perturbations of Earth, Jupiter, and Venus, which may in turn be applied to an efficient evaluation of Mars Local True Solar Time (LTST) to within the uncertainty of the inertial position of the Mars prime meridian. For specific applications to the data archives for landed Mars spacecraft, simple conversion formulae are given for the determination of the Viking “Local Lander Time” and the Pathfinder “Local True Solar Time” in terms of the terrestrial calendar date and UTC.