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NASA NTRS1979

M2, S2, K1 models of the global ocean tide

Ocean tidal signals appear in many geophysical measurements. Geophysicists need realistic tidal models to aid in interpretation of their data. Because of the closeness to resonance of dissipationless ocean tides, it is difficult for numerical models to correctly represent the actual open ocean tide. As an approximate solution to this problem, test functions derived by solving Laplace's Tidal Equations with ocean loading and self gravitation are used as a basis for least squares dynamic interpolation of coastal and island tidal data for the constituents M2, S2, and Kl. The resulting representations of the global tide are stable over at least a ?5% variation in the mean depth of the model basin, and they conserve mass. Maps of the geocentric tide, the induced free space potential, the induced vertical component of the solid earth tide, and the induced vertical component of the gravitational field for each contituent are presented.

Parke, M. E.↗

NASA NTRS1984

A reappraisal of Darwin's fission hypothesis and a possible limit to the primordial angular momentum of the Earth

G. H. Darwin proposed that the primordial Earth may have rotated fast enough that the solar tidal period was nearly resonant with the fundamental free oscillation period of a fluid Earth and that a large and unstable tidal oscillation split off to become the moon. Jeffreys argumented that dissipation during resonance would be sufficient to prevent such an unstable oscillation greater than the tidal frequency (period - 2.68 hr). It is considered that solar tides have extracted angular momentum from the Earth-Moon system over 4.5 b.y. The correspondence of the primordial tidal and resonant frequencies is nearly exact. (The effect of central condensation of the proto earth is to increase both frequencies by a similar amount, though the resonance is not precisely known. This result, was unknown to Darwin or Jeffreys. The effects of resonance were evaluated. The resonance is likely to be too damped for fission. This argument is more general than Jeffreys', who considered friction between the oscillating mantle and a rigid core. It is argued that the fact that Q must be so great for fission that equilibrium can not be maintained; the fluid proto Earth passes so quickly through resonance that maximum amplitude is not reached. It is suggested that solar resonant tides acted as a brake on the spin of the primordial partially molten Earth. Certain proposed origins for the Moon do not necessarily involve addition of substantial amounts of angular momentum to the Earth-Moon system. The primordial Earth-Moon system may have had nearly the same angular momentum as it has today.

Mckinnon, W. B.↗

NASA NTRS1989

Gravitational model improvement at the Goddard Space Flight Center

Major new computations of terrestrial gravitational field models were performed by the Geodynamics Branch of Goddard Space Flight Center (GSFC). This development has incorporated the present state of the art results in satellite geodesy and have relied upon a more consistent set of reference constants than was heretofore utilized in GSFC's GEM models. The solutions are complete in spherical harmonic coefficients out to degree 50 for the gravity field parameters. These models include adjustment for a subset of 66 ocean tidal coefficients for the long wavelength components of 12 major ocean tides. This tidal adjustment was made in the presence of 550 other fixed ocean tidal terms representing 32 major and minor ocean tides and the Wahr frequency dependent solid earth tidal model. In addition 5-day averaged values for Earth rotation and polar motion were derived for the time period of 1980 onward. Two types of models were computed. These are satellite only models relying exclusively on tracking data and combination models which have incorporated satellite altimetry and surface gravity data. The satellite observational data base consists of over 1100 orbital arcs of data on 31 satellites. A large percentage of these observations were provided by third generation laser stations (less than 5 cm). A calibration of the model accuracy of the GEM-T2 satellite only solution indicated that it was a significant improvement over previous models based solely upon tracking data. The rms geoid error for this field is 110 cm to degree and order 36. This is a major advancement over GEM-T1 whose errors were estimated to be 160 cm. An error propagation using the covariances of the GEM-T2 model for the TOPEX radial orbit component indicates that the rms radial errors are expected to be 12 cm. The combination solution, PGS-3337, is a preliminary effort leading to the development of GEM-T3. PGS-3337 has incorporated global sets of surface gravity data and the Seasat altimetry to produce a model complete to (50,50). A solution for the dynamic ocean topography to degree and order 10 was included as part of this adjustment.

Marsh, J. G.↗

NASA NTRS1990

Stellar kinematics of elliptical galaxies in pairs

In both galaxy pairs Arp 166 and 3C 278 the authors find radially increasing velocity dispersions indicating a perturbed, non-equilibrium state of the galaxies after the tidal interaction. In all galaxies, the increase is most pronounced in the regions which correspond to the centers of the outer isophotes. The authors suggest a scenario in which the galaxies are strongly decelerated on their orbits during the encounter. The deceleration depends on the radial position in the perturbed galaxy and vanishes in the center of the perturbed galaxy (Spitzer, 1958). In addition, the crossing time of the stars near the center is very short, implying that the tidal perturbations can be averaged over several orbital periods (e.g., Binney and Tremaine, 1987). In consequence, the central parts are not affected by the tidal interaction while the outer parts are strongly decelerated. This leads to a displacement of the central parts of the galaxies with respect to their envelopes in an anti-symmetrical way for the two components of each galaxy pair. The motions of the central parts subsequently are opposed by dynamical friction with the surrounding envelopes. Due to dynamical friction, the density of the stars increases in the wakes of the moving central parts (Mulder, 1983). The overdensity of stars in the wakes of the moving central parts efficiently decelerates the motions of the central parts. The reaction of the stars in the overdensity regions leads to an increase of the velocity dispersion mainly along the orbits of the moving central parts. The presented observations, especially the asymmetrical luminosity profiles and the radially increasing velocity dispersions support consistently the above scenario of tidal interaction between galaxies. Further spectroscopic observations are necessary in order to investigate the degree of anisotropy in the kinematically perturbed regions.

Madejsky, Rainer↗

NASA NTRS1990

The complex nature of the Seyfert galaxy NGC 7592

Long slit spectra of NGC 7592 were taken on Sep. 26 to 30, 1989 at the 1.52 cm European Southern Observatory (ESO) telescope, equipped with a Boller and Chivens spectrograph and an RCA High Resolution charge coupled device (CCD) camera. The problem of the nature of Region C is addressed at first. C shows an heliocentric radial velocity very similar to that of Regions A and B. Moreover, the arm departing from C is most probably a tidal tail, because its extension is large and its orientation is peculiar. The high H alpha luminosity of C is typical of a starburst nucleus. These facts argue in favor of C being the nucleus of a third galactic component (southern component S) physically interacting with the SE component of NGC 7592. The directions of the velocity vectors in various regions of NGC 7592 are marked. It is noteworthy that the SE component rotates clockwise, if the radial velocity difference delta v sub r from its nucleus B is due to rotation. Under the same assumption for the delta v sub r = v sub r-v sub r, A, the NW component seems to rotate counterclockwise. Thus, the gas in the regions where the two galactic bodies are in contact moves in the same way, suggesting that a prograde encounter is occurring. It is known (e.g., Toomre and Toomre, 1972) that prograde encounters have the most disruptive effects on the interacting galaxies, leading to the formation of tidal tails. The interpretation of the wing of the NW component in terms of a tidal tail thus appears very likely. A similar situation holds for the interaction between SE and S too, where S rotates counterclockwise. The interpretation of the arm departing from C as a tidal tail is supported also in this case. The difference in radial velocity between A and B (delta v sub r approx. equal - 40 km s(exp-1)) and the morphology of NGC 7592 suggests that the NW component is beyond the SE one and is approaching it. The most heavily reddened regions (E(B - V) approx. equals 0.7, derived from the H alpha/H beta ratio) are B2 and B4, namely the regions between A and B. Moreover, the small difference in radial velocity between A,B,C suggests that the three galaxies form a bound system, and their kinematics hints that they are most probably in the early stage of a merging phenomenon.

Rafanelli, Piero↗

NASA NTRS1993

The effect of ocean tides on the earth's rotation as predicted by the results of an ocean tide model

The published ocean tidal angular momentum results of Seiler (1991) are used to predict the effects of the most important semidiurnal, diurnal, and long period ocean tides on the earth's rotation. The separate, as well as combined, effects of ocean tidal currents and sea level height changes on the length-of-day, UT1, and polar motion are computed. The predicted polar motion results reported here account for the presence of the free core nutation and are given in terms of the motion of the celestial ephemeris pole so that they can be compared directly to the results of observations. Outside the retrograde diurnal tidal band, the summed effect of the semidiurnal and diurnal ocean tides studied here predict peak-to-peak polar motion amplitudes as large as 2 mas. Within the retrograde diurnal tidal band, the resonant enhancement caused by the free core nutation leads to predicted polar motion amplitudes as large as 9 mas.

Gross, Richard S.↗

NASA NTRS1994

Modulating terrestrial impacts from Oort cloud comets by the adiabatically changing galactic tides

Time modulation of the flux of new Jupiter-dominated Oort cloud comets is the subject of interest here. The major perturbation of these comets during the present epoch is due to the tidal field of the relatively smooth distribution of matter in the galactic disk. A secondary source of the near-parabolic comet flux are stars penetrating the inner Oort cloud and providing impulses that create brief comet showers. Substantial stellar-induced showers occur approximately every 100 m.y. Less frequent (but stronger) impulses due to giant molecular clouds can also perturb comets from the inner cloud. These occur on timescales of approximately equal to 500 m.y. In contrast to these infrequent stochastic shower phenomena is the continuously varying tidal-induced flux due to the galaxy. As the Sun orbits the galactic center it undergoes quasiharmonic motion about the galactic midplane, which is superimposed on the small eccentricity, near-Keplerian motion in the plane having epicycle period approximately equal to 150 m.y. In the process the galactic tidal field on the Sun/cloud system will vary causing a modulation of the observable Oort cloud flux. We have created a model of the galactic matter distribution as it affects the solar motion over a time interval ranging from 300 m.y. in the past to 100 m.y. into the future. As constraints on the disk's compact dark matter component we require consistency with the following: (1) the observed galactic rotation curve, (2) today's flux distribution of new comets, (3) the studies of K-giant distributions, and (4) the periodicity found in the terrestrial cratering record. The adiabatically varying galactic tidal torque is then determined and used to predict the time dependence of the flux. We find that a model in which approximately half the disk matter is compact is consistent with these constraints. Under such circumstances the peak-to-trough flux variation will be approx. equal to 5:1 with a full width of 9 m.y. This variability will manifest in the terrestrial cratering record and is consistent with the observed cratering periodicity, if over half of the impacts on Earth are caused by comets or asteroids that originate in the outer Oort cloud.

Matese, J. J.↗

NASA NTRS1994

Observation model and parameter partials for the JPL VLBI parameter estimation software MODEST, 19 94

This report is a revision of the document Observation Model and Parameter Partials for the JPL VLBI Parameter Estimation Software 'MODEST'---1991, dated August 1, 1991. It supersedes that document and its four previous versions (1983, 1985, 1986, and 1987). A number of aspects of the very long baseline interferometry (VLBI) model were improved from 1991 to 1994. Treatment of tidal effects is extended to model the effects of ocean tides on universal time and polar motion (UTPM), including a default model for nearly diurnal and semidiurnal ocean tidal UTPM variations, and partial derivatives for all (solid and ocean) tidal UTPM amplitudes. The time-honored 'K(sub 1) correction' for solid earth tides has been extended to include analogous frequency-dependent response of five tidal components. Partials of ocean loading amplitudes are now supplied. The Zhu-Mathews-Oceans-Anisotropy (ZMOA) 1990-2 and Kinoshita-Souchay models of nutation are now two of the modeling choices to replace the increasingly inadequate 1980 International Astronomical Union (IAU) nutation series. A rudimentary model of antenna thermal expansion is provided. Two more troposphere mapping functions have been added to the repertoire. Finally, corrections among VLBI observations via the model of Treuhaft and lanyi improve modeling of the dynamic troposphere. A number of minor misprints in Rev. 4 have been corrected.

Sovers, O. J.↗

NASA NTRS1994

An estimate of the momentum deposition in the lower thermosphere by the observed diurnal tide

This paper reports a calculation of the acceleration of the zonal mean flow induced by dissipating tides in the equatorial lower thermosphere. Estimates of the gravest symmetric gravitiational Hough mode (1,1) of the migrating diurnal tide are obtained from monthly composites of global winds observed by the Upper Atmosphere Research Satellite (UARS) High Resolution Doppler Imager (HRDI). Using the principles of classical tidal theory, the tidal momentum flux divergence is computed for a series of monthly mean (1,1) fields from January 1992 to May 1993. The contribution to the mean flow by the leading mode of the migrating tide ranges between -5 and -20 (easterly) m/s/day in the equatorial lower thermosphere. A semiannual variation is noted in the tidal amplitudes and the inferred tidal accelerations. These variations are consistent with observed trends in the zonal mean flow of the lower thermosphere.

Lieberman, Ruth S.↗

NASA NTRS1995

Ventilatory inhomogeneity determined from multiple-breath washouts during sustained microgravity on Spacelab SLS-1

We used multiple-breath N2 washouts (MBNW) to study the homogeneity of ventilation in four normal humans (mean age 42.5 yr) before, during, and after 9 days of exposure to microgravity on Spacelab Life Sciences-1. Subjects performed 20-breath MBNW at tidal volumes of approximately 700 ml and 12-breath MBNW at tidal volumes of approximately 1,250 ml. Six indexes of ventilatory inhomogeneity were derived from data from (1) distribution of specific ventilation (SV) from mixed-expired and (2) end-tidal N2, (3) change of slope of N2 washout (semilog plot) with time, (4) change of slope of normalized phase III of successive breaths, (5) anatomic lead dead space, and (6) Bohr dead space. Significant ventilatory inhomogeneity was seen in the standing position at normal gravity (1 G). When we compared standing 1 G with microgravity, the distributions of SV became slightly narrower, but the difference was not significant. Also, there were no significant changes in the change of slope of the N2 washout, change of normalized phase III slopes, or the anatomic and Bohr dead spaces. By contrast, transition from the standing to supine position in 1 G resulted in significantly broader distributions of SV and significantly greater changes in the changes in slope of the N2 washouts, indicating more ventilatory inhomogeneity in that posture. Thus these techniques can detect relatively small changes in ventilatory inhomogeneity. We conclude that the primary determinants of ventilatory inhomogeneity during tidal breathing in the upright posture are not gravitational in origin.

Prisk, G. Kim↗

NASA NTRS1995

Application of Space Shuttle photography to studies of upper ocean dynamics

Three studies have been conducted using space shuttle imagery to explain the dynamics behavior of internal waves in the Atlantic and Indian Oceans and to derive tide-related parameters for Delaware Bay. By interpreting space shuttle photographs taken during mission STS-40, a total of 34 internal wave packets on the continental shelf of the Middle Atlantic Bight have been recognized. Using the finite-depth theory we derived that the maximum amplitude of solitons is 5.6 m, the phase speed 0.42 m/s, and the period 23.8 min. Deep-ocean internal waves in the western equatorial Indian Ocean on photographs taken during mission STS-44 were also interpreted and analyzed. The internal waves occurred in the form of a multisoliton packet in which there are about a dozen solitons. The average wavelength of the solitons is 1.8 +/- 0.5 km. The crest lines are mostly straight and reach as long as 100 km. The distance between two adjacent packets is about 66 km. Using the deepwater soliton theory, we derived that the mean amplitude of the solitons is 25 m, the nonlinear phase speed 1.7 m/s, and the average period 18 min. For both cases, the semidiural tides are the principal generating mechanism. The tide-related parameters of Delaware Bay were derived from space shuttle time-series photographs taken during mission STS-40. The water area in the bay were measured from interpretation maps of the photographs. The corresponding tidal levels were calculated using the exposure time. From these data, an approximate function relating the water area to the tidal level at a reference point was determined. Then, the water areas of the Delaware Bay at mean high water (MHW) and mean low water (MLW), below 0 m, for the tidal zone, and the tidal flux were inferred. All parameters derived were reasonable and compared well with results of previous investigations.

Zheng, Quanan↗

NASA NTRS1995

Nocturnal Observations of the Semidiurnal Tide at a Midlatitude Site

Fabry-Perot interferometer observations of the mesospheric hydroxyl emission and the lower thermospheric OI (5577A) emission have been conducted from an airglow observatory at a dark field site in southeastern Michigan for the past several years. The primary functions of the observatory are to provide a database for correlative observations with the UARS satellite and to provide a synoptic measurement program for the coupling energetics and dynamics of atmospheric regions effort, An intensive operational effort between May 1993 and July 1994 has resulted in a substantial data set from which neutral winds have been determined from the bifilter acquisition sequence. A 'best fit' analysis in the least squares sense of the simultaneous measurements of the neutral winds to a 12-hour periodicity has provided amplitude and phase parameters for the semidiurnal tide as well as a measure of the mean wind. The measured tidal amplitude is greater at the higher altitude, though the seasonal behavior at both altitudes is similar with greater amplitudes during August/September and April/May. Both meridional and zonal wind components are consistent with a semidiurnal tidal description during the entire observational sequence except for the May to July 1993 period. The mean winds show annual variation in the meridional flow, being equatorward from May to October and poleward during the winter. The zonal flow is primarily eastward during the entire observational window with higher speed flows during May/June at the higher attitude and June/July at the lower altitude. A comparison with a semidiurnal tidal model indicates that the measured tidal amplitudes are a factor of 2 times greater, while the phases show similar equinoctial transitions.

Niciejewski, R. J.↗

NASA NTRS1995

Venus' Free Obliquity

The predicted orientation of Venus' rotation axis relative to its orbit ran be uniquely determined given knowledge of its J2 gravity coefficient and polar moment of inertia C if its free obliquity is fully damped. This assumption seems warranted given the dominant damping mechanism: turbulent fluid friction at a core mantle boundary (CMB). This skin friction results from differential obliquity of mantle and core spin axes, and the associated damping rate could be as short as 1/10(exp 6) year. However, the observed pole orientation indicates a free obliquity amplitude epsilon approx. = 2.1 deg compared with a nominal forced amplitude of 0.5 deg. There are two plausible explanations. The most likely is that the observed obliquity is a tidally evolved end state in which core friction, modulated by CMB ellipticity and core obliquity amplitude, counterbalances solid and atmospheric tidal torques. This concept is similar to the explanation for the retrograde spin omega as an end state in which solid and atmospheric thermal tidal torques balance at the present spin rate because of the omega(sup -1) dependence of the axial thermal torque. Weaker effects such as solid and thermal tides can then compete with core friction and for plausible models, their sum tends to increase free obliquity. The obliquity balance is controlled by the nonlinear (and nearly quadratic) dependence of the CMB turbulent "skin friction" torque on obliquity.Core ellipticity also has a profound effect on tidal evolution of Venus obliquity.

Yoder, Charles F.↗

NASA NTRS1999

Stratospheric Tides and Data Assimilation

In the upper stratosphere, the atmosphere exhibits significant diurnal and semi-diurnal tidal variations, with typical amplitude of about 2K in mid-latitudes. In this paper we examine how well the tidal variations in temperature are represented by the Goddard Geodesic Earth Orbiting Satellite (GEOS-2) data assimilation system. We show that the GEOS-2 atmospheric model is quite successful at simulating the tidal temperature variations. However, the assimilation of satellite temperature soundings significantly damps the simulated tides. The reason is because the tides are not well represented by the satellite retrievals used by the assimilation system (which have a typical tidal amplitude of around 1K). As a result of this study, we suggest improvements that should be made to the treatment of satellite soundings by the assimilation system.

Swinbank, R.↗

NASA NTRS2001

Orbital, Rotational and Climatic Interactions: Energy Dissipation and Angular Momentum Exchange in the Earth-Moon System

A numerical ocean tide model has been developed and tested using highly accurate TOPEX/Poseidon (T/P) tidal solutions. The hydrodynamic model is based on time stepping a finite difference approximation to the non-linear shallow water equations. Two novel features of our implementation are a rigorous treatment of self attraction and loading (SAL), and a physically based parameterization for internal tide (IT) radiation drag. The model was run for a range of grid resolutions, and with variations in model parameters and bathymetry. For a rational treatment of SAL and IT drag, the model run at high resolution (1/12 degree) fits the T/P solutions to within 5 cm RMS in the open ocean. Both the rigorous SAL treatment and the IT drag parameterization are required to obtain solutions of this quality. The sensitivity of the solution to perturbations in bathymetry suggest that the fit to T/P is probably now limited by errors in this critical input. Since the model is not constrained by any data, we can test the effect of dropping sea-level to match estimated bathymetry from the last glacial maximum (LGM). Our results suggest that the 100 m drop in sea-level in the LGM would have significantly increased tidal amplitudes in the North Atlantic, and increased overall tidal dissipation by about 40%. However, details in tidal solutions for the past 20 ka are sensitive to the assumed stratification. IT drag accounts for a significant fraction of dissipation, especially in the LGM when large areas of present day shallow sea were exposed, and this parameter is poorly constrained at present.

Egbert, Gary D.↗

NASA NTRS2002

Tides and Modern Geodesy

In modem high-precision geodesy, and especially in modem space geodesy, every measurement that one makes contains tidal signals. Generally these signals are considered noise and must somehow be eliminated. The stringent requirements of the latest space geodetic missions place severe demands on tidal models. On the other hand, these missions provide the strongest data for improving tidal models. In particular, TOPEX/POSEIDON altimetry and LAGEOS laser ranging have improved models to such an extent that new geophysical information about the ocean and the solid Earth are coming to light. Presumably GRACE intersatellite ranging data will also add to this information. This paper discusses several of these new geophysical results, with special emphasis given to the dissipation of tidal energy. Strong constraints have recently been placed on the partitioning of energy dissipation among the ocean, atmosphere, and solid earth and between the deep and shallow ocean. The dissipation in deep water is associated with internal tides and has potentially important implications for understanding the ocean's thermohaline circulation.

Ray, Richard D.↗

NASA NTRS2003

Lunar Rotation and the Lunar Interior

Variations in rotation and orientation of the Moon are sensitive to solid-body tidal dissipation, dissipation due to relative motion at the fluid-core/ solid-mantle boundary, and tidal Love number k2. There is weaker sensitivity to flattening of the core-mantle boundary (CMB) and fluid core moment of inertia. Accurate Lunar Laser Ranging (LLR) measurements of the distance from observatories on the Earth to four retroreflector arrays on the Moon are sensitive to lunar rotation and orientation variations and tidal displacements. Past solutions using the LLR data have given results for dissipation due to solid-body tides and fluid core plus Love number. Past detection of CMB flattening has been marginal but is improving, while direct detection of the core moment has not yet been achieved. Three decades of Lunar Laser Ranging (LLR) data are analyzed using a weighted least-squares approach. The lunar solution parameters include dissipation at the fluid-core/solid-mantle boundary, tidal dissipation, dissipation-related coefficients for rotation and orientation terms, potential Love number k2, a correction to the constant term in the tilt of the equator to the ecliptic which is meant to approximate the influence of core-mantle boundary flattening, and displacement Love numbers h2 and l2. Several solutions, with different combinations of solution parameters and constraints, are considered.

Williams, J. G.↗

NASA NTRS2004

Improved Estimate of Phobos Secular Acceleration from MOLA Observations

We report on new observations of the orbital position of Phobos, and use them to obtain a new and improved estimate of the rate of secular acceleration in longitude due to tidal dissipation within Mars. Phobos is the inner-most natural satellite of Mars, and one of the few natural satellites in the solar system with orbital period shorter than the rotation period of its primary. As a result, any departure from a perfect elastic response by Mars in the tides raised on it by Phobos will cause a transfer of angular momentum from the orbit of Phobos to the spin of Mars. Since its discovery in 1877, Phobos has completed over 145,500 orbits, and has one of the best studied orbits in the solar system, with over 6000 earth-based astrometric observations, and over 300 spacecraft observations. As early as 1945, Sharpless noted that there is a secular acceleration in mean longitude, with rate (1.88 + 0.25) 10(exp -3) degrees per square year. In preparation for the 1989 Russian spacecraft mission to Phobos, considerable work was done compiling past observations, and refining the orbital model. All of the published estimates from that era are in good agreement. A typical solution (Jacobson et al., 1989) yields (1.249 + 0.018) 10(exp -3) degrees per square year. The MOLA instrument on MGS is a laser altimeter, and was designed to measure the topography of Mars. However, it has also been used to make observations of the position of Phobos. In 1998, a direct range measurement was made, which indicated that Phobos was slightly ahead of the predicted position. The MOLA detector views the surface of Mars in a narrow field of view, at 1064 nanometer wavelength, and can detect shadows cast by Phobos on the surface of Mars. We have found 15 such serendipitous shadow transit events over the interval from xx to xx, and all of them show Phobos to be ahead of schedule, and getting progressively farther ahead of the predicted position. In contrast, the cross-track positions are quite close to the predicted values. Assuming that the along-track discrepancy is small enough that we can linearize the corrections, we model the mean orbital longitude as a quadratic function of time, and solve for corrections to the constant, linear, and quadratic terms. The time span of the recent observations is insufficient to properly resolve this issue alone, but when the 127 years of prior observations are added, we find a solution which reduces misfit to the new observations considerably, and makes no significant change to the fit to earlier observations. Our estimate for the secular acceleration term is (1.367 + 0.006) degrees per square year. The corresponding rate of energy dissipation is 3.34 MW. From a geophysical perspective, a more interesting parameter than the secular acceleration itself is the tidal lag angle, or tidal quality factor Q, for Mars. Unfortunately, the limiting error source in that determination is remaining uncertainty in the tidal Love numbers at harmonic degrees 2,3, and even 4. Until those parameters are better constrained, improvement in the orbital model of Phobos will not provide corresponding benefits for understanding the interior of Mars.

Bills, Bruce↗