Anomalistic Period Research Articles

When the Anomalistic, Draconitic and Sidereal Orbital Periods Do Not Coincide: The Impact of Post-Keplerian Perturbing Accelerations

In a purely Keplerian picture, the anomalistic, draconitic and sidereal orbital periods of a test particle orbiting a massive body coincide with each other. Such degeneracy is removed when post-Keplerian perturbing acceleration enters the equations of motion, yielding generally different corrections to the Keplerian period for the three aforementioned characteristic orbital timescales. They are analytically worked out in the case of the accelerations induced by the general relativistic post-Newtonian gravitoelectromagnetic fields and, to the Newtonian level, by the oblateness of the central body. The resulting expressions hold for completely general orbital configurations and spatial orientations of the spin axis of the primary. Astronomical systems characterized by extremely accurate measurements of orbital periods like transiting exoplanets and binary pulsars may offer potentially viable scenarios for measuring such post-Keplerian features of motion, at least in principle. As an example, the sidereal period of the brown dwarf WD1032 + 011 b is currently known with an uncertainty as small as ≃10−5s, while its predicted post-Newtonian gravitoelectric correction amounts to 0.07s; however, the accuracy with which the Keplerian period can be calculated is just 572 s. For double pulsar PSR J0737–3039, the largest relativistic correction to the anomalistic period amounts to a few tenths of a second, given a measurement error of such a characteristic orbital timescale as small as ≃10−6s. On the other hand, the Keplerian term can be currently calculated just to a ≃9 s accuracy. In principle, measuring at least two of the three characteristic orbital periods for the same system independently would cancel out their common Keplerian component, provided that their difference is taken into account.

Quantum and post-Newtonian effects in the anomalistic period and the mean motion of celestial bodies

We study the motion of a secondary celestial body under the influence of the corrected gravitational force of a primary. We study the effect of quantum and relativistic corrections to the gravitational potential of a primary body acting on the orbiting body. More specifically, two equations are derived to approximate the perigee/perihelion/periastron time rate of change and its total variation over one revolution (i.e., the difference between the anomalistic period and the Keplerian period) under the influence of the quantum as well as post- Newtonian accelerations. Numerical results have been obtained for the artificial Earth satellite Molnya, Mercury, and, finally, the for the HW Vir c, planetary companion.

Heterogeneity and exchange: Safe-conducts in Medieval Spain

During a time when religious animosities were reaching their peak in Medieval Europe leading up to the violent crusades, Christians, Jews, and Muslims residing in close proximity in Medieval Spain experienced a period of relative peace, prosperity, and cultural exchange. This historical episode, referred to as La Convivencia, or the coexistence, is a puzzle to many period historians. This paper argues that this anomalistic period arose due to the emergence of commerce, which generated cooperation and mitigated conflict among these heterogeneous groups, a la the Doux-Commerce Thesis of the Scottish Enlightenment. Commercial exchange between Christians, Jews, and Muslims was established through the use of safe-conducts, a promise of protection sold to foreign merchants. This paper details the operation of safe-conducts and the civilizing role that commerce played among Christians, Jews, and Muslims in Medieval Spain.

Effects in the anomalistic period of celestial bodies due to a logarithmic correction to the Newtonian gravitational potential

We study the motion of a secondary celestial body under the influence of the logarithmic corrected gravitational force of a primary one. This kind of correction was introduced by Fabris and Campos (Gen. Relativ. Gravit. 41(1):93, 2009). We derive two equations to compute the rate of change of the anomalistic period with respect to the eccentric anomaly and its total variation over one revolution. In a kinematical sense, this influence produces an apsidal motion. We perform numerical estimations for some celestial bodies. We also compare our results to those obtained by considering a Yukawa correction.

THE LENSE-THIRRING EFFECT IN THE ANOMALISTIC PERIOD OF CELESTIAL BODIES

In the weak field and slow motion approximation, the general relativistic field equations are linearized, resembling those of the electromagnetic theory. In a way analogous to that of a moving charge generating a magnetic field, a mass-energy current can produce a gravitomagnetic field. In this contribution, the motion of a secondary celestial body is studied under the influence of the gravitomagnetic force generated by a spherical primary. More specifically, two equations are derived to approximate the periastron time rate of change and its total variation over one revolution (i.e., the difference between the anomalistic period and the Keplerian period). Kinematically, this influence results to an apsidal motion. The aforementioned quantities are numerically estimated for Mercury, the companion star of the pulsar PSR 1913+16, the companion planet of the star HD 80606 and the artificial Earth satellite GRACE-A. The case of the artificial Earth satellite GRACE-A is also considered, but the results present a low degree of reliability from a practical standpoint.

Leveraging Reputation: Long-Distance Trade in Medieval Spain

During a time when religious animosities were reaching their peak in Medieval Europe leading up to the violent crusades, Christians, Jews, and Muslims residing in close proximity in Medieval Spain experienced a period of relative peace, prosperity, and cultural exchange. This historical episode, referred to as La Convivencia, or the coexistence, is a puzzle to many period historians. This paper argues that this anomalistic period arose due to the emergence of commerce, which generated cooperation and mitigated conflict among these heterogeneous groups, a la the Doux-Commerce Thesis of the Scottish Enlightenment. Commercial exchange between Christians, Jews, and Muslims was established through the use of safe-conducts, a promise of protection sold to foreign merchants. This paper details the operation of safe-conducts and the civilizing role that commerce played among Christians, Jews, and Muslims in Medieval Spain.

Yukawa-type potential effects in the anomalistic period of celestial bodies

Several contemporary modified models of gravity predict the existence of a non-Newtonian Yukawa-type correction to the classical gravitational potential. We study the motion of a secondary celestial body under the influence of the corrected gravitational force of a primary. We derive two equations to approximate the periastron time rate of change and its total variation over one revolution (i.e., the difference between the anomalistic period and the Keplerian period) under the influence of the non-Newtonian radial acceleration. Kinematically, this influence produces apsidal motion. We performed numerical estimations for Mercury, for the companion star of the pulsar PSR 1913+16, and for the extrasolar Planet b of the star HD 80606. We also considered the case of the artificial Earth satellite GRACE-A, but the results present a low degree of reliability from a practical standpoint.

Orbital effects of Sun’s mass loss and the Earth’s fate

I calculate the classical effects induced by an isotropic mass loss of a body on the orbital motion of a test particle around it; the present analysis is also valid for a variation of the Newtonian constant of gravitation. I perturbatively obtain negative secular rates for the osculating semimajor axis a, the eccentricity e and the mean anomaly , while the argument of pericenter ω does not undergo secular precession, like the longitude of the ascending node Ω and the inclination I. The anomalistic period is different from the Keplerian one, being larger than it. The true orbit, instead, expands, as shown by a numerical integration of the equations of motion in Cartesian coordinates; in fact, this is in agreement with the seemingly counter-intuitive decreasing of a and e because they only refer to the osculating Keplerian ellipses which approximate the trajectory at each instant. By assuming for the Sun it turns out that the Earth's perihelion position is displaced outward by 1.3 cm along the fixed line of apsides after each revolution. By applying our results to the phase in which the radius of the Sun, already moved to the Red Giant Branch of the Hertzsprung-Russell Diagram, will become as large as 1.20 AU in about 1 Myr, I find that the Earth's perihelion position on the fixed line of the apsides will increase by AU (for ); other researchers point towards an increase of AU. Mercury will be destroyed already at the end of the Main Sequence, while Venus should be engulfed in the initial phase of the Red Giant Branch phase; the orbits of the outer planets will increase by AU. Simultaneous long-term numerical integrations of the equations of motion of all the major bodies of the solar system, with the inclusion of a mass-loss term in the dynamical force models as well, are required to check if the mutual N-body interactions may substantially change the picture analytically outlined here, especially in the Red Giant Branch phase in which Mercury and Venus may be removed from the integration.

Global repeating events in the history of the Earth and the motion of the Sun in the Galaxy

Chronological analyses of correlations between certain global repeating events (mass extinctions of marine organisms, meteorite impacts, and flashes in the frequency of geomagnetic reversals) during the Phanerozoic Eon and the motion of the solar system in the Galaxy are presented for five rotationally symmetrical models for the regular Galactic gravitational field. Thirteen of sixteen mass-extinction events can be described by a repetition interval of 183±3 million years. This is in agreement with the anomalistic period (interval between two subsequent passages of the Sun through the apocenter of its Galactic orbit) in the model of Allen and Martos. The positions of the minima and maxima in Gaussian functions approximating the frequency distribution for geomagnetic reversals also agree with the times of passage of the Sun through the apocenter and pericenter, respectively, of its Galactic orbit in this model. The maximum in the distribution of the deviations of the dates of mass extinctions from the nearest dates of impacts of large, crater-forming bodies is close to zero, providing evidence that many such events are correlated. As a rule, extinctions follow impact events. The impacts of large bodies have occurred most often when the solar system passes through the Galactic plane, while mass extinctions occur more often at some distance from the Galactic plane (about 40 pc). As a rule, intervals of increases in the frequency of geomagnetic reversals coincide with dates of impacts of large bodies. At the same time, these intervals do not show a clear correlation with the dates of mass extinctions. The intensity of mass extinctions, like the energy released by impacts, is consistently higher in periods when the Sun is moving from the apocenter toward the pericenter of its orbit, than when it is moving from the pericenter toward the apocenter. Thus, there is evidence for a variety of relationships between repeating global events in the Phanerozoic and the motion of the Sun in the Galaxy. Long-period variations in the frequency of geomagnetic reversals are correlated with the orbital motion of the Sun, and increases in the frequency of geomagnetic reversals are correlated with impacts. Mass extinctions are correlated with the impacts of large bodies, whose motions may have been perturbed by clouds of interstellar material concentrated toward the Galactic plane and by the shock front associated with the Perseus spiral arm, through which the solar system passes. The velocity of the Sun relative to the spiral pattern is estimated.

Reliability of lunar orbital periods extracted from ancient cyclic tidal rhythmites

Ancient tidal periods extracted from the geologic record are useful to constrain and calibrate theoretical models of the evolution of the earth-moon system. Analysis of rocks that contain such tidal periodicities, termed cyclic rhythmites, have tended to concentrate on long series and have focused on the more obvious periods. For example, neap-spring periods have been the most widely discussed; such information can be reliably extracted from bedform thickness series measured within laterally accreted tidal bundles and vertically accreted tidal laminae. Other lunar orbital data is also encoded in cyclic rhythmites; however this additional information has been much less extensively applied to earth-moon models. Herein, short segments of well-formed cyclic rhythmites are examined to determine how reliably the various tidal periods can be extracted via harmonic analysis. Rhythmite series analyzed here exhibit well-formed neap-spring tidal periods and also exhibit a prominent fortnightly inequality or a variation in the height of successive spring tides. This inequality is related to the elliptical nature of the lunar orbit (anomalistic month) and it is demonstrated that this period can also be reliably extracted from cyclic rhythmites. Tropical periods, which reflect the changing lunar declination that occurs during a tropical month, have also been reported from cyclic rhythmites. Unlike synodic and anomalistic periods, the apparent tropical periods extracted from some cyclic rhythmites do not always accurately correspond to the expected values. In only a few cases can this period be reliably extracted from short segments of cyclic tidal-rhythmite data. This period is commonly only poorly encoded in rhythmites because: (1) it tends to be masked by higher amplitude synodic periods, and (2) it is not well expressed within predominantly semidiurnal tidal systems. Samples selected from the Proterozoic and Phanerozoic tend to be more common during specific intervals of geologic time. These intervals were characterized by glacio-eustatic flux and, in some cases, by unique continental configurations. Understanding these basic controls on rhythmite occurrences should aid in the discovery of additional examples.

The change in satellite mean anomaly due to a rotating oblate atmosphere with a diurnal variation in density

The effect of atmospheric drag on satellite orbits of small eccentricity e ≲ 0.2, is considered. The atmospheric model allows for oblateness, and has a density profile that approximates to the observed day-to-night variation. The equation governing the changes due to drag in the mean anomaly M , during one revolution of the satellite is integrated, assuming that H , the density scale height is constant. Two particular cases are detailed. In the first, the change ∆ M in M over one anomalistic period is given for eccentricity e in the range 0.01 ≲ e ≲ 0.2, and in the second the change in ( M + ω) over one draconic period, ∆( M + ω) is given for eccentricity e in the range 0 ≼ e ≲ 0.01, where ω is the argument of perigee. Finally a compact expression for e ∆ M is derived for an atmosphere with a linear variation of H with altitude, if 0.01 ≲ e ≲ 0.2. This is the final paper in a series describing the secular and long-periodic effects of drag on the six Keplerian elements that determine the orbit of a satellite.

Satellite orbital precession caused by a rotating oblate atmosphere with a diurnal variation in density

The effect of the motion of the upper atmosphere on satellite orbits of small eccentricity, e < 0.2, is considered. The atmospheric model allows for oblateness, and has a density profile that approximates to the observed day-to-night variation. The equations governing the changes due to zonal (west-to-east) and meridional (south-to-north) winds in the right ascension of the ascending node of the orbital plane Ω during one anomalistic period of the satellite are integrated, assuming that H , the density scale-height, is constant. The resulting expressions for ∆ Ω ω , due to zonal winds, and ∆ Ω Φ , due to meridional winds, are given. It is shown that there can be appreciable changes in ∆ Ω /∆ T D , where ∆ T D is the corresponding change in orbital period, that persist for several months for certain resonant orbits if the diurnal variation in density is allowed for. Compact expressions for ∆ Ω and ∆ Ω /∆ T D are given when H varies linearly with height. It is demonstrated that in general there is no constant value of H that gives an exact equivalence between the variable- H and constant- H equations in the presence of a diurnal density variation.

The change in satellite orbital inclination due to a rotating oblate atmosphere with a diurnal variation in density

The effect of the motion of the upper atmosphere on satellite orbits of small eccentricity, e < 0.2, is considered. The atmospheric model allows for oblateness, and has a density profile that approximates to the observed day-to-night variation. The equations governing the changes due to zonal (west to east) and meridional (south to north) winds in the inclination of the orbital plane i during one anomalistic period of the satellite are integrated, with H , the density scale height, assumed to be constant. The resulting expressions for ∆ i w , due to zonal winds, and ∆ i ϕ , due to meridional winds, are given. Compact expressions for ∆ i and the ratio ∆ i /∆ T D , where ∆ T D is the corresponding change in orbital period, are given when H varies linearly with height. An equivalence between the variable- H equation and the constant- H equation is demonstrated for ∆ i w , when the value of H used in the latter is appropriately chosen. It is shown that there is no such equivalence for ∆ i ϕ and ∆ i /∆ T D .

On the perturbation of the anomalistic period of a highly eccentric orbit satellite due to the zonal harmonics

In this note a simple formula is given for the perturbation of the anomalistic period of a highly eccentric orbit due to the zonal harmonics. This perturbation depends essentially only on the semi-major axisa, the eccentricitye (or pericentre radius rπ =a(1-e)) and the latitude β of the pericentre.

Keplerian Representation of a Non-Keplerian Orbit

The properties of a mean reference orbit are developed through the introduction of an augmented radial potential proportional to c/r2. The arbitrary constant c is chosen to satisfy the perturbed energy level over an oblate Earth. This produces a mean orbit with not only the correct first order anomalistic period, but one which is also functionally identical to a Kepler orbit. Explicit formulas are obtained for the mean elements in terms of the osculating initial values. A comparison of the solution with a numerically integrated 12 h highly eccentric orbit shows that most of the error in the mean orbit is contained within the neglected periodic terms. HE fundamental problem in Keplerian representations of real orbits is the failure to account for the correct energy in the orbit. This leads to secular in-track errors (or equivalently timing errors) of the form, 5M=(nk—n)t, where nk is the Kepler mean motion and n the real mean motion. As a consequence of 6M, perturbations with respect to a reference orbit are not uniformly valid over long intervals of time. A classical approach in perturbation theory is to in- corporate as much as possible of the perturbative influence into a reference orbit that is still expressible in closed form. Garfinkel,1 Sterne,2 and Aksnes3 employ an augmented reference potential that is a function of both radius and latitude. Vinti4 utilizes oblate-spheroidal coordinates that lead to an exact solution for J2 and part of J4. Unfortunately, these solutions introduce elliptic functions and integrals and are somewhat more complicated than pure Kepler motion. If the augmented potential is restricted to a term proportional to c/r2 (where c is a constant) then the solution retains the character of the Kepler motion. By suitable choice of this constant, we bring the orbit to nearly the correct energy level and remove the objectionable secular in-track drift. The use of the energy integral for in-track stabilization is not new. Both Nacozy5 and Baumgarte6 have considered these ideas from the viewpoint of time transformations that incorporate the energy integral. Nacozy7 outlines a method which constrains the numerical solution of a system of dif- ferential equations to remain on an integral surface by ap- plying corrections at each integration step. This procedure is shown to reduce the errors in a two body numerical in- tegration by several orders of magnitude. There are a series of papers on this topic by Baumgarte and Nacozy that generally treat the numerical aspects of stabilization of dynamical systems. The intent here is to present a simple closed form incorporation of the energy integral within the structure of Keplerian mechanics. This solution should prove to be of great utility in orbit transfer and targeting problems.

Perturbations in the Anomalistic Period of Artificial Satellites Caused by the Direct Solar Radiation Pressure Centre for Astronomy and Space Sciences, Satellite Tracking Station No. 1132

AbstractApproximate formulae for the evaluation of the influence of direct solar radiation pressure (neglecting the POYNTING‐ROBERTSON effect) on the anomalistic period of artificial satellites with quasi‐circular and completely illuminated orbits are derived.

Lunar transient phenomena (LTP): Manifestations, site distribution, correlations and possible causes

The author has been compiling a catalog of LTP reports (temporary changes observed on the moon). More than 1,400 of these observations, of which 1,353 have ancillary data, were analyzed in an attempt to determine the possible causes of LTP. There were 201 sites reported at least once; about 1 2 had two or more reports. One dozen sites contain 70% of all observations, and one site, Aristarchus, provides 30%. Of the dozen most reported sites, 1 2 are rayed and 1 2 are dark flat-floored craters. The distribution of sites strongly favors the borders of both the terra and marial sides of the maria. Many are within the maria, and a very few are inland; yet most of these are associated with dark flat areas. The phenomena manifest themselves in five categories, viz., Brightenings, or Darkenings, or as Gaseous, Reddish and Bluish events. Among the hypotheses proposed for their causes are tidal, low-illumination/thermoluminescence, magnetic-tail and solar-flare effects. Analyses were conducted to see if different phenomena had different causes. There is some suggestion that they do. As concerns the tidal effects. the strongest peaks are at 0.5 (apogee) for Gaseous and Darkenings phenomena, 0.6 for Reddish events, and 0.7 for Brightenings. Reddish LTP have the strongest correlation with sunrise, while Aristarchus, Plato, Ross D area, and Bluish phenomena have the strongest correlations with solar-flare activity that produced magnetic storms on earth. “All” observations, the ones labeled “Best” (probable true anomalies), and Aristarchus, showed minima in the first half and maxima in the last half of the anomalistic (tidal) period. Histograms of several individual sites, including neighboring ones, behave differently, e.g. Aristarchus and Herodotus. When observed data are compared with expected observations (assumed to be evenly distributed) there were various correlations. For the Best data, 12 and 10% of the LTP fall close to perigee and apogee, respectively, and 10% would be expected for each. Seventeen percent occur within one day after sunrise when 3% would be expected; 20% occur while the moon is in the earth's magnetopause where 14% would be expected, and 12% occurred the same day the earth had a magnetic storm where 3% would be expected. Charts of albedo vs. age of several points for ten features were constructed. From these the normal behavior of the features throughout a lunation period was obtained. Measures that depart 2 or more full steps in Elger's albedo scale, are considered to be anomalies. Several cases of anomalous measures show up; e.g., for points on the south wall of Eimmart an albedo of 3.5 was reported once at age 10 days while for age 9 days the average albedo was 8, as it was afterward at age 11 days. The 3.5 may have been an anomalous darkening but unnoticed by the observer. Most of the features remained stable. A few exceptions were found, with Dawes showing the most anomalies. These amounted to 12% by nights or 2% by individual measures. Thus, monitoring the moon may yield an LTP once in ten nights, or 50 observations. All hypotheses show correlations with some categories and some features. Sunrise correlation is the most frequent correlation. Few correlations involve as many as 50% of the observations. The distribution of all LTP sites is different from and unique compared with deep- and shallow-focus moonquake epicenters. Routine albedo measures reveal unobserved variations which amount to about 10% in nights of observation bu 2% of individual albedo measures.

Comparative analyses of observations of lunar transient phenomena

From the author's collection of more than 900 reports of Lunar Transient Phenomena (LTP) covering the period 1540–1970, 771 positive plus 112 negative observations (several times more than any previously published analyses) with sufficient ancillary data were analyzed for five hypotheses of causes. Approximately one third were of Aristarchus, divided almost equally between one observer, Bartlett, and all others. Treated as two groups they were divided into four categories: (1) gaseous, (2) reddish, (3) bluish, and (4) brightenings, and analyzed separately and combined with respect to the hypotheses. Six other individual LTP sites also were analyzed. The five hypotheses involved effects of tides, sunrise, low-angle illumination, Earth's magnetic tail, and solar particles. Novel analyses were made by [1] histograms with respect to (a) phase (φ) of anomalistic period, (b) Moon's age; [2] plot on a representative cycle of anomalistic orbits; [3] ratios of observed to expected percentages ( O E ); [4] plot of distributions of (a) LTP sites (<100), (b) dark, flat-floored craters, and (c) the author's (1967) ringdike candidates. The histograms show: [1] caprice in analytical behavior between (a) the sites individually, even between near neighbors, and collectively, (b) the different categories, and (c) Bartlett and All Others, especially in φ: [2] any tidal correlations found were almost exclusively with perigee; [3] in lunar age, rather persistent correlations were found with feature sunrise and with the magnetic tail; [4] differences between Bartlett and All Others for the reddish phenomena, and similarities between the Bartlett bluish and All Others' brightenings suggest variant optical physiological responses, indicating higher blue-sensitivity and and lower(?) red-sensitivity for Bartlett; [5] inverse relationships between Bartlett's gaseous and reddish phenomena in both tidal and age diagrams, and the same inverse relation between his reddish and All Others' suggest that Bartlett glimpsed true ground color in the absence of obscuring matter; [6] the sharp perigee and apogee correlations found by Burley and Middlehurst (1966) smeared out or disappeared in the much larger number of observations of all sites analyzed here. Most perigee or apogee correlations clustered more at times of greater orbital eccentricity than at lesser eccentricity, as shown by the orbital plot; [7] absent and present (especially the onset) phenomena correlated similarly, instead of oppositely as expected, and both were found to advance around the orbits on the orbital plot, which weakens any positive correlations found; [8] phenomena have been reported for all ages, from O d to 28 d with >50% within ±4 d of full Moon ( < 1 4 near perigee). The ratios ( O E ) revealed: [1] universely high correlations with sunrise, not all accountable by observational selection; [2] rather high correlation with Earth's magnetopause, but not the bow-shock front turbulence; [3] decline of perigee correlation with increasing number of observations, from ( O E ) = 1.9 to 1.1 ; [4] little correlation with direct bombardment of solar flare particles but apparent enhancement by the magnetic tail when the Moon was within it at time of arrival of the particles in the Earth-Moon system. Some facts indicate that some of the brightenings result from unknown atmospheric, instrumental, and geometric effects, but many puzzling aspects still remain. Comparison of the distributions (nonrandom) of the LTP sites, dark craters, and the author's proposed ringdikes shows strong affinities for the mare edges for each, implying internal activity (probably degassing) as the source of most LTP. Considering everything the data show, the author is skeptical that there are any decisive external (even tidal) influences on this internal activity released at LTP sites.

The orbital periods of revolution of a satellite

The relationship between the anomalistic and nodal periods of revolution of a satellite is investigated and it is shown that the anomalistic period of revolution can change even in a conservative system of forces and it is suggested that the mean nodal period represents the energy of the satellite.

Attitude Determination and Prediction of Spin-Stabilized Satellites

Techniques for both attitude determination and prediction for spin-stabilized satellites are developed. Their use is demonstrated using Telstar I and II satellite data. It is shown that an inclined dipole model of the earth's magnetic field and the method of averaging the gravitational and magnetic torques over each anomalistic period of the satellite permits attitude predictions to within a few tenths of a degree of determined values in most instances. In those few cases where departures are above one degree, explanations are presented to show the reason for such discrepancies. The usefulness of combining optical flash and solar sensor data for attitude determination and their inherent accuracy are demonstrated. Optical flash data can provide loci with a resolution of 0.1°. Solar sensor loci are resolved to within 1°. All techniques described have been consolidated into working computer programs which follow closely the mathematical analysis presented. A number of important supporting calculations such as the solar position, sidereal time, orbit updating, etc. are also developed. Because of the complexities of the mean torque and gyroscopic equations, the precessional techniques presented are most useful in computer embodiments.