True Longitude Research Articles

Table Text Jagadbhūṣaṇa of Haridatta

In the seventeenth century, the astronomer Haridatta of Mewar, Rājasthān, produced a table text named the Jagadbhūṣaṇa (epoch Śaka 1560, or 1638 CE). This table text provided calendar makers with a complete set of data and associated procedures for the computation of the annual calendar known in Sanskrit as a pañcāṅga. The tables are huge and represent an enormous computational effort, and the astronomical structure that underlies them is somewhat akin to the Babylonian Goal Year texts and similar cyclic schemes set out by Ptolemy and al-Zarqālī. The accompanying text consists of around one hundred and thirty verses organised into five chapters. This article is the first in a series that presents the Sanskrit text of the Jagadbhūṣaṇa along with a translation and a detailed technical commentary of the text and analysis of the associated tabular material, chapter by chapter. In the opening chapter, Haridatta begins the work with a lengthy encomium to his patron, Mewar Rajput Jagatsiṃha, before describing the procedures by which the true longitudes and motions of the sun and the moon can be determined using the accompanying tabulated data.

«Combustion Tables» in Twelfth-Century Latin Europe: A Preliminary Study

The Latin manuscript sources studied in this article jointly document the existence and erstwhile circulation of a highly atypical set of computational tables for planetary longitudes, coupled with extensive tables for ascensions, which served astrologers in Latin Europe as early as the 1130s. An associated text of c.1220 refers to the tables for the five planets as « combustion tables » (tabule combustionis), which reflects the way the tables in question use the time and position of the last conjunction between a planet and the Sun - known in medieval terminology as combustio - as an anchor for calculating the planet’s true ecliptic longitude at a later date within its synodic period. Multiple lines of evidence suggest that the combustion tables as well as the associated solar and ascension tables originally circulated alongside the pseudo-Ptolemaic Iudicia, an astrological work of probable Arabic origin (pre-1138). Overall, the surviving manuscript material raises the possibility that the tables and the Iudicia were at one point a single work that supported astrological computations and judgments at an early stage of their respective development in Latin Europe.

Development and Evaluation of Deep Learning Models for Forecasting Gas Production and Flowback Water in Shale Gas Reservoirs

In this paper, deep learning models are developed based on a multilayer perceptron to forecast 12 month cumulative produced shale gas and 90 day produced flowback water using a study area within the Eagle Ford Formation as a database. These models can help decision-makers have important references when drilling new wells. Latitude, longitude, true vertical depth, lateral longitude, total proppant, and total fracture water are used as input variables. The trained models are evaluated in a study area within the Burgos Basin to analyze the energy benefits, freshwater consumption, and flowback water produced through well drilling for the first time in Mexico. A non-dominated sorting genetic algorithm is used for the minimization of total freshwater consumption and maximization of cumulative produced gas. The multilayer perceptron models showed good results. The coefficient of determination and the mean absolute error of the single-output models remained above 0.93 and below 3.00 for the validation set, respectively.

Solution space exploration of low-thrust minimum-time trajectory optimization by combining two homotopies

Solving the low-thrust trajectory optimization problem with indirect methods has two main challenges including initial guessing and global optimum searching. To overcome these difficulties, two homotopies, including the thrust-homotopy and the longitude-homotopy, are combined to investigate the solution space of the low-thrust minimum-time trajectory optimization problem. The thrust-homotopy can be interpreted as a continuation on the maximum thrust magnitude, and the longitude-homotopy is proposed via a continuation on the final cumulative true longitude. Afterwards, the structure of the solution space is revealed by simulating a transfer scenario from the geostationary transfer orbit to the geostationary orbit. A bounded and smooth solution surface, where each point represents a solution with specific thrust and final cumulative true longitude, is formed by intersecting longitude-homotopy paths and thrust-homotopy paths. Based on the solution surface, the thrust-longitude-combined homotopic approach and a new hybrid homotopy are proposed to obtain local solutions by tracking homotopy paths from a solution of an easy problem, avoiding initial guessing. Finally, the longitude-homotopy is applied to search the global minimum-time solution from an obtained local solution.

Approximate time-optimal low-thrust rendezvous solutions between circular orbits

Traditional planetocentric low-thrust solutions mainly focus on orbital rephasing or transfer maneuvers, while the rendezvous maneuver is more challenging to obtain and cannot be approximated by existing solutions in some mission scenarios. In this paper, approximate solutions to the time-optimal rendezvous problem between two close circular orbits are presented. First, the optimal control problem is established in terms of the true longitude by using the Sundman transformation, and a set of unified linearized equations of motion is derived to reduce the number of key parameters characterizing this problem. Then, we show that two key parameters can characterize the rendezvous problem, and that the rephasing (or transfer) corresponds to the case when one of the key parameters equals zero. Thus, a set of piecewise functions is developed to approximate the rephasing and transfer solutions. The rendezvous solutions are finally obtained by a traversal method and summarized in a set of contour maps. Numerical simulations demonstrate that the proposed solutions provide good initial guesses for solving the problem with nonlinear dynamics and accurately approximate the performance index for the preliminary mission design.

Observed tidal evolution of Kleopatra’s outer satellite

Aims. The orbit of the outer satellite Alexhelios of (216) Kleopatra is already constrained by adaptive-optics astrometry obtained with the VLT/SPHERE instrument. However, there is also a preceding occultation event in 1980 attributed to this satellite. Here, we try to link all observations, spanning 1980–2018, because the nominal orbit exhibits an unexplained shift by + 60° in the true longitude. Methods. Using both a periodogram analysis and an ℓ = 10 multipole model suitable for the motion of mutually interacting moons about the irregular body, we confirmed that it is not possible to adjust the respective osculating period P2. Instead, we were forced to use a model with tidal dissipation (and increasing orbital periods) to explain the shift. We also analysed light curves spanning 1977–2021, and searched for the expected spin deceleration of Kleopatra. Results. According to our best-fit model, the observed period rate is Ṗ2 = (1.8 ± 0.1) × 10−8 d d−1 and the corresponding time-lag Δt2 = 42 s of tides, for the assumed value of the Love number k2 = 0.3. This is the first detection of tidal evolution for moons orbiting 100 km asteroids. The corresponding dissipation factor Q is comparable with that of other terrestrial bodies, albeit at a higher loading frequency 2|ω − n|. We also predict a secular evolution of the inner moon, Ṗ1 = 5.0 × 10−8, as well as a spin deceleration of Kleopatra, Ṗ0 = 1.9 × 10−12. In alternative models, with moons captured in the 3:2 mean-motion resonance or more massive moons, the respective values of Δt2 are a factor of between two and three lower. Future astrometric observations using direct imaging or occultations should allow us to distinguish between these models, which is important for our understanding of the internal structure and mechanical properties of (216) Kleopatra.

A generalization of the equinoctial orbital elements

We introduce six quantities that generalize the equinoctial orbital elements when some or all the perturbing forces that act on the propagated body are derived from a disturbing potential. Three of the elements define a non-osculating ellipse on the orbital plane, other two fix the orientation of the equinoctial reference frame, and the last allows one to determine the true longitude of the body. The Jacobian matrices of the transformations between the new elements and the position and velocity are explicitly given. As a possible application we investigate their use in the propagation of Earth's artificial satellites showing a remarkable improvement compared to the equinoctial orbital elements.

Practical Optimization of Low-Thrust Minimum-Time Orbital Rendezvous in Sun-Synchronous Orbits

High-specific-impulse electric propulsion technology is promising for future space robotic debris removal in sun-synchronous orbits. Such a prospect involves solving a class of challenging problems of low-thrust orbital rendezvous between an active spacecraft and a free-flying debris. This study focuses on computing optimal low-thrust minimum-time many-revolution trajectories, considering the effects of the Earth oblateness perturbations and null thrust in Earth shadow. Firstly, a set of mean-element orbital dynamic equations of a chaser (spacecraft) and a target (debris) are derived by using the orbital averaging technique, and specifically a slow-changing state of the mean longitude difference is proposed to accommodate to the rendezvous problem. Subsequently, the corresponding optimal control problem is formulated based on the mean elements and their associated costate variables in terms of Pontryagin’s maximum principle, and a practical optimization procedure is adopted to find the specific initial costate variables, wherein the necessary conditions of the optimal solutions are all satisfied. Afterwards, the optimal control profile obtained in mean elements is then mapped into the counterpart that is employed by the osculating orbital dynamics. A simple correction strategy about the initialization of the mean elements, specifically the differential mean true longitude, is suggested, which is capable of minimizing the terminal orbital rendezvous errors for propagating orbital dynamics expressed by both mean and osculating elements. Finally, numerical examples are presented, and specifically, the terminal orbital rendezvous accuracy is verified by solving hundreds of rendezvous problems, demonstrating the effectiveness of the optimization method proposed in this article.

Application of the Angular Independent Variable and Its Regularizing Transformation in the Problems of Optimizing Low-Thrust Trajectories

The expediency of using the true longitude or the associated angle as an independent variable in the problems of optimizing multi-revolution, low-thrust trajectories of spacecraft is considered. The auxiliary longitude is introduced, which is used further as a new independent variable instead of time. The advantage of using the auxiliary longitude as an independent variable in the problem of optimizing the trajectories with a fixed angular range and optimum transfer time is demonstrated. The problems of optimizing the trajectories of spacecraft with an ideally controlled, limited-power engine and a limited-thrust engine with a constant exhaust velocity are considered. The possibility of improving the convergence of the methods of solving the problems of optimizing the trajectories with a limited-thrust engine is considered. This improvement is achieved by joint utilization of the smoothing approximation of a relay thrust-switching function and the regularizing transformation of an independent variable, which extends the switching function values in the vicinity of thrust switching instants. Based on the results, the method for solving the problems of optimizing multi-revolution transfers is proposed; the numerical examples of its application are presented.

Islamic Astronomical Tables: Mathematical Analysis and Historical Investigation

Contents: Preface Part 1 Methods for Analysing Astronomical Tables: A statistical method for recovering unknown parameters from medieval astronomical tables On Ptolemy's table for the equation of time A table for the true solar longitude in the Jami' Zij Al-Khwarizmi's astronomical tables revisited: analysis of the equation of time Origin of the mean motion tables of Jai Singh. Part 2 Studies of Zijes: The Zij-i Nasiri by Mahmud ibn 'Umar. The earliest Indian-Islamic astronomical handbook with tables and its relation the Ala i Zij A second manuscript of the Mumtahan Zij Re-editing the tables in the Sabi' Zij by al-Battani (ca. AD 900) (with Fritz S. Pedersen) Dates and eras in the Islamic world: era chronology in astronomical handbooks Indexes.

PERIPHERIES OF EPICYCLES IN THE <italic>GRAHALĀGHAVA</italic>

For finding the true positions of the Sun, the Moon and the five planets the Indian classical astronomical texts use the concept of the <italic>manda</italic> epicycle which accounts for the equation of the centre. In addition, in the case of the five planets (Mercury, Venus, Mars, Jupiter and Saturn) another equation called <italic>śīghraphala</italic> and the corresponding <italic>śīghra</italic> epicycle are adopted. This correction corresponds to the transformation of the true heliocentric longitude to the true geocentric longitude in modern astronomy. In some of the popularly used handbooks (<italic>karaṇa</italic>) instead of giving the mathematical expressions for the above said equations, their discrete numerical values, at intervals of 15°, are given. In the present paper using the data of discrete numerical values we build up continuous functions of periodic terms for the <italic>manda</italic> and <italic>śīghra</italic> equations. Further, we obtain the critical points and the maximum values for these two equations.

Secrets of South Asia from Fra Mauro (1459) to Later Maps

This evaluation of Fra Mauro's world map is due to coincidence. After studying Marco Polo's route in Asia for a decade, in early 2016, I discovered Professor Piero Falchetta's important book about the subject, which every major library should have. Considering his 2,921 entries, one may assume that Falchetta has identified about 90% of Fra Mauro's place names. Our aim was to add about another 3%, including some proposed revisions. Myanmar, Thailand, Central Asia, and China offered many new cartographical fix points. Some of these are confirmed by the identifications and conclusions of the Thai scholar Thavatchai Tangsirivanich. The accuracy of the Mediterranean and the Black Sea on Fra Mauro's map invites a map-maker to trace their shorelines onto vellum and overlay it on a modern map of Eurasia in the Mercator projection. This methodology may seem too simple, but such visual comparison yields invaluable conclusions, including a table of geographical points with true latitudes and longitudes compared with those obtained from Fra Mauro. Many maps from past centuries are compared with his across South Asia from Turkey to Vietnam.

The Moon in the Oxford Tables of 1348

Shortly after being recast in Paris in the 1320s, the Alfonsine Tables reached England. There, the otherwise unknown Oxford scholar William Batecombe compiled a set of tables based on them, some of which display a clever and compact presentation. The author composed double argument tables for the five planets and the Moon whose entries depend on two variables. In the case of the planets, the entries are their true longitudes and latitudes, whereas for the Moon the entries are the true elongation of the Moon from the Sun. Thus, the user is able to avoid the long and cumbersome calculations that are required when using tables where the entries in each column depend on a single variable. Batecombe’s tables are extant in many Latin manuscripts, where they are usually called Tabule anglicane, as well as in medieval Hebrew versions. In this paper, we analyze one of these double argument tables not addressed previously, that for the Moon.

Effects of mass transfer between Martian satellites on surface geology

Impacts on planetary bodies can lead to both prompt secondary craters and projectiles that reimpact the target body or nearby companions after an extended period, producing so-called “sesquinary” craters. Here we examine sesquinary cratering on the moons of Mars. We model the impact that formed Voltaire, the largest crater on the surface of Deimos, and explore the orbital evolution of resulting high-velocity ejecta across 500years using four-body physics and particle tracking.The bulk of mass transfer to Phobos occurs in the first 102years after impact, while reaccretion of ejecta to Deimos is predicted to continue out to a 104year timescale (cf. Soter, S. [1971]. Studies of the Terrestrial Planets. Cornell University). Relative orbital geometry between Phobos and Deimos plays a significant role; depending on the relative true longitude, mass transfer between the moons can change by a factor of five. Of the ejecta with a velocity range capable of reaching Phobos, 25–42% by mass reaccretes to Deimos and 12–21% impacts Phobos. Ejecta mass transferred to Mars is <10%.We find that the characteristic impact velocity of sesquinaries on Deimos is an order of magnitude smaller than those of background (heliocentric) hypervelocity impactors and will likely result in different crater morphologies. The time-averaged flux of Deimos material to Phobos can be as high as 11% of the background (heliocentric) direct-to-Phobos impactor flux. This relatively minor contribution suggests that spectrally red terrain on Phobos (Murchie, S., Erard, S. [1996]. Icarus 123, 63–86) is not caused by Deimos material. However the high-velocity ejecta mass reaccreted to Deimos from a Voltaire-sized impact is comparable to the expected background mass accumulated on Deimos between Voltaire-size events. Considering that the high-velocity ejecta contains only 0.5% of the total mass sent into orbit, sesquinary ejecta from a Voltaire-sized impact could feasibly resurface large parts of the Moon, erasing the previous geological record. Dating the surface of Deimos may be more challenging than previously suspected.

Minimum-Time Trajectory Optimization of Multiple Revolution Low-Thrust Earth-Orbit Transfers

The problem of determining high-accuracy minimum-time Earth-orbit transfers using low-thrust propulsion is considered. The optimal orbital transfer problem is posed as a constrained nonlinear optimal control problem and is solved using a variable-order Legendre–Gauss–Radau quadrature orthogonal collocation method. Initial guesses for the optimal control problem are obtained by solving a sequence of modified optimal control problems where the final true longitude is constrained and the mean square difference between the specified terminal boundary conditions and the computed terminal conditions is minimized. It is found that solutions to the minimum-time low-thrust optimal control problem are only locally optimal, in that the solution has essentially the same number of orbital revolutions as that of the initial guess. A search method is then devised that enables computation of solutions with an even lower cost where the final true longitude is constrained to be different from that obtained in the original locally optimal solution. A numerical optimization study is then performed to determine optimal trajectories and control inputs for a range of initial thrust accelerations and constant specific impulses. The key features of the solutions are then determined, and relationships are obtained between the optimal transfer time and the optimal final true longitude as a function of the initial thrust acceleration and specific impulse. Finally, a detailed postoptimality analysis is performed to verify the close proximity of the numerical solutions to the true optimal solution.

Limitations of Methods: The Accuracy of the Values Measured for the Earth's/Sun's Orbital Elements in the Middle East, A.D. 800–1500, Part 1

(ProQuest: ... denotes formulae omitted.)(ProQuest: ... denotes non-USASCII text omitted.)The present paper deals with the methods proposed and the values achieved for the eccentricity and the longitude of apogee of the (apparent) orbit of the Sun in the Ptolemaic context in the Middle East during the medieval period. The main goals of this research are as follows: first, to determine the accuracy of the historical values in relation to the theoretical accuracy and/or the intrinsic limitations of the methods used; second, to investigate whether medieval astronomers were aware of the limita- tions, and if so, which alternative methods (assumed to have a higher accuracy) were then proposed; and finally, to see what was the fruit of the substitution in the sense of improving the accuracy of the values achieved.In Section 1, the Ptolemaic eccentric orbit of the Sun and its parameters are introduced. Then, its relation to the Keplerian elliptical orbit of the Earth, which will be used as a criterion for comparing the historical values, is briefly explained. In Section 2, three standard methods of measurement of the solar orbital elements in the medieval period found in the primary sources are reviewed. In Section 3, more than twenty values for the solar eccentricity and longitude of apogee from the medieval period will be classified, provided with historical comments. Discus- sion and conclusions will appear in Section 4 (in Part 2), followed there by two discussions of the medieval astronomers' considerations of the motion of the solar apogee and their diverse interpretations of the variation in the values achieved for the solar eccentricity.1. IntroductionPtolemy, in Almagest ΠΙ, presented a solar model consisting of an eccentric circle whose centre is displaced from the centre of the Earth by an amount of eccentricity e = TO expressed in terms of an arbitrary length R-60 for the radius of the eccentre (Figure 1). The Sun revolves on the eccentre around the Earth but its motion is uniform with respect to the centre of eccentre, and completes one revolution in a tropical year TY (counted in days), i.e., with a mean motion of ? = 360/7T. Due to the eccentricity of its orbit, the true longitude of the Sun (i.e., its longitude as seen from the Earth T) does not match its mean longitude (its longitude as it appears from the centre O of uniform motion) except at apogee (A) and perigee (Π). The difference between the two, called the 'equation of centre' q, is defined as a one-variable function of the solar mean anomaly c, which is the difference between its mean longitude and the longitude Aap of the apogee:Therefore, the solar eccentric model has three parameters (TY or ?, e, and Aap) but only one anomaly due to its eccentricity.Ptolemy assumes that all three parameters are constant.' Since, at least, the latter part of the ninth century, the Middle Eastern astronomers found (cf. Tables 1 and 2) that the longitude of the solar apogee increases with the passage of time and they (seemingly, first of all, the Banu Musa or Thabit b. Qurra and Habash) assumed its rate of change to be the same as the rate of precession, as is the case with the motion of the apogee of the planets in the Ptolemaic context (the relevant discussion will be presented in Section 4). Also, different values for e and TY were observed during the medieval period. The variety in the values obtained for the two was so large that, for example, Copernicus in the prologue of his De revolutionibus complains that [the astronomers] are uncertain of the motion of the Sun and Moon to such an extent that they cannot demonstrate or observe a constant magnitude for the tropical year,2 while BirunT was forced to resolve his readers' worries about the huge differences in the values measured for the solar eccentricity over a short period (see Section 4).In fact both e and TY are changing with the passage of time, e decreases slowly by the average amount of 4. …

Bīrūnī's Four-Point Method for Determining the Eccentricity and the Direction of the Apsidal Lines of the Superior Planets

(ProQuest: ... denotes formulae omitted.)In the Ptolemaic models for the superior planets and Venus, the centre of the epicycle is located on a circle (the so-called deferent), the centre of which is displaced from that of the Earth by an eccentricity e, but its motion is uniform with respect to the so-called equant point, removed from the Earth by an eccentricity 2e from the Earth on the side of the centre of the deferent. The derivation of the planet's orbital elements (eccentricity and direction of the apsidal line) requires observations of three oppositions of the planet to the mean sun, when the planet points to the centre of epicycle. A difficulty arises since the angles of mean motion of the planet between two successive mean oppositions are known with respect to the equant point while the angles of the true motions between two consecutive mean oppositions are measured with respect to the Earth, the centre of the deferent being located exactly between the two. Ptolemy, in the Almagest (X.7, XI. 1 , XL5), explains an iterative algorithmic solution for finding the two parameters.1N. M. Swerdlow, in 1987, published a detailed analysis of another method proposed by Jabir b. Aflah of Seville (fl. the first quarter of the twelfth century), based on the Latin translation of his Islah al-majisfi (Improvement of the Almagest), made by Gerard of Cremona in 1 175. The method is a theoretical alternative to Ptolemy's but, as Swerdlow shows, it is practically inapplicable.2 1 will here call it the four-point method, because it makes use of four oppositions of the planet with the mean sun. Hitherto it has remained unnoticed that this method is already found in al-Qanun al-mas'udi of Abu al-Rayhan al-Biruni (973-1 048), 3 composed about one century earlier (apparently, during the reign of Mas 'Ud I, the ninth ruler of the Ghaznavid dynasty of Iran, 1030-41). What follows is a complete description of the method as explained by Biruni supplied with some mathematical explanations.Biruni states that Ptolemy was able to determine the eccentricity and the direction of the apsidal Une of a superior planet by seeking out four oppositions of the planet with the mean sun satisfying the condition below. Figure 1, as drawn in the edited text of al-Qanun with the original lettering (except those bearing primes) transcribed according to the standard proposed by E. S. Kennedy,4 shows the deferent ABCGK of the planet about the centre D; E represents the Earth and T, the centre of uniform motion, the equant [point]. TE is then the eccentricity of the equant [point] and DE, that of the deferent; in the Ptolemaic context, TE = 2DE = 2e. In the four mean oppositions, the planet and thus the centre of the epicycle are located, respectively, at the points A, B, G, and K. Our author first explains the essential condition in the application of the method: the difference between the true longitudes of the planet in each pair of oppositions (A-B and G-K) should be identical: angle AEB = angle GEK. The trajectories travelled by the epicycle's centre on the deferent in each pair of oppositions should also be identical (arc AB = arc GK) so that the mean motions of the planet in each pair of oppositions are equal: angle ATB = angle KTG. These conditions may be formulated as follows: in the two pairs of oppositions, the planet's centre of the epicycle describes the equal arcs, AB and GK, in equal periods of time. Biruni adds that what we mentioned is the property of the two arcs on the deferent equidistant from the diameter of the deferent passing through the apogee and perigee. Then the point C that marks one of the two apses (here, the apogee) is at the middle of the arc BG between the two arcs AB and GK (the point O, diametrically opposed to C, is then the perigee). Our author had already proved (al-Qanun VL8)5 the two particular theorems treating of the relation between the true motion in longitude and the corresponding equation of centrum in the solar eccentric orbit. …

Displaced tables in Latin: the Tables for the Seven Planets for 1340

The anonymous set of astronomical tables preserved in Paris, Bibliotheque nationale de France, MS lat. 10262, is the first set of displaced tables to be found in a medieval Latin text. These tables are a reworking of the standard Alfonsine tables and yield the same results. However, the mean motions are defined differently, the presentation of the tables is unprecedented, and some new functions are introduced for computing true planetary longitudes. The absence of any instructions as well as unusual technical terms in the headings make it difficult to appreciate the cleverness that went into the construction of these tables that are extant in a unique copy. In this article we provide a detailed analysis of these tables and their underlying parameters. The displaced tables are typical of a pervasive tendency in Islamic science to provide extensive and elegant numerical tables for the convenience of practitioners. The underlying astronomical theory is neither questioned nor affected. Edward S. Kennedy

CONSTRAINTS ON CHARON'S ORBITAL ELEMENTS FROM THE DOUBLE STELLAR OCCULTATION OF 2008 JUNE 22

Pluto and its main satellite, Charon, occulted the same star on 2008 June 22. This event was observed from Australia and La Reunion Island, providing the east and north Charon Plutocentric offset in the sky plane (J2000): X= + 12,070.5 ± 4 km (+ 546.2 ± 0.2 mas), Y= + 4,576.3 ± 24 km (+ 207.1 ± 1.1 mas) at 19:20:33.82 UT on Earth, corresponding to JD 2454640.129964 at Pluto. This yields Charon's true longitude L= 153.483 ± 0fdg071 in the satellite orbital plane (counted from the ascending node on J2000 mean equator) and orbital radius r= 19,564 ± 14 km at that time. We compare this position to that predicted by (1) the orbital solution of Tholen & Buie (the solution), (2) the PLU017 Charon ephemeris, and (3) the solution of Tholen et al. (the solution). We conclude that (1) our result rules out solution TB97, (2) our position agrees with PLU017, with differences of ΔL= + 0.073 ± 0fdg071 in longitude, and Δr= + 0.6 ± 14 km in radius, and (3) while the difference with the T08 ephemeris amounts to only ΔL= 0.033 ± 0fdg071 in longitude, it exhibits a significant radial discrepancy of Δr= 61.3 ± 14 km. We discuss this difference in terms of a possible image scale relative error of 3.35 × 10-3in the 2002-2003 Hubble Space Telescope images upon which the T08 solution is mostly based. Rescaling the T08 Charon semi-major axis, a = 19, 570.45 km, to the TB97 value, a = 19636 km, all other orbital elements remaining the same (T08/TB97 solution), we reconcile our position with the re-scaled solution by better than 12 km (or 0.55 mas) for Charon's position in its orbital plane, thus making T08/TB97 our preferred solution.

Total irradiation during any time interval of the year using elliptic integrals

The calculation of the total solar energy received during a given interval of time over the year requires more attention than the calculation of the daily incoming solar radiation (daily insolation). It depends indeed upon whether the time interval is defined in terms of the true longitude or of the calendar date. The details of such a calculation based on elliptic integrals and on the sum of the daily irradiations are given in this paper. Numerical examples show the very high accuracy of both methods. The analytical expression and the spectral analysis of the long-term variations of such irradiation received during any time interval over the year show that they depend almost exclusively upon obliquity with a very small contribution of eccentricity, not at all upon precession. The eccentricity signal comes from the variation of the so-called solar constant through the variation of the mean distance from the Earth to the Sun. The precession signal is eliminated through the second law by Kepler. The correlation between total irradiation and obliquity reverses its sign at a specific latitude of the summer hemisphere which is a function of the selected time interval. At this latitude the obliquity signal disappears and only a pure eccentricity signal is left in the total irradiation, but with a very weak amplitude. Amplitude of the continuous wavelet transforrm shows that the amplitude variation of both the daily irradiance (mostly precession) and of the total irradiation (obliquity) vanishes around the Mid-Pleistocene Transition.