Kepler`s Laws

"Johannes Kepler (1571-1630) é admirado como um dos principais agentes de mudança do pensamento científico, responsável por introduzir o método científico no ocidente e, juntamente com Galileu Galilei (1564 – 1642), de fundamentar a teoria da gravidade, mais tarde formalizada por Isaac Newton (1643 – 1727). Kepler, reformulando paradigmas, alterou profundamente a forma como percebemos o mundo e a nós mesmos.

Thomas Kuhn propôs como o marco limítrofe da Idade Média, a elaboração do sistema heliocêntrico por Nicolau Copérnico (1473 – 1543), que corrigia as idéias de Aristóteles e Ptolomeu (≅ 90 – 168), consideradas até então como verdades absolutas. Tal evento alterou o equilíbrio de forças entre religião e ciência, entre dogma e conhecimento, ao destronar a Terra, e por conseguinte o homem, do centro do universo. Kuhn denominou a obra de Copérnico como a Revolução Copernicana. Kepler, tomaria para si a tarefa de comprovar as idéias de seu antecessor, colocando por terra uma crença estabelecida por quinze séculos.

Το διασημότερο απόσπασμα του Γενικού Σχολίου ξεκινά με την περίφημη εξαγγελία «hypotheses non fingo», αναφορικά με τον «λόγο (reason), την εξήγηση για τις ιδιότητες της βαρύτητας στις οποίες έχει αναφερεθεί αμέσως προηγουμένως ο Νεύτων: κατά κύριο λόγο, πρόκειται για την συμπεριφορά της βαρύτητας σύμφωνα με τον νόμο αντιστρόφου τετραγώνου, συμπεριλαμβανομένου του κεντρικού («κεντρομόλου») και παγκόσμιου-καθολικού χαρακτήρα της, και της αναλογίας της με τις ποσότητες ύλης (μάζες) των αλληλεπιδρώντων σωμάτων. Αυτό διότι, κατά τον ίδιο, ό,τι δεν «παράγεται (is deduced) από «φαινόμενα», θα πρέπει να αποκαλείται υπόθεση. Και δεν έχει μπορέσει -«μέχρι στιγμής»- να παραγάγει από φαινόμενα έναν τέτοιο «λόγο». Ενώ οι «υποθέσεις», ακόμη και αν είναι «φυσικές» ή «μηχανικές», δεν έχουν θέση στην «πειραματική φιλοσοφία». Στην (φυσική) φιλοσοφία αυτού του είδους, όπου ο ίδιος αυτοτοποθετείται, οι προτάσεις «παράγονται» από «φαινόμενα» και γενικεύονται μέσω επαγωγής. Κοινά παραδεκτές (τουλάχιστον για τότε) ποιότητες όπως η αδιαπερατότητα ή η κινητικότητα των σωμάτων, καθώς και νόμοι που τις διέπουν -όπως οι νόμοι της κίνησης που ο ίδιος διατύπωσε, καθώς και ο επίμαχος νόμος της βαρύτητας- έχουν ανακαλυφθεί με τέτοια μέθοδο, σημειώνει. Τελειώνοντας το απόσπασμα, βεβαιώνει πως είναι αρκετό ότι η βαρύτητα υπάρχει πραγματικά, δρα σύμφωνα με τους νόμους που έχει ο ίδιος παρουσιάσει, και επαρκεί ως εξήγηση όλων των κινήσεων, τόσο στην ουράνια όσο και στη γήινη σφαίρα -τις οποίες με αυτό τον τρόπο ενοποίησε. Η εξαγγελία του Νεύτωνα περί «υποθέσεων», και γενικότερα το υπό συζήτηση απόσπασμα, θα πρέπει ίσως, μέχρι ορισμένου βαθμού, να ιδωθεί υπό το φως μιας διάκρισης μεταξύ θεωρησιακής (speculative) και πειραματικής (experimental) φυσικής φιλοσοφίας. Η συνήθης κατανόηση της εναντίωσης του Νεύτωνα στις «υποθέσεις» τον παρουσιάζει ως κάποιον που αποφεύγει να βασιστεί στον θεωρησιακό και θεωρούμενο ως γνω-σιολογικά προβληματικό (στο πεδίο της φυσικής φιλοσοφίας, τουλάχιστον) a priori στοχασμό, και αντ΄ αυτού προχωρά με αυστηρό τρόπο από εμπειρικά παρατηρησιακά και πειραματικά δεδομένα («φαι-νόμενα») σε συμπεράσματα καλά θεμελιωμένα σε τέτοια δεδομένα, αποκλείοντας και απορρίπτοντας επισφαλείς εικασίες -ακόμη και αν αυτές θα εξηγούσαν τα δεδομένα, αν ήταν αληθείς. Προτείνεται ένα ιδεώδες εμπειρικής «απόδειξης», όπως φαίνεται να αξιώνει ο όρος «παραγωγή (deduction)». Ενώ ό,τι δεν πληροί τις προδιαγραφές που τίθενται από ένα τέτοιο ιδεώδες, θα πρέπει να εξοβελίζεται από την («πειραματική») φιλοσοφία. Θα δούμε ότι υπάρχουν περιορισμοί σε μια τέτοια αξίωση, καθώς και σε μια αντίστοιχη ανάγνωση του Νεύτωνα. Ωστόσο, δεν θα παρεκκλίνουμε πολύ από μια παρόμοια ερμηνευτική κατεύθυνση, καθότι, εκτός του ότι είναι αρκετά συνήθης, δίνει μια καλή αφετηρία για στοχασμό γύρω από τη μέθοδο, και οδηγεί με φυσικό τρόπο στις κατάλληλες γνωσιολογικές αλλά και ιστορικές «διορθώσεις» της απλουστευτικής εικόνας.

In the Solar System most orbits are circular, but there are some exceptions. The paper addresses results from a two-step linearisation of the Kepler laws, to model non-circular orbits, at Newtonian gravity and other interactions with adjacent bodies. The orbit will then be characterised by a generalised eccentricity and a secondary frequency denoted L-frequency, ωL (and considered proportional to the angular velocity). The path will be that of a circle, superimposed by small vibrations with the L-frequency. Hereby, the amplitude corresponds to an eccentricity, such that the radius varies, with time. When the ratio between the L-frequency and angular velocity is a non-integer, ‘perihelion’ moves. Bounds are derived and resulting orbits are generated and visualized.
For the integer ratio 2, results are compared with an ellipsoidal, and a tidal wave. For a non-integer ratio, the orbit is related to data for Mercury. Methods for detecting and measuring the secondary frequency are discussed, in terms of transfer orbits in Spaceflight dynamics.

In the past, Kepler painstakingly derived laws of planetary motion using difficult to understand and hard to follow techniques. In 1843 William Hamilton created and described the quaternions, which extend the complex numbers and can easily describe rotations in three dimensional space. In this article, we will harness this system to provide a new and intuitive way to derive Kepler’s laws. This will include using a quaternionic version of the spatial Kepler problem differential equation, and using the general solution to describe the motion of planets orbiting a central body. We use the standard method for regularizing celestial mechanics, but this article will be solely focused on showing the validity of Kepler’s laws.

Interactive whiteboards (IWBs) have become commonplace in classrooms in the western world. However, using IWBs productively in ways that leverage their unique possibilities for student engagement is more challenging. I propose a theoretical perspective that recognizes the IWB-based learning environment as a case of a mixed-reality (a combination of real and virtual worlds). I show how the ideas of embodied cognition and distributed cognition were used to guide the design of a learning sequence on the topic of the orbital motion of planets and I analyse its implementation. I also analyse a short video excerpt that illustrates the way in which embodiment enters high-school student discourse in the context of small-group collaborative inquiry, supported by the IWB. Students’ gestures that draw on their physical experience within the IWB-based environment can help students fluently communicate ideas that would be much more difficult to express verbally.

In the past, Kepler painstakingly derived laws of planetary motion using difficult to understand and hard to follow techniques. In 1843 William Hamilton created and described the quaternions, which extend the complex numbers and can easily describe rotations in three dimensional space. In this article, we will harness this system to provide a new and intuitive way to derive Kepler's laws. This will include solving the spatial Kepler problem using a quaternionic version of the differential equation, and using the general solution to describe the motion of planets orbiting a central body. We use the standard method for regularizing celestial mechanics, but this article will be solely focused on showing the validity of Kepler's laws.

Спустя 412 лет от сформулировки и публикации (в Astronomia Nova) уравнения Кеплера, которое связывает эксцентрическую (и, посредова-телъно, истинную) аномалию траекторий планет с равномерно текущим временем, в соответствии с его Вторым законом площадей, впоследствии-в процессе развития орбитальной механики-с вторим законом Кепле-ра связанное и формально выведенно как несуществующее (нулевое) по-перечное ускорение- пересматривается. Кратко указуегся на некоторые по-следствия для эллиптической интеграции, симплектической интеграции, симплектической геометрии/топологии, а также на связь между физи-ческими и математическими континуумами в контексте многоуровневой масштабно-инвариантной механики/динамики, с прибавленными истинно центробежной к притягивающей и соответствующей крутящией сылами. Ключевые слова: Уравнение Кеплера, Закон площадей, Первые интегра-лы, Орбитальная механика, Неконсервативность. Введение Помимо полной энергии, состоящей из суммы кинетической и потен-циальной энергий, это также момент количества движения (L = r 2φ) ор-битального тела, который считается неизменным в течении орбитально-го периода (T), таким образом формируя так называемые (два) Первые интегралы для процесса решения с ними связанных (по сути, не коле-бательных) нелинейных дифференциальных уравнений движения. Хотя относительно легко показать, что интеграл энергии является неоснова-тельным, то есть противоречивым постулатом 1-либо путем прямой аналитической и численной оценки и/или формально применяя полную производную по времени от соответствующего дифференциального выра-жения (Приложение A в расширенной версии статьи, со ссылкой в сноске 4), для углового момента-и его аналога в форме второго закона Кеплера-это становится очень субтильной и достаточно сложной задачей. Одна-ко, вследствие самого присутствиа ненулевого тангенциального ускорения 1 Довольно редкие авторы учебников пытаются доказать его достоверность либо через предполагаемую коллинеарность векторов положения и скорости-по существу невыполненных либо в начальных условиях либо в полученных решениях, или полагаясь на полностью нулевое значение-иначе справедливо только на апсидах-поперечного ускорениа, что на самом деле опровергается в этой статье.

Here we compute for a two-body system (two spherical bodies) the dependency of the annual insolation on the ellipticity of the orbit of the celestial planetary body. This will be the basis for a simple Python progam which demonstrates the insolation as function of orbital ellipticity. This text might also be introductory for undergraduate students. This study is mostly based on the work of A.Berger [2a]

It has been found that the Kepler’s (un)fortunate and essentially unallowable, but literally “life-rescuing” systematic modification of the Mars’ trajectory measured positions, despite his subsequent explanation that the Area law – valid for the angle(s) time-dependencies determined by his Equation – pertains to the solely Sun-governed circular motion, has led to the combining of circular (through the angular momentum  constancy and the resulting annulling of the transverse accelerationand rectilinear motions (through the constancy of total energy, which is nothing else than the definitional relation of the potential energy as the acquired and/or lost kinetic energy in the potential field)  as orbital motion’s two Prime integrals.  However, since the very presence of non-zero tangential acceleration in theobserved trajectories of orbital bodies as well as in the produced solutions, thevalidity of the related First integral ‘status’ of both of these (otherwise, interdependent) quantities should have long beenquestioned.
The above expression for transverse acceleration and the corresponding one for the total energy are given in the so-called “non-inertial coordinate system”, which has turned out to be rotationally-invariant to the rectangular one, so that the numerical integration in both, as well as the analytic-numerical evaluations, produce the same results corroborating the outgoing integrals. However, when evaluating in the tangential-normal generalized coordinates (numerical integration will be pursued, too) the following is realized:
1. One gets the non-zero tangential acceleration, the projection of which on the transverse direction produces non-zero transverse acceleration;
2. Integrating the gravitational force projection on tangential direction for the vertically scaled nominal ellipse with the pertinent KE, one gets the positive energy-well.

Based on this, as well as the fact that there are numerous situations where the (closed) elliptic trajectories with two central or non-central forces do not obey the area-law and Kepler’s equation, the principle of conservativeness should be reconsidered and possibly be altogether abandoned.

On the other hand, the “Kepler’s Physical Astronomy” based on his own principle that every motion has its geometrical representation while the latter has to be ‘supported’ by the truly physical processes, Kepler has very early – at the very stage of the eccentric trajectory hypothesis – realized the necessity of presence of the mechanisms of both the attraction and repulsion, through his “Physical Astronomy”, rooted on Gilbert’s book De Magnete, assigning to the Sun’s virtus-motrics causation of purely circular motion and to a Planet – vis-insitia, the intrinsic ability to modify the very circular motion. Although this has been largely unaccounted for in the subsequent development of mechanics and dynamics, it had been followed by Leibniz with his by distance’s inverse cube explicit centrifugal force, Boscovich’s intermittent attraction-repulsion Curve, (Kepler-)Ermakov’s Equation, and by this author’s extension of the Thermo-Dynamical Oscillator concept of professor VujoGordić  - all this justifying Kepler’s insights.

Kepler's second law of general form is derived and the Newton's law of gravity from the Kepler's third law. References are made to known moments and potential energies to introduce the pseudo scalar product of vectors into the topic. This is a continuation of my work on elementary so-called commutators to facilitate the calculation of surfaces, which have been generalized here to relate celestial mechanics, quantum physics, and (my) information theory. It is also a contribution to (my future) book.