José Monteiro da Rocha (1734–1819) and His 1782 Work on the Determination of Comet Orbits

In 1782 José Monteiro da Rocha, astronomer and professor at the University of Coimbra, presented, in a public session of the Royal Academy of Sciences of Lisbon, a memoire on the problem of the determination of the orbits of comets. Only in 1799 would the “ Determinação das Orbitas dos Cometas” (Determination of the orbits of comets) be published in the Academy’s memoires. In that work, Monteiro da Rocha presents a method for solving the problem of the determination of the parabolic orbit of a comet making use of three observations. Monteiro da Rocha’s method is essentially the same as the method proposed by Olbers and published under von Zach’s sponsorship 2 years before, in 1797. Having been written and published in Portuguese was certainly a hindrance for its dissemination among the international astronomical community. In this paper, we intend to present Monteiro da Rocha’s method and try to explain to what extent we can justify Gomes Teixeira’s assertion that Monteiro da Rocha and Olbers must figure together in the history of astronomy, as the first inventors of a practical and easy method for the determination of parabolic orbits of comets.

Galileo’s Parabola Observed from Pappus’ Directrix, Apollonius’ Pedal Curve (Line), Galileo’s Empty Focus, Newton’s Evolute, Leibniz’s Subtangent and Subnormal, Ptolemy’s Circle (Hodograph), and Dürer-Simon Parabola (16.03.2019)

Galileo’s Parabola describing the projectile motion passed through hands of all scholars of the classical mechanics. Therefore, it seems to be impossible to bring to this topic anything new. In our approach we will observe the Galileo’s Parabola from Pappus’ Directrix, Apollonius’ Pedal Curve (Line), Galileo’s Empty Focus, Newton’s Evolute, Leibniz’s Subtangent and Subnormal, Ptolemy’s Circle (Hodograph), and Dürer-Simon Parabola. For the description of events on this Galileo’s Parabola (this conic section parabola was discovered by Menaechmus) we will employ the interplay of the directrix of parabola discovered by Pappus of Alexandria, the pedal curve with the pedal point in the focus discovered by Apollonius of Perga (The Great Geometer), and the Galileo’s empty focus that plays an important function, too. We will study properties of this MAG Parabola with the aim to extract some hidden parameters behind that visible parabolic orbit in the Aristotelian World. For the visible Galileo’s Parabola in the Aristotelian World, there might be hidden curves in the Plato’s Realm behind the mechanism of that Parabola. The analysis of these curves could reveal to us hidden properties describing properties of that projectile motion. The parabolic path of the projectile motion can be described by six expressions of projectile speeds. In the Dürer-Simon’s Parabola we have determined tangential and normal accelerations with resulting acceleration g = 9.81 msec-2 directing towards to Galileo’s empty focus for the projectile moving to the vertex of that Parabola. When the projectile moves away from the vertex the resulting acceleration g = 9.81 msec-2 directs to the center of the Earth (the second focus of Galileo’s Parabola in the “infinity”). We have extracted some additional properties of Galileo’s Parabola. E.g., the Newtonian school correctly used the expression for “kinetic energy E = ½ mv2 for parabolic orbits and paths, while the Leibnizian school correctly used the expression for “vis viva” E = mv2 for hyperbolic orbits and paths. If we will insert the “vis viva” expression into the Soldner’s formula (1801) (e.g., Fengyi Huang in 2017), then we will get the right experimental value for the deflection of light on hyperbolic orbits. In the Plato’s Realm some other curves might be hidden and have been waiting for our future research. Have we found the Arriadne’s Thread leading out of the Labyrinth or are we still lost in the Labyrinth?

Newton’s Parabola Observed from Pappus’ Directrix, Apollonius’ Pedal Curve (Line), Newton’s Evolute, Leibniz’s Subtangent and Subnormal, Castillon’s Cardioid, and Ptolemy’s Circle (Hodograph) (09.02.2019)

Johannes Kepler and Isaac Newton inspired generations of researchers to study properties of elliptic, hyperbolic, and parabolic paths of planets and other astronomical objects orbiting around the Sun. The books of these two Old Masters “Astronomia Nova” and “Principia…” were originally written in the geometrical language. However, the following generations of researchers translated the geometrical language of these Old Masters into the infinitesimal calculus independently discovered by Newton and Leibniz. In our attempt we will try to return back to the original geometrical language and to present several figures with possible hidden properties of parabolic orbits. For the description of events on parabolic orbits we will employ the interplay of the directrix of parabola discovered by Pappus of Alexandria, the pedal curve with the pedal point in the focus discovered by Apollonius of Perga (The Great Geometer), and the focus occupied by our Sun discovered in several stages by Aristarchus, Copernicus, Kepler and Isaac Newton (The Great Mathematician). We will study properties of this PAN Parabola with the aim to extract some hidden parameters behind that visible parabolic orbit in the Aristotelian World. In the Plato’s Realm some curves carrying hidden information might be waiting for our research. One such curve - the evolute of parabola - discovered Newton behind his famous gravitational law. We have used the Castillon’s cardioid as the curve describing the tangent velocity of objects on the parabolic orbit. In the PAN Parabola we have newly used six parameters introduced by Gottfried Wilhelm Leibniz - abscissa, ordinate, length of tangent, subtangent, length of normal, and subnormal. We have obtained formulae both for the tangent and normal velocities for objects on the parabolic orbit. We have also obtained the moment of tangent momentum and the moment of normal momentum. Both moments are constant on the whole parabolic orbit and that is why we should not observe the precession of parabolic orbit. We have discovered the Ptolemy’s Circle with the diameter a (distance between the vertex of parabola and its focus) where we see both the tangent and normal velocities of orbiting objects. In this case the Ptolemy’s Circle plays a role of the hodograph rotating on the parabolic orbit without sliding. In the Plato’s Realm some other curves might be hidden and have been waiting for our future research. Have we found the Arriadne’s Thread leading out of the Labyrinth or are we still lost in the Labyrinth?

Model of the Galaxy with Hot Dark Matter

The model of the galaxy is considered as a structure of the baryonic matter embedded into the hot dark matter. The dark matter is supposed to come into being from the decaying matter after the epoch of structure formation. The galaxy is divided into two regions. In the inner region, the baryonic matter predominates over the hot dark matter while in the outer region, the hot dark matter predominates over the baryonic matter. The motion of the test particle is bounded in the inner region (elliptic orbit) and unbounded in the outer region (parabolic orbit). Observational constraints on the proposed model are considered from the rotation curves of the galaxies: Milky Way, M33, NGC 2366 and IC 2574.

Euler’s and Barker’s equations: A geometric derivation of the time of flight along parabolic trajectories

The parabolic orbit is rarely found in nature although the orbits of some comets have been observed to be very close to parabolic. The parabola is of interest mathematically because it represents the boundary between the open and closed orbit forms. An object moving along a parabolic path is on a oneway trip to infinity never being able to retrace the same orbit again. The velocity of such an object is the escape velocity and its total energy is zero.