Why Rockets Don't Get Left Behind (and Where the Train Idea Falls Short)

"Alle Beobachtungen von Strecken, Winkeln und Zeitdifferenzen stimmen in beiden Bezugssystemen überein, alle Beschleunigungen und Kräfte ebenso."

"Das bedeutet: Jedes Bezugssystem, das relativ zu einem Inertialsystem ruht oder sich mit konstanter Geschwindigkeit gegenüber ihm bewegt, ist ebenfalls ein Inertialsystem."

"Die Gesetze der klassischen Mechanik haben die besondere Eigenschaft, in jedem Inertialsystem gleichermaßen zu gelten. Dieses Relativitätsprinzip hatte bereits Galileo Galilei formuliert."

"Das Relativitätsprinzip besagt jedoch, dass es keinen physikalischen Grund gibt, warum man ein bestimmtes Bezugssystem herausgreifen und gegenüber anderen Systemen bevorzugen sollte. Sämtliche physikalischen Gesetze, die in einem Inertialsystem gelten, gelten auch in jedem anderen."

"wählt werden darf und dass es keinen grundlegenden Prozess gibt, durch den man ein bestimmtes Bezugssystem vor allen anderen auszeichnen könnte."

"Nur die beobachteten Geschwindigkeiten unterscheiden sich alle um die konstante Relativgeschwindigkeit der beiden Bezugssysteme gegeneinander."

"Ein Inertialsystem ist ein Bezugssystem, in dem der Trägheitssatz gilt. Das bedeutet, dass sich jeder kräftefreie Körper geradlinig gleichförmig bewegt oder im Zustand der Ruhe verharrt."

"Im Fall einer konstanten Geschwindigkeit der Bezugssysteme gegeneinander ist für kartesische Koordinaten in der klassischen Mechanik die Galilei-Transformation anzuwenden."

"Welche Bewegungen man als „absolut“ ansieht, ist also vollkommen willkürlich."

"Die Naturgesetze ändern sich nicht unter Galilei-Transformation."

"Eine Verschiebung des Orts oder der Zeit, eine Drehung der Koordinatenachsen sowie die Beschreibung durch einen Beobachter mit konstanter Relativgeschwindigkeit, ändern nichts am Ablauf der beschriebenen physikalischen Prozesse und Ereignisse."

"This work considers only orbital perturbations due to aerodynamic forces, which spacecraft experience due to the presence of a rarefied atmosphere, causing tumbling motion and orbital decay."

"Space debris orbiting in space experiences orbital decay due to various orbital perturbations."

"Geomagnetic storms enhance thermospheric density and increase atmospheric drag on low-Earth-orbit (LEO) satellites, accelerating their orbital decay, as illustrated by the loss of 38 Starlink satellites following the modest February 2022 storm."

"because the mean free path of the gas molecules is large compared to the size of the satellite"

"as illustrated by the loss of 38 Starlink satellites following the modest February 2022 storm"

"Eleven values of density scale height have been determined from the decrease in perigee height, with a 2 per cent error"

"The satellite would also experience a spin decay due to the aerodynamic torque acting to slow the satellite spinning."

"Although the overall drop in inclination over the 16 days was only 0.02°"

"The effect of drag on a circular orbit can be shown to cause a decay in orbital radius per orbit"

"The polar orbit of Heos 2 second-stage rocket, 1972-05B, has been determined on each of the final 16 days before its decay in September 1978"

"The first Soviet satellite, launched Oct. 4, weighs 184 pounds, has an orbit reaching out 360 miles, moves at 18.000 miles an hour and circles the earth every 96.2 min minutes."

"An Apollo TEI burn lasted approximately 150 seconds, providing a posigrade velocity increase of 1,000 m/s (3,300 ft/s)."

"Apollo 8 was the first crewed mission to leave low Earth orbit and to orbit another celestial body, and Apollo 11 was the first crewed mission to land humans on one."

"On the Apollo missions, it was performed by the restartable Service Propulsion System (SPS) engine on the Service Module after the undocking of the (LM) Lunar Module if provided."

"The first three lunar missions (Apollo 8, Apollo 10 , and Apollo 11) used a free return trajectory , keeping a flight path coplanar with the lunar orbit, which would allow a return to Earth in case the SM engine failed to make lunar orbit insertion."

"It was first performed by the Apollo 8 mission on December 25, 1968."

"Landing site lighting conditions on later missions dictated a lunar orbital plane change, which required a course change maneuver soon after TLI, and eliminated the free-return option."

"Uncrewed, second flight of Saturn V, attempted demonstration of trans-lunar injection, and direct-return abort using SM engine; three engine failures, including failure of S-IVB restart."

"Apollo 6 was intended to demonstrate the ability of the Saturn V's third stage, the S-IVB , to propel itself and the Apollo spacecraft to lunar distances."

"Trans-Earth injection The SM engine fires to send the CSM back to Earth."

"Then, after the standard two orbits to assess the vehicle's readiness for trans-lunar injection (TLI), the S-IVB was ordered to restart, but failed to do so."

"The SPS engine burned for 442 seconds to get to the planned"

"There was now, however, not enough propellant to speed up the atmospheric reentry with a second SPS engine burn"

"Los astronautas sienten el efecto de la ingravidez porque ellos van en una constante caída libre alrededor del planeta, fueron lanzados al espacio con una inclinación y una velocidad predeterminada para poder conseguir quedar en órbita."

"But if the satellite’s tangential speed is sufficient, the force of gravity acts as an unopposed turning force (acting perpendicular to the tangential velocity) such that sufficient centripetal acceleration is produced by the turning of the tangential velocity vector to keep the satellite moving in a circular path around the earth."

"are two opposing forces at work on an orbiting object: gravitational force and tangential velocity (13)."

"Si se da más velocidad al sistema, los astronautas se alejarán en el espacio; si son lanzados con menos fuerza, terminarán por caer a tierra."

"As the projectile moves horizontally in a direction tangent to the earth, the force of gravity would pull it downward if the tangential speed of the projectile is too small and it would eventually fall toward the earth."

"Para permanecer en una órbita baja, un satélite debe tener una velocidad horizontal muy alta con respecto a la Tierra."

"Ese estado se logra durante una caída libre (en un precipicio, por ejemplo) o en una órbita alrededor de un planeta (como la órbita de las estaciones espaciales), que es una especie de caída perpetua."

"The fundamental principle to be understood concerning satellites is that a satellite is a projectile. That is to say, a satellite is an object upon which the only force acting is the force of gravity."

"Para permanecer en una órbita circular 300 km por encima de la superficie de la Tierra, el satélite debe tener una velocidad de 7,8 km/s o 28000 km/h que corresponde a una órbita completa de la Tierra en 90 minutos."

"El peso, fuerza con que un astro atrae a un vehículo espacial , es inversamente proporcional al cuadrado de la distancia y, por consiguiente, disminuye con rapidez cuando el aparato se aleja de aquel"

"Now, we know that gravitational force plays a vital role in maintaining an object in its orbit."

"Esto es precisamente por estar en órbita, pues si un cuerpo estuviese estático a esa altura, rápidamente la gravedad lo haría caer)."

"Lo mismo ocurre con el ocupante de un ascensor si se rompiera el cable y si no existiera ningún roce de la cabina con el aire ni con las guías."

"Note that in an inertial frame (the earth), the satellite is in free fall and thus weightless because the only force acting on it is the force of gravity."

"Origina los movimientos a gran escala que se observan en el universo : la órbita de la Luna alrededor de la Tierra , las órbitas de los planetas alrededor del Sol , etcétera."

"Isaac Newton fue el primero en exponer que es de la misma naturaleza la fuerza que hace que los objetos caigan con aceleración constante en la Tierra (gravedad terrestre) y la fuerza que mantiene en movimiento los planetas y las estrellas ."

"es directamente proporcional al producto de las masas, e inversamente proporcional al cuadrado de la distancia que las separa:"

"La gravedad newtoniana tiene un alcance teórico infinito; la fuerza es mayor si los objetos están próximos pero a mayor distancia dicha fuerza pierde intensidad."

"He noted that a freefalling observer in a closed box would not be able to detect the force of gravity; hence, freefalling reference frames are equivalent to inertial reference frames (the equivalence principle )."

"For objects in space , the equations of motion that describe orbital motion are simpler in a non-rotating frame such as ECI."

"In its flight at 18,000 to 19,000 miles an hour, its egg-shaped or orbit - bit orbit will swing it as far as 2,500 miles out into space and as close to the earth as 400 miles."

"Observers inside a closed box that is moving with a constant velocity cannot detect their own motion; however, observers within an accelerating reference frame can detect that they are in a non-inertial reference frame from the fictitious forces that arise."

"Thus the speed of ro rotation - tation rotation of the earth is being utilized, and this results in substantial advantage in re respect - spect respect of the capacity of the rocket."

"ECI: inertial , not rotating , with respect to the stars; useful to describe motion of celestial bodies and spacecraft."

"He said the object was now nowtraveling - traveling nowtraveling around the earth at the rate of once every 103.6 minutes."

"The extent to which an ECI frame is actually inertial is limited by the non-uniformity of the surrounding gravitational field ."

"Tass reported that both Sputnik I, launched October 4, and Sputnik n, which went up last Sunday, were slowing down slightly."

"decreasing by some something - thing something like one second a day due to atmospheric drag at the low"

"point in Sputnik's orbit—about 120 or 130 miles above the"

"positively and without question, we have not —repeat not—solved this prob"

"The time for Sputnik II to make an orbit now was given as 103.52 minutes"

"The reason that gravity assist is important for these missions is that they allow the spacecraft to reach outer planets which otherwise would not be possible because of the large amount of energy required."

"Incorporating gravity assists provides highly efficient interplanetary trajectories, enabling otherwise infeasible spacecraft missions."

"An interesting way to reduce such a requirement is to resort to Gravity Assist (GA) maneuvers as shown by Langevin2 in 1999."

"Gravity assist trajectories allow the spacecraft to gain momentum from the planets."

"Preliminary interplanetary trajectory designs frequently use simplified two-body orbital mechanics and linked conics methodology to model the complex trajectories in multi-body systems."

"The simplest way to model a gravity assist maneuver is to resort to link-conic approximation: the sphere of influence of a planet is assumed to have zero radius and the gravity maneuver is considered instantaneous."

"During a close approach of a gravitational body momentum is exchanged between the spacecraft and the gravitational body."

"Gravity assist (GA) proposed by Michael A. Minovitch is an indispensable technique for present solar system exploration."

"For an ideal hyperbolic orbit, not subject to perturbations or ∆v maneuvers, the modulus of the incoming relative velocity must be equal to the modulus of the outgoing relative velocity:"

"In reality the planet loses momentum but due to the huge mass difference the change in velocity of the gravitational body is negligible compared to the gain in velocity of the spacecraft."

"In solar system exploration, it is important to optimize the maneuver and flight time to the target planet."

"An interesting alternative approach is to resort to direct collocation as demonstrated by Betts6, who efficiently optimized a transfer trajectory to Mars combining low-thrust with two swing-bys of Venus."

"Low-thrust propulsion is then used to shape trajectory arcs between two subsequent encounters and to meet the best incoming conditions for a swing-by."

"Einige Trägerraketen fliegen zuerst eine niedrige Parkbahn an und starten von dort meist einen Hohmann-Transfer , dessen elliptischer Teil die geostationäre Transferbahn ist."

"Normalerweise setzt eine Rakete den Satelliten am (oder in der Nähe des) erdnächsten Punkts (dem Perigäum ) der Ellipsenbahn aus."