Falling Sideways: Why a Spacecraft Isn't Left Behind
A satellite is a projectile whose only force is gravity 42, so the real puzzle is not how it catches up to Earth but how it keeps falling sideways fast enough to keep missing the ground. Sputnik I, launched on October 4, 1957, circled the Earth every 96.2 minutes 22.
In short
A satellite is a projectile whose only force is gravity 4248, so in Earth's frame it is in free fall and weightless 48. If its sideways speed is too small, gravity pulls it down toward Earth 39; staying in low orbit requires a very high horizontal velocity relative to Earth 40. Observed velocity differs between frames moving at constant relative velocity, shifting by that constant relative velocity 6. A posigrade burn gave the Apollo trans-Earth injection, which sent the spacecraft back toward Earth from lunar orbit, a speed increase of about 1,000 m/s over roughly 150 seconds 23. Gravity assist trajectories let a spacecraft gain momentum from planets it passes 69. In low Earth orbit, aerodynamic forces from the rarefied atmosphere cause tumbling and orbital decay 12, and geomagnetic storms increase that drag by heating and expanding the thermosphere 14. The cited facts do not describe the outbound trans-lunar injection that first left Earth orbit 23.
Velocity depends on who's looking
Galileo Galilei had already formulated the principle that the same laws of classical mechanics hold in every inertial frame 3. Any frame moving at constant velocity relative to an inertial frame is also inertial 2, and no reference frame is privileged in physics 5.
The bookkeeping for this is the Galilean transformation, which adds the frame's velocity to all relative velocities 8. The only quantity that differs between such frames is the observed velocity, which shifts by that constant relative velocity 6. Observers in a closed box moving at constant velocity cannot detect their own motion 56.
A satellite is a projectile whose only force acting on it is gravity 42. An elevator analogy illustrates the same free-fall effect: an occupant of a cabin whose cable breaks, with no air friction or guide friction, would feel weightless 47. In that sense, astronauts in orbit are in constant free fall around the planet, because they were launched with a specific inclination and velocity so that they stay in orbit 35.
Sputnik and the art of falling sideways
Sputnik I was launched on October 4, 1957, and its orbit reached 360 miles out, circling the Earth every 96.2 minutes 22. Dr. Paul Herget reported that Sputnik II circled the Earth once every 103.6 minutes 59.
The orbit stays up because gravity pulls inward while the sideways velocity carries the craft forward 37. Without enough sideways speed, a satellite falls back toward Earth 40, and at about 300 km a circular orbit needs roughly 7.8 km/s 43. Sputnik's orbit was about 120 or 130 miles up, low enough for residual atmosphere to slow it, and its orbital period was shrinking by roughly one second per day 6263.
By March 1958 the Vanguard satellite was travelling at about 18,000 to 19,000 miles per hour, on an egg-shaped orbit that reached as close to Earth as 400 miles 55. A Soviet scientist also said that the Explorer satellite's low-inclination orbit used Earth's rotational speed for a substantial advantage in rocket capacity 57.
Apollo 6 and Apollo 8
Apollo 6, flying on April 4, 1968, was meant to test whether the Saturn V's third stage, the S-IVB, could propel itself and the Apollo spacecraft to lunar distances 30. Three engine failures occurred during the flight, including the failure of the S-IVB restart needed for trans-lunar injection 2932. Because the restart failed, mission control used the service module's SPS engine to raise the orbit to a high apogee instead 33, and the craft came back into the atmosphere at 10,000 m/s rather than the planned 11,000 m/s 34.
Apollo 8 then made history. It was the first crewed mission to leave low Earth orbit and to orbit another celestial body 24, and the first Apollo trans-Earth injection was performed by Apollo 8 on December 25, 1968 27. The first three lunar missions used a free return trajectory, which would bring the crew home if the service module engine failed to insert them into lunar orbit 26.
For the trip home, the trans-Earth injection burn gave a posigrade speed increase of about 1,000 m/s over roughly 150 seconds, performed by the restartable SPS engine after the Lunar Module was undocked 2325. The source for that burn does not describe the outbound injection that first left Earth orbit 23, so the outbound burn is a gap in this evidence. Later missions needed a plane change that eliminated the free-return option 28.
A satellite's slow fade
The rocket body Heos 2 1972-05B was tracked on each of its final 16 days before it decayed in September 1978 21. Air drag progressively lowered its perigee, and that steady drop was used to measure the atmospheric density scale height 17. Over those 16 days the inclination fell by only about 0.02 degrees, yet that small change was enough to measure upper-atmosphere winds accurately 19.
At the satellite's ejection altitude, the gas flow is free-molecular because the mean free path of the gas molecules is large compared with the satellite's size 15. Drag on a satellite in a circular orbit causes its orbital radius to decay on each orbit 20, and aerodynamic torque from gas-surface interaction also slows a spinning satellite, producing spin decay 18.
Borrowing a planet's motion
Gravity assists let spacecraft reach outer planets that would otherwise be out of reach because of the large energy required 66. During a close approach, momentum is exchanged between the spacecraft and the planet 72, and the planet loses momentum too, but because of its huge mass its change in velocity is negligible next to the spacecraft's gain 75. Gravity assist trajectories let a spacecraft gain momentum from the planets it passes 69.
In an ideal hyperbolic flyby with no perturbations or delta-v maneuvers, the spacecraft's incoming and outgoing speeds relative to the planet are equal in magnitude, so the flyby changes only the direction of the velocity 74. In this idealized case, the speed relative to the planet is unchanged; only the direction of the velocity changes 74.
Preliminary interplanetary trajectory designs commonly model complex multi-body trajectories using simplified two-body orbital mechanics and linked conics 70, and gravity assists are commonly modeled with a linked-conic approximation, in which the planet's sphere of influence has zero radius and the flyby happens instantaneously 71. Betts optimized a low-thrust transfer to Mars that used two swing-bys of Venus 77, and low-thrust propulsion is used to shape the trajectory between successive encounters 78.
Where things stand now
Aerodynamic forces from the rarefied atmosphere cause a spacecraft in low Earth orbit to tumble and undergo orbital decay 12. In February 2022, a modest geomagnetic storm was linked to the loss of 38 Starlink satellites, which fell back because of the extra drag 16. Geomagnetic storms heat and expand the thermosphere, which increases atmospheric drag on low-Earth-orbit satellites and speeds up their orbital decay 14.
In Earth's frame, a satellite is in free fall and therefore weightless, because gravity is the only force acting on it 48. Air drag from the rarefied atmosphere, however, causes a spacecraft in low Earth orbit to tumble and undergo orbital decay 12. Geomagnetic storms make this worse: they heat and expand the thermosphere, which increases atmospheric drag on low-Earth-orbit satellites and speeds up their orbital decay 14.
Still unknown
Where the extra energy of a gravity assist comes from in the wider frame is not settled by the checked evidence, which describes the planet's momentum change only as negligible.
The checked list has no numerical rotation speed at the equator or launch sites, so the size of Earth's spin help is still an open gap in this evidence.
Present-day missions are thin: the checked evidence stops at a 2026 analysis of 2022 storms and does not describe the current status of Artemis II or any 2026 probe trajectories.
The outbound trans-lunar burn that first left Earth orbit is not described in the checked Apollo documents.
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Sources
"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."Simulation of the orbital decay of a spacecraft in low Earth orbit due to aerodynamic drag (2021)
"Space debris orbiting in space experiences orbital decay due to various orbital perturbations."Simulation of the orbital decay of a spacecraft in low Earth orbit due to aerodynamic drag (2021)
"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."Impact of geomagnetic storm intensity on thermospheric density and orbital decay of LEO satellites: a G1–G5 per-event analysis (2026)
"because the mean free path of the gas molecules is large compared to the size of the satellite"NASA Technical Reports Server (NTRS) 19860006782: Analysis of Satellite Drag and Spin Decay Data (1985)
"as illustrated by the loss of 38 Starlink satellites following the modest February 2022 storm"Impact of geomagnetic storm intensity on thermospheric density and orbital decay of LEO satellites: a G1–G5 per-event analysis (2026)
"Eleven values of density scale height have been determined from the decrease in perigee height, with a 2 per cent error"DTIC ADA097696: The Orbit of 1972-05B in its Final Phase, with Geophysical Inferences. (1980)
"The satellite would also experience a spin decay due to the aerodynamic torque acting to slow the satellite spinning."NASA Technical Reports Server (NTRS) 19860006782: Analysis of Satellite Drag and Spin Decay Data (1985)
"Although the overall drop in inclination over the 16 days was only 0.02°"DTIC ADA097696: The Orbit of 1972-05B in its Final Phase, with Geophysical Inferences. (1980)
"The effect of drag on a circular orbit can be shown to cause a decay in orbital radius per orbit"NASA Technical Reports Server (NTRS) 19860006782: Analysis of Satellite Drag and Spin Decay Data (1985)
"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"DTIC ADA097696: The Orbit of 1972-05B in its Final Phase, with Geophysical Inferences. (1980)
"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."Image 5 of The Key West citizen (Key West, Fla.), November 4, 1957
"An Apollo TEI burn lasted approximately 150 seconds, providing a posigrade velocity increase of 1,000 m/s (3,300 ft/s)."Trans-Earth injection
"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."Apollo program
"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."Trans-Earth injection
"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."Apollo program
"It was first performed by the Apollo 8 mission on December 25, 1968."Trans-Earth injection
"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."Apollo program
"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 program
"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."Apollo 6
"Trans-Earth injection The SM engine fires to send the CSM back to Earth."Apollo program
"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."Apollo 6
"The SPS engine burned for 442 seconds to get to the planned"Apollo 6
"There was now, however, not enough propellant to speed up the atmospheric reentry with a second SPS engine burn"Apollo 6
"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."Satellite Orbit (2023)
"are two opposing forces at work on an orbiting object: gravitational force and tangential velocity (13)."The Orbital Properties of Black Holes: Exploring the Relationship between Orbital Velocity and Distance (2023)
"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."Satellite Orbit (2023)
"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."Satellite Orbit (2023)
"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."The Orbital Properties of Black Holes: Exploring the Relationship between Orbital Velocity and Distance (2023)
"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."Satellite Orbit (2023)
"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 )."Fictitious force
"For objects in space , the equations of motion that describe orbital motion are simpler in a non-rotating frame such as ECI."Earth-centered inertial
"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."Image 1 of The Key West citizen (Key West, Fla.), March 17, 1958
"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."Fictitious force
"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."Image 8 of Evening star (Washington, D.C.), February 2, 1958
"ECI: inertial , not rotating , with respect to the stars; useful to describe motion of celestial bodies and spacecraft."Earth-centered inertial
"He said the object was now nowtraveling - traveling nowtraveling around the earth at the rate of once every 103.6 minutes."Image 8 of Evening star (Washington, D.C.), November 8, 1957
"The extent to which an ECI frame is actually inertial is limited by the non-uniformity of the surrounding gravitational field ."Earth-centered inertial
"Tass reported that both Sputnik I, launched October 4, and Sputnik n, which went up last Sunday, were slowing down slightly."Image 11 of Evening star (Washington, D.C.), November 10, 1957
"decreasing by some something - thing something like one second a day due to atmospheric drag at the low"Image 8 of Evening star (Washington, D.C.), November 8, 1957
"point in Sputnik's orbit—about 120 or 130 miles above the"Image 8 of Evening star (Washington, D.C.), November 8, 1957
"positively and without question, we have not —repeat not—solved this prob"Image 11 of Evening star (Washington, D.C.), November 10, 1957
"The time for Sputnik II to make an orbit now was given as 103.52 minutes"Image 11 of Evening star (Washington, D.C.), November 10, 1957
"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."Multi-gravity assist design tool for interplanetary trajectory optimisation (2009)
"Incorporating gravity assists provides highly efficient interplanetary trajectories, enabling otherwise infeasible spacecraft missions."Preliminary interplanetary trajectory design tools using ballistic and powered gravity assists (2015)
"An interesting way to reduce such a requirement is to resort to Gravity Assist (GA) maneuvers as shown by Langevin2 in 1999."Optimizing Low-Thrust and Gravity Assist Maneuvers to Design Interplanetary Trajectories (2003)
"Gravity assist trajectories allow the spacecraft to gain momentum from the planets."Multi-gravity assist design tool for interplanetary trajectory optimisation (2009)
"Preliminary interplanetary trajectory designs frequently use simplified two-body orbital mechanics and linked conics methodology to model the complex trajectories in multi-body systems."Preliminary interplanetary trajectory design tools using ballistic and powered gravity assists (2015)
"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."Optimizing Low-Thrust and Gravity Assist Maneuvers to Design Interplanetary Trajectories (2003)
"During a close approach of a gravitational body momentum is exchanged between the spacecraft and the gravitational body."Multi-gravity assist design tool for interplanetary trajectory optimisation (2009)
"Gravity assist (GA) proposed by Michael A. Minovitch is an indispensable technique for present solar system exploration."Design of Interplanetary Trajectories Using Aero-Gravity Assist (2019)
"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:"Optimizing Low-Thrust and Gravity Assist Maneuvers to Design Interplanetary Trajectories (2003)
"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."Multi-gravity assist design tool for interplanetary trajectory optimisation (2009)
"In solar system exploration, it is important to optimize the maneuver and flight time to the target planet."Design of Interplanetary Trajectories Using Aero-Gravity Assist (2019)
"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."Optimizing Low-Thrust and Gravity Assist Maneuvers to Design Interplanetary Trajectories (2003)
"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."Optimizing Low-Thrust and Gravity Assist Maneuvers to Design Interplanetary Trajectories (2003)
"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."