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45-563: Explorer 3 (Harvard designation 1958 Gamma ) was an American artificial satellite launched into medium Earth orbit in 1958. It was the second successful launch in the Explorer program , and was nearly identical to the first U.S. satellite Explorer 1 in its design and mission. Explorer 3 was the third satellite in the Explorer small satellite series, which started with Explorer 1 , America's first artificial satellite. The Explorer program

90-547: A central mass . In control theory , a trajectory is a time-ordered set of states of a dynamical system (see e.g. Poincaré map ). In discrete mathematics , a trajectory is a sequence ( f k ( x ) ) k ∈ N {\displaystyle (f^{k}(x))_{k\in \mathbb {N} }} of values calculated by the iterated application of a mapping f {\displaystyle f} to an element x {\displaystyle x} of its source. A familiar example of

135-699: A free fall frame which happens to be at ( x , y ) = (0,0) at  t  = 0. The equation of motion of the projectile in this frame (by the equivalence principle ) would be y = x tan ⁡ ( θ ) {\displaystyle y=x\tan(\theta )} . The co-ordinates of this free-fall frame, with respect to our inertial frame would be y = − g t 2 / 2 {\displaystyle y=-gt^{2}/2} . That is, y = − g ( x / v h ) 2 / 2 {\displaystyle y=-g(x/v_{h})^{2}/2} . Now translating back to

180-517: A latency (delay) five times lower than on the existing platform based on geostationary orbit satellites. In September 2023, satellite operator SES announced the first satellite internet service to use satellite constellations in both MEO and Low Earth Orbit (LEO) . The SES Cruise mPOWERED + Starlink service will use SES's O3b mPOWER MEO satellites and SpaceX 's Starlink LEO system to provide cruise ship passengers with internet, social media and video calls at up to 3 Gbps per ship anywhere in

225-508: A complete radiation history for each orbit, Ludwig having had time to accommodate for the spin-stabilization of the satellite.<Vanguard/> Its total weight was 14.1 kg (31 lb), of which 8.4 kg (19 lb) was instrumentation. The instrument section at the front end of the satellite and the empty scaled-down fourth-stage rocket casing orbited as a single unit, to be spun around its long axis at 750 revolutions per minute . Data from these instruments would be transmitted to

270-597: A futile hypothesis by practical-minded investigators all through the Middle Ages in Europe . Nevertheless, by anticipating the existence of the vacuum , later to be demonstrated on Earth by his collaborator Evangelista Torricelli , Galileo was able to initiate the future science of mechanics . In a near vacuum, as it turns out for instance on the Moon , his simplified parabolic trajectory proves essentially correct. In

315-678: A given range d h {\displaystyle d_{h}} . The angle θ {\displaystyle \theta } giving the maximum range can be found by considering the derivative or R {\displaystyle R} with respect to θ {\displaystyle \theta } and setting it to zero. which has a nontrivial solution at 2 θ = π / 2 = 90 ∘ {\displaystyle 2\theta =\pi /2=90^{\circ }} , or θ = 45 ∘ {\displaystyle \theta =45^{\circ }} . The maximum range

360-537: A high inclination of 63.4° and high eccentricity of 0.722 with a period of 12 hours, so a satellite spends most of its orbit above the chosen area in high latitudes. This orbit was used by the (now defunct) North American Sirius Satellite Radio and XM Satellite Radio satellites and the Russian Molniya military communications satellites, after which it is named. Communications satellites in MEO include

405-429: A higher apogee and lower perigee than planned: 1,735 mi (2,792 km) and 125 mi (201 km), respectively. The ensuing orbital decay made for a comparatively short lifespan: initial estimates placed it at 4.6 months. In fact, the satellite reentered the atmosphere on 28 June 1958, after just 93 days of operation; by the week before reentry, Explorer 3's apogee had dropped to 375 mi (604 km) and

450-411: A particle of mass m {\displaystyle m} , moving in a potential field V {\displaystyle V} . Physically speaking, mass represents inertia , and the field V {\displaystyle V} represents external forces of a particular kind known as "conservative". Given V {\displaystyle V} at every relevant position, there

495-499: A player is positioned so as to catch it as it descends, he sees its angle of elevation increasing continuously throughout its flight. The tangent of the angle of elevation is proportional to the time since the ball was sent into the air, usually by being struck with a bat. Even when the ball is really descending, near the end of its flight, its angle of elevation seen by the player continues to increase. The player therefore sees it as if it were ascending vertically at constant speed. Finding

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540-463: A reasonably good approximation, although if a comet passes close to the Sun, then it is also influenced by other forces such as the solar wind and radiation pressure , which modify the orbit and cause the comet to eject material into space. Newton's theory later developed into the branch of theoretical physics known as classical mechanics . It employs the mathematics of differential calculus (which

585-484: A rocket based on its Viking rocket ( Project Vanguard ) for the International Geophysical Year , which would start 1 July 1957. Nevertheless, ABMA hoped Redstone-Orbiter could still be used as a backup orbital system. Reentry tests that year conducted with the newly developed, Redstone-based Jupiter-C , further strengthened ABMA confidence in their vehicle as an orbital launcher. Following

630-554: A trajectory is the path of a projectile, such as a thrown ball or rock. In a significantly simplified model, the object moves only under the influence of a uniform gravitational force field . This can be a good approximation for a rock that is thrown for short distances, for example at the surface of the Moon . In this simple approximation, the trajectory takes the shape of a parabola . Generally when determining trajectories, it may be necessary to account for nonuniform gravitational forces and air resistance ( drag and aerodynamics ). This

675-490: A uniform downwards gravitational force we consider two bodies orbiting with the mutual gravitation between them, we obtain Kepler's laws of planetary motion . The derivation of these was one of the major works of Isaac Newton and provided much of the motivation for the development of differential calculus . If a projectile, such as a baseball or cricket ball, travels in a parabolic path, with negligible air resistance, and if

720-411: Is a way to infer the associated force that would act at that position, say from gravity. Not all forces can be expressed in this way, however. The motion of the particle is described by the second-order differential equation On the right-hand side, the force is given in terms of ∇ V {\displaystyle \nabla V} , the gradient of the potential, taken at positions along

765-607: Is an arbitrary altitude chosen by accepted convention, whereas the boundary between MEO and HEO is the particular altitude of a geosynchronous orbit , in which a satellite takes 24 hours to circle the Earth, the same period as the Earth’s own rotation. All satellites in MEO have an orbital period of less than 24 hours, with the minimum period (for a circular orbit at the lowest MEO altitude) about 2 hours. Satellites in MEO orbits are perturbed by solar radiation pressure, which

810-543: Is perpendicular to it ( parallel to the gravitational field lines ). Let g {\displaystyle g} be the acceleration of gravity . Relative to the flat terrain, let the initial horizontal speed be v h = v cos ⁡ ( θ ) {\displaystyle v_{h}=v\cos(\theta )} and the initial vertical speed be v v = v sin ⁡ ( θ ) {\displaystyle v_{v}=v\sin(\theta )} . It will also be shown that

855-839: Is sometimes called mid Earth orbit or intermediate circular orbit ( ICO ). Two medium Earth orbits are particularly significant. A satellite in the semi-synchronous orbit at an altitude of approximately 20,200 kilometres (12,600 mi) has an orbital period of 12 hours and passes over the same two spots on the equator every day. This reliably predictable orbit is used by the Global Positioning System (GPS) constellation . Other navigation satellite systems use similar medium Earth orbits including GLONASS (with an altitude of 19,100 kilometres, 11,900 mi), Galileo (with an altitude of 23,222 kilometres, 14,429 mi) and BeiDou (with an altitude of 21,528 kilometres, 13,377 mi). The Molniya orbit has

900-422: Is the acceleration due to gravity). The range , R , is the greatest distance the object travels along the x-axis in the I sector. The initial velocity , v i , is the speed at which said object is launched from the point of origin. The initial angle , θ i , is the angle at which said object is released. The g is the respective gravitational pull on the object within a null-medium. The height , h ,

945-468: Is the dominating non-gravitational perturbing force. Other perturbing forces include: Earth's albedo , navigation antenna thrust, and thermal effects related to heat re-radiation. The MEO region includes the two zones of energetic charged particles above the equator known as the Van Allen radiation belts , which can damage satellites’ electronic systems without special shielding. A medium Earth orbit

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990-509: Is the focus of the discipline of ballistics . One of the remarkable achievements of Newtonian mechanics was the derivation of Kepler's laws of planetary motion . In the gravitational field of a point mass or a spherically-symmetrical extended mass (such as the Sun ), the trajectory of a moving object is a conic section , usually an ellipse or a hyperbola . This agrees with the observed orbits of planets , comets , and artificial spacecraft to

1035-464: Is the greatest parabolic height said object reaches within its trajectory In terms of angle of elevation θ {\displaystyle \theta } and initial speed v {\displaystyle v} : giving the range as This equation can be rearranged to find the angle for a required range Note that the sine function is such that there are two solutions for θ {\displaystyle \theta } for

1080-447: Is the path that an object with mass in motion follows through space as a function of time. In classical mechanics , a trajectory is defined by Hamiltonian mechanics via canonical coordinates ; hence, a complete trajectory is defined by position and momentum , simultaneously. The mass might be a projectile or a satellite . For example, it can be an orbit — the path of a planet , asteroid , or comet as it travels around

1125-403: Is then R max = v 2 / g {\displaystyle R_{\max }=v^{2}/g\,} . At this angle sin ⁡ ( π / 2 ) = 1 {\displaystyle \sin(\pi /2)=1} , so the maximum height obtained is v 2 4 g {\displaystyle {v^{2} \over 4g}} . To find

1170-400: Is zero when θ = π / 2 = 90 ∘ {\displaystyle \theta =\pi /2=90^{\circ }} . So the maximum height H m a x = v 2 2 g {\displaystyle H_{\mathrm {max} }={v^{2} \over 2g}} is obtained when the projectile is fired straight up. If instead of

1215-809: The O3b and O3b mPOWER constellations for low- latency broadband and data backhaul to maritime, aero and remote locations (with an altitude of 8,063 kilometres, 5,010 mi). Communications satellites to cover the North and South Pole are also put in MEO. Telstar 1, an experimental communications satellite launched in 1962, orbited in MEO. In May 2022, Kazakhstani mobile network operator, Kcell , and satellite owner and operator, SES used SES's O3b MEO satellite constellation to demonstrate that MEO satellites could be used to provide high-speed mobile internet to remote regions of Kazakhstan for reliable video calling, conferencing and streaming, and web browsing, with

1260-455: The range is 2 v h v v / g {\displaystyle 2v_{h}v_{v}/g} , and the maximum altitude is v v 2 / 2 g {\displaystyle v_{v}^{2}/2g} . The maximum range for a given initial speed v {\displaystyle v} is obtained when v h = v v {\displaystyle v_{h}=v_{v}} , i.e.

1305-798: The Earth's magnetic field. Explorer 4 , equipped with a lead-shielded counter, flew in July and confirmed the existence of the radiation fields subsequently known as the Van Allen Belts . On 7 May 1958, micrometeorites associated with the Eta Aquariids meteor shower ruptured two of Explorer 3's micrometeorite erosion gauges. A replica of the spacecraft is currently located in the Smithsonian Institution 's National Air and Space Museum , Milestones of Flight Gallery. Medium Earth orbit The boundary between MEO and LEO

1350-676: The World. Subsequently, in February 2024, SES announced that Virgin Voyages will be the first cruise line to deploy the service. Space debris in medium Earth orbit stays practically permanently orbiting Earth. Most space debris extends to the lowest high Earth orbits just beyond the edge of medium Earth orbit, where geostationary satellites are and where after their end of use they are parked in similar orbits, so-called graveyard orbits . Trajectory A trajectory or flight path

1395-405: The analysis that follows, we derive the equation of motion of a projectile as measured from an inertial frame at rest with respect to the ground. Associated with the frame is a right-hand coordinate system with its origin at the point of launch of the projectile. The x {\displaystyle x} -axis is tangent to the ground, and the y {\displaystyle y} axis

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1440-687: The angle giving the maximum height for a given speed calculate the derivative of the maximum height H = v 2 sin 2 ⁡ ( θ ) / ( 2 g ) {\displaystyle H=v^{2}\sin ^{2}(\theta )/(2g)} with respect to θ {\displaystyle \theta } , that is d H d θ = v 2 2 cos ⁡ ( θ ) sin ⁡ ( θ ) / ( 2 g ) {\displaystyle {\mathrm {d} H \over \mathrm {d} \theta }=v^{2}2\cos(\theta )\sin(\theta )/(2g)} which

1485-449: The ground by a 60 milliwatt transmitter operating on 108.03 MHz and a 10 milliwatt transmitter operating on 108.00 MHz. Transmitting antennas consisted of two fiberglass slot antennas in the body of the satellite itself. The four flexible whip antennas of Explorer 1 were removed from the design. The external skin of the instrument section was painted in alternate strips of white and dark green to provide passive temperature control of

1530-425: The inertial frame the co-ordinates of the projectile becomes y = x tan ⁡ ( θ ) − g ( x / v h ) 2 / 2 {\displaystyle y=x\tan(\theta )-g(x/v_{h})^{2}/2} That is: (where v 0 is the initial velocity, θ {\displaystyle \theta } is the angle of elevation, and g

1575-399: The initial angle is 45 ∘ {\displaystyle ^{\circ }} . This range is v 2 / g {\displaystyle v^{2}/g} , and the maximum altitude at the maximum range is v 2 / ( 4 g ) {\displaystyle v^{2}/(4g)} . Assume the motion of the projectile is being measured from

1620-562: The job of modifying the Jupiter-C to the Juno 1 and building Explorer 1 in 84 days. An experiment developed for Vanguard by George Ludwig, comprising an Anton 314 omnidirectional Geiger tube detector for measuring the flux of high energy charged protons and electrons, was adapted for Explorer 1. Because of the high spin rate of the Explorer 1 rocket, the experiment's tape recorder had to be omitted, which meant that data could only be collected when

1665-453: The launch of the Soviet satellite Sputnik 1 on 4 October 1957, Project Orbiter was revived, with two shots authorized as a back-up to Vanguard in early November. The failure of America's first attempted Vanguard launch on 9 December 1957, cleared the way for an "Explorer" (as the crash program was dubbed) to be the first American satellite.' Working closely together, ABMA and JPL completed

1710-407: The limited space available and the requirements for low weight, the Explorer 3 instrumentation was designed and built with simplicity and high reliability in mind. Explorer 3 was launched at 17:38:03 GMT on 26 March 1958 from Cape Canaveral Launch Complex 5 via the same type of modified Jupiter-C ( Juno 1 ) as the prior two Explorers. A guidance system orbit placed the satellite into an orbit with

1755-425: The perigee to 99 mi (159 km). Explorer 3 largely confirmed the findings of Explorer 1, with the same zero count returned above an altitude of around 1,000 km (620 mi) to 1,200 km (750 mi). Because of the consistent results, Van Allen hypothesized that the satellites' equipment might have been saturated by unexpectedly high radiation concentrations, trapped in a belt of charged particles by

1800-441: The satellite was in sight and range of a ground station. Explorer 1 took off 31 January 1958, becoming America's first satellite. Its Geiger tube worked properly, but acted contrary to expectations. As the satellite ascended in its orbit, the radiation count increased, then abruptly dropped to zero. When the satellite was descending, the tube abruptly began detecting charged particles again. As data could only be received about 15% of

1845-437: The satellite. The proportions of the light and dark strips were determined by studies of shadow-sunlight intervals based on firing time, trajectory , orbit, and orbital inclination . Electrical power was provided by Mallory type RM Mercury batteries that made up approximately 40% of the payload weight. These provided power that operated the high power transmitter for 31 days and the low-power transmitter for 105 days. Because of

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1890-425: The time, it was yet impossible to determine the phenomenon Explorer had detected. The objective of this spacecraft was a continuation of experiments started with Explorer 1. The payload consisted of a micrometeorite detector (a wire grid array and acoustic detector) and the same cosmic ray counter (a Geiger-Müller tube ) experiment included on Explorer 1, but this time with an on-board tape recorder to provide

1935-402: The trajectory. This is the mathematical form of Newton's second law of motion : force equals mass times acceleration, for such situations. The ideal case of motion of a projectile in a uniform gravitational field in the absence of other forces (such as air drag) was first investigated by Galileo Galilei . To neglect the action of the atmosphere in shaping a trajectory would have been considered

1980-573: Was a direct successor to the Army Ballistic Missile Agency (ABMA)'s Project Orbiter , initiated in November 1954 to use a slightly modified Redstone missile combined with solid-propellant rocket cluster upper stage to put a satellite into orbit. In 1955, the "Stewart Committee", under the chairmanship of Homer J. Stewart of Jet Propulsion Laboratory (JPL), chose an Naval Research Laboratory (NRL) satellite plan using

2025-401: Was also initiated by Newton in his youth). Over the centuries, countless scientists have contributed to the development of these two disciplines. Classical mechanics became a most prominent demonstration of the power of rational thought, i.e. reason , in science as well as technology. It helps to understand and predict an enormous range of phenomena ; trajectories are but one example. Consider

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