Physics
Satellite Orbits
Satellite orbits refer to the paths followed by satellites as they revolve around celestial bodies. These orbits are determined by the gravitational pull of the body being orbited and the satellite's velocity. Common types of satellite orbits include low Earth orbit (LEO), geostationary orbit (GEO), and polar orbit. Understanding satellite orbits is crucial for the successful deployment and operation of satellites for communication, navigation, and scientific purposes.
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eBook - ePubSatellite Technology
Principles and Applications
- Anil K. Maini, Varsha Agrawal(Authors)
- 2014(Publication Date)
- Wiley(Publisher)
The motion of different planets of the solar system around the sun and the motion of artificial satellites around Earth (Figure 2.1) are examples of orbital motion. Figure 2.1 Example of orbital motion – satellites revolving around Earth The term ‘trajectory’, on the other hand, is associated with a path that is not periodically revisited. The path followed by a rocket on its way to the right position for a satellite launch (Figure 2.2) or the path followed by orbiting satellites when they move from an intermediate orbit to their final destined orbit (Figure 2.3) are examples of trajectories. Figure 2.2 Example of trajectory –path followed by a rocket on its way during satellite launch Figure 2.3 Example of trajectory – motion of a satellite from the intermediate orbit to the final orbit 2.2 Orbiting Satellites – Basic Principles The motion of natural and artificial satellites around Earth is governed by two forces. One of them is the centripetal force directed towards the centre of the Earth due to the gravitational force of attraction of Earth and the other is the centrifugal force that acts outwards from the centre of the Earth (Figure 2.4). It may be mentioned here that the centrifugal force is the force exerted during circular motion, by the moving object upon the other object around which it is moving. In the case of a satellite orbiting Earth, the satellite exerts a centrifugal force. However, the force that is causing the circular motion is the centripetal force. In the absence of this centripetal force, the satellite would have continued to move in a straight line at a constant speed after injection. The centripetal force directed at right angles to the satellite's velocity towards the centre of the Earth transforms the straight line motion to the circular or elliptical one, depending upon the satellite velocity
No longer available |Learn more- (Author)
- 2014(Publication Date)
- College Publishing House(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter 6 Orbit Two bodies of different mass orbiting a common barycenter. The relative sizes and type of orbit are similar to the Pluto–Charon system. In physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet around the center of a star system, such as the solar system. Orbits of planets are typically elliptical. Current understanding of the mechanics of orbital motion is based on Albert Einstein's general theory of relativity, which accounts for gravity as due to curvature of space-time, with orbits following geodesics; though in common practice an approximate force-based theory of universal gravitation based on Kepler's laws of planetary motion is often used instead for ease of calculation. History Historically, the apparent motions of the planets were first understood geometrically (and without regard to gravity) in terms of epicycles, which are the sums of numerous circular motions. Theories of this kind predicted paths of the planets moderately well, until Johannes Kepler was able to show that the motions of planets were in fact (at least approximately) elliptical motions. ________________________ WORLD TECHNOLOGIES ________________________ In the geocentric model of the solar system, the celestial spheres model was originally used to explain the apparent motion of the planets in the sky in terms of perfect spheres or rings, but after the planets' motions were more accurately measured, theoretical mechanisms such as deferent and epicycles were added. Although it was capable of accurately predicting the planets' position in the sky, more and more epicycles were required over time, and the model became more and more unwieldy. The basis for the modern understanding of orbits was first formulated by Johannes Kepler whose results are summarised in his three laws of planetary motion.
eBook - PDFRemote Sensing Physics
An Introduction to Observing Earth from Space
- Rick Chapman, Richard Gasparovic(Authors)
- 2022(Publication Date)
- American Geophysical Union(Publisher)
Geosta-tionary satellites orbit at 36,000 km. Medium Earth orbits are between these two cate-gories, with global navigation satellite systems (e.g., GPS (USA), GLONASS (Russia), Galileo (Europe), BeiDou (China)) at 20,000 km as prime examples. Remote Sensing Physics: An Introduction to Observing Earth from Space, Advanced Textbook 3 , First Edition. Rick Chapman and Richard Gasparovic. © 2022 American Geophysical Union. Published 2022 by John Wiley & Sons, Inc. Companion website: www.wiley.com/go/chapman/physicsofearthremotesensing 12 Remote Sensing Physics Geostationary International Space Station Low Earth Orbit Medium Earth Orbit 88 min Lower Van Allen Belt Orbital Periods Upper Van Allen Belt Navigation Satellites Sun Synchronous Satellites Other Polar Orbit Satellites High Earth Orbit 100 min 2 hr 6 hr 12 hr 24 hr Figure 2.2 Orbital altitudes drawn to scale. Kepler’s law describes Satellite Orbits around a central mass like the Earth. As illustrated in Figure 2.3, all orbits follow an elliptical path, with the Earth at one focus. The range r from the center of the Earth to the satellite is then given by: r = a ( 1 − e 2 ) 1 + e cos 𝜃 (2.1) Important definitions in this diagram include: • length of the semimajor axis ( a ). • length of the semiminor axis ( b ). • eccentricity ( e = √ 1 − b 2 a 2 ) . • apogee, the peak altitude of the satellite = a ( 1 + e ) . • perigee, the minimum altitude of the satel-lite = a ( 1 − e ) . Apogee = a (1 + e ) Perigee = a (1 – e ) Satellite Earth ae ae 2 a r θ Focus b = a √ 1 – e 2 Figure 2.3 Elliptical orbit. • true anomaly ( 𝜃 ) defined to be the angle to the satellite relative to perigee in an Earth-centered coordinate system. As illustrated in Figure 2.4, Kepler’s law implies that the position of any satellite in ellip-tical orbit, as measured in a coordinate system relative to the center of the Earth, is specified by six variables called Kepler elements. In Figure 2.4: Ω is the longitude of the ascending node of the orbit.
- Ian Poole(Author)
- 2003(Publication Date)
- Newnes(Publisher)
For a very low orbit of around 100 miles a velocity of about 17 500 miles per hour is needed and this means that the satellite will orbit the earth in about 90 minutes. At an altitude of 22 000 Figure 8.1 Satellite Orbits Satellites 223 miles a velocity of just less than 7000 miles per hour is needed giving an orbit time of about 24 hours. A satellite can orbit the earth in one of two basic types of orbit. The most obvious is a circular orbit where the distance from the earth remains the same at all times. A second type of orbit is an elliptical one. Both types of orbit are shown in Figure 8.1, where the main characteristics are shown. When a Satellite Orbits the earth, either in a circular or elliptical orbit it forms a plane. This passes through the centre of gravity of the earth or the geocentre. The rotation around the earth is also categorized. It may be in the same direction as the earth’s rotation when it is said to be posigrade, or it may be in the opposite direction when it is retrograde. At any given time there is a point on the earth at which the satellite is directly overhead. As the satellite moves so does this point, and the track that this traces out on the earth’s surface is known as the groundtrack. The groundtrack is a great circle, i.e. the centre of the circle is at the geocentre. Geostationary satellites are a special case as they appear directly over the same point of the earth all the time and accordingly their groundtrack consists of a single point on the earth’s equator. For satellites with equatorial orbits the groundtrack is along the equator. Satellites may also be in other orbits. These will cross the equator twice, once in a northerly direction and once in a southerly direction. The point at which the groundtrack crosses the equator is known as a node. There are two, and the one where the groundtrack passes from the southern hemisphere to the northern hemisphere is called the ascending node.
- Ranjan Vepa(Author)
- 2019(Publication Date)
- Cambridge University Press(Publisher)
2 Space Vehicle Orbit Dynamics 2.1 Orbit Dynamics: An Introduction All space vehicle flights occur under the influence of the gravitational force of a multitude of planetary objects in the Solar system. In particular, the motion of an artificial Earth satellite is primarily influenced by the gravitational pull of the Earth and the gravitational forces exerted on it by the Moon and the Sun. Given that the Moon is at distance of 384,440 km (semi-major axis), orbits the Earth in a near circular orbit (eccentricity, e = 0.0549), and has a mass of only 0.01226 of the Earth’s mass, it is conceivable that for artificial Earth satellites orbiting the Earth at a distance of less than approximately 38,444 km the influence of the Moon can be neglected. Furthermore, the Sun is at a distance of 149, 599 10 3 km, which is 400 times further away than the Moon, although it is 332,952 times heavier than the Earth. For an artificial satellite orbiting the Earth at a distance of less than approximately 38,440 km, the influence of the Sun can be assumed to be no different than its influence on the Earth. 2.2 Planetary Motion: The Two-Body Problem As far as motion around the Earth is concerned, any orbiting artificial satellite and the Earth may be considered in isolation as two interacting independent celestial bodies as long as the artificial satellite is sufficiently close to Earth. This leads to the classical two-body approximation problem that serves as a valuable paradigm for developing the theory of planetary motion. The key question is: How close is sufficiently close? That is a question we shall not seek to answer yet, although it is indeed a fundamental one. 2.2.1 Kepler’s Laws The German astronomer Johann Kepler (1571–1630) formulated three empirical laws of planetary motion based on astronomical data provided to him by the Danish astronomer Tycho Brahe in the late 1590s.
eBook - PDF- Wolfgang Torge(Author)
- 2011(Publication Date)
- De Gruyter(Publisher)
Satellites may reflect incident light only (passive satellites), or they may carry on board subsystems such as transmitters/receivers, different type sensors, clocks, and computers (active satellites). In the latter case, an energy supply is required, and lifetime is rather limited. The mean orbital velocity of a satellite moving in an approximately circular orbit (r = a) is given from (5.29) by (5.37) For a satellite close to the earth (h = 1000 km), we obtain, with r = R + h= 7370 km, a velocity of 7.4 km/s. Kepler's third law yields the period of revolution U = 2xr/v = 104 min. The intersection of the orbital plane with a non-rotating earth represents a great circle on the earth's surface: subsatellite track. The rotation of the earth causes a western displacement of subsequent Satellite Orbits (Fig. 5.6), with a shift on the equator given by = 15 0 ·ί/Γη] = 0.25°·ί/Γπύη]. (5.38) sidereal day 5.2 Satellite Observations 137 Fig. 5.6. Subsatellite tracks The latitude range of the subsatellite tracks is determined by the inclination of the satellite. The following aspects have to be considered during the design (choice of orbital parameters) of satellite missions for geodetic applications: For positioning, the network geometry of the ground stations and the satellites plays a primary role. Simultaneous direction measurements from two ground stations to a satellite form a plane, and the intersection of planes provides positions. Range measurements utilize the intersection of spheres, whereas range differences, derived from Doppler-frequency shifts, use the intersection of hyperboloids. Satellites at high altitudes are less influenced by gravitational and air drag perturbations and therefore preferred. Low altitude satellites are required for the determination of the gravitational field. This is mainly due to the attenuation factor (ajr) in the spherical harmonic expansion of the gravitational potential ( a e = semimajor axis of the earth ellipsoid), cf.
eBook - PDF- Ray E. Sheriff, Y. Fun Hu(Authors)
- 2003(Publication Date)
- Wiley(Publisher)
The orbital e~uations derived in the previous section are based on two basic assumptions: The only force that acts upon the satellite is that due to the Earth’s gravitati The satellite and the E are considered as point masses with the shape of th In practise, the above assumptions do not hold. The shape of the arth is far from spherical. In addition, apart from the gravitational force due to the Earth, th satellite will also experi- tational fields due to other planets, and more noticeably, those due to the g~avitational field related factors including the solar radiation also contri~ute to the satellite orbit perturbing around its el1ipti techni~ues have been derived to include the perturbing forces in the orbital y as sum in^ that the perturb in^ forces cause a constant drift, to th its Keplerian orbit, which varies linearly with time, the satellite9s terms of the six orbital parameters (see Section 3.2.1) at any instant of time tl, can be written as: dl-2 di d o de da du dt dt dt dt dt dt + -St, io + -St, + -St, eo + -at, ao + -St, vo + -6t where ( ~ 0 , io, oo, eo, ao, uo) are the satellite’s orbital parameters at time to; d ( ) / ~ ~ is the linear drift in the orbital parameter with respect to time; and St is (tl - to). n order to counteract the pe~urbation effect, station-keep in^ of the satellite has to be c d e d out periodicalls during the satellite’s mission lifetime, 3.3.2 ~ ~ ~ c t ~ ~ ~ t h ~ on an^ the Sun ~ravitational ~erturbation is inversely propo~ional to the cube of the distance between two ct of the gravitationa1pull from planets, other than the Earth, has a cant effect on geostationa~ satellites than that on Low Earth Orbit these planets, the eEect of the Sun and the Moon are more notice the mass of the Sun is approximatels 30 times that of the Moon, the pe~urbation effect of the Sun on. a ~ e o s t a t i o n ~ satellite is only half that of the Moon due to the inverse cube law.
eBook - PDF- Andrew Fraknoi, David Morrison, Sidney C. Wolff(Authors)
- 2016(Publication Date)
- Openstax(Publisher)
To illustrate how a satellite is launched, imagine a gun firing a bullet horizontally from the top of a high mountain, as in Figure 3.11, which has been adapted from a similar diagram by Newton. Imagine, further, that the friction of the air could be removed and that nothing gets in the bullet’s way. Then the only force that acts on the bullet after it leaves the muzzle is the gravitational force between the bullet and Earth. 88 Chapter 3 Orbits and Gravity This OpenStax book is available for free at http://cnx.org/content/col11992/1.13 Figure 3.11 Firing a Bullet into Orbit. (a) For paths a and b, the velocity is not enough to prevent gravity from pulling the bullet back to Earth; in case c, the velocity allows the bullet to fall completely around Earth. (b) This diagram by Newton in his De Mundi Systemate, 1731 edition, illustrates the same concept shown in (a). If the bullet is fired with a velocity we can call v a , the gravitational force acting upon it pulls it downward toward Earth, where it strikes the ground at point a. However, if it is given a higher muzzle velocity, v b , its higher speed carries it farther before it hits the ground at point b. If our bullet is given a high enough muzzle velocity, v c , the curved surface of Earth causes the ground to remain the same distance from the bullet so that the bullet falls around Earth in a complete circle. The speed needed to do this—called the circular satellite velocity—is about 8 kilometers per second, or about 17,500 miles per hour in more familiar units. Each year, more than 50 new satellites are launched into orbit by such nations as Russia, the United States, China, Japan, India, and Israel, as well as by the European Space Agency (ESA), a consortium of European nations (Figure 3.12). Today, these satellites are used for weather tracking, ecology, global positioning systems, communications, and military purposes, to name a few uses.
eBook - PDFSatellite Geodesy
Foundations, Methods, and Applications
- Günter Seeber(Author)
- 2008(Publication Date)
- De Gruyter(Publisher)
3 Satellite Orbital Motion Precise time-dependent satellite positions in a suitable reference frame are required for nearly all tasks in satellite geodesy. The computation and prediction of precise Satellite Orbits, together with appropriate observations and adjustment techniques is, for example, essential for the determination of − geocentric coordinates of observation stations [12.1], − field parameters for the description of the terrestrial gravity field as well as for the determination of a precise and high resolution geoid [12.2] − trajectories of land-, sea-, air-, and space-vehicles in real-time navigation [12.3] − Earth’s orientation parameters in space [12.4]. Essentially, the accuracy of the final results depends on the accuracy of the available Satellite Orbits. This is increasingly true for tasks in applied geodesy, such as the determination of relative coordinates with the Global Positioning System [7]. The requirement for 1 cm relative accuracy in coordinates implies the requirement for the knowledge of Satellite Orbits on the few meter accuracy level or even better [7.4.3]. Those who apply satellite methods in geodesy, navigation and adjacent fields, must have a basic knowledge of satellite orbital motion, including the major perturbations, in order to assess the appropriate requirements for orbit determinations. Chapter 3 aims to provide this basic knowledge. Starting with the undisturbed Keplerian motion in a central force field [3.1] the major perturbations, as well as an elementary perturbation theory are discussed [3.2]. The effects of perturbations on Satellite Orbits are also treated [3.2.4]. A section on the integration and representation of orbits [3.3] follows because algorithms for orbit improvement are included in modern software packages for applied satellite geodesy. The appropriate use of satellite ephemerides is discussed together with the corresponding observation methods (e.g. [7.1.5]).
eBook - PDFVistas in Astronomy
Volume 11
- Arthur Beer(Author)
- 2016(Publication Date)
- Pergamon(Publisher)
Therefore, the amplitudes of the long-periodic perturbations are small for close satellites, even if the eccentricity and the inclination are large except for the critical inclination case. However, if the semimajor axis is very large, due to the perturbations by the Moon and the Sun, the orbital elements can change very much as for the lunar satellites. Therefore, high satellites have a greater probability to hit the Earth surface than lower satellites, if air-drag effects are not consid-ered. The longperiodic perturbations for Earth satellites are discussed in Section II. II. SATELLITES AROUND AN OBLATE PLANET For artificial satellites moving around the Earth a general theory of motion cannot be formulated easily, since the motion is disturbed by various kinds of forces whose intensities depend on the orbital elements and/or characteristics of the satellites. For example: air-drag effects are predominant for satellites with very low perigee height ; luni-solar perturba-tions are important for very high satellites; and the solar radiation pressure is the main factor to disturb orbits of Echo-type satellites with large area-to-mass ratios. However, for most of the satellites, in which we are interested, perturbations due to the figure of the Earth, particularly due to the oblateness, are predominant. The equations of motion for satellites moving under the attraction of an oblate planet, or the Earth, which is assumed to have a potential of rotational symmetry, have been solved by many authors. The potential of the Earth for this case can be expressed as a function of the geocentric radius r, expressed in units of the equatorial radius, and the geo-centric latitude (p: (i) where G is the gravitation constant, M is the mass of the Earth, P k is Legendre polynomial of the &th order. Coefficients J k in (1) are small quantities of the second order (10~ 6 ) or less, except for J 2 which is of the first order.
Available until 25 Jan |Learn more- Tai L. Chow(Author)
- 2013(Publication Date)
- CRC Press(Publisher)
The calculated value of the precessional rate for Mercury by relativity is 43.03 + 0.03 second of arc per century. The observed value (corrected for the influence of the other planets) is 43.11 + 0.45 second of arc. This striking agreement is one of the major triumphs of the theory of relativity. The gravitational field near the Earth departs slightly from the inverse-square law. This is because the Earth is not a perfect sphere. Consequently, the perigee of a satellite whose orbit lies near the Earth’s equatorial plane will advance steadily in the direction of the satellite’s motion as the satellite moves around the Earth. We can use the observation of this advance to determine the shape of the 192 Classical Mechanics © 2010 Taylor & Francis Group, LLC Earth. Such observations have shown that the Earth is slightly pear-shaped. The oblateness of the Earth also causes the orbital plane of a satellite to process if it is not in the Earth equatorial plane. 6.11 LAPLACE–RUNGE–LENZ VECTOR AND THE KEPLER ORBIT (OPTIONAL) The equation of the orbit for the Kepler problem can be obtained by a simple algebraic method. This method requires only vector analysis techniques and does not involve solving a differential equa-tion or performing any integration. Apart from its simplicity, the method is intrinsically interesting because it makes use of an unusual conserved quantity, the Laplace–Rung–Lenz (LRL) vector. In the corresponding quantum-mechanical case, the hydrogen atom, eigenvalues, and eigenfunctions are readily found from knowledge of the LRL vector. Knowledge of the LRL vector also allows one to study the symmetry characteristics in a simple way. Thus, the reader may encounter generaliza-tions of the ideas considered here in more advanced courses. We first show that the LRL vector is a constant of the motion and then use this fact to obtain the equation of the orbit.
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