what information will you need to determine an orbital period during each experiment

ORBITAL MECHANICS

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• Conic Sections
• Orbital Elements
• Types of Orbits
• Newton's Laws of Move and Universal Gravitation
• Uniform Circular Move
• Motions of Planets and Satellites
• Launch of a Infinite Vehicle
• Position in an Elliptical Orbit
• Orbit Perturbations
• Orbit Maneuvers
• The Hyperbolic Orbit

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Orbital mechanics, also called flight mechanics, is the study of the motions of artificial satellites and infinite vehicles moving nether the influence of forces such as gravity, atmospheric elevate, thrust, etc. Orbital mechanics is a mod adjunct of angelic mechanics which is the report of the motions of natural celestial bodies such as the moon and planets. The root of orbital mechanics tin can be traced dorsum to the 17th century when mathematician Isaac Newton (1642-1727) put forward his laws of motion and formulated his police of universal gravitation. The engineering science applications of orbital mechanics include rise trajectories, reentry and landing, rendezvous computations, and lunar and interplanetary trajectories.

Conic Sections

A conic section, or just conic, is a curve formed by passing a plane through a right circular cone. As shown in Figure 4.one, the athwart orientation of the plane relative to the cone determines whether the conic section is a circumvolve, ellipse, parabola, or hyperbola. The circle and the ellipse arise when the intersection of cone and plane is a bounded curve. The circumvolve is a special case of the ellipse in which the plane is perpendicular to the axis of the cone. If the plane is parallel to a generator line of the cone, the conic is called a parabola. Finally, if the intersection is an unbounded curve and the aeroplane is non parallel to a generator line of the cone, the effigy is a hyperbola. In the latter instance the aeroplane will intersect both halves of the cone, producing two separate curves.

We tin can define all conic sections in terms of the eccentricity. The type of conic department is too related to the semi-major axis and the energy. The table below shows the relationships between eccentricity, semi-major centrality, and free energy and the type of conic section.

Conic Department	Eccentricity, e	Semi-major axis	Energy
Circle	0	= radius	< 0
Ellipse	0 < e < 1	> 0	< 0
Parabola	ane	infinity	0
Hyperbola	> 1	< 0	> 0

Satellite orbits can be any of the four conic sections. This page deals mostly with elliptical orbits, though we conclude with an test of the hyperbolic orbit.

Orbital Elements

To mathematically describe an orbit ane must define vi quantities, chosen orbital elements. They are

An orbiting satellite follows an oval shaped path known equally an ellipse with the torso being orbited, called the primary, located at one of two points called foci. An ellipse is defined to be a curve with the post-obit belongings: for each point on an ellipse, the sum of its distances from ii stock-still points, called foci, is constant (see Figure 4.two). The longest and shortest lines that can exist drawn through the center of an ellipse are called the major centrality and pocket-size axis, respectively. The semi-major centrality is i-half of the major axis and represents a satellite's mean distance from its master. Eccentricity is the distance between the foci divided past the length of the major axis and is a number between goose egg and i. An eccentricity of zero indicates a circumvolve.

Inclination is the angular distance between a satellite'south orbital plane and the equator of its primary (or the ecliptic plane in the case of heliocentric, or sunday centered, orbits). An inclination of zero degrees indicates an orbit about the principal'due south equator in the same direction equally the primary'southward rotation, a management chosen prograde (or directly). An inclination of ninety degrees indicates a polar orbit. An inclination of 180 degrees indicates a retrograde equatorial orbit. A retrograde orbit is one in which a satellite moves in a direction contrary to the rotation of its primary.

Periapsis is the bespeak in an orbit closest to the master. The opposite of periapsis, the farthest point in an orbit, is called apoapsis. Periapsis and apoapsis are usually modified to apply to the body being orbited, such equally perihelion and aphelion for the Sun, perigee and apogee for Globe, perijove and apojove for Jupiter, perilune and apolune for the Moon, etc. The statement of periapsis is the angular distance between the ascending node and the indicate of periapsis (see Figure 4.3). The fourth dimension of periapsis passage is the fourth dimension in which a satellite moves through its point of periapsis.

Nodes are the points where an orbit crosses a plane, such as a satellite crossing the Globe's equatorial airplane. If the satellite crosses the plane going from south to northward, the node is the ascending node; if moving from north to south, it is the descending node. The longitude of the ascending node is the node'due south celestial longitude. Celestial longitude is analogous to longitude on Earth and is measured in degrees counter-clockwise from nix with zero longitude beingness in the management of the vernal equinox.

In general, three observations of an object in orbit are required to calculate the six orbital elements. Ii other quantities often used to describe orbits are period and true anomaly. Period, P, is the length of time required for a satellite to complete one orbit. True anomaly, , is the angular distance of a point in an orbit past the point of periapsis, measured in degrees.

Types Of Orbits

For a spacecraft to accomplish World orbit, it must be launched to an meridian above the Globe'due south atmosphere and accelerated to orbital velocity. The most energy efficient orbit, that is 1 that requires the least amount of propellant, is a direct low inclination orbit. To reach such an orbit, a spacecraft is launched in an due east direction from a site near the World's equator. The reward being that the rotational speed of the World contributes to the spacecraft's final orbital speed. At the United states of america' launch site in Greatcoat Canaveral (28.5 degrees north latitude) a due east launch results in a "free ride" of 1,471 km/h (914 mph). Launching a spacecraft in a direction other than east, or from a site far from the equator, results in an orbit of higher inclination. Loftier inclination orbits are less able to take advantage of the initial speed provided by the Globe'southward rotation, thus the launch vehicle must provide a greater part, or all, of the free energy required to attain orbital velocity. Although loftier inclination orbits are less energy efficient, they do have advantages over equatorial orbits for certain applications. Below we describe several types of orbits and the advantages of each:

Geosynchronous orbits (GEO) are round orbits around the Earth having a flow of 24 hours. A geosynchronous orbit with an inclination of zero degrees is called a geostationary orbit. A spacecraft in a geostationary orbit appears to hang motionless above one position on the Earth's equator. For this reason, they are ideal for some types of communication and meteorological satellites. A spacecraft in an inclined geosynchronous orbit will appear to follow a regular figure-eight pattern in the sky one time every orbit. To reach geosynchronous orbit, a spacecraft is first launched into an elliptical orbit with an apogee of 35,786 km (22,236 miles) chosen a geosynchronous transfer orbit (GTO). The orbit is then circularized by firing the spacecraft'south engine at apogee.

Polar orbits (PO) are orbits with an inclination of 90 degrees. Polar orbits are useful for satellites that carry out mapping and/or surveillance operations considering as the planet rotates the spacecraft has access to nigh every betoken on the planet'southward surface.

Walking orbits : An orbiting satellite is subjected to a great many gravitational influences. Showtime, planets are non perfectly spherical and they have slightly uneven mass distribution. These fluctuations take an effect on a spacecraft'south trajectory. Besides, the sunday, moon, and planets contribute a gravitational influence on an orbiting satellite. With proper planning it is possible to design an orbit which takes advantage of these influences to induce a precession in the satellite's orbital plane. The resulting orbit is chosen a walking orbit, or precessing orbit.

Dominicus synchronous orbits (SSO) are walking orbits whose orbital airplane precesses with the same period as the planet's solar orbit period. In such an orbit, a satellite crosses periapsis at about the same local fourth dimension every orbit. This is useful if a satellite is conveying instruments which depend on a certain angle of solar illumination on the planet's surface. In order to maintain an exact synchronous timing, it may be necessary to conduct occasional propulsive maneuvers to adjust the orbit.

Molniya orbits are highly eccentric Earth orbits with periods of approximately 12 hours (2 revolutions per twenty-four hours). The orbital inclination is chosen then the rate of change of perigee is nil, thus both apogee and perigee tin can be maintained over fixed latitudes. This condition occurs at inclinations of 63.iv degrees and 116.6 degrees. For these orbits the statement of perigee is typically placed in the southern hemisphere, so the satellite remains above the northern hemisphere near apogee for approximately xi hours per orbit. This orientation can provide skilful ground coverage at loftier northern latitudes.

Hohmann transfer orbits are interplanetary trajectories whose advantage is that they swallow the least possible amount of propellant. A Hohmann transfer orbit to an outer planet, such equally Mars, is accomplished by launching a spacecraft and accelerating it in the direction of Earth'southward revolution around the sun until it breaks free of the Earth's gravity and reaches a velocity which places it in a dominicus orbit with an aphelion equal to the orbit of the outer planet. Upon reaching its destination, the spacecraft must decelerate so that the planet'southward gravity tin can capture information technology into a planetary orbit.

To transport a spacecraft to an inner planet, such as Venus, the spacecraft is launched and accelerated in the management opposite of Earth'due south revolution around the dominicus (i.e. decelerated) until information technology achieves a sun orbit with a perihelion equal to the orbit of the inner planet. Information technology should be noted that the spacecraft continues to movement in the same direction as Earth, only more slowly.

To attain a planet requires that the spacecraft be inserted into an interplanetary trajectory at the correct time so that the spacecraft arrives at the planet'south orbit when the planet will be at the bespeak where the spacecraft volition intercept it. This task is comparable to a quarterback "leading" his receiver so that the football and receiver go far at the aforementioned point at the same time. The interval of time in which a spacecraft must be launched in social club to complete its mission is called a launch window.

Newton's Laws of Motility and Universal Gravitation

Newton's laws of movement describe the relationship betwixt the motion of a particle and the forces acting on it.

The first law states that if no forces are acting, a body at rest will remain at rest, and a torso in motion volition remain in motion in a straight line. Thus, if no forces are acting, the velocity (both magnitude and direction) will remain constant.

The second police tells the states that if a strength is practical there will be a change in velocity, i.eastward. an acceleration, proportional to the magnitude of the force and in the direction in which the force is applied. This constabulary may be summarized by the equation

where F is the forcefulness, yard is the mass of the particle, and a is the acceleration.

The third police force states that if trunk one exerts a force on body ii, then torso 2 will exert a forcefulness of equal strength, but opposite in direction, on body 1. This law is commonly stated, "for every activeness there is an equal and opposite reaction".

In his law of universal gravitation, Newton states that two particles having masses one thousand_i and m_two and separated by a distance r are attracted to each other with equal and opposite forces directed along the line joining the particles. The common magnitude F of the two forces is

where K is an universal constant, chosen the constant of gravitation, and has the value 6.67259x10^-eleven N-m²/kg^two (three.4389x10^-viii lb-ft^two/slug²).

Allow'due south now wait at the force that the Earth exerts on an object. If the object has a mass chiliad, and the World has mass Chiliad, and the object's distance from the middle of the Globe is r, then the force that the World exerts on the object is GmM /rⁱⁱ . If nosotros drop the object, the Globe's gravity volition cause it to accelerate toward the eye of the Earth. By Newton's 2nd law (F = ma), this dispatch thou must equal (GmM /r²)/thou, or

At the surface of the Earth this acceleration has the valve 9.80665 m/s^two (32.174 ft/s²).

Many of the upcoming computations will be somewhat simplified if we express the product GM equally a constant, which for Earth has the value 3.986005x10¹⁴ m^three/south² (1.408x10^xvi ft³/s²). The product GM is frequently represented past the Greek letter of the alphabet .

For boosted useful constants delight see the appendix Basic Constants.

For a refresher on SI versus U.Southward. units see the appendix Weights & Measures.

Uniform Round Motility

In the unproblematic case of free autumn, a particle accelerates toward the center of the Earth while moving in a straight line. The velocity of the particle changes in magnitude, but not in direction. In the case of uniform circular motion a particle moves in a circumvolve with abiding speed. The velocity of the particle changes continuously in management, merely not in magnitude. From Newton's laws we meet that since the direction of the velocity is changing, there is an acceleration. This acceleration, called centripetal dispatch is directed inward toward the centre of the circle and is given by

where v is the speed of the particle and r is the radius of the circle. Every accelerating particle must accept a strength interim on it, defined by Newton's second police force (F = ma). Thus, a particle undergoing uniform circular move is under the influence of a force, called centripetal force, whose magnitude is given by

The direction of F at any instant must be in the direction of a at the aforementioned instant, that is radially inward.

A satellite in orbit is acted on simply past the forces of gravity. The in dispatch which causes the satellite to move in a circular orbit is the gravitational acceleration acquired by the trunk around which the satellite orbits. Hence, the satellite's centripetal dispatch is g, that is g = five²/r. From Newton's police of universal gravitation we know that g = GM /r^two . Therefore, past setting these equations equal to one some other we find that, for a round orbit,

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Motions of Planets and Satellites

Through a lifelong study of the motions of bodies in the solar system, Johannes Kepler (1571-1630) was able to derive three basic laws known every bit Kepler'south laws of planetary move. Using the information compiled by his mentor Tycho Brahe (1546-1601), Kepler found the post-obit regularities after years of laborious calculations:

ane.  All planets movement in elliptical orbits with the sun at one focus.
ii.  A line joining any planet to the dominicus sweeps out equal areas in equal times.
three.  The foursquare of the menses of whatever planet about the sun is proportional to the cube of the planet'due south hateful distance from the sun.

These laws can be deduced from Newton's laws of move and law of universal gravitation. Indeed, Newton used Kepler's piece of work as basic information in the formulation of his gravitational theory.

Equally Kepler pointed out, all planets move in elliptical orbits, however, nosotros tin can learn much about planetary motion by considering the special case of circular orbits. We shall neglect the forces between planets, because just a planet's interaction with the lord's day. These considerations utilise equally well to the movement of a satellite about a planet.

Allow'south examine the instance of ii bodies of masses One thousand and m moving in circular orbits under the influence of each other's gravitational attraction. The center of mass of this arrangement of two bodies lies forth the line joining them at a point C such that mr = MR. The large body of mass M moves in an orbit of constant radius R and the small torso of mass m in an orbit of constant radius r, both having the aforementioned angular velocity . For this to happen, the gravitational force acting on each body must provide the necessary centripetal dispatch. Since these gravitational forces are a simple activeness-reaction pair, the centripetal forces must be equal only opposite in management. That is, m ²r must equal M ^twoR. The specific requirement, and so, is that the gravitational forcefulness acting on either trunk must equal the centripetal force needed to keep it moving in its circular orbit, that is

If 1 body has a much greater mass than the other, as is the case of the sun and a planet or the Earth and a satellite, its altitude from the centre of mass is much smaller than that of the other body. If we presume that m is negligible compared to Grand, then R is negligible compared to r. Thus, equation (iv.vii) and so becomes

If we express the athwart velocity in terms of the period of revolution, = 2/P, we obtain

where P is the period of revolution. This is a basic equation of planetary and satellite motion. It also holds for elliptical orbits if we define r to be the semi-major axis (a) of the orbit.

A significant effect of this equation is that it predicts Kepler's third police of planetary move, that is P²~r^three .

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In celestial mechanics where we are dealing with planetary or stellar sized bodies, it is ofttimes the case that the mass of the secondary trunk is significant in relation to the mass of the main, every bit with the Moon and Earth. In this case the size of the secondary cannot be ignored. The distance R is no longer negligible compared to r and, therefore, must be carried through the derivation. Equation (four.9) becomes

More commonly the equation is written in the equivalent course

where a is the semi-major axis. The semi-major axis used in astronomy is always the primary-to-secondary distance, or the geocentric semi-major axis. For example, the Moon's mean geocentric distance from World (a) is 384,403 kilometers. On the other paw, the Moon's distance from the barycenter (r) is 379,732 km, with Earth's counter-orbit (R) taking up the difference of four,671 km.

Kepler's second constabulary of planetary motion must, of course, concord true for circular orbits. In such orbits both and r are constant so that equal areas are swept out in equal times by the line joining a planet and the dominicus. For elliptical orbits, however, both and r will vary with time. Permit's now consider this case.

Figure 4.5 shows a particle revolving around C along some arbitrary path. The area swept out by the radius vector in a curt fourth dimension interval t is shown shaded. This area, neglecting the small triangular region at the end, is one-half the base times the height or approximately r(r t)/2. This expression becomes more exact as t approaches cipher, i.e. the pocket-size triangle goes to aught more apace than the large i. The rate at which expanse is being swept out instantaneously is therefore

For any given body moving under the influence of a central force, the value r² is constant.

Let's now consider two points P₁ and P₂ in an orbit with radii r_one and r₂ , and velocities 5₁ and v_two . Since the velocity is ever tangent to the path, information technology can exist seen that if is the angle between r and v, then

where vsin is the transverse component of five. Multiplying through by r, nosotros have

or, for two points P_one and P₂ on the orbital path

Note that at periapsis and apoapsis, = 90 degrees. Thus, letting P₁ and P_two exist these two points we get

Let's now look at the free energy of the in a higher place particle at points P₁ and P₂ . Conservation of free energy states that the sum of the kinetic energy and the potential free energy of a particle remains abiding. The kinetic energy T of a particle is given by mv²/2 while the potential energy of gravity V is calculated by the equation -GMm/r. Applying conservation of energy nosotros accept

From equations (4.14) and (iv.15) we obtain

Rearranging terms nosotros become

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The eccentricity eastward of an orbit is given by

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If the semi-major centrality a and the eccentricity e of an orbit are known, and then the periapsis and apoapsis distances tin can be calculated by

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Launch of a Space Vehicle

The launch of a satellite or infinite vehicle consists of a catamenia of powered flight during which the vehicle is lifted above the Earth's atmosphere and accelerated to orbital velocity by a rocket, or launch vehicle. Powered flight concludes at burnout of the rocket's last stage at which fourth dimension the vehicle begins its free flight. During free flying the space vehicle is assumed to exist subjected merely to the gravitational pull of the World. If the vehicle moves far from the World, its trajectory may be affected by the gravitational influence of the sunday, moon, or another planet.

A space vehicle'due south orbit may be determined from the position and the velocity of the vehicle at the beginning of its free flight. A vehicle's position and velocity can be described by the variables r, v, and , where r is the vehicle'south altitude from the heart of the World, v is its velocity, and is the angle between the position and the velocity vectors, called the zenith angle (run across Figure 4.7). If we letr₁, v₁ , and _ane be the initial (launch) values ofr, five, and , then we may consider these as given quantities. If we let indicate P_two represent the perigee, so equation (four.13) becomes

Substituting equation (4.23) into (iv.xv), we tin can obtain an equation for the perigee radius R_p .

Multiplying through by -R_p ^two/(r_one ²v₁ ²) and rearranging, we get

Note that this is a simple quadratic equation in the ratio (R_p/r₁) and that 2GM /(r_one × five₁ ²) is a nondimensional parameter of the orbit.

Solving for (R_p/r_i) gives

Similar whatsoever quadratic, the above equation yields two answers. The smaller of the 2 answers corresponds to R_p , the periapsis radius. The other root corresponds to the apoapsis radius, R_a .

Delight note that in practice spacecraft launches are usually terminated at either perigee or apogee, i.e. = 90. This condition results in the minimum use of propellant.

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Equation (iv.26) gives the values of R_p and R_a from which the eccentricity of the orbit tin can exist calculated, still, information technology may be simpler to summate the eccentricity due east directly from the equation

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To pivot down a satellite'due south orbit in space, we need to know the angle , the true anomaly, from the periapsis signal to the launch point. This angle is given by

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In most calculations, the complement of the zenith angle is used, denoted by . This angle is called the flying-path angle, and is positive when the velocity vector is directed away from the primary as shown in Effigy 4.8. When flight-path angle is used, equations (four.26) through (four.28) are rewritten as follows:

The semi-major axis is, of grade, equal to (R_p+R_a)/2, though it may be easier to calculate information technology directly equally follows:

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If e is solved for directly using equation (4.27) or (4.30), and a is solved for using equation (4.32), R_p and R_a can be solved for simply using equations (4.21) and (4.22).

Orbit Tilt, Rotation and Orientation

Higher up we adamant the size and shape of the orbit, but to determine the orientation of the orbit in space, we must know the latitude and longitude and the heading of the space vehicle at burnout.

Figure four.9 above illustrates the location of a space vehicle at engine exhaustion, or orbit insertion. is the azimuth heading measured in degrees clockwise from due north, is the geocentric latitude (or declination) of the burnout bespeak, is the athwart distance between the ascending node and the exhaustion point measured in the equatorial plane, and is the athwart distance between the ascending node and the burnout bespeak measured in the orbital airplane. ₁ and _two are the geographical longitudes of the ascending node and the burnout point at the instant of engine burnout. Figure 4.10 pictures the orbital elements, where i is the inclination, is the longitude at the ascending node, is the statement of periapsis, and is the true bibelot.

If , , and _two are given, the other values can be calculated from the following relationships:

In equation (4.36), the value of is found using equation (iv.28) or (4.31). If is positive, periapsis is west of the burnout point (as shown in Figure 4.10); if is negative, periapsis is e of the exhaustion signal.

The longitude of the ascending node, , is measured in celestial longitude, while ₁ is geographical longitude. The celestial longitude of the ascending node is equal to the local apparent sidereal time, in degrees, at longitude _i at the time of engine burnout. Sidereal time is defined as the hour angle of the vernal equinox at a specific locality and fourth dimension; it has the same value as the correct rise of any celestial torso that is crossing the local meridian at that same instant. At the moment when the vernal equinox crosses the local meridian, the local apparent sidereal time is 00:00. See this sidereal fourth dimension calculator.

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Geodetic Latitude, Geocentric Latitude, and Declination

Latitude is the athwart distance of a betoken on Earth's surface north or south of Earth's equator, positive north and negative south. The geodetic latitude (or geographical latitude), , is the angle defined by the intersection of the reference ellipsoid normal through the betoken of involvement and the true equatorial plane. The geocentric latitude, ', is the angle between the true equatorial plane and the radius vector to the signal of intersection of the reference ellipsoid and the reference ellipsoid normal passing through the point of interest. Declination, , is the athwart distance of a celestial object north or south of Earth'due south equator. It is the angle between the geocentric radius vector to the object of interest and the true equatorial plane.

R is the magnitude of the reference ellipsoid's geocentric radius vector to the bespeak of interest on its surface, r is the magnitude of the geocentric radius vector to the angelic object of interest, and the altitude h is the perpendicular distance from the reference ellipsoid to the celestial object of interest. The value of R at the equator is a, and the value of R at the poles is b. The ellipsoid's flattening, f, is the ratio of the equatorial-polar length difference to the equatorial length. For Earth, a equals 6,378,137 meters, b equals vi,356,752 meters, and f equals 1/298.257.

When solving problems in orbital mechanics, the measurements of greatest usefulness are the magnitude of the radius vector, r, and declination, , of the object of interest. Even so, we are oft given, or required to written report, data in other forms. For instance, at the time of some specific event, such as "orbit insertion", we may exist given the spacecraft's altitude along with the geodetic latitude and longitude of the sub-vehicle point. In such cases, it may exist necessary to convert the given data to a form more suitable for our calculations.

The relationship between geodetic and geocentric latitude is,

The radius of the reference ellipsoid is given by,

The length r tin can be solved from h, or h from r, using ane of the following,

And declination is calculated using,

For spacecraft in low earth orbit, the difference between and ' is very small, typically not more than most 0.00001 degree. Even at the altitude of the Moon, the difference is not more than about 0.01 degree. Unless very high accuracy is needed, for operations about Earth we tin can assume ≈ ' and r ≈ R + h.

It is important to annotation that the value of h is non ever measured every bit described and illustrated in a higher place. In some applications information technology is customary to limited h as the perpendicular altitude from a reference sphere, rather than the reference ellipsoid. In this example, R is considered constant and is ofttimes assigned the value of World'due south equatorial radius, hence h = r � a. This is the method typically used when a spacecraft'due south orbit is expressed in a class such as "180 km × 220 km". The example issues presented in this web site also assume this method of measurement.

Position in an Elliptical Orbit

Johannes Kepler was able to solve the problem of relating position in an orbit to the elapsed fourth dimension, t-t_o , or conversely, how long it takes to go from one point in an orbit to another. To solve this, Kepler introduced the quantity M, chosen the mean anomaly, which is the fraction of an orbit period that has elapsed since perigee.  The mean anomaly equals the truthful anomaly for a circular orbit. By definition,

where G_o is the hateful anomaly at time t_o and north is the hateful motion, or the average angular velocity, adamant from the semi-major axis of the orbit as follows:

This solution volition give the average position and velocity, but satellite orbits are elliptical with a radius constantly varying in orbit. Because the satellite's velocity depends on this varying radius, information technology changes equally well. To resolve this problem nosotros can define an intermediate variable East, called the eccentric anomaly, for elliptical orbits, which is given past

where is the true anomaly. Mean bibelot is a function of eccentric anomaly by the formula

For small eccentricities a good approximation of truthful anomaly can be obtained by the post-obit formula (the fault is of the order e^three):

The preceding five equations can exist used to (1) discover the fourth dimension information technology takes to go from one position in an orbit to another, or (two) find the position in an orbit after a specific menstruum of time. When solving these equations it is important to work in radians rather than degrees, where two radians equals 360 degrees.

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At any fourth dimension in its orbit, the magnitude of a spacecraft'southward position vector, i.due east. its distance from the principal body, and its flight-path angle tin can be calculated from the post-obit equations:

And the spacecraft's velocity is given by,

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Orbit Perturbations

The orbital elements discussed at the beginning of this section provide an excellent reference for describing orbits, however there are other forces interim on a satellite that adjy information technology away from the nominal orbit. These perturbations, or variations in the orbital elements, can be classified based on how they bear on the Keplerian elements. Secular variations stand for a linear variation in the element, short-period variations are periodic in the element with a menstruation less than the orbital period, and long-period variations are those with a period greater than the orbital period. Because secular variations have long-term effects on orbit prediction (the orbital elements affected go along to increase or decrease), they will be discussed hither for Earth-orbiting satellites. Precise orbit determination requires that the periodic variations be included too.

Third-Body Perturbations

The gravitational forces of the Sun and the Moon cause periodic variations in all of the orbital elements, but simply the longitude of the ascending node, argument of perigee, and mean bibelot feel secular variations. These secular variations arise from a gyroscopic precession of the orbit about the ecliptic pole. The secular variation in mean bibelot is much smaller than the mean motility and has little effect on the orbit, however the secular variations in longitude of the ascending node and argument of perigee are important, especially for loftier-distance orbits.

For almost round orbits the equations for the secular rates of change resulting from the Sun and Moon are

Longitude of the ascending node:

Argument of perigee:

where i is the orbit inclination, northward is the number of orbit revolutions per twenty-four hour period, and and are in degrees per mean solar day. These equations are only approximate; they fail the variation caused by the irresolute orientation of the orbital plane with respect to both the Moon'southward orbital plane and the ecliptic plane.

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Perturbations due to Not-spherical Earth

When developing the two-body equations of motion, we assumed the Earth was a spherically symmetrical, homogeneous mass. In fact, the Globe is neither homogeneous nor spherical. The most dominant features are a bulge at the equator, a slight pear shape, and flattening at the poles. For a potential part of the Earth, we tin find a satellite'south acceleration by taking the gradient of the potential function. The most widely used form of the geopotential role depends on latitude and geopotential coefficients, J_northward , called the zonal coefficients.

The potential generated past the non-spherical Globe causes periodic variations in all the orbital elements. The dominant effects, still, are secular variations in longitude of the ascending node and argument of perigee because of the Earth'due south oblateness, represented by the J₂ term in the geopotential expansion. The rates of change of and due to J₂ are

where due north is the mean motion in degrees/day, J₂ has the value 0.00108263, R_Eastward is the Earth's equatorial radius, a is the semi-major centrality in kilometers, i is the inclination, e is the eccentricity, and and are in degrees/day. For satellites in GEO and below, the J_ii perturbations dominate; for satellites to a higher place GEO the Sun and Moon perturbations dominate.

Molniya orbits are designed so that the perturbations in statement of perigee are zero. This atmospheric condition occurs when the term 4-5sin²i is equal to nix or, that is, when the inclination is either 63.4 or 116.vi degrees.

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Perturbations from Atmospheric Elevate

Drag is the resistance offered by a gas or liquid to a body moving through information technology. A spacecraft is subjected to drag forces when moving through a planet's atmosphere. This drag is greatest during launch and reentry, nevertheless, fifty-fifty a space vehicle in depression Earth orbit experiences some drag equally information technology moves through the Earth's sparse upper atmosphere. In fourth dimension, the action of drag on a space vehicle will cause it to spiral dorsum into the atmosphere, eventually to disintegrate or burn up. If a infinite vehicle comes within 120 to 160 km of the Earth'south surface, atmospheric elevate will bring it down in a few days, with final disintegration occurring at an altitude of nearly 80 km. In a higher place approximately 600 km, on the other mitt, drag is so weak that orbits usually final more than x years - beyond a satellite'southward operational lifetime. The deterioration of a spacecraft's orbit due to drag is called decay.

The drag force F_D on a torso acts in the opposite direction of the velocity vector and is given by the equation

where C_D is the elevate coefficient, is the air density, v is the trunk's velocity, and A is the area of the body normal to the flow. The drag coefficient is dependent on the geometric form of the body and is generally determined past experiment. World orbiting satellites typically accept very high elevate coefficients in the range of virtually 2 to four. Air density is given by the appendix Atmosphere Properties.

The region to a higher place 90 km is the Earth'due south thermosphere where the absorption of extreme ultraviolet radiation from the Sunday results in a very rapid increase in temperature with altitude. At approximately 200-250 km this temperature approaches a limiting value, the boilerplate value of which ranges between about 700 and i,400 K over a typical solar cycle. Solar activity also has a significant affect on atmospheric density, with high solar activity resulting in high density. Below virtually 150 km the density is not strongly afflicted past solar activity; all the same, at satellite altitudes in the range of 500 to 800 km, the density variations between solar maximum and solar minimum are approximately 2 orders of magnitude. The large variations imply that satellites volition decay more quickly during periods of solar maxima and much more slowly during solar minima.

For circular orbits we tin can approximate the changes in semi-major axis, period, and velocity per revolution using the post-obit equations:

where a is the semi-major axis, P is the orbit flow, and V, A and grand are the satellite'southward velocity, area, and mass respectively. The term m/(C_DA), called the ballistic coefficient, is given every bit a constant for well-nigh satellites. Drag effects are strongest for satellites with low ballistic coefficients, this is, calorie-free vehicles with large frontal areas.

A crude approximate of a satellite's lifetime, L, due to drag can be computed from

where H is the atmospheric density scale height. A substantially more accurate estimate (although still very judge) can exist obtained by integrating equation (4.53), taking into account the changes in atmospheric density with both altitude and solar activity.

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Perturbations from Solar Radiation

Solar radiation pressure causes periodic variations in all of the orbital elements. The magnitude of the dispatch in m/s² arising from solar radiations pressure level is

where A is the cantankerous-sectional area of the satellite exposed to the Dominicus and m is the mass of the satellite in kilograms. For satellites beneath 800 km altitude, dispatch from atmospheric elevate is greater than that from solar radiation pressure; in a higher place 800 km, acceleration from solar radiation pressure is greater.

Orbit Maneuvers

At some point during the lifetime of most infinite vehicles or satellites, nosotros must change one or more of the orbital elements. For instance, we may demand to transfer from an initial parking orbit to the last mission orbit, rendezvous with or intercept another spacecraft, or correct the orbital elements to accommodate for the perturbations discussed in the previous department. Nigh often, we must modify the orbit altitude, plane, or both. To change the orbit of a space vehicle, we take to alter its velocity vector in magnitude or direction. Nearly propulsion systems operate for only a brusque fourth dimension compared to the orbital period, thus we can treat the maneuver every bit an impulsive change in velocity while the position remains fixed. For this reason, any maneuver irresolute the orbit of a space vehicle must occur at a point where the onetime orbit intersects the new orbit. If the orbits practise not intersect, we must employ an intermediate orbit that intersects both. In this case, the total maneuver volition require at to the lowest degree two propulsive burns.

Orbit Altitude Changes

The about common blazon of in-aeroplane maneuver changes the size and energy of an orbit, commonly from a depression-altitude parking orbit to a college-altitude mission orbit such every bit a geosynchronous orbit. Because the initial and final orbits practise not intersect, the maneuver requires a transfer orbit. Figure 4.eleven represents a Hohmann transfer orbit. In this instance, the transfer orbit'due south ellipse is tangent to both the initial and concluding orbits at the transfer orbit's perigee and apogee respectively. The orbits are tangential, then the velocity vectors are collinear, and the Hohmann transfer represents the almost fuel-efficient transfer between two circular, coplanar orbits. When transferring from a smaller orbit to a larger orbit, the change in velocity is applied in the direction of motion; when transferring from a larger orbit to a smaller, the change of velocity is reverse to the direction of movement.

The total modify in velocity required for the orbit transfer is the sum of the velocity changes at perigee and apogee of the transfer ellipse. Since the velocity vectors are collinear, the velocity changes are simply the differences in magnitudes of the velocities in each orbit. If we know the initial and final orbits, r_A and r_B , nosotros tin calculate the total velocity change using the following equations:

Annotation that equations (4.59) and (4.60) are the aforementioned as equation (4.6), and equations (four.61) and (four.62) are the aforementioned as equation (four.45).

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Usually we want to transfer a infinite vehicle using the smallest corporeality of energy, which unremarkably leads to using a Hohmann transfer orbit. All the same, sometimes we may need to transfer a satellite between orbits in less fourth dimension than that required to complete the Hohmann transfer. Figure iv.12 shows a faster transfer chosen the One-Tangent Burn. In this instance the transfer orbit is tangential to the initial orbit. It intersects the final orbit at an angle equal to the flight path angle of the transfer orbit at the bespeak of intersection. An infinite number of transfer orbits are tangential to the initial orbit and intersect the final orbit at some angle. Thus, we may choose the transfer orbit past specifying the size of the transfer orbit, the angular change of the transfer, or the time required to complete the transfer. We tin can then define the transfer orbit and calculate the required velocities.

For example, we may specify the size of the transfer orbit, choosing any semi-major axis that is greater than the semi-major centrality of the Hohmann transfer ellipse. Once we know the semi-major axis of the ellipse, a_tx , we tin can calculate the eccentricity, angular distance traveled in the transfer, the velocity change required for the transfer, and the fourth dimension required to complete the transfer. We do this using equations (iv.59) through (four.63) and (iv.65) higher up, and the following equations:

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Another selection for changing the size of an orbit is to use electric propulsion to produce a constant low-thrust burn, which results in a spiral transfer. We can approximate the velocity change for this type of orbit transfer by

where the velocities are the round velocities of the two orbits.

Orbit Plane Changes

To change the orientation of a satellite's orbital aeroplane, typically the inclination, nosotros must modify the direction of the velocity vector. This maneuver requires a component of V to exist perpendicular to the orbital airplane and, therefore, perpendicular to the initial velocity vector. If the size of the orbit remains constant, the maneuver is chosen a simple aeroplane change. Nosotros can find the required change in velocity by using the law of cosines. For the case in which V_f is equal to V_i , this expression reduces to

where V_i is the velocity earlier and later on the burn down, and is the bending change required.

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From equation (four.73) nosotros see that if the athwart modify is equal to 60 degrees, the required change in velocity is equal to the current velocity. Plane changes are very expensive in terms of the required change in velocity and resulting propellant consumption. To minimize this, we should change the airplane at a point where the velocity of the satellite is a minimum: at apogee for an elliptical orbit. In some cases, it may even be cheaper to boost the satellite into a higher orbit, alter the orbit plane at apogee, and return the satellite to its original orbit.

Typically, orbital transfers require changes in both the size and the plane of the orbit, such every bit transferring from an inclined parking orbit at low distance to a zero-inclination orbit at geosynchronous distance. We tin do this transfer in two steps: a Hohmann transfer to alter the size of the orbit and a elementary aeroplane change to make the orbit equatorial. A more efficient method (less total change in velocity) would be to combine the plane change with the tangential burn at apogee of the transfer orbit. As nosotros must modify both the magnitude and direction of the velocity vector, nosotros can detect the required change in velocity using the law of cosines,

where V_i is the initial velocity, V_f is the final velocity, and is the angle alter required. Note that equation (4.74) is in the same form as equation (4.69).

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Every bit tin can be seen from equation (4.74), a pocket-sized plane change can exist combined with an altitude alter for almost no toll in V or propellant. Consequently, in practice, geosynchronous transfer is washed with a small airplane alter at perigee and nigh of the aeroplane change at apogee.

Another pick is to complete the maneuver using three burns. The outset burn is a coplanar maneuver placing the satellite into a transfer orbit with an apogee much higher than the final orbit. When the satellite reaches apogee of the transfer orbit, a combined plane change maneuver is done. This places the satellite in a second transfer orbit that is coplanar with the final orbit and has a perigee distance equal to the altitude of the final orbit. Finally, when the satellite reaches perigee of the second transfer orbit, some other coplanar maneuver places the satellite into the final orbit. This three-burn maneuver may salve propellant, simply the propellant savings comes at the expense of the full time required to complete the maneuver.

When a plane alter is used to modify inclination just, the magnitude of the bending alter is only the difference between the initial and final inclinations. In this example, the initial and final orbits share the same ascending and descending nodes. The plane change maneuver takes places when the space vehicle passes through ane of these ii nodes.

In some instances, all the same, a aeroplane change is used to modify an orbit's longitude of ascending node in add-on to the inclination. An case might exist a maneuver to correct out-of-aeroplane errors to make the orbits of two infinite vehicles coplanar in preparation for a rendezvous. If the orbital elements of the initial and final orbits are known, the airplane change bending is determined by the vector dot product. If i_i and _i are the inclination and longitude of ascending node of the initial orbit, and i_f and _f are the inclination and longitude of ascending node of the concluding orbit, then the angle between the orbital planes, , is given past

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The plane change maneuver takes identify at i of two nodes where the initial and final orbits intersect. The breadth and longitude of these nodes are determined by the vector cantankerous product. The position of one of the two nodes is given by

Knowing the position of one node, the 2nd node is just

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Orbit Rendezvous

Orbital transfer becomes more complicated when the object is to rendezvous with or intercept some other object in space: both the interceptor and the target must get in at the rendezvous signal at the same time. This precision demands a phasing orbit to accomplish the maneuver. A phasing orbit is any orbit that results in the interceptor achieving the desired geometry relative to the target to initiate a Hohmann transfer. If the initial and final orbits are circular, coplanar, and of different sizes, then the phasing orbit is simply the initial interceptor orbit. The interceptor remains in the initial orbit until the relative motion betwixt the interceptor and target results in the desired geometry. At that point, we would inject the interceptor into a Hohmann transfer orbit.

Launch Windows

Similar to the rendezvous problem is the launch-window problem, or determining the appropriate time to launch from the surface of the Globe into the desired orbital plane. Considering the orbital plane is stock-still in inertial space, the launch window is the fourth dimension when the launch site on the surface of the Earth rotates through the orbital plane. The time of the launch depends on the launch site's latitude and longitude and the satellite orbit's inclination and longitude of ascending node.

Orbit Maintenance

Once in their mission orbits, many satellites need no additional orbit adjustment. On the other hand, mission requirements may demand that we maneuver the satellite to correct the orbital elements when perturbing forces have changed them. Two detail cases of note are satellites with repeating basis tracks and geostationary satellites.

Later the mission of a satellite is complete, several options exist, depending on the orbit. We may allow low-altitude orbits to decay and reenter the atmosphere or use a velocity change to speed up the process. We may too boost satellites at all altitudes into benign orbits to reduce the probability of collision with active payloads, peculiarly at synchronous altitudes.

V Upkeep

To an orbit designer, a space mission is a series of different orbits. For example, a satellite might be released in a low-Globe parking orbit, transferred to some mission orbit, become through a series of resphasings or alternate mission orbits, and and so move to some concluding orbit at the end of its useful life. Each of these orbit changes requires energy. The V budget is traditionally used to business relationship for this energy. Information technology sums all the velocity changes required throughout the space mission life. In a wide sense the V budget represents the toll for each mission orbit scenario.

The Hyperbolic Orbit

The discussion thus far has focused on the elliptical orbit, which volition result whenever a spacecraft has bereft velocity to escape the gravity of its primary. There is a velocity, chosen the escape velocity, Five_esc , such that if the spacecraft is launched with an initial velocity greater than V_esc , information technology will travel away from the planet and never render. To attain escape velocity we must requite the spacecraft enough kinetic energy to overcome all of the negative gravitational potential free energy. Thus, if m is the mass of the spacecraft, Thousand is the mass of the planet, and r is the radial distance between the spacecraft and planet, the potential free energy is -GmM /r. The kinetic energy of the spacecraft, when it is launched, is mv²/2. Nosotros thus have

which is contained of the mass of the spacecraft.

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A space vehicle that has exceeded the escape velocity of a planet will travel a hyperbolic path relative to the planet. The hyperbola is an unusual and interesting conic section because it has two branches. The arms of a hyperbola are asymptotic to two intersecting straight line (the asymptotes). If we consider the left-hand focus, f, as the prime focus (where the eye of our gravitating body is located), so simply the left branch of the hyperbola represents the possible orbit. If, instead, we assume a force of repulsion between our satellite and the body located at f (such as the strength between ii like-charged electric particles), then the right-mitt branch represents the orbit. The parameters a, b and c are labeled in Figure 4.14. Nosotros can see that c² = a²+ b² for the hyperbola. The eccentricity is,

The angle between the asymptotes, which represents the angle through which the path of a space vehicle is turned past its run into with a planet, is labeled . This turning angle is related to the geometry of the hyperbola equally follows:

If we let equal the angle between the periapsis vector and the departure asymptote, i.eastward. the true bibelot at infinity, we have

If we know the radius, r, velocity, v, and flight path bending, , of a point on the orbit (see Figure 4.15), we can summate the eccentricity and semi-major centrality using equations (four.30) and (iv.32) every bit previously presented. Notation that the semi-major axis of a hyperbola is negative.

The truthful anomaly corresponding to known valves of r, v and can be calculated using equation (4.31), however special care must be taken to assure the angle is placed in the correct quadrant. It may be easier to kickoff summate e and a, and then calculate truthful anomaly using equation (4.43), rearranged every bit follows:

Whenever is positive, should exist taken equally positive; whenever is negative, should be taken every bit negative.

The bear upon parameter, b, is the distance of closest arroyo that would consequence between a spacecraft and planet if the spacecraft trajectory was undeflected past gravity. The impact parameter is,

Closet approach occurs at periapsis, where the radius distance, r_o, is equal to

p is a geometrical constant of the conic called the parameter or semi-latus rectum, and is equal to

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At whatever known true anomaly, the magnitude of a spacecraft's radius vector, its flight-path angle, and its velocity can be calculated using equations (iv.43), (4.44) and (four.45).

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Early we introduced the variable eccentric anomaly and its utilise in deriving the fourth dimension of flight in an elliptical orbit. In a similar style, the analytical derivation of the hyperbolic time of flight, using the hyperbolic eccentric anomaly, F, tin be derived as follows:

where,

Whenever is positive, F should be taken as positive; whenever is negative, F should be taken as negative.

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Hyperbolic Excess Velocity

If you give a space vehicle exactly escape velocity, it will just barely escape the gravitational field, which means that its velocity will be approaching zero every bit its altitude from the strength center approaches infinity. If, on the other hand, we requite our vehicle more escape velocity at a point near Earth, we would expect the velocity at a great distance from Earth to exist approaching some finite constant value. This residual velocity the vehicle would accept left over even at infinity is called hyperbolic excess velocity. Nosotros tin can calculate this velocity from the free energy equation written for two points on the hyperbolic escape trajectory – a point near Globe called the burnout indicate and a point at infinite distance from World where the velocity will be the hyperbolic excess velocity, five_∞ . Solving for v_∞ nosotros obtain

Annotation that if v_∞ = 0 (every bit it is on a parabolic trajectory), the exhaustion velocity, v_bo , becomes only the escape velocity.

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It is, of course, absurd to talk about a space vehicle "reaching infinity" and in this sense it is meaningless to talk about escaping a gravitational field completely. It is a fact, still, that once a space vehicle is a great distance from World, for all practical purposes it has escaped. In other words, it has already slowed down to very near its hyperbolic excess velocity. It is convenient to define a sphere around every gravitational body and say that when a probe crosses the edge of this sphere of influence it has escaped. Although it is difficult to become agreement on exactly where the sphere of influence should exist drawn, the concept is convenient and is widely used, especially in lunar and interplanetary trajectories. For most purposes, the radius of the sphere of influence for a planet tin exist calculated every bit follows:

where D_sp is the distance between the Sunday and the planet, M_p is the mass of the planet, and Chiliad_south is the mass of the Sun. Equation (4.89) is also valid for calculating a moon'south sphere of influence, where the moon is substituted for the planet and the planet for the Lord's day.

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Compiled, edited and written in part by Robert A. Braeunig, 1997, 2005, 2007, 2008, 2011, 2012, 2013.
Bibliography

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Source: http://www.braeunig.us/space/orbmech.htm

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