Orbital mechanics - Citizendia

Orbital mechanics or astrodynamics is the study of the motion of rockets and other spacecraft. A rocket or rocket vehicle is a Missile, Aircraft or other Vehicle which obtains Thrust by the reaction of the A spacecraft is a Vehicle or machine designed for Spaceflight. The motion of these objects is usually calculated from Newton's laws of motion and Newton's law of universal gravitation, collectively known as classical mechanics. Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the Newton 's law of universal Gravitation is a physical law describing the gravitational attraction between bodies with mass Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects

Celestial mechanics focuses more broadly on the orbital motions of artificial and natural astronomical bodies such as planets, moons, and comets. Celestial mechanics is the branch of Astrophysics that deals with the motions of Celestial objects The field applies principles of Physics, historically A planet, as defined by the International Astronomical Union (IAU is a celestial body Orbiting a Star or stellar remnant that is A comet is a small Solar System body that orbits the Sun and when close enough to the Sun exhibits a visible coma (atmosphere or a tail — Orbital mechanics is a subfield which focuses on spacecraft trajectories, including orbital maneuvers, orbit plane changes, and interplanetary transfers, and is used by mission planners to predict the results of propulsion. Trajectory is the path a moving object follows through space The object might be a Projectile or a Satellite, for example In Spaceflight, an orbital maneuver is the use of propulsion systems to change the Orbit of a Spacecraft. Spacecraft propulsion is any method used to change the velocity of Spacecraft and artificial Satellites There are many different methods

General relativity provides more exact equations for calculating orbits, sometimes necessary for greater accuracy or high-gravity situations (such as orbits close to the Sun). General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916

Rules of thumb

The following rules of thumb are useful for situations approximated by classical mechanics under the standard assumptions of astrodynamics. Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects For most of the problems in Astrodynamics involving two bodies m_1\ and m_2\ standard assumptions are usually the following A1 The specific example discussed is of a satellite orbiting a planet, but the rules of thumb could also apply to other situations, such as orbits of small bodies around a star such as the Sun.

Kepler's laws of planetary motion, which can be mathematically derived from Newton's laws, hold in the absence of thrust:
- Orbits are either circular, with the planet at the center of the circle, or form an ellipse, with the planet at one focus. In Astronomy, Kepler's Laws of Planetary Motion are three mathematical laws that describe the motion of Planets in the Solar System. In Mathematics, an ellipse (from the Greek ἔλλειψις literally absence) is a Conic section, the locus of points in a In Geometry, the foci (singular focus) are a pair of special points used in describing Conic sections The four types of conic sections are the Circle
- A line drawn from the planet to the satellite sweeps out equal areas in equal times no matter which portion of the orbit is measured.
- The square of a satellite's orbital period is proportional to the cube of its average distance from the planet.
Without firing a rocket engine (generating thrust), the height and shape of the satellite's orbit won't change, and it will maintain the same orientation with respect to the fixed stars. Thrust is a reaction force described quantitatively by Newton 's Second and Third Laws.
A satellite in a low orbit (or low part of an elliptical orbit) moves more quickly with respect to the surface of the planet than a satellite in a higher orbit (or a high part of an elliptical orbit), due to the stronger gravitational attraction closer to the planet.
If a brief rocket firing is made at only one point in the satellite's orbit, it will return to that same point on each subsequent orbit, though the rest of its path will change. Thus to move from one circular orbit to another, at least two brief firings are needed.
From a circular orbit, a brief firing of a rocket in the direction which slows the satellite down, will create an elliptical orbit with a lower perigee (lowest orbital point) at 180 degrees away from the firing point, which will be the apogee (highest orbital point). In Celestial mechanics, an apsis, plural apsides (ˈæpsɨdɪːz is the point of greatest or least distance of the Elliptical orbit of an object from If the rocket is fired to speed the rocket, it will create an elliptical orbit with a higher apogee 180 degrees away from the firing point (which will become the perigee).

The consequences of the rules of orbital mechanics are sometimes counter-intuitive. For example, if two spacecraft are in the same circular orbit and wish to dock, unless they are very close, the trailing craft cannot simply fire its engines to go faster. This will change the shape of its orbit, causing it to gain altitude and miss its target. One approach is to actually fire a reverse thrust to slow down, and then fire again to re-circularize the orbit at a lower altitude. Because lower orbits are faster than higher orbits, the trailing craft will begin to catch up. A third firing at the right time will put the trailing craft in an elliptical orbit which will intersect the path of the leading craft, approaching from below.

To the degree that the assumptions do not hold, actual trajectories will vary from those calculated. Atmospheric drag is one major complicating factor for objects in Earth orbit. In Fluid dynamics, drag (sometimes called fluid resistance) is the force that resists the movement of a Solid object through a Fluid (a The differences between classical mechanics and general relativity can become important for large objects like planets. Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 These rules of thumb are decidedly inaccurate when describing two or more bodies of similar mass, such as a binary star system. A binary star is a Star system consisting of two Stars orbiting around their Center of mass.

Laws of astrodynamics

The fundamental laws of astrodynamics are Newton's law of universal gravitation and Newton's laws of motion, while the fundamental mathematical tool is his differential calculus. Newton 's law of universal Gravitation is a physical law describing the gravitational attraction between bodies with mass Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives

Standard assumptions in astrodynamics include non-interference from outside bodies, negligible mass for one of the bodies, and negligible other forces (such as from the solar wind, atmospheric drag, etc. For most of the problems in Astrodynamics involving two bodies m_1\ and m_2\ standard assumptions are usually the following A1 ). More accurate calculations can be made without these simplifying assumptions, but they are more complicated. The increased accuracy often does not make enough of a difference in the calculation to be worthwhile.

Kepler's laws of planetary motion may be derived from Newton's laws, when it is assumed that the orbiting body is subject only to the gravitational force of the central attractor. In Astronomy, Kepler's Laws of Planetary Motion are three mathematical laws that describe the motion of Planets in the Solar System. When an engine thrust or propulsive force is present, Newton's laws still apply, but Kepler's laws are invalidated. When the thrust stops, the resulting orbit will be different but will once again be described by Kepler's laws.

Escape velocity

The formula for escape velocity is easily derived as follows. In Physics, escape velocity is the speed where the Kinetic energy of an object is equal to the magnitude of its Gravitational potential energy The specific energy (energy per unit mass) of any space vehicle is composed of two components, the specific potential energy and the specific kinetic energy. Specific energy is defined as the Energy per unit Mass: J/kg or in basic SI units m2/s2 Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object Potential energy can be thought of as Energy stored within a physical system The kinetic energy of an object is the extra Energy which it possesses due to its motion The specific potential energy associated with a planet of mass M is given by

$- G M / r \,$

while the specific kinetic energy of an object is given by

$v^2/2 \,$

Since energy is conserved, the total specific orbital energy

$v^2/2 - G M / r \,$

does not depend on the distance, $r$ , from the center of the central body to the space vehicle in question. Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object Specific kinetic energy is Kinetic energy per unit mass ( J /kg In Physics, the law of conservation of energy states that the total amount of Energy in an isolated system remains constant and cannot be created although it may In Astrodynamics the specific Orbital energy \epsilon\\! (or vis-viva energy) of an Orbiting body traveling through Space Therefore, the object can reach infinite $r$ only if this quantity is nonnegative, which implies

$v\geq\sqrt{2 G M / r}$

The escape velocity from the Earth's surface is about 11 km/s, but that is insufficient to send the body an infinite distance because of the gravitational pull of the Sun. To escape the solar system from the vicinity of the Earth requires around 42 km/s velocity, but there will be "part credit" for the Earth's orbital velocity for spacecraft launched from Earth, if their further acceleration (due to the propulsion system) carries them in the same direction as Earth travels in its orbit.

Formulae for free orbits

Orbits are conic sections, so, naturally, the formula for the distance of a body for a given angle corresponds to the formula for that curve in polar coordinates, which is:

$r = {a \over (1 + e \cos \theta) }$ . In Mathematics, a conic section (or just conic) is a Curve obtained by intersecting a cone (more precisely a circular Conical surface In Mathematics, the polar coordinate system is a two-dimensional Coordinate system in which each point on a plane is determined by

The parameters are given by the orbital elements. The elements of an orbit are the parameters needed to specify that Orbit uniquely given a model of two point-masses obeying the Newtonian laws of motion and the

Circular orbits

Although most orbits are elliptical in nature, a special case is the circular orbit, which is an ellipse of zero eccentricity. The formula for the velocity of a body in a circular orbit at distance r from the center of gravity of mass M is

$\ v = \sqrt{\frac{GM} {r}\ }$

where $G$ is the gravitational constant, equal to

6. The gravitational constant, denoted G, is a Physical constant involved in the calculation of the gravitational attraction between objects with mass 672 598 × 10⁻¹¹ m³/(kg·s²)

To properly use this formula, the units must be consistent; for example, M must be in kilograms, and r must be in meters. The answer will be in meters per second.

The quantity GM is often termed the standard gravitational parameter, which has a different value for every planet or moon in the solar system. Small body orbiting a central body Under Standard assumptions in astrodynamics we have m where m \ is the mass The Solar System consists of the Sun and those celestial objects bound to it by Gravity.

Once the circular orbital velocity is known, the escape velocity is easily found by multiplying by the square root of 2:

$\ v = \sqrt 2\sqrt{\frac {GM} {r}\ } = \sqrt{\frac {2GM} {r}\ }.$

Historical approaches

Until the rise of space travel in the twentieth century, there was little distinction between orbital and celestial mechanics. In Physics, escape velocity is the speed where the Kinetic energy of an object is equal to the magnitude of its Gravitational potential energy Spaceflight is the use of Space technology to fly a Spacecraft into and through Outer space. The twentieth century of the Common Era began on The fundamental techniques, such as those used to solve the Keplerian problem, are therefore the same in both fields. In Astronomy, Kepler's Laws of Planetary Motion are three mathematical laws that describe the motion of Planets in the Solar System. Furthermore, the history of the fields is almost entirely shared.

Kepler's equation

Kepler was the first to successfully model planetary orbits to a high degree of accuracy.

Derivation

To compute the position of a satellite at a given time (the Keplerian problem) is a difficult problem. The opposite problem—to compute the time-of-flight given the starting and ending positions—is simpler. We present a derivation for the time-of-flight equation here.

Kepler's construction for deriving the time-of-flight equation. The bold ellipse is the satellite's orbit, with the star or planet at one focus Q. The goal is to compute the time required for a satellite to travel from periapsis P to a given point S. In Celestial mechanics, an apsis, plural apsides (ˈæpsɨdɪːz is the point of greatest or least distance of the Elliptical orbit of an object from Kepler circumscribed the blue auxiliary circle around the ellipse, and used it to derive his time-of-flight equation in terms of eccentric anomaly. The eccentric anomaly is the angle between the direction of Periapsis and the current position of an object on its Orbit, projected onto the ellipse's circumscribing

The problem is to find the time $t$ at which the satellite reaches point $S$ , given that it is at periapsis $P$ at time $t = 0$ . In Celestial mechanics, an apsis, plural apsides (ˈæpsɨdɪːz is the point of greatest or least distance of the Elliptical orbit of an object from We are given that the semimajor axis of the orbit is $a$ , and the semiminor axis is $b$ ; the eccentricity is $e$ , and the planet is at $Q$ , at a distance of $a e$ from the center $C$ of the ellipse. In Geometry, the semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolae In Geometry, the semi-minor axis (also semiminor axis) is a Line segment associated with most Conic sections (that is with ellipses and In Astrodynamics, under standard assumptions, any Orbit must be of Conic section shape

The key construction that will allow us to analyse this situation is the auxiliary circle (shown in blue) circumscribed on the orbital ellipse. This circle is taller than the ellipse by a factor of $a / b$ in the direction of the minor axis, so all area measures on the circle are magnified by a factor of $a / b$ with respect to the analogous area measures on the ellipse.

Any given point on the ellipse can be mapped to the corresponding point on the circle that is $a / b$ further from the ellipse's major axis. If we do this mapping for the position $S$ of the satellite at time $t$ , we arrive at a point $R$ on the circumscribed circle. Kepler defines the angle $P C R$ to be the eccentric anomaly angle $E$ . The eccentric anomaly is the angle between the direction of Periapsis and the current position of an object on its Orbit, projected onto the ellipse's circumscribing (Kepler's terminology often refers to angles as "anomalies". ) This definition makes the time-of-flight equation easier to derive than it would be using the true anomaly angle $P Q S$ . In Astronomy, the true anomaly \nu\\! (Greek nu also written \theta\ or f\) is the angle between the direction z-s of

To compute the time-of-flight from this construction, we note that Kepler's second law allows us to compute time-of-flight from the area swept out by the satellite, and so we will set about computing the area $P Q S$ swept out by the satellite. In Astronomy, Kepler's Laws of Planetary Motion are three mathematical laws that describe the motion of Planets in the Solar System.

First, the area $P Q R$ is a magnified version of the area $P Q S$ :

$PQR = \frac{a}{b} PQS$

Furthermore, area $P Q S$ is the area swept out by the satellite in time $t$ . We know that, in one orbital period $T$ , the satellite sweeps out the whole area $π a b$ of the orbital ellipse. $P Q S$ is the $t / T$ fraction of this area, and substituting, we arrive at this expression for $P Q R$ :

$PQR = \frac{t}{T} \pi a^2$

Second, the area $P Q R$ is also formed by removing area $Q C R$ from $P C R$ :

$PQR = PCR - QCR \;$

Area $P C R$ is a fraction of the circumscribed circle, whose total area is $π a 2$ . The fraction is $E / 2π$ , thus:

$PCR = \frac{a^2}{2}E$

Meanwhile, area $Q C R$ is a triangle whose base is the line segment $Q C$ of length $a e$ , and whose height is $a sin E$ :

$QCR = \frac{a^2}{2} e \sin E$

Combining all of the above:

$PQR = \frac{t}{T} \pi a^2 = \frac{a^2}{2}E - \frac{a^2}{2} e \sin E$

Dividing through by $a 2 / 2$ :

$\frac{2 \pi}{T}t = E - e \sin E$

To understand the significance of this formula, consider an analogous formula giving an angle $M$ during circular motion with constant angular velocity $n$ :

$nt = M \;$

Setting $n = 2π / T$ and $M = E - e sin E$ gives us Kepler's equation. In Geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its end points Kepler referred to $n$ as the mean motion, and $E - e sin E$ as the mean anomaly. The term "mean" in this case refers to the fact that we have "averaged" the satellite's non-constant angular velocity over an entire period to make the satellite's motion amenable to analysis. All satellites traverse an angle of $2π$ per orbital period $T$ , so the mean angular velocity is always $2π / T$ .

Substituting $n$ into the formula we derived above gives this:

$nt = E - e \sin E \;$

This formula is commonly referred to as Kepler's equation.

Application

With Kepler's formula, finding the time-of-flight to reach an angle (true anomaly) of $θ$ from periapsis is broken into two steps:

Compute the eccentric anomaly $E$ from true anomaly $θ$
Compute the time-of-flight $t$ from the eccentric anomaly $E$

Finding the angle at a given time is harder. Kepler's equation is transcendental in $E$ , meaning it cannot be solved for $E$ analytically, and so numerical approaches must be used. A transcendental function is a function that does not satisfy a Polynomial equation whose Coefficients are themselves polynomials in contrast to an In effect, one must guess a value of $E$ and solve for time-of-flight; then adjust $E$ as necessary to bring the computed time-of-flight closer to the desired value until the required precision is achieved. Usually, Newton's method is used to achieve relatively fast convergence. In Numerical analysis, Newton's method (also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson) is perhaps the

The main difficulty with this approach is that it can take prohibitively long to converge for the extreme elliptical orbits. For near-parabolic orbits, eccentricity $e$ is nearly 1, and plugging $e = 1$ into the formula for mean anomaly, $E - sin E$ , we find ourselves subtracting two nearly-equal values, and so accuracy suffers. For near-circular orbits, it is hard to find the periapsis in the first place (and truly circular orbits have no periapsis at all). Furthermore, the equation was derived on the assumption of an elliptical orbit, and so it does not hold for parabolic or hyperbolic orbits at all. These difficulties are what led to the development of the universal variable formulation, described below.

Perturbation theory

One can deal with perturbations just by summing the forces and integrating, but that is not always best. Historically, variation of parameters has been used which is easier to mathematically apply with when perturbations are small.

Practical techniques

Further information: List of orbits

Transfer orbits

Transfer orbits allow spacecraft to move from one orbit to another. The following is a list of types of orbits Centric classifications Galactocentric orbit: An orbit about the center of a Galaxy Usually they require a burn at the start, a burn at the end, and sometimes one or more burns in the middle. The Hohmann transfer orbit typically requires the least delta-v, but any orbit that intersects both the origin orbit and destination orbit may be used. In Astronautics and Aerospace engineering, the Hohmann transfer orbit is an Orbital maneuver using two engine impulses which under standard assumptions In Astrodynamics, the term delta-v, literally "change in velocity" (see symbol delta) has a specific meaning it is a Scalar which takes

Gravity assist and the Oberth effect

In a gravity assist, a spacecraft swings by a planet and leaves in a different direction, at a different velocity. In Orbital mechanics and Aerospace engineering, a gravitational slingshot, gravity assist or swing-by is the use of the relative movement and This is useful to speed or slow a spacecraft instead of carrying more fuel.

This maneuver can be approximated by an elastic collision at large distances, though the flyby does not involve any physical contact. elastic collision is a collision in which the total Kinetic energy of the colliding bodies after collision is equal to their total kinetic energy before collision Due to Newton's Third Law (equal and opposite reaction), any momentum gained by a spacecraft must be lost by the planet, or vice versa. However, because the planet is much, much more massive than the spacecraft, the effect on the planet's orbit is negligible.

The Oberth effect can be employed, particularly during a gravity assist operation. The Oberth effect is a feature of Astronautics where using a Rocket engine close to a gravitational body gives a higher final speed than the same burn executed further This effect is that use of a propulsion system works better at high speeds, and hence course changes are best done when close to a gravitating body; this can multiply the effective delta-v. In Astrodynamics, the term delta-v, literally "change in velocity" (see symbol delta) has a specific meaning it is a Scalar which takes

Interplanetary Transport Network and fuzzy orbits

Main article: Interplanetary Transport Network

See also: Low energy transfers

It is now possible to use computers to search for routes using the nonlinearities in the gravity of the planets and moons of the solar system. The Interplanetary Transport Network (ITN is a collection of Gravitationally determined pathways through the Solar system that require very little Energy Low energy transfers, or low energy trajectories, are routes in space which allow spacecraft to change orbits using very little fuel For example, it is possible to plot an orbit from high earth orbit to Mars, passing close to one of the Earth's Trojan points. Collectively referred to as the Interplanetary Transport Network, these highly perturbative, even chaotic, orbital trajectories in principle need no fuel (in practice keeping to the trajectory requires some course corrections). The Interplanetary Transport Network (ITN is a collection of Gravitationally determined pathways through the Solar system that require very little Energy The biggest problem with them is they are usually exceedingly slow, taking many years to arrive. In addition launch windows can be very far apart.

They have, however, been employed on projects such as Genesis. The Genesis spacecraft was the first ever attempt to collect a sample of Solar wind, and the first " Sample return mission " to return from beyond the This spacecraft visited Earth's lagrange L1 point and returned using very little propellant.

Modern mathematical techniques

Conic orbits

For simple things like computing the delta-v for coplanar transfer ellipses, traditional approaches work pretty well. But time-of-flight is harder, especially for near-circular and hyperbolic orbits.

The patched conic approximation

The transfer orbit alone is not a good approximation for interplanetary trajectories because it neglects the planets' own gravity. Planetary gravity dominates the behaviour of the spacecraft in the vicinity of a planet, so it severely underestimates delta-v, and produces highly inaccurate prescriptions for burn timings.

One relatively simple way to get a first-order approximation of delta-v is based on the patched conic approximation technique. Orders of approximation have been used not only in Science, Engineering, and other quantitative disciplines to make Approximations with various degrees The idea is to choose the one dominant gravitating body in each region of space through which the trajectory will pass, and to model only that body's effects in that region. For instance, on a trajectory from the Earth to Mars, one would begin by considering only the Earth's gravity until the trajectory reaches a distance where the Earth's gravity no longer dominates that of the Sun. EARTH was a short-lived Japanese vocal trio which released 6 singles and 1 album between 2000 and 2001 The Sun (Sol is the Star at the center of the Solar System. The spacecraft would be given escape velocity to send it on its way to interplanetary space. In Physics, escape velocity is the speed where the Kinetic energy of an object is equal to the magnitude of its Gravitational potential energy Next, one would consider only the Sun's gravity until the trajectory reaches the neighbourhood of Mars. During this stage, the transfer orbit model is appropriate. Finally, only Mars's gravity is considered during the final portion of the trajectory where Mars's gravity dominates the spacecraft's behaviour. The spacecraft would approach Mars on a hyperbolic orbit, and a final retrograde burn would slow the spacecraft enough to be captured by Mars.

The size of the "neighborhoods" (or spheres of influence) vary with radius $r S O I$ :

$r_{SOI} = a_p\left(\frac{m_p}{m_s}\right)^{2/5}$

where $a p$ is the semimajor axis of the planet's orbit relative to the Sun; $m p$ and $m s$ are the masses of the planet and Sun, respectively. A sphere of influence (SOI in Astrodynamics and Astronomy is the spherical region around a Celestial body where the primary gravitational In Geometry, the semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolae The Sun (Sol is the Star at the center of the Solar System. Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object

This simplification is sufficient to compute rough estimates of fuel requirements, and rough time-of-flight estimates, but it is not generally accurate enough to guide a spacecraft to its destination. For that, numerical methods are required.

The universal variable formulation

To address the shortcomings of the traditional approaches, the universal variable approach was developed. It works equally well on circular, elliptical, parabolic, and hyperbolic orbits; and also works well with perturbation theory. The differential equations converge nicely when integrated for any orbit.

Perturbations

The universal variable formulation works well with the variation of parameters technique, except now, instead of the six Keplerian orbital elements, we use a different set of orbital elements: namely, the satellite's initial position and velocity vectors $x 0$ and $v 0$ at a given epoch $t = 0$ . In a two-body simulation, these elements are sufficient to compute the satellite's position and velocity at any time in the future, using the universal variable formulation. Conversely, at any moment in the satellite's orbit, we can measure its position and velocity, and then use the universal variable approach to determine what its initial position and velocity would have been at the epoch. In perfect two-body motion, these orbital elements would be invariant (just like the Keplerian elements would be).

However, perturbations cause the orbital elements to change over time. Hence, we write the position element as $x 0 (t)$ and the velocity element as $v 0 (t)$ , indicating that they vary with time. The technique to compute the effect of perturbations becomes one of finding expressions, either exact or approximate, for the functions $x 0 (t)$ and $v 0 (t)$ .

Non-ideal orbits

The following are some effects which make real orbits differ from the simple models based on a spherical earth. Most of them can be handled on short timescales (perhaps less than a few thousand orbits) by perturbation theory because they are small relative to the corresponding two-body effects.

Equatorial bulges cause precession of the node and the perigee
Tesseral harmonics [1] of the gravity field introduce additional perturbations
lunar and solar gravity perturbations alter the orbits
Atmospheric drag reduces the semi-major axis unless make-up thrust is used

Over very long timescales (perhaps millions of orbits), even small perturbations can dominate, and the behaviour can become chaotic. Precession refers to a change in the direction of the axis of a rotating object Chaos (derived from the Ancient Greek, Chaos) typically refers to Unpredictability, and is the antithesis of Cosmos. On the other hand, the various perturbations can be orchestrated by clever astrodynamicists to assist with orbit maintenance tasks, such as station-keeping, ground track maintenance or adjustment, or phasing of perigee to cover selected targets at low altitude. A ground track or ground trace is the projection of a Satellite 's Orbit onto the surface of the Earth (or whatever body the satellite is orbiting

References

Bate, Roger R. Spacecraft propulsion is any method used to change the velocity of Spacecraft and artificial Satellites There are many different methods Tsiolkovsky's rocket equation, or ideal rocket equation is named after Konstantin Tsiolkovsky, who independently derived it and published in his 1903 work considers Astrophysics is the branch of Astronomy that deals with the Physics of the Universe, including the physical properties ( Luminosity, Celestial mechanics is the branch of Astrophysics that deals with the motions of Celestial objects The field applies principles of Physics, historically In Mathematics, chaos theory describes the behavior of certain dynamical systems – that is systems whose state evolves with time – that may exhibit dynamics that The n -body problem is the problem of finding given the initial positions masses and velocities of n bodies their subsequent motions as determined by In Physics, an orbit is the gravitationally curved path of one object around a point or another body for example the gravitational orbit of a planet around a star Determining the Roche limit The Roche limit depends on the rigidity of the satellite In Astrodynamics, A canonical unit is a Unit of measurement defined in terms of an object's reference Orbit. ; Mueller, Donald D. , and White, Jerry E. (1971). Fundamentals of Astrodynamics. Dover Publications. ISBN 0-486-60061-0.

Sellers, Jerry J. ; Astore, William J. , Giffen, Robert B. , Larson, Wiley J. (2004). in Kirkpatrick, Douglas H. : Understanding Space: An Introduction to Astronautics, 2, McGraw Hill, 228. ISBN 0072424680.

Air University Space Primer, Chapter 8 - Orbital Mechanics. USAF.

External links

ORBITAL MECHANICS (Rocket and Space Technology)
Java Astrodynamics Toolkit

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