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Encyclopedia > Astrodynamics

Astrodynamics is the study of the motion of rockets, missiles, and space vehicles, as determined from Sir Isaac Newton's laws of motion and his law of universal gravitation. It is a specific and distinct branch of celestial mechanics, which focuses more broadly on Newtonian gravitation and includes the orbital motions of artificial and natural astronomical bodies such as planets, moons, and comets. Astrodynamics is principally concerned with spacecraft trajectories, from launch to atmospheric re-entry, including all orbital maneuvers, orbit plane changes, and interplanetary transfers. For a less technical treatment, see the article on space mathematics. A Redstone rocket, part of the Mercury program The traditional definition of a rocket is a vehicle, missile or aircraft which obtains thrust by the reaction to the ejection of fast moving exhaust gas from within a rocket engine. ... A missile (CE pronunciation: ; AmE: ) is, in general, a projectileâ€”that is, something thrown or otherwise propelled. ... Sir Isaac Newton, President of the Royal Society, (4 January 1643 â€“ 31 March 1727) [OS: 25 December 1642 â€“ 20 March 1727] was an English mathematician, physicist, astronomer, alchemist, chemist, inventor, and natural philosopher who is generally regarded as one of the most influential scientists and mathematicians in history. ... Newtons First and Second laws, in Latin, from the original 1687 edition of the Principia Mathematica. ... This article covers the physics of gravitation. ... Celestial mechanics is a division of astronomy dealing with the motions and gravitational effects of celestial objects. ... A planet is generally considered to be a relatively large mass of accreted matter in orbit around a star. ... Bulk composition of the moons mantle and crust estimated, weight percent Oxygen 42. ... Comet Hale-Bopp For other uses, see Comet (disambiguation). ... A spacecraft is designed to leave Earths atmosphere and operate beyond the surface of the Earth in outer space. ... A trajectory is an imagined trace of positions followed by an object moving through space. ... The purpose of this article is to introduce non-scientists, non-engineers, and other laymen to the basic mathematics of space exploration. ...

## Laws of astrodynamics GA_googleFillSlot("encyclopedia_square");

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. Kepler's laws of planetary motion may be derived from these laws, when it is assumed that the orbiting body is subject only to the gravitational force of the central attractor. When an engine thrust or propulsive force is present, Newton's second law of motion applies, and Kepler's laws are temporarily invalidated. Gravity is a force of attraction that acts between bodies that have mass. ... Newtons First and Second laws, in Latin, from the original 1687 edition of the Principia Mathematica. ... Calculus is a central branch of mathematics, developed from algebra and geometry. ... Johannes Keplers primary contributions to astronomy/astrophysics were his three laws of planetary motion. ...

The formula for escape velocity is easily derived as follows. 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. The specific potential energy associated with a planet of mass M is given by In physics, for a given gravitational field and a given position, the escape velocity is the minimum speed an object without propulsion, at that position, needs to have to move away indefinitely from the source of the field, as opposed to falling back or staying in an orbit within a... The fuel value or relative energy density is the quantity of potential energy in fuel, food or other substance. ... Mass is a property of a physical object that quantifies the amount of matter and energy it contains. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Kinetic jkljfkdffmdklcjenergy (SI unit: the [[klof its motion. ... Mass is a property of a physical object that quantifies the amount of matter and energy it contains. ...

$- G M / r ,$

while the specific kinetic energy of an object is given by Specific kinetic energy is kinetic energy per unit mass (J/kg). ...

$v^2/2 ,$

Since energy is conserved, the total specific orbital energy For the physical concepts, see conservation of energy and energy efficiency. ... In astrodynamics the specific orbital energy (or vis-viva energy) of an orbiting body traveling through space under standard assumptions is the sum of its potential energy () and kinetic energy () per unit mass. ...

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

does not depend on the distance, r, from the center of the central body to the space vehicle in question. Therefore, the object can reach infinite r only if this quantity is nonnegative, which implies

$vgeqsqrt{2 G M / r}$

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 ellipse

Orbits are ellipses, so, naturally, the formula for the distance of a body for a given angle corresponds to the formula for an ellipse in polar coordinates. The parameters of the ellipse are given by the orbital elements. The ellipse and some of its mathematical properties. ... This article describes some of the common coordinate systems that appear in elementary mathematics. ... The elements of an orbit are the parameters needed to specify that orbit uniquely, given a model of two ideal masses obeying the Newtonian laws of motion and the inverse-square law of gravitational attraction. ...

## Historical approaches

Until the rise of space travel in the twentieth century, there was little distinction between astrodynamics and celestial mechanics. The fundamental techniques, such as those used to solve the Keplerian problem, are therefore the same in both fields. Furthermore, the history of the fields is essentially identical. Space exploration is the physical exploration of outer-Earth objects and generally anything that involves the technologies, science, and politics regarding space endeavors. ... (19th century - 20th century - 21st century - more centuries) Decades: 1900s 1910s 1920s 1930s 1940s 1950s 1960s 1970s 1980s 1990s The 20th century lasted from 1901 to 2000 in the Gregorian calendar (often from (1900 to 1999 in common usage). ... To compute the position of a satellite at a given time using Keplers laws of planetary motion (the Keplerian problem) is a difficult problem. ...

### 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. 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 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. 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 ae from the center C of the ellipse. Diagram for derivation of Keplers equation for time-of-flight. ... This article is about several astronomical terms (apogee & perigee, aphelion & perihelion, generic equivalents based on apsis, and related but rarer terms. ... The eccentric anomaly is the angle between the direction of periapsis and the current position of an object on its orbit, projected onto the ellipses circumscribing circle perpendicularly to the major axis, measured at the centre of the ellipse. ... This article is about several astronomical terms (apogee & perigee, aphelion & perihelion, generic equivalents based on apsis, and related but rarer terms. ... In geometry, the semi-major axis (also semimajor axis) a applies to ellipses and hyperbolas. ... In geometry, the semi-minor axis (also semiminor axis) applies to ellipses and hyperbolas. ... 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 PCR to be the eccentric anomaly angle E. (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 PQS. The eccentric anomaly is the angle between the direction of periapsis and the current position of an object on its orbit, projected onto the ellipses circumscribing circle perpendicularly to the major axis, measured at the centre of the ellipse. ... In astronomy, the true anomaly (, also written ) is the angle between the direction z-s of periapsis and the current position p of an object on its orbit, measured at the focus s of the ellipse (the point around which the object orbits). ...

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 PQS swept out by the satellite. Johannes Keplers primary contributions to astronomy/astrophysics were his three laws of planetary motion. ...

First, the area PQR is a magnified version of the area PQS:

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

Furthermore, area PQS is the area swept out by the satellite in time T. We know that, in one orbital period τ, the satellite sweeps out the whole area πab of the orbital ellipse. PQS is the T / τ fraction of this area, and substituting, we arrive at this expression for PQR:

Second, the area PQR is also formed by removing area QCR from PCR:

$PQR = PCR - QCR ;$

Area PCR is a fraction of the circumscribed circle, whose total area is πa2. The fraction is E / 2π, thus:

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

Meanwhile, area QCR is a triangle whose base is the line segment QC of length ae, and whose height is asinE: In mathematics, a line segment is a part of a line that is bounded by two end points. ...

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

Combining all of the above:

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

Dividing through by a2 / 2:

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

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

$MT = theta ;$

Setting M = 2π / τ and θ = EesinE gives us Kepler's equation. Kepler referred to M as the mean motion, and EesinE 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 per orbital period τ, so the mean angular velocity is always 2π / τ.

Substituting M into the formula we derived above gives this:

$MT = E - e sin E ;$

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:

1. Compute the eccentric anomaly E from true anomaly θ
2. 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. 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. Transcendental in philosophical contexts In philosophy, transcendental experiences are experiences of an exclusively human nature that are other-worldly or beyond the human realm of understanding. ... In numerical analysis, Newtons method (or the Newton-Raphson method) is an efficient algorithm for finding approximations to the zeros (or roots) of a real-valued function. ...

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 − sinE, 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

You 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.

## Modern techniques

Today, we do not use the same techniques that Kepler used, in general.

### 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.

### Transfer orbits

Transfer orbits get you from one orbit to another. Usually they require a burn at the start, a burn at the end, and sometimes a burn in the middle. The Hohmann transfer orbit typically requires the least delta-v, but any orbit that intersects both your origin and destination will work. In astronautics and aerospace engineering, the Hohmann transfer orbit is an orbital maneuver that moves a spacecraft from one orbit to another using the lowest possible delta-v for the specific transfer. ...

### 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. 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. The spacecraft would be given escape velocity to send it on its way to interplanetary space. 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' gravity is considered during the final portion of the trajectory where Mars' 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. Often in science, engineering, or other quantitative disciplines, it is necessary to make approximations with various degrees of precision. ... Earth (often referred to as The Earth) is the third planet in the solar system in terms of distance from the Sun, and the fifth in order of size. ... Mars is the fourth planet from the Sun in the solar system, named after the Roman god of war (the counterpart of the Greek Ares), on account of its blood red color as viewed in the night sky. ... The Sun is the star at the center of Earths solar system. ... In physics, for a given gravitational field and a given position, the escape velocity is the minimum speed an object without propulsion, at that position, needs to have to move away indefinitely from the source of the field, as opposed to falling back or staying in an orbit within a...

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 x0 and v0 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 x0(t) and the velocity element as v0(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 x0(t) and v0(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. 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 Precession refers to a change in the direction of the axis of a rotating object. ... This article or section is missing references or citation of sources. ...

Many of the options, procedures, and supporting theory are covered in standard works such as:

1. Bate, R.R., Mueller, D.D., White, J.E., Fundamentals of Astrodynamics, Dover Publications, New York, 1971.

2. Vallado, D. A., Fundamentals of Astrodynamics and Applications, 2nd Edition, McGraw-Hill, 2001

3. Battin, R.H., An Introduction to the Mathematics and Methods of Astrodynamics, New York, 1987.

4. Chobotov, V.A. (ed.), Orbital Mechanics, 3rd Edition, AIAA, Washington, DC, 2002.

5. Herrick, S. Astrodynamics, Van Nostrand Reinhold, London, 1971 (two volumes).

6. Kaplan, M.H., Modern Spacecraft Dynamics and Controls, Wiley, New York, 1976.

7. Logsdon, T., Orbital Mechanics, Wiley-Interscience, New York, 1997.

8. Prussing, J.E., and B.A. Conway, Orbital Mechanics, Oxford University Press, New York, 1993.

9. Sidi, M.J., Spacecraft Dynamics and Control, Cambridge University Press, New York, 1997.

10. Wiesel, W.E., Spaceflight Dynamics, McGraw-Hill, New York, 1996, 2nd edition.

11. Vinti, J.P., Orbital and Celestial Mechanics, AIAA, Reston, VA, 1998.

or, on line:

[2] and [3]

The most elementary but very widely used reference is Bate, Mueller and White. It has several useful graphs off which one can read the rates of change of perigee and node due to earth oblateness, but there are typographical errors in a few equations. For example, in Eq. (9.7.5) the term in (3/2) J2 needs (re/r) squared and the term in J3 needs it cubed. The coefficient 315 in the J6 term, Eq.(9.7.6.) should be 245 (but the 315 in the J5 term is just fine). Battin's book may be too mathematical for many users.

### Interplanetary Transport Network and fuzzy orbits

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. 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 biggest problem with them is they are usually exceedingly slow, taking many years to arrive. In addition launch windows can be very far apart. In celestial mechanics, the Lagrangian points, (also Lagrange point, L-point, or libration point) are the five stationary solutions of the circular restricted three-body problem. ... Artists concept of the Interplanetary Transport Network. ...

They have, however, been employed on projects such as Genesis. This spacecraft visited Earth's lagrange L1 point and returned using very little propellant. In its collecting configuration, the Genesis spacecraft exposed collecting wafers to the solar wind. ...

## See also

A remote camera captures a close-up view of a Space Shuttle Main Engine during a test firing at the John C. Stennis Space Center in Hancock County, Mississippi Spacecraft propulsion is used to change the velocity of spacecraft and artificial satellites, or in short, to provide delta-v. ... Tsiolkovskys rocket equation, named after Konstantin Tsiolkovsky who independently derived it, considers the principle of a rocket: a device that can apply an acceleration to itself (a thrust) by expelling part of its mass with high speed in the opposite direction, due to the conservation of momentum. ... This article is about the branch of Physics. ... Spiral Galaxy ESO 269-57 // Astrophysics is the branch of astronomy that deals with the physics of the universe, including the physical properties (luminosity, density, temperature and chemical composition) of astronomical objects such as stars, galaxies, and the interstellar medium, as well as their interactions. ... Celestial mechanics is a division of astronomy dealing with the motions and gravitational effects of celestial objects. ... A plot of the trajectory Lorenz system for values r = 28, Ïƒ = 10, b = 8/3 In mathematics and physics, chaos theory describes the behavior of certain nonlinear dynamical systems that under certain conditions exhibit a phenomenon known as chaos. ... A contour plot of the effective potential of a two-body system (the Sun and Earth here), showing the 5 Lagrange points. ... 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 classical mechanics, i. ... In physics, an orbit is the path that an object makes, around another object, whilst under the influence of a source of centripetal force, such as gravity. ... The Roche limit is the distance within which a celestial body held together only by its own gravity will disintegrate due to a second celestial bodys tidal forces exceeding the first bodys gravitational self-attraction. ...

## Reference

• Bate, Roger R., Mueller, Donald D., and White, Jerry E. (1971). Fundamentals of Astrodynamics. Dover Publications. ISBN 0-48-660061-0

Results from FactBites:

 Astrodynamics - Wikipedia, the free encyclopedia (2415 words) Astrodynamics is the study of the motion of rockets, missiles, and space vehicles, as determined from Sir Isaac Newton's laws of motion and his law of universal gravitation. Astrodynamics is principally concerned with spacecraft trajectories, from launch to atmospheric re-entry, including all orbital maneuvers, orbit plane changes, and interplanetary transfers. 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.
 The Infography about Astrodynamics (170 words) The following sources are recommended by a professor whose research specialty is astrodynamics. R.H. Battin, "An Introduction to the Mathematics and Methods of Astrodynamics," AIAA Education Series, 1987. R.R. Bate, D.D. Mueller, and J.E. White, "Fundamentals of Astrodynamics," Dover, 1971.
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