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Keplerian Problem

Keplerian problem

To compute the position of a satellite at a given time using Kepler's laws of planetary motion (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, also called Kepler's equation here. Kepler's laws of planetary motion.]] The problem is as follows: 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. The satellite is at periapsis

P

at time

t = 0

. The goal is to find the time

T

at which the satellite reaches point

S

. The key construction that will allow us to analyse this situation is the 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

. 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. First, the area

PQR

is a magnified version of the area

PQS

: :

PQR = \frac PQS

Furthermore, area

PQS

is the area swept out by the satellite in time

T

. We know that, in one orbital period

\tau

, the satellite sweeps out the whole area

\pi a b

of the orbital ellipse.

PQS

is the

T / \tau

fraction of this area, and substituting, we arrive at this expression for

PQR

: :

PQR = \frac \pi a^2

Another expression for

PQR

is found by a simple conglomeration of adjacent areas: :

PQR = PCR - QCR \;

Area

PCR

is a fraction of the circumscribed circle, whose total area is

\pi a^2

. The fraction is

E / 2 \pi

, thus: :

PCR = \fracE

Meanwhile, area

QCR

is a triangle whose base is the line segment

QC

of length

ae

, and whose height is

a \sin E

: :

QCR = \frac e \sin E

Combining all of the above: :

PQR = \frac \pi a^2
 = \fracE - \frac e \sin E

Dividing through by

a^2 / 2

: :

\fracT = E - e \sin E

To understand the significance of this formula, consider an analogous formula giving an angle

\theta

during circular motion with constant angular velocity

M

: :

MT = \theta \;

Setting

M = 2 \pi / \tau

and

\theta = E - e \sin E

gives us Kepler's equation. Kepler referred to

M

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

2 \pi

per orbital period

\tau

, so the mean angular velocity is always

2 \pi / \tau

. 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

\theta

from periapsis is broken into two steps: # Compute the eccentric anomaly

E

from true anomaly

\theta

# 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. 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 doesn't hold for parabolic or hyperbolic orbits at all. These difficulties are what led to the development of the universal variable formulation. Category:Celestial mechanics Category:Astrodynamics

Satellite

A satellite is any object that orbits another object (which is known as its primary). All masses that are part of the solar system, including the Earth, are satellites either of the Sun, or satellites of those objects, such as the Moon. It is not always a simple matter to decide which is the 'satellite' in a pair of bodies. Because all objects exert gravity, the motion of the primary object is also affected by the satellite. If two objects are sufficiently similar in mass, they are generally referred to as a binary system rather than a primary object and satellite; an extreme example is the 'double asteroid' 90 Antiope. The general criterion for an object to be a satellite is that the center of mass of the two objects is inside the primary object. In popular usage, the term 'satellite' normally refers to an artificial satellite (a man-made object that orbits the Earth or another body). However, scientists may also use the term to refer to natural satellites, or moons. This article is primarily concerned with artificial satellites. See natural satellite for information on moons.

Artificial satellites

History of artificial satellites

natural satellite In May, 1946, the Preliminary Design of an Experimental World-Circling Spaceship stated, "A satellite vehicle with appropriate instrumentation can be expected to be one of the most potent scientific tools of the Twentieth Century. The achievement of a satellite craft would produce repercussions comparable to the explosion of the atomic bomb..." (see: Project RAND) The space age began in 1946, as scientists began using captured German V-2 rockets to make measurements in the upper atmosphere. Before this period, scientists used balloons that went up to 30 km and radio waves to study the ionosphere. From 1946 to 1952, upper-atmosphere research was conducted using V-2s and Aerobee rockets. This allowed measurements of atmospheric pressure, density, and temperature up to 200 km. (see also: magnetosphere, Van Allen radiation belt) The U.S. had been considering launching orbital satellites since 1945 under the Bureau of Aeronautics of the United States Navy. The Air Force's Project RAND eventually released the above report, but did not believe that the satellite was a potential military weapon; rather they considered it to be a tool for science, politics, and propaganda. In 1954, the Secretary of Defense stated, "I know of no American satellite program." Following pressure by the American Rocket Society, the National Science Foundation, and the International Geophysical Year, military interest picked up and in early 1955 the Air Force and Navy were working on Project Orbiter, which involved using a Jupiter C rocket to launch a small satellite called Explorer 1 on January 31, 1958. On July 29, 1955, the White House announced that the U.S. intended to launch satellites by the spring of 1958. This became known as Project Vanguard. On July 31, the Soviets announced that they intended to launch a satellite by the fall of 1957 and on October 4, 1957 Sputnik I was launched into orbit, which triggered the Space Race between the two nations. The largest artificial satellite currently orbiting the earth is the International Space Station, which can sometimes be seen with the unaided human eye.

Types of satellites

Astronomical satellites are satellites used for observation of distant planets, galaxies, and other outer space objects. Communications satellites are artificial satellites stationed in space for the purposes of telecommunications using radio at microwave frequencies. Most communications satellites use geosynchronous orbits or near-geostationary orbits, although some recent systems use low Earth-orbiting satellites. Earth observation satellites are satellites specifically designed to observe Earth from orbit, similar to reconnaissance satellites but intended for non-military uses such as environmental monitoring, meteorology, map making etc. (See especially Earth Observing System.) Navigation satellites are satellites which use radio time signals transmitted to enable mobile receivers on the ground to determine their exact location. The relatively clear line of sight between the satellites and receivers on the ground, combined with ever-improving electronics, allows satellite navigation systems to measure location to accuracies on the order of a few metres in real time. Reconnaissance satellites are Earth observation satellite or communications satellite deployed for military or intelligence applications. Little is known about the full power of these satellites, as governments who operate them usually keep information pertaining to their reconnaissance satellites classified. Solar power satellites are proposed satellites built in high Earth orbit that use microwave power transmission to beam solar power to very large antenna on Earth where it can be used in place of conventional power sources. Space stations are man-made structures that are designed for human beings to live on in outer space. A space station is distinguished from other manned spacecraft by its lack of major propulsion or landing facilities — instead, other vehicles are used as transport to and from the station. Space stations are designed for medium-term living in orbit, for periods of weeks, months, or even years. Weather satellites are satellites that primarily are used to monitor the weather and/or climate of the Earth. Miniaturized satellites are satellites of unusually low weights and small sizes. New classifications are used to categorize these satellites: minisatellite (500–200 kg), microsatellite (below 200 kg), nanosatellite (below 10 kg).

Orbit types

Many times satellites are characterized by their orbit. Although a satellite may orbit at almost any height, satellites are commonly categorized by their altitude:
- Low Earth Orbit (LEO: 200 - 1200km above the Earth's surface)
- Medium Earth Orbit (ICO or MEO: 1200 - 35286 km)
- Geosynchronous Orbit (GEO: 35786 km above Earth's surface)
- Geostationary Orbit (GSO: zero inclination geosynchronous orbit)
- High Earth Orbit (HEO: above 35786 km) The following orbits are special orbits that are also used to categorize satellites:
- Molniya orbits
- Heliosynchronous or sun-synchronous orbit
- Polar orbit
- LTO lunar transfer orbit
- Hohmann transfer orbit For this particular orbit type, it is more common to identify the satellite as a spacecraft.
- Supersynchronous orbit or drift orbit - orbit above GEO. Satellites will drift in a westerly direction.
- (GEO + 235 km + (1000 × CR × A/m) km)
- where CR is the solar pressure radiation coefficient (typically between 1.2 and 1.5) and A/m is the aspect area [m²] to dry mass [kg] ratio
- Subsynchronous orbit or drift orbit - orbits close to but below GEO. Used for satellites undergoing station changes in an eastern direction. Satellites can also orbit libration points.

Countries with satellite launch capability

This list includes counties with an independent capability to place satellites in orbit, including production of the necessary launch vehicle. Many more countries have built satellites that were launched with the aid of others. The French and British capabilities are now subsumed by the European Union under the European Space Agency. In 1998, North Korea claimed to have launched a satellite, but this was never confirmed, and widely believed to be a cover for the test launch of the Taepodong-1 missile over Japan (See Kwangmyongsong).

Reference

External links

- [http://science.nasa.gov/Realtime/JPass/20/ J-Pass] NASA site for satellite-watching
- [http://www.stoff.pl Orbitron - Satellite Tracking System] Free satellite tracking software
- [http://ilectric.com/glance/Recreation/Radio/Amateur/Satellite_Tracking/ Satellite Tracking in Recreation Radio Amateur] an excellent link to many links
- [http://www.oosa.unvienna.org UN Office for Outer Space Affairs] ensures all countries benefit from satellites
- [http://www.satellite-service-providers.com/ Satellite Service Providers] Compare and review on top satellite tv, radio and internet service providers] Category:Satellites Category:Unmanned vehicles ko:인공 위성 ja:人工衛星

Kepler's laws of planetary motion

Johannes Kepler's primary contributions to astronomy/astrophysics were his three laws of planetary motion. Kepler, a nearly blind though brilliant German mathematician, derived these laws, in part, by studying the observations of the keen-sighted Danish astronomer Tycho Brahe. The article on Johannes Kepler gives a less mathematical description of the laws, as well as a treatment of their historical and intellectual context. Sir Isaac Newton's later discovery of the laws of motion and universal gravitation depended strongly on Kepler's work. Although from the modern point of view, Kepler's laws can be seen as a consequence of Newton's laws, historically, it was the other way around: Kepler provided a mathematically distilled description of the empirical observations, which Newton then interpreted.

Kepler's first law

The orbit of a planet about a star is an ellipse with the star at one focus. There is no object at the other focus of a planet's orbit. The semimajor axis, a, is half the major axis of the ellipse. In some sense it can be regarded as the average distance between the planet and its star, but it is not the time average in a strict sense, as more time is spent near apocentre than near pericentre.

Kepler's second law

A line joining a planet and its star sweeps out equal areas during equal intervals of time. This is also known as the law of equal areas. Suppose a planet takes 1 day to travel from points A to B. During this time, an imaginary line, from the Sun to the planet, will sweep out a roughly triangular area. This same amount of area will be swept every day. As a planet travels in its elliptical orbit, its distance from the Sun will vary. As an equal area is swept during any period of time and since the distance from a planet to its orbiting star varies, one can conclude that in order for the area being swept to remain constant, a planet must vary in velocity. The physical meaning of the law is that the planet moves faster when it is closer to the sun. This is because the sun's gravity accelerates the planet as it falls toward the sun, and decelerates it on the way back out.

Kepler's third law

The square of the sidereal period of an orbiting planet is directly proportional to the cube of the orbit's semimajor axis. :

P^2 \propto a^3

:P = object's sidereal period :a = orbit's semimajor axis Thus, not only does the length of the orbit increase with distance, also the orbital speed decreases, so that the increase of the sidereal period is more than proportional. See the actual figures: attributes of major planets. This law is also known as the harmonic law.

Accuracy and limitations

As discussed below, Kepler's third law needs to be modified when the orbiting body's mass is not negligible compared to the mass of the body being orbited. However, the correction is fairly small in the case of the planets orbiting the Sun. A more serious limitation of Kepler's laws is that they assume a two-body system. This is a particularly bad approximation in the case of the Earth-Sun-Moon system, and for calculations of the Moon's orbit, Kepler's laws are far less accurate than the empirical method invented by Ptolemy hundreds of years before. Newton generalized Kepler's first law with the realization that an object moving at greater than escape velocity (e.g., some comets) would have an open parabolic or hyperbolic orbit rather than a closed elliptical one. Thus, all of the conic sections are possible orbits. The second law is still valid for open orbits (since angular momentum is still conserved), but the third law is inapplicable because the motion is not periodic. Because electrical forces also obey an inverse square law, Kepler's laws also apply to bodies interacting electrically. If the interaction is repulsive, as in Rutherford scattering, then the orbit is hyperbolic, with the center of repulsion outside the hyperbola. In the planetary model of the atom, the electrons were imagined to orbit the nucleus exactly like the planets orbiting the Sun, and Niels Bohr even went so far as to attempt to assign specific atomic energy levels to specific elliptical orbits. The Keplerian approximation is successful when the quantum-mechanical wavelengths are small compared to the size of the orbit (as in Rutherford scattering, and in some unusual artificially populated electron orbitals with very high values of the principal quantum number). Kepler's laws do not consider the emission of radiation. The emission of gravitational radiation is negligible in our solar system, but important in some stellar systems containing black holes or neutron stars (any system involving large masses combined with large accelerations). In the planetary model of the atom, the emission of electromagnetic radiation should have led to the collapse of all atoms, and this was a hint that classical physics was in need of modification. Kepler's laws don't incorporate relativity, so, for example, they don't correctly predict the precession of Mercury's orbit. The successful explanation of this non-Keplerian behavior was one of the most dramatic early successes of general relativity. Kepler's laws fail even more dramatically in the case of an object orbiting a black hole; here, general relativity predicts that many orbits —those that pass through the event horizon— are one-way streets to a dead-end region of spacetime.

Connection to Newton's laws and conservation laws

Kepler did not understand why his laws were correct; it was Isaac Newton who discovered the answer to this more than fifty years later. The second law can also be seen as a statement of conservation of angular momentum, which is a logical consequence of Newton's laws in the special case of a force that acts along the line connecting two objects.

Kepler's first law

Newton proposed that "every object in the universe attracts every other object along a line of the centres of the objects, proportional to each object's mass, and inversely proportional to the square of the distance between the objects." The following assumes that acceleration is of the form :

\frac = f(r)\hat

. Recall that in polar coordinates: :

\frac = \dot r\hat + r\dot\theta\hat

\frac = (\ddot r - r\dot\theta^2)\hat + (r\ddot\theta + 2\dot r \dot\theta)\hat

In component form, we have: :

\ddot r - r\dot\theta^2 = f(r)

r\ddot\theta + 2\dot r\dot\theta = 0

Since

\dot r = dr/dt

and

\ddot\theta=/

, the latter equation is equivalent to :

\frac = -2\frac

. When integrated, this yields :

\log\dot\theta = -2\log r + \log\ell

, :

\ell = r^2\dot\theta

, for some constant

\ell

, which can be shown to be the specific angular momentum. Now we substitute. Let: :

r = \frac

\dot r = -\frac\dot u = -\frac\frac\frac= -\ell\frac

\ddot r = -\ell\frac\frac = -\ell\dot\theta\frac= -\ell^2u^2\frac

The equation of motion in the

\hat

direction becomes: :

\frac + u = - \fracf\left(\frac\right)

f(r)=-k/r^2

, as Newton's law of gravitation claims, then: :

\frac + u = \frac

where -k is our proportionality constant. This differential equation has the general solution: :

u = A\cos(\theta-\theta_0) + \frac.

Replacing u with r and letting θ₀=0: :

r = \frac

. This is indeed the equation of a conic section with the origin at one focus. Thus, Kepler's first law is a direct result of Newton's law of gravitation and Newton's second law of motion.

Kepler's second law

Assuming Newton's laws of motion, we can show that Kepler's second law is consistent. By definition, the angular momentum

\mathbf

of a point mass with mass

m

and velocity

\mathbf

is : :

\mathbf \equiv m \mathbf \times \mathbf

. where

\mathbf

is the position vector of the particle. Since

\mathbf = \frac

, we have: :

\mathbf = \mathbf \times m\frac

taking the time derivative of both sides: :

\frac = \mathbf \times \mathbf = 0

since the cross product of parallel vectors is 0. We can now say that

|\mathbf|

is constant. The area swept out by the line joining the planet and the sun is half the area of the parallelogram formed by

\mathbf

and

d\mathbf

. :

dA = \begin\frac\end |\mathbf \times d\mathbf| = \begin\frac\end \left|\mathbf \times \fracdt\right| = \fracdt

Since

|\mathbf|

is constant, the area swept out by is also constant. Q.E.D.

Kepler's third law

Newton used the third law as one of the pieces of evidence used to build the conceptual and mathematical framework supporting his law of gravity. If we take Newton's laws of motion as given, and consider a hypothetical planet that happens to be in a perfectly circular orbit of radius r, then we have

F=mv^2/r

for the sun's force on the planet. The velocity is proportional to r/T, which by Kepler's third law varies as one over the square root of r. Substituting this into the equation for the force, we find that the gravitational force is proportional to one over r squared. (Newton's actual historical chain of reasoning is not known with certainty, because in his writing he tended to erase any traces of how he had reached his conclusions.) Reversing the direction of reasoning, we can consider this as a proof of Kepler's third law based on Newton's of gravity, and taking care of the proportionality factors that were neglected in the argument above, we have: :

T^2 = \frac \cdot r^3

where:
- T = planet's sidereal period
- r = radius of the planet's circular orbit
- G = the gravitational constant
- M = mass of the sun The same arguments can be applied to any object orbiting any other object. This discussion implicitly assumed that the planet orbits around the stationary sun, although in reality both the planet and the sun revolve around their common center of mass. Newton recognized this, and modified this third law, noting that the period is also affected by the orbiting body's mass. However typically the central body is so much more massive that the orbiting body's mass may be ignored. Newton also proved that in the case of an elliptical orbit, the semimajor axis could be substituted for the radius. The most general result is: :

T^2 = \frac \cdot a^3

where:
- T = object's sidereal period
- a = object's semimajor axis
- G = 6.67 × 10⁻¹¹ N · m²/kg² = the gravitational constant
- M = mass of one object
- m = mass of the other object For objects orbiting the sun, it can be convenient to use units of years, AU, and solar masses, so that G, 4π² and the various conversion factors cancel out. Also with m<<M we can set m+M = M, so we have simply

T^2=a^3

Proving Kepler's third law

Define point A to be the periapsis, and point B as the apoapsis of the planet when orbiting the sun. Kepler's second law states that the orbiting body will sweep out equal areas in equal quantities of time. If we now look at a very small periods of time at the moments when the planet is at points A and B, then we can approximate the area swept out as a triangle with an altitude equal to the distance between the planet and the sun, and the base equal to the time times the speed of the planet. :

\frac\cdot(1-\epsilon)a\cdot V_A\,dt=\frac\cdot(1+\epsilon)a\cdot V_B\,dt

(1-\epsilon)\cdot V_A=(1+\epsilon)\cdot V_B

V_A=V_B\cdot\frac

Using the law of conservation of energy for the total energy of the planet at points A and B, :

\frac-\frac=\frac-\frac

\frac-\frac=\frac-\frac

\frac=\frac\cdot \left ( \frac-\frac \right )

\frac=\frac\cdot \left ( \frac \right )

V_B^2 \cdot \left ( \frac\right ) ^2-V_B^2=\frac\cdot \left ( \frac \right )

V_B^2 \cdot \left ( \frac\right )=\frac

V_B^2 \cdot \left ( \frac\right )=\frac

V_B^2 \cdot 4\epsilon=\frac

V_B=\sqrt.

Now that we have

V_B

, we can find the rate at which the planet is sweeping out area in the ellipse. This rate remains constant, so we can derive it from any point we want, specifically from point B. :

\frac=\frac=\frac\cdot(1+\epsilon)a\cdot V_B=\frac\cdot(1+\epsilon)a\cdot\sqrt=
\frac\cdot\sqrt

However, the total area of the ellipse is equal to

\pi a \sqrta

. (That's the same as

\pi a b

, because

b=\sqrta

). The time the planet take out to sweep out the entire area of the ellipse equals the ellipse's area, so, :

T\cdot \frac=\pi a \sqrta

T\cdot \frac\cdot\sqrt=\pi \sqrta^2

T=\frac=\frac=
\frac\sqrt

T^2=\fraca^3.

However, if the mass m is not negligible in relation to M, then the planet will orbit the sun with the exact same velocity and position as a very small body orbiting an object of mass

M+m

(see reduced_mass). To integrate that in the above formula, M must be replaced with

M+m

, to give

T^2=\fraca^3

. Q.E.D.

Solution for the motion as a function of time

The Keplerian problem assumes an orbit with semimajor axis a, semiminor axis b, and eccentricity ε. To convert the laws into predictions, Kepler began by adding the orbit's auxiliary circle (that with the major axis as a diameter) and defined these points:
- c center of auxiliary circle and ellipse
- s sun (at one focus of ellipse);

\mboxcs=a\varepsilon

- p the planet
- z perihelion
- x is the projection of the planet to the auxiliary circle; then

\mboxsxz=\frac ba\mboxspz

- y is a point on the circle such that

\mboxcyz=\mboxsxz=\frac ba\mboxspz

and three angles measured from perihelion:
- true anomaly

T=\angle zsp

, the planet as seen from the sun
- eccentric anomaly

E=\angle zcx

, x as seen from the centre
- mean anomaly

M=\angle zcy

, y as seen from the centre image:kepler's-equation-scheme.png Then :

\mboxcxz=\mboxcxs+\mboxsxz

=\mboxcxs+\mboxcyz

\frac2E=a\varepsilon\frac a2\sin E+\frac2M

giving Kepler's equation :

M=E-\varepsilon\sin E

. To connect E and T, assume

r=\mboxsp

then :

a\varepsilon+r\cos T=a\cos E

and

r\sin T=b\sin E

r=\frac=\frac

\tan T=\frac=\frac ba\frac=\frac

which is ambiguous but useable. A better form follows by some trickery with trigonometric identities: :

\tan\frac T2=\sqrt\frac\tan\frac E2

(So far only laws of geometry have been used.) Note that

\mboxspz

is the area swept since perihelion; by the second law, that is proportional to time since perihelion. But we defined

\mboxspz=\frac ab\mboxcyz=\frac ab\frac2M

and so M is also proportional to time since perihelion—this is why it was introduced. We now have a connection between time and position in the orbit. The catch is that Kepler's equation cannot be rearranged to isolate E; going in the time-to-position direction requires an iteration (such as Newton's method) or an approximate expression, such as :

E\approx M+\left(\varepsilon-\frac18\varepsilon^3\right)\sin M+\frac12\varepsilon^2\sin 2M+\frac38\varepsilon^3\sin 3M

via the Lagrange reversion theorem. For the small ε typical of the planets (except Pluto) such series are quite accurate with only a few terms; one could even develop a series computing T directly from M.[http://info.ifpan.edu.pl/firststep/aw-works/fsII/mul/mueller.html]

External links

- Crowell, Benjamin, Conservation Laws, [http://www.lightandmatter.com/area1book2.html http://www.lightandmatter.com/area1book2.html], an online book that gives a proof of the first law without the use of calculus. Category:Celestial mechanics Category:Equations Category:Eponymous laws ko:케플러 법칙 ja:ケプラーの法則

Semimajor axis

In geometry, the term semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolas.

Ellipse

The major axis of an ellipse is its longest diameter, a line that runs through the centre and both foci, its ends being at the widest points of the shape. The semi-major axis is one half of the major axis, and thus runs from the centre, through a focus, and to the edge of the ellipse. It is related to the semi-minor axis

b\,\!

through the eccentricity

e\,\!

and the semi-latus rectum

\ell\,\!

, as follows: :

b = a \sqrt\,\!

\ell=a(1-e^2)\,\!

. :

a\ell=b^2\,\!

. A parabola can be obtained as the limit of a sequence of ellipses where one focus is kept fixed as the other is allowed to move arbitrarily far away in one direction, keeping

\ell\,\!

fixed. Thus

a\,\!

and

b\,\!

tend to infinity,

a\,\!

faster than

b\,\!

. The semi-major axis is the mean value of the smallest and largest distances from one focus to the points on the ellipse. Now consider the equation in polar coordinates, with one focus at the origin and the other on the positive x-axis, :

r(1-e\cos\theta)=l\,\!

The mean value of

r=\,\!

and

r=\,\!

, is

a=\,\!

Hyperbola

The semi-major axis of a hyperbola is one half of the distance between the two branches; if this is in the x-direction the equation is:

\frac - \frac = 1

In terms of the semi-latus rectum and the eccentricity we have

a=

Astronomy

Orbital period

In astrodynamics the orbital period

T\,

of a small body orbiting a central body in a circular or elliptical orbit is: :

T = 2\pi\sqrt

where: :

a\,

is the length of the orbit's semi-major axis :

\mu

is the standard gravitational parameter Note that for all ellipses with a given semi-major axis, the orbital period is the same, regardless of eccentricity. In astronomy, the semi-major axis is one of the most important orbital elements of an orbit, along with its orbital period. For solar system objects, the semi-major axis is related to the period of the orbit by Kepler's third law (originally empirically derived), :

P^2=a^3\,

where P is the period in years, and a is the semimajor axis in astronomical units. This form turns out to be a simplification of the general form for the two-body problem, as determined by Newton: :

P^2= \fraca^3\,

where G is the gravitational constant, and M is the mass of the central body, and m is the mass of the orbiting body. Typically, the central body's mass is so much greater than the orbiting body's, that m may be ignored. Making that assumption and using typical astronomy units results in the simpler form Kepler discovered. Remarkably, the orbiting body's path around the barycentre and its path relative to its primary are both ellipses. The semi-major axis used in astronomy is always the primary-to-secondary distance; thus, the orbital parameters of the planets are given in heliocentric terms. The difference between the primocentric and "absolute" orbits may best be illustrated by looking at the Earth-Moon system. The mass ratio in this case is 81.30059. The Earth-Moon characteristic distance, the semi-major axis of the geocentric lunar orbit, is 384,400 km. The barycentric lunar orbit, on the other hand, has a semi-major axis of 379,700 km, the Earth's counter-orbit taking up the difference, 4,700 km. The Moon's average barycentric orbital speed is 1.010 km/s, whilst the Earth's is 0.012 km/s. The total of these speeds gives the geocentric lunar average orbital speed, 1.022 km/s; the same value may be obtained by considering just the geocentric semi-major axis value.

Average distance

It is often said that the semi-major axis is the "average" distance between the primary (the focus of the ellipse) and the orbiting body. This is not quite accurate, as it depends over what the average is taken.
- averaging the distance over the eccentric anomaly (q.v.) indeed results in the semi-major axis.
- averaging over the true anomaly (the true orbital angle, measured at the focus) results, oddly enough, in the semi-minor axis

b = a \sqrt\,\!

.
- averaging over the mean anomaly (the fraction of the orbital period that has elapsed since pericentre, expressed as an angle), finally, gives the time-average (which is what "average" usually means to the layman):

a (1 + \frac)\,\!

. The time-average of the inverse of the radius,

r^\,\!

, is

a^\,\!

Energy; calculation of semi-major axis from state vectors

In astrodynamics semi-major axis

a \,

can be calculated from orbital state vectors:

a = \,

for an elliptical orbit and

a = \,

for a hyperbolic trajectory and

\epsilon =  -

(specific orbital energy) and

\mu = GM \,

(standard gravitational parameter), where:
-

v\,

is orbital velocity from velocity vector of an orbiting object,
-

\mathbf\,

is cartesian position vector of an orbiting object in coordinates of a reference frame with respect to which the elements of the orbit are to be calculated (e.g. geocentric equatorial for an orbit around Earth, or heliocentric ecliptic for an orbit around the Sun),
-

G \,

is the gravitational constant,
-

M \,

the mass of the central body. Note that for a given central body and total specific energy, the semi-major axis is always the same, regardless of eccentricity. Conversely, for a given central body and semi-major axis, the total specific energy is always the same.

Example

The International Space Station has an orbital period of 91.74 minutes, hence the semi-major axis is 6738 km [http://www.google.com/search?num=100&hl=en&lr=&newwindow=1&safe=off&q=%28%2891.74
- 60%2F2%2Fpi%29%5E2
- 398600%29%5E%281%2F3%29]. Every minute more corresponds to ca. 50 km more: the extra 300 km of orbit length takes 40 seconds, the lower speed accounts for an additional 20 seconds.

References

- [http://orca.phys.uvic.ca/~tatum/celmechs/celm9.pdf Jeremy B. Tatum, Celestial Mechanics, Chapter 9 - The Two Body Problem in Two Dimensions (2004)]
- [http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=AJPIAS000071000011001198000001&idtype=cvips&gifs=yes Darren M. Williams, Average distance between a star and planet in an eccentric orbit, American Journal of Physics, November 2003, Volume 71, Issue 11, pp. 1198-1200] Category:Conic sections Category:Astrodynamics Category:Celestial mechanics

Semiminor axis

In geometry, the semi-minor axis (also semiminor axis) applies to ellipses and hyperbolas.

Ellipse

The semi-minor axis of an ellipse is one half of the minor axis, running from the center, halfway between and perpendicular to the line running between the foci, and to the edge of the ellipse. The minor axis is the longest line that runs perpendicular to the major axis. It is related to the semi-major axis

a

through the eccentricity

e

and the semi-latus rectum

l

, as follows: :

b = a \sqrt

al=b^2

. A parabola can be obtained as the limit of a sequence of ellipses where one focus is kept fixed as the other is allowed to move arbitrarily far away in one direction, keeping l fixed. Thus a and b tend to infinity, a faster than b.

Hyperbola

The semi-minor axis of a hyperbola is the distance from a top, along the tangent line, to each asymptote; if this is in the y-direction it is b in this equation of the hyperbola:

\frac - \frac = 1

It is related to the semi-major axis through the eccentricity, as follows: :

b = a \sqrt

Note that in a hyperbola b can be larger than a! Category:Conic sections Category:Astrodynamics Category:Celestial mechanics

Eccentricity (orbit)

__NOTOC__ :This page refers to eccentricity in astrodynamics. For other uses, see the disambiguation page eccentricity. eccentricity In astrodynamics, under standard assumptions any orbit must be of conic section shape. The eccentricity of this conic section, the orbit's eccentricity, is an important parameter of the orbit that defines its absolute shape. Eccentricity may be interpreted as a measure of how much this shape deviates from a circle. Under standard assumptions eccentricity (

e\,\!

) is strictly defined for all circular, elliptic, parabolic and hyperbolic orbits and may take following values:
- for circular orbits:

e=0\,\!

,
- for elliptic orbits:

0,

- for parabolic trajectories:

e=1\,\!

,
- for hyperbolic trajectories:

e>1\,\!

Calculation

Eccentricity of an orbit can be calculated from orbital state vectors as a magnitude of eccentricity vector: :

e= \left | \mathbf \right |

where:
-

\mathbf\,\!

is eccentricity vector. ---- For elliptic orbits it can also be calculated from distance at periapsis and apoapsis: :

e==1-\frac=\frac-1

where:
-

d_p\,\!

is distance at periapsis,
-

d_a\,\!

is distance at apoapsis.

Examples

For example, the eccentricity of the Earth's orbit today is 0.0167. Through time, the eccentricity of the Earth's orbit slowly changes from nearly 0 to almost 0.05 as a result of gravitational attractions between the planets (see graph [http://www.museum.state.il.us/exhibits/ice_ages/eccentricity_graph.html]). Other values: Pluto 0.2488 (largest value among the planets of the Solar System), Mercury 0.2056, Moon 0.0554. For the values for all planets in one table, see :de:Planet (Tabelle).

External links

- [http://scienceworld.wolfram.com/physics/Eccentricity.html World of Physics: Eccentricity] Category:Astrodynamics Category:Celestial mechanics

Periapsis

: This article is about several astronomical terms (apogee & perigee, aphelion & perihelion, generic equivalents based on apsis, and related but rarer terms). In architecture, apsis is a synonym for apse; Apogee is also the name of a video game publisher. video game publisher In astronomy, an apsis (plural apsides "ap-si-deez") is the point of greatest or least distance of the elliptical orbit of a celestial body from its center of attraction (the center of mass of the system). The point of closest approach is called the periapsis or pericentre and the point of farthest approach is the apoapsis (Greek απο, from), apocentre or apapsis (the latter term, although etymologically more correct, is much less used). A straight line drawn through the periapsis and apoapsis is the line of apsides. This is the major axis of the ellipse, the line through the longest part of the ellipse. Related terms are used to identify the body being orbited. The most common are perigee and apogee, referring to Earth orbits, and perihelion and aphelion, referring to orbits around the Sun (Greek ἡλιος).

Formulae

We have:
- Periapsis: maximum speed

v_\mathrm = \sqrt \,

at minimum distance

r_\mathrm=(1-e)a\!\,

(periapsis distance)
- Apoapsis: minimum speed

v_\mathrm = \sqrt \,

at maximum distance

r_\mathrm=(1+e)a\!\,

(apoapsis distance) where one easily verifies :

h = \sqrt

\epsilon=-\frac

(each the same for both points, like they are for the whole orbit, in accordance with Kepler's laws of planetary motion (conservation of angular momentum) and the conservation of energy) where:
-

a\!\,

is the semi-major axis
-

e\!\,

is the eccentricity
-

h\!\,

is the specific relative angular momentum
-

\epsilon\!\,

is the specific orbital energy
-

\mu\!\,

is the standard gravitational parameter Properties: :

e=\frac=1-\frac=\frac-1

Note that for conversion from heights above the surface to distances, the radius of the central body has to be added, and conversely. The arithmetic mean of the two distances is the semi-major axis

a\!\,

. The geometric mean of the two distances is the semi-minor axis

b\!\,

. The geometric mean of the two speeds is

\sqrt

, the speed corresponding to a kinetic energy which, at any position of the orbit, added to the existing kinetic energy, would allow the orbiting body to escape (the square root of the sum of the squares of the two speeds is the local escape velocity).

Terminology

Various related terms are used for other celestial objects. The '-gee', '-helion' and '-astron' and '-galacticon' forms are frequently used in the astronomical literature, while the other listed forms are occasionally used, although '-saturnium' has very rarely been used in the last 50 years. The '-gee' form is commonly (although incorrectly) used as a generic 'closest approach to planet' term instead of specifically applying to the Earth. The term peri/apomelasma was used by Geoffrey A. Landis in 1998 before peri/aponigricon appeared in the scientific literature in 2002. ⁽¹⁾ Properly pronounced 'affelion', although 'ap-helion' is commonly heard. Since "peri" and "apo" are Greek, it is considered by purists more correct to use the Greek form for the body, giving forms such as '-zene' for Jupiter and '-krone' for Saturn. For Venus, the alternate form '-krition' (from Kritias, an older name for Aphrodite) has also been suggested. In the Moon's case, in practice all three forms are used, albeit very infrequently. The '-cynthion' form is, according to some, reserved for artificial bodies, whilst others reserve '-lune' for an object launched from the Moon and '-cynthion' for an object launched from elsewhere. For Jupiter, the '-jove' form is occasionally used by astronomers whilst the '-zene' form is never used, like the other pure Greek forms ('-cytherion' (Venus), '-areion' (Mars), '-hermion' (Mercury), '-krone' (Saturn), '-uranion' (Uranus), '-poseidion' (Neptune) and '-hadion' (Pluto)). The daunting prospect of having to maintain a different word for every orbitable body in the solar system (and beyond) is the main reason why the generic '-apsis' has become the almost universal norm.

True anomaly

In astronomy, the true anomaly (

T\,\!

, also written

v\

) 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). In the diagram below, true anomaly is the angle z-s-p. Image:Kepler's-equation-scheme.png

Calculation from state vectors

For elliptic orbits true anomaly

T\,\!

can be calculated from orbital state vectors as: :

T = \arccos

(if

\mathbf \cdot \mathbf < 0

then replace T by 2π − T) where:
-

\mathbf\,

is orbital velocity vector of the orbiting body,
-

\mathbf\,

is eccentricity vector,
-

\mathbf\,

is orbital position vector (segment sp) of the orbiting body. : ---- For circular orbits this can be simplified to: :

T = \arccos

(if

\mathbf \cdot \mathbf >0

then replace T by 2π − T) where:
-

\mathbf

is vector pointing towards the ascending node (i.e. the z-component of

\mathbf

is zero). : ---- For circular orbits with the inclination of zero this can be simplified further to: :

T = \arccos

(if

v_x\ > 0

then replace T by 2π − T) where:
-

r_x \,

is x-component of orbital position vector

\mathbf

,
-

v_x \,

is x-component of orbital velocity vector

\mathbf

Other relations

The relation between T and E, the eccentric anomaly, is: :

\cos = ,\,

or equivalently :

\tan = \sqrt \tan.

The relations between the radius (position vector magnitude) and the anomalies are: :

r = a \left ( 1 - e \cdot \cos \right )\,\!

and :

r = a\,\!

where a is the orbit's semi-major axis (segment cz).

Kepler's laws of planetary motion

Kepler's first law

Kepler's second law

Kepler's third law

The square of the sidereal period of an orbiting planet is directly proportional to the cube of the orbit's semimajor axis. :

P^2 \propto a^3

Accuracy and limitations

Connection to Newton's laws and conservation laws

Kepler's first law

\frac = f(r)\hat

. Recall that in polar coordinates: :

\frac = \dot r\hat + r\dot\theta\hat

\frac = (\ddot r - r\dot\theta^2)\hat + (r\ddot\theta + 2\dot r \dot\theta)\hat

In component form, we have: :

\ddot r - r\dot\theta^2 = f(r)

r\ddot\theta + 2\dot r\dot\theta = 0

Since

\dot r = dr/dt

and

\ddot\theta=/

, the latter equation is equivalent to :

\frac = -2\frac

. When integrated, this yields :

\log\dot\theta = -2\log r + \log\ell

, :

\ell = r^2\dot\theta

, for some constant

\ell

, which can be shown to be the specific angular momentum. Now we substitute. Let: :

r = \frac

\dot r = -\frac\dot u = -\frac\frac\frac= -\ell\frac

\ddot r = -\ell\frac\frac = -\ell\dot\theta\frac= -\ell^2u^2\frac

The equation of motion in the

\hat

direction becomes: :

\frac + u = - \fracf\left(\frac\right)

f(r)=-k/r^2

, as Newton's law of gravitation claims, then: :

\frac + u = \frac

where -k is our proportionality constant. This differential equation has the general solution: :

u = A\cos(\theta-\theta_0) + \frac.

Replacing u with r and letting θ₀=0: :

r = \frac

. This is indeed the equation of a conic section with the origin at one focus. Thus, Kepler's first law is a direct result of Newton's law of gravitation and Newton's second law of motion.

Kepler's second law

Assuming Newton's laws of motion, we can show that Kepler's second law is consistent. By definition, the angular momentum

\mathbf

of a point mass with mass

m

and velocity

\mathbf

is : :

\mathbf \equiv m \mathbf \times \mathbf

. where

\mathbf

is the position vector of the particle. Since

\mathbf = \frac

, we have: :

\mathbf = \mathbf \times m\frac

taking the time derivative of both sides: :

\frac = \mathbf \times \mathbf = 0

since the cross product of parallel vectors is 0. We can now say that

|\mathbf|

is constant. The area swept out by the line joining the planet and the sun is half the area of the parallelogram formed by

\mathbf

and

d\mathbf

. :

dA = \begin\frac\end |\mathbf \times d\mathbf| = \begin\frac\end \left|\mathbf \times \fracdt\right| = \fracdt

Since

|\mathbf|

is constant, the area swept out by is also constant. Q.E.D.

Kepler's third law

F=mv^2/r

T^2 = \frac \cdot r^3

T^2 = \frac \cdot a^3

T^2=a^3

Proving Kepler's third law

\frac\cdot(1-\epsilon)a\cdot V_A\,dt=\frac\cdot(1+\epsilon)a\cdot V_B\,dt

(1-\epsilon)\cdot V_A=(1+\epsilon)\cdot V_B

V_A=V_B\cdot\frac

Using the law of conservation of energy for the total energy of the planet at points A and B, :

\frac-\frac=\frac-\frac

\frac-\frac=\frac-\frac

\frac=\frac\cdot \left ( \frac-\frac \right )

\frac=\frac\cdot \left ( \frac \right )

V_B^2 \cdot \left ( \frac\right ) ^2-V_B^2=\frac\cdot \left ( \frac \right )

V_B^2 \cdot \left ( \frac\right )=\frac

V_B^2 \cdot \left ( \frac\right )=\frac

V_B^2 \cdot 4\epsilon=\frac

V_B=\sqrt.

Now that we have

V_B

, we can find the rate at which the planet is sweeping out area in the ellipse. This rate remains constant, so we can derive it from any point we want, specifically from point B. :

\frac=\frac=\frac\cdot(1+\epsilon)a\cdot V_B=\frac\cdot(1+\epsilon)a\cdot\sqrt=
\frac\cdot\sqrt

However, the total area of the ellipse is equal to

\pi a \sqrta

. (That's the same as

\pi a b

, because

b=\sqrta

). The time the planet take out to sweep out the entire area of the ellipse equals the ellipse's area, so, :

T\cdot \frac=\pi a \sqrta

T\cdot \frac\cdot\sqrt=\pi \sqrta^2

T=\frac=\frac=
\frac\sqrt

T^2=\fraca^3.

However, if the mass m is not negligible in relation to M, then the planet will orbit the sun with the exact same velocity and position as a very small body orbiting an object of mass

M+m

(see reduced_mass). To integrate that in the above formula, M must be replaced with

M+m

, to give

T^2=\fraca^3

. Q.E.D.

Solution for the motion as a function of time

\mboxcs=a\varepsilon

- p the planet
- z perihelion
- x is the projection of the planet to the auxiliary circle; then

\mboxsxz=\frac ba\mboxspz

- y is a point on the circle such that

\mboxcyz=\mboxsxz=\frac ba\mboxspz

and three angles measured from perihelion:
- true anomaly

T=\angle zsp

, the planet as seen from the sun
- eccentric anomaly

E=\angle zcx

, x as seen from the centre
- mean anomaly

M=\angle zcy

, y as seen from the centre image:kepler's-equation-scheme.png Then :

\mboxcxz=\mboxcxs+\mboxsxz

=\mboxcxs+\mboxcyz

\frac2E=a\varepsilon\frac a2\sin E+\frac2M

giving Kepler's equation :

M=E-\varepsilon\sin E

. To connect E and T, assume

r=\mboxsp

then :

a\varepsilon+r\cos T=a\cos E

and

r\sin T=b\sin E

r=\frac=\frac

\tan T=\frac=\frac ba\frac=\frac

which is ambiguous but useable. A better form follows by some trickery with trigonometric identities: :

\tan\frac T2=\sqrt\frac\tan\frac E2

(So far only laws of geometry have been used.) Note that

\mboxspz

is the area swept since perihelion; by the second law, that is proportional to time since perihelion. But we defined

\mboxspz=\frac ab\mboxcyz=\frac ab\frac2M

E\approx M+\left(\varepsilon-\frac18\varepsilon^3\right)\sin M+\frac12\varepsilon^2\sin 2M+\frac38\varepsilon^3\sin 3M

External links

Transcendental

See:
- Transcendence (mathematics)
- Transcendence (philosophy)
- Transcendence (religion)
- Transcendence (novel) by Charles Sheffield th:อุตรภาพ

Newton's method

In numerical analysis, Newton's method (or the Newton-Raphson method) is an efficient algorithm for finding approximations to the zeros (or roots) of a real-valued function. As such, it is an example of a root-finding algorithm. It can also be used to find a minimum or maximum of such a function, by finding a zero in the function's first derivative, see Newton's method as an optimization algorithm.

Description of the method

The idea of the method is as follows: one starts with a value which is reasonably close to the true zero, then replaces the function by its tangent (which can be computed using the tools of calculus) and computes the zero of this tangent (which is easily done with elementary algebra). This zero of the tangent will typically be a better approximation to the function's zero, and the method can be iterated. Suppose f : [a, b] -> R is a differentiable function defined on the interval [a, b] with values in the real numbers R. We start with an arbitrary value x₀ (the closer to the zero the better) and then define for each natural number n: :

x_ = x_n - \frac.

Here, f ' denotes the derivative of the function f. One can prove that, if f ' is continuous, and if the unknown zero x is isolated, then there exists a neighborhood of x such that for all starting values x₀ in that neighborhood, the sequence will converge towards x. Furthermore, if f '(x) ≠ 0, then the convergence is quadratic, which intuitively means that the number of correct digits roughly doubles in every step. ::alt Illustration of Newton's method An illustration of Newton's method. We see that

x_

is a better guess than

x_n

for the zero

x

of the function f.

Example

Consider the problem of finding the positive number x with cos(x) = x³. We can rephrase that as finding the zero of f(x) = cos(x) - x³. We have f '(x) = -sin(x) - 3x². Since cos(x) ≤ 1 for all x and x³ > 1 for x>1, we know that our zero lies between 0 and 1. We try a starting value of x₀ = 0.5. :

\begin
  x_1 & = & x_0 - \frac & = & 0.5 - \frac & = & 1.112141637097 \\
  x_2 & = & x_1 - \frac & & \vdots & = & \underline.909672693736 \\
  x_3 & & \vdots & & \vdots & = & \underline7263818209 \\
  x_4 & & \vdots & & \vdots & = & \underline7135298 \\
  x_5 & & \vdots & & \vdots & = & \underline11 \\
  x_6 & & \vdots & & \vdots & = & \underline
\end

The correct digits are underlined in the above example. In particular, all digits of x₆ are correct. We see that the number of correct digits after the decimal point increases from 2 (for x₃) to 5 and 10, illustrating the quadratic convergence. Consider the following example in pseudocode. function newtonIterationFunction(x) var x := 0.5 for i from 0 to 99 Here is the same using a calculator.

\begin
\mbox & & & \\
0.5\;\mbox & = & x_0 & = & 0.5 \\
Ans - \frac\;\mbox & = & x_1 & = & 1.1121416371 \\
\mbox & = & x_2 & = & 0.909672693736 \\
\mbox & = & x_3 & = & 0.867263818209 \\
\vdots & & \vdots & & \vdots
\end

History

Newton's method was described by Isaac Newton in De analysi per aequationes numero terminorum infinitas (written in 1669, published in 1711 by William Jones) and in De metodis fluxionum et serierum infinitarum (written in 1671, translated and published as Method of Fluxions in 1736 by John Colson). However, his description differs substantially from the modern description given above: Newton applies the method only to polynomials. He does not compute the successive approximations x_n, but computes a sequence of polynomials and only at the end, he arrives at an approximation for the root x. Finally, Newton views the method as purely algebraic and fails to notice the connection with calculus. Isaac Newton probably derived his method from a similar but less precise method by François Viète. The essence of Viète's method can be found in the work of the Persian mathematician Sharaf al-Din al-Tusi. Heron of Alexandria used essentially the same method in book 1, chapter 8, of his Metrica to determine the square root of 720. Newton's method was first published in 1685 in A Treatise of Algebra both Historical and Practical by John Wallis. In 1690, Joseph Raphson published a simplified description in Analysis aequationum universalis. Raphson again viewed Newton's method purely as an algebraic method and restricted its use to polynomials, but he describes the method in terms of the successive approximations x_n instead of the more complicated sequence of polynomials used by Newton. Finally, in 1740, Thomas Simpson described Newton's method as an iterative method for solving general nonlinear equations using fluxional calculus, essentially giving the description above. In the same publication, Simpson also gives the generalization to systems of two equations and notes that Newton's method can be used for solving optimization problems by setting the gradient to zero.

Practical considerations

In general the convergence is quadratic: the error is essentially squared at each step (that is, the number of accurate digits doubles in each step). There are some caveats, however. First, Newton's method requires that the derivative be calculated directly. If instead the derivative is approximated by the slope of the line through two points on the function's graph, the secant method results — though depending on how one measures computational effort, the secant method may be more efficient. Second, if the starting value is too far removed from the true zero, Newton's method can fail to converge at all. Because of this, all practical implementations of Newton's method put an upper limit on the number of iterations and perhaps on the size of the iterates. Third, if the root being sought has multiplicity greater than one, the convergence rate is reduced to linear (errors reduced by a constant factor at each step) unless special steps are taken.

Generalizations

One may use Newton's method also to solve systems of k (non-linear) equations, which amounts to finding the zeros of continuously differentiable functions F : R^k -> R^k. In the formulation given above, one then has to multiply with the inverse of the k-by-k Jacobian matrix J_F(x_n) instead of dividing by f '(x_n). Rather than actually computing the inverse of this matrix, one can save time by solving the system of linear equations :

J_F(x_n) (x_ - x_n) = -F(x_n)

for the unknown x_n+1 - x_n. Again, this method only works if the initial value x₀ is close enough to the true zero. Typically, a region which is well-behaved is located first with some other method and Newton's method is then used to "polish" a root which is already known approximately.

Complex functions

well-behaved well-behaved When dealing with complex functions, however, Newton's method can be directly applied to find their zeros. For many complex functions, the set (also know as the basin of attraction) of all starting values that cause the method to converge to the true zero is a fractal.

References

- Tjalling J. Ypma, Historical development of the Newton-Raphson method, SIAM Review 37 (4), 531–551, 1995.
- P. Deuflhard, Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms. Springer Series in Computational Mathematics, Vol. 35. Springer, Berlin, 2004. ISBN 3-540-21099-7.
- C. T. Kelley, Solving Nonlinear Equations with Newton's Method, no 1 in Fundamentals of Algorithms, SIAM, 2003. ISBN 0-89871-546-6.
- J. M. Ortega, W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables. Classics in Applied Mathematics, SIAM, 2000. ISBN 0-898-71461-3.

Category:Celestial mechanics

Celestial mechanics is an application of physics, particularly Newtonian mechanics, to astronomical objects such as stars and planets. Category:Classical mechanics Category:Astronomy ko:분류:천체역학

Category:Astrodynamics

Astrodynamics is a term application of physics, particularly Newtonian mechanics, to space objects such as stars, planets and satellites. Category: Celestial mechanics Category: Astronomy

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Hellenistic
--------- HELLENISMO (n) [hel-le-nì-smo] --------- Etymologia: Ab le lat. hellenismu(m), ab le gr. hellìnismós 'modo, locution grec', deriv. de héllìn -inos 'hellenico', Definition:
- Le phenomeno de diffusion del civilisation grec in le mundo mediterranee e asiatic.
- Le periodo historic in le qual tal phenomeno eveniva, que se colloca conventionalmente inter le morte de Alexandro le grande (323 a. C.) e le battalia de Atio (31 a. C.).
- Le complexo de factos politic e cultural que derivava de

Keplerian problem

Application

Satellite

Artificial satellites

History of artificial satellites

Types of satellites

Orbit types

Countries with satellite launch capability

See also

Reference

External links

Kepler's laws of planetary motion

Kepler's first law

Kepler's second law

Kepler's third law

Accuracy and limitations

Connection to Newton's laws and conservation laws

Kepler's first law

Kepler's second law

Kepler's third law

Proving Kepler's third law

Solution for the motion as a function of time

See also

External links

Semimajor axis

Ellipse

Hyperbola

Astronomy

Orbital period

Average distance

Energy; calculation of semi-major axis from state vectors

Example

References

Semiminor axis

Ellipse

Hyperbola

Eccentricity (orbit)

Calculation

Examples

See also

External links

Periapsis

Formulae

Terminology

See also

True anomaly

Calculation from state vectors

Other relations

See also

Kepler's laws of planetary motion

Kepler's first law

Kepler's second law

Kepler's third law

Accuracy and limitations

Connection to Newton's laws and conservation laws

Kepler's first law

Kepler's second law

Kepler's third law

Proving Kepler's third law

Solution for the motion as a function of time

See also

External links

Transcendental

Newton's method

Description of the method

Example

History

Practical considerations

Generalizations

Complex functions

References

See also

Category:Celestial mechanics

Category:Astrodynamics

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