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Eccentricity (orbit)

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

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.

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. 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. 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

- 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. Therefore, the object can reach infinite

r

only if this quantity is nonnegative, which implies

v\geq\sqrt

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.

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.

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. Keplerian problem.]] 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. 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

. 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

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

\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 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.

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. 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 [http://mathworld.wolfram.com/TesseralHarmonic.html] 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 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: [http://www.psatellite.com/products/manuals/SCTUsersGuideBook1.pdf] and [http://www.psatellite.com/products/manuals/SCTUsersGuideBook2.pdf] 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 (r_e/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 superhighway 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 Superhighway, 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. They have, however, been employed on projects such as Genesis. This spacecraft visited Earth's lagrange L1 point and returned using very little propellant.

Reference

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

External links

- [http://www.braeunig.us/space/orbmech.htm ORBITAL MECHANICS] (Rocket and Space Technology)
- [http://jat.sourceforge.net Java Astrodynamics Toolkit]
-

Standard assumptions in astrodynamics

For most of the problems in astrodynamics involving two bodies

m_1\,

and

m_2\,

standard assumptions are usually the following:
- A1:

m_1\,

and

m_2\,

are the only objects in the universe and thus influence of other objects is disregarded,
- A2: The orbiting body (

m_2\,

) is far smaller than central body (

m_1\,

), i.e.: :

<<1

Results:
- A3: As the result of disparities in masses between

m_1\,

and

m_2\,

standard gravitational parameter (

\mu\,

) includes only the mass of the central body, i.e.: :

\mu=G\simeq(m_1+m_2)

where

G\,

is a gravitational constant.
- A4: Orbit of orbiting body is not perturbed in any way, so the only orbits allowed are circular, elliptic, parabolic or hyperbolic.
- A5: One focus of orbiting body's orbit coincides with the center of the central body, The center of the central body can be taken as the origin of an inertial frame of reference for the orbiting body,

Examples where those assumptions do not hold

- A1:
- although escape velocity is described as a velocity that should allow an orbiting body to coast to infinity and arrive there with zero velocity for most cases this will not be. E.g. even if the spacecraft is launched with escape velocity with respect to Earth it will not escape to infinity (e.g. leave the Solar system) because it will eventually succumb to the gravitational influence of the Sun.
- a rocket applying thrust
- in the case of atmospheric drag
- A2: a binary star

Two bodies orbiting each other

If A2 is not fulfilled, many results still apply with a small modification, see the two-body problem in astrodynamics.

Conic section

In mathematics, a conic section (or just conic) is a curved locus of points, formed by intersecting a cone with a plane. The conic sections were named and studied as long ago as 200 BC, when Apollonius of Perga undertook a systematic study of their properties.

Types of conics

Two well-known conics are the circle and the ellipse. These arise when the intersection of cone and plane is a closed curve. The circle is a special case of the ellipse in which the plane is perpendicular to the axis of the cone. If the plane is parallel to a generator line of the cone, the conic is called a parabola. Finally, if the intersection is an open curve and the plane is not parallel to a generator line of the cone, the figure is a hyperbola. (In this case the plane will intersect both halves of the cone, producing two separate curves, though often one is ignored.) The degenerate cases, where the plane passes through the apex of the cone, resulting in an intersection figure of a point, a straight line, or a pair of intersecting lines, are often excluded from the list of conic sections. In the Cartesian coordinate system, the graph of a quadratic equation in two variables is always a conic section, and all conic sections arise in this way. If the equation is of the form :

ax^2 + 2hxy + by^2 +2gx + 2fy + c = 0\;

quadratic equation then:
- if h² = ab, the equation represents a parabola;
- if h² < ab and a

\ne

b and/or h

\ne

0 , the equation represents an ellipse;
- if h² > ab, the equation represents a hyperbola;
- if h² < ab and a = b and h = 0, the equation represents a circle;
- if a + b = 0, the equation represents a rectangular hyperbola.

Eccentricity

An alternative definition of conic sections starts with a point F (the focus), a line L (the directrix) not containing F and a positive number e (the eccentricity). The corresponding conic section consists of all points whose distance to F equals e times their distance to L. For 0 < e < 1 we obtain an ellipse, for e = 1 a parabola, and for e > 1 a hyperbola. For an ellipse and a hyperbola, two focus-directrix combinations can be taken, each giving the same full ellipse or hyperbola. The distance from the center to the directrix is

\over

, where

a

is the semi-major axis of the ellipse, or the distance from the center to the tops of the hyperbola. The distance from the center to a focus is

ae

. In the case of a circle e = 0 and one imagines the directrix to be infinitely far removed from the center. However, the statement that the circle consists of all points whose distance is e times the distance to L is not useful, because we get zero times infinity. The eccentricity of a conic section is thus a measure of how far it deviates from being circular. For a given

a

, the closer

e

is to 1, the smaller is the semi-minor axis.

Semi-latus rectum and polar coordinates

semi-minor axis The semi-latus rectum of a conic section, usually denoted l, is the distance from the single focus, or one of the two foci, to the conic section itself, measured along a line perpendicular to the major axis. It is related to a and b by the formula

al=b^2\,\!

, or

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

. In polar coordinates, a conic section with one focus at the origin and, if any, the other on the positive x-axis, is given by the equation :

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

Properties

Conic sections are always "smooth". More precisely, they never contain any inflection points. This is important for many applications, such as aerodynamics, where a smooth surface is required to ensure laminar flow/prevent turbulence.

Applications

Conic sections are important in astronomy: the orbits of two massive objects that interact according to Newton's law of universal gravitation are conic sections if their common center of mass is considered to be at rest. If they are bound together, they will both trace out ellipses; if they are moving apart, they will both follow parabolas or hyperbolas. See two-body problem. In projective geometry, the conic sections in the projective plane are equivalent to each other up to projective transformations. For specific applications of each type of conic section, see the articles circle, ellipse, parabola, and hyperbola.

Dandelin spheres

See Dandelin spheres for a short elementary argument showing that the characterization of these curves as intersections of a plane with a cone is equivalent to the characterization in terms of foci, or of a focus and a directrix.

Derivation

Let there be a cone whose axis is the z-axis. Let its vertex be the origin. The equation for the cone is :

x^2 + y^2 - a^2 z^2 = 0 \qquad \qquad (1)

where :

a = \tan \theta > 0 \;

and

\theta

is the angle which the generators of the cone make with respect to the axis. Notice that this cone is actually a pair of cones: one cone standing upside down on the vertex of the other cone—or, as mathematicians say, this cone consists of two "nappes." Let there be a plane with a slope running along the x direction but which is level along the y direction. Its equation is :

z = mx + b \qquad \qquad (2)

where :

m = \tan \phi > 0 \;

and

\phi

is the angle of the plane with respect to the x-y plane. We are interested in finding the intersection of the cone and the plane, which means that equations (1) and (2) shall be combined. Both equations can be solved for z and then equate the two values of z. Solving equation (1) for z yields :

z = \sqrt

therefore :

\sqrt = m x + b.

Square both sides and expand the squared binomial on the right side, :

= m^2 x^2 + 2 m b x + b^2. \;

Grouping by variables yields :

x^2 \left(  - m^2 \right) +  - 2 m b x - b^2 = 0. \qquad \qquad (3)

Note that this is the equation of the projection of the conic section on the xy-plane, hence contracted in the x-direction compared with the shape of the conic section itself.

Derivation of the parabola

The parabola is obtained when the slope of the plane is equal to the slope of the generators of the cone. When these two slopes are equal, then the angles

\theta

and

\phi

become complementary. This implies that :

\tan \theta = \cot \phi \;

therefore :

m = . \qquad \qquad (4)

Substituting equation (4) into equation (3) makes the first term in equation (3) vanish, and the remaining equation is :

-  b x - b^2 = 0.

Multiply both sides by a², :

y^2 - 2 a b x - a^2 b^2 = 0 \;

then solve for x, :

x =  y^2 - . \qquad \qquad (5)

Equation (5) describes a parabola whose axis is parallel to the x-axis. Other versions of equation (5) can be obtained by rotating the plane around the z-axis.

Derivation of the ellipse

An ellipse happens when the angles

\theta

and

\phi

, when added together, do not measure up to a right angle: :

\theta + \phi <  \qquad \qquad \mbox

which implies that the tangent of the sum of these two angles is positive. :

\tan (\theta + \phi) > 0. \;

But a trigonometric identity states that :

\tan (\theta + \phi) =

therefore :

\tan (\theta + \phi) =  > 0 \qquad \qquad (6)

but m + a is positive, since the summands are given to be positive, so inequality (6) is positive if the denominator is also positive: :

1 - m a > 0. \qquad \qquad (7)

From inequality (7) we can deduce :

m a < 1, \;

m^2 a^2 < 1, \;

1 - m^2 a^2 > 0, \;

> 1,

- 1 > 0,

- m^2 > 0 \qquad \qquad \mbox.

Let us start out again from equation (3), :

x^2 \left(  - m^2 \right) +  - 2 m b x - b^2 = 0, \qquad \qquad (3)

but this time the coefficient of the x² term does not vanish but is instead positive. Solve for y, :

y = a \sqrt. \qquad \qquad (8)

This would clearly describe an ellipse were it not for the second term under the radical, the 2 m b x: it would be the equation of a circle which has been stretched proportionally along the directions of the x-axis and the y-axis. Equation (8) is an ellipse but it is not obvious, so it will be rearranged further until it is obvious. Complete the square under the radical, :

y = a \sqrt.

Group together the b² terms, :

y = a \sqrt.

Divide by a then square both sides, :

+ \left( x \sqrt -  \right)^2 = b^2 \left( 1 +  \right).

The x has a coefficient. It is desired to pull this coefficient out by factoring it out of the second term which is a square, :

+ \left(  - m^2 \right) \left( x -  \right)^2 = b^2 \left( 1 +  \right).

Further rearrangements of constants finally leads to :

+ \left( x -  \right)^2 = .

The coefficient of the y term is positive (for an ellipse). Renaming of coefficients and constants leads to :

+ (x - C)^2 = R^2 \qquad \qquad (9)

which is clearly the equation of an ellipse. That is, equation (9) describes a circle of radius R and center (C,0) which is then stretched vertically by a factor of

\sqrt

. The second term on the left side (the x term) has no coefficient but is a square, so that it must be positive. The radius is a product of squares, so it must also be positive. The first term on the left side(the y term) has a coefficient which is positive (one of the inequalities derived earlier), so the equation describes an ellipse.

Derivation of the hyperbola

The hyperbola happens when the angles

\theta

and

\phi

add up to an obtuse angle, which is greater than a right angle. The tangent of an obtuse angle is negative. All the inequalities which were valid for the ellipse become reversed. Therefore :

1 - a^2 m^2 < 0 \qquad \qquad \mbox.

Otherwise the equation for the hyperbola is the same as equation (9) for the ellipse, except that the coefficient A of the y term is negative. The sign change is enough to convert an ellipse into a hyperbola. This is because the equation of a real ellipse contains an imaginary hyperbola, and the equation of a real hyperbola contains an imaginary ellipse (see imaginary number). The sign change of coefficient A causes real and imaginary values of the function y=f(x) equivalent to equation (9) to swap.

External links

- [http://xahlee.org/SpecialPlaneCurves_dir/ConicSections_dir/conicSections.html Special plane curves: Conic sections]
- http://mathworld.wolfram.com/Focus.html
- [http://ccins.camosun.bc.ca/~jbritton/jbconics.htm Occurrence of the conics] in nature and elsewhere
- [http://fishrock.com/conics/default.htm Conic structures] in architecture
- Category:Euclidean solid geometry ja:円錐曲線

Eccentricity (mathematics)

(This page refers to eccentricity in mathematics. For other uses, see the disambiguation page eccentricity.) In mathematics, eccentricity is a parameter associated with every conic section, see Conic section#Eccentricity. It can be thought of as a measure of how much the conic section deviates from being circular. In particular,
- The eccentricity of a circle is zero.
- The eccentricity of an ellipse is greater than zero and less than 1.
- The eccentricity of a parabola is 1.
- The eccentricity of a hyperbola is greater than 1 and less than infinity.
- The eccentricity of a straight line is infinity. It is given by: :

e = \sqrt

Where a is the length of the semimajor axis of the section, b the length of the semiminor axis, and k is equal to +1 for an ellipse, 0 for a parabola, and -1 for a hyperbola. It is also called the first eccentricity when necessary to distinguish it from the second eccentricity, e', which is sometimes used for algebraic convenience. The second eccentricity is defined as: :

e' = \sqrt

And is related to the first eccentricity by the equation: :

1 = (1 - e^2)(1 + e'^2)\,\!

Ellipse

semiminor axis For any ellipse, where the length of the semi-major axis is a, and where the same of the semi-minor axis is b, the eccentricity is given by: :

e = \sqrt

The eccentricity is the ratio of the distance between the foci (

F_1

and

F_2

) to the major axis; i.e.

\left ( \frac \right )

. The term linear eccentricity is used for

Hyperbola

For any hyperbola, where the length of the semi-major axis is a, and where the same of the semi-minor axis is b, eccentricity is given by: :

e = \sqrt

Surfaces

The eccentricity of a surface is the eccentricity of a designated section of the surface. For example, on a triaxial ellipsoid, the meridional eccentricity is that of the ellipse formed by a section containing both the longest and the shortest axes (one of which will be the polar axis), and the equatorial eccentricity is the eccentricity of the ellipse formed by a section through the centre, perpendicular to the polar axis (i.e. in the equatorial plane).

External links

- [http://mathworld.wolfram.com/Eccentricity.html MathWorld: Eccentricity] Category:Conic sections als:Exzentrizität (Mathematik)

Standard assumptions in astrodynamics

For most of the problems in astrodynamics involving two bodies

m_1\,

and

m_2\,

standard assumptions are usually the following:
- A1:

m_1\,

and

m_2\,

are the only objects in the universe and thus influence of other objects is disregarded,
- A2: The orbiting body (

m_2\,

) is far smaller than central body (

m_1\,

), i.e.: :

<<1

Results:
- A3: As the result of disparities in masses between

m_1\,

and

m_2\,

standard gravitational parameter (

\mu\,

) includes only the mass of the central body, i.e.: :

\mu=G\simeq(m_1+m_2)

where

G\,

Examples where those assumptions do not hold

Two bodies orbiting each other

If A2 is not fulfilled, many results still apply with a small modification, see the two-body problem in astrodynamics.

Elliptic orbit

In astrodynamics or celestial mechanics a elliptic orbit is an orbit with the eccentricity greater than 0 and less than 1. Specific energy of an elliptical orbit is negative. An orbit with a eccentricity of 0 is a circular orbit. Examples of elliptic orbits include: Hohmann transfer orbit, Molniya orbit and tundra orbit.

Velocity

Under standard assumptions the orbital velocity (

v\,

) of a body traveling along elliptic orbit can be computed as: :

v=\sqrt

where:
-

\mu\,

is standard gravitational parameter,
-

r\,

is radial distance of orbiting body from central body,
-

a\,\!

is length of semi-major axis. Conclusion:
- Velocity does not depend on eccentricity but is determined by length of semi-major axis (

a\,\!

),
- Velocity equation is similar to that for hyperbolic trajectory with the difference that for the latter,

is positive.

Orbital period

Under standard assumptions the orbital period (

T\,\!

) of a body traveling along elliptic orbit can be computed as: :

T=a^

where:
-

\mu\,

is standard gravitational parameter,
-

a\,\!

is length of semi-major axis. Conclusions:
- The orbital period is equal to that for a circular orbit with the orbit radius equal to the semi-major axis (

a\,\!

),
- The orbital period does not depend on the eccentricity (See also: Kepler's third law).

Energy

Under standard assumptions, specific orbital energy (

\epsilon\,

) of elliptic orbit is negative and the orbital energy conservation equation for this orbit takes form: :

-=-=\epsilon<0

where:
-

v\,

is orbital velocity of orbiting body,
-

r\,

is radial distance of orbiting body from central body,
-

a\,

is length of semi-major axis,
-

\mu\,

is standard gravitational parameter. Conclusions:
- Specific energy for elliptic orbits is independent of eccentricity and is determined only by semi-major axis of the ellipse. Using the virial theorem we find:
- the time-average of the specific potential energy is equal to 2ε
- the time-average of r^-1 is a^-1
- the time-average of the specific kinetic energy is equal to -ε

Flight path angle

Equation of motion

See orbit equation.

Orbital parameters

Solar system

In the solar system planets, asteroids, comets and space debris have elliptical orbits around the Sun. Moons have an elliptic orbit around their planet. Many artificial satellites have various elliptic orbits around the Earth.

Parabolic trajectory

In astrodynamics or celestial mechanics a parabolic trajectory is an orbit with the eccentricity equal to 1. When moving away from the source it is called an escape orbit, otherwise a capture orbit. Under standard assumptions a body traveling along an escape orbit will coast to infinity, with velocity relative to the central body tending to zero, and therefore will never return. Parabolic trajectory is a minimum-energy escape trajectory.

Velocity

Under standard assumptions the orbital velocity (

v\,

) of a body traveling along parabolic trajectory can be computed as: :

v=\sqrt

where:
-

r\,\!

is radial distance of orbiting body from central body,
-

\mu\,\!

is standard gravitational parameter. At any position the orbiting body has the escape velocity for that position. If the body has the escape velocity with respect to the Earth, this is not enough to escape the Solar System, so near the Earth the orbit resembles a parabola, but further away it bends into an elliptical orbit around the Sun. ---- This velocity (

v\,

) is closely related to the orbital velocity of a body in a circular orbit of the radius equal to the radial position of orbiting body on the parabolic trajectory: :

v=\sqrt\cdot v_O

where:
-

v_O\,\!

is orbital velocity of a body in circular orbit.

Equation of motion

Under standard assumptions, for a body moving along this kind of trajectory an orbital equation becomes: :

r=

where:
-

r\,

is radial distance of orbiting body from central body,
-

h\,

is specific angular momentum of the orbiting body,
-

\theta\,

is a true anomaly of the orbiting body,
-

\mu\,

is standard gravitational parameter.

Energy

Under standard assumptions, specific orbital energy (

\epsilon\,

) of parabolic trajectory is zero, so the orbital energy conservation equation for this trajectory takes form: :

\epsilon=-=0

where:
-

v\,

is orbital velocity of orbiting body,
-

r\,

is radial distance of orbiting body from central body,
-

\mu\,

is standard gravitational parameter.

Flight path angle

Orbit

.]] :For other meanings of the term "orbit", see orbit (disambiguation) In physics, an orbit is the path that an object makes around another object while under the influence of a source of centripetal force, such as gravity.

History

Orbits were first analysed mathematically by Johannes Kepler who formulated his results in his laws of planetary motion. He found that the orbits of the planets in our solar system are elliptical, not circular (or epicyclic), as had previously been believed. Isaac Newton demonstrated that Kepler's laws were derivable from his theory of gravitation and that, in general, the orbits of bodies responding to the force of gravity were conic sections. Newton showed that a pair of bodies follow orbits of dimensions that are in inverse proportion to their masses about their common center of mass. Where one body is much more massive than the other, it is a convenient approximation to take the center of mass as coinciding with the center of the more massive body.

Planetary orbits

Within a planetary system, planets, asteroids, comets and space debris orbit the central star in elliptical orbits. Any comet in a parabolic or hyperbolic orbit about the central star is not gravitationally bound to the star and therefore is not considered part of the star's planetary system. To date, no comet has been observed in our solar system with a distinctly hyperbolic orbit. Bodies which are gravitationally bound to one of the planets in a planetary system, either natural or artificial satellites, follow orbits about that planet. Due to mutual gravitational perturbations, the eccentricities of the orbits of the planets in our solar system vary over time. Pluto and Mercury have the most eccentric orbits. At the present epoch, Mars has the next largest eccentricity while the smallest eccentricities are those of the orbits of Venus and Neptune. As an object orbits another, the periapsis is that point at which the two objects are closest to each other and the apoapsis is that point at which they are the farthest from each other. In the elliptical orbit, the centre of mass of the orbiting-orbited system will sit at one focus of both orbits, with nothing present at the other focus. As a planet approaches periapsis, the planet will increase in velocity. As a planet approaches apoapsis, the planet will decrease in velocity. See also: Kepler's laws of planetary motion

Understanding orbits

There are a few common ways of understanding orbits.
- As the object moves sideways, it falls toward the orbited object. However it moves so quickly that the curvature of the orbited object will fall away beneath it.
- A force, such as gravity, pulls the object into a curved path as it attempts to fly off in a straight line.
- As the object falls, it moves sideways fast enough (has enough tangential velocity) to miss the orbited object. This understanding is particularly useful for mathematical analysis, because the object's motion can be described as the sum of the three one-dimensional coordinates oscillating around a gravitational center. As an illustration of the orbit around a planet (eg Earth), the much-used cannon model may prove useful (see image below). Imagine a cannon sitting on top of a (very) tall mountain, which fires a cannonball horizontally. The mountain needs to be very tall, so that the cannon will be above the Earth's atmosphere and we can ignore the effects of air friction on the cannon ball. 300px If the cannon fires its ball with a low initial velocity, the trajectory of the ball will curve downwards and hit the ground (A). As the firing velocity is increased, the cannonball will hit the ground further (B) and further (C) away from the cannon, because while the ball is still falling towards the ground, the ground is curving away from it (see first point, above). If the cannonball is fired with sufficient velocity, the ground will curve away from the ball at the same rate as the ball falls - it is now in orbit (D). The orbit may be circular like (D) or if the firing velocity is increased even more, the orbit may become more (E) and more (F) elliptical. At a certain even faster velocity (called the escape velocity) the motion changes from an elliptical orbit to a parabola.

Newton's laws of motion

For a system of only two bodies that are only influenced by their mutual gravity, their orbits can be exactly calculated by Newton's laws of motion and gravity. Briefly, the sum of the forces will equal the mass times its acceleration. Gravity is proportional to mass, and falls off proportionally to the square of distance. To calculate, it is convenient to describe the motion in a coordinate system that is centered on the heavier body, and we can say that the lighter body is in orbit around the heavier body. An unmoving body that's far from a large object has more energy than one that's close. This is because it can fall farther. This is called "potential energy" because it is not yet actual. With two bodies, an orbit is a flat curve. The orbit can be open (so the object never returns) or closed (returning), depending on the total kinetic + potential energy of the system. In the case of an open orbit, the speed at any position of the orbit is at least the escape velocity for that position, in the case of a closed orbit, always less. The path of a free-falling (orbiting) body is always a conic section. An open orbit has the shape of a hyperbola (or in the limiting case, a parabola); the bodies approach each other for a while, curve around each other around the time of their closest approach, and then separate again forever. This is often the case with comets that occasionally approach the Sun. A closed orbit has the shape of an ellipse (or in the limiting case, a circle). The point where the orbiting body is closest to Earth is the perigee, called periapsis (less properly, "perifocus" or "pericentron") when the orbit is around a body other than Earth. The point where the satellite is farthest from Earth is called apogee, apoapsis, or sometimes apifocus or apocentron. A line drawn from periapsis to apoapsis is the line-of-apsides. This is the major axis of the ellipse, the line through its longest part. Orbiting bodies in closed orbits repeat their path after a constant period of time. This motion is described by the empirical laws of Kepler, which can be mathematically derived from Newton's laws. These can be formulated as follows: # The orbit of a planet around the Sun is an ellipse, with the Sun in one of the focal points of the ellipse. Therefore the orbit lies in a plane, called the orbital plane. The point on the orbit closest to the attracting body is the periapsis. The point farthest from the attracting body is called the apoapsis. There are also specific terms for orbits around particular bodies; things orbiting the Sun have a perihelion and aphelion, things orbiting the Earth have a perigee and apogee, and things orbiting the Moon have a perilune and apolune (or, synonymously, periselene and aposelene). An orbit around any star, not just the Sun, has a periastron and an apastron # As the planet moves around its orbit during a fixed amount of time, the line from Sun to planet sweeps a constant area of the orbital plane, regardless of which part of its orbit the planet traces during that period of time. This means that the planet moves faster near its perihelion than near its aphelion, because at the smaller distance it needs to trace a greater arc to cover the same area. This law is usually stated as "equal areas in equal time." # For each planet, the ratio of the 3rd power of its semi-major axis to the 2nd power of its period is the same constant value for all planets. Except for special cases like Lagrangian points, no method is known to solve the equations of motion for a system with four or more bodies. The 2-body solutions were published by Newton in Principia in 1687. In 1912, K. F. Sundman developed a converging infinite series that solves the 3-body problem, however it converges too slowly to be of much use. Instead, orbits can be approximated with arbitrarily high accuracy. These approximations take two forms. One form takes the pure elliptic motion as a basis, and adds perturbation terms to account for the gravitational influence of multiple bodies. This is convenient for calculating the positions of astronomical bodies. The equations of motion of the moon, planets and other bodies are known with great accuracy, and are used to generate tables for celestial navigation. The differential equation form is used for scientific or mission-planning purposes. According to Newton's laws, the sum of all the forces will equal the mass times its acceleration (F = ma). Therefore accelerations can be expressed in terms of positions. The perturbation terms are much easier to describe in this form. Predicting subsequent positions and velocities from initial ones corresponds to solving an initial value problem. Numerical methods calculate the positions and velocities of the objects a tiny time in the future, then repeat this. However, tiny arithmetic errors from the limited accuracy of a computer's math accumulate, limiting the accuracy of this approach. Differential simulations with large numbers of objects perform the calculations in a hierarchical pairwise fashion between centers of mass. Using this scheme, galaxies, star clusters and other large objects have been simulated.

Analysis of orbital motion

(see also orbit equation and Kepler's first law) To analyse the motion of a body moving under the influence of a force which is always directed towards a fixed point, it is convenient to use polar coordinates with the origin coinciding with the centre of force. In such coordinates the radial and transverse components of the acceleration are, respectively: :

\frac - r\left( \frac \right)^2

and :

\frac\frac\left( r^2\frac \right)

. Since the force is always radial, the transverse acceleration is zero, and it follows that: :

\frac = hu^2

, where h is a constant of integration and we have introduced the auxiliary variable u defined as 1/r. If magnitude of the radial force is f(r) per unit mass of the orbiting body, then the elimination of the time variable from the radial component of the equation of motion yields: :

\frac + u = \frac

. In the case of an inverse square force law the right hand side of the equation becomes a constant and the equation is seen to be the harmonic equation (up to a shift of origin of the dependent variable). The equation of the orbit described by the particle is thus: :

r = \frac = \frac

, where φ and e are constants of integration and L is the Semi-latus rectum. This can be recognised as the equation of a conic section in polar coordinates.

Orbital parameters

See: Orbital elements For a general elliptic orbit, the relations between the axis, eccentricity, and least and largest distance are: :Semimajor axis = (periapsis + apoapsis)/2 = mean of the extreme radii :Periapsis = semimajor axis × (1 - eccentricity) = least distance :Apoapsis = semimajor axis × (1 + eccentricity) = largest distance Note that there are alternative definitions for a "mean radius" or "average distance": if you average the radius over time for one orbit (mean anomaly), or over the orbital angle as observed by the primary (true anomaly), then you get a different result. See here for details.

Orbital period

See: orbital period

Orbital decay

If some part of a body's orbit enters an atmosphere, its orbit can decay because of drag. At each periapsis, the object scrapes the air, losing energy. Each time, the orbit grows less eccentric (more circular) because the object loses kinetic energy precisely when that energy is at its maximum. Eventually, the orbit circularises and then the object spirals into the atmosphere. The bounds of an atmosphere vary wildly. During solar maxima, the Earth's atmosphere causes drag up to a hundred kilometres higher than during solar minimums. Some satellites with long conductive tethers can also decay because of electromagnetic drag from the Earth's magnetic field. Basically, the wire cuts the magnetic field, and acts as a generator. The wire moves electrons from the near vacuum on one end to the near-vacuum on the other end. The orbital energy is converted to heat in the wire. Another method of artificially influencing an orbit is through the use of solar sails or magnetic sails. These forms of propulsion require no propellant or energy input, and so can be used indefinitely. See statite for one such proposed use. Orbital decay can also occur due to tidal forces for objects below the synchronous orbit for the body they're orbiting. The gravity of the orbiting object raises tidal bulges in the primary, and since below the synchronous orbit the orbiting object is moving faster than the body's surface the bulges lag a short angle behind it. The gravity of the bulges is slightly off of the primary-satellite axis and thus has a component along the satellite's motion. The near bulge slows the object more than the far bulge speeds it up, and as a result the orbit decays. Conversely, the gravity of the satellite on the bulges applies torque on the primary and speeds up its rotation. Artificial satellites are too small to have an appreciable tidal effect on the planets they orbit, but several moons in the solar system are undergoing orbital decay by this mechanism. Mars' innermost moon Phobos is a prime example, and is expected to either impact Mars' surface or break up into a ring within 50 million years. Finally, orbits can decay via the emission of gravitational waves. This mechanism is extremely weak for most stellar objects, only becoming significant in cases where there is a combination of extreme mass and extreme acceleration, such as with black holes or neutron stars that are orbiting each other closely.

Earth orbits

See Earth orbit for more details.
- Low Earth orbit
- High Earth Orbit
- Intermediate circular orbit
- Geostationary orbit
- Geosynchronous orbit
- Geostationary transfer orbit
- Molniya orbit
- Polar orbit
- Polar Sun Synchronous Orbit (this is not a complete list).

Scaling in gravity

The gravitational constant G is defined to be:
- 6.6742 × 10⁻¹¹ N·m²/kg²
- 6.6742 × 10⁻¹¹ m³/(kg·s²)
- 6.6742 × 10⁻¹¹(kg/m³)^-1s^-2. Thus the constant has dimension density^-1 time^-2. This corresponds to the following properties. Scaling of distances (including sizes of bodies, while keeping the densities the same) gives similar orbits without scaling the time: if for example distances are halved, masses are divided by 8, gravitational forces by 16 and gravitational accelerations by 2. Hence orbital periods remain the same. Similarly, when an object is dropped from a tower, the time it takes to fall to the ground remains the same with a scale model of the tower on a scale model of the earth. When all densities are multiplied by four, orbits are the same, but with orbital velocities doubled. When all densities are multiplied by four, and all sizes are halved, orbits are similar, with the same orbital velocities. These properties are illustrated in the formula :

GT^2 \sigma = 3\pi \left( \frac \right)^3,

for an elliptical orbit with semi-major axis a, of a small body around a spherical body with radius r and average density σ, where T is the orbital period.

Role in the evolution of atomic theory

When atomic structure was first probed experimentally early in the twentieth century, an early picture of the atom portrayed it as a miniature solar system bound by the coulomb force rather than by gravity. This was inconsistent with electrodynamics and the model was progressively refined as quantum theory evolved, but there is a legacy of the picture in the term orbital for the wave function of an energetically bound electron state.

External links

- An on-line orbit plotter: http://www.bridgewater.edu/departments/physics/ISAW/PlanetOrbit.html
- [http://www.braeunig.us/space/orbmech.htm Orbital Mechanics] (Rocket and Space Technology) Category:Celestial mechanics Category:Solar System als:Umlaufbahn ja:軌道 (力学) simple:Orbit th:วงโคจร

Elliptic orbit

Velocity

Under standard assumptions the orbital velocity (

v\,

) of a body traveling along elliptic orbit can be computed as: :

v=\sqrt

where:
-

\mu\,

is standard gravitational parameter,
-

r\,

is radial distance of orbiting body from central body,
-

a\,\!

is length of semi-major axis. Conclusion:
- Velocity does not depend on eccentricity but is determined by length of semi-major axis (

a\,\!

),
- Velocity equation is similar to that for hyperbolic trajectory with the difference that for the latter,

is positive.

Orbital period

Under standard assumptions the orbital period (

T\,\!

) of a body traveling along elliptic orbit can be computed as: :

T=a^

where:
-

\mu\,

is standard gravitational parameter,
-

a\,\!

is length of semi-major axis. Conclusions:
- The orbital period is equal to that for a circular orbit with the orbit radius equal to the semi-major axis (

a\,\!

),
- The orbital period does not depend on the eccentricity (See also: Kepler's third law).

Energy

Under standard assumptions, specific orbital energy (

\epsilon\,

) of elliptic orbit is negative and the orbital energy conservation equation for this orbit takes form: :

-=-=\epsilon<0

where:
-

v\,

is orbital velocity of orbiting body,
-

r\,

is radial distance of orbiting body from central body,
-

a\,

is length of semi-major axis,
-

\mu\,

Flight path angle

Equation of motion

See orbit equation.

Orbital parameters

Solar system

Hyperbolic trajectory

In astrodynamics or celestial mechanics a hyperbolic trajectory is an orbit with the eccentricity greater than 1. Under standard assumptions a body traveling along this trajectory will coast to infinity, arriving there with hyperbolic excess velocity relative to the central body. Similarly to parabolic trajectory all hyperbolic trajectories are also escape trajectories. Specific energy of hyperbolic trajectory orbit is positive.

Hyperbolic excess velocity

Under standard assumptions the body traveling along along hyperbolic trajectory will attain in infinity an orbital velocity called hyperbolic excess velocity (

v_\infty\,\!

) that can be computed as: :

v_\infty=\sqrt\,\!

where:
-

\mu\,\!

is standard gravitational parameter,
-

a\,\!

is length of semi-major axis of orbit's hyperbola. The hyperbolic excess velocity is related to the specific orbital energy or characteristic energy by :

2\epsilon=2C_3=v_^2\,\!

Velocity

Under standard assumptions the orbital velocity (

v\,

) of a body traveling along hyperbolic trajctory can be computed as: :

v=\sqrt

where:
-

\mu\,

is standard gravitational parameter,
-

r\,

is radial distance of orbiting body from central body,
-

a\,\!

is length of semi-major axis. Under standard assumptions, at any position in the orbit the following relation holds for orbital velocity (

v\,

), local escape velocity(

\,

) and hyperbolic excess velocity (

v_\infty\,\!

): :

v^2=^2+^2

Note that this means that a relatively small extra delta-v above that needed to accelerate to the escape speed, results in a relatively large speed at infinity.

Energy

Under standard assumptions, specific orbital energy (

\epsilon\,

) of a hyperbolic trajectory is greater than zero and the orbital energy conservation equation for this kind of trajectory takes form: :

\epsilon=-=

where:
-

v\,

is orbital velocity of orbiting body,
-

r\,

is radial distance of orbiting body from central body,
-

a\,

is length of semi-major axis,
-

\mu\,

is standard gravitational parameter.

External links

http://www.cix.co.uk/~sjbradshaw/msc/traject.html http://www.go.ednet.ns.ca/~larry/orbits/ellipse.html Category:Astrodynamics Category:Celestial mechanics

Orbit

History

Planetary orbits

Understanding orbits

Newton's laws of motion

Analysis of orbital motion

\frac - r\left( \frac \right)^2

and :

\frac\frac\left( r^2\frac \right)

. Since the force is always radial, the transverse acceleration is zero, and it follows that: :

\frac = hu^2

\frac + u = \frac

r = \frac = \frac

, where φ and e are constants of integration and L is the Semi-latus rectum. This can be recognised as the equation of a conic section in polar coordinates.

Orbital parameters

Orbital period

See: orbital period

Orbital decay

Earth orbits

Scaling in gravity

GT^2 \sigma = 3\pi \left( \frac \right)^3,

for an elliptical orbit with semi-major axis a, of a small body around a spherical body with radius r and average density σ, where T is the orbital period.

Role in the evolution of atomic theory

External links

Orbital state vectors

In astrodynamics or celestial dynamics orbital state vectors (sometimes State Vectors) are vectors of position (

\mathbf

) and velocity (

\mathbf

) that together with their time (

t\,

) ( epoch) uniquely determine the state of an orbiting body. State vectors are excellent for pre-launch orbital predictions when combined with time (epoch) expressed as an offset to the launch time. This makes the state vectors time-independent and good general prediction for orbit.