Two bodies of different mass
orbiting a common barycenter
. The relative sizes and type of orbit are similar to the Pluto
– Charon
system.
In
physics
, an orbit is the gravitationally curved path of an object around a
point in space, for example the orbit of a planet
around the center
of a star system, such as the solar system
.
Current
understanding of the mechanics of orbital motion is based on Albert
Einstein 's general theory of relativity
, which accounts for gravity as due to curvature of space-time
, with orbits following geodesics
; though in common practice an approximate force-based theory of universal gravitation
based on Kepler's
laws of planetary motion is often used instead for ease of
calculation.
History
Historically,
the apparent motions of the planets were first understood geometrically (and
without regard to gravity) in terms of epicycles
, which are the sums of numerous circular motions.Theories of this kind predicted paths
of the planets moderately well, until Johannes
Kepler was able to show that the motions of
planets were in fact (at least approximately) elliptical motions
In
the geocentric
model of the solar system
, the celestial
spheres model was originally used to explain
the apparent motion of the planets in the sky in terms of perfect spheres or
rings, but after the planets' motions were more accurately measured,
theoretical mechanisms such as deferent and epicycles
were added. Although it was capable of accurately predicting the planets'
position in the sky, more and more epicycles were required over time, and the
model became more and more unwieldy.
The
basis for the modern understanding of orbits was first formulated by Johannes
Kepler whose results are summarised in his
three laws of planetary motion
. First, he found that the orbits of the planets in our solar system
are elliptical , not circular
(or epicyclic
), as had previously been believed, and that the sun is not located at the
center of the orbits, but rather at one focus . [
6 ] Second, he found that the orbital
speed of each planet is not constant, as had previously been thought, but
rather that the speed depends on the planet's distance from the sun.
The lines traced out by orbits
dominated by the gravity of a central source are conic sections
: the shapes of the curves of intersection between a plane and a cone. Parabolic
(1) and hyperbolic
(3) orbits are escape
orbits, whereas elliptical
and circular
orbits (2) are captive.
Isaac
Newton demonstrated that Kepler's laws were
derivable from his theory of gravitation
and that, in general, the orbits of bodies subject to gravity were conic sections
, if the force of gravity propagated instantaneously. Newton showed that, for a pair of
bodies, the orbits' sizes are in inverse proportion to their masses , and that the bodies revolve 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.
Albert
Einstein was able to show that gravity was due
to curvature of space-time
, and thus he was able to remove Newton's assumption that changes propagate
instantaneously. In relativity theory
, orbits follow geodesic
trajectories which approximate very well to the Newtonian predictions. However there are
differences that can be used to determine which theory describes reality more
accurately. Essentially all experimental evidence that
can distinguish between the theories agrees with relativity theory to within
experimental measuremental accuracy, but the differences from Newtonian
mechanics are usually very small (except where there are very strong gravity
fields and very high speeds). However,
the Newtonian solution is still used for most purposes since it is significantly
easier to use. Namun, solusi Newton masih digunakan untuk sebagian besar tujuan
karena lebih mudah digunakan.
Planetary orbits
Within
a planetary
system , planets , dwarf planets
, asteroids
(aka minor planets), comets , and space
debris orbit the barycenter
in elliptical orbits
. A comet in a parabolic
or hyperbolic
orbit about a barycenter is not gravitationally bound to the star and therefore
is not considered part of the star's planetary system. Bodies which are
gravitationally bound to one of the planets in a planetary system, either natural
or artificial satellites
, follow orbits about a barycenter near that planet.
Owing
to mutual gravitational perturbations
, the eccentricities
of the planetary orbits vary over time. Mercury
, the smallest planet in the Solar System, has the most eccentric orbit.At the
present epoch , Mars
has the next largest eccentricity while the smallest orbital eccentricities are
seen in Venus
and Neptune
.
As
two objects orbit each other, 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. (More specific
terms are used for specific bodies. For example, perigee and apogee are the
lowest and highest parts of an Earth orbit.)
In
the elliptical orbit, the center
of mass of the orbiting-orbited system is at
one focus of both orbits, with
nothing present at the other focus. As a planet approaches periapsis, the
planet will increase in speed, or velocity
. As a planet approaches apoapsis, its velocity will decrease.
Understanding orbits
There
are a few common ways of understanding orbits: Ada beberapa cara umum orbit
pemahaman:
- As the object moves sideways, it falls toward the central body. However, it moves so quickly that the central body will curve 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 moves sideways (tangentially), it falls toward the central body. However, it has enough tangential velocity to miss the orbited object, and will continue falling indefinitely. 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 an orbit around a planet, the Newton's
cannonball model may prove useful (see image
below).This is a 'thought experiment', in which a cannon on top of a
tall mountain is able to fire a cannonball horizontally at any chosen muzzle
velocity. The effects of air friction on the cannonball are ignored (or perhaps
the mountain is high enough that the cannon will be above the Earth's
atmosphere, which comes to the same thing.)
Newton's
cannonball , an illustration of how objects can
"fall" in a curve
If
the cannon fires its ball with a low initial velocity, the trajectory of the
ball curves downward and hits the ground (A). As the firing velocity is increased, the cannonball hits the ground
farther (B) away from the cannon, because while the ball is still falling
towards the ground, the ground is increasingly curving away from it (see first
point, above). All these motions are
actually "orbits" in a technical sense — they are describing a
portion of an elliptical path around the center of
gravity — but the orbits are interrupted by striking the Earth.
If
the cannonball is fired with sufficient velocity, the ground curves away from
the ball at least as much as the ball falls — so the ball never strikes the
ground. It is now in what could be called a non-interrupted, or
circumnavigating, orbit. For any specific combination of height above the
center of gravity and mass of the planet, there is one specific firing velocity
(unaffected by the mass of the ball, which is assumed to be very small relative
to the Earth's mass) that produces a circular
orbit , as shown in (C).
As
the firing velocity is increased beyond this, elliptic orbits
are produced; one is shown in (D).If the initial firing is above the
surface of the Earth as shown, there will also be elliptical orbits at slower
velocities; these will come closest to the Earth at the point half an orbit
beyond, and directly opposite, the firing point.
At
a specific velocity called escape
velocity , again dependent on the firing height
and mass of the planet, an open orbit such as (E) results — a parabolic
trajectory .At even faster velocities the object
will follow a range of hyperbolic trajectories
. In a practical sense, both of these trajectory types mean the object is
"breaking free" of the planet's gravity, and "going off into
space".
The
velocity relationship of two moving objects with mass can thus be considered in
four practical classes, with subtypes:
- No orbit
- Suborbital trajectories
- Range of interrupted elliptical paths
- Orbital trajectories (or simply "orbits")
- Range of elliptical paths with closest point opposite firing point
- Circular path
- Range of elliptical paths with closest point at firing point
- Open (or escape) trajectories
- Parabolic paths
- Hyperbolic paths
Newton's laws of motion
In
many situations relativistic effects can be neglected, and Newton's laws
give a highly accurate description of the motion.The acceleration of each body is
equal to the sum of the gravitational forces on it, divided by its mass, and
the gravitational force between each pair of bodies is proportional to the
product of their masses and decreases inversely with the square of the distance
between them. To this Newtonian
approximation, for a system of two point masses or spherical bodies, only
influenced by their mutual gravitation (the two-body
problem ), the orbits can be exactly
calculated. If the heavier body is much more massive than the smaller, as for a satellite
or small moon orbiting a planet or for the Earth orbiting the Sun, it is
accurate and convenient to describe the motion in a coordinate
system that is centered on the heavier body,
and we say that the lighter body is in orbit around the heavier. For the
case where the masses of two bodies are comparable, an exact Newtonian solution
is still available, and qualitatively similar to the case of dissimilar masses,
by centering the coordinate system on the center
of mass of the two.
Energy
is associated with gravitational fields.Since work is required to separate
two bodies against the pull of gravity, their gravitational potential energy
increases as they are separated, and decreases as they approach one another.
For point masses the
gravitational energy decreases without limit as they approach zero separation,
and it is convenient and conventional to take the potential energy as zero when
they are an infinite distance apart, and then negative (since it decreases from
zero) for smaller finite distances.
With
two bodies, an orbit is a conic
section . The orbit can be open (so the object
never returns) or closed (returning), depending on the total energy
( 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. Since the kinetic energy is never negative, if
the common convention is adopted of taking the potential energy as zero at
infinite separation, the bound orbits have negative total energy, parabolic
trajectories have zero total energy, and hyperbolic orbits have positive total
energy.
An
open orbit has the shape of a hyperbola
(when the velocity is greater than the escape velocity), or a parabola
(when the velocity is exactly the escape velocity). The bodies approach each other for a
while, curve around each other around the time of their closest approach, and
then separate again foreverThis may be the case with some comets if they come from outside the solar
system.
A
closed orbit has the shape of an ellipse
. In the special case that the orbiting body is always the same distance from
the center, it is also the shape of a circle
. Otherwise, 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.he point
where the satellite is farthest from Earth is called apogee , apoapsis, or sometimes apifocus or
apocentron. 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. 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 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 a given orbit, the ratio of the cube of its semi-major axis to the square of its period is constant.
Note
that that while bound orbits around a point mass or around a spherical body
with an ideal Newtonian gravitational field are closed ellipses, which repeat
the same path exactly and indefinitely, any non-spherical or non-Newtonian
effects (as caused, for example, by the slight oblateness of the Earth, or by relativistic effects
, changing the gravitational field's behavior with distance) will cause the
orbit's shape to depart from the closed ellipses characteristic of Newtonian
two-body motion. The two-body solutions were published by Newton in Principia
in 1687. Tubuh-solusi dua diterbitkan oleh Newton dalam Principia
pada tahun 1687. In 1912 Karl Fritiof Sundman
developed a converging infinite series that solves the three-body problem;
however, it converges too slowly to be of much use. Except for special
cases like the Lagrangian points
, no method is known to solve the equations of motion for a system with four or
more bodies.
Instead,
orbits with many bodies 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 . Still there are secular
phenomena that have to be dealt with by post-newtonian
methods.
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 values corresponds to solving an initial
value problem . Numerical methods calculate the
positions and velocities of the objects a short time in the future, then repeat
the calculation. However, tiny arithmetic
errors from the limited accuracy of a computer's math are cumulative, which
limits 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
Note
that the following is a classical ( Newtonian
) analysis of orbital
mechanics , which assumes that the more subtle
effects of general
relativity , such as frame dragging
and gravitational
time dilation are negligible. Relativistic effects
cease to be negligible when near very massive bodies (as with the precession of
Mercury's orbit about the Sun), or when extreme
precision is needed (as with calculations of the orbital
elements and time signal references for GPS satellites.
To
analyze 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 center of force.
and
Since
the force is entirely radial, and since acceleration is proportional to force,
it follows that the transverse acceleration is zero. As a result,
After
integrating, we have
which
is actually the theoretical proof of Kepler's second law
(A line joining a planet and the sun sweeps out equal areas during equal
intervals of time). The constant of integration, h , is
the angular momentum per
unit mass .
where
we have introduced the auxiliary variable
The
radial force ƒ ( r ) per unit mass is the radial acceleration a
r defined above. Solving the above differential equation with respect to time. See also Binet
equation ) yields:
In
the case of gravity
, Newton's
law of universal gravitation states that the
force is proportional to the inverse square of the distance:
where
G is the constant of universal gravitation
, m is the mass of the orbiting body (planet), and M is the mass
of the central body (the Sun). Substituting into the prior equation, we have:
So
for the gravitational force — or, more generally, for any 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). JThe solution is:
where
A and θ 0 are arbitrary constants.
The
equation of the orbit described by the particle is thus:
where
e is:
In
general, this can be recognized as the equation of a conic
section in polar coordinates
( r , θ ). We can make a further connection with the classic description of
conic section with:
Orbital planes
The
analysis so far has been two dimensional; it turns out that an unperturbed
orbit is two-dimensional in a plane fixed in space, and thus the extension to
three dimensions requires simply rotating the two-dimensional plane into the
required angle relative to the poles of the planetary body involved. The
rotation to do this in three dimensions requires three numbers to uniquely
determine; traditionally these are expressed as three angles.
Orbital period
The
orbital period is simply how long an orbiting body takes to complete one orbit.
Specifying orbits
Six
parameters are required to specify an orbit about a body. For example, the 3 numbers which describe
the body's initial position, and the 3 values which describe its velocity will
describe a unique orbit that can be calculated forwards (or backwards).
The
traditionally used set of orbital elements is called the set of Keplerian elements
, after Johannes
Kepler and his Kepler's laws
. The Keplerian elements are six:
- Inclination
- Longitude of the ascending node
- Argument of periapsis
- Eccentricity
- Semimajor axis
- Mean anomaly at epoch
In
principle once the orbital elements are known for a body, its position can be
calculated forward and backwards indefinitely in time. However, in practice, orbits are affected or perturbed
, by other forces than simple gravity from an assumed point source (see the
next section), and thus the orbital elements change over time.
Orbital perturbations
An
orbital perturbation is when a force or impulse which is much smaller than the
overall force or average impulse of the main gravitating body and which is
external to the two orbiting bodies causes an acceleration, which changes the
parameters of the orbit over time.
Radial, prograde and transverse perturbations
A
small radial impulse given to a body in orbit changes the eccentricity, but not
the orbital period (to first order).A prograde or retrograde impulse (ie an impulse applied
along the orbital motion) changes both the eccentricity and the orbital period.
Notably, a prograde impulse given at periapsis
raises the altitude at apoapsis
, and vice versa, and a retrograde impulse does the opposite. A transverse
impulse (out of the orbital plane) causes rotation of the orbital plane without
changing the period or eccentricity. In all instances, a closed orbit will still intersect the
perturbation point.
Orbital decay
If
an orbit is about a planetary body with significant atmosphere, its orbit can
decay because of drag . Particularly at
each periapsis, the object experiences atmospheric drag, 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.
This
is similar to the effect of slowing a pendulum at its lowest point; the highest
point of the pendulum's swing becomes lower. With each successive slowing more of the orbit's path
is affected by the atmosphere and the effect becomes more pronounced. Eventually, the effect becomes so great
that the maximum kinetic energy is not enough to return the orbit above the
limits of the atmospheric drag effect. When this happens the body will rapidly spiral down
and intersect the central body.
The
bounds of an atmosphere vary wildly. During a solar
maximum , the Earth's atmosphere causes drag
up to a hundred kilometres higher than during a solar minimum.
Some
satellites with long conductive tethers can also experience orbital decay
because of electromagnetic drag from the Earth's
magnetic field . As the wire cuts the magnetic field
it acts as a generator, moving electrons from one end to the otherThe orbital energy is converted to heat in the wire.
Orbits
can be artificially influenced through the use of rocket motors which change
the kinetic energy of the body at some point in its path. This is the conversion of chemical or
electrical energy to kinetic energy. In this way changes in the orbit shape or
orientation can be facilitated.
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 other than
that of the sun, and so can be used indefinitely. See statite
for one such proposed use.
Orbital
decay can 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.
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.
Oblateness
The
standard analysis of orbiting bodies assumes that all bodies consist of uniform
spheres, or more generally, concentric shells each of uniform density. It can be shown that such bodies are gravitationally equivalent to point
sources.
However,
in the real world, many bodies rotate, and this introduces oblateness
and distorts the gravity field, and gives a quadrupole
moment to the gravitational field which is significant at distances comparable
to the radius of the body . The
general effect of this is to change the orbital parameters over time;
predominantly this gives a rotation of the orbital plane around the rotational
pole of the central body (it perturbs the argument of perigee
) in a way that is dependent on the angle of orbital plane to the equator as
well as altitude at perigee.
Multiple gravitating bodies
The
effects of other gravitating bodies can be significant.For example, the orbit
of the Moon cannot be accurately described without
allowing for the action of the Sun's gravity as well as the Earth's.
When
there are more than two gravitating bodies it is referred to as an n-body
problem.Most n-body problems have no closed form solution, although some
special cases have been formulated.
Light radiation and stellar wind
For
smaller bodies particularly, light and stellar wind can cause significant
perturbations to the attitude and direction of motion of the body, and over
time can be significant. Of the planetary
bodies, the motion of asteroids
is particularly affected over large periods when the asteroids are rotating
relative to the Sun.
Astrodynamics
Orbital
mechanics or astrodynamics is the
application of ballistics
and celestial
mechanics to the practical problems concerning
the motion of rockets and other spacecraft
. The motion of these objects is
usually calculated from Newton's
laws of motion and Newton's
law of universal gravitation . Celestial mechanics treats more broadly the orbital dynamics of
systems under the influence of gravity
, including spacecraft and natural astronomical bodies such as star systems, planets , moons , and comets . Orbital mechanics focuses on
spacecraft trajectories
, including orbital maneuvers
, orbit plane changes, and interplanetary transfers, and is used by mission
planners to predict the results of propulsive maneuvers
. General
relativity is a more exact theory than Newton's
laws for calculating orbits, and is sometimes necessary for greater accuracy or
in high-gravity situations (such as orbits close to the Sun).
Earth orbits
Scaling in gravity
- (6.6742 ± 0.001) × 10 −11 N·m 2 /kg 2 (6,6742 ± 0,001) × 10 -11 N m 2 / kg 2
- (6.6742 ± 0.001) × 10 −11 m 3 /(kg·s 2 ) (6,6742 ± 0,001) × 10 -11 m 3 / (· kg s 2)
- (6.6742 ± 0.001) × 10 −11 (kg/m 3 ) −1 s −2 . (6,6742 ± 0,001) × 10 -11 (kg / m 3) -1 s -2.
Thus
the constant has dimension density −1 time −2 .
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 velocities are halved
and 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.
Scaling
of distances while keeping the masses the same (in the case of point masses, or
by reducing the densities) gives similar orbits; if distances are multiplied by
4, gravitational forces and accelerations are divided by 16, velocities are
halved and orbital periods are multiplied by 8.
When
all densities are multiplied by 4, orbits are the same; gravitational forces
are multiplied by 16 and accelerations by 4, velocities are doubled and orbital
periods are halved.
When
all densities are multiplied by 4, and all sizes are halved, orbits are
similar; masses are divided by 2, gravitational forces are the same,
gravitational accelerations are doubled.
In
all these cases of scaling. if densities are
multiplied by 4, times are halved; if velocities are doubled, forces are
multiplied by 16.
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. untuk orbit elips dengan sumbu
semi-major , See also Kepler's Third Law . Lihat juga Kepler Hukum Ketiga
