# Parabolic trajectory

In astrodynamics or celestial mechanics a parabolic trajectory is a Kepler orbit with the eccentricity equal to 1. When moving away from the source it is called an escape orbit, otherwise a capture orbit. It is also sometimes referred to as a C3 = 0 orbit (see Characteristic energy).

Under standard assumptions a body traveling along an escape orbit will coast along a parabolic trajectory to infinity, with velocity relative to the central body tending to zero, and therefore will never return. Parabolic trajectories are minimum-energy escape trajectories, separating positive-energy hyperbolic trajectories from negative-energy elliptic orbits.

The green path in this image is an example of a parabolic trajectory.
A parabolic trajectory is depicted in the bottom-left quadrant of this diagram, where the gravitational potential well of the central mass shows potential energy, and the kinetic energy of the parabolic trajectory is shown in red. The height of the kinetic energy decreases asymptotically toward zero as the speed decreases and distance increases according to Kepler's laws.

## Velocity

The orbital velocity (${\displaystyle v\,}$) of a body travelling along parabolic trajectory can be computed as:

${\displaystyle v={\sqrt {2\mu \over r}}}$

where:

• ${\displaystyle r\,}$ is the radial distance of orbiting body from central body,
• ${\displaystyle \mu \,}$ is the 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 (${\displaystyle v\,}$) 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:

${\displaystyle v={\sqrt {2}}\,v_{o}}$

where:

## Equation of motion

For a body moving along this kind of trajectory an orbital equation becomes:

${\displaystyle r={h^{2} \over \mu }{1 \over {1+\cos \nu }}}$

where:

## Energy

Under standard assumptions, the specific orbital energy (${\displaystyle \epsilon \,}$) of a parabolic trajectory is zero, so the orbital energy conservation equation for this trajectory takes the form:

${\displaystyle \epsilon ={v^{2} \over 2}-{\mu \over r}=0}$

where:

• ${\displaystyle v\,}$ is orbital velocity of orbiting body,
• ${\displaystyle r\,}$ is radial distance of orbiting body from central body,
• ${\displaystyle \mu \,}$ is the standard gravitational parameter.

This is entirely equivalent to the characteristic energy (square of the speed at infinity) being 0:

${\displaystyle C_{3}=0}$

## Barker's equation

Barker's equation relates the time of flight to the true anomaly of a parabolic trajectory.[1]

${\displaystyle t-T={\frac {1}{2}}{\sqrt {\frac {p^{3}}{\mu }}}\left(D+{\frac {1}{3}}D^{3}\right)}$

Where:

• D = tan(ν/2), ν is the true anomaly of the orbit
• t is the current time in seconds
• T is the time of periapsis passage in seconds
• μ is the standard gravitational parameter
• p is the semi-latus rectum of the trajectory ( p = h2/μ )

More generally, the time between any two points on an orbit is

${\displaystyle t_{f}-t_{0}={\frac {1}{2}}{\sqrt {\frac {p^{3}}{\mu }}}\left(D_{f}+{\frac {1}{3}}D_{f}^{3}-D_{0}-{\frac {1}{3}}D_{0}^{3}\right)}$

Alternately, the equation can be expressed in terms of periapsis distance, in a parabolic orbit rp = p/2:

${\displaystyle t-T={\sqrt {\frac {2r_{p}^{3}}{\mu }}}\left(D+{\frac {1}{3}}D^{3}\right)}$

Unlike Kepler's equation, which is used to solve for true anomalies in elliptical and hyperbolic trajectories, the true anomaly in Barker's equation can be solved directly for t. If the following substitutions are made[2]

{\displaystyle {\begin{aligned}A&={\frac {3}{2}}{\sqrt {\frac {\mu }{2r_{p}^{3}}}}(t-T)\\[3pt]B&={\sqrt[{3}]{A+{\sqrt {A^{2}+1}}}}\end{aligned}}}

then

${\displaystyle \nu =2\arctan \left(B-{\frac {1}{B}}\right)}$

A radial parabolic trajectory is a non-periodic trajectory on a straight line where the relative velocity of the two objects is always the escape velocity. There are two cases: the bodies move away from each other or towards each other.

There is a rather simple expression for the position as function of time:

${\displaystyle r={\sqrt[{3}]{4.5\mu t^{2}}}}$

where

• μ is the standard gravitational parameter
• ${\displaystyle t=0\!\,}$ corresponds to the extrapolated time of the fictitious starting or ending at the center of the central body.

At any time the average speed from ${\displaystyle t=0\!\,}$ is 1.5 times the current speed, i.e. 1.5 times the local escape velocity.

To have ${\displaystyle t=0\!\,}$ at the surface, apply a time shift; for the Earth (and any other spherically symmetric body with the same average density) as central body this time shift is 6 minutes and 20 seconds; seven of these periods later the height above the surface is three times the radius, etc.

## References

1. ^ Bate, Roger; Mueller, Donald; White, Jerry (1971). Fundamentals of Astrodynamics. Dover Publications, Inc., New York. ISBN 0-486-60061-0. p 188
2. ^ Montenbruck, Oliver; Pfleger, Thomas (2009). Astronomy on the Personal Computer. Springer-Verlag Berlin Heidelberg. ISBN 978-3-540-67221-0. p 64
Areocentric orbit

An areocentric orbit is an orbit around the planet Mars.

The areo- prefix is derived from the ancient Greek word Ares which is the personification of the planet Mars in Greek mythology. The name is an analogue to the term "geocentric orbit" for an orbit around Earth.

The first artificial satellites in areocentric orbit and the first orbiters of another celestial body (other than the Moon) were the U.S. Mariner 9 probe and Soviet Mars 2 and Mars 3 orbiters in 1971, 14 November and 27 November respectively. Later they were followed by many probes.

Box orbit

In stellar dynamics, a box orbit refers to a particular type of orbit that can be seen in triaxial systems, i.e. systems that do not possess a symmetry around any of its axes. They contrast with the loop orbits that are observed in spherically symmetric or axisymmetric systems.

In a box orbit, a star oscillates independently along the three different axes as it moves through the system. As a result of this motion, it fills in a (roughly) box-shaped region of space. Unlike loop orbits, the stars on box orbits can come arbitrarily close to the center of the system. As a special case, if the frequencies of oscillation in different directions are commensurate, the orbit will lie on a one- or two-dimensional manifold and can avoid the center. Such orbits are sometimes called "boxlets".

C/2000 W1

C/2000 W1 (Utsunomiya-Jones) is a long-period comet discovered on November 18, 2000, by Syogo Utsunomiya and Albert F. A. L. Jones.The comet has an observation arc of 58 days allowing a reasonable estimate of the orbit. But the near-parabolic trajectory with an osculating perihelion eccentricity of 0.9999996 generates an extreme unperturbed aphelion distance of 2,034,048 AU (32 light-years). The orbit of a long-period comet is properly obtained when the osculating orbit is computed at an epoch after leaving the planetary region and is calculated with respect to the center of mass of the solar system. Using JPL Horizons, the barycentric orbital elements for epoch 2020-Jan-01 generate a semi-major axis of 832 AU, an aphelion distance of 1660 AU, and a period of approximately 24,000 years.C/2000 W1 disappeared in February 2001.

Characteristic energy

In astrodynamics, the characteristic energy (${\displaystyle C_{3}}$) is a measure of the excess specific energy over that required to just barely escape from a massive body. The units are length2 time−2, i.e. velocity squared or twice the energy per mass.

Every object in a 2-body ballistic trajectory has a constant specific orbital energy ${\displaystyle \epsilon }$ equal to the sum of its specific kinetic and specific potential energy:

${\displaystyle \epsilon ={\frac {1}{2}}v^{2}-{\frac {\mu }{r}}={\text{constant}}={\frac {1}{2}}C_{3},}$

where ${\displaystyle \mu =GM}$ is the standard gravitational parameter of the massive body with mass ${\displaystyle M}$, and ${\displaystyle r}$ is the radial distance from its center. As an object in an escape trajectory moves outward, its kinetic energy decreases as its potential energy (which is always negative) increases, maintaining a constant sum.

Note that C3 is twice the specific orbital energy ${\displaystyle \epsilon }$ of the escaping object.

Geocentric orbit

A geocentric orbit or Earth orbit involves any object orbiting Planet Earth, such as the Moon or artificial satellites. In 1997 NASA estimated there were approximately 2,465 artificial satellite payloads orbiting the Earth and 6,216 pieces of space debris as tracked by the Goddard Space Flight Center. Over 16,291 previously launched objects have decayed into the Earth's atmosphere.A spacecraft enters orbit when its centripetal acceleration due to gravity is less than or equal to the centrifugal acceleration due to the horizontal component of its velocity. For a low Earth orbit, this velocity is about 7,800 m/s (28,100 km/h; 17,400 mph); by contrast, the fastest manned airplane speed ever achieved (excluding speeds achieved by deorbiting spacecraft) was 2,200 m/s (7,900 km/h; 4,900 mph) in 1967 by the North American X-15. The energy required to reach Earth orbital velocity at an altitude of 600 km (370 mi) is about 36 MJ/kg, which is six times the energy needed merely to climb to the corresponding altitude.Spacecraft with a perigee below about 2,000 km (1,200 mi) are subject to drag from the Earth's atmosphere, which decreases the orbital altitude. The rate of orbital decay depends on the satellite's cross-sectional area and mass, as well as variations in the air density of the upper atmosphere. Below about 300 km (190 mi), decay becomes more rapid with lifetimes measured in days. Once a satellite descends to 180 km (110 mi), it has only hours before it vaporizes in the atmosphere. The escape velocity required to pull free of Earth's gravitational field altogether and move into interplanetary space is about 11,200 m/s (40,300 km/h; 25,100 mph).

High Earth orbit

A high Earth orbit is a geocentric orbit with an altitude entirely above that of a geosynchronous orbit (35,786 kilometres (22,236 mi)). The orbital periods of such orbits are greater than 24 hours, therefore satellites in such orbits have an apparent retrograde motion – that is, even if they are in a prograde orbit (90° > inclination ≥ 0°), their orbital velocity is lower than Earth's rotational speed, causing their ground track to move westward on Earth's surface.

Inclined orbit

A satellite is said to occupy an inclined orbit around Earth if the orbit exhibits an angle other than 0° to the equatorial plane. This angle is called the orbit's inclination. A planet is said to have an inclined orbit around the Sun if it has an angle other than 0° to the ecliptic plane.

Mean longitude

Mean longitude is the ecliptic longitude at which an orbiting body could be found if its orbit were circular and free of perturbations. While nominally a simple longitude, in practice the mean longitude does not correspond to any one physical angle.

Near-equatorial orbit

A near-equatorial orbit is an orbit that lies close to the equatorial plane of the object orbited. This orbit allows for rapid revisit times (for a single orbiting spacecraft) of near-equatorial ground sites.

Non-inclined orbit

A non-inclined orbit is an orbit coplanar with a plane of reference. The orbital inclination is 0° for prograde orbits, and π (180°) for retrograde ones. If the plane of reference is a massive spheroid body's equatorial plane, these orbits are called equatorial; if the plane of reference is the ecliptic plane, they are called ecliptic.

As these orbits lack nodes, the ascending node is usually taken to lie in the reference direction (usually the vernal equinox), and thus the longitude of the ascending node is taken to be zero. Also, the argument of periapsis is undefined.

Geostationary orbit is a geosynchronous example of an equatorial orbit.

Orbit equation

In astrodynamics an orbit equation defines the path of orbiting body ${\displaystyle m_{2}\,\!}$ around central body ${\displaystyle m_{1}\,\!}$ relative to ${\displaystyle m_{1}\,\!}$, without specifying position as a function of time. Under standard assumptions, a body moving under the influence of a force, directed to a central body, with a magnitude inversely proportional to the square of the distance (such as gravity), has an orbit that is a conic section (i.e. circular orbit, elliptic orbit, parabolic trajectory, hyperbolic trajectory, or radial trajectory) with the central body located at one of the two foci, or the focus (Kepler's first law).

If the conic section intersects the central body, then the actual trajectory can only be the part above the surface, but for that part the orbit equation and many related formulas still apply, as long as it is a freefall (situation of weightlessness).

Orbital eccentricity

The orbital eccentricity of an astronomical object is a parameter that determines the amount by which its orbit around another body deviates from a perfect circle. A value of 0 is a circular orbit, values between 0 and 1 form an elliptic orbit, 1 is a parabolic escape orbit, and greater than 1 is a hyperbola. The term derives its name from the parameters of conic sections, as every Kepler orbit is a conic section. It is normally used for the isolated two-body problem, but extensions exist for objects following a Klemperer rosette orbit through the galaxy.

Parabolic

Parabolic usually refers to something in a shape of a parabola. It is also the adjectival form of parable.

Parabolic may refer to:

In mathematics:

In elementary mathematics, especially elementary geometry:

Parabolic coordinates

Parabolic cylindrical coordinates

parabolic Möbius transformation

Parabolic geometry (disambiguation)

Parabolic spiral

Parabolic line

Parabolic cylinder function

Parabolic induction

Parabolic Lie algebra

Parabolic partial differential equation

In physics:

Parabolic trajectory

In technology:

Parabolic antenna

Parabolic microphone

Parabolic reflector

Parabolic trough - a type of solar thermal energy collector

Parabolic flight - a way of achieving weightlessness

Parabolic action, or parabolic bending curve - a term often used to refer to a progressive bending curve in fishing rods.

In commodities and stock markets:

Parabolic SAR - a chart pattern in which prices rise or fall with an increasingly steeper slope

Polar orbit

A polar orbit is one in which a satellite passes above or nearly above both poles of the body being orbited (usually a planet such as the Earth, but possibly another body such as the Moon or Sun) on each revolution. It therefore has an inclination of (or very close to) 90 degrees to the body's equator. A satellite in a polar orbit will pass over the equator at a different longitude on each of its orbits.

In astrodynamics and celestial mechanics a radial trajectory is a Kepler orbit with zero angular momentum. Two objects in a radial trajectory move directly towards or away from each other in a straight line.

Sagittal plane

A sagittal plane (), or longitudinal plane, is an anatomical plane which divides the body into right and left parts. The plane may be in the center of the body and split it into two halves (mid-sagittal) or away from the midline and split it into unequal parts (para-sagittal).

Semi-synchronous orbit

A semi-synchronous orbit is an orbit with a period equal to half the average rotational period of the body being orbited, and in the same direction as that body's rotation.

For Earth, a semi-synchronous orbit is considered a medium Earth orbit, with a period of just under 12 hours. For circular Earth orbits, the altitude is approximately 20,200 kilometres (12,600 mi).Semi-synchronous orbits are typical for GPS satellites.

Subsynchronous orbit

A subsynchronous orbit is an orbit of a satellite that is nearer the planet than it would be if it were in synchronous orbit, i.e. the orbital period is less than the sidereal day of the planet.

True longitude

In celestial mechanics true longitude is the ecliptic longitude at which an orbiting body could actually be found if its inclination were zero. Together with the inclination and the ascending node, the true longitude can tell us the precise direction from the central object at which the body would be located at a particular time.

Gravitational orbits
Types
Parameters
Maneuvers
Orbital mechanics

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