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Circular Orbit

 
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Circular orbit
 
 
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In astrodynamics
Astrodynamics

Orbital mechanics or astrodynamics is the application of celestial mechanics to the practical problems concerning the motion of rockets and other spacecraft....
 or celestial mechanics
Celestial mechanics

Celestial mechanics is the branch of astronomy that deals with the motion s of celestial objects. The field applies principles of physics, historically classical mechanics, to astronomical objects such as stars and planets to produce ephemeris data....
 a circular orbit is an elliptic orbit
Elliptic orbit

In astrodynamics or celestial mechanics an elliptic orbit is a Kepler orbit with the eccentricity greater than 0 and less than 1. In a gravitational two-body problem with the eccentricity in this range both bodies follow Similarity elliptic orbits with the same orbital period around their common barycenter....
 with the eccentricity equal to 0. It is an example of a rotation around a fixed axis
Rotation around a fixed axis

Rotation around a fixed axis is a special case of rotational motion. It does not involve rotation around more than one axis, and cannot describe such phenomena as wobbling or precession....
: this axis is the line through the center of mass
Center of mass

The center of mass of a system of wiktionary:Particles is a specific point at which, for many purposes, the system's mass behaves as if it were concentrated....
 perpendicular to the plane of motion.

lass="link1" onMouseover='showByLink("m2801448",this)' onMouseout='hide("m2801448")'href="http://www.absoluteastronomy.com/topics/Transverse">Transverse
Transverse

Transverse may refer to:*Transversality, a concept related to the intersection of manifolds in topology*Transverse City, an album by Warren Zevon...
 acceleration (perpendicular
Perpendicular

In geometry, two line or plane , are considered perpendicular to each other if they form congruence adjacent angles angles . The term may be used as a noun or adjective....
 to velocity) causes change in direction.






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Orbit2
In astrodynamics
Astrodynamics

Orbital mechanics or astrodynamics is the application of celestial mechanics to the practical problems concerning the motion of rockets and other spacecraft....
 or celestial mechanics
Celestial mechanics

Celestial mechanics is the branch of astronomy that deals with the motion s of celestial objects. The field applies principles of physics, historically classical mechanics, to astronomical objects such as stars and planets to produce ephemeris data....
 a circular orbit is an elliptic orbit
Elliptic orbit

In astrodynamics or celestial mechanics an elliptic orbit is a Kepler orbit with the eccentricity greater than 0 and less than 1. In a gravitational two-body problem with the eccentricity in this range both bodies follow Similarity elliptic orbits with the same orbital period around their common barycenter....
 with the eccentricity equal to 0. It is an example of a rotation around a fixed axis
Rotation around a fixed axis

Rotation around a fixed axis is a special case of rotational motion. It does not involve rotation around more than one axis, and cannot describe such phenomena as wobbling or precession....
: this axis is the line through the center of mass
Center of mass

The center of mass of a system of wiktionary:Particles is a specific point at which, for many purposes, the system's mass behaves as if it were concentrated....
 perpendicular to the plane of motion.

Circular acceleration

Transverse
Transverse

Transverse may refer to:*Transversality, a concept related to the intersection of manifolds in topology*Transverse City, an album by Warren Zevon...
 acceleration (perpendicular
Perpendicular

In geometry, two line or plane , are considered perpendicular to each other if they form congruence adjacent angles angles . The term may be used as a noun or adjective....
 to velocity) causes change in direction. If it is constant in magnitude and changing in direction with the velocity, we get a circular motion
Circular motion

In physics, circular motion is rotation along a circle: a circular path or a circular orbit. It can be uniform circular motion, that is, with constant angular rate of rotation, or non-uniform circular motion, that is, with a changing rate of rotation....
. For this centripetal acceleration we have

where:
  • is orbital velocity of orbiting body,
  • is radius
    RADIUS

    Remote Authentication Dial In User Service is a networking protocol that provides centralized access, authorization and accounting management for people or computers to connect and use a network service....
     of the circle
  • is angular frequency
    Angular frequency

    In physics , angular frequency ? is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity....
    , measured in radian per second.


Velocity

Under standard assumptions
Standard assumptions in astrodynamics

For most of the problems in astrodynamics involving two bodies and standard assumptions in astrodynamics are usually the following:*A1: and are the only objects in the universe and thus influence of other objects is disregarded,...
 the orbital velocity of a body traveling along circular orbit, can be computed as: where:
  • is radius
    RADIUS

    Remote Authentication Dial In User Service is a networking protocol that provides centralized access, authorization and accounting management for people or computers to connect and use a network service....
     of orbit
    ORBit

    ORBit is a Common Object Request Broker Architecture 2.4 compliant Object Request Broker . It features mature C , C++ and Python bindings, and less developed bindings for Perl, Lisp , Pascal , Ruby , and Tcl....
     equal to radial distance of orbiting body
    Orbiting body

    In astrodynamics, an orbiting body is a body that orbits central body . Under standard assumptions in astrodynamics:* it is less massive than the central body by several orders of magnitude ....
     from central body
    Central body

    In astrodynamics a central body is a body that is being orbited by an orbiting body . Under standard assumptions in astrodynamics:* it is more massive than the orbiting body by several orders of magnitude ,...
    ,
  • is the standard gravitational parameter
    Standard gravitational parameter

    In astrodynamics, the standard gravitational parameter of a celestial body is the product of the gravitational constant and the mass :The units of the standard gravitational parameter are km3s-2...
    , which is the product of the gravitational constant and the central mass .


Conclusion:
  • Velocity is constant along the path.


Orbital period

Under standard assumptions
Standard assumptions in astrodynamics

For most of the problems in astrodynamics involving two bodies and standard assumptions in astrodynamics are usually the following:*A1: and are the only objects in the universe and thus influence of other objects is disregarded,...
 the orbital period
Orbital period

The orbital Periodicity is the time taken for a given object to make one complete orbit about another object.When mentioned without further qualification in astronomy this refers to the sidereal period of an astronomical object, which is calculated with respect to the stars....
  of a body traveling along circular orbit can be computed as: where:
  • is the semi-major axis
    Semi-major axis

    In geometry, the semi-major axis is used to describe the dimensions of ellipses and hyperbolae....
     of the orbit; radius of the central body
    Central body

    In astrodynamics a central body is a body that is being orbited by an orbiting body . Under standard assumptions in astrodynamics:* it is more massive than the orbiting body by several orders of magnitude ,...
     plus altitude
    Altitude

    Altitude has multiple uses depending on the context in which it is used . As a general definition, altitude is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object....
     of the orbit
  • is standard gravitational parameter
    Standard gravitational parameter

    In astrodynamics, the standard gravitational parameter of a celestial body is the product of the gravitational constant and the mass :The units of the standard gravitational parameter are km3s-2...
    .


Energy

Under standard assumptions
Standard assumptions in astrodynamics

For most of the problems in astrodynamics involving two bodies and standard assumptions in astrodynamics are usually the following:*A1: and are the only objects in the universe and thus influence of other objects is disregarded,...
, specific orbital energy
Specific orbital energy

In astrodynamics the specific orbital energy of an orbiting body traveling through space under standard assumptions in astrodynamics is the sum of its potential energy and kinetic energy per unit mass....
  is negative for a closed orbit and the orbital energy conservation equation (the Vis-viva equation
Vis-viva equation

In astrodynamics, the vis viva equation, also referred to as orbital energy conservation equation, is one of the fundamental and useful equations that govern the motion of orbiting bodies....
) can take the form: where:
  • is orbital velocity of orbiting body,
  • is radius
    RADIUS

    Remote Authentication Dial In User Service is a networking protocol that provides centralized access, authorization and accounting management for people or computers to connect and use a network service....
     of orbit
    ORBit

    ORBit is a Common Object Request Broker Architecture 2.4 compliant Object Request Broker . It features mature C , C++ and Python bindings, and less developed bindings for Perl, Lisp , Pascal , Ruby , and Tcl....
     equal to radial distance of orbiting body
    Orbiting body

    In astrodynamics, an orbiting body is a body that orbits central body . Under standard assumptions in astrodynamics:* it is less massive than the central body by several orders of magnitude ....
     from central body
    Central body

    In astrodynamics a central body is a body that is being orbited by an orbiting body . Under standard assumptions in astrodynamics:* it is more massive than the orbiting body by several orders of magnitude ,...
    ,
  • is standard gravitational parameter
    Standard gravitational parameter

    In astrodynamics, the standard gravitational parameter of a celestial body is the product of the gravitational constant and the mass :The units of the standard gravitational parameter are km3s-2...
    .


The boundary case is which corresponds to escape from the primary (parabolic orbit), with .

The virial theorem
Virial theorem

In mechanics, the virial theorem provides a general equation relating the average over time of the total kinetic energy, , of a stable system, bound by potential forces, with that of the total potential energy, , where angle brackets represent the average over time of the enclosed quantity....
 applies even without taking a time-average:

  • the potential energy of the system is equal to twice the total energy
  • the kinetic energy of the system is equal to minus the total energy


Thus the escape velocity
Escape velocity

In physics, escape velocity is the speed where the kinetic energy of an object is equal to the magnitude of its gravitational potential energy, as calculated by the equation,...
 from any distance is v2 times the speed in a circular orbit at that distance: the kinetic energy is twice as much, hence the total energy is zero.

Equation of motion

Under standard assumptions
Standard assumptions in astrodynamics

For most of the problems in astrodynamics involving two bodies and standard assumptions in astrodynamics are usually the following:*A1: and are the only objects in the universe and thus influence of other objects is disregarded,...
, the orbital equation becomes: where:
  • is radial distance of orbiting body from central body
    Central body

    In astrodynamics a central body is a body that is being orbited by an orbiting body . Under standard assumptions in astrodynamics:* it is more massive than the orbiting body by several orders of magnitude ,...
    ,
  • is specific angular momentum of the orbiting body
    Orbiting body

    In astrodynamics, an orbiting body is a body that orbits central body . Under standard assumptions in astrodynamics:* it is less massive than the central body by several orders of magnitude ....
    ,
  • is standard gravitational parameter
    Standard gravitational parameter

    In astrodynamics, the standard gravitational parameter of a celestial body is the product of the gravitational constant and the mass :The units of the standard gravitational parameter are km3s-2...
    .


Delta-v to reach a circular orbit

Maneuvering into a large circular orbit, e.g. a geostationary orbit
Geostationary orbit

A geostationary orbit is a geosynchronous orbit directly above the Earth's equator , with a period equal to the Earth's rotational period and an orbital eccentricity of approximately zero....
, requires a larger delta-v
Delta-v

In astrodynamics, the term delta-v, literally "change in velocity" , has a specific meaning: it is a scalar which takes units of speed that measures the amount of "effort" needed to carry out an orbital maneuver, i.e., to change from one trajectory to another....
 than an escape orbit
Escape orbit

An escape orbit is a high-energy parabolic orbit around the central body. A body in this orbit has at each position the escape velocity with respect to this central body, for this position....
, although the latter implies getting arbitrarily far away and having more energy than needed for the orbital speed
Orbital speed

The orbital speed of a body, generally a planet, a natural satellite, an satellite, or a multiple star, is the speed at which it orbits around the barycenter of a system, usually around a more massive body....
 of the circular orbit. It is also a matter of maneuvering into the orbit. See also Hohmann transfer orbit
Hohmann transfer orbit

In orbital mechanics, the Hohmann transfer orbit is an orbital maneuver using two engine impulses which, under Standard assumptions in astrodynamics, move a spacecraft between two coplanar circular orbits....
.

See also

  • Orbit
    ORBit

    ORBit is a Common Object Request Broker Architecture 2.4 compliant Object Request Broker . It features mature C , C++ and Python bindings, and less developed bindings for Perl, Lisp , Pascal , Ruby , and Tcl....
  • Elliptic orbit
    Elliptic orbit

    In astrodynamics or celestial mechanics an elliptic orbit is a Kepler orbit with the eccentricity greater than 0 and less than 1. In a gravitational two-body problem with the eccentricity in this range both bodies follow Similarity elliptic orbits with the same orbital period around their common barycenter....
  • List of orbits
    List of orbits

    The following is a list of types of Orbit s:...
  • Two-body problem
    Two-body problem

    In classical mechanics, the two-body problem is to determine the motion of two point particles that interact only with each other. Common examples include a satellite orbiting a planet, a planet orbiting a star, two stars orbiting each other , and a classical electron orbiting an atomic nucleus....