The perifocal coordinate (PQW) system is a frame of reference for an
orbit
In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as ...
. The frame is centered at the focus of the orbit, i.e. the celestial body about which the orbit is centered. The unit vectors
and
lie in the plane of the orbit.
is directed towards the
periapsis
An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary body. For example, the apsides of the Earth are called the aphelion and perihelion.
General description
There are two apsides in any elli ...
of the orbit and
has a
true anomaly (
) of 90 degrees past the periapsis. The third unit vector
is the
angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
vector and is directed orthogonal to the orbital plane such that:
:
And, since
is the angular momentum vector, it may also be expressed as:
:
where h is the specific relative angular momentum.
The position and velocity vectors can be determined for any location of the orbit. The position vector, r, can be expressed as:
:
where
is the true anomaly and the radius ''r'' may be calculated from the
orbit equation
In astrodynamics, an orbit equation defines the path of orbiting body m_2\,\! around central body m_1\,\! relative to m_1\,\!, without specifying position as a function of time. Under standard assumptions, a body moving under the influence of a f ...
.
The velocity vector, v, is found by taking the
time derivative
A time derivative is a derivative of a function with respect to time, usually interpreted as the rate of change of the value of the function. The variable denoting time is usually written as t.
Notation
A variety of notations are used to denote th ...
of the position vector:
:
A derivation from the orbit equation can be made to show that:
:
where
is the
gravitational parameter
In celestial mechanics, the standard gravitational parameter ''μ'' of a celestial body is the product of the gravitational constant ''G'' and the mass ''M'' of the bodies. For two bodies the parameter may be expressed as G(m1+m2), or as GM when ...
of the focus, ''h'' is the specific relative angular momentum of the orbital body, ''e'' is the
eccentricity
Eccentricity or eccentric may refer to:
* Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal"
Mathematics, science and technology Mathematics
* Off-Centre (geometry), center, in geometry
* Eccentricity (g ...
of the orbit, and
is the true anomaly.
is the radial component of the velocity vector (pointing inward toward the focus) and
is the tangential component of the velocity vector. By substituting the equations for
and
into the velocity vector equation and simplifying, the final form of the velocity vector equation is obtained as:
: