Astrodynamics/Coordinate Systems

Astrodynamics

Coordinate Systems

Before orbits can be discussed, it is necessary to define the coordinate systems that will be used to describe them. Multiple systems are used in the study of space, and by extension astrodynamics.

Every coordinate system consists of three mutually perpendicular axes that describe the three-dimensional coordinates of all points in space. Each particular system is defined by three geometric objects: an origin, a reference direction and a fundamental plane. The origin defines the center from which the axes originate, three options for origins will be discussed in detail here: topocentric, geocentric and heliocentric. The reference direction defines the x-axis. The fundamental plane contains the x-axis and defines the z-axis as the direction normal to the plane. The y-axis completes the right-hand system defined by the other two vectors:

$\mathbf {\hat {y}} =\mathbf {\hat {z}} \times \mathbf {\hat {x}}$

Positions in these systems may be expressed either as spherical or rectangular coordinates. Generally, spherical coordinates are used in astronomy and other ground-based observations where objects are sufficiently far away to not have discernible parallax.

Coordinate System	Inertial?	Origin	Fundamental Plane	Reference Direction	Horizontal Angle	Vertical Angle
Topocentric Horizon	No	Topocentric	Local Horizon	North/South point	Azimuth	Elevation
Earth-centered, Earth-Fixed (ECEF)	No	Geocentric	Equator	Prime Meridian	Geodetic Longitude/ hour angle	Geodetic Latitude/ declination
Earth-centered inertial (ECI)	Yes	Geocentric	Equatorial plane	Vernal Equinox	Right Ascension (α)	Declination (δ)
Ecliptic	Yes	Geocentric	Ecliptic	Vernal Equinox	Ecliptic longitude(λ)	Ecliptic Latitude (β)
	Yes	Heliocentric	Ecliptic	Vernal Equinox	Ecliptic longitude(l)	Ecliptic Latitude (b)
Perifocal	Yes	Central body	orbital plane	Periapsis (eccentricity vector)	True anomaly	-

One can define coordinate systems centered on planets and other large bodies, but these are analogous to geocentric coordinate systems and will not be discussed in great detail. Additional astronomical coordinate systems have been defined, such as galactic and supergalactic, but these are currently of little practical use in astrodynamics.

Topocentric-Horizon

The topocentric-horizon system, also known as the "SEZ" system is a system of coordinates for use by observers on the surface of the earth. The surface of the observer forms the fundamental plane, that is tangent to the surface of the earth. The positive horizontal vector S is due south , the positive horizontal vector E is east, and the vector Z normal to the surface of the earth (up) is the third axis. Notice that the unit vectors change with time, which means that the topocentric-horizon system is a non-inertial system.

We can introduce a new vector, the vector r_e, which is the position vector from the center of the earth to a point on the surface of the earth. It is important to note that the length of this vector is not a constant, because the Earth is not a perfect sphere. The Z vector, which is normal to the surface of the earth does not necessarily pass through the center of the earth, and so it cannot be used to define the r_e vector.

Geocentric-Equatorial

The geocentric-equatorial coordinate system uses the center of the earth as the origin, and uses the circle of the Earth's equator as the fundamental plane with the positive vertical axis is the north pole of the earth. Two different coordinate frames are generally employed: the Earth-centered, Earth fixed (ECEF) frame, which is non-inertial; and the Earth-centered Inertial (ECI) frame, which is inertial.

Earth-centered inertial (ECI)

The ECI system is fixed in space, and does not rotate with the earth. The unit vectors that we will be using with the geocentric-equatorial system are I (for the vernal direction), J for the second direction in the equatorial plane), and K (for the direction of the north pole).

Ecliptic

The heliocentric-ecliptic coordinate system uses the center of the sun as the origin, and is supremely useful for discussing orbits where the sun is the prime focus. The Earth's orbit forms the fundamental plane, and the vertical axis is the normal vector to that fundamental plane.

Perifocal

The perifocal coordinate system is used to describe a satellite's position within its orbit. In a completely unperturbed orbit, the motion of the satellite is completely confined to one plane of motion, the perifocal system reduces orbital motion from three dimensions to two dimensions, greatly simplifying computations. This is most useful in the process of orbit propagation. Coordinates for the perifocal system are generally expressed as PQW. The fundamental plane is the orbital plane. The P vector points from the prime focus to the periapsis point. The Q vector is in the fundamental plane, rotated 90° from the P vector, in the direction of the orbital motion. The W vector is normal to the orbital plane, and points in the direction of the angular momentum, h. Position and velocity vectors exist entirely within the orbital plane and thus can be described exclusively in terms of P and Q.

Noninertial Derivatives

A problem arises when we have two coordinate systems, the first is a fixed system (a, b, c) and the second is rotating with respect to the first and is (d, e, f). We know that (d, e, f) is rotating in (a, b, c) with an angular velocity vector ω. That is, we can say:

$\mathbf {D} '=\mathbf {\omega } \times \mathbf {D}$

$\mathbf {E} '=\mathbf {\omega } \times \mathbf {E}$

$\mathbf {F} '=\mathbf {\omega } \times \mathbf {F}$

We have a vector X that is defined in both systems:

$\mathbf {X} =X_{a}\mathbf {A} +X_{b}\mathbf {B} +X_{c}\mathbf {C} =X_{d}\mathbf {D} +X_{e}\mathbf {E} +X_{f}\mathbf {F}$

The time derivative of A can be defined as:

${\frac {d}{dt}}\mathbf {A} |_{(a,b,c)}={\frac {d}{dt}}\mathbf {A} |_{(d,e,f)}+\mathbf {\omega } \times \mathbf {A}$