Astrodynamics/Coordinate Systems

Coordinate Systems

The earth rotates around the sun, and the sun travels around the galaxy, and the galaxy is drifting through the universe. Physical equations need a coordinate system, with a set origin, to make any sense. In this chapter we will define some of the coordinate systems that are commonly used in the study of space.

Heliocentric-Ecliptic

Template:Wikipedia 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.

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. The positive vertical axis is the north pole of the earth. The positive x direction is in the direction of the vernal equinox.

Notice that the geocentric-equatorial 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).

Right-Ascension-Declination

Right-Ascension, Declination is a polar style coordinate system that is used to find the locations of bodies from earth. We define angle values α (the right-ascention) and δ (the declination).

The Right-Ascention Declination system is a polar form of the geocentric-horizon system. The right-ascention is the angle, in the equatorial plane, from the I vector, and the declination is the angle from the equatorial plane to the position vector of the object.

The right-ascention declination system does not include a distance, because it is used primarily for optical observations.

Perifocal

There perifocal coordinate system is very useful for describing a satellite's orbit. 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.

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 postive 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.

Transformation to Geocentric-Equatorial

L is the geodetic latitude, the angle that the Z vector makes with the equatorial plane. θ is the angle, on the equator, from the I vector to the position vector r_e. From these quantities, we can define x to be the distance from the point on the earth's surface to the K vector:

$x=\left|\frac{a_e}{\sqrt{1-e^2{\sin }^2(L)}}\right|\cos (L)$

We can define z to be the distance from the equatorial plane to the point on the earth's surface:

$z=\left|\frac{a_e(1-e^2)}{\sqrt{1-e^2{\sin }^2(L)}}\right|\sin (L)$

We can now define our position vector in terms of x and z:

${𝐫}_e=x\cos (\theta )𝐈+\sin (\theta )𝐉+z𝐊$

These equations seem complicated because the Earth is not spherical.

Determining θ

The equations above use the angle θ, which is the angle between the I vector and the position vector, in the equatorial plane. However, we know that the topocentric-equatorial coordinate system is an inertial system, and the geocentric-equatorial system is not. This means that the angle θ is a function of time. We have to be talking about siderial time here, because 1 siderial day is a complete 360° revolution.

We can define θ as:

$\theta ={\theta }_{g0}+{\omega }_e(t-t_0)+\lambda$

Where θ_g is the angle from the I vector to the prime meridian, and θ_g0 is the angle from the I vector to the prime meridian since a particular known time, t₀. λ is the longitude (angle from the prime meridian) of the point on the surface of the earth. ω_e is the angular velocity of the earth.

Azimuth and Elevation

the topocentric-horizon system has a polar variant that uses angles Azimuth and Elevation, and a distance ρ.

The azimuth, Az is the angle on the surface tangent plane from the negative S vector (from the north direction). The elevation El is the angle from the surface tangent plane to the line of sight. ρ is the distance from the surface of the earth to the satellite.

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:

${𝐃}^{\prime }=\mathit{\omega }\times 𝐃$

${𝐄}^{\prime }=\mathit{\omega }\times 𝐄$

${𝐅}^{\prime }=\mathit{\omega }\times 𝐅$

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

$𝐗=X_a𝐀+X_b𝐁+X_c𝐂=X_d𝐃+X_e𝐄+X_f𝐅$

The time derivative of A can be defined as:

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