13.5: Coordinates

We need to make use of several coordinate systems, and I reproduce here the descriptions of them from section 10.7 of chapter 10. You may wish to refer back to that chapter as a further reminder.

1. Heliocentric plane-of-orbit. $\odot xyz$ with the $\odot x$ axis directed towards perihelion. The polar coordinates in the plane of the orbit are the heliocentric distance $r$ and the true anomaly $v$. The $z$-component of the asteroid is necessarily zero, and $x = r \cos v$ and $y = r \sin v$.
2. Heliocentric ecliptic. $\odot XYZ$ with the $\odot X$ axis directed towards the First Point of Aries $\Upsilon$, where Earth, as seen from the Sun, will be situated on or near September 22. The spherical coordinates in this system are the heliocentric distance $r$, the ecliptic longitude $λ$, and the ecliptic latitude $β$, such that $X = r \cos β \cos λ$, $Y = r \cos β \sin λ$ and $Z = r \sin β$.
3. Heliocentric equatorial coordinates. $\odot ξηζ$ with the $\odot ξ$ axis directed towards the First Point of Aries and therefore coincident with the $\odot X$ axis . The angle between the $\odot Z$ axis and the $\odot ζ$ axis is $ε$, the obliquity of the ecliptic. This is also the angle between the $XY$-plane (plane of the ecliptic, or of Earth’s orbit) and the $ξη$-plane (plane of Earth’s equator). See figure $\text{X.4}$.
4. Geocentric equatorial coordinates. $\oplus \mathfrak{xyz}$ with the $\oplus \mathfrak{x}$ axis directed towards the First Point of Aries. The spherical coordinates in this system are the geocentric distance $∆$, the right ascension $α$ and the declination $δ$, such that $\mathfrak{x} = ∆ \cos δ \cos α$, $\mathfrak{y} = ∆ \cos δ \sin α$ and $\mathfrak{z} = ∆ \sin δ$.

A summary of the relations between them is as follows

$$\mathfrak{x} = ∆ \cos α \cos δ = l ∆ = \mathfrak{x}_o + ξ, \label{13.5.1} \tag{13.5.1}$$

$$\mathfrak{y} = ∆ \sin α \cos δ = m ∆ = \mathfrak{y}_o + η , \label{13.5.2} \tag{13.5.2}$$

$$\mathfrak{z} = ∆ \sin δ = n ∆ = \mathfrak{z}_o + ζ . \label{13.5.3} \tag{13.5.3}$$

Here, $(l , \ m , \ n)$ are the direction cosines of the planet’s geocentric radius vector. They offer an alternative way to $(α , \ δ)$ for expressing the direction to the planet as seen from Earth. They are not independent but are related by

$$l^2 + m^2 + n^2 = 1 . \label{13.5.4} \tag{13.5.4}$$

The symbols $\mathfrak{x}_o , \ \mathfrak{y}_o$ and $\mathfrak{z}_o$ are the geocentric equatorial coordinates of the Sun.