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# 13.5: Coordinates

• • Contributed by Jeremy Tatum
• Emeritus Professor (Physics & Astronomy) at University of Victoria

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.