Celestial Coordinate Systems

This work is from part of UCB CS289A 2020 Fall Project S Final. See Github repo for more information.

A physical model is evaluated so that we can understand the problem more properly.

Solar-centered ecliptic coordinate system

Solar-centered ecliptic coordinate system is centered at the sun, using spring equinox as x+x+ or polar axis, ecliptic as xyxy plane.

Earth location (polar)

\mathbf{x}_e = (r_e, \theta_e, 0)

where

\theta_e = \frac{2 \pi}{T_{ey}} t + \theta_{e0}

or (dirichlet)

\mathbf{x}_e = \begin{bmatrix} r_e \cos{(\cfrac{2 \pi}{T_{ey}} t + \theta_{e0})}\\ r_e \sin{(\cfrac{2 \pi}{T_{ey}} t + \theta_{e0})}\\ 0\\ \end{bmatrix}

other planets can have a similar definition.

Earth-centered ecliptic coordinate system

Solar-centered ecliptic coordinate system is centered at the earth, using spring equinox as x+x+ or polar axis, ecliptic as xyxy plane.

Equatorial coordinate system

Equatorial coordinate system is centered at the earth, using spring equinox as x+x+ or polar axis, equator as xyxy plane.

Planets location in such system is defined as right ascension α\alpha and declination δ\delta,

as (polar)

\mathbf{X}_p = (R_p, \frac{\pi}{2} - \delta, \alpha)

as we normally do with spherical coordinate system (r,θ,ϕ)(r,\theta,\phi)

or (dirichlet)

\mathbf{X}_p = \begin{bmatrix} R_p \cos\delta \cos\alpha\\ R_p \cos\delta \sin\alpha\\ R_p \sin\delta\\ \end{bmatrix}

Horizontal coordinate system

Horizontal coordinate system is centered at the observer, using local north as x+x+ or polar axis, local vertical up direction as z+z+.

Planets location in such system is defined as azimuth AA and altitude aa,

as (polar)

\hat \mathbf{X}_p = (R_p, \frac{\pi}{2} - a, A)

as we normally do with spherical coordinate system (r,θ,ϕ)(r,\theta,\phi)

or (dirichlet)

\hat \mathbf{X}_p = \begin{bmatrix} R_p \cos a \cos A\\ R_p \cos a \sin A\\ R_p \sin a\\ \end{bmatrix}

Coordinate transformation

It is easy to transform between the solar-centered ecliptic coordinate system and the earth-centered ecliptic coordinate system. A planet with coordinate xp\mathbf{x}_p in solar-centered ecliptic coordinate system is at xpxe\mathbf{x}_p - \mathbf{x}_e in earth-centered ecliptic coordinate system.

\mathbf{x}_p - \mathbf{x}_e = \begin{bmatrix} r_p \cos{(\cfrac{2 \pi}{T_{py}} t + \theta_{p0})} - r_e \cos{(\cfrac{2 \pi}{T_{ey}} t + \theta_{e0})}\\ r_p \sin{(\cfrac{2 \pi}{T_{py}} t + \theta_{p0})} - r_e \sin{(\cfrac{2 \pi}{T_{ey}} t + \theta_{e0})}\\ 0\\ \end{bmatrix}

transformation from the earth-centered ecliptic to equatorial coordinate system 1

\begin{bmatrix} x_{\text{equatorial}}\\ y_{\text{equatorial}}\\ z_{\text{equatorial}}\\ \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0\\ 0 & \cos \varepsilon & -\sin \varepsilon \\ 0 & \sin \varepsilon & \cos \varepsilon \\ \end{bmatrix} \begin{bmatrix} x_{\text{ecliptic}}\\ y_{\text{ecliptic}}\\ z_{\text{ecliptic}}\\ \end{bmatrix}

where ecliptic obliquity

ε=23\degree26'20.512''

so we have

\begin{aligned} \begin{bmatrix} R_p \cos\delta \cos\alpha\\ R_p \cos\delta \sin\alpha\\ R_p \sin\delta\\ \end{bmatrix} &= \begin{bmatrix} 1 & 0 & 0\\ 0 & \cos \varepsilon & -\sin \varepsilon \\ 0 & \sin \varepsilon & \cos \varepsilon \\ \end{bmatrix} \begin{bmatrix} r_p \cos{(\cfrac{2 \pi}{T_{py}} t + \theta_{p0})} - r_e \cos{(\cfrac{2 \pi}{T_{ey}} t + \theta_{e0})}\\ r_p \sin{(\cfrac{2 \pi}{T_{py}} t + \theta_{p0})} - r_e \sin{(\cfrac{2 \pi}{T_{ey}} t + \theta_{e0})}\\ 0\\ \end{bmatrix} \\ &= \begin{bmatrix} r_p \cos{(\cfrac{2 \pi}{T_{py}} t + \theta_{p0})} - r_e \cos{(\cfrac{2 \pi}{T_{ey}} t + \theta_{e0})}\\ \cos \varepsilon (r_p \sin{(\cfrac{2 \pi}{T_{py}} t + \theta_{p0})} - r_e \sin{(\cfrac{2 \pi}{T_{ey}} t + \theta_{e0})})\\ \sin \varepsilon (r_p \sin{(\cfrac{2 \pi}{T_{py}} t + \theta_{p0})} - r_e \sin{(\cfrac{2 \pi}{T_{ey}} t + \theta_{e0})})\\ \end{bmatrix} \end{aligned}

transformation from equatorial to horizontal coordinate system 2

\cos A\cdot \cos a=-\cos \phi \cdot \sin \delta +\sin \phi \cdot \cos \delta \cdot \cos H
\sin A\cdot \cos a=\cos \delta \cdot \sin H
\sin a=\sin \phi \cdot \sin \delta +\cos \phi \cdot \cos \delta \cdot \cos H

or

\begin{aligned} \begin{bmatrix} \cos A\cdot \cos a\\ \sin A\cdot \cos a\\ \sin a\\ \end{bmatrix} &= \begin{bmatrix} \sin \phi & 0 & -\cos \phi\\ 0 & 1 & 0\\ \cos \phi & 0 & \sin \phi \\ \end{bmatrix} \begin{bmatrix} \cos\delta \cos H\\ \cos\delta \sin H\\ \sin\delta\\ \end{bmatrix} \\ \end{aligned}

where hour angle3

H(t,\alpha) = GST(t) + \lambda - \alpha

One of the final goal of this project is to predict AA and aa with tt, given longitude λ\lambda and latitude ϕ\phi under specific model, which we are to try explaing.

Note

  • Planets are actually in ellipse orbits instead of circle ones and zz is not to be precise, here we made some simplification just to show the complexity of the problem
  • A slow motion of Earth's axis, precession, causes a slow, continuous turning of the coordinate system westward about the poles of the ecliptic, completing one circuit in about 26,000 years.

Reference


  1. Ecliptic coordinate system - Wikipedia, https://en.wikipedia.org/wiki/Ecliptic_coordinate_system 

  2. Horizontal coordinate system - Wikipedia, https://en.wikipedia.org/wiki/Horizontal_coordinate_system 

  3. Hour angle - Wikipedia, https://en.wikipedia.org/wiki/Hour_angle