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+ or polar axis, ecliptic as xy 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+ or polar axis, ecliptic as xy plane.

Equatorial coordinate system

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

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

as (polar)

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

as we normally do with spherical coordinate system (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+ or polar axis, local vertical up direction as z+.

Planets location in such system is defined as azimuth A and altitude a,

as (polar)

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

as we normally do with spherical coordinate system (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 \mathbf{x}_p in solar-centered ecliptic coordinate system is at \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 A and a with t, given longitude \lambda and latitude \phi under specific model, which we are to try explaing.

Note

  • Planets are actually in ellipse orbits instead of circle ones and z 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 ↩