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 angle^[3]

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

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 $0$ 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