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 α and declination δ,
as (polar)
\mathbf{X}_p = (R_p, \frac{\pi}{2} - \delta, \alpha)
as we normally do with spherical coordinate system (r,θ,ϕ)
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,θ,ϕ)
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 in solar-centered ecliptic coordinate system is at xp−xe 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
\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
\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
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 λ and latitude ϕ 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