Calculating topocentric and geocentric positions and distances

The geometry for the calculations of positions and distances to the requested planet is shown in Figure .

Figure: Simplified geometry for planetary calculations

The topocentric vector of a planet, $\bf {V_t}$, or the geocentric vector of the planet, $\bf {V_g}$, are calculated with reference to the geocentric vector of the Sun, $\bf {VSG}$, the heliocentric vector of the planet, $\bf {VSP}$, and the geocentric vector of the observer, $\bf {VGO}$. Also used by the current version of FLUXES are the geocentric vector to the Moon, $\bf {VGM}$, the vector from the Sun to the Earth-Moon barycentre, $\bf {VSE}$, and the knowledge that the geocentric vector of the Earth-Moon barycentre is given by ( $0.012150581 \times \bf {VGM}$).

When the directions of these vectors are as shown in Figure , the geocentric position and velocity vector of the planet is given by

$$\bf {V_g} = - \bf {VSG} + \bf {VSP}$$

where the geocentric vector of the Sun, $\bf {VSG}$, may be given as

$${\bf {VSG}} = {\bf {VSE}} - 0.01215081 ({\bf {VGM}}).$$

Similarly, the topocentric position and velocity vector of the planet is given by

$$\begin{eqnarray*} \bf {V_t} & = & \bf {V_g} - \bf {VGO}\\ & = & \bf {VSP} - \bf... ...{VSP}} - ({\bf {VSE}} - 0.012150581 ({\bf {VGM}})) - {\bf {VGO}} \end{eqnarray*}$$

All these vector quantities must be precessed to the appropriate date. The prime symbol ($\prime$) will be used in the following equations to indicate precessed positions.

Once the necessary vector is known, the actual distance is calculated as the square root of the summed square of each component, e.g.

$$d_g = \sqrt{{{{V^\prime}_{g_x}}^2} + {{{V^\prime}_{g_y}}^2} + {{{V^\prime}_{g_z}}^2}}$$

and hence the light travel time to the planet is given by

$$T_l = \frac{d_g}{c}$$

where $c$ is in the appropriate units. The position vector of the planet as viewed from the Earth should then be corrected for planetary aberration, e.g.

$${V^\prime}_{g_x, abb.} = {V^\prime}_{g_x} - (T_l \times {V^\prime}_{g_{\dot{x}}})$$

where ${V^\prime}_{g_{\dot{x}}}$ is the (uncorrected) velocity vector component in the $x$ direction. The corrected position vector ${V^\prime}_{g, abb.}$ or ${V^\prime}_{t, abb.}$ can then be converted into standard R.A. and Dec coordinates, and the airmass can be calculated from the topocentric coordinates.