Orbit Grid

In order to study the visibility of each single sky direction as a function of time, we analysed all the BeppoSAX ephemerides in the time interval: July 3, 1996 - October 3, 2001. Let (X,Y,Z) the BeppoSAX position vector referred to a geocentric frame of reference; let $\rho = \sqrt{X^2 + Y^2}$ be the XY distance from the Earth center, $\psi$ the angle between the (X,Y,0) and the (1,0,0) directions; the cylindrical variables ($\rho, \psi, z$) have been found to vary within the following ranges:

$$6875.7 \, \mbox{Km} \ \leq \rho \ \leq \ 6978.1 \, \mbox{Km}, \ \ \quad \Delta \rho = 102.4 \, \mbox{Km}$$

$$0\rm ^{\circ}\ \leq \ \psi \ \leq \ 360\rm ^{\circ}$$

$$-483.0 \, \mbox{Km} \ \leq z \ \leq \ 483.0 \, \mbox{Km}, \ \ \quad \Delta z = 966.0 \, \mbox{Km}$$

The rectangular section torus containing all the BeppoSAX orbits has been split into a grid of cells of coordinates ( $\rho_0, \psi_i, z_j$), $i=1$,...,360, $j=1$,...,8. Taking a fixed $\rho_0 = 6926.9$ Km, i.e. the mean value of the $\rho$ range, an angular step $\Delta \psi_i = 1\rm ^{\circ}$, and a $z$ step $\Delta z_j = 124$ Km, the resulting grid consists of $1\times 360\times 8 = 2880$ cells, that can be approximated to parallelepipedal (nearly cubic) ($102, 124, 120$) Km$^3$ blocks ( $\delta \rho, \rho_0 \, \delta\psi, \delta z$).

Whenever the spacecraft is inside a given cell, its position is approximated to the cell center, whose coordinates are expressed as follows:

$$\psi_i \ = \ i\rm ^{\circ}, \qquad i=1,\ldots,360$$

$$z_j = \Big ( -434.0 \ + \ (j - 1) \times 124.0 \Big ) \, \mbox{Km}, \qquad j=1,\ldots,8$$

The step between two adjacent cells ($\sim$ 120 Km) subtends an angle $\theta_{min} \simeq (120$ Km) $/ \rho_0 \simeq 1\rm ^{\circ}$ with respect to the Earth center; hence, the angular resolution of the sky portion which is calculated to be Earth-blocked is $\sim 1\rm ^{\circ}$, when using such an orbit grid.