Earth Eclipse of Server Sky Arrays

If the Earth was perfectly round, and the poles were not inclined, arrays in the 12789km, 17280 second radius equatorial orbit would spend 2868 seconds per orbit shaded by the 6371km radius Earth ( = 17280 \times asin( 6371 / 12789 ) / 180^\circ ~ ).

In fact, the Earth has an equatorial radius of 6378.1 km, a polar radius of 6356.8 km, and an axial tilt of \phi = 23.439281° . The sun has an angular size of 0.53 degrees, and the Earth's atmosphere refracts light, meaning that the light dims gradually over approximately 30 seconds entering eclipse. For the rest of this analysis, we will ignore these gradual effects, pretend the sun is a point source at infinity, and calculate the hard cutoff time as a function of time of year.

The variable \beta represents the time of year in the northern hemisphere, from 0° in spring, 90° in summer, 180° in the fall, and 270° in winter.

Oblate Earth

The equatorial plane is tilted towards the sun by angle \theta_{eq} defined by:

\sin( \theta_{eq} ) = \sin( \beta ) \sin( \phi ) ~ ~ ~ see [Precession]

The earth can be approximated as an elliptical disk, a projection of a ellipsoidal spheroid with an equatorial radius R_E = 6,378,137 meters and a polar radius R_P = 6,356,752 meters. The edge of this elliptical disk follows the equation:

y = \sqrt{ ( R_E^2 - x^2 ) ( ( 1 - ( R_P / R_E )^2 ) \sin( \theta_{eq} )^2 + ( R_P / R_E )^2 } ~ ~ ~ see [TiltingOblate]

The m288 orbit is a circle in the equatorial plane with a radius of R_{m288} . This circle projects into the X,Z plane as

y = \sin( \theta_{eq} ) \sqrt{ R_{m288}^2 - x^2 }

Two of the four points where these y values are equal are the points were the orbit enters or leaves the eclipse, so:

y_e = \sin( \theta_{eq} ) \sqrt{ R_{m288}^2 - x_e^2 } = \sqrt{ ( R_E^2 - x_e^2 ) ( ( 1 - ( R_P / R_E )^2 ) \sin( \theta_{eq} )^2 + ( R_P / R_E )^2 }

Let's solve for x_e :

\sin( \theta_{eq} )^2 ( R_{m288}^2 - x_e^2 ) = ( R_E^2 - x_e^2 ) \left( ( 1 - ( R_P / R_E )^2 ) \sin( \theta_{eq} )^2 + ( R_P / R_E )^2 \right)

x_e^2 \left( \sin( \theta_{eq} )^2 - \left( ( 1 - ( R_P / R_E )^2 ) \sin( \theta_{eq} )^2 + ( R_P / R_E )^2 \right) \right) = \sin( \theta_{eq} )^2 R_{m288}^2 - R_E^2 \left( ( 1 - ( R_P / R_E )^2 ) \sin( \theta_{eq} )^2 + ( R_P / R_E )^2 \right)

$ x_e ~ = ~ sqrt{ { \sin( \theta_{eq} )2 R_{m288}2 - R_E2 \left( ( 1 - ( R_P / R_E )2 ) \sin( \theta_{eq} )2 + ( R_P / R_E )2 \right) } \over{ \sin( \theta_{eq} )2 - \left( ( 1 - ( R_P / R_E )2 ) \sin( \theta_{eq} )2 + ( R_P / R_E )2 \right) } }