EXAMPLE $3\mathrm{.}3$

Let a satellite be in a $500$ km by $5000$ km orbit with its apse line parallel to the line from the earth to the sun, as shown in Fig. $3\mathrm{.}9.$ Find the time that the satellite is in the earth's shadow if

$($a$)$ the apogee is toward the sun

$($b$)$ the perigee is toward the sun.

FIG. $3\mathrm{.}9$

Satellite passing in and out of the earth's shadow.

ra$=27400$ km$,$ rp$=8410$ km