An observer is trying to determine an approximate value of the orbital eccentricity of a man-made satellite. When the satellite was at apogee, it was observed to have moved by $\Delta \theta_2 = 2'44"$ in a short time. When the radius vector connecting Earth and the satellite is perpendicular to the major axis, within the same duration of time, it was observed to have moved by $\Delta \theta_2 = 21'17"$. Assume that the observer is located at the center of the Earth. Find an approximate value of the eccentricity of the satellite’s orbit.

Source: IOAA