This is the Arecibo Observatory in Arecibo, Puerto Rico. Its reflector is spherical, measuring 1,001 ft. in diameter. It is considered the most sensitive radio telescope on Earth, but the fact that its reflector is spherical and not parabolic makes me wonder how much more sensitive it could be if the reflector were parabolic. What are the pros and cons of a sphere vs. parabolic?
It's spherical because the main dish cannot be steered; steering is done by moving the receiver (the big thing hanging over the center of the reflector). A parabolic reflector would produce varying errors when aimed in different directions; a spherical reflector has the same error for all directions. Presumably the receiver is designed to compensate for this.
its always been my understanding that a parabolic dish focuses all reflected signals onto one specific point on the antenna
No. They only focus signals that come parallel to the axis on the focus. Signals that come into the dish from other directions are not focused on a single point. With a parabolic reflector the more off axis the signal is coming from the greater the aberration.
A spherical reflector induces spherical aberration but this is independent of the direction of the signal and hence can be compensated for.