The challenge now is to compute the actual integral. Not for the faint-hearted.
]]>I have re-computed them. They should be
and
And you want to find the following integral for the solid angle:
]]>Supposed the trajectory is angled
horizontally from the vertical plane through the y-axis, taking to be positive to the right and negative to the left of the y-axis. The straight line on the ground through the origin making this angle with the y-axis intersects the circle at two points. The distances from the origin to these two points can be calculated; these are the minimum and maximum ranges respectively at the angle . Let be the vertical angle from the ground at which the particle must be projected to cover the minimum range, and be the vertical angle to reach the maximum range. (This can be computed from the formula where is the acceleration due to gravity and the desired range.) The maximum value of the horizontal angle is .Then the solid angle you want is:
]]>