For rudder deflection angles of 5, 10, 20, 30, and 45 degrees, at what (negative) angle-of-attack does a fin-rudder combination (including dorsal fin if present) create exactly the same amount of sideforce as would be the case if the fin somehow vanished, leaving the rudder to act as an all-moving vertical tail, with no change in its geometrical orientation relative to the aircraft?
For the purpose of this question, define angle-of-attack in relation to the chord line of the fin alone, not including the deflected rudder. If we ignore things like spiral slipstream, deflection of airflow around cabin, etc, it would be the same as the sideslip angle.
Assume a symmetrical airfoil on the vertical fin.
Assume the direction of sideslip (if sideslip angle is non-zero) is such that the airflow is striking the side of the fin that is opposite the direction of rudder deflection-- i.e. the rudder deflection is in the pro-slip direction. The deflected rudder is promoting, not opposing, the sideslip.
Implicit in this question is the idea that at an angle-of-attack of zero relative to the fin, the fin-rudder combination creates slightly more sideforce than the deflected rudder would by itself, for any given rudder deflection angle. I.e. at zero degrees angle-of-attack, and presumably at other very small angles-of-attack, the fin is actually helping the rudder to create sideforce. This question is about the point where that stops being true. However you may answer "zero degrees" if you feel that is the correct answer. If you feel that the fin is actually reducing the net sideforce produced by a deflected rudder even at zero degrees angle-of-attack relative to the fin, then please make a note of that in your answer; in this case the question would have no correct answer.
To allow for a definitive answer, I'll specify to use the fin (including dorsal fin) and rudder from a current -production Cessna 172 S, but feel to post an answer addressing the fin and rudder from any other aircraft.
Approximations are fine-- obviously an exact answer would involve a complicated computer modelliing project, or actual wind tunnel experiments. My hope is that existing information relating to the lift curves of symmetrical airfoils with deployed flaps may be adapted to give an approximation of an answer to this question.
Here is the basic purpose of this question: in general, during a strong sideslip (such as an intentional cross-controlled sideslip for a crosswind landing), the fixed vertical fin is creating a yaw torque that is opposing the yaw torque from the deflected rudder, and the sideslip angle would increase if the fixed fin were somehow to suddenly vanish. However it seems likely that at very small sideslip angles, this is no longer true. At a sideslip angle of zero-- which would still be compatible with a deflected rudder if some other yaw torque were acting on the aircraft-- it seems likely that the fin-rudder combination forms a better airfoil-- generating more sideways lift-- than the deflected rudder would by itself if the fin were absent. This question is asking at what slip angle this no longer is true, for a given rudder deflection. In other words, for a given rudder deflection, at what slip angle is the fixed fin neither adding to, nor subtracting from, the aerodynamic sideforce created by the deflected rudder.
Related questions and answers:
Note that the present question is different than the related question At what angle-of-attack (sideslip angle) would a symmetric vertical fin plus a deflected rudder have a lift coefficient of exactly zero? . The related question asks about the conditions where the fin-rudder combination creates no lift at all. Obviously in that case, if the rudder is deflected, the fin is producing a sideforce that acts against the sideforce form the deflected rudder. The present question acts about the conditions where the sideforce created by the fin-rudder combination is exactly the same as the sideforce that would be created by the deflected rudder alone if the fin were to vanish.