Oil tanker must turn her rudder by 10° to make turn of radius 10 km?

Question

Assuming that curvature of ship path is proportional to angle of her rudder,

at what angle must the rudder be turned to follow the path strictly along the line of 45° latitude?

Answer 1

I'd suppose the factor relating turn radius Rt to circle radius Rc on a sphere is 1/sin(latitude), i.e.,1 at the poles, infinite at the equator, so Rt = Rc/sin(latitude). Defining Re as earth radius, Rc = Re*cos(latitude), so Rt = Re*cos(latitude)/sin(latitude) = Re/tan(latitude). With latitude = 45 deg, tan(latitude) = 1, Rt = Re.
Rudder angle theta (deg) is directly proportional to curvature, inversely proportional to turn radius in km.
theta = k/turn radius = 100/10; k = 100
At 45 deg latitude, theta = 100/Re deg, Re in km.

Answer 2

curvature is INVERSELY proportional (larger rudder angle implies smaller turn radius)

The 45 degree lattitude line represents a turn of radius 25,000 km (approx) so:

x/10 =10/25000 so x=100/25000 = 1/250 of a degree