I have been giving some more thought to how the form of the equation that Peter used in his talk to describe solar differential rotation was arrived at. I think Peter used the equation ω = A + B sin²θ + C sin⁴θ where ω is angular velocity of the sun at the observable surface in degrees per day, θ is the solar latitude, A is the angular velocity at the equator and B and C are constants that control how the velocity decreases with increasing latitude (see this reference in wikipedia).
What I think is that the physics tells us that ω is a function of sin²θ, that is w = f(sin²θ). If that is the case then writing x = sin²θ we can expand w = f(x) using a Taylor series about x = 0, namely
f(x) = f(0) + f'(0)x + (f”(0)/2)x² + …
where f'(0) and f”(0) are the first and second derivatives of f with respect to x at x = 0. Replacing x by sin²θ we get
f(θ) = f(0) + f'(0)sin²θ + (f”(0)/2)sin⁴θ + …
Now x = 0 when θ = 0, so f(0) = A, the angular velocity at the equator. Also if x is small, which is true if θ is small then we can neglect higher order terms (those involving powers of x greater than 2) in the expansion. Equating constants B and C with constants f'(0) and (f”(0)/2) we get
w = f(θ) = A + B sin²θ + C sin⁴θ
as Peter quoted. I hope this makes sense.
