Elevation Speed

The computational discovery that the speed of the antenna in elevation is independent of the declination of the source required a rigourous determination of same from the spherical geometry equations used in the computations.

The diagram above shows the terms used :

O is the origin of the coordinate systems.
Z is the local zenith
P is the (north) pole
X is the location of a star

The distance XZ = 90 - E , where E is the elevation
The distance XP = 90 - D , where D is the declination
The distance ZP = 90 - L , where L is the latitude of the observer
The angle ZPX = H (the Hour Angle) (-ve in the east)
The angle PZX = A (the azimuth, reflexive in the west)

Then, in the triangle ZPX

          cos(XZ) = cos(XP)cos(ZP) + sin(XP)sin(ZP)cos(ZPX) . . . . . (cosine rule), i.e.

          sin(E) = sin(D)sin(L) + cos(D)cos(L)cos(H)

which, by differentiation, gives

          dE/dH = -sin(H)cos(D)cos(L) / cos(E) . . . . . . . . . . . . . . (1)

Now, by the sine rule in the same triangle

          sin(90-D) / sin(A) = sin(90-D) / sin(H) , i.e.

          sin(H)cos(D)/cos(E) = sin(A)

so, in (1)           dE/dH = -sin(A)cos(L)

with the sign being -ve for positive HA and vice-versa.

Q.E.D.

Again, and for completeness, here's the relevant plot for the case of an observatory on Mauna Kea :