Radial and angular acceleration

A spot moves on the xy plane. It is at R radius and T angle in radians.
y = R * Sin[T]

Take first derivative with respect to time.
y' = first derivative      y'' = second derivative
Use radians
y' = R' * Sin[T] + R * Cos[T] * T'

Take second derivative
y'' = R''* Sin[T] + R' * Cos[T] * T'
     + R' * Cos[T] * T''
     - R  * Sin[T] *(T')^2
     + R  * Cos[T] * T''


Collect Sin and Cos terms

y''=  Cos[T] * ( R *  T'' + 2 * R' * T' )
    + Sin[T] * ( R'' - R * (T')^2       )

Angular acceleration = ( R *  T'' + 2 * R' * T' )
Radial acceleration  = ( R'' - R * (T')^2       )
R*T^2=v^2/R

In an orbit, angular acceleration = zero since no force exists
other than gravity, which is in a radial direction.
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p= semilatus rectum = q * ( 1 + e ) =  0.5106 AU
R' = dR/dt = k * e * Sin[T]  AU/day
T' = k *SQRT[p]/R^2 radians/day