What is the math relationship between the earth's orbital and escape velocity?
The magnitude of the escape velocity varies a little, either way, from its average 11.2 km/s. It depends on time and location of launching of the rocket. See the details in the explanation.
My discussion is on changes about the average, relating to the nuances in the orbital acceleration. Changes in orbital velocity is attributed to changes in this acceleration. ..
Changes in the centripetal orbital acceleration is responsible for changes in the escape velocity. It might reduce or increase the escape velocity. There are maximums and minimums.
The direction of this acceleration is nearly opposite to the direction of midnight-launching of the rocket. It is in the like direction, for noon-launching, .
Also, the change of distance from the Sun changes the centripetal acceleration. At aphelion, it is least in magnitude. At perihelion, it is maximum.
Latitude of the launching site also has some effect on the escape velocity
On about Jan 2nd at perihelion noon, the fuel required might be less for attaining the escape velocity.
Using rigorous Mathematics, considering all aspects, it is possible to reveal that around April 1 or about October 3, the advantage in escape velocity is about 0.5 km/s, for midnight launch. Of course, for noon launch about this time, the advantage might be more.