Question

state the rate of change of time with longitude?

Answer 1

As the earth turns, the sun appears to move across the sky. This apparent motion gives us day and night. A sundial uses the position of the sun in the sky to tell the time. The time measured by a sundial is solar time. Solar time is 12 noon when the sun is on the (observer's local) meridian. Note that we don't say "when the sun is directly overhead" -- why not?

Various Kinds of Time

Sundials do not keep time at a uniform rate the way mechanical clocks do. If you set a sundial to be on time at one time of the year then it will run fast or slow at other times of the year. When people started depending more on mechanical clocks than sundials it became convenient to switch to using time that, on average, matched sundial time, but ran at a constant rate. This is mean solar time. The difference between mean solar time and solar time is called the "equation of time."