You literally are trying to disprove that the latitudinal rotational speed of the Earth does not influence rapid transition from daylight to darkness at the Equator and slower towards the geographical polesUnless you have not forgottenit is daily rotation which causes the transition from daylight to darkness while the spherical characteristics of a rotating Earth generate the latitudinal specifics of that transitionto believe otherwise is truly remarkableIt is that twilight effect and the table of values which indicate the Earths rotation through  degrees in  hours and until you find a way to incorporate the   hour  minute  second value for rotation through  degreesyou will have to consider that sidereal time value and the reasoning behind it as invalid