For largely historical reasons, our time-keeping system is reckoned by the position of the Sun. We can also choose to keep time by reckoning how long it takes for a specific star on the Celestial Sphere to get back to the same point in the sky night after night. Since one rotation of the Earth is all it takes to do this for distant stars, that is the Sidereal Day. NOTE Since one rotation of the Earth also moves the Earth $~1/365^{th}$ of the way around the Earth's orbit around the Sun, the Earth must rotate a little further to get the Sun back into the same position.

That $~1^\circ$ extra (did you catch why $~1^\circ$?) amounts to 4 minutes of extra rotation. It is on this 4 extra minutes of rotation that we base our 24h clock. So the Sidereal Day is 4 minutes shorted than the Solar Day (23h 56m). We still use the 24 hour clock, so we must account for this discrepancy.

Since RA($\alpha$) is divided into $24^h$, one might assume that stars travel through our LHCS sphere in $360/24 = 15^\circ/hr$ or (equivalently) $15''/s$ along the CEq. However, the $24^h$ of RA is not quite right. It should be $23^h56^m$ for the Sidereal Day. This comes out to $23*3600 + 56*60 = 86160s$. So our earlier estimate of $15''/s$ is off by a factor of $(24*3600)/86160$.

Accounting for this discrepancy, we find that the rate at which objects move through our LHCS along the CEq is $15*(86400/86160) = 15.0417 ''/s$.

Since DEC($\delta$) is measured from the CEq, we find that lines of constant DEC trace out concentric circles on the CNP and CSP. The radius of these circles of constant DEC decrease with $\cos{\delta}$. So at $\delta=\pm 90^\circ$ (at either CNP or CSP) the circle of constant DEC is a point and at $\delta=0^\circ$ (along the CEq), the circumference of the circle of constant DEC is maximized. In the same way, the rate of motion along these circles of constant DEC also decreases off of the Celestial Equator with the $\cos{\delta}$.

The next time you are participating in an Observing Session, calculate how long it will take for a star (not near the CNP) to move from one side of the frame to the other side.

Written by Roy Prouty 20240318
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