coordinate - Need Simple equation for Rise, Transit, and Set time

I'm not sure it qualifies as "simple", but, using
http://idlastro.gsfc.nasa.gov/ftp/pro/astro/hadec2altaz.pro (and some
additional calculations/simplifications):

$begin{array}{|c|c|c|c|} hline text{Event} & text{Time} & phi & Z \ hline text{Any} & text{t} & tan ^{-1}(cos (lambda ) sin (delta )-cos (delta ) cos (alpha -t) sin (lambda ),cos (delta ) sin (alpha -t)) & tan ^{-1}left(sqrt{(cos (lambda ) sin (delta )-cos (delta ) cos (alpha -t) sin (lambda ))^2+cos ^2(delta ) sin ^2(alpha -t)},cos (delta ) cos (lambda ) cos (alpha -t)+sin (delta ) sin (lambda )right) \ hline text{Rise} & alpha -cos ^{-1}(-tan (delta ) tan (lambda )) & tan ^{-1}left(sec (lambda ) sin (delta ),cos (delta ) sqrt{1-tan ^2(delta ) tan ^2(lambda )}right) & 0 \ hline text{Transit} & alpha & begin{cases} delta >lambda & 0 \ delta =lambda & text{Zenith} \ delta <lambda & pi end{cases} & frac{pi }{2}-left| delta -lambda right| \ hline text{Set} & alpha +cos ^{-1}(-tan (delta ) tan (lambda )) & tan ^{-1}left(sec (lambda ) sin (delta ),-cos (delta ) sqrt{1-tan ^2(delta ) tan ^2(lambda )}right) & 0 \ hline text{Lowest Point} & alpha +pi & begin{cases} delta >-lambda & 0 \ delta =-lambda & text{Nadir} \ delta <-lambda & pi end{cases} & left| delta +lambda right|-frac{pi }{2} \ hline end{array}$

where:

• $phi$ is the azimuth of the object




• $Z$ is the altitude of the object above the horizon




• $alpha$ is the right ascension of the object




• $delta$ is the declination of the object




• $lambda$ is the latitude of the observer




• $t$ is the current local sidereal time

Note the two-argument form of arctangent is required so that the
results are in the correct quadrant:
https://en.wikipedia.org/wiki/Inverse_trigonometric_functions#Two-argument_variant_of_arctangent

Additional caveats:

• If $left| delta -lambda right|>frac{pi }{2}$, the object is
  always below the horizon, and the equations for rising time and
  setting time will not work.




• If $left| delta +lambda right|>frac{pi }{2}$, the object is
  always above the horizon (circumpolar), and the equations for rising
  and setting time will also not work.




• The measurements above are in radians. You can convert $pi to 180 {}^{circ}$ for degrees.




• Because we use the local sidereal time, the longitude doesn't
  appear in any of the formulas above. However, we do need it to find
  the local sidereal time, as below.




• To find the local sideral time $t$ in radians, we use
  http://aa.usno.navy.mil/faq/docs/GAST.php and make some substitions
  to get:

$t = 4.894961212735792 + 6.30038809898489 d + psi$

where $psi$ is your longitude in radians, and $d$ is the number of
days (including fractional days) since "2000-01-01 12:00:00 UTC". Traditionally, we use $phi$ for longitude, but I'm already using it in the formulas above for azimuth.

If you combine the formula for local sidereal time and
azimuth/altitude and assume excessive precision, you get my answer to
http://astronomy.stackexchange.com/a/8415/21

Additional computations for these results at:
https://github.com/barrycarter/bcapps/blob/master/STACK/bc-rst.m

I was going to add some graphs to show how the altitude is NOT a sine wave and how the azimuth is NOT a straight line (although you might expect them to be), but they turned out not to be terribly instructive/helpful.

You might also be able to get simpler formulas if you set $t$ to be the "hour angle" (which is $alpha-t$ in the current setup).

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