Need Simple equation for Rise, Transit, and Set time

Question

I've been looking, unsuccessfully, for hours for a simple set of equations:

Input:
RA and Dec of an object
Observers Lat and Lng
Current Time

Output:
Rise Time
Transit Time
Set Time
(and, hopefully) max elevation at transit
(and, hopefully) current alt and az of object

It seems like this should be a simple relationship, but I haven't been able to find it.
If an equation simply gives the current alt/az for an object, I guess I could solve for az and then find the maximum for a given time range.

Questions:
1. Anybody know of such equations? Or, even better, some sample code.
2. If not, can anybody point me in the right direction?

Thanks

Answer 1

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 https://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).

Answer 2

The calculations are not trivial, but they are encapsulated in software libraries such as pyephem, which has examples of finding rise, set and transit If you want to understand how these are calculated, you can readnthe source, which is based on the xephem application.

One detail which complicates the calculation is the refraction caused by the atmosphere, which can change the moment of setting by several minutes.

Answer 3

If you want a website rather than a software package like James suggests, I recommend use the INGT object visibility tool which has all 3 inputs you request and produces an elevation plot with rise and set times, maximum altitude etc. An example is shown below:

Answer 4

Meeus provides a fairly simple algorithm in Astronomical Algorithms. An example implementation is available in Rise and Set Algorithm. The code is fairly simple and I have reproduced it below.

A couple of warnings. The first is that the algorithm assumes longitudes to the West are positive, which is the opposite of how most GPS APIs return it. And, if the object moves considerably through the day, you may want to recompute the position for the given times, then recompute the rise and set times again.

//By Greg Miller [email protected] www.astrogreg.com
//Released as public domain
const toRad=Math.PI/180.0;
const toDeg=180.0/Math.PI;

//Corrects values to make them between 0 and 1
function constrain(v){
    if(v<0){return v+1;}
    if(v>1){return v-1;}
    return v;
}

//All angles must be in radians
//Outputs are times in hours GMT (not accounting for daylight saving time)
//From Meeus Page 101
function getRiseSet(jd,lat,lon,ra,dec){
    const h0=-0.8333 //For Sun
    //const h0=-0.5667 //For stars and planets
    //const h0=0.125   //For Moon

    const cosH=(Math.sin(h0*Math.PI/180.0)-Math.sin(lat)*Math.sin(dec)) / (Math.cos(lat)*Math.cos(dec));
    const H0=Math.acos(cosH)*180.0/Math.PI;

    const gmst=GMST(Math.floor(jd)+.5);

    const transit=(ra*toDeg+lon*toDeg-gmst)/360.0;
    const rise=transit-(H0/360.0);
    const set=transit+(H0/360.0);

    return [constrain(transit)*24.0,constrain(rise)*24.0,constrain(set)*24.0];
}

//Greenwhich mean sidreal time from Meeus page 87
//Input is julian date, does not have to be 0h
//Output is angle in degrees
function GMST(jd){
    const T=(jd-2451545.0)/36525.0;
    let st=280.46061837+360.98564736629*(jd-2451545.0)+0.000387933*T*T - T*T*T/38710000.0;
    st=st%360;
    if(st<0){st+=360;}

    return st;
    //return st*Math.PI/180.0;
}