This seems like an important topic, so I've posted a longer than typical answer: if this algorithm is to be used by others in the future, I think it's important that it be accompanied by references to the literature from which it has been derived.

The short answer

As you've noted, your posted code does not work properly for locations near the equator, or in the southern hemisphere.

To fix it, simply replace these lines in your original code:

elc <- asin(sin(dec) / sin(lat))az[el >= elc] <- pi - az[el >= elc]az[el <= elc & ha > 0] <- az[el <= elc & ha > 0] + twopi

with these:

cosAzPos <- (0 <= sin(dec) - sin(el) * sin(lat))sinAzNeg <- (sin(az) < 0)az[cosAzPos & sinAzNeg] <- az[cosAzPos & sinAzNeg] + twopiaz[!cosAzPos] <- pi - az[!cosAzPos]

It should now work for any location on the globe.

Discussion

The code in your example is adapted almost verbatim from a 1988 article by J.J. Michalsky (Solar Energy. 40:227-235). That article in turn refined an algorithm presented in a 1978 article by R. Walraven (Solar Energy. 20:393-397). Walraven reported that the method had been used successfully for several years to precisely position a polarizing radiometer in Davis, CA (38° 33' 14" N, 121° 44' 17" W).

Both Michalsky's and Walraven's code contains important/fatal errors. In particular, while Michalsky's algorithm works just fine in most of the United States, it fails (as you've found) for areas near the equator, or in the southern hemisphere. In 1989, J.W. Spencer of Victoria, Australia, noted the same thing (Solar Energy. 42(4):353):

Dear Sir:

Michalsky's method for assigning the calculated azimuth to the correct quadrant, derived from Walraven, does not give correct values when applied for Southern (negative) latitudes. Further the calculation of the critical elevation (elc) will fail for a latitude of zero because of division by zero. Both these objections can be avoided simply by assigning the azimuth to the correct quadrant by considering the sign of cos(azimuth).

My edits to your code are based on the corrections suggested by Spencer in that published Comment. I have simply altered them somewhat to ensure that the R function sunPosition() remains 'vectorized' (i.e. working properly on vectors of point locations, rather than needing to be passed one point at a time).

Accuracy of the function sunPosition()

To test that sunPosition() works correctly, I've compared its results with those calculated by the National Oceanic and Atmospheric Administration's Solar Calculator. In both cases, sun positions were calculated for midday (12:00 PM) on the southern summer solstice (December 22nd), 2012. All results were in agreement to within 0.02 degrees.

testPts <- data.frame(lat = c(-41,-3,3, 41),                       long = c(0, 0, 0, 0))# Sun's position as returned by the NOAA Solar Calculator,NOAA <- data.frame(elevNOAA = c(72.44, 69.57, 63.57, 25.6),                   azNOAA = c(359.09, 180.79, 180.62, 180.3))# Sun's position as returned by sunPosition()sunPos <- sunPosition(year = 2012,                      month = 12,                      day = 22,                      hour = 12,                      min = 0,                      sec = 0,                      lat = testPts$lat,                      long = testPts$long)cbind(testPts, NOAA, sunPos)#   lat long elevNOAA azNOAA elevation  azimuth# 1 -41    0    72.44 359.09  72.43112 359.0787# 2  -3    0    69.57 180.79  69.56493 180.7965# 3   3    0    63.57 180.62  63.56539 180.6247# 4  41    0    25.60 180.30  25.56642 180.3083

Other errors in the code

There are at least two other (quite minor) errors in the posted code. The first causes February 29th and March 1st of leap years to both be tallied as day 61 of the year. The second error derives from a typo in the original article, which was corrected by Michalsky in a 1989 note (Solar Energy. 43(5):323).

This code block shows the offending lines, commented out and followed immediately by corrected versions:

# leapdays <- year %% 4 == 0 & (year %% 400 == 0 | year %% 100 != 0) & day >= 60  leapdays <- year %% 4 == 0 & (year %% 400 == 0 | year %% 100 != 0) &               day >= 60 & !(month==2 & day==60)# oblqec <- 23.429 - 0.0000004 * time  oblqec <- 23.439 - 0.0000004 * time

Corrected version of sunPosition()

Here is the corrected code that was verified above:

sunPosition <- function(year, month, day, hour=12, min=0, sec=0,                    lat=46.5, long=6.5) {    twopi <- 2 * pi    deg2rad <- pi / 180    # Get day of the year, e.g. Feb 1 = 32, Mar 1 = 61 on leap years    month.days <- c(0,31,28,31,30,31,30,31,31,30,31,30)    day <- day + cumsum(month.days)[month]    leapdays <- year %% 4 == 0 & (year %% 400 == 0 | year %% 100 != 0) &                 day >= 60 & !(month==2 & day==60)    day[leapdays] <- day[leapdays] + 1    # Get Julian date - 2400000    hour <- hour + min / 60 + sec / 3600 # hour plus fraction    delta <- year - 1949    leap <- trunc(delta / 4) # former leapyears    jd <- 32916.5 + delta * 365 + leap + day + hour / 24    # The input to the Atronomer's almanach is the difference between    # the Julian date and JD 2451545.0 (noon, 1 January 2000)    time <- jd - 51545.    # Ecliptic coordinates    # Mean longitude    mnlong <- 280.460 + .9856474 * time    mnlong <- mnlong %% 360    mnlong[mnlong < 0] <- mnlong[mnlong < 0] + 360    # Mean anomaly    mnanom <- 357.528 + .9856003 * time    mnanom <- mnanom %% 360    mnanom[mnanom < 0] <- mnanom[mnanom < 0] + 360    mnanom <- mnanom * deg2rad    # Ecliptic longitude and obliquity of ecliptic    eclong <- mnlong + 1.915 * sin(mnanom) + 0.020 * sin(2 * mnanom)    eclong <- eclong %% 360    eclong[eclong < 0] <- eclong[eclong < 0] + 360    oblqec <- 23.439 - 0.0000004 * time    eclong <- eclong * deg2rad    oblqec <- oblqec * deg2rad    # Celestial coordinates    # Right ascension and declination    num <- cos(oblqec) * sin(eclong)    den <- cos(eclong)    ra <- atan(num / den)    ra[den < 0] <- ra[den < 0] + pi    ra[den >= 0 & num < 0] <- ra[den >= 0 & num < 0] + twopi    dec <- asin(sin(oblqec) * sin(eclong))    # Local coordinates    # Greenwich mean sidereal time    gmst <- 6.697375 + .0657098242 * time + hour    gmst <- gmst %% 24    gmst[gmst < 0] <- gmst[gmst < 0] + 24.    # Local mean sidereal time    lmst <- gmst + long / 15.    lmst <- lmst %% 24.    lmst[lmst < 0] <- lmst[lmst < 0] + 24.    lmst <- lmst * 15. * deg2rad    # Hour angle    ha <- lmst - ra    ha[ha < -pi] <- ha[ha < -pi] + twopi    ha[ha > pi] <- ha[ha > pi] - twopi    # Latitude to radians    lat <- lat * deg2rad    # Azimuth and elevation    el <- asin(sin(dec) * sin(lat) + cos(dec) * cos(lat) * cos(ha))    az <- asin(-cos(dec) * sin(ha) / cos(el))    # For logic and names, see Spencer, J.W. 1989. Solar Energy. 42(4):353    cosAzPos <- (0 <= sin(dec) - sin(el) * sin(lat))    sinAzNeg <- (sin(az) < 0)    az[cosAzPos & sinAzNeg] <- az[cosAzPos & sinAzNeg] + twopi    az[!cosAzPos] <- pi - az[!cosAzPos]    # if (0 < sin(dec) - sin(el) * sin(lat)) {    #     if(sin(az) < 0) az <- az + twopi    # } else {    #     az <- pi - az    # }    el <- el / deg2rad    az <- az / deg2rad    lat <- lat / deg2rad    return(list(elevation=el, azimuth=az))}

References:

Michalsky, J.J. 1988. The Astronomical Almanac's algorithm for approximate solar position (1950-2050). Solar Energy. 40(3):227-235.

Michalsky, J.J. 1989. Errata. Solar Energy. 43(5):323.

Spencer, J.W. 1989. Comments on "The Astronomical Almanac's Algorithm for Approximate Solar Position (1950-2050)". Solar Energy. 42(4):353.

Walraven, R. 1978. Calculating the position of the sun. Solar Energy. 20:393-397.

Using "NOAA Solar Calculations" from one of the links above I have changed a bit the final part of the function by using a slighly different algorithm that, I hope, have translated without errors. I have commented out the now-useless code and added the new algorithm just after the latitude to radians conversion:

# -----------------------------------------------# New code# Solar zenith anglezenithAngle <- acos(sin(lat) * sin(dec) + cos(lat) * cos(dec) * cos(ha))# Solar azimuthaz <- acos(((sin(lat) * cos(zenithAngle)) - sin(dec)) / (cos(lat) * sin(zenithAngle)))rm(zenithAngle)# -----------------------------------------------# Azimuth and elevationel <- asin(sin(dec) * sin(lat) + cos(dec) * cos(lat) * cos(ha))#az <- asin(-cos(dec) * sin(ha) / cos(el))#elc <- asin(sin(dec) / sin(lat))#az[el >= elc] <- pi - az[el >= elc]#az[el <= elc & ha > 0] <- az[el <= elc & ha > 0] + twopiel <- el / deg2radaz <- az / deg2radlat <- lat / deg2rad# -----------------------------------------------# New codeif (ha > 0) az <- az + 180 else az <- 540 - azaz <- az %% 360# -----------------------------------------------return(list(elevation=el, azimuth=az))

To verify azimuth trend in the four cases you mentioned let's plot it against time of day:

hour <- seq(from = 0, to = 23, by = 0.5)azimuth <- data.frame(hour = hour)az41S <- apply(azimuth, 1, function(x) sunPosition(2012,12,22,x,0,0,-41,0)$azimuth)az03S <- apply(azimuth, 1, function(x) sunPosition(2012,12,22,x,0,0,-03,0)$azimuth)az03N <- apply(azimuth, 1, function(x) sunPosition(2012,12,22,x,0,0,03,0)$azimuth)az41N <- apply(azimuth, 1, function(x) sunPosition(2012,12,22,x,0,0,41,0)$azimuth)azimuth <- cbind(azimuth, az41S, az03S, az41N, az03N)rm(az41S, az03S, az41N, az03N)library(ggplot2)azimuth.plot <- melt(data = azimuth, id.vars = "hour")ggplot(aes(x = hour, y = value, color = variable), data = azimuth.plot) +     geom_line(size = 2) +     geom_vline(xintercept = 12) +     facet_wrap(~ variable)

Image attached:

Here's a rewrite in that's more idiomatic to R, and easier to debug and maintain. It is essentially Josh's answer, but with azimuth calculated using both Josh and Charlie's algorithms for comparison. I've also included the simplifications to the date code from my other answer. The basic principle was to split the code up into lots of smaller functions that you can more easily write unit tests for.

astronomersAlmanacTime <- function(x){  # Astronomer's almanach time is the number of   # days since (noon, 1 January 2000)  origin <- as.POSIXct("2000-01-01 12:00:00")  as.numeric(difftime(x, origin, units = "days"))}hourOfDay <- function(x){  x <- as.POSIXlt(x)  with(x, hour + min / 60 + sec / 3600)}degreesToRadians <- function(degrees){  degrees * pi / 180}radiansToDegrees <- function(radians){  radians * 180 / pi}meanLongitudeDegrees <- function(time){  (280.460 + 0.9856474 * time) %% 360}meanAnomalyRadians <- function(time){  degreesToRadians((357.528 + 0.9856003 * time) %% 360)}eclipticLongitudeRadians <- function(mnlong, mnanom){  degreesToRadians(      (mnlong + 1.915 * sin(mnanom) + 0.020 * sin(2 * mnanom)) %% 360  )}eclipticObliquityRadians <- function(time){  degreesToRadians(23.439 - 0.0000004 * time)}rightAscensionRadians <- function(oblqec, eclong){  num <- cos(oblqec) * sin(eclong)  den <- cos(eclong)  ra <- atan(num / den)  ra[den < 0] <- ra[den < 0] + pi  ra[den >= 0 & num < 0] <- ra[den >= 0 & num < 0] + 2 * pi   ra}rightDeclinationRadians <- function(oblqec, eclong){  asin(sin(oblqec) * sin(eclong))}greenwichMeanSiderealTimeHours <- function(time, hour){  (6.697375 + 0.0657098242 * time + hour) %% 24}localMeanSiderealTimeRadians <- function(gmst, long){  degreesToRadians(15 * ((gmst + long / 15) %% 24))}hourAngleRadians <- function(lmst, ra){  ((lmst - ra + pi) %% (2 * pi)) - pi}elevationRadians <- function(lat, dec, ha){  asin(sin(dec) * sin(lat) + cos(dec) * cos(lat) * cos(ha))}solarAzimuthRadiansJosh <- function(lat, dec, ha, el){  az <- asin(-cos(dec) * sin(ha) / cos(el))  cosAzPos <- (0 <= sin(dec) - sin(el) * sin(lat))  sinAzNeg <- (sin(az) < 0)  az[cosAzPos & sinAzNeg] <- az[cosAzPos & sinAzNeg] + 2 * pi  az[!cosAzPos] <- pi - az[!cosAzPos]  az}solarAzimuthRadiansCharlie <- function(lat, dec, ha){  zenithAngle <- acos(sin(lat) * sin(dec) + cos(lat) * cos(dec) * cos(ha))  az <- acos((sin(lat) * cos(zenithAngle) - sin(dec)) / (cos(lat) * sin(zenithAngle)))  ifelse(ha > 0, az + pi, 3 * pi - az) %% (2 * pi)}sunPosition <- function(when = Sys.time(), format, lat = 46.5, long = 6.5) {      if(is.character(when)) when <- strptime(when, format)  when <- lubridate::with_tz(when, "UTC")  time <- astronomersAlmanacTime(when)  hour <- hourOfDay(when)  # Ecliptic coordinates    mnlong <- meanLongitudeDegrees(time)     mnanom <- meanAnomalyRadians(time)    eclong <- eclipticLongitudeRadians(mnlong, mnanom)       oblqec <- eclipticObliquityRadians(time)  # Celestial coordinates  ra <- rightAscensionRadians(oblqec, eclong)  dec <- rightDeclinationRadians(oblqec, eclong)  # Local coordinates  gmst <- greenwichMeanSiderealTimeHours(time, hour)    lmst <- localMeanSiderealTimeRadians(gmst, long)  # Hour angle  ha <- hourAngleRadians(lmst, ra)  # Latitude to radians  lat <- degreesToRadians(lat)  # Azimuth and elevation  el <- elevationRadians(lat, dec, ha)  azJ <- solarAzimuthRadiansJosh(lat, dec, ha, el)  azC <- solarAzimuthRadiansCharlie(lat, dec, ha)  data.frame(      elevation = radiansToDegrees(el),       azimuthJ  = radiansToDegrees(azJ),      azimuthC  = radiansToDegrees(azC)  )}