Introduction

Solar altitude is a function of time, longitude and latitude, and so it may be possible to infer location based on measurements of solar altitude over time.

I have explored this idea for a school-based project I call “SkyView” involving light sensors and Arduino microcontrollers, and readers could well imagine a range of other applications.

I will illustrate the method and its accuracy based on sunrise and sunset times on Remembrance Day in Halifax, Nova Scotia, a city where respect for the fallen is very strong.

Methods

According to various websites (e.g. ref. 1), the Halifax on Remembrance Day of 2014 is at 7:06AM (11:06 UTC), and sunset at 4:51PM (20:51 UTC).

Rather than devising an inverse formula to infer location from time and solar altitude, the R function optim() may be used to find the longitude and latitude that minimize angle the sun makes to the horizon. That angle is given by the altitude component of the list returned by oce::solarAngle().

The first step is to define a function that returns the absolute value of the solar angle, which of course is a minimum at sunrise and sunset.

library(oce)
misfit <- function(lonlat) {
    riseAlt <- sunAngle(rise, longitude=lonlat[1], latitude=lonlat[2], useRefraction=TRUE)$altitude
    setAlt <- sunAngle(set, longitude=lonlat[1], latitude=lonlat[2], useRefraction=TRUE)$altitude
    0.5 * (abs(riseAlt) + abs(setAlt))
}

Here, useRefraction is set to account for the bending of the sunlight near the horizon, which perturbs the observed sunrise and sunset times. Note that the sunrise and sunset times (rise and set) must be defined in order for misfit to work. Use help(sunAngle) for more about how this function works.

The goal is to use optim() to find values of longitude and latitude that yield an optimal match to specified sunrise and sunset times. This function needs an initial value, or guess, and for that we pick 50W and 50N.

start <- c(-50, 50) # set this vaguely near the expected location
rise <- as.POSIXct("2014-11-11 11:06:00", tz="UTC")
set <- as.POSIXct("2014-11-11 20:51:00", tz="UTC")
bestfit <- optim(start, misfit)

An examination of the fit

str(bestfit)

yields as follows.

## List of 5
##  $ par        : num [1:2] -63.7 44.5
##  $ value      : num 0.000541
##  $ counts     : Named int [1:2] 203 NA
##   ..- attr(*, "names")= chr [1:2] "function" "gradient"
##  $ convergence: int 0
##  $ message    : NULL

Notice that the function value to be very low, indicating a sun just on the horizon. The optimal values for longitude and latitude are stored in par. See help("optim") to learn more about the return value.

It can be helpful to show the results on a map. To explore the dependence on sunrise and sunset times, random values can be added to those times and resultant positions plotted. In the present example, the times are reported to the nearest minute, so random values with standard deviation of 30 seconds will be used.

data(coastlineWorldFine, package = "ocedata")
plot(coastlineWorldFine, clon = -64, clat = 44.5, span = 800)
grid()

n <- 25                                # use 25 perturbations
rises <- rise + rnorm(n, sd = 30)
sets <- set + rnorm(n, sd = 30)
for (i in 1:n) {
    rise <- rises[i]
    set <- sets[i]
    fit <- optim(start, misfit)
    points(fit$par[1], fit$par[2], pch = 21, bg = "pink", cex = 1.4)
}
# Show the unperturbed fit
points(bestfit$par[1], bestfit$par[2], pch=21, cex=2, bg="red")
# Show the actual location of Halifax
points(-63.571, 44.649, pch = 22, cex = 2, bg = 'yellow')
# A legend summarizes all this work
legend("bottomright", bg = "white", 
       pch = c(21, 21, 22), pt.bg = c("red", "pink", "yellow"),
       legend = c("Best", "Range", "True"))

center

Results

The diagram indicates that reported sunrise and sunset times on Remembrance Day, 2014, are sufficient to locate Halifax to within about 10km. (Note that 1 degree of latitude is 111km.)

Adjusting sunrise and sunset times by half a minute (corresponding to the rounding interval in public forecasts of sunrise and sunset times) yields approximately 25km uncertainty in geographical position.

Discussion and conclusions

Since it is easy to observe sunrise and sunset times to a precision of 1 minute, the method outlined here might be the basis for school projects that combine computer work with field work, with proper instruction on observing the sun.

Plastic sextants are available for a few tens of dollars, and this opens up many more possibilities for exploring solar (and lunar) navigation using the Oce package to do the celestial calculations.

Further discussion of matters relating to solar angles can be found in my book (ref. 2). For example, the method can be used to test a phrase about an eclipse in a famous Carly Simon song.

Exercises

  1. Alter the initial guess to see what effect it has.

  2. Evaluate the relationship between positional uncertainty and temporal uncertainty.

  3. Try similar exercises at various times of the year, and map the uncertainty as a function of season.

  4. Buy a cheap sextant, and try extending the analysis to times other than sunrise and sunset.

References and resources

  1. A website providing the requisite sunrise and sunset times is http://www.timeanddate.com/astronomy/canada/halifax.

  2. Kelley, Dan E. Oceanographic Analysis with R. 1st ed. 2018. New York, NY: Springer New York : Imprint: Springer, 2018. https://doi.org/10.1007/978-1-4939-8844-0.