Analemmas

Last week I listened to an episode of The Sceptics' Guide to the Universe where the word of the week was "analemma". An analemma is a diagram showing the position of the Sun in the sky over the course of a year, as viewed at a fixed time of day from the same location on Earth. I once tried to make such a diagram when I was still living in Norway from a series of photos, but the weather wasn't consistent enough to make that work.

But as I am currently starting to update the Guide to Date and Time Programming for a second edition, I was wondering whether I could create an analemma from existing PHP functions. Unfortunately, PHP only provides functionality to calculate when the Sun is at its highest point, through date_sun_info():

<?php
$sunInfo = date_sun_info(
        (new DateTimeImmutable())->getTimestamp(), // Unix timestamp
        51.53,                                     // latitude
        -0.19                                      // longitude
);

$zenith = new DateTimeImmutable( "@{$sunInfo['transit']}" );
echo $zenith->format( DateTimeImmutable::ISO8601 ), "\n";
?>

Which on February 26th, was at 2018-02-26T12:13:38+0000 in London.

Then I remembered that a few years ago I wrote Where is the Sun?. There I features a new hobby library "astro" that I was working on. This library implements a few astronomical calculations. I wrote a little PHP extension around it too: php-solarsystem. Neither library or extension have really been released.

The php-solarsystem extension implements just one function: earth_sunpos(), which fortunately does exactly what I needed for drawing an analemma: it gives you the position of the Sun in the sky for a specific location on Earth at a specific time.

With this function, all I had to do is calculate the position of the Sun in the sky at the same time-of-day for a whole year. With the DatePeriod class in PHP, I can easily create an iterator that does just that:

<?php
date_default_timezone_set( "UTC" );

$dateStart = new DateTimeImmutable( "2018-01-01 09:00" );
$dateEnd   = $dateStart->modify( "+1 year 1 day" );
$dateInterval = new DateInterval( "P1D" );

foreach ( new DatePeriod( $dateStart, $dateInterval, $dateEnd ) as $date )
{
        …
}
?>

We don't really want Daylight Saving Time to be in the way, so we set the time zone to just UTC, which works fine for London for which we'll draw the analemma.

We start at the start of the year (2018-01-01 09:00) and iterate for a year and a day (+1 year 1 day) so we can create a closed loop. Each iteration increases the returned DateTimeImmutable by exactly one day (P1D).

After defining the latitude and longitude of London, all we need to do is to use the earth_sunpos() function to calculate the azimuth and altitude inside the loop. Azimuth is the direction of where the Sun is, with 180° being due South. And altitude is the height of the Sun above the horizon.

$lat = 51.53;
$lon = -0.09;

foreach ( new DatePeriod( $dateStart, $dateInterval, $dateEnd ) as $date )
{
        $ts = $date->format( 'U' );
        $position = earth_sunpos( $ts, $lat, $lon );
        echo $ts, "\n";
        echo $position['azimuth'], ",";
        echo $position['altitude'], "\n";
}

The script outputs the calculation as a "CSV", which we should redirect to a file:

php tests/analemma.php > /tmp/analemma.csv

To plot we use the following gnuplot script:

set style line 1 lt 1 lw 2 pt 0 ps 0 linecolor rgb "orange"
set style line 2 lt 1 lw 1 pt 0 ps 0 linecolor rgb "grey"

set datafile separator comma
set xrange [100:150]
set yrange [0:50]

set grid linestyle 2
set terminal png size 640,640 enhanced font "Helvetica,12"
set output '/tmp/analemma.png'

plot "/tmp/analemma.csv" using 2:3 title "London @ 9 am" with linespoints linestyle 1

With this script, we can then draw the analemma:

gnuplot /tmp/analemma.plot

The result:

Credits