Sun Path and Shadow Simulator

The shade that creeps across a vegetable bed in October is not bad luck. It was settled the day the fence went up, and it could have been read off in advance, to the minute.

The sun keeps the strictest schedule in nature. Its height above the horizon and its compass bearing at any instant follow from where you stand, what day it is and what time it is, and from nothing else. NOAA publishes a two-page summary of the calculation, and says its Solar Calculator rests on equations from Jean Meeus’s book Astronomical Algorithms. What turns those angles into something a homeowner can use is one piece of school trigonometry. Know the sun’s height and you know the length of every shadow on level ground. Know its bearing and you know which way each one points.

Three numbers set the sun’s place

Describing the sun’s position takes two angles. Elevation is measured upward from the horizon. Azimuth is measured clockwise from true north to the point on the horizon directly below the sun, so east is 90 degrees, south is 180 and west is 270. Three inputs decide both. The first is your latitude. The second is the solar declination, the angle between the plane of Earth’s equator and a line from the center of the Earth to the center of the sun. The third is the hour angle, which says how far the sun has swung east or west of your meridian, the north-south line overhead.

Declination is the season, written as an angle. Earth’s axis is tilted about 23.4 degrees from the perpendicular to its orbit and stays pointed the same way all year, so each hemisphere leans toward the sun for half the year and away from it for the other half. It is that tilt, and not the changing distance to the sun, that makes summer and winter. At the June solstice the sun stands directly over the Tropic of Cancer. At the December solstice it stands over the Tropic of Capricorn, and at the two equinoxes it is overhead at the equator at noon.

The hour angle is the time of day, written as an angle. Earth turns once in about a day, so the hour angle grows by roughly 15 degrees an hour. It is zero at solar noon, the moment the sun crosses your meridian and reaches its highest point. Mornings are negative. Afternoons are positive.

At that noon moment the arithmetic shrinks to something you can do in your head. Outside the tropics, the sun’s noon elevation is 90 degrees minus your latitude plus the declination. At 40 degrees north that gives about 73 degrees in late June and about 27 in late December, a swing of nearly 47 degrees that no weather can touch. Between the tropics the sun can pass north or south of overhead depending on the month. Beyond the polar circles there are winter days when it never clears the horizon at noon at all.

The equinoxes have their own signature. Only then does the sun rise due east and set due west. For the rest of the year sunrise and sunset slide north or south along the horizon, and the day stretches or shrinks with them.

Why noon is rarely at twelve

A sundial and a wristwatch seldom agree. The Naval Observatory calls the gap the equation of time: apparent solar time minus mean solar time. It has two causes. The tilt of the axis makes the sun’s east-west progress uneven through the year, and the slightly elliptical orbit carries Earth fastest in early January and slowest in early July. Together they let the true sun run ahead of or behind a steady clock by as much as 16 minutes. Plot that difference against the sun’s noon height for a year and the result is the figure-eight called the analemma.

Geography adds a second, constant offset. Civil time is kept in zones about 15 degrees of longitude wide, while the sun reaches each meridian four minutes later for every degree you move west. The Naval Observatory’s example is Omaha, at 96 degrees west in a zone defined by the 90 degree meridian, where mean solar time always runs 24 minutes behind the clock. Daylight saving time then shifts the clock a full hour. In the United States it begins on the second Sunday of March and ends on the first Sunday of November, and Hawaii and most of Arizona do not observe it.

So anything that works from clock time has to be told the offset from UTC, and told again when the clocks change. Get that wrong by an hour and the sun is 15 degrees of hour angle out of place.

A shadow is a tangent

On level ground, the top of a vertical object, its base and the tip of its shadow make a right triangle, and the sun’s elevation is the angle at the shadow’s tip. The shadow length is therefore the height divided by the tangent of the elevation. At 45 degrees a shadow is exactly as long as the thing that casts it. At 30 degrees it is 1.73 times as long, at 10 degrees 5.67 times, and at 5 degrees more than 11 times. The curve is gentle when the sun is high and savage when it is low, which is why winter shade reaches farther than intuition suggests.

Direction is simpler still. A shadow points directly away from the sun, so its bearing is the sun’s azimuth turned by 180 degrees. A southwest sun throws shadows to the northeast.

Take New York City, 40.71 degrees north and 74.01 degrees west, on 21 June 2026 at 15:00 Eastern Standard Time, five hours behind UTC. The equations put the sun 48.3 degrees above the horizon at a bearing of 259.8 degrees, a little south of due west. A 1 m pole therefore casts a shadow of 1 divided by the tangent of 48.3 degrees, which is 0.89 m, pointing just north of east. A house with a ridge 6 m high throws its shadow about 5.3 m. Sunrise comes at 04:25, solar noon at 11:58 and sunset at 19:31 on the same standard clock, and the Naval Observatory’s data service returns the same three times.

Now change only the date, to 21 December. At the same clock time the sun is 12.8 degrees up, and the same pole casts 4.41 m of shadow. The 6 m ridge now reaches about 26.5 m, well into the next lot. Noon tells the same story: the sun peaks at 72.7 degrees in June and 25.9 degrees in December, so the pole’s noon shadow grows from 0.31 m to 2.06 m.

Latitude stretches the numbers further. A University of Florida note works the case of Moscow, where the noon sun on 21 December stands less than 11 degrees high and a 1 m post casts a shadow more than 5 m long.

What the arc is good for

Gardeners can use this before they dig. Iowa State University’s extension service says a vegetable garden should receive at least 6 hours of direct sun each day and should avoid shady sites near large trees and shrubs. Whether a bed clears that bar depends on the sun’s arc on particular dates, not on how bright the spot looks on the afternoon you visit. A fence that shades nothing in June can cover the same bed in March, when the seedlings go in.

Solar panels turn the same question into output. The U.S. Energy Information Administration notes that panels collect sunlight most efficiently when the rays strike them at a right angle, that most panels in the Northern Hemisphere face south, and that tilting a panel toward the south at an angle equal to its latitude maximizes its yearly exposure to direct sun. Shallower tilts favor summer at the expense of fall and winter. None of that helps if a tree stands between the panel and the low winter sun, so the December arc is the one to check a site against.

Buildings can be shaped around the two solstice arcs. The U.S. Department of Energy’s guide to passive solar homes puts the idea plainly: use the sun’s low rays in winter and deflect its high rays in summer. It asks for windows that face within 30 degrees of true south and stay unshaded from 9 a.m. to 3 p.m. in the winter months, with roof overhangs or trees to shade the glass in summer. An overhang works because of the same tangent. The high June sun is cut off at the eave, and the low December sun slips in underneath.

Photographers chase the other end of the curve. The golden hour, shortly after sunrise or before sunset, is prized because the light is redder and softer than at midday. The tangent supplies the rest: at 10 degrees of elevation every post, person and ripple in a lawn carries a shadow nearly six times its own height.

Where the geometry stops

The formulas describe the sun. They know nothing about what stands between it and you. A simple model treats the site as a flat plane with a clear horizon, and real lots have slopes, hills, trees and the building next door. The Naval Observatory makes the same point about its own tables: local topography and the height of the observer change the times at which the sun actually appears and disappears, and at high latitudes the change can be large.

The horizon itself is soft. Air bends sunlight, so the sun stays visible when it is geometrically below the horizon. NOAA and the Naval Observatory both count sunrise and sunset from the moment the sun’s center is 0.833 degrees down, an allowance for average refraction plus the sun’s own radius. Real air does not stay average. NOAA rates its sunrise and sunset times as accurate to within a minute between 72 degrees north and south and within 10 minutes nearer the poles, and warns that observed values can differ. Within a few degrees of the horizon, treat any computed shadow length as approximate.

The research algorithm published by the National Renewable Energy Laboratory states an uncertainty of 0.0003 degrees for years from -2000 to 6000. The shorter equations behind an everyday calculator give some of that up, which rarely matters at garden scale. The sun’s disc is about half a degree wide, so the edge of a long shadow is a blur and not a line.

The last limit is the clock. A sun position is only as right as the UTC offset fed into it, and no formula can tell whether your region was on daylight time on a given date. Check that first when a result looks an hour wrong.

Within those limits the schedule holds. Pick the place, the day and the hour, and the light has only one place to be. The shadow follows, exactly as long as the tangent says, and it will be back at the same spot next year.