I really wish I could remember what prompted this questions. I suspect it had to do with one of our conversations about time, and how “day” is from sunrise to sunset. There was probably something we were doing in the evening, something he was looking forward to doing, and the sun seemed to be just crawling through the sky. Whatever the reason, the question certainly seems to make sense. The sun takes all day to cross the sky, so it looks slow. It comes up gradually, takes four to six hours to reach noon, and then slowly sinks into the west.
Of course, appearances are deceiving.
How Fast Does The Sun Move Across The Sky?
This is actually sort of tricky, because the Sun isn’t actually moving around the Earth. To begin with, let’s refer back to the technical definition of sunrise and sunset:
Sunrise and sunset. For computational purposes, sunrise or sunset is defined to occur when the geometric zenith distance of center of the Sun is 90.8333 degrees. That is, the center of the Sun is geometrically 50 arcminutes below a horizontal plane. For an observer at sea level with a level, unobstructed horizon, under average atmospheric conditions, the upper limb of the Sun will then appear to be tangent to the horizon. The 50-arcminute geometric depression of the Sun’s center used for the computations is obtained by adding the average apparent radius of the Sun (16 arcminutes) to the average amount of atmospheric refraction at the horizon (34 arcminutes).
Now, an arcminute is 1/60th of a degree, so the day begins when the center of the sun is 5/6ths of a degree below the eastern horizon and ends when the center of the sun is 5/6ths of a degree below the western horizon. Assuming perfect viewing conditions, et cetera, et cetera. So, that means that the sun has to cover 181 2/3 degrees in a single day. Here in Cincinnati, sunrise on the day this article is published is(August 19, 2016) is 6:55 am, and sunset is 8:27 pm. So, it will require 13 hours and 33 minutes (813 minutes) to cover that distance. That works out to, let’s see… 181.6666 / 813 = 0.22345 degrees per minute, or 13.407 degrees per hour.
Now, let’s do some more math. Cincinnati is at 39.1031 degrees north. According to Ask Dr. Math, you “just multiply the equatorial circumference by the cosine of the latitude, and you will have the circumference at that latitude.” The equatorial circumference of the earth is 24,901 miles, so 24,901 * cos(39.1031) = 19323.48 miles. The Sun crosses 181.666/360 = 0.5046 of that distance in 813 minutes, so that’s 9751.197 miles in 813 minutes. That’s essentially 12 miles a minute or 719.65 miles an hour.
But that’s the speed at which the sun passes over the Earth at that latitude on August 19. It’s not how fast it appears to an observer on the ground! To an observer standing on the surface of the Earth, the distance to the horizon is approximately 2.9 miles. That means that my son, observing the motion of the Sun, is standing at the center of a perceptual circle with a circumference of 2(π)2.9 = 18.22 miles. Which means that he sees the Sun appear to take 813 minutes to cover 18.22 * 0.5046 = 9.19 miles. That works out to a perceived speed of 0.0113 miles per minute, or 0.678 miles an hour.
No wonder it looks so slow to him. When we’re out on walks, my son and I hit a pace almost four and a half times faster!
How Fast Does The Sun Move Through The Galaxy?
Of course, the speed of the sun gets even trickier. Because, although it doesn’t move through the sky (we move, creating the illusion), it still orbits Sagittarius A*, the supermassive black hole at the center of the Milky Way. This monster is some 26,000 light years from earth and weighs in at around 4,000,000 M☉. Our Sun travels in a roughly circular orbit around this distant behemoth at a speed between 217 amd 250 kilometers per second – let’s take the average of those five figures and call it 230.4 kilometers per second (143.16392 miles per second). That’s 829,440 kph (515,390.112 mph).
That sounds impressive, doesn’t it?
Here’s something to consider, though: the speed of light is 299,792,458 meters per second, or 299,792.458 kilometers per second. That means the Sun (and it’s attendant planets and dwarf planets and other detritus) are moving at 0.07685% of the speed of light. Remember that line above, the one that reads “this monster is some 26,000 light years from earth”? That means that our Sun orbits Sagittarius A* in a circle approximately 163,360 light years in circumference. As a result, it will take about 163.360/0.07685% = 212.5 million years to complete an orbit. (Actual calculations from real astronomers come in at between 225 and 250 million years, which makes sense – they have access to more accurate figures, and the sun would actually describe an ellipse instead of a circle.)
So, why is the sun moving so slow? It isn’t. It’s tearing through space at a pace five times faster than the New Horizons probe at it’s maximum velocity – the fastest ship ever built by humanity (although the Sun’s gravity had slowed it to ‘only’ 14 kilometers per second by the time it passed Pluto). We just don’t notice, because it’s a huge universe and we have a very small frame of reference.