# How fast does the Sun move?

How fast does the Sun move? I don’t mean though the galaxy, or the universe, but simply how fast does it move across the sky?

I’ve seen it stated categorically that the Sun moves at 15 degrees per hour. But, for one thing, 15.000000° can’t be perfectly correct at all times, because the Earth’s orbit isn’t a perfect circle. For another thing, 15 degrees of what?

## Thought experiments

Suppose you’re standing on the North pole, and it’s the equinox. The Sun hugs the horizon, and in 24 hours it travels (almost exactly) 360° around you, in a great circle. 360° divided by 24 hours is 15° per hour. So far, so good.

Now suppose it’s the summer solstice. The Sun is about 23.5° above the horizon, and it stays at that angle all day as it traces out a circular path. In 24 hours, it has moved through 360° of that circle, so it still moves 15° per hour. But this circle is smaller (in your field of view) than the equinox one, so the Sun is moving slower in terms of absolute speed. It’s moving slower by a factor of the cosine of 23.5°, which is 0.917, so 13.76 great-circle degrees per hour.

Being on a pole simplifies the thought experiment, but your location on Earth has no significant effect on the Sun’s apparent speed. It only affects your orientation, and whether you can see the Sun.

Consider the night sky. The field of stars appears to rotate around the celestial pole (approximately the star Polaris, if you’re in the northern hemisphere). The stars all rotate at 360° per sidereal day, which is 15.041° per hour. Unlike the Sun, this angular motion is essentially perfectly constant. But only the stars on the celestial equator actually move at 15.041 great-circle degrees per hour. The stars near the pole obviously move much slower, in terms of absolute speed. Polaris barely moves at all.

If we could see the stars during the day, we’d see that the Sun pretty much moves along with the star field. But it moves a little slower, falling behind the nearby stars by very roughly one degree per day.

The Sun is only on the celestial equator at the equinoxes. At the solstices, it’s 23.5° from it. To a first approximation, its speed varies between 13.76 great-circle degrees per hour at the solstices, and 15.00 at the equinoxes.

## Methodology

I decided to calculate two things, for a number of sample days throughout a year:

1. What the Sun’s speed would be, as a fraction of the maximum of 15°/hour, if we consider only its distance from the celestial equator.
2. A corrective factor, based on the Earth’s actual angular speed in its orbit on that day, compared to its average.

The Sun’s actual speed will be the product of these two factors, times 15°/hour.

The second of these numbers is far less significant than the first. There are other corrective factors as well that I’m ignoring, but they should be even less significant.

For (1), I used the latitude of the sub-solar point on Earth. I think this is a good enough approximation, though it might not be quite perfect, because the Earth isn’t a perfect sphere. I looked up the data, and took the cosine of it.

I had some trouble figuring out (2), but I think I finally worked out the right formula, starting from a table of Earth-Sun distances. The relevant “transverse” motion of the Earth is inversely proportional to the distance, as per Kepler’s sweeps-out-equal-areas law. I’m not sure I did it right, but the numbers look plausible to me.

## Results

Multiplying it out, I got this graph:

The graph is too low resolution to see the effect of the elliptical orbit (factor (2)).

Note that factor (1) always reaches a maximum of exactly 1.0, at the moment of an equinox. Its minimum is 0.91748, on the solstices.

Factor (2) varies from about 1.000090 in early July (when Earth is farthest from the Sun), down to 0.999906 in early January (when Earth is closest to the Sun).

Factor (2) is greater than 1.0 for about half the year, so we can expect one equinox to occur during that time. For some window of time containing that equinox, the Sun’s speed will be (ever so slightly) greater than 15° per hour. In the current era, that’s the September equinox. The Sun’s speed across the sky reaches a maximum of maybe 15.0003° per hour, shortly before the September equinox. (I didn’t do enough calculation to figure out exactly when; it might be just a few hours before.) It’s above 15° for maybe a couple of days.

Near the March equinox, it doesn’t quite get up to 15°, having a local maximum of about 14.9997°.

The minimum is about 13.7610°/hour, shortly after the December solstice.