The following is an excerpt from my new book that will be released on August 21, 2017, the day of the next US eclipse: “Total Solar Eclipse Countdown: Totals and Annulars Through 2024.”
Question 2: Which direction and at what speed does the eclipse shadow move across the Earth and why?
Things are moving fast up there, and while we hardly notice it on normal days, it all suddenly becomes very interesting on the day of the eclipse. Consider that the shadow itself is moving through space while at the same time we are spinning like a top.
Even though the Moon moves from east to west across the sky (rises in the east and sets in the west), the combined motions of the spinning Earth and the orbiting Moon cause the eclipse shadow to move from west to east. This is not intuitive.
It is hard to get your head around this, so I will explain with another thought experiment. The Earth rotates counterclockwise as seen from above the North Pole, the same direction as the orbit of the Moon, and hence it’s shadow. Now imagine the Earth is standing still and see yourself hovering over the North Pole looking down at Earth. The Moon is moving counterclockwise in its orbit and the shadow sweeping over the Earth follows along in a counterclockwise direction. Now, you ask yourself the question… is that shadow moving towards the east or west? To figure this last part out, imagine you are standing looking head-on at your standard high school globe. On your globe, the North Pole is at the top and the outline of the USA is directly in front of you with Los Angeles is on your left and NY on your right. Now, if the globe is still and the shadow moves counterclockwise as seen from above the North Pole, you can ‘see’ the shadow moving across the US from west to east!
But you point out that we assumed that the Earth was not rotating. OK, let’s add the motion of the spinning Earth and see what happens. Both the Moon’s orbit and the Earth’s rotation is in the same direction, counterclockwise. Continue to imagine that shadow moving across your globe from west to east. Now start to spin the globe in the counterclockwise direction (from west to east). Next, we need some simple math.
It turns out that when you are standing on the counterclockwise spinning Earth’s equator, you are being hurled to the east at about 1,200 miles per hour. See the breakout box for this calculation.
The Moon and its shadow is moving to the east at about 2,300 miles per hour and you are moving to the east at about 1,000 miles per hour. At what speed and in which direction will the shadow pass you? The answer is the shadow is moving to the east at about 1,300 miles per hour (2,300-1,000).
This is the same thing as saying you are in the slow lane on the freeway and moving east at 1000 miles per hour and a red Ferrari passes you in the fast lane moving 2,300 miles per hour. How fast is the Ferrari passing you and in what direction? It is passing you at 1,300 miles per hour and is moving to the east.
The simple explanation is that during the few hours of the eclipse, the Moon’s shadow overtakes the Earth spin. The Moon’s orbit is really fast and the shadow also moves at the orbital speed of 2,300 miles per hour, overtaking the slower earth rotation.
On the other hand, although the Moon is moving really fast in its orbit, it is so far away that it takes a full Month for it to go completely around the Earth. In fact, during one month (actually 27.3216 days or 655.7184 hours) the Moon will travel 1.5 million miles to complete its orbit.
The calculations are as follows: 1) Distance Moon travels in a month: The circumference of a circle (we know it is not a circle, but close enough for this) is 2 x pi x radius, and the radius for Moon is 240,000 miles. 2 x 3.14 x 240,000 = 1.5 million miles traveled per orbit. 2) Speed of Moon: An orbit takes a month or 656 hours. Therefor the speed of the Moon = distance/time; speed = 1,500,000 / 656 hours = 2,286 miles per hour.
So, in one Earth day, the Moon appears to have shifted only slightly in its orbit (about 12 degrees which is 360 degrees divided by 30 days). It’s as if the Moon practically stood still and at the next Sun-moon-rise we see it again rise in the east.
For you physics majors, this is the difference between angular and linear velocity.