Of Particular Significance

Hey! Where are you Headed Today?

POSTED BY Matt Strassler

POSTED BY Matt Strassler

ON 02/14/2022

Even if you’re working from home, so that you’re spending the day at a fixed location on the Earth’s surface, you’re not at a fixed location relative to the Earth’s center. As the Earth turns daily, it carries you around with it. So where are you headed today? Presumably Earth’s spin takes you around in a big circle, right?

That’s great. Which circle?

Point to it, right now.

Let me ask that again, in case that wasn’t clear. With your feet on the ground, looking whichever direction you choose, please show me the circle you’ll be taking today on your travels.

Most people who hazard a guess imagine that if they face east (toward the rising Sun, which here is into the plane of your screen), they are traveling on a circle that cuts vertically into the ground. But this is true for very few of us.

No idea? In my experience, many people have never even thought about it. Those who are willing to hazard a guess have to think for a moment to figure out that the Earth is rotating west to east — that’s why the Sun appears to rise in the east and set in the west. Once they are clear on that point, many people face east, and then indicate a circle that goes straight ahead, which would be combination of east and then down, as you can see in the figure.

To say that another way, if you imagine the circle of travel as being the edge of a disk, that disk would face east-west and slice directly down into the ground.

For the vast majority of us, it turns out this guess is not correct.

So where are we headed? People located at the equator or the poles can answer this more easily than the rest of us, so let’s start with them.

An Aside on Implicit Approximation

By the way, as is common for physicists, I’ll be implicitly describing how things would work on a perfect Earth. Earth’s surface of course is not a perfect sphere. It’s a little flattened at the poles, and it has valleys, mountains and slopes of all sorts. So when I say, for instance, that “down is perpendicular to the ground”, I’m implicitly assuming that the ground is smooth and without any slopes.

The reason this kind of approximate talk is legitimate is that the Earth is almost perfect. It’s a sphere to better than 1%; for example, its tallest mountain is 13 mi (20 km) higher than its deepest valley, and that’s nothing compared to its diameter of about 8000 miles (13000 km). So Earth’s imperfections are merely a distraction; what exactly true of a perfectly spherical Earth is almost exactly true of an almost perfectly spherical Earth.

Ok, back to the main points.

Travel at the Equator

The equator is an example of a “great circle.” A great circle is the largest circle you can draw on a sphere. As for any great circle, its length is the full circumference of the Earth, it divides the Earth into two equal halves, and the exact center of the circle is also the exact center of the Earth. To say it another way, if you view the equator as the edge of a disk, that disk cuts right across the Earth, perpendicular to the Earth’s spherical surface right the equator. The direction of gravity’s pull, perpendicular to the Earth’s surface is what we call “down”.

If you’re standing on the equator and face east, your perspective on this disk is that it stretches out in front of you and cuts straight down. For people on the equator, the guess in the earlier figure is actually correct, as you can see in the figure below.

Left: a person at the equator goes around the Earth on the equator. The disk for which the equator is the edge cuts the Earth exactly in half. Right: from the point of view of someone on the equator, this disk points east-west and straight down, so the person is traveling straight ahead as the Earth rotates. It is perpendicular to the Earth’s rotation axis, which points north and south from this person’s perspective.

Travel at the Poles

But this simple picture clearly is not always right, because at the north or south pole, the word “east” has no meaning. All directions from the north pole take you south, and all directions from the south pole take you north. And a person standing at the north pole or south pole doesn’t go anywhere at all. They stay at the same location as the Earth rotates underneath them; all they do is change their orientation, going round once a day as though on a slowly rotating platform. In other words, they’re not on traveling any circle at all.

Left: A person standing exactly at the Earth’s north pole doesn’t change location (relative to the Earth’s center) as the Earth rotates. Instead this person will simply rotate, as though standing on a turntable. Right: for them, standing at the point where the Earth’s axis meets the surface of the Earth, all they do is turn, without a change of position.

Travel for the Rest of Us

Most of us live neither at the equator nor at a pole. So what happens with us? I’ll focus first on the Earth’s northern hemisphere, north of the equator, since that’s where 90% of us live.

Let’s look at the experience of someone about halfway between the pole and the equator, such as someone living in Seattle, Montreal or Venice, at 45 degrees latitude. To do this, it’s essential to be very clear in one’s head about the precise meanings of down, north, east, and also the Earth’s spin axis and how it is oriented.

  • Down is the direction that gravity pulls, and that is toward the Earth’s center.
  • North is the direction along the Earth’s surface that would take us, if we walked, to the north pole.
  • East is the direction along the Earth’s surface that the Earth is rotating toward.
  • The axis around which Earth spins is not oriented north-south along the Earth’s surface, except at the equator. (In fact, as we saw, at the poles it is oriented up-down!) In the northern hemisphere, it points north-up and south-down; at 45 degrees latitude, it is 45 degrees off the horizontal.

Now if you look at the left side of the figure below, you can see that the during the day each of us travels a circle of constant latitude. I’ve marked the one at 45 degrees latitude. If you made that circle into the edge of a disk, that disk does not pass through Earth’s center. It’s perpendicular to the Earth’s axis, but it is not perpendicular to the ground.

To say this another way, down is the direction toward the Earth’s center, but the disk whose edge is the trajectory of Venice or Montreal does not point downward!

This is perhaps easiest to see in my north-pole-at-the-top drawing at left, but now look at it from the perspective of the citizen of one of these cities, which is drawn at right, with up-at-the-top. Everything gets tilted over a bit. And what do we learn? When we face east, the circle that Venetians, Montrealers, and Seattlites will be traveling starts straight ahead but quickly moves to the left as it moves forward.

Left: Each of us circles the Earth’s axis daily on a fixed line of latitude; a disk whose edge is this circle does not pass through the Earth’s center, and is not perpendicular to the Earth’s surface. Right: tilting the image so that the ground is horizontal, we see that the Earth’s axis and the circle of daily rotation are both tilted relative to the ground at this latitude.

Relative to the ground that this person is standing on, the Earth’s axis tilts up-and-north to down-and-south. We move on a circle that is perpendicular to this axis, and that direction is not straight ahead around the Earth. Walking straight ahead without turning would take us on a great circle, whose center would be the Earth’s center. But the Earth, via the friction that keeps us from sliding on the ground, pulls us around to the left. Our daily circle is easy to see on a north-pole-at-the-top globe, but it’s not so easy for us to visualize in ordinary life because of our strong preference to orient ourselves with up pointing, ahem, up.

The Southern Hemisphere

Everything I’ve just said applies in the southern hemisphere too, except with directions flipped around. North and south are switched (but not east and west, and not up and down. [Wherever you live, if you’re standing then down is toward your feet.]) So for someone facing east in the southern hemisphere, their circle of travel takes them to the right.

Comments: Orbits and Coriolis

I hope these answers are satisfying and clear, even if they might be initially a bit disorienting. Let me finish with a couple of answers to questions some readers may have.

Notice how different our motion is from an orbit! We do not orbit the Earth. A circular orbit of the Earth has to be a great circle and go round its center, not its axis, so a line of latitude is the wrong type of circle. We’re also moving far too slowly. If the ground disappeared underneath us and the Earth became a tiny lump with the same mass, we would find ourselves traveling on elliptical orbits that would bring us far closer to the Earth’s center.

Left: The daily path of a person at a northern latitude, a circle whose center is the Earth’s axis but not the center of the Earth. Middle: just as for near-Earth satellites, to orbit the Earth would mean traveling on a great circle whose center is the Earth’s center, and which crosses from the northern to the southern hemisphere; also it would require a speed of 17000 mph (27000 kph) relative to the Earth’s center, whereas our speed due to Earth’s rotation is less than 10% of this. Right: if the Earth was suddenly compressed into a tiny ball, each of us would fall mostly downward, due to insufficient forward speed, and begin to orbit the Earth’s center in a highly elliptical path.

Conversely, if something passes directly over our heads, heading east, whether it is in orbit or moving in some other way that keeps it from being dragged by the Earth on the constant latitude circle, that object will proceed on a great circle. Since (in the northern hemisphere) the Earth drags us to the left, the object will look to us as though it is turning to the right. This is a simple example of the Coriolis effect (clarified by Coriolis, but known two centuries before, and key to understanding both Earth’s weather patterns and Foucault’s pendulum.) This was one of the key insights which convinced skeptics that the Earth really does rotate and that we go round on weird circles daily, even though it doesn’t feel that way.

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4 Responses

  1. Potsdam Gravity Potato withstanding, the almost perfectly spherical Earth shrunk down to fit in the palm of one’s hand, the beautiful distractions would be imperfections of less than a millimeter which is too small to detect with our tactile sense. The oceans would make it feel sweaty. Spoiler alert! The Earth is not likely to shrink to that size (anytime within the lifetime of a proton not at CERN.)
    OK, now to the main point.
    Reviewing Earth’s (or any body’s) gravity and thinking about dynamic interactions within (unsmashed) particles seems to be a path leading to UFT, GUT and TOE. (Which are all the same thing.) And you’re so close. The Mexican hat of the Higgs boson points toward the local gravity, as the particle spins, perturbations on the surface seemly search for other sources of gravity. Perhaps in future posts, the subject of why particles in the ocean’s water seem attracted to the moon will be discussed.
    Anyone comoving with the Earth is being flung through the galaxy at 237km/s and it certainly doesn’t feel that way, huh?
    Thanks for the posts and keep thinking.

  2. the earth in turning drags the space with it and if so up to what altitude does it have an important influence?

    1. To get any significant (meaning “easily noticeable”) dragging would require the Earth be a black hole or an object not much larger, i.e. about 1 cm in size, in which case I believe frame dragging would be significant out to a few centimeters, or maybe a few meters, depending on what you mean by “important”. And this dragging would not be noticeable unless the Earth were rotating very fast compared to its current rotation rate. If you want to see frame-dragging with important influence, go look at the fast-spinning huge black holes in the centers of galaxies, not at Earth.

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