The rotating Earth

Viewed from a reference plane tangent to the surface of the earth, the sun rises and the sun sets. Viewed from the star polaris, the reference plane attached to the earth revolves around the axis of rotation through the earth, this causes to sun to rise and set on this plane. Both views are used by physicists, but often elementary school teachers label the view that the sun rises or sets as wrong.

How long is a day?

How fast does the earth rotate? Linear speed versus angular speed.

Using approximate measurememts, the earth's equatorial circumference is 24,000 miles and the earth rotates once in 24 hours so the speed of rotation at the equator is

24,000 mi./24 hours = 1,000 mi/hr.

But we can also measure the angular speed of rotation.

360 degrees/24 hr = 15 degrees per hour.

The sun subtends is 1/2 degree how long does it take for the sun to move through an angle equal to its angular diameter?

(The angle subtended by the sun is the angle from one edge of the sun to the opposite edge)

(The sun is about the same angular size as a pea held at arms length.) (The angular size of the moon)

1/2 degree/15 degrees/hr = 1/30 hr = 2 minutes.

So when the bottom of the sun touches the horizon as it sets near the equator it will vanish in 2 minutes.

You can measure the rotation of the earth by looking at a distant star, and using a watch to time how long it takes the star to circle the sky and return to the same place I usually observe a star as it passes behind a north-south power line. This is known as a sidereal day.

Measure the rotation of the earth

You may be surprised by the answer. It is not 24 hours.

So why do we say the length of the day is 24 hours?

The dance of the earth and the sun

You have to time the motion of the sun across a north south line.If you do that using shadows or a solar viewer and a north-south power line you will find that it takes 24 hours (plus or minus 20 seconds depending on the time of the year)

The dance of the earth and the sun. Have a person point an arm out directly in front of them and rotate around one full time, starting with the arm pointing at a nearby person and ending with the arm pointing at the same person. They rotate through a complete circle of 360 degrees. This rotation represents one day.

Next have the person representing the earth walk in a circle around a stationary person. This is called a revolution, not a rotation, and represents one year.

Now combine the motions walk in a circle around the central person, the sun, and rotate at the same time. Start with one arm pointing toward the sun walk around counterclockwise while rotating counterclockwise. Complete one full rotation while revolving 1/4 of the way around the circle. In this model solar system how many days are there in a year?

Now measure the angle the earth person rotates through when their arm points to the sun the first time. The answer is more than 360 degrees. It is actually 360 + 90 or 450 degrees. So the length of the solar day is greater than the length of the sidereal day. The solar day is 1 + 1/4 rotation which takes 125% of a sidereal day.

The earth rotates 365 times each year. So in one day the earth has to rotate 1 + 1/365 rotation. which takes about 1/365 of 24 hours which happens to be 4 minutes. Which is why the solar day is longer than the sidereal day.

Switching things around if you look at the sun against the background of stars (OK to do this you have to wait for a total solar eclipse which will happen every 300 years if you just stand and wait.) You will see the sun appear to move around you in my model by 1/4 of the circle each day.

The Ancient Babylonians knew that the sun moved around in the star field about 1/365 of a circle each day.

The Sumerians and Babylonians used a sexigesimal number system based on the number 60.

They seemed to hate fractions and they loved12, because it can be divided by 2,3,4,6, without any fractional remainders, they also loved 60 because it can be evenly divided by 2,3,4,6,10,12... and they loved 360 figure its dividers out yourself.

They chose 360 degrees for the circle, which is an approximation to the motion of the sun through the stars each day. The 360 degree circle was passed down to the Greeks and to us. We use it for dividing circles and time.

When they wanted to divide one of the 360 degrees of a circle into smaller , or minute, parts they broke it into 60 parts called arc minutes.
Consider day and night, the Egyptians divided days into 12 parts and nights into 12 parts, giving us the modern 24 jours per day. Our clock faces are divided into 12 equal parts.

And we want to divide a degree of arc into smaller pieces we divide it into 60 arcminutes. We also divide an hour into 60 minutes.

We divide an arcminute into 60 arc seconds, and divide one minute of time into 60 second-minutes, which we have shortened to seconds.

There are third minites but they are seldom used, interestingly enough a third minute is 1/60 of a second. The frequency of the electric powerlines in the U.S. is 60 hertz.

The true solar day

The sky is crossed by a meridian which runs north-south directly over your head, we can start our day length timer as the sun crosses the meridian. Before the sun crosses the meridian it is AM, ante meridian and after it crosses it is PM post meridian. As it crosses it is solar noon.The time from noon to noon is te length of the true solar day.

The earth's orbit is not circular, it is elliptical with the sun at one focus. On January 4, the earth is closest to the sun, at perihelion, on july 4 it is furthest from the sun , at aphelion. In an elliptical orbit, planets travel faster when they are closer to the sun, and when planets are closer to the sun a fixed distance of arc spans more of an angle. So at perihelion the earth moves through a greater angle in its orbit, this means that it must rotate a little angle to make the sun cross the meridian again. So one day at perihelion is longer than one day at aphelion.

The mean solar day

Each day of the year has a slightly different length true solar day. Making a clock is hard enough, but making one that speeds up and slows down during the year is even harder. So clock makers make a clock rotate so that one clokc day has a length that is the average over an entire year of the true solar day. This is known as the mean solar day.

Tilt is the reason for the seasons

It gets worse. The earths axis of rotation is not perpendicular to the plane of its revolution. It is tilted 23.5 degrees from the perpendicular. Over the course of a year the rotation axis of the earth remains pointed at the north celestial pole, near the star polaris. As the earth revolvesabout the sun it keeps its axis pointed near polaris. This means that for part of the year the northern hemisphere is tipped toward the sun, summer, and for another part of the year it is tipped away, winter.

This means that the rising and setting points of the sun move around on the horizon, and the height of the sun above the horizon changes from day to day. The length of the day from sunrise to sunset changes over the year. Ancient cultures tracked the rising and setting points of the sun to make a calendar. Stonehenge, for example, can be used as a calendar.

Build your own Stonehenge

It gets even worse.

The sun and the moon exert twisting forces on the equatorial bulge of the earth. These twisting forces, known as torques, cause the axis of the rotating gyroscope called earth to precess, to change its direction through a large circle over the course of 19,600 years. During this times other stars are the north star. Much of the time there is no north star at all, just as right now we have no south star.

Synchronous rotation

The moon circles the earth keeping one face toward the earth. Describe how the moon rotates as it revolves around the earth.

Roll quarters around each other, roll magnets.

rotation of the sun

rotation of other planets

rotation of millisecond pulsars.


Scientific Explorations with Paul Doherty

© 2013

12 July 2013