As we gaze across a beautiful valley or stare in awe at a distant mountain, it is easy to forget that we are on a spinning platform that is traveling on an elliptical orbit around the sun at an average speed of 66,600 miles per hour. I find this seemly unending journey truly amazing. In this post, I would like to take a look at some of the facts that mankind has learned about this journey.

Before Nicholas Copernicus (1474 – 1543), many people thought that the Sun, planets, and stars rotated about the Earth, and each planet in turn rotates on its own private circular arc. This complicated Earth centered view of nature became so entrenched that it became an article of faith in the Catholic Church. In fact, the Catholic Inquisition threated Galileo (1564 – 1642) with torture on the rack unless he publicly retracted his belief in the Sun centered circular orbit Copernican world system. Galileo publicly retracted his belief in the Copernican world view and was spared torture on the rack, but spent the remaining years of his life under house arrest.

Johannes Kepler (1571 – 1630) discovered three laws of planetary motion which is a relatively simple description of planetary motion. (You may find it helpful to read my post Demonstrating Dynamics in a Mathematical Model.) Kepler’s first law stated that the orbit of a planet around the Sun is an ellipse where the Sun is located at one of the two foci of the ellipse. An ellipse is a very special curve where every point **P** on the ellipse, the distance from **P** to one focus point plus the distance from **P** to the other focus point, is a constant. The diagram below shows an ellipse with foci at **F _{1}** and

**F**, length of major axis = 10 units, length of minor axis = 6 units, and center point at (0, 0). For very point P on an ellipse, the sum of the distances from point

_{2}**P**to the two focus points equals the length of the major axis. As indicated in the diagram below, an ellipse can be drawn by first anchoring the endpoints of a length of string on a piece of paper or cardboard. Use a pencil to make the string taunt, and then trace the curve by keeping the string taunt as you move the pencil along the elliptical curve.

To better understand planetary orbits, it’s necessary to understand what we mean by the **eccentricity** of an ellipse. If **a** = half the length of the major axis, and **c** = the distance from the center to a focus point, then the eccentricity **e** of the ellipse = **c/a**. Thus elliptical eccentricity **e** ranges from 0 to 1. If **e** = 0, the ellipse is a circle, and if **e** = 1, the ellipse degenerates to a line segment with foci at the endpoints of the major axis. (By definition, the eccentricity of a parabola equals 1, and the eccentricity of a hyperbola is greater than 1.) The two diagrams below show eccentricity values for five ellipses where the ellipse and foci have the same color. Note that eccentricity approaches 1 as the foci approach the endpoints of the major axis. **The eccentricity of the Earth’s orbit = 0.0167086.** This is the reason, I suspect, that Copernicus thought the Earth’s orbit was circular, not elliptical. Since half the length of the major axis of the Earth’s elliptical orbit equals **149.6 million km,** it follows that the Sun is 0.0167086*149.6 million km = **2.4996 million km** from the center of the Earth’s orbit.

The two diagrams below show an exaggerated oval shape of the Earth’s yearly orbit around the Sun; the purpose is to draw your attention to key time periods in a year. Orbital dates can vary slightly from year to year, and therefore the dates shown in the diagrams are approximate. The following points describe the key time periods in Earth’s orbit:

- At the point of
**perihelion**, the Earth is at its closest point of 147.1 million km from the Sun. In northern latitudes, the direction of the Earth’s polar axis is tilted away from the Sun, which results is less direct sunlight and cooler average temperatures. - At the point of
**aphelion**, the Earth is at its farthest point of 152.1 million km from the Sun. In northern latitudes, the direction of the Earth’s polar axis is tilted towards the Sun, which results in more direct sunlight and warmer average temperatures. - The equinoxes and solstices divide a year into approximately four equal time periods or seasons. At the fall and spring equinoxes, the Earth’s polar axis is perpendicular to the plane of the Earth’s orbit which results in equal periods of daylight and darkness. At the summer and winter solstices, the Earth’s polar axis is tilted towards or away from the Sun which results the longest and shortest days of the year.
- At the point of perihelion, the Earth reaches its fastest orbital speed of 109,080 km/hour.
- At the point of aphelion, the Earth reaches its slowest orbital speed of 105,480 km/hour.
- The average or mean orbital speed of the Earth equals 107,200 km/hour or 66,600 mph.
**It takes the Earth 365.256 363 004 days to orbit the Sun.**Because of the extra 0.256 363 004 days in a year, it’s necessary to add an extra day to our calendar every four years in February. To be more specific,**leap years occur in years that are multiples of 4 or 400, but not multiples of 100.**Hence the years 2000 and 2400 are leap years, but the years 1800, 1900, 2100, 2200 and 2300 are not leap years. All other years that are multiples of 4 such as 1868, 1936 and 2016 are leap years.- In the diagrams below, note that seasons in the northern and southern hemispheres occur at opposite times of the year.

Everyone knows that the Earth does a daily rotation about its polar axis. Here are a few facts about the Earth’s rotation.

- The Earth rotates in about 24 hours with respect to the Sun and once every 23 hours, 56 minutes and 4 seconds with respect to the stars.
- The Earth’s rate of rotation rate is slowing with time. Atomic clocks have demonstrated that a modern-day is about 1.7 milliseconds longer than a day in 1900. (I doubt that this fact will be reported in the national news any time soon.)
- In the northern hemisphere, the Earth rotates east towards the Sun in the morning hours and away from the Sun in the west in the evening hours. This is the reason that the folks in New York see the Sun about 4 hours before the folks in California.
- Technically speaking, there is no such thing as sunrise and sunset. The Sun only appears to rise and set in the sky because of the rotation of the Earth. Buckminster Fuller who was an American architect (geodesic domes) and systems theorist suggested that we should the terms
*sunsight*and*sunclipse*because the terms sunrise and sunset do not accurately describe what we observe. - The Earth’s rate of rotation is not constant. The true solar day is about 10 seconds longer at the point of perihelion and 10 seconds shorter at the point of aphelion.
- At the equator, the Earth’s linear speed of rotation is 465.1 m/s, 1,674.4 km/h or 1,040.4 mph. At higher latitudes, the linear rate of rotation is reduced by a factor of Cos(angle of latitude). Example: The Kennedy Space Center is located 28.59° North latitude and has a linear rotation rate of 1,674.4Cos(28.59°) = 1,470.23 km/h = 913.56 mph.

I will close this post about an epiphany I experienced many years ago. As I recall, it was about March of 1975 when my neighbor Chuck Beck invited me into his back yard to view Sun spot activity. Chuck had placed his expensive Celestron telescope with an attached power cord and lens filter on his picnic table. As I adjusted a knob on the Celestron in order to keep the Sun in view, I had the same physical sensation in my legs as if I was riding a merry-go-round. I thought to myself, “Johnson you really ARE on a moving and spinning platform in space!”

This is exactly what I’ve been looking for! Great balance of math with astronomical detail. I appreciate the careful use of language, and thus ask: Is it not accurate to consider eccentricity as the ratio of center-to-focus to half-major-axis?