Celestial Mechanics, Vector Analysis and
the Geometry of the Path of Solar Ejections

Looking at Spaceweather.com we see on the left hand side, a section entitled Coronal Holes. The text here can read "There are no coronal holes on the Earth-facing side of the sun.", or it can read that solar wind from a coronal hole is expected to hit the earth on a given date, or it can read that a wind stream will miss the earth. I would like, on this page, to look at the geometry involved in making a decision as to whether the stream might hit the earth.

Celestial Mechanics is the study of the gravi- and electro-mechanical relationships (forces) among the celestial bodies (macro).Vector Analysis is a method for analyzing these forces. Consider the notion of ocean tides. The main components are the sun and the moon, and the main data are the masses of these, the square of the distance to them at any given time, and the angle between them. Vector analysis is the tug-of-war model of the solar system.

As the moon orbits the earth, it comes into line with the sun twice a month producing higher tides. When they are at 90 degrees, 'pulling' across one another, the tides are lower. From Kepler we learned that orbiting bodies don't orbit in circles but in ellipses (two focii), with the parent body at one focal point. As the force of gravity varies with the square of the distance, we have variations in both the solar and lunar component to keep up with.

We also learn that the planets don't orbit the sun in the same plane. We will call the plane that the earth orbits the sun in, the ecliptic. The ecliptic is also understood to be the path of the sun against the background of stars. As we orbit, we are tilted 23.5 degrees right, so that the ecliptic slices the globe from 23.5 north to 23.5 south.

The moon orbits the earth in a plane that shifts and can be as much as 5 degrees above or below the ecliptic in a 19 year pattern. As the lunar plane shifts the days for eclipses are earlier every year, and occur where the moon's path crosses the ecliptic at a new or full moon.

Next we add the notion that the moon doesn't really orbit the earth in an ellipse, because the earth moves 1 degree a day through its orbit. Imagine the earth moving through space with the sun on the left. At the full moon, the moon is on the right side moving in the same direction as the earth (same instantaneous velocity). As it approaches the quarter, it is ahead of us, and at the new moon, it is on our left moving in the opposite direction. At the last quarter, it is 'behind' the earth, crossing its trail.

(Electical engineers take note of the implications for the induction between bodies in different directions. The electrical charge existing between bodies in free space is a function of their relative velocities, just as the mutual induction of their surrounding fields is a function of their relative directions or planes of motion. I direct interested readers to the 1929 work of L E Johndro.)

Now add the notion of ejections from the sun which when presented, is usually accompanied by a graphic like this:

Which brings us to the issue with which we started, how can we decide which ejections might impact us. Note that the image above is not to scale, and suggests a more or less instantaneous connection, when in actuallity it takes several days for ejections to reach the earth. Another thing to consider is that the sun is rotaing on its axis every 27 days, and that it's axis of rotation is tilted almost 7.5 degrees out of perpendicular to Earth's orbital plane.The sun's equitorial plane is inclined to the ecliptic.

Imagine that you are on a rotating merry-go-round with a paint ball gun, and want to hit a target; where do you aim? Or when do you fire? Remember that at the edge, as the ball leaves the gun it is traveling straight away from the center of the wheel, but will look to move to the right (given counter clockwise motion of wheel). Also, if you shoot straight up the arc will be different. My point is that where ejections from the sun go depend on what latitude on the sun they occur, and the longitude and rotation rate of the sun.

This image is taken from a Scientific American article by James van Allen, showing the projected path of an equitorial ejection from the sun. Arrows indicate that the instantaneous velocity of the particles is pointed away from the center of the sun as they moves outward. This shows that in order for an ejection to impact nthe earth, it needs to be ejected ahead of us in our orbit ... if all ejections were equitorial and the sun wasn't moving through space too.

Example

Today (Jan 18th, 2010) we read at spaceweather.com "A solar wind stream flowing from the indicated coronal hole should reach the Earth on or about Jan. 19th". An image of the sun shows the hole.

Here we see how the earth's orbit around the sun and the season's are usually depicted. This picture of things leaves out the fact that the sun is tilted on it's axis, the sun rotates every 27 days, and it is moving at 19 Km/sec through space.

When you were introduced to Johannes Kepler in high school, it was in either a physics (celestial mechanics) class or a mathematics (calculus) class, using the terms ellipse and equal area rule, and some version of this image.The idea is that while the shape of the path of orbiting bodies is elliptical and they move more quickly when they are close to the parent body, and more slowly when they are far from it, we are told that the area 'under' the curves of same time periods (a week or month) will be the same.

We we taught with models that showed closed loops which simply don't exist in celstial mechanics. Adding the element of time below we see a better picture of the orbiting earth.

The plane of the ecliptic is inclined to the direction of travel of the sun. In December, we are behind the sun; on Jan 3rd closest ot it. In March we are beside it. In June we pass in front of and below it, and are farthest from it July 4. In Sept as we are beside it again, our instantaneous velocity is 90 degrees to that of the sun.

I hope that clears some things up.