This is another confession. Shannon Templeton wanted to know how to find what she called the Earth’s Direction of Travel, EDOT, and whether there is another name for it, and how it is affected by complications such as the eccentricity of the orbit. I told her that it might, on the analogy of the Apex of the Sun’s Way among the stars, be called the Apex of the Earth’s Way (it is, as Alastair McBeath later told us) and that, being tangent to the orbit, it is simply the Sun’s longitude minus 90 degrees.
I went ahead and incorporated it into diagram-making. Shannon said a humble thankyou, but she was – rightly – dissatisfied and did some more thinking, and these words are worth quoting: “One of the greatest tools my professors taught me to use was when I am struggling with a concept to take it to extremes.” She made this sketch:
Earth’s orbit is nearly circular, but suppose it were extremely eccentric, like this long ellipse (of which I show only half). At points along it (all points except for the two ends, the perihelion and aphelion), a tangent line would clearly not be perpendicular to a line pointing to the Sun (the star).
So what is the true formula for that tangent direction? I haven’t yet found it (it is surely extractable from the elaborate answers that you can get online from “Doctor Math”) but have approximated it by brute force. That is, I’ve made the program do some transforming between rectangular and polar coordinates and find the direction in space from the Earth’s position to its position a short while later. (Actually, from a position some minutes before to a position some minutes after.)
In this picture re-done,
the symbol marking the Edot has shifted just 0.15 of a degree relative to the Jan. 2 First Quarter Moon [correction: Last Quarter], which is handily nearby for me to make this measurement! The Earth is only just past its perihelion position (also Jan. 2, but that coincidence is an irrelevant one), where the difference would be zero. So we’ll expect the difference to grow more significant in the spring and autumn months, when the slight outward and inward trends of the orbit are greater.
And here is the other picture for the Quadrantids slightly re-done.
When I gave it a longer flight-of-Earth arrow so as to make clearer the angle between this direction of travel and the approach angle of the meteors, it occurred to me to amplify the arrow further. Now there are additional segments of arrow, or, there are several arrows superimposed: the distance each protrudes past the last is the distance Earth moves in one minute.
Do you feel the eight-thousand-mile-wide Earth pushing us along, at more than a thousand miles each minute? No, but you believe it; Copernicus’s and Galileo’s critics did not feel it and therefore did not believe it.
Believe that’s a last (or third) quarter moon in the diagram rather than a first quarter.
You’re right. Raises (again) my quandary of whether it’s ethical to go back and correct the text of a post.
Maybe do like some other blogs I’ve seen and just insert (correction: third quarter) after the mention in the text? That acknowledges the oops but doesn’t leave the incorrect info standing alone.
Yes, I think I’ll do that, using square brackets {} to show the insertion.
In other instances I’ve retrospectively changed a post, but only in the sense of improving the wording.
I have truly enjoyed having these discussions with you about the edot, or more correctly, the Apex of the Earth’s Way as I now know. It always surprises me how difficult I can make a simple problem! You set me on the correct path with a casual mention of using ecliptic coordinates instead of celestial, I had forgotten one of my prime rules of writing astronomy programs – choose your coordinate system wisely, you can always convert later. I was able to find the formula for solving for the slope of a tangent to an ellipse and again felt silly that I had forgotten it as well; it is such a beautifully simple and elegant formula. Once the slope of the tangent is found, trig will give the exact angle of the Apex of the Earth’s Way (edot) in relation to the Sun.
This bit of mental gymnastics began when I was studying astronomy and learned the direction the solar system is moving. I enjoyed pointing this spot in the sky out to people as I was showing them constellations, etc, and it got me wondering exactly where was the Earth racing towards at any given moment. I knew it would be the point where the meridian crosses the ecliptic exactly at sunrise, but that took some thinking and an ephemeris to figure out at times other than sunrise. I could work it out in my head roughly where the point would be, but I enjoy precision and the challenge of writing out a nice algorithm to use in programming.
Thank you again for tackling this task with me, after all these years I can finally put this puzzle to bed.
Alas here in Wisconsin the only showers of late left a half inch of water on my driveway. It seems the better the astronomical event the more likely it is to be cloudy.