Fall Equinox

The autumnal equinox is at 4:21am tomorrow.  What does that even mean?

The earth circles the sun once each year.  The earth's axis is tilted 23 1/2 degrees out of the vertical.  (Recall every globe you've seen).  Since the direction of that tilt does not change, but the earth's position around the sun does change, the sun shines most on the northern hemisphere from late June to late September, and on the southern hemisphere from late September to late June.  At a moment between these two periods the sun shines directly on the equator, as the earth transitions from one hemisphere to the other: this is the equinox.  So tomorrow fall officially begins in the northern hemisphere, while spring officially begins in the southern hemisphere.  (Here's a satellite's eye view.)

(NOT to scale: sun is roughly 100 earth's wide, and roughly 100 suns away!)

Image by Kevin Niewood

At the equinox day and night are equal in length everywhere in the world--hence the name.  Only at the equinox does the sun REALLY rise due east and set due west.  It also roughly marks the time when sunrise and sunset are at their shortest: in takes only a few minutes at our latitude for the sun to completely appear at dawn, or disappear beneath the horizon at sunset.  Most importantly, the equinoxes are equal in the amount of solar radiation at any given place.

That means the sun is giving us the same amount of daily energy as it did on March 21st!  Why the difference in temperature?  Think of the daily solar energy input as a day's pay.  The heat held by Earth's surface (mainly in the enormous storage capacity of ocean and lake water) is your bank account.  Our income is the same, but we have a big bank balance of heat thanks to all the bonuses we got during the summer!  That is why you are probably dressed differently now than you were last March 21!

Everything you need to know is here!

How far has the earth gone?

A question in my physics text led me to a fact I could hardly believe.  The question was: "How far has the earth traveled in its orbit around the sun since it first formed?  Assume the earth is 4.5 billion years old and has a circular orbit with a radius of 149,597,870km.  Ignore the solar system's movement through space."  The math is simple, but the result amazing.  Every year the earth goes 2π149,597,870km=939,953,337km.  Over its lifetime, that's (939,953,337km/y)X(4,500,000,000yrs) =4.3X10¹⁸km.  

Now that number, ending in 17 zeroes, is so large it’s almost meaningless.  Let's use the light year to put it in perspective.  Light, traveling at 300,000,000m/s, is the fastest thing in the universe.* In a year's time, light can travel 3.00X10⁸m/sX365X24X60X60 =9.46X10¹²km (about 6 trillion miles).  That distance is a light year.**  A light year is unimaginably large: it would take you 10 trillion years to cover that distance at highway speed.  Yet even the nearest star to our sun is several light years away, and many of those visible in the night sky are tens or hundreds of LY away.  Our whole galaxy, the Milky Way, is about 100,000LY across.
 
The earth, moving much faster than highway speed, covers more than 900 million km each year, for a speed of 939,953,337km/(365X24X60X60) =29.806km/s.  (That's about 67,000 mph.  Don't challenge the earth to a race.)  That cumulative distance is equal to 455,000LY!  Over its life, the sun could have made 1½ laps around our entire galaxy!

The earth takes about 10,000 years to travel 1LY in its revolution about the sun. 

*In its broadest sense, light includes all forms of electromagnetic radiation: x-rays, radio, etc, as well as visible light.  All travel at that magical speed of 3.00X10⁸m/s.  Oh, and of course light isn't a "thing" at all.  Actual things (matter) can't go that fast.

**Right: the LY is a measure of distance, not time.

Earth is nearest the sun in winter

In fact, it passed perihelion (position nearest the sun) on Jan 4.  

The summer-winter difference of 3 million miles (out of a total average distance of about 93 million miles isn't enough to overwhelm the axial tilt that causes our seasons, but I was surprised to learn that it DOES affect the length of our seasons!  Since Earth gradually speeds up in its orbit, reaching its highest speed at perihelion, the northern hemisphere winter (and southern hemisphere summer) is almost 5 days longer than the opposite seasons.

(Poor southern hemispherians!)

Solar Calendar, again

I will call it a shadow calendar from now on, since solar calendar really means something like this. 

My first attempt at a shadow clock ended in disappointment, when it became clear that the corner of a roof gutter (my gnomon) was too far from the wall the shadow fell on.  Because the noonday sun moves (seems to, from our point of view) a full 47 degrees from solstice to solstice, my wall was not nearly tall enough to contain the whole range of the shadow.  (A little simple math would have saved me wasted time, had I not been overconfident.)  In addition, the fact that the wall was not really east-west meant that the shadow did not reach its highest point at solar noon, as would be expected.*  This is more an aesthetic than a practical point, since it’s hard to stand there long enough to confirm anyway.

My next, more modest attempt used the shadow cast by the house eave on the wall of the house, only a few feet away.  One fortuitous advantage of this “calendar” was that the shadow of the downspout falls “just so,” telling you when it is really solar noon without need for a clock.  But in this case, the distance was too short to show much shadow movement from season to season.  (It amounted to less than a full clapboard in height over several months.)  I did not even get the satisfaction of seeing the shadow on Dec 21, since it was cloudy.

The math involves trigonometry of a right triangle, and relies on having the shadow falling on a vertical wall (opposite side) a known horizontal distance (adjacent side) from the gnomon.  Then you need the angle above the horizon of the noonday sun at each solstice (angle theta).  These angles can be found from your latitude: summer solstice theta = 90 degrees – latitude + 23.4 degrees, while winter solstice theta is the same, but minus those 23.4 degrees.  (These angles, by the way, are the height of the noonday sun above the horizon on those days.)  Opposite side = Tan(theta) X distance.

Then minimum height of wall needed can be found by working out the opposite sides, and subtracting them.  IF the height of your wall is no smaller than this difference, AND IF the noonday shadow falls at the bottom of this wall on the summer solstice, THEN the winter solstice shadow will fall at (or short of) the top of the wall.  (Phew!)

To save you the trouble, here at 42 degrees north latitude, the noonday sun is at 24.6 degrees at the winter solstice, 48 degrees at the equinoxes, and 71.4 degrees at the summer solstice.  My gnomon was about 20 feet from the wall, so the opposite side would be about 59 feet and 9 feet: a difference of 50 feet!  That result surprises me even now.  Needless to say, my house isn’t that tall.  (So great a height is needed partly because of the downward slant of the rays: near the north or south pole, with the rays shining nearly horizontally, the height needed would only be about 20 feet.  –while at the equator the sun couldn’t shine on the same wall at both solstices at all—it would hop to the other side, shining in the southern sky at the end of December, but the northern sky at the end of June.)

In the meantime, I realized that even a flat, vertical, and perfectly east-west wall would distort: the sun’s path from the gnomon would be changing continuously through the day, and also be different lengths at different times of year—that means the position of the shadow would not change in even increments week by week.  In fact, the ONLY way to give the shadow a steady march would be to project it on a semicircle whose radius was the length of the shadow.  (Got anything like that outside of YOUR house?)  So much for my plan of having a “found” shadow calendar!

On the other hand, a length of heavy aluminum bar would be pretty easy to bend into the necessary quarter-circle.  The main problem would be adjusting it and holding it solidly in place.  And I wonder how expansion and contraction with temperature would affect it?  Hmmm…

Here, since I missed posting for so long due to Life, computer death, etc, is Everything You Need to Know About the Winter Solstice.

*An email reply from the folks at NOAA made it clear to me: the shadow moves left to right on the wall, but since the sun’s rays slant downward, and the wall is tilted so the path of those rays gets longer, the shadow continues to move downward a long while even after the sun has passed its high point for the day.

Misconceptions of the Seasons

 

From Wikimedia, via EarthSky.org

I was collapsing a cereal box last Wednesday for recycling when I noticed an explanation of the seasons on the back.  It had been quite a while since I'd noticed "educational packaging" of this sort, and here it was on a box of Shaws store brand (Essential Everyday) cold cereal. 

Does anyone pay attention to these things?  In this case I hope not.  Alas, Shaws disappointed: they got the importance of the axial tilt, but explained that the northern hemisphere warms in summer because it is "closer to the sun" than the southern hemisphere.  Out of a total earth-sun distance of 93,000,000 miles, that's only a difference of 0.0086% from one side of the earth to the other.  And that's the maximum possible distance; the actual difference between distances to the poles would be less than a tenth of that.

Here's a better explanation. 

The rest of the article--which isn't really about the seasons--interests me more.  It first explains the slow wobble in earth's axis, and its consequences.  Then it mentions seasons elsewhere in the solar system.  Finally it discusses the common misconception that our changing distance from the sun causes seasons. 

Three comments on these.

First, Uranus has extreme seasons because its axis is tipped nearly 90 degrees, so that its poles alternately face the sun directly--well and good; but what does it mean that Venus (with no seasons) has a tilt of 177 degrees?  Although Venus' axis is nearly upright, we know it was tipped completely over early in its history because it rotates in a direction opposite that of Earth and other planets.  That is, Earth, Mars, etc., both rotate and revolve in the same direction (counterclockwise viewed from above the north pole), while Venus rotates clockwise! 

Second, although Mars has a similar tilt to earth, unlike Earth its seasons really are caused by changing distance from the sun!  The reason is Mars' orbit is less circular than ours, a longer ellipse, so that the greater difference in distance overwhelms any seasonality related to axial tilt. 

Finally, two facts help refute the idea that earth's seasons are related to its distance from the sun: (1) the seasons are opposite in the northern and southern hemispheres, and (2) the earth is actually closest to the sun in early January.  The difference in energy received is simply overwhelmed by the difference in angle and daylength. 

It does make me wonder, though, whether southern hemisphere seasons are more extreme--with the effects added together--than those in similar latitudes and situations in the north.

Happy Equinox! (12:57 EDT)

In what direction does the sun rise?  In what direction does it set?  I often run into those who are not sure whether the sun rises in the east or west!  But for the rest of us, this can be a bit of a trick question, since it depends on the season.  It is tied up also with the path the sun takes across the sky, and helps to explain seasons from an earthly point of view.

Let's unpack it by considering different situations.  In the simplest case, we are standing on the equator, and it is the beginning of spring.  The sun will rise directly east, pass directly over our heads at noon, and set directly west.  As the year lengthens towards summer, the sun rises and sets more and more northerly.  As summer turns toward fall, the sun returns to rising straight east and west, then after the beginning of fall the sun begins rising and setting south of east and west. 

We here in the temperate zone (outside of the tropics, which are bounded by lines running 23½ degrees* north and south of the equator) see a similar pattern even though we never see the sun directly overhead: it is low in the south in winter, of course, but even in summer never comes near to being overhead.  But the direction it rises and sets still depends on the season.  At the beginning of winter (the winter solstice, on or about December 21st) the sun rises south of east, climbs to low in the southern sky at noon, and sets south of west.  This is true everywhere.  At the beginning of summer (summer solstice, June 21st or so), the sun rises NORTH of east, crosses all the way to a point high in the southern sky at noon, and then sets NORTH of west.  In other words, the sun makes a much longer path across the sky on June 21 than on December 21, so that it is in the sky much longer.  Of course, we know that early summer days are the longest and early winter days the shortest, giving us more or less heat in that time period, but now you can see the reason in terms of the sun's path across the sky.  (In addition, that path looks down much more directly in summer, and those more direct rays heat the ground more effectively.) 

 

From our point of view, the sky is a dome!  Explaining solstices, equinoxes (and seasons) in one graphic.

Now to the original question: in what direction does the sun rise?  Since today is the vernal equinox--exactly between the solstices, at the point in earth's orbit when the equator is aimed directly at the sun, giving us equal night ("equi-nox") of twelve hours all over the globe--today all over the earth the sun rose east and will set west.**

A final note: with the equinox occurring  almost exactly at local noon means that we actually "saw" the sun cross the "celestial equator" and move into the north "celestial hemisphere."  And today is divided almost exactly into a winter morning and a spring afternoon.  

A pity its overcast!

 

Here is the situation in space.  Imagine the universe projected on a sphere that surrounds the earth (the celestial sphere).  The ecliptic is the plane of our orbit around the sun.  The celestial equator is a projection of the equator on the celestial sphere.   Remember that the earth's axis always points in the same direction (coincidentally, at the north star).  In the model above, the earth spins, and the sun sloowwwly makes its way along the yellow line.

 

"Star trails" made on a long-exposure photograph.  The star near the center is the north star,

aka Polaris; that it does not appear to move shows that Polaris is in line with the earth's axis of rotation.

(No, you're right: I don't suppose the photo was taken here in Massachusetts.)

*which is the earth's tilt on its axis relative to the plane of its orbit around the sun.

**Okay, really the equinox is a point in time, not a whole day, but this is east and west near enough not to matter!