Calculations
Here's a picture I took on a foggy morning just after sunrise a few months back in the Village of Villa Park. I posted a different picture of that morning on the day I took it, and I meant to come back and post more but never did. So here's this one.
I post this one tonight because those calculations I did for the last space station post got me to thinking about math and the passage of time. I mentioned when we moved to Villa Park a month before I took this photo of a sunrise that we'd moved west, so sunrise would find us just a bit later. The question I always meant to answer but never did, though, was, how much later?
That's a relatively easy question to answer if you answer two other questions first: 1.) How much farther west are we? 2.) How fast is my particular patch of planet rotating? Or, what's my speed as the planet spins me toward sunrise?
That second question is the harder one to answer, because I have to track down my latitude and then do a bunch of trigonometry to figure out how fast the planet at this latitude is spinning. But I did that math for the last space station post and therefore know the answer. I live at 41.9°N latitude, and am therefore being spun east at a speed of 783 miles per hour.
That leaves the first question, which is easy. I just have to go measure the distance from the old place to the new place, and then do a little dance because of an oddly coincidental little quirk of geography and random real estate transactions that simplifies this entire thought experiment tremendously. When we moved from Oak Park to Villa Park, we moved a distance of 10.52 miles almost exactly due west. And when I say almost exactly due west, I mean that a line running exactly west from our old place misses our new place by about 600 feet. Draw a line due east from our new place, and you hit the house on the corner of our Oak Park street 600 feet from our apartment that I always called the "scary house."
This is the kind of thing that gets me excited.
The reason this is significant for this thought exercise, though, is that our position almost exactly due west of the old place means the seasonal changes that move the Sun's position in the sky don't affect how long it takes sunrise to get from there to here. Our house in Villa Park is always going to be 10.52 miles behind our apartment in Oak Park at 783 miles an hour. That works out to 48.4 seconds. Sunrise hits our big back window 48.4 seconds later than it hits the big front windows on the old apartment.
Want to complicate this up, though? We used to live near the corner of Fullerton and Maplewood in Chicago's Logan Square neighborhood. That's farther east than the Oak Park place. How much earlier does Maplewood see sunrise?
Unfortunately, our departure from Maplewood didn't take us due west. There was a southward component to that vector. The straight line distance from Maplewood to Oak Park was 5.52 miles, but the heading was 32.78° south of due west. So we moved 4.66 miles west, but 2.98 miles south. So that means calculations for sunrise timing are only easy on the equinoxes. On the mornings of March 22 and September 22, Maplewood will see the sun 21.4 seconds before Oak Park, and 69.8 seconds before Villa Park. But only on those two days. Any other day, the tilting of the Earth on its axis throws Maplewood either a little closer to the sunrise than Oak and Villa Parks, or a little farther away, and calculating how much involves an enormous amount of math that changes for every day.
I'm not entirely sure how you'd do that math ... but for fun, I'll try it for the summer solstice. I think what I need to do is shift the sunrise line to match the maximum tilt of the Earth on its axis, which is 23.5°. Then I just plug in some trigonometry, calculate an adjustment, and if my thinking's right I find that on June 22, sunrise hits Maplewood 52.9 seconds before it hits Oak Park, and a minute and 41.3 seconds before it hits Villa Park. And what about the winter solstice? On December 22, sunrise hits Maplewood only 16.9 seconds before it hits Villa Park, and it actually beats its way into Oak Park 10.1 seconds ahead of Maplewood, even though Maplewood is further east.
So ... did you make it all the way through all that? I'm not sure I did. That's enough math for tonight, I think. I might have cured my insomnia.
Calculations
Here's a picture I took on a foggy morning just after sunrise a few months back in the Village of Villa Park. I posted a different picture of that morning on the day I took it, and I meant to come back and post more but never did. So here's this one.
I post this one tonight because those calculations I did for the last space station post got me to thinking about math and the passage of time. I mentioned when we moved to Villa Park a month before I took this photo of a sunrise that we'd moved west, so sunrise would find us just a bit later. The question I always meant to answer but never did, though, was, how much later?
That's a relatively easy question to answer if you answer two other questions first: 1.) How much farther west are we? 2.) How fast is my particular patch of planet rotating? Or, what's my speed as the planet spins me toward sunrise?
That second question is the harder one to answer, because I have to track down my latitude and then do a bunch of trigonometry to figure out how fast the planet at this latitude is spinning. But I did that math for the last space station post and therefore know the answer. I live at 41.9°N latitude, and am therefore being spun east at a speed of 783 miles per hour.
That leaves the first question, which is easy. I just have to go measure the distance from the old place to the new place, and then do a little dance because of an oddly coincidental little quirk of geography and random real estate transactions that simplifies this entire thought experiment tremendously. When we moved from Oak Park to Villa Park, we moved a distance of 10.52 miles almost exactly due west. And when I say almost exactly due west, I mean that a line running exactly west from our old place misses our new place by about 600 feet. Draw a line due east from our new place, and you hit the house on the corner of our Oak Park street 600 feet from our apartment that I always called the "scary house."
This is the kind of thing that gets me excited.
The reason this is significant for this thought exercise, though, is that our position almost exactly due west of the old place means the seasonal changes that move the Sun's position in the sky don't affect how long it takes sunrise to get from there to here. Our house in Villa Park is always going to be 10.52 miles behind our apartment in Oak Park at 783 miles an hour. That works out to 48.4 seconds. Sunrise hits our big back window 48.4 seconds later than it hits the big front windows on the old apartment.
Want to complicate this up, though? We used to live near the corner of Fullerton and Maplewood in Chicago's Logan Square neighborhood. That's farther east than the Oak Park place. How much earlier does Maplewood see sunrise?
Unfortunately, our departure from Maplewood didn't take us due west. There was a southward component to that vector. The straight line distance from Maplewood to Oak Park was 5.52 miles, but the heading was 32.78° south of due west. So we moved 4.66 miles west, but 2.98 miles south. So that means calculations for sunrise timing are only easy on the equinoxes. On the mornings of March 22 and September 22, Maplewood will see the sun 21.4 seconds before Oak Park, and 69.8 seconds before Villa Park. But only on those two days. Any other day, the tilting of the Earth on its axis throws Maplewood either a little closer to the sunrise than Oak and Villa Parks, or a little farther away, and calculating how much involves an enormous amount of math that changes for every day.
I'm not entirely sure how you'd do that math ... but for fun, I'll try it for the summer solstice. I think what I need to do is shift the sunrise line to match the maximum tilt of the Earth on its axis, which is 23.5°. Then I just plug in some trigonometry, calculate an adjustment, and if my thinking's right I find that on June 22, sunrise hits Maplewood 52.9 seconds before it hits Oak Park, and a minute and 41.3 seconds before it hits Villa Park. And what about the winter solstice? On December 22, sunrise hits Maplewood only 16.9 seconds before it hits Villa Park, and it actually beats its way into Oak Park 10.1 seconds ahead of Maplewood, even though Maplewood is further east.
So ... did you make it all the way through all that? I'm not sure I did. That's enough math for tonight, I think. I might have cured my insomnia.