The orbit of the moon around the sun doesn't look like what you'd expect.
(Although now that I've told you this, you might think a little harder about what the orbit looks like.)
Showing posts with label astronomy. Show all posts
Showing posts with label astronomy. Show all posts
15 November 2008
21 March 2008
Easter's early this year. Deal with it.
Family holidays ruined by earliest Easter in 90 years (from the Daily Mail).
About halfway down, a formula is given:
Um, do you understand that formula? I think I know why some of the numbers are there -- the 31 at the end probably has something to do with the length of months, the 7 with the length of weeks, and the 19 with the Metonic cycle. Also, any sane mathematician wouldn't write the formula like that. First, there are repeated subexpressions like that ((u + 8) \ 25 + 1); I'd just call that by some other name and be done with it. Second, the formula just sits there in the middle of the article; this gives people the idea that mathematicians are freaks of nature who think in formula. What do the variables mean?
If you're curious, there is an algorithm at the Calendar FAQ. Easter is the first Sunday after the first (computed) full moon on or after the vernal equinox (calculated, and assumed to be March 21). The algorithm reflects this. First, assume that the Metonic cycle, which says that lunar phases repeat every 19 solar years, is exactly correct in the Julian calendar. (The algorithm was invented back when the Julian calendar was used.) Then make two corrections, one for the fact that the Julian calendar includes leap years that the Gregorian doesn't (years divisible by 100 but not 400) and one for the fact that the Metonic cycle's a bit off. (The expression "(u+8)\25" in the formula above comes from the second correction.) This gives the date of the full moon. Presumably if you've gotten this far you already know what the days of the week are.
Anyway, the cycle of Easter dates repeat themselves every 5,700,000 years. The cycle of epacts (which encode the date of the full moon) in the Julian calendar repeat every nineteen years. There are two corrections made to the epact, each of which depend only on the century; one repeats (modulo 30, which is what matters) every 120 centuries, the other every 375 centuries, so the air of them repeat every 300,000 years. The days of the week are on a 400-year cycle, which doesn't matter because that's a factor of 300,000. So the Easter cycle has length the least common multiple of 19 and 300,000, which is 5,700,000.
This whole computation is known as the computus (Latin for "computation"; I guess it was just that important at the time). Not surprisingly, Gauss had an algorithm which is much easier. Let Y be the current year. Then take:
a = Y mod 19
b = Y mod 4
c = Y mod 7
d = (19a + M) mod 30
e = (2b + 4c + 6d + N) mod 7
where M and N are constants depending on the century that don't look that hard to calculate, and which I assume are the corrections I alluded to above; the Wikipedia article gives them in a table. Then Easter falls on the d+e+22 of March or the d+e-9 of April, with certain exceptions which move it up a week when this algorithm gives a very late date for Easter. Basically, d finds the date of the full moon (so M is something like the epact) and e find the day of the week. In the case of this year you get a = 13, b = 0, c = 6; a table gives M = 24, N = 5 for this century, so d = 1, e = 0, and Easter is on the 23rd of March.
As for when Easter usually falls, well, go back to the original description: Easter is the date of the first Sunday after the first full moon on or after March 21. To me this seems like adding two random variables -- the number of days between March 21 and the first full moon, which is roughly uniformly distributed over [0, 29], and the number of days between that moon and the next Sunday, which is uniformly distributed over [1, 7]. There are 210 ordered pairs in ([0, 29] × [1, 7]). One of them sums to 1, giving an Easter date of March 22 in about one year out of 210. Two sum to 2, giving an Easter date of March 23 in two years out of 210. Three sum to 3 (March 24), ..., six sum to 6 (March 27). Seven sum to each of 7 through 30, giving Easter dates of each of March 28 through April 20 in seven years out of 210. Six sum to 31, giving April 21 in six years out of 210, ..., one sums to 36, giving April 26 in one year out of 210.
Indeed, this is basically what computations show, except that for some reason, when the methods given above call for Easter to be on April 26 it gets moved up to April 19. But basically the distribution of Easter dates is just a convolution of two uniform distributions! The Wikipedia article on the computus has a nice graph.
And I have no sympathy for the people quoted in that article. They've known this was coming since 1752, when the UK changed over to the Gregorian calendar. (It perhaps says something about me that I have more sympathy for the bakeries with lots of Irish patrons that are unhappy because Easter was only six days after St. Patrick's day this year.)
About halfway down, a formula is given:
It may look daunting to non-mathematicians but the fiendishly complex formula used to work out when Easter actually falls is:
((19*t+u-w-(u-(u+8)\25)+1)\3)+15)mod30)+(32+2*x+2*y-(19*t+u-w- (u-(u+8)\25)+1)\3)+15)mod30)-z)mod7)-7*(t+11*(19*t+u-w(u- (u+8)\25)+1)\3)+15)mod30)+22*(32+2*x+2*y-(19*t+u-w-(u- (u+8)\25)+1)\3)+15)mod30)-g)mod7)+114)\31
Um, do you understand that formula? I think I know why some of the numbers are there -- the 31 at the end probably has something to do with the length of months, the 7 with the length of weeks, and the 19 with the Metonic cycle. Also, any sane mathematician wouldn't write the formula like that. First, there are repeated subexpressions like that ((u + 8) \ 25 + 1); I'd just call that by some other name and be done with it. Second, the formula just sits there in the middle of the article; this gives people the idea that mathematicians are freaks of nature who think in formula. What do the variables mean?
If you're curious, there is an algorithm at the Calendar FAQ. Easter is the first Sunday after the first (computed) full moon on or after the vernal equinox (calculated, and assumed to be March 21). The algorithm reflects this. First, assume that the Metonic cycle, which says that lunar phases repeat every 19 solar years, is exactly correct in the Julian calendar. (The algorithm was invented back when the Julian calendar was used.) Then make two corrections, one for the fact that the Julian calendar includes leap years that the Gregorian doesn't (years divisible by 100 but not 400) and one for the fact that the Metonic cycle's a bit off. (The expression "(u+8)\25" in the formula above comes from the second correction.) This gives the date of the full moon. Presumably if you've gotten this far you already know what the days of the week are.
Anyway, the cycle of Easter dates repeat themselves every 5,700,000 years. The cycle of epacts (which encode the date of the full moon) in the Julian calendar repeat every nineteen years. There are two corrections made to the epact, each of which depend only on the century; one repeats (modulo 30, which is what matters) every 120 centuries, the other every 375 centuries, so the air of them repeat every 300,000 years. The days of the week are on a 400-year cycle, which doesn't matter because that's a factor of 300,000. So the Easter cycle has length the least common multiple of 19 and 300,000, which is 5,700,000.
This whole computation is known as the computus (Latin for "computation"; I guess it was just that important at the time). Not surprisingly, Gauss had an algorithm which is much easier. Let Y be the current year. Then take:
a = Y mod 19
b = Y mod 4
c = Y mod 7
d = (19a + M) mod 30
e = (2b + 4c + 6d + N) mod 7
where M and N are constants depending on the century that don't look that hard to calculate, and which I assume are the corrections I alluded to above; the Wikipedia article gives them in a table. Then Easter falls on the d+e+22 of March or the d+e-9 of April, with certain exceptions which move it up a week when this algorithm gives a very late date for Easter. Basically, d finds the date of the full moon (so M is something like the epact) and e find the day of the week. In the case of this year you get a = 13, b = 0, c = 6; a table gives M = 24, N = 5 for this century, so d = 1, e = 0, and Easter is on the 23rd of March.
As for when Easter usually falls, well, go back to the original description: Easter is the date of the first Sunday after the first full moon on or after March 21. To me this seems like adding two random variables -- the number of days between March 21 and the first full moon, which is roughly uniformly distributed over [0, 29], and the number of days between that moon and the next Sunday, which is uniformly distributed over [1, 7]. There are 210 ordered pairs in ([0, 29] × [1, 7]). One of them sums to 1, giving an Easter date of March 22 in about one year out of 210. Two sum to 2, giving an Easter date of March 23 in two years out of 210. Three sum to 3 (March 24), ..., six sum to 6 (March 27). Seven sum to each of 7 through 30, giving Easter dates of each of March 28 through April 20 in seven years out of 210. Six sum to 31, giving April 21 in six years out of 210, ..., one sums to 36, giving April 26 in one year out of 210.
Indeed, this is basically what computations show, except that for some reason, when the methods given above call for Easter to be on April 26 it gets moved up to April 19. But basically the distribution of Easter dates is just a convolution of two uniform distributions! The Wikipedia article on the computus has a nice graph.
And I have no sympathy for the people quoted in that article. They've known this was coming since 1752, when the UK changed over to the Gregorian calendar. (It perhaps says something about me that I have more sympathy for the bakeries with lots of Irish patrons that are unhappy because Easter was only six days after St. Patrick's day this year.)
21 December 2007
When asteroids attack
Asteroid May Hit Mars in Next Month (AP).
If they expect the odds to diminish, why haven't they set them lower in the first place? I think this may be another issue of mean-median confusion -- say, there's a 1 in 3 chance that the odds will go up to 1 in 25 next month and a 2 in 3 chance it'll go away completely. But the statement seems kind of silly. Apparently efficient market theory doesn't apply to asteroids.
Scientists tracking the asteroid, currently halfway between Earth and Mars, initially put the odds of impact at 1 in 350 but increased the chances this week. Scientists expect the odds to diminish again early next month after getting new observations of the asteroid's orbit, Chesley said.
If they expect the odds to diminish, why haven't they set them lower in the first place? I think this may be another issue of mean-median confusion -- say, there's a 1 in 3 chance that the odds will go up to 1 in 25 next month and a 2 in 3 chance it'll go away completely. But the statement seems kind of silly. Apparently efficient market theory doesn't apply to asteroids.
04 December 2007
How much land is in the tropics?
In Tropics widen, fringe areas drier, experts say, from the Associated Press, I found the following quote:
In fact, the tropics make up about two-fifths of the globe. To be more precise, I mean that about two-fifths of the total area of the surface of the earth is between the two tropics. The tropics are at latitudes +23.5 and -23.5 degrees (I'll use + and - for north and south here.) So it's easy to see where the "one-quarter" figure might come from -- the tropics span a total of 47 degrees of latitude, out of the full range of 180, and 47/180 is essentially 1/4.
But there's an interesting fact about the surface area of a sphere. Take a sphere of radius r. Cut it with two parallel planes of distance h apart. Then the area of your slice will be h/(2r) times the surface area of the sphere, or 2πrh, regardless of the way the sphere is cut; notice that this is also the surface area of a cylinder of height h and radius r. I saw this demonstrated once, when I first saw this fact in a calculus class, by cutting a spherical loaf of bread into slices of equal thickness; the slices get varying amounts of the interior of the bread but all get the same amount of crust.
The thickness of the slice in question, for the tropics, is (2 sin(23.5o))r, where r is the radius of the earth. Thus this slice makes up sin(23.5o) = 0.398 of the earth, which is just under two-fifths. I knew from the moment I read this fact that one-quarter was an underestimate, but I suspected that perhaps it was more accurate than, say, "one-third". But it's not.
(The actual point of the article is not that the tropics make up one-fourth, or two-fifths, or whatever fraction of the earth, but that they are "widening"; this uses an atmospheric definition of the tropics which is different from the astronomical one implied by the quote above.)
Geographically, the tropical region is a wide swath around Earth's middle stretching from the Tropic of Cancer, just south of Miami, to the Tropic of Capricorn, which cuts Australia almost in half. It's about one-quarter of the globe and generally thought of as hot, steamy and damp, but it also has areas of brutal desert.
In fact, the tropics make up about two-fifths of the globe. To be more precise, I mean that about two-fifths of the total area of the surface of the earth is between the two tropics. The tropics are at latitudes +23.5 and -23.5 degrees (I'll use + and - for north and south here.) So it's easy to see where the "one-quarter" figure might come from -- the tropics span a total of 47 degrees of latitude, out of the full range of 180, and 47/180 is essentially 1/4.
But there's an interesting fact about the surface area of a sphere. Take a sphere of radius r. Cut it with two parallel planes of distance h apart. Then the area of your slice will be h/(2r) times the surface area of the sphere, or 2πrh, regardless of the way the sphere is cut; notice that this is also the surface area of a cylinder of height h and radius r. I saw this demonstrated once, when I first saw this fact in a calculus class, by cutting a spherical loaf of bread into slices of equal thickness; the slices get varying amounts of the interior of the bread but all get the same amount of crust.
The thickness of the slice in question, for the tropics, is (2 sin(23.5o))r, where r is the radius of the earth. Thus this slice makes up sin(23.5o) = 0.398 of the earth, which is just under two-fifths. I knew from the moment I read this fact that one-quarter was an underestimate, but I suspected that perhaps it was more accurate than, say, "one-third". But it's not.
(The actual point of the article is not that the tropics make up one-fourth, or two-fifths, or whatever fraction of the earth, but that they are "widening"; this uses an atmospheric definition of the tropics which is different from the astronomical one implied by the quote above.)
05 September 2007
Yet more fun with phase lag
It's still warm here in Philadelphia. Highs are in the eighties this week, which is slightly above average for early September.
Walking home from campus yesterday, a little before 7 PM, the sun was low in the sky. I found myself digging for my sunglasses. And I remembered the circumstances under which I bought the sunglasses -- in late March and early April I was having the same problem. (I live west of campus, so I could actually have this problem twice a day, as I walked east in the mornings and west in the evenings.)
But wasn't it cold then, I thought?
But the position of the sun doesn't depend on the cold. The position on the horizon at which the sun rises and sets depends only on the distance of the current day from the summer solstice. Yesterday was the 75th day after the summer solstice (I'm using June 21 here); the 75th day before the solstice was April 7. The high temperature in Philadelphia that day was 41 degrees. (The average high then is 59.) (The declination, which is the celestial analogue of latitude, varies essentially sinuisoidally, but that doesn't mean the position of sunrise varies sinusoidally because there's another circle to deal with.)
In short: in terms of insolation right now is like early April, but in terms of temperature it's like early June. Maybe this sort of asymmetry explains why spring and fall feel different, since we're sensitive to both temperature and amount of light.
Of course, it's hard to know for sure, because you also have to take into account the derivatives of insolation and temperature; in the spring they're both increasing, and in the fall they're both decreasing. (A Jewish friend of mine once said that he doesn't think it's coincidence that Yom Kippur, when Jews are obliged to fast from sundown to sundown, falls near the autumn equinox; that makes the fast a few minutes less than it would be if it fell near the spring equinox.) And there's a big psychological factor, as well. Right now, an hour and ten minutes before my first class of the year, I feel exhilirated intellectually; in, say, May I'll feel exhausted. There's not much of a way to control for the cycles we humans have imposed on the year.
But what would it be like to live in a world where the season with the most sunlight was the cold season? I suspect we'll never know, because the only way to create that sort of world is to isolate people from the natural world for a very long time, and if you're willing to pay people to live in your cave for a while you can probably think of more worthwhile things to find out by using them as test subjects.
Walking home from campus yesterday, a little before 7 PM, the sun was low in the sky. I found myself digging for my sunglasses. And I remembered the circumstances under which I bought the sunglasses -- in late March and early April I was having the same problem. (I live west of campus, so I could actually have this problem twice a day, as I walked east in the mornings and west in the evenings.)
But wasn't it cold then, I thought?
But the position of the sun doesn't depend on the cold. The position on the horizon at which the sun rises and sets depends only on the distance of the current day from the summer solstice. Yesterday was the 75th day after the summer solstice (I'm using June 21 here); the 75th day before the solstice was April 7. The high temperature in Philadelphia that day was 41 degrees. (The average high then is 59.) (The declination, which is the celestial analogue of latitude, varies essentially sinuisoidally, but that doesn't mean the position of sunrise varies sinusoidally because there's another circle to deal with.)
In short: in terms of insolation right now is like early April, but in terms of temperature it's like early June. Maybe this sort of asymmetry explains why spring and fall feel different, since we're sensitive to both temperature and amount of light.
Of course, it's hard to know for sure, because you also have to take into account the derivatives of insolation and temperature; in the spring they're both increasing, and in the fall they're both decreasing. (A Jewish friend of mine once said that he doesn't think it's coincidence that Yom Kippur, when Jews are obliged to fast from sundown to sundown, falls near the autumn equinox; that makes the fast a few minutes less than it would be if it fell near the spring equinox.) And there's a big psychological factor, as well. Right now, an hour and ten minutes before my first class of the year, I feel exhilirated intellectually; in, say, May I'll feel exhausted. There's not much of a way to control for the cycles we humans have imposed on the year.
But what would it be like to live in a world where the season with the most sunlight was the cold season? I suspect we'll never know, because the only way to create that sort of world is to isolate people from the natural world for a very long time, and if you're willing to pay people to live in your cave for a while you can probably think of more worthwhile things to find out by using them as test subjects.
21 June 2007
The summer solstice and the longest days
Quick, what's the longest day of the year? (In the northern hemisphere.)
If you answered June 21 (today!), you're probably right. That's the date of the summer solstice, at least in most years and in most time zones. (You may have thought that the solstice was an entire day, but in fact it's just a moment in time, the moment when the sun is furthest north. The sun doesn't move, of course, at least not in the usual treatment of astronomy -- but the Earth moves around the sun, so sometimes its northern part is pointed more towards the sun and sometimes its southern part is.
But on what day does the sun rise the earliest? Or set the latest?
This is a trickier question. "Trickier", here, means "I don't remember the answer". But the U. S. Naval Observatory makes available a sun or moon rise/set table for one year. You can enter your location and it'll tell you when the sun rises and sets on each day in, say, 2007. The patterns don't change from year to year, because the Gregorian calendar is what we call a "solar calendar" and is pretty well correlated with the seasons. Its predecessor, the Julian calendar, didn't have this property -- it slipped relative to the seasons by a bit under a day per century. For more than you ever wanted to know about calendars, see Claus Tondering's calendar FAQ.
If I enter my location -- Philadelphia -- into the table, it tells me that the day the sun rises the earliest is any day between June 10 and June 18, when it rises at 5:31 am. Let's say that the actual earliest sunrise is in the middle of this period, June 14. Similarly, the latest sunset is on any day between June 26 and June 29; let's call it June 27. These are a week earlier and later than the solstice. On the winter side of things, the shortest day is December 20 (only nine hours and nineteen minutes - sunrise is at 7:19, sunset at 4:38), but the earliest sunset is around December 7 (4:35) and the latest sunrise is around January 5, 2008 (7:23).
What's the cause of this? It's a little something known as the equation of time, which basically says that the earth runs "fast" in some seasons and "slow" in others. In December it's running faster than in January, and in early June it's running faster than in late June.
I first really became aware of this phenomenon when I lived in Boston. In Boston winters, night comes very early -- it's not uncommon to see the pink and purple shades of sunset at around 3:30 on a December afternoon. The actual earliest sunset comes at 4:12 on the 8th of December; as you head further north the earliest sunset and latest sunrise both move towards December 21, because the "equation of time" becomes less significant with respect to the variation in day length. (In Miami, for example, they're November 29 and January 14; in Anchorage, they're December 15 and December 26.) You end up getting some strange asymmetries. You'd think that on dates equidistant from the winter solstice, you'd have the same time of sunset.
But you don't. In mid-November and late January, the sun sets in alignment with MIT's Infinite Corridor, which is a very long hallway running through the center of campus. In November it happens around November 12 (thirty-nine days before the winter solstice), at 4:20 pm; in January, it happens around January 29 (thirty-nine days after), at 4:50 pm. That actually helps in the Boston winters, believe it or not -- by the time it's getting really cold at least it feels like the sunlight is starting to come back. At least if you were someone like me who was never awake for sunrise.
Finally, in the summer the sun lines up with Manhattan streets at sunset, on May 28 and July 12. This is called "Manhattanhenge", and some people claim that the alignments are cosmic signs of Memorial Day and baseball's All-Star break. Of course, they're not; Manhattan just isn't aligned with the "north" that we usually call by that name. Most places with a regular grid of streets will have a day like this, although I haven't seen references to it happening in places other than Manhattan.
If you answered June 21 (today!), you're probably right. That's the date of the summer solstice, at least in most years and in most time zones. (You may have thought that the solstice was an entire day, but in fact it's just a moment in time, the moment when the sun is furthest north. The sun doesn't move, of course, at least not in the usual treatment of astronomy -- but the Earth moves around the sun, so sometimes its northern part is pointed more towards the sun and sometimes its southern part is.
But on what day does the sun rise the earliest? Or set the latest?
This is a trickier question. "Trickier", here, means "I don't remember the answer". But the U. S. Naval Observatory makes available a sun or moon rise/set table for one year. You can enter your location and it'll tell you when the sun rises and sets on each day in, say, 2007. The patterns don't change from year to year, because the Gregorian calendar is what we call a "solar calendar" and is pretty well correlated with the seasons. Its predecessor, the Julian calendar, didn't have this property -- it slipped relative to the seasons by a bit under a day per century. For more than you ever wanted to know about calendars, see Claus Tondering's calendar FAQ.
If I enter my location -- Philadelphia -- into the table, it tells me that the day the sun rises the earliest is any day between June 10 and June 18, when it rises at 5:31 am. Let's say that the actual earliest sunrise is in the middle of this period, June 14. Similarly, the latest sunset is on any day between June 26 and June 29; let's call it June 27. These are a week earlier and later than the solstice. On the winter side of things, the shortest day is December 20 (only nine hours and nineteen minutes - sunrise is at 7:19, sunset at 4:38), but the earliest sunset is around December 7 (4:35) and the latest sunrise is around January 5, 2008 (7:23).
What's the cause of this? It's a little something known as the equation of time, which basically says that the earth runs "fast" in some seasons and "slow" in others. In December it's running faster than in January, and in early June it's running faster than in late June.
I first really became aware of this phenomenon when I lived in Boston. In Boston winters, night comes very early -- it's not uncommon to see the pink and purple shades of sunset at around 3:30 on a December afternoon. The actual earliest sunset comes at 4:12 on the 8th of December; as you head further north the earliest sunset and latest sunrise both move towards December 21, because the "equation of time" becomes less significant with respect to the variation in day length. (In Miami, for example, they're November 29 and January 14; in Anchorage, they're December 15 and December 26.) You end up getting some strange asymmetries. You'd think that on dates equidistant from the winter solstice, you'd have the same time of sunset.
But you don't. In mid-November and late January, the sun sets in alignment with MIT's Infinite Corridor, which is a very long hallway running through the center of campus. In November it happens around November 12 (thirty-nine days before the winter solstice), at 4:20 pm; in January, it happens around January 29 (thirty-nine days after), at 4:50 pm. That actually helps in the Boston winters, believe it or not -- by the time it's getting really cold at least it feels like the sunlight is starting to come back. At least if you were someone like me who was never awake for sunrise.
Finally, in the summer the sun lines up with Manhattan streets at sunset, on May 28 and July 12. This is called "Manhattanhenge", and some people claim that the alignments are cosmic signs of Memorial Day and baseball's All-Star break. Of course, they're not; Manhattan just isn't aligned with the "north" that we usually call by that name. Most places with a regular grid of streets will have a day like this, although I haven't seen references to it happening in places other than Manhattan.
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