The Winter Solstice Eclipse and Statistics

A lot is being made of tonight’s “rare” lunar eclipse that occurs on the Winter Solstice.  Most news reports state there hasn’t been a lunar eclipse on the solstice since 1638 to make the point that this is a rare event.

Which got me to thinking, how often would a lunar eclipse occur on any given date.  Heck, I know one has not occurred on my birthday during my lifetime.

I admittedly don’t know all the intricacies of calculating eclipse frequencies, so this is an exercise in estimation.  Eclipses can occur any month of the year as seen from this list.  Not every eclipse is total however, so let’s restrict ourselves to total lunar eclipses since that is what everyone is making such a big deal about.  They occur about every 18 months on average.  Every 100 years there are about 67 total lunar eclipses.  Let’s assume for a moment (I know this is a bad assumption) that they all occur on  different calendar date.  It would take about 550 years (well 544, but this is an estimate so close enough) for every day of the year to have on eclipse occur on that date.  In reality it will take a lot longer since many dates will have two eclipses occur on them long before the 550 years is up.  So to me, the fact that this hasn’t happened in 372 years just doesn’t seem like that big of a deal to me (although the solstice can vary couple of days adding another layer to the mix, it still seems well within statistical norms).

Now I have made some assumptions here…the most questionable of which is that eclipses are equally likely on any given day.  If this assumption is bad, it could make the winter solstice eclipse a more or less rare event…would take some number crunching to find out.  I found a cool catalog of 5000 years of eclipses from NASA but it’s not in nice form to easily analyze the dates.  However, during 5000 years, there were an average of 69.5 total solar eclipses per century, so my guess was pretty close!

So before you think this is a big deal, take ANY random date of the year and try and find when the last time an eclipse occurred on that date.  I bet it won’t be hard to find a date that has an interval greater than 372 years!  So enjoy it, but remember, statistically speaking, it sure looks like nothing out of the ordinary.

A Couple of Good Reads

It’s well known that I am a big fan of Science Friday.  Last week’s show had interviews with two authors that intrigued me.

The first interview was with Jeff Potter, author of the soon to be released Cooking for Geeks: Real Science, Great Hacks and Good Food. There is a Cooking for Geeks website where you can learn things like how to make ice cream in 30 seconds.

Be sure to check out the blog for all the latest geek cooking tips.

The second interview is with Danica McKellar and covers the next book in her series of math books for girls, Hot X: Algebra Exposed.  McKellar first gained fame as Winnie on the Wonder Years before earning a degree in math and appearing on the West Wing.  She keeps chugging along and I am already trying to avoid coming up with semi-crude names for the inevitable Calculus Books (can anyone say Calculus Coed?  That’s one of the milder things that crossed my mind!)

They are both entertaining interviews.  If you’ll excuse me, I am hungry!

Mathemagic!

I just saw that mathmatician Arthur Benjamin is the guest on the Colbert Report tonight.  Bejamin is one of those masters of doing mental math in his head faster than you can even punch it into a calculator.  I have heard interviews with him before and he is very entertaining.  Here is a good one from the O’Reilly Media Emerging Technology Conference in 2007.  Give it a listen (for free!)

I’ll post a link to the video here tomorrow when it becomes available.

And here is the video…

How Many Licks and Other Fermi Problmes

You might remember the classic old commercial that poses the question “How many licks does it take to get to the center of a Tootsie Pop?”  The wise old owl licks twice, bites one and proclaims the answer is three.

Now author Aaron Santos has taken a stab at the question in his new book How Many Licks? Or How to Estimate Damn Near Anything. He takes on what are commonly called Fermi problems.  Fermi problems are basically exercises in estimation.  You take a problem, make a few quick reasonable assumptions, and try to figure out a reasonable (although not exact) answer.  The legendary Fermi problem is Fermi estimating the strength of the atomic blast at Trinity by observing how far the blast wave blew some paper.  Knowing how far away he was, he got a pretty good estimate of the strength of the blast.

I heard about this book on the podcast the Naked Scientists.  Naturally, they had to estimate how many people are having sex in the world at any given moment (a question sure to get the attention of high school students!)

I used these when teaching.  They are great exercises in creative thinking.  You can probably do some if you give it a shot. How many gallons of gasoline are used each year in the U.S. by automobiles?  Well, you have an idea of how many cars there are, how far the average U.S. driver drives per year (my car insurance statement tells me this!) about what the average fuel economy is and you can get a reasonable number with a basic four function calculator (and if you are only interested in the nearest power of 10, you might be able to do it in your head).

One of my favorites was when my students asked me if there was a google (a 1 followed by 100 zeroes) of atoms in the universe.  I didn’t know the answer but quickly started jotting down some figures on the board (average density of the universe, size of the universe), got out my four function calculator, and quickly pronounced no.  Off by many powers of 10.  I got about 10^79 atoms which puts me nicely in the range of accepted estimates.

Fermi problems also serve as a good B.S. detector.  Earlier this year with Obama’s stimulus plan, a bunch of chain emails proposing other solutions that were allegedly cheaper.  One such email proposed giving everyone over 50 $1 million to retire to solve unemployment.  Well, a quick check of the math (that I could do without a calculator) put the cost of this proposal at about $40 trillion (which is a lot more than $787 billion).  For reference, the entire U.S. economy is about $14 trillion per year.  Obviously people need to be able to estimate things a little bit better before passing on chain emails!  Another one claimed that the stimulus package amounted to more than $1 million per day every day since Jesus was born.  I heard this, estimated the number of days in 2,000 years, and that one turned out to be true.

Fermi problems have a wide variety of applications.  You can use them to figure out if you are being charged too much at the store or to see if the chain email your crazy uncle sent you is total nonsense. They are easy and one of the more powerful and useful tools that everyone should have in their mathematical toolbox.