Tag Archives: sun

Scaling the Solar System

Too often, people throw around numbers without even thinking about the tremendous differences in the scales of those numbers.  Here’s the result of an exploration I did this week with my  afternoon math club of 4th and 5th graders to get a perspective on just how big our solar system really is.  They call themselves $mc^2$.

My school has an three foot diameter iGlobe in the middle of its lower school entry hall.  The group decided to see if they could use the school’s iGlobe to create a scaled model of the solar system.  After some quick research, they learned that the earth was about 8000 miles in diameter and the sun was about 875,000 miles wide.  I challenged them to put that in perspective.

They first tried to think in terms of the iGlobe model they already had.  Some quick division convinced them that a relatively sized sun would be about 109 iGlobes wide.  When I asked them how they could get a handle on just how big that was, one suggested measuring off 109 iGlobes across the field outside.  That’s when another realized that 109 iGlobes = 109 yards = about one football field plus an end zone.  Eyes widened.  After all, the sun doesn’t look nearly that big in the sky!

Since we couldn’t move the iGlobe from its perch, I asked them to come up with a different way to relate these sizes.  That’s when one student realized that if we could call the iGlobe our unit, why not any other object?  He grabbed some Cuisenaire rods, called the Earth one of the square centimeter singles, and started explaining.

OK, that was a nice connection and it gave the students a chance to get some perspective on the size of the Earth.  But I still wasn’t convinced they really got just how enormous the scales were here.  That’s when I asked if they could use the same Sun-Earth model to place the Earth cube in its proper relative position to their Sun diameter.  They found the average distance from the Earth to the Sun (93 million miles), converted that to their new scale, and paced off the distance. NOTE:  It gets pretty small at the end, so I recommend a full-screen view, but it will still be pretty small.

At the end of the video, it’s very difficult to see the running student throw his hands in the air, much less the little tiny Earth cube he’s still holding.  But that’s the point.  Distances across our solar system are mind-bogglingly large.  Sending probes from our little Earth cube to any other object requires a tremendous feat of engineering.  Maybe this is a good first step in helping these students come to grips with the enormity of the scaling of the universe.

This twister was reported at nearly 186,000 mph with temperatures “between 90,000 and 3.6 million degrees Fahrenheit.”  Pretty stunning.

I thought the twisting winds might make a really interesting multivariable calculus problem, but a physics colleague, John Burk (@occam98) asked, “I wonder what the F-number is on a tornado with 186,000 mi/hr winds?”

OK, a new direction.  I started by finding the Fujita Scale for tornado categories:

 F0 F1 F2 F3 F4 F5 F6 40-72 mph 73-112 mph 113-157 mph 158-206 mph 207-260 mph 261-318 mph 319-379 mph

Reminding me that the damage caused by a tornado is connected to the square of the velocity of the winds.  John found a “phenomenal quadratic fit” to the Fujita Scale for tornadoes by using the midpoint of each range.  I repeated John’s analysis and found basically the same results.  I also added a residual analysis.

There are very few data points here, so I shouldn’t have been all that surprised by a residual pattern.  As John’s analysis suggested, the quadratic is very close to the data points–$residuals\in{[-0.607,0.476]}$–a very small interval relative to the dependent range.  But as a math teacher, I began to wonder.

1. How you would set the range of each category if you use midpoints?
2. Can the fit be any better?
3. Once you get an equation, do the coefficients tell you anything?

I’m not sure how to answer the first question in any sort of non-arbitrary way, so I turned my focus to two other approaches:  using the minimum and maximum wind speeds for each F-category.  Those graphs and their residuals follow.

So all three appear to fit the F-scale data very well, with maybe a slightly less obvious residual pattern in the min and max curves (although I wouldn’t stake any deep claims on that assertion based on so few data points). The residuals range for the minimum speeds is $residuals_{min}\in{[-0.929,0.667]}$, and is $residuals_{max}\in{[-0.429,0.571]}$ for the maximum speeds.  If forced to make a call, perhaps the maximum speeds are better for their smaller overall residual range and possibly less defined pattern.  Also, using the equation for the maximum (or minimum) wind speeds avoids the vague endpoints issue for the Fujita scalings the mean wind speed approach encountered.

I don’t recognize anything about the coefficient values, so I tried converting the three equations into factored and vertex forms with nothing really enlightening there either.  I don’t know what I was hoping to find, but none of the coefficients in any of the forms seem to be sharing any secrets.

For now, I think I’ve run to the end of my reverse-engineering of the Fujita Scale.

Returning to the solar tornado that inspired all of this, I used my CAS to solve for the minimum and maximum Fujita scales based on these data, getting that the solar tornado would be rated somewhere between an F-270 and an F-285 tornado if it happened on Earth.  Wow.

In the end, perhaps it’s time for me to study the development of the Fujita Scale, but I’m pretty convinced from these tight fits that it was not a purely random equation.

In the meantime, I hope you find here a (not so surprising) connection to quadratic functions and possibly something to provide a deeper connection between mathematics and science–something woefully underrepresented for too many students.