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How are the planets arranged around the sun?
Kepler's third law implies that a planet closer to the sun will
have a shorter "year". That is, the closer the planet is to the sun
the sorter the time needed to go around the sun. The periods of the
planets are given below (given in years — so the period of the earth is
1 year.)
- Mercury = 0.2408
- Venus = 0.61521
- Earth = 1.0000
- Mars = 1.8809
- Jupiter = 11.8622
- Saturn = 29.4577
- Uranus = 84.0133
- Neptune = 164.7934
- Pluto = 247.686
Odd coincidences have been identified in this list of numbers. For
example, the period of Jupiter times 5/2 is very close to the period of
Saturn. (Or 11.8622/29.4577 is very close to 2/5.)
This coincidence has implications for the precise prediction
of the positions of these planets and is called the "Great Inequality".
Another way to say this is that 5 times 11.8622 is approximately
2 times 29.4577, or 5*11.8622-2*29.4577 is approximately zero.
Getting Started
- Find other pairs of numbers where the ratio of the lengths of
the years is close to a fraction with a small denominator. (Note that
"close" and "small" are relative terms.)
- Can you find a similar relationship between 3 of the numbers
above? That is, three entries in the table where the multiplying by
small integers then adding and subtracting gives approximately zero.
Going Deeper
If you picked two numbers at random between 0 and 30, what would
be the chance that the ratio of the two numbers is close to a fraction
with small denominator? (How close would it have to be before it would
be a surprise?) What if you picked three numbers--what's the chance that
two of them would have ratio close to a small fraction.
Communicated by the Chelsea High School Mathematics Department.
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