The Bully Mnemonic

The Bully Mnemonic is a technique for remembering the exact number of seconds that occur in Earth's sidereal year, and the approximate relationships that exist between Earth's sidereal year, tropical Year, Great Year, and the Solar System's galactic year.

Bully Mnemonic Steps

Initial Definitions

Step 1

The first step is to write down the first five digits:

$$\begin{array}{ccccc}1& 2& 3& 4& 5\end{array}$$

Step 2

The second step is to select odd digits and intersperse them with zeros into integers a) and b) as shown below: (important to remember that the first integer ends with an extra 0, whereas the second integer ends with 5)

$$\begin{array}{ccccc}1& \text{2}& 3& \text{4}& 5\end{array}$$

$$a)10330$$ $$b)3055$$

Step 3

The third step is to select even digits and define numbers c) and d) as shown below:

$$\begin{array}{ccccc}\text{1}& 2& \text{3}& 4& \text{5}\end{array}$$

$$c)2$$ $$d)0.40$$

Sidereal & Tropical Years

Step 4

Multiply integers a) and b) from Step 2 to get the total number of seconds in a sidereal year.

$$10330\times 3055=31558150=\frac{1SiderealYear}{1Second}$$

Step 5

The tropical year has a slightly shorter duration than the sidereal year. Reduce integer a) by amount d), and multiply by b) from Step 2 to get the total number of seconds in a tropical year.

$$(10330-0.40)\times 3055=31556928\approx \frac{1TropicalYear}{1Second}$$

Great Years

Step 6

The Great Year is, by definition, a least common multiple of the sidereal year and the tropical year.

$$1GreatYear\approx \frac{10330}{0.40}TropicalYears\approx 25825TropicalYears$$

$$1GreatYear\approx \frac{10330}{0.40}SiderealYears\approx 25824SiderealYears$$

Proof:

$$25824SiderealYears\approx 25825TropicalYears$$

$$25824(10330\times 3055sec)=25825((10330-0.40)\times 3055sec)$$

$$25824(10330\times 3055sec)=\frac{10330}{0.40}((10330-0.40)\times 3055sec)$$

$$25824(10330\times 3055sec)=10330((\frac{10330-0.40}{0.40})\times 3055sec)$$

$$25824(10330\times 3055sec)=10330((25825-1)\times 3055sec)$$

Galactic Years

Step 6

Multiply integer c) by the square of integer a) to get the approximate number of tropical years required for the Solar System to orbit around the galactic center.

$$2\times {10330}^2=213417800\approx \frac{1GalacticYear}{1TropicalYear}$$

$$2\times 10330\times (10330-0.40)=213409536\approx \frac{1GalacticYear}{1SiderealYear}$$

Step 7

This step is an immediate consequence of steps 4, 5, and 6 above:

$$2\times {10330}^2\times (10330-0.40)\times 3055=6.7348101\times {10}^{15}\approx \frac{1GalacticYear}{1Second}$$

Step 8

This step is an immediate consequence of steps 4 and 7 above:

$$1GalacticYear\approx (2\times {10330}^2)SiderealYears$$

$$1GalacticYear\approx (2\times 10330\times 0.40)\times (\frac{10330}{0.40})SiderealYears$$

$$1GalacticYear\approx 8264\times 25825SiderealYears$$

$$1GalacticYear\approx 8264GreatYears$$

Summary

The Bully Mnemonic can be represented as follows:

$$\frac{1GalacticYear}{1Second}\approx 2\times {10330}^3\times 3055$$

$$=(2\times 0.40\times 10330)\cdot (\frac{10330}{0.40})\cdot ((10330-0.40)\times 3055)=(8264)\cdot (25825)\cdot (31556928)$$ $$=(2\times 0.40\times 10330)\cdot (\frac{10330}{0.40}-1)\cdot (10330\times 3055)=(8264)\cdot (25824)\cdot (31558150)$$

The following relationships are encoded in the Bully Mnemonic:

$$1GalacticYear\approx 8264GreatYears$$

$$1GreatYear\approx 25824SiderealYears\approx 25825TropicalYears$$

$$1TropicalYear\approx 31556928Seconds$$

$$1SiderealYear=31558150Seconds$$

Julian Years & Gregorian Years

There are exactly 86400 seconds in a day:

$$1Day=24\times 60\times 60seconds=86400seconds$$