The Bully Mnemonic

The Bully Mnemonic is a technique for remembering the exact number of seconds that occur in Earth's sidereal year, good approximations of the relationships that exist between Earth's sidereal, tropical, and Great Year, and a rough approximation of 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.

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

$$1GreatYear\approx \frac{10330-0.04}{0.40}SiderealYears=25824SiderealYears$$

Proof:

$$\frac{10330}{0.40}TropicalYears=\frac{10330}{0.40}\times (10330-0.40)\times 3055seconds$$

$$\frac{10330-0.04}{0.40}SiderealYears=\frac{10330-0.04}{0.40}\times 10330\times 3055seconds$$

Galactic Years

Step 7

Multiply integer c) by the square of integer a) to get a rough 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 8

This step is an immediate consequence of steps 6 and 7 above:

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

Summary

The Bully Mnemonic can be represented as follows:

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

$$\frac{1GalacticYear}{1Second}\approx (2\times 0.40\times 10330)\cdot (\frac{10330}{0.40})\cdot ((10330-0.40)\times 3055)=(8264)\cdot (25825)\cdot (31556928)$$

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

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:

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