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:

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

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)

$${\begin{matrix}{\color {Red}1}&\scriptstyle {\text{2}}&{\color {Red}3}&\scriptstyle {\text{4}}&{\color {Red}5}\end{matrix}}$$

$$a)\,{\color {Red}1}0{\color {Red}33}0$$

$$b)\,{\color {Red}3}0{\color {Red}55}$$

Step 3

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

$${\begin{matrix}\scriptstyle {\text{1}}&{\color {Red}2}&\scriptstyle {\text{3}}&{\color {Red}4}&\scriptstyle {\text{5}}\end{matrix}}$$

$$c)\,{\color {Red}2}$$

$$d)\,0.{\color {Red}4}0$$

Sidereal & Tropical Years

Step 4

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

$${\color {Red}1}0{\color {Red}33}0\times {\color {Red}3}0{\color {Red}55}=31558150={\frac {1\,Sidereal\,Year}{1\,Second}}$$

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.

$$({\color {Red}1}0{\color {Red}33}0-0.{\color {Red}4}0)\times {\color {Red}3}0{\color {Red}55}=31556928\approx {\frac {1\,Tropical\,Year}{1\,Second}}$$

Great Years

Step 6

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

$$1\,Great\,Year\approx {\frac {10330}{0.40}}\,Tropical\,Years=25825\,Tropical\,Years$$

$$1\,Great\,Year\approx {\frac {10330-0.04}{0.40}}\,Sidereal\,Years=25824\,Sidereal\,Years$$

Proof:

$${\frac {10330}{0.40}}\,Tropical\,Years={\frac {10330}{0.40}}\times ({\color {Red}1}0{\color {Red}33}0-0.{\color {Red}4}0)\times {\color {Red}3}0{\color {Red}55}\,seconds$$

$${\frac {10330-0.04}{0.40}}\,Sidereal\,Years={\frac {10330-0.04}{0.40}}\times {\color {Red}1}0{\color {Red}33}0\times {\color {Red}3}0{\color {Red}55}\,seconds$$

Galactic Years

Step 7

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.

$${\color {Red}2}\times {{\color {Red}1}0{\color {Red}33}0}^{2}=213417800\approx {\frac {1\,Galactic\,Year}{1\,Tropical\,Year}}$$

$${\color {Red}2}\times {\color {Red}1}0{\color {Red}33}0\times ({\color {Red}1}0{\color {Red}33}0-0.{\color {Red}4}0)=213409536\approx {\frac {1\,Galactic\,Year}{1\,Sidereal\,Year}}$$

Step 8

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

$${\color {Red}2}\times {{\color {Red}1}0{\color {Red}33}0}^{2}\times ({\color {Red}1}0{\color {Red}33}0-0.{\color {Red}4}0)\times {\color {Red}3}0{\color {Red}55}=6.7348101\times {10}^{15}\approx {\frac {1\,Galactic\,Year}{1\,Second}}$$

Summary

The Bully Mnemonic can be represented as follows:

$${\frac {1\,Galactic\,Year}{1\,Second}}\approx {\color {Red}2}\times {{\color {Red}1}0{\color {Red}33}0}^{3}\times {\color {Red}3}0{\color {Red}55}$$

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

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

The following relationships are encoded in the Bully Mnemonic:

$${{1}\,Galactic\,Year}\approx {8264}\,Great\,Years$$

$$1\,Great\,Year\approx 25824\,Sidereal\,Years\approx 25825\,Tropical\,Years$$

$${1\,Tropical\,Year}\approx {31556928\,Seconds}$$

$${1\,Sidereal\,Year}={31558150\,Seconds}$$

Julian Years & Gregorian Years

There are exactly 86400 seconds in a day:

$${1\,Day}={24\times 60\times 60\,seconds}=86400\,seconds$$