The Bully Mnemonic

The Bully Mnemonic is a technique for remembering the exact number of seconds that occur in Earth's sidereal year; a good approximation 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.

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

Using Long Multiplication:

       3055
×     10330
————————————
   3055
    0000
     9165
      9165
+      0000
————————————
   31558150

Step 5

The tropical year has a slightly shorter duration than the sidereal year. The approximate number of seconds in a tropical year is obtained by reducing integer a) by amount d), and then multiplying by b).

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

Using the Distributive Property of Multiplication:

(10330 - 0.40) × 3055 = (10330 × 3055) - (0.40 × 3055)
                      =    31558150    -     1222
                      =    31556928

Great Years

Step 6

The Great Year is, by definition, a least common multiple of the sidereal year and the tropical year. From steps 4 and 5 above, we have that the ratio of tropical years to sidereal years is:

$${\frac {1\,Tropical\,Year}{1\,Sidereal\,Year}}\approx {\frac {(10330-0.40)\times 3055\,sec}{10330\times 3055\,sec}}$$

Divide top and bottom by amount d) and use the Distributive Property of Multiplication to obtain:

$${\frac {1\,Tropical\,Year}{1\,Sidereal\,Year}}\approx {\frac {({\frac {10330}{0.40}}-{\frac {0.40}{0.40}})\times 3055\,sec}{({\frac {10330}{0.40}})\times 3055\,sec}}$$

From whence:

$${\frac {1\,Tropical\,Year}{1\,Sidereal\,Year}}\approx {\frac {(25825-1)\times 3055\,sec}{(25825)\times 3055\,sec}}$$

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

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.

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

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

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:

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

$${\frac {1\,Galactic\,Year}{1\,Second}}\approx (2\times {0.40}\times {10330})\cdot ({\frac {10330}{0.40}}-1)\cdot ({10330}\times {3055})=(8264)\cdot (25824)\cdot (31558150)$$

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 86400 seconds in a day:

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

The exact number of days in a tropical year; however, is not a whole number. To keep the seasons synchronized with human calendars, it is necessary to round up or round down the number of days included in each specific year. To stay synchronized, the majority of years (about three out of four) must be rounded down to have 365 days, whereas a smaller percentage (about one in four) must be rounded up to have 366 days.