## Modelling Phases of the Moon and hiding planets

Queensland based T3 Instructor and hobbyist astronomer, Stephen Broderick, captured this amazing image of the occulation of Saturn last month. (Photo taken 12^{th} August at 6:10 pm AEST)

For many years now, Stephen has shared his astronomical expertise with fellow maths and science teachers at various conference presentation both in Australia and abroad. His astrophotography skills are, without doubt, reaching to professional levels, yet Stephen has also demonstrated how school students and other budding amateur astronomers can easily and inexpensively start into this field. One simple example of this is pin-hole camera Solargraph photography, which Stephen has promoted in his presentations.

Stephen’s own students are most fortunate to have this expert guidance available to them and he has also developed a range of activities and record-sheets to help students discover the maths and science behind real ‘out of this-world’ study.

See below, one such example where Stephen uses circular functions for modelling the phases of the moon:

__Modelling the Phases of the Moon__ Author: Stephen Broderick

The following data was obtained from the U.S. Naval Observatory.

https://aa.usno.navy.mil/data/docs/MoonFraction.php

The fraction of the Moon illuminated for January 2019 (in the orange box) was plotted to obtain a sinusoidal model for the phases of the Moon. The model can be used to determine the period of the lunar phases, predict when full Moons occur throughout the year and determine when the next blue Moon occurs. (A blue Moon is when two full Moons appear in the same month)

The sinusoidal Moon data model for the month of January 2019 is graphed below in Figure 1.

Figure 1

The model is represented by the function: f(x) = 0.495267 sin(0.213405 x – 3.0094) + 0.469134

From the model, the period for the phases of the Moon = = 29.44 days. The accepted value is 29.55 days, so the percentage error is 0.37%. ((29.55-29.44)/29.55) x 100 =0.37%)

How good is the model? A full Moon occurs on the 19^{th} May which is day 139. Subbing 139 into the model:

f(139) = 0.495267 sin(0.213405 x 139 – 3.0094) + 0.469134

= 0.96, which means the Moon is 96% illuminated and so it still appears as a full Moon.

In 2019, there will be 12 Full Moons and 13 New Moons. The two New Moons occur in August 2019. When will the next blue Moon occur. (A blue Moon happens when two Full Moons occur in the same month) Thirteen Full Moons occur in 2020, so 2 Full Moons occur in the same month. The month in question is September 2020.