f80119…: “As regards the blue building”

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The Earth travels in an orbit around the Sun that can be approximated by a circle. The distance from the Earth to the Sun is (on average) $1$ au. We are going to make some models of the Earth’s orbit in the $(x,y)$ -plane. For all of our models, we will make the following assumptions:

The Sun will be at the origin.
At the starting time, $t=0$ , the Earth will be at the point $(1,0)$ in the $(x,y)$ -plane.
The Earth will travel in a counterclockwise direction around the Sun.

Give a parametrization of the Earth’s orbit that will model the Earth’s position in terms of $t$ , where the units are years. $\vec {y}(t) = \vector {\answer {\cos (2\pi t)},\answer {\sin (2\pi t)}}$

Give a parametrization of the Earth’s orbit that will model the Earth’s position in terms of $t$ , where the units are months. $\vec {m}(t) = \vector {\answer {\cos (2\pi t/12)},\answer {\sin (2\pi t/12)}}$

Give a parametrization of the Earth’s orbit that will model the Earth’s position in terms of $t$ , where the units are days. $\vec {d}(t) = \vector {\answer {\cos (2\pi t/365)},\answer {\sin (2\pi t/365)}}$

Given that the Earth’s orbit has a circumference of $940$ million miles, give a parametrization of the Earth’s orbit that will model the Earth’s position in terms of $s$ , the distance traveled in millions of miles. $\vec {e}(s) = \vector {\answer {\cos (2\pi s/940)},\answer {\sin (2\pi s/940)}}$

True or False: This is an arc length parameterization.

True False

Compute: $\vec {e}'(s) \begin {prompt} = \vector {\answer {(-\pi /470) \sin (2\pi s/940)},\answer {(\pi /470) \cos (2\pi s/940)}} \end {prompt}$

Compute: $|\vec {e}'(s)| = \answer {\pi /470}$

Note this will not be $1$ !

Why is the magnitude of the derivative not $1$ ?

We made a mistake. The units are the key.

Note, $1\unit {au} = 150\unit {million~miles}$ . Multiplying $\pi /470\cdot 150 \approx 1$ .