A new look at orbits

Diagrams similar to figure 1 appear in almost all first-year university physics text books that teach students about planetary orbits and the first two laws of Kepler, (1) orbits are elliptical and, (2) a line joining the center of the Sun and the center of a planet sweeps out equal area in equal time.

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In figure 1, the difference between the semi-major and semi-minor axes is greatly exaggerated, as is the distance of the Sun from the center of the ellipse. Similar diagrams are used to show the Earth at perihelion, aphelion, and at both ends of the semi-minor axis between these extremes. These diagrams are intended to show the circular orbit of the Earth at an oblique angle, but the perception is that the Earth's orbit is much more elliptical than it actually is. This leads to the impression that the Earth is much closer to the Sun at perihelion compared to aphelion, which is not the case.

Several papers in the education literature discuss the relation between teaching students about the elliptical orbits of the planets and the origin of the seasons (Aravind 1987, Lee 2010, Oostra 2014, 2015, De Paor et al 2017). A common theme discussed is the relation between how orbits are drawn and the very common misconception that it is warmer in summer because the Earth is closer to the Sun.

A correctly scaled diagram of the Earth's orbit is shown in figure 2, which looks quite different from figure 1. The Earth's orbit around the Sun, and the other planets are far more circular than commonly supposed.

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By definition, it is of course true that the semi-minor axis (a) is smaller than the semi-major axis (b), and perihelion is smaller than aphelion, but due to the Sun being at the focus of an ellipse the ratio a/b is much closer to unity.

An analogy to the difference between figures 1 and 2 is that figure 1 is equivalent to exaggerated stage acting where emotions and movements are amplified so they can be seen by the audience seated out in the theatre.

Table 1 shows the orbital characteristics of the Moon and planets. The value in the 'f' column is the distance of the focus from the center of the ellipse in millions of km. The last column shows the tolerances of the orbits applied to a 26 inch bike wheel. For the tol. (mm) column, the semi-major axis is taken as exactly 26 inches (660.4 mm) for the Moon and eight planets. This value of 660.4 is then multiplied by the ratio of the semi-minor axis to the semi-major axes, i.e. $b/a$. In the case of mercury, $b/a =$ 0.978 651 49 and when multiplied $660.4$ gives 646.3 mm. The difference is $660.4 - 646.3 = 14.01{\text{ mm}}$ as seen in the table.

Bicycle wheels are considered to be 'true' if the variation is less than 1 mm. Using this criterion, table 1 shows that the orbits of all the planets, except Mercury and Mars are practically perfect circles. Venus and Neptune have the most circular orbits with respective 26-inch wheel equivalent tolerances of just 6 and 10 $\mu {\text{m}}$–much less than the thickness of the paint on most bicycles. Even the orbit of the Moon is close to a perfect circle.

The orbit of the Earth is equivalent to the rim of a 26 inch bike wheel with a diameter deviation less than 0.1 mm–about the same as the thickness of a thin coat of paint. Figure 3 shows a schematic comparison of the orbit of Mercury compared to the other seven planets. Mercury has by far the most elliptical orbit. However, even this is not far from circular.

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Figure 4 shows a photo of a truing stand in the Uni Bike Shop (www.unibikeshop.com.au/) at the University of Queensland, St. Lucia Campus, which is used for making wheels more circular and the plane flatter, a process known as truing. The tolerance of the wheel on the stand was 2–3 mm, i.e. about the same as the orbit of Mars.

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Studies have shown that it is extremely difficult to dislodge misconceptions about the orbit of the Earth, which has led some authors, for example De Paor et al (2017) to advocate teaching seasons before elliptical orbits.

The bike wheel analogy of planetary orbits is a good one to start with since even young children are familiar with bike wheels. The orbits of all the planets, with the exception of Mars and Mercury, are analogous to near perfect bicycle wheels. Mars is like a wheel slightly out of true and Mercury a wheel after it has had a major collision with a curb or rock–but you could probably still ride a bike with a wheel like this.

If students are initially taught that the orbit of the Earth is essentially circular, like a bike wheel, when they eventually come across diagrams like figure 1, they will realise that the shape of the orbit has been exaggeration for educational purposes, or they are viewing the circular orbit of the Earth at an oblique angle. The adage prevention is better than cure is relevant here.

The Greeks believed that heavenly bodies travelled in perfect circles. Paradoxically the common idea of the Earth's orbit being highly elliptical is further from the truth. We have now come full circle if you will excuse the deliberate pun.

The author would like to thank Gavin Young of the Uni Bike shop at the University of Queensland St. Lucia campus for allowing the figure 4 photo to be taken and explaining the truing of bike wheels, and cycling aficionado Ash Moor of UQ College for a discussion about the riding experience when wheels are untrue.

All data that support the findings of this study are included within the article (and any supplementary files).

The aphelion and perihelion values for the Moon and planets were extracted from the NASA planet fact sheet, https://nssdc.gsfc.nasa.gov/planetary/factsheet/ and the semi axes, focus and eccentricities calculated as follows.

The semi-major axis (a) is defined as the mean of aphelion (ah) and perihelion (ph),

$$\begin{equation*}a = \frac{{ah + ph}}{2}.\end{equation*}$$

The distance of each focus from the semi-major axis is half the difference between aphelion and perihelion.

$$\begin{equation*}f = \frac{{ah - ph}}{2}.\end{equation*}$$

In terms of the semi-major axis (a) and semi-minor axis (b) the focus is defined as

$$\begin{equation*}{f^2} = {a^2} + {b^2}.\end{equation*}$$

Which enables us to find the semi-minor axis (b),

$$\begin{equation*}b = \sqrt {{a^2} - {f^2}} .\end{equation*}$$

The eccentricity (e) of an ellipse is defined as

$$\begin{equation*}e = \sqrt {1 - \frac{{{b^2}}}{{{a^2}}}} \end{equation*}$$