Plotting Log graphs of planetary patterns

Desmos regression tool is used to plot all graphs representing the data values from the tables of the raw data (IV)section of the Internal Assessment. Mean distance in Astronomical Units (AU) from each planet to its moon (d) –lying on the y axis and log(d) values being put on the x-axis.

German astronomer Johann Daniel Titius and Johann Elert Bode both suggested there might be a pattern in the separations between the planets of the Solar System in the late 1760s and early 1770s, respectively. Mathematically, it seemed to be true despite the lack of any physical support for their hypothesis or any explanation for why there would only be a relationship.

The orbital separations of the planets from the Sun, measured in astronomical units (AU), are as follows:

Mercury: 0.39 AU

Venus: 0.72 AU

Earth: 1.00 AU

Mars: 1.52 AU

Jupiter: 5.20 AU

Saturn: 9.54 AU

The average distance between the Sun's and Earth's centres is measured in astronomical units (AU) (149.6million kilometres). However, the amount was rounded to 150 million kilometres for the Internal Assessment.

When examining the values listed above, it becomes clear that there is either no link between the numbers or a clear linear relationship. To see the relationship suggested by Titius ad Bode, one should examine the pattern from the perspective of geometric sequence. By dividing the term above by the word below, one can determine the r-value (for instance\(\frac{0.72}{0.39} \)= 1.8).

This way, we get r values of:

1.8, 1.4, 1.5, 3.4, 1.8

The four values of 1.8, 1.4, 1.5, and 1.8 are remarkably comparable when one considers the numbers. There is a 3.4 outlier, though. There was first thought to be Planet X, an unidentified planet, between Mars and Jupiter, but it was later proposed and discovered to be Ceres.

There is still no explanation for the worlds' remarkably precise conformity to the rule. Along with the theoretical recommendations, a formula is provided in addition to the geometric patterns of the values. However, the specific format of the recipe has no theoretical support. The Titius-Bode rule is explained using the theorem's technique, however the study only focuses on the linear relationship that now exists between planetary entities and their moons.

d = a x bⁿ⁵

One should apply the rules of the geometric sequence, and AU values of Mars and Jupiter are 1.52 AU and 5.20 AU, respectively, to get \(\frac{X}{1.52}=\frac{5.20}{x}.\)

The missing planet would be located at a distance of 2.8 AU from the Sun, which is the value obtained after calculating the equation. There are no obvious outliers because this would result in a list of new r values of 1.8, 1.4, 1.5, 1.8, 1.9, and 1.8. The exact distance from the Sun at which Ceres was found in 1801—2.8 AU—was predicted by the Titius-Bode theorem. Ceres is now categorised as a minor planet that lies between Jupiter and Mars.

The data below shows the distance between the following planetary entities and the Sun in AU and the log(d) value, respectively:

Mercury: d = 0.39 AU. log(d) = -0.41

Venus: d = 0.72 AU. log(d) = -0.14

Earth: d = 1.00 AU. log(d) = 0

Mars: d = 1.52 AU. log(d) = 0.18

Ceres (dwarf): d = 2.8 AU. log(d) = 0.45

Jupiter: d = 5.20 AU. log(d) = 0.72

Saturn: d = 9.54 AU. log(d) = 0.98

Uranus: d = 19.2 AU. log(d) = 1.28

Given that d = a × \(b^n\) with a and b assumed as constants; one can use apply the laws of logs to give:

log d = log a + n log b.

Plotting these values is the best way to represent a relationship that is to be proved visually. log d is put on the y-axis, while n (mean distance in Astronomical Units (AU) is on the x-axis. A linear graph will prove the geometric relationship, as log a is the y-intercept, and log b is the gradient.

The equation will not apply to Neptune, according to its mean distance being 38.8 AUs, but it is 30 AUs, showing a strong disagreement between the values; therefore, it is not applicable.

DESMOS Regression Tool was used to create an exemplary linear graph of the results above:

I have spent the majority of my free time reading and rereading chapters on our solar system, the planets, and space in general ever since my parents gave me my first encyclopaedia. Astronomy is the science that linked all of the fascinating things together, as I subsequently discovered. My passion evolved into reading about it and discovering new information about anything astronomical, and as I got older, I began exploring Steven Hawking's works on space. I read essay after essay and then started looking into additional possibilities that might or might not have been proven. When the variables of AU and the log of the value were given on a graph, the article I came across one day discussed the linear patterns of the planets in relation to their orbital centres. When I studied more about this occurrence later, I learned that it is known as the Titus-Bode theory. Titus and Bode, two astronomers, made this notion known in the late 1760s and early 1770s. The fact that no one has been able to explain this pattern since it was discovered is extremely astounding. There is a wealth of knowledge about the interactions between the Solar System's planets and the Sun, but the hypothesis contends that this rule holds true for all planetary bodies with moons. Therefore, the purpose of my internal assessment is to determine whether the AU (distance between planetary entities) and the log of the value seen in the planets and their moons, respectively, follow the same pattern of positive linear connections.

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By employing the log of each distance in astronomical units, the Titius-Bode relation is an empirical statement that shows a pattern of a linear positive correlation of how far the planetary entities are from one another (AU).

Interestingly, it is difficult to ascribe any meaning to the link unless one can find a scientific explanation. Long before any other answer was found in physics or any other branch of study, the Titius-Bode rule was recognized as the way solar systems created their orbits and controlled the distances between them. The rule itself was pretty compelling because it allowed one to predict the positions of every planet known at the time—Mercury, Venus, Earth, Mars, Jupiter, and Saturn—by using it. Even planetary bodies that are today categorised as dwarf planets, like Pluto and Ceres, partially complied with the criteria and were therefore regarded as planets at the time. The two 'planets' were later reclassified as dwarf planets since it was discovered that their orbital shapes made them the exception to the norm rather than the rule.

Despite the fact that it has yet to be refuted, the Titius-Bode rule has many limits and is questioned. The relationship cannot be fully established because the rule only applies to fixed planet distances (i.e., distances that do not significantly change at any point along the planets' orbits). Therefore, the community of planetary scientists suggests that the Titius-Bode rule is really a coincidence. For instance, Ceres, the object classified as a planet by the Titius-Bode rule, is really the largest asteroid in the belt of asteroids between Mars and Jupiter 2. The Titius-Bode relation has been modified for many years in an effort to make it applicable to all planets, but the current understanding of this physical mechanism is quite limited.