Topic 1 – Our Solar System's Planets Distances from the Sun or the Titus-Bode Calculation (to be used during the study of inferential statistics; typically, second half of a statistics course).
We have all seen sequences in logic puzzles and exams. Some of these sequences help us determine the number of days in a month. For example, you can "play" a simple counting game to determine whether a month has 31 or 30 days by utilizing the knuckles and spaces between the knuckles of your hand. Start by labeling the first knuckle as January, February as the space between your first and second knuckle, March as the second knuckle and so. Since we know that February is the shortest month, and it is a space between the knuckles, it follows that all of the months that land on knuckles have 31 days and the months in between the knuckles have 30, except for February, which has 28 or 29 days. (Augustus Caesar insisted on 31 days in his month.)
In 1766, a German physicist, Johann Titus, predicted the mean (average) distance of each planet from the sun by using a relatively simple mathematical progression of numbers. The number progression is as follows: 0 (Mercury), 3 (Venus), 6 (Earth), 12 (Mars), 24 (Asteroid Belt), 48 (Jupiter), 96 (Saturn), 192 (Uranus), 384 (Neptune) and 768 (Pluto). Now add 4 to each number in the progression and then divide the result by 10. The third number in the progression (Earth) has a value of 1 and all other numbers (planets) are ratios of that 1, thus determining their mean distance from the sun. His discovery didn't garner much attention until Johann Bode, another German astronomer, published the calculation in the late 1770's. This set of ratios became known as the "Titus-Bode Law." Utilizing this expression, Bode predicted the existence of another planet between Mars and Jupiter. At the time of the development of the Titus-Bode Law, only six planets were known. However, this prediction was later borne out by the discovery of the asteroid belt in 1801. It took more than 100 years after Bode to discover the remaining planets: Uranus (1871), Neptune (1846) and Pluto (1930).
The question we are looking to answer is how accurate is the Titus-Bode Law? The work sheet and chart below offer a tabular and graphic representation of this law as well as one measure of its accuracy. To create this table, the planet names are in column 1, with Titus's numbers of 0,3,6 etc. in the second column. The third column is the constant 4 to be added to each of Titus's numbers; we add the two columns in column four. We divide by 10 and we get column five. Column five is then copied to column six as we wish to compare Titus's number to our current knowledge, which is in column seven. Just as we use one foot as a unit of length, astronomy has a standard unit of measure, the astronomical unit or AU. An AU is the average distance between the Earth and the Sun, which is approximately 1.4960 x 10 8 km. To convert this distance to our American unit of miles, multiply the 1.4960 x 10 8 by .621 (1 km = .621 miles) for 92,901,600 miles.
At this point, you might want to give the students a little refresher in metric to English conversion. We start with 1.4960 x 10 8 km (kilometers or thousand meters) which translates into 149,600,000 km or 149,600,000,000 meters. If you have a meter stick available, have the students note that there are about 39.4 inches per meter (or have them look it up in the dictionary). We then multiply the 149,600,000,000 times 39.4 for a product of 5,894,240,000,000 inches. Since there are 12 inches in a foot we divide by 12 to get 491,186,667,000 feet, and since there are 5,280 feet in a mile we divide again and get 93,027,777 miles. Thus, one AU is approximately 93 million miles –; the distance from the Sun to the Earth.
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