Chapter 9.1, Problem 91AE

91. Bode s Law In 1772, Johann Bode published the following formula for predicting the mean distances, in astronomical units $\left(AU\right)$, of the planets from the sun:

$\left(AU\right)$ $a_n\text{ = 0}\text{.4 + 0}\text{.3 }\cdot {\text{ 2}}^{n-\text{ 2}}$

where $n\ge 2$ is the number of the planet from the sun.

a. Determine the first eight terms of this sequence.

b. At the time of Bode's publication, the known planets were Mercury $\left(0.39AU\right)$, Venus $\left(0.72AU\right)$, Earth $\left(1AU\right)$, Mars $(1.52AU)$, Jupiter $\left(5.20AU\right)$ and Saturn $\left(9.54AU\right)$. How do the actual distances compare to the terms of the sequence?

c. The planet Uranus was discovered in 1781, and the asteroid Ceres was discovered in 1801. The mean orbital distances from the sun to Uranus and Ceres* are $19.2AU$ and $2.77AU$, respectively. How well do these values fit within the sequence?

* Ceres, Haumea, Make make, Pluto, and Eris are referred to as dwarf planets.

d. Determine the ninth and tenth terms of Bode's sequence.

e. The planets Neptune and Pluto* were discovered in 1846 and 1930, respectively. Their mean orbital distances from the sun are $30.07AU$ and $39.44AU$, respectively How do these actual distances compare to the terms of the sequence?

f. On July 29. 2005, NASA announced the discovery of a dwarf planet* $(n=11)$, which has been named Eris. Use Bode’s Law to predict the mean orbital distance of Eris from the sun. Its actual mean distance is not yet known, but Eris is currently about 97 astronomical units from the sun.