## Graphs Help

Ref: NASA/JPL Solar System Dynamics pages URL: http://ssd.jpl.nasa.gov/?histo_a_ast

Ref: Chicago Manual of Style online URL: http://www.chicagomanualofstyle.org/home.html

1. The URL above links to a histogram of number of asteroids versus their average distance from the Sun (“semi-major axis” is the average distance). Clearly, there are populated distances and unpopulated distances. The unpopulated distances are called Kirkwood gaps. Your task is to

(i) identify all the Kirkwood gaps in this histogram,

(ii) calculate how long an asteroid inside the gap would take to orbit the Sun once, and

(iii) check that this asteroid’s orbit time is in resonance with Jupiter.

(iv) Lastly, write a Chicago-style reference for this web page.

(i) Identify all Kirkwood gaps: list the distances from the Sun at which zero asteroids are found.

(ii) Calculate orbit times: you can use a simple rule for calculating orbit times from orbit distances, (orbit time in years) = (orbit distance in AU ) 3/2

(iii) Check for resonance: divide Jupiter’s orbit time by the asteroid’s orbit time (both in years); if in resonance then the resultant division will produce a simple ratio as an answer. A simple ratio is a ratio of small integers

. Example calculation: The first gap is at a distance of 2.5 AU. Using the formula, I calculate an orbit time = 2.5 3/2 = 3.95 years, in other words, a hypothetical asteroid at a distance of 2.5 AU from the Sun would orbit the Sun once every 3.95 years. To check for resonance, I divide into Jupiter’s orbit time of 11.86 years; 11.86 years / 3.95 years = 3.00 (years divided by years produces a unitless quantity). Or, I could write 11.86:3.95 = 3.00:1.00. The simple ratio is therefore 3:1. In other words, for every three asteroid orbits, Jupiter orbits exactly once. I have thus verified the 3:1 orbit resonance.