General instructions for assignments
All responses must be typed.
All calculations must be shown in full.
All graphs must be electronically produced.
All references must be cited using Chicago Manual of Style conventions.
no title page
Ref: NASA/JPL Solar system dynamics pages: Asteroid Main Belt Distribution
https://ssd.jpl.nasa.gov/?histo_a_astLinks to an external site.
Ref: Chicago Manual of Style
https://www.chicagomanualofstyle.org/home.htmlLinks to an external site.
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 (1) identify six Kirkwood gaps in this histogram, (2) calculate how long a hypothetical object inside the gap would take to orbit the Sun once, and (3) check that this object’s orbit time is in resonance with Jupiter. (4) Lastly, write a Chicago-style reference for this web page.
Identify (at least) six Kirkwood gaps: list the distances from the Sun at which zero asteroids are found. This list must include the four labeled gaps and at least two unlabeled gaps.
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
Check for resonance: divide Jupiter’s orbit time by the asteroid’s orbit time (both in years); express your result to at least 3 significant figures. If in resonance then your decimal result can be converted to a simple ratio. A simple ratio is a fraction of small integers. This task is not simple: your simple ratio should agree with your decimal result to at least 2 sig figs, and ideally three.
Finally, for this web page write down the full citation using Chicago Style.
Here is an example calculation and response you are allowed to imitate:
The first gap is at a distance of 2.5 AU. Using the formula, I calculate an orbit time = 2.53/2 = 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.
Grading: (1) [6 points] Identify at least 6 Kirkwood gaps, 4 of which are labeled on the diagram and at least two of which are not.
(2) [12 points] Calculate the orbital period of a hypothetic asteroid in each of the 6 gaps.
(3) [6 points] Reduce the Jupiter-to-asteroid ratio to a simple ratio.
(4) [6 points] Write a full Chicago style citation, including the author’s name.
