What does that have to do with comets or asteroids?

In the above diagram we can find some basic realities about the dynamics of our solar system. In it, we can also find many questions.

• Where do asteroids fit in? Why are there two major clumps?
• Why do the points of one cluster group around a vertical line?
• Why is there a big gap between the two major clumps?
• Is there any significance to the empty regions of this "phase diagram"?
• Which comets are the outliers and is there anything else special about them?

Well, when we add in the asteroids, and reorient the diagram, here is what we get:

Very interesting! It seems that the asteroids and the comets are dynamically quite distinguishable! In this and all subsequent diagrams, points represented by comets are in red, and asteroids are in green.

How are orbital elements related?

The graph above, and more, can be constructed from orbital element databases and celestial mechanics level mathematics.

Take, for instance, Kepler's 3rd relationship between period and orbit size,

1. G M (T/2pi)² = a³

   (where G is Newton's constant, M is the sum of the masses, T the orbit period, and 'a' the semimajor axis), and combine it with the linear relation between periapsis, eccentricity and orbit size,

2. a = q/(1 - e)

   where 'q' is periapsis, and 'e' eccentricity, then, we get:

3. G M (T/2pi)² (1 - e)³ = q³.

   This equation is linear in "eccentric complement" and periapsis, and the lines of constant-period pass through the origin. If we construct a q versus 1-e diagram and put a few lines of constant-period on it along with all comets which have only been seen once, it looks like this:

   

   Are the groupings of this graph significant, and if so, of what? Why do there appear to be one or two 'bands' emanating from the origin out in the 10 o'clock direction?

   Although all those objects falling along the same line may have the same period, (and therefore the same semimajor axes) they have different eccentricities, and therefore different aphelions.

   ---

   The lines of constant period are also lines of constant energy as can be seen from (1). Stated without proof, it can be shown that the energy per unit mass is

4. E = G M / (2 a)

   which can be combined with (2) to form

5. 2 q E = - G M (e - 1).

   The lines of 'greater' energy per unit mass (or longer periods) are those rotated about the origin in the clockwise direction. If we follow such a moving "clock hand" as it rises from the 9 o'clock position approaching the y-axis, we find the clock hand to pass through clumps seeming to lie along lines drawn from the origin. What is the nature of these clumps? The implication is that there are groups of objects with a rather few and narrow range of periods for a large area of the eccentricity-perihelion phase diagram.

   Part of the reason is that beyond a certain perihelion distance, objects become more difficult to detect as comets. The low-e, low-q region of the graph is the realm of the The Belt. Comets haven't survived as comets in this part of the solar system. But what about the lack of objects in the vicinity of (e=0.8,q=3.5)? Where and what are these objects, and why don't we see many of them (at least, as comets)? From the left hand side of the graph, we can see there certainly is a population of comets recognizable as such at least out to 4a.u. perihelion distance. Undoubtedly, part of the reason is that this is the Jupiter clearance zone.

   ---

   Likewise, angular momentum per unit mass can be extracted from the Kepler relation using conic geometry to give

6. h² = p G M

   where

   p = semilatus rectum, and
   h = angular momentum per unit mass of an orbiting body.

   Angular momentum can be related to orbit size by using

   p = a (1 - e²)

   Substitution into (6), and using (3) gives

   h² = a G M (1 - e²)

   = q G M (1 + e)

   h²/(q G M) = 2 + (e - 1)

   Lines of constant angular momentum are "1/X" functions on a q vs (1+e) graph, but a little more complex if drawn on the above graph. The usefulness of q vs 1+e graphs can be seen on the diagram below. Unlike all of the other plots on this page, the graph below shows all comet apparitions, including the multi-apparition ones.

   

   ---

   Below are two related graphs showing eccentricity plotted against energy and angular momentum per unit mass.

   Eccentricity vs Angular Momentum Per Unit Mass

   When we add in the asteroids and rotate the graph, we get:

   Note above the curious parallel band structure in the middle of the graph. Real, or apparent? Also note the apparent gap between the parabolic comets (clustered along a vertical line at the right) and the long-period elliptical comets.

   ---

   Eccentricity vs Energy Per Unit Mass

   Note again the gap between the cluster of parabolic comets (upper right) and the other comets. Note also the apparent "forbidden" regions along the top, to the right and below the parabolics, and in the lower left (main asteroid belt).

   And when we add in the asteroids, we get:

   

   ---

   Differnces between comets and asteroids show up dynamically in other ways as well. Note below the plot showing Specific Angular Momentum plotted against perihelion distance.

   

   Along the upper curve the aperiodic comets lie, while the periodic comets and the asteroids are grouped closer together, with the comets on top because the comets in general have a higher eccentricity.

   Note below that long-period comets also show themselves in yet another way.

   

   Below we see that the periodic comet apparitions are generally of low inclination to the ecliptic

   

   ---

   Below we see another interesting comparison. Why the gap between the periodic comets and the asteroids? Why the bigger gap between the periodic comets and the aperiodic ones?

   

   ---

   Correlations exist between the raw orbital elements as well.

   

   ---

   What do these graphs look like when we plot only a single point for each periodic comet and show them distinct from the aperiodic comets?