about TNOs

I downloaded all TNO asteroids with data-arc span >= 180 days from JPL Small-Body Database Search Engine:

JPL Small-Body Database Search Engine

[ Refine Search ]    Results:	2132 matching objects
Constraints:	asteroids and orbital class (TNO) and data-arc span >= 180 (d)

This is a summary table for all oribital parameters (this table and all plots below have been done using the R programming language plus ggplot2 and ggtern packages and linux imagemagick-6 ):

	a	e	i	om	w
	Min. : 30.11	Min. :0.002734	Min. : 0.07095	Min. : 0.2482	Min. : 0.5456
	1st Qu.: 40.95	1st Qu.:0.086505	1st Qu.: 3.71255	1st Qu.: 68.5043	1st Qu.: 85.2858
	Median : 44.04	Median :0.178993	Median : 10.08412	Median :142.0243	Median :173.6868
	Mean : 56.04	Mean :0.241073	Mean : 14.25290	Mean :155.5471	Mean :175.3624
	3rd Qu.: 48.93	3rd Qu.:0.312872	3rd Qu.: 19.59501	3rd Qu.:225.2598	3rd Qu.:265.1186
	Max. :1603.44	Max. :0.991355	Max. :170.98832	Max. :359.3391	Max. :359.9533

The minimum value of the semi-major axis is about 30 AU: for reference, Pluto's semi-major axis is about 39 AU.
The maximum value for the semi-major axis is a striking 1600 AU achieved by asteroid 2012 DR30.
The famous asteroid Sedna has a semi-major axis about 480 AU and the recently announced 2015 TG387 (the Goblin) has a semi-major axis about 1094 AU.

For every TNO, I calculated its Tisserand parameter (Tap) with reference to a generic planet with semi-major axis ap in the range [5.2 - 1000] AU, step 0.1 AU.

This calculation was done in two ways:

• all TNO together 
• TNO divided in 20 semi-major axis intervals

TNO (all together)

For every value of ap, I counted the number of TNO with Tap in the range [2 - 3].

The result is plotted here:

The above plot takes into account all TNO ranging from semi-major axis about 30 AU to about 1600 AU as shown in the previous table.
It seems that the value of ap that maximizes the count of TNO with Tap in the range [2 - 3] is ap = 43.5 AU (green vertical line).
 

TNO analyzed in various semi-major axis intervals

The semi-major axis was divided into 20 intervals such that in every interval you have about 5% of the population (actually about 107 asteroid per interval).

These are the intervals ( AU ):

	intervals
1	[30.1,38.3]
2	(38.3,39.2]
3	(39.2,39.5]
4	(39.5,39.8]
5	(39.8,41]
6	(41,42.2]
7	(42.2,42.9]
8	(42.9,43.4]
9	(43.4,43.7]
10	(43.7,44]
11	(44,44.4]
12	(44.4,45.3]
13	(45.3,46.3]
14	(46.3,47.4]
15	(47.4,48.9]
16	(48.9,54.7]
17	(54.7,58.2]
18	(58.2,68.8]
19	(68.8,91.6]
20	(91.6,1.6e+03]

For every interval I generated the same plot as seen before: thus, in total I created 20 plots that for ease of reference are shown here as an animated gif:

As you switch from one interval to the next, the ap value steadily slowly increases according to the approximated sequence 36, 39, 39, 40 ... 42, ..., 45,

...50, 57, 68, 90 AU and then, on the last 20th interval [91.6 - 1600 ] AU, something strange happens: the ap value goes back again to about 40 AU.

It is not clear to me why the 20th interval is showing this behaviour (I even thought this could be due to a bug in the program but I could not find it).

For ease of reference the "strange" 20th interval is again shown here below:

It is also interesting to note that probably due to the fact that in this interval we have extreme values of semi-major axis , there are also other relative maxima at bout ap=170 AU and maybe (much lower) at about ap=500 AU and ap=700 AU.

Curiosity: this last value (700 AU) is the estimated value for the semi-major axis of hypothetical Planet IX.

There are 14 TNO that contribute to the count for ap=700 AU including Sedna:

	a	e	i	om	w	Tap
90377 Sedna (2003 VB12)	479.904866256856	0.841319584894813	11.9299242495342	144.327455233507	311.537354629566	2.3344160106886
336756 (2010 NV1)	290.079238904053	0.967604322181253	140.749066145312	136.12865174322	132.653401330885	2.16141987021466
418993 (2009 MS9)	368.487142381523	0.970179216772268	68.0660878375908	220.219433266459	128.567138515509	2.03104187948606
474640 (2004 VN112)	319.030257013192	0.851750858061802	25.59923139738	65.9764025946988	326.79999934344	2.8321379285744
523622 (2007 TG422)	472.770400320918	0.924843603208057	18.6197089456926	112.840026264392	285.537555783418	2.07306646432227
523719 (2014 LM28)	277.831093719718	0.939635865307078	84.7387523405941	246.178631712983	38.3643481614205	2.55905094271
(1996 PW)	253.360837265531	0.990159010876786	29.9564761499396	144.38344890968	181.599728786903	2.90875109210892
(2010 GB174)	350.594295543395	0.86096080378873	21.5863384533732	130.834994060552	347.447039201188	2.6660937467626
(2011 OR17)	267.62292506094	0.988371756092761	110.50391097494	271.443208707997	14.0658781504528	2.54975571325885
(2013 FT28)	310.616250286609	0.860083663649765	17.3373719033496	217.770979685592	40.5386236787914	2.90236805429801
(2013 RF98)	357.875680246011	0.89920589784544	29.5866288844056	67.5780554516926	311.60258271302	2.50008216999172
(2014 SR349)	303.688812847671	0.843030035231188	17.9784555874619	34.7970603621366	340.9130466591	2.97894344497346
(2015 GT50)	324.376802633332	0.881380238783511	8.7830884236707	46.0926336967559	129.328691760594	2.79360748411466
(2015 RX245)	416.893254246754	0.890708576281222	12.1442299283332	8.59935840940333	65.1236515168313	2.36500035898143

TNO (all together) - dependency on w

For the value ap=43.5, I calculated their Tap values in the range [2 - 3].

Then, I divided the Tap values in tertiles.

By definition, in a ternary diagram, the whole asteroids are displayed like this:

In the above triangle, every vertex is associated to one of the three Tap tertiles.

The dot represents all asteroids taken together independently by their w (argument of perihelion) parameter: by definition they belong to the tertiles according to the proportion (1/3,1/3,1/3), so the dot is in the barycenter of the triangle.

I was curious to see what happens to this diagram if you investigate all parameter w (argument of perihelion) ranges (0-10],(10-20], ... (350-360].

This is what happens:

As expected, the various dots (ranges of 10 degrees) are no longer perfectly located in the barycenter.

Not sure of this but: I have the impression that for most of them the difference in the ideal proportion  (1/3,1/3,1/3) is likely to be not statistically significant (blue dots - labeled as "Same Proportion" = TRUE).

However, there might be some ranges (red dots, labeled as "SameProportion"=FALSE) where there is a greater difference that is probably really significant (to be understood better).

The suspect ranges and proportions are:

	[2.01,2.85]	(2.85,2.96]	(2.96,3]
(10,20]	0.51	0.28	0.22
(90,100]	0.2	0.58	0.22
(170,180]	0.22	0.49	0.29

TNO (all together) - dependency on om

For the value ap=43.5, I calculated their Tap values in the range [2 - 3].

Then, I divided the Tap values in tertiles.

Same analysis as before but this time I investigated the ascending node (parameter om).

The most "suspect" om ranges where the proportion is very different from the ideal (1/3,1/3,1/3) are the following:

	[2.01,2.85]	(2.85,2.96]	(2.96,3]
(40,50]	0.19	0.41	0.4
(80,90]	0.18	0.21	0.61
(110,120]	0.15	0.3	0.55
(200,210]	0.39	0.43	0.18
(210,220]	0.6	0.33	0.063
(220,230]	0.46	0.43	0.12
(280,290]	0.26	0.56	0.19

Kind Regards,
Alessandro Odasso