PROCEDIMIENTO SEGÚN EL CÓDIGO ORGÁNICO GENERAL DE 1.

The analyses presented here result in a fairly large array of numbers. How can they be summarized and assessed in some compact fashion so that we can make sense of the results? Our approach to answering this question is to present our results in terms of the probability that an angular difference between the perturbed and unperturbed cases is statistically significant as a function of observation arc length. Then, for whatever level of significance is appropriate, one can figuratively read off the length of observation arc required.

Figure 6.3 shows probability of a significant angular difference as a function of time since perihelion for the case with a semimajor axis of 20 AU. This case corresponds to that shown in Figure 6.1. Again, all objects on this graph have their perihelia at 2000 Jan 1, which is the origin of the time axis. Once again, we note the noise at the beginning of the abscissa axis; however, the level of significance to which this noise rises is never more than about 20 percent, a level of significance that would never, in practice, be quoted in a hypothesis test. Similarly, Figure 6.4 shows the probability of a significant angular difference as a function of elapsed time for the case with a semimajor axis of 40 AU. This case corresponds to Figure 6.2. Again, the initial noise is of shorter duration in this figure; however, the two families seen in Figure 6.2 remain.

In both Figures 6.3 and 6.4, the inverse correlation between the time required to attain a given level of significance and the orbital eccentricity is noted.

In order to make sense of the numbers associated with these figures and our other cases, we choose to fit the probability of a significant difference in position to time from perihelion

0.0 0.2 0.4 0.6 0.8 1.0 0 5 10 15 20 25 30 35 40

Probability of a Significant Angular Difference

Time (yr) e = 0.01 e = 0.1 e = 0.3 e = 0.5 e = 0.7 e = 0.9

Figure 6.3 Probability of a statistically significant angular difference as a function of time from perihelion for an object with a semimajor axis of 20 AU.

0.0 0.2 0.4 0.6 0.8 1.0 0 5 10 15 20 25 30 35 40

Probability of a Significant Angular Difference

Time (yr) e = 0.01 e = 0.1 e = 0.3 e = 0.5 e = 0.7 e = 0.9

Figure 6.4 Probability of a statistically significant angular difference as a function of time from perihelion for an object with a semimajor axis of 40 AU.

Table 6.1 The mean (in years), standard deviation (in years), and rms residual (in arcsec) for each case described by a semimajor axis and an eccentricity.

Semimajor Axis (AU)

e Parameter 15 20 25 30 35 40 45 0.01 Mean – 28.68/6.34 7.56 8.52 9.44 10.90 10.91 Std. Deviation – 1.34/1.34 2.02 2.54 3.02 4.36 3.27 rms Residual – 0.10/0.07 0.05 0.06 0.04 0.05 0.04 0.05 Mean – 24.06/1.78 7.35 8.22 9.47 10.03 9.97 Std. Dev. – 1.03/0.99 1.44 2.18 2.14 2.86 1.70 rms Residual – 0.09/0.01 0.04 0.04 0.04 0.03 0.04 0.10 Mean – 23.05/2.12 6.84 7.59 8.57 9.42 10.27 Std. Dev. – 1.02/1.00 2.03 2.50 2.30 2.91 3.82 rms Residual – 0.09/0.03 0.03 0.05 0.03 0.03 0.04 0.20 Mean – 21.23/1.86 6.33 7.29 8.14 8.52 9.62 Std. Dev. – 1.51/1.05 1.72 2.31 2.11 2.02 3.55 rms Residual – 0.07/0.02 0.02 0.05 0.05 0.03 0.04 0.30 Mean – 19.39/1.66 14.46/2.21 6.37 7.04 8.34 8.45 Std. Dev. – 1.96/1.06 1.20/1.11 2.56 2.14 1.78 2.65 rms Residual – 0.08/0.01 0.05/0.02 0.04 0.02 0.06 0.04 0.40 Mean 23.88/2.29 18.19/1.94 16.12/2.18 12.01/2.48 6.54 7.08 8.12 Std. Dev. 0.71/0.71 0.12/0.36 0.96/0.95 0.85/0.85 1.92 2.02 1.78 rms Residual 0.11/0.02 0.09/0.04 0.06/0.02 0.05/0.02 0.03 0.04 0.04 0.50 Mean 19.85/2.02 17.39/2.15 16.13/2.19 14.34/2.29 11.71/2.37 6.23 6.61 Std. Dev. 0.77/0.78 0.57/0.57 1.00/0.99 2.09/0.80 0.83/0.82 1.78 2.00 rms Residual 0.07/0.05 0.08/0.00 0.05/0.03 0.08/0.02 0.05/0.02 0.03 0.02 0.60 Mean 17.78/2.42 15.79/1.97 15.57/2.34 14.76/2.14 13.75/2.06 12.58/2.35 9.94/2.21 Std. Dev. 0.68/0.68 0.93/0.95 0.62/0.62 1.08/1.06 1.34/1.26 1.05/1.03 1.20/1.16 rms Residual 0.07/0.03 0.06/0.03 0.06/0.03 0.05/0.02 0.06/0.03 0.04/0.01 0.05/0.01 0.70 Mean 15.98/2.56 14.11/1.81 14.11/1.97 14.11/2.12 14.08/2.27 13.42/1.97 13.24/2.27 Std. Dev. 0.88/0.87 1.87/1.64 1.14/1.13 0.91/0.85 0.90/0.90 1.42/1.26 1.02/0.93 rms Residual 0.07/0.04 0.06/0.04 0.05/0.03 0.06/0.00 0.04/0.02 0.05/0.02 0.05/0.02 0.80 Mean 14.37/2.66 13.58/2.61 13.33/2.52 13.22/2.40 13.22/2.34 13.48/2.54 13.23/2.24 Std. Dev. 1.75/1.57 1.15/1.13 0.59/0.59 0.88/0.88 1.07/1.05 0.67/0.67 0.91/0.89 rms Residual 0.07/0.04 0.05/0.02 0.06/0.02 0.05/0.02 0.06/0.03 0.06/0.03 0.03/0.01 0.90 Mean 13.82/3.53 12.65/3.14 12.63/3.29 12.01/2.73 12.15/2.82 12.03/2.59 11.96/2.36 Std. Dev. 1.64/1.62 1.51/1.39 0.88/0.88 1.38/1.34 1.16/1.13 1.19/1.17 1.21/1.18 rms Residual 0.06/0.02 0.06/0.04 0.05/0.05 0.03/0.01 0.05/0.02 0.04/0.03 0.04/0.02

Notes: Bolded cells are those for which the orbit is always in the perturbed region, with heliocentric distance greater than 20 AU through the entire orbit. The unbolded cells that describe objects that move from within 20 AU to outside that distance. The entries for these objects also show the mean and standard deviation in years of a fit to a model in terms of time in years from crossing the 20 AU boundary. These are shown to the right of the slash mark. Cells marked with a dash indicate objects that never move into the perturbation region.

to some reasonable functional form. We choose to fit the data to the standard error function; thus, we fit each curve of probability versus time to a two parameter family of curves. Table 6.1 shows the results of this process, along with the rms residual associated with each fit. Much of the residual is ascribable to the stochastic noise discussed above; the fits achieved by this error model are generally quite satisfactory. However, as a general proposition, we may say that the fits for the cases where the object moves across the perturbation boundary is a factor of two or more better for the model based on time from passing the 20 AU boundary, at least as measured by the rms residual of the individual fits.

The result of this process is a somewhat smoothed set of parameters that describe the length of time required before observations will succeed in showing a statistically signifi- cantly different position on the sky if the additional gravitational perturbation actually acts

on the object.

The utility of this approach of fitting to an error function is that we can formulate a confidence limit at any level we choose. Picking 95 percent, the average shown in Table 6.1 plus 1.965 standard deviations gives that confidence limit (2-sided test). Thus, after that number of years, the objects subject to the peturbation and those not perturbed will show a significant difference in sky position at the 95 percent significance level.

A corollary to this approach is that we can plot the probability of a significant positional difference as a function of time. Representative samples of this type of data are shown in Figure 6.5. In this figure, the dashed lines show the 95 percent significance level and the associated time. The dotted lines, read on the right hand axis, show the orbital anomaly as a function of time. In Panel A, the first part of the line shows the true anomaly, measured from perihelion, while te disjoint second part of the line shows the anomaly measured from the angle at which the object passes the 20 AU. Since the object portrayed in Panel B never comes within 20 AU, the dotted line shows only true anomaly, measured from perihelion. Panel A, on the top, shows a typical case for an object that moves from within the 20 AU boundary to a point outside that distance, while the bottom panel (Panel B) shows an object that is always outside the 20 AU boundary. Also shown in these figures is the anomaly as a function of time. In Panel A, corresponding to an object moving from within to outside 20 AU, we show one curve showing true anomaly from perihelion out to the 20 AU boundary and another outside 20 AU showing the anomaly measured from the true anomaly at that distance. Panel B shows only the true anomaly. A general feature of these figures is that an arc of approximately 10-40 degrees in the perturbed region (e.g., outside 20 AU) is necessary to detect a positional difference at the 95 percent significance level.