El exequátur en el proyecto de Código Orgánico General de Procesos

Thus far, we have discussed the degree to which orbits fit to observations subject to normal gravity differ in predicted sky position from those fit to perturbed observations, and when

0.0 0.2 0.4 0.6 0.8 1.0 0 20 40 60 80 100 0 30 60 90 120 150 180

Probability of a Significant Angular Difference

Anomaly (deg)

Time From Perihelion (yr)

Panel A Probability 95% Significance Level True Anomaly 0.0 0.2 0.4 0.6 0.8 1.0 0 20 40 60 80 100 0 30 60 90 120 150 180

Probability of a Significant Angular Difference

Anomaly (deg)

Time From Perihelion (yr)

Panel B Probability

95% Significance Level True Anomaly

Figure 6.5 Probability of a significant sky position difference between the perturbed and the unperturbed case for selected objects as a function of time from perihelion. The object shown on the top (Panel A) is representative of one that sometimes passes inside 20 AU and sometimes is outside that distance (semimajor axis = 25 AU, eccentricity = 0.3) while the bottom panel (Panel B) shows a representative object that is always outside 20 AU (semimajor axis = 30 AU, eccentricity = 0.3).

that difference becomes statistically significant. Another measure of the quality of the or- bital fits is found in the total rms deviation between the observations and the corresponding fits. This value provides a standard benchmark for fit quality.

There are four cases to consider. We have two sets of observations, one resulting from the existence of the gravitational perturbation and one not. Correspondingly, we have two possible models, one containing purely Newtonian gravity and one including the small, as- sumed perturbation. We now consider each mix of these two pairs of factors. Thus, we have one situation where observations generated from an unperturbed orbit are compared with the fit obtained from the unperturbed model and the corresponding “matched set” consist- ing of observations generated from the perturbed ephemeris compared with the perturbed model. The “cross terms” form the other pair of results. Thus, observations generated from unperturbed motion compared with a perturbed model and observations developed from perturbed motion compared to unperturbed model form “mismatches” that we now investigate to determine the quality of fit that can be obtained.

Figure 6.6 shows representative sets of these results. Panel A, on the top, shows a typical case for an object that moves from outside the perturbation region (e.g., from within 20 AU) to within the perturbed region. Specifically, this object has a semimajor axis of 25 AU and an eccentricity of 0.3. The interesting thing about this graph is that for the “matched” sets of data; that is, when the gravitational model matches the origin of the synthetic observations there is a uniformly good fit with an rms residual of about 0.3 arc seconds. This level of error matches quite nicely the initial astrometric error that we assumed at the beginning of the analysis.

Similarly, the “mismatched” cases start to show a declining quality of fit that is mani- fested through a systematic increase in the rms value, which appears roughly quadratic in time. Thus, attempting to fit the wrong gravitational model with eventually be manifested through a decline in the quality of the orbital fit.

The bottom panel (Panel B) of Figure 6.6 shows similar data for the case of an object that has a semimajor axis of 30 AU and an eccentricity of 0.3 and is therefore always in

0.0 0.2 0.4 0.6 0.8 1.0 0 20 40 60 80 100 rms residual (arcsec)

Arc Length (yr)

Panel A

Ephermeris without/Gravity without Ephermeris without/Gravity with Ephermeris with/Gravity without Ephermeris with/Gravity with

0.0 0.2 0.4 0.6 0.8 1.0 0 20 40 60 80 100 rms residual (arcsec)

Arc Length (yr)

Panel B

Ephermeris without/Gravity without Ephermeris without/Gravity with Ephermeris with/Gravity without Ephermeris with/Gravity with

Figure 6.6 Rms residual for various combinations of observations and gravity model as the observation arc lengthens. Panel A shows the case for an object with a semimajor axis of 25 AU and an eccentricity of 0.3. Panel B shows the case of an object that is always in the perturbation region beyond 20 AU. Its semimajor axis is 30 AU and its eccentricity is 0.3. Note: in the legend, “ephemeris with/without” means the synthetic observations were generated with/without a Pioneer perturbation; “gravity with/without” means the object’s motion is calculated with/without the additional perturbing force.

the assumed perturbation region. The general characteristics outlined above are found here as well, except for a slightly slower rate of decrease in the quality of the orbital fit as time progresses for the “mismatched” cases. The overall quality of the fit for the “matched” cases is slightly better as well, This is probably due to this case being in an area where the gravitational force is smoothly varying. Thus, the orbital fit does not need to accommodate the discontinuity in acceleration present at 20 AU due to our assumed perturbation model. The result is that while acceptable orbital fits might be obtained with “matched” ob- servations and models, the “mismatched” cases provide inferior fits. However, it should be noted that a sophisticated analysis of astrometric errors might be necessary to determine that the achieved fit quality is actually inferior. All “mismatched” cases shown in Figure 6.6 still have sub-arcsecond rms residuals even after an observation arc a century long.

This buttresses the observation made at the beginning of the paper that an adjustment of orbital parameters can conceal the existence of small perturbations; the quality of the fit with and without the perturbation is almost indistinguishable on the basis of the degree to which observations can match an orbital model to which they are fit for an arc of 20 years length or more.

Figure 6.6 might be compared with Figure 3.4 in Chapter 3 and one might ask why the rms values here are monotonically increasing while the referenced curves decrease to a minimum before increasing and why the rate of increase in rms errors here seem so much smaller than found previously for “mismatched” cases. There are two answers to each of these questions. First, the earlier work dealt with an object whose orbit was already known and which possessed a number of existing observations. Thus, new observations had to overcome the statistical weight of the pre-existing observations and the corresponding observation arcs. Since they were relatively few and at an uneven cadence, the rms fit was at first improved by new observations. In the current case, the observations from the start were evenly spaced and at a high cadence. Thus, the initial fit quality could hardly be improved as the arc length increased.

Secondly, the observation cadence investigated here is much higher than that in the ear- lier work. Thus, the statistical weight of a relatively large number of observations needed to be overcome before the fit started to degrade significantly. Even then, the rate of degra- dation was slower than that seen in the earlier work.

6.3

Discussion