Angular Separation

Our approach compares sets of sky positions for objects with the same nominal orbital elements subjected to two different gravity models. One of the first issues we must confront deals with the fact that the angular differences between the nominal case and the perturbed case are quite small. The magnitude of the angular differences can be estimated by consid- ering that the heliocentric distance in the perturbed case is diminished by an amount equal to one half the product of the perturbing acceleration and the square of the time under which the object is subjected to the perturbation. This diminished radius, together with conservation of angular momentum, implies an angular difference. If we assume a radial distance of 25 AU and a time interval of 5 years, the resulting angular difference is less than a quarter of a degree, even for a circular orbit. For noncircular orbits the angular difference would be even less, as it would be for larger values of the semimajor axis.

Although previous work (Iorio and Giudice, 2006; Tangen, 2007) bears a superficial re- semblance to the information presented below, the earlier work is significantly different. Iorio and Giudice (2006) shows the differences in projected sky positions resulting from the same orbital elements projected forward in unperturbed and perturbed paths. Our calcu- lations, although starting from known initial conditions, are based upon elements resulting from fitting sets of noisy observations and thus reflect more clearly the unavoidable errors in elements. Tangen (2007) shows work that is based on a subspace of the full space of orbital elements that consists of four elements. Our approach represents a fuller and more complete fitting approach to finding orbital elements and thus portrays the effect of errors

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Angular Separation (deg)

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

Figure 6.1 Angular separation as a function of time from perihelion for an object with a semimajor axis of 20 AU.

in orbital elements more clearly as well.

Figure 6.1 shows angular separation as a function of time for a representative case, with a semimajor axis of 20 AU. All the objects portrayed in this graph have their perihelion on 2000 January 1; thus, the abscissa shows elapsed time from perihelion. There are several interesting aspects of this figure. First, at small times there is a significant amount of noise with a maximum value of approximately a microdegree (about three milliseconds of arc). This noise originates in our monte carlo treatment of the variation in predictions due to different ensembles of observations. Its level is far below the level of observational detectability and has no effect on our conclusions.

Another interesting aspect of the figure is the increase in separation, whose magnitude depends slightly upon eccentricity at an elapsed time of approximately 90 years, approxi- mately an orbital period for this object. Objects with large eccentricities move very quickly near perihelion. Small differences in angular position are thus magnified as one object moves through perihelion and the other has not yet reached it. After both have passed perihelion,

the objects angular separation narrows once again to a small value, although this is be- yond the times investigated here. Objects with smaller eccentricities (more circular orbits) move with more nearly constant angular velocity and do not suffer the differential angular separation described above.

Finally, it should be noted that although the separate curves appear close together, they are fairly substantially separated in time at a given angular separation. This time interval is roughly correlated with the time required for the objects to cross the 20 AU boundary and enter our defined perturbation zone.

Figure 6.2 shows a similar curve for an object with a semimajor axis of 40 AU. As above, perihelion for these objects occurs at 2000 Jan 1 and the abscissa shows elapsed time from perihelion. The noise seen in Figure 6.1 at early times is present here for a diminished length of time because the action of the assumed perturbation is smooth and we do not need to overcome as great a statistical weight of observations early in the arc within the 20 AU boundary. Similarly, the increase in separation seen previously at one orbital period is not seen here because the period for this object is approximately 250 years, more than the range of times shown on the abscissa. The time difference between similar angular separations is actually larger than a cursory inspection of the figure would indicate. The time interval is correlated with the time required to the objects to cross the 20 AU boundary where we define our perturbation to be active.

Most interesting in this figure is the two families of curves that appear at early times. These correspond to cases where the objects are always within the perturbation region (e.g., their heliocentric distance is always greater than 20 AU) and when they are only in the perturbation region sometimes (e.g., they move across the 20 AU boundary). The families both exhibit an inverse correlation between the rise of the curves and the eccentricity of the associated orbit. However, the orbits that are always in the perturbation region begin their rise earlier that the orbits that move from within the perturbation boundary to outside 20 AU. These orbits also exhibit an inverse correlation between the rise of the curves and the eccentricity, but the rise begins at greater times than for the other case.

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Angular Separation (deg)

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

Figure 6.2 Angular separation as a function of time from perihelion for an object with a semimajor axis of 40 AU.

The origin of these families is related to the method by which orbital elements are developed. Since elements are the result of a least squares fit to observational data, objects which begin their path inside the region where the perturbation takes effect will have their initial orbital elements produced on the basis of a non-perturbed gravitational field. This is the classical case. However, when these objects first enter the perturbed region, the slowly accruing new observations that reflect the existence of the perturbation will have to overcome the prior observations before they can alter the value of orbital elements and the associated position on the sky. Thus, in this situation, and given that we begin all our objects at perihelion, we would expect that a longer interval of observation would be required before a statistically significant angular position difference would be observed.

On the other hand, objects whose paths are always in the perturbation region do not have a statistical weight of unperturbed observations to overcome. Their elements are derived subject to the perturbation, and more quickly can show a statistically significant angular position difference.

It is in this part of our analysis that variation in the magnitude of our assumed grav- itational perturbation can be introduced. Since the perturbation is very small, we are operating in a linear regime of the governing equations. The angular differences observed between the perturbed and unperturbed cases are linear in the product of the perturbing acceleration and the time. Thus, the time at which a given angular separation occurs can be scaled inversely with the magnitude of the perturbing acceleration.