Pole Position and Axis Ratios

In order to find the pole position and axis ratios, we used the amplitude variation model described in Magnusson (1986)

∆mpψi, αiq “ 1.25logr pb{cq

2 cos2

ψi`sin2 ψi

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where ψi is the aspect angle (the angle between the asteroid’s c-axis and line of sight to Earth) for observation i, a{b and b{c are axial ratios greater than or equal to 1, βA is the amplitude phase coefficient described below, andαi is the phase angle of the asteroid at the time of observation i.

The aspect angle ψi can be defined for each epoch as

ψi “90´arcsinrsinβC,isinβp`cosβC,icosβpcospλC,i´λpqs (4.2)

where λp and βp are the heliocentric ecliptic longitude and latitude of the pole of Anchises, and λC,i and βC,i are the ecliptic longitude and latitude for Earth at observation i, all in a

reference frame centered at Anchises.

We used a grid search to simultaneously fit the pole position and axial ratios by mini- mizing the difference between the observed amplitudes and the model for a tri-axial ellipsoid with axes a ě b ě c. The results of this grid search can be seen in Figure 4.3. There is a 4-fold degeneracy in the pole orientation with this model, but the axis ratios were found to be consistent for all pole solutions.

When no bounds were placed on the minimization function we found axial ratios of

a{b “ 1.62˘0.02 and b{c “ 0.91˘0.1. This is consistent with a b{c value of 1. When a lower bound of 1 is placed on the b{c ratio, a best fit for a{b of 1.61 is found. This is well within the error determined by a 10% level above the best fit χ2

.

We found the pole position to be best fit at ecliptic longitude and latitude (198, ´29) and (18, 29), with 1 sigma errors of about˘1.5˝ _{in longitude and ˘5}˝ _{in latitude. The two}

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The pole coordinates (198, 6) and (18,´6) are also well fit, but have a best fitχ2

more than 10% higher than the pole solution given above. It should be noted that Horner et al.(2012) found that a retrograde rotation was best fit to their infrared (IR) models, perhaps leaving the solutions with north pole positions at negative latitudes as more probable options, thus leaving us with a preferred orientation of (198, ´29) ecliptic longitude and latitude.

The amplitude phase coefficientβA, a linear correction factor to the amplitude model, was calculated simultaneously as part of the fit parameters. This value accounts for variation of the object’s amplitude with solar phase angle4

and was found to be best fit at about 0.002 mag/deg. The amplitude phase coefficient found here is similar to values found in the literature for other objects. For example, Magnusson (1986) found values for this phase coefficient for 18 main belt objects with results ranging between 0.01 and 0 mag/deg with an average value of about 0.003 mag/deg.

These well-defined pole orientation and axial ratio results found using this amplitude- aspect model agree well with the first-order approximations that can be determined from the simple ∆m-longitude model shown in Figure 4.2.

4

βA is a low-phase linear approximation of the non-linear effect that a changing phase angle has on an

object’s photometric amplitude. Phase angle affects the apparent amplitude of an object because the ratio of the projected surface area of the visible, gibbous portion of the object to the full projected area of the facing side is not a constant, but instead depends on the size of the full projected area. The concept is explored in much more detail byZappala et al.(1990).

111 194 196 198 200 202 Pole Longitude (o ) -40 -30 -20 -10 0 10 20 Pole Latitude ( o) 14 16 18 20 22 Pole Longitude (o ) -20 -10 0 10 20 30 40 Pole Latitude ( o ) 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 Axis Ratio b/c 1.61 1.62 1.63

Axis Ratio a/b

0.000 0.002 0.004 0.006 0.008 Phase Coefficient (mag/deg) 0 5 10 15 20 25 % χ

2 Above Best Fit

Best Fit

χ

2: 1.24

Figure 4.3: Results of a grid search for parameters inEquation 4.1 for an amplitude model

with red indicating preferred results and green representing aχ2

about 10% higher than the best fit value (top right). We find Anchises’ axis ratios a/b and b/c (top left) to be best fit at 1.62˘0.002 and 0.91˘0.1 respectively. A preferred phase coefficient of 0.002 mag/deg is indicated by the top middle plot. Pole longitude and latitude (bottom) were found to be (198,´29) or (18,29) with errors of˘1.5˝ _{in longitude and ˘5}˝ _{in latitude. The two bottom}

frames are clipped to show only the best fit regions, which are reflections of each other (180˝

in longitude and about the ecliptic in latitude.)