Dunham

About the recent and upcoming Kleopatra occultation events.
By Roger Venable

I have projected the DAMIT shape model against the path as drawn by Occult Watcher for 4 occultation events by 216 Kleopatra. The shape model is available on the DAMIT website at
http://astro.troja.mff.cuni.cz/projects/asteroids3D/web.php?page=project_main_page . By entering the accurate Julian date of the event, the web program creates a view of the shape model projected as it should appear on the sky at that time.

The DAMIT models do not model the size of the asteroid, but only the shape and spin axis. The size can be estimated in several ways, but occultations are probably the most accurate way. A recent article reviewed Kleopatra's light curve inversion models and occultation data together, and derived a new shape model incorporating both. See Shepard MK, Timerson B, et al. (2018). "A revised shape model of asteroid (216) Kleopatra." Icarus 311:197-209. These researchers concluded that the long axis of Kleopatra is 276 km, and the two orthogonal axes are 94 and 76 km. These represent the maximum dimensions along those axes, which in the case of Kleopatra means that the two smaller axial dimensions are the measurements of one or both bulbs at the ends of the asteroid. They estimated a maximal possible error of 15% for each of the three dimensions. Interestingly, these authors estimated the rotational pole uncertainty at +/- 5 degrees.

In looking at the DAMIT model sky projections, I find that it is beyond my skill to decide which of the end bulbs and which dimension of them are respectively 94 and 76 km. So, I used the mean of the two (85 km) as the size to which I would adjust the sky projected model in projecting it onto the path. Note that, in order to show the shape model projected onto an Earth map rather than onto the sky, the shape model must be flipped horizontally.

Here is the path of the shadow across Georgia, USA, on September 21st, 2018, with the shape model incorporated into it at the same scale as the map. The green lines parallel to the path, except the centerline's green line, are the occultation periods that I recorded, though the length of the 3rd one from the top may have to be adjusted after further analysis. The small reds spots are the locations of my recording sites. Note that my southernmost station recorded a miss, represented here as a thinner green line. The path width is 142 km.
By Roger Venable

There may be a slight path shift to the north of the predicted path, amounting to about one tenth of a path width. However, the striking thing about this diagram is that the asteroid as documented by the occultation chords is tilted with respect to the prediction. I estimate that the shape model is rotated counterclockwise by 15 degrees, compared to the shape that the occultation suggests. As you will see below, the tilts of two the shape models for upcoming events are similarly nearly parallel to the direction of motion along the path, while the asteroid's projection onto Scotland for the November 7th event is nearly perpendicular to the path.

When I first looked at these projections, I made a mistake in the Julian date of the occultation of November 7th, and therefore mistakenly thought that for that event, in addition to the October 28th and December 7th events, the orientation of the long axis would be nearly parallel to the direction of motion along the occultation path. This caused me to think that we were looking at the asteroid from near the plane of its rotational equator. David Herald noticed that the shape model for the November 7th event was quite different from that which I found, and this is a corrected version of my analysis. I no longer think that these shape models support the notion that we are looking at the asteroid from near its equatorial plane.

Nevertheless, this is among the most interesting of asteroids by virtue of its shape and its moons. Occultation observations are highly desirable.

On Sunday AM, Oct 28, 2018, the path width at the Catskill Mountains is 147 km. In the figure, the projected shape of the model is superimposed onto the path, but the suspected one-tenth path width shift northward and the 15 degree counterclockwise rotation have NOT been applied. The asteroid's shape, which includes its orientation, has been deformed to illustrate the angle of projection onto the surface of the Earth:

Observers may wish to apply a one-tenth pathwidth northward shift as well as a 15 degree clockwise rotation to the projection. Regardless, it appears that, on October 28th, observers should concentrate on the path and not the uncertainty zones, despite the 276 km length of this asteroid.

I have been asked by an observer in Scotland to give advice for the planning of observing locations. On Tuesday night, Nov 6th-7th, 2018, the path width at Dundee, Scotland is drawn by Occult Watcher at about 132 km. However, the shape model projected onto the path requires substantial deformation due to the 64 degree distance of Dundee from the sub-asteroid point. The corrected view shows that the asteroid is NOT oriented with its long axis parallel to the path. This refutes the thesis that I previously made about our viewing angle being near the equatorial plane of the asteroid:

There are many persons who have already claimed chords for this event. Observers should cover the one- and two-sigma error zones, and even a wider path than that. Also, observers may care to apply a one-tenth path width shift to the north, and a slight counterclockwise rotation, due to the asteroid's presentation on September 21 as described above. These two corrections are not incorporated into the diagram.

The next event, on Thursday PM, Dec 6, 2018, has a path width at Dundee, Scotland, of 188 km. Similarly, the suspected 15 degree counterclockwise rotation and one-tenth pathwidth northward shift have NOT been applied, though the deformation of the shape due to projection onto Earth's surface at Scotland has been applied:

Note that the long axis of the asteroid is oriented roughly parallel to the path. Due to the elongation by the projection onto Earth's surface, a counterclockwise rotation should be 9 degrees instead of 15.

It will be very interesting to see how the occultaton results fit the shape model projections, when plotted onto Earth's surface. These, of course, will be quite different from the plots onto the sky, both because of the distortion that the Earth's surface causes to the shape, and because of the horizontal flipping that must be done when plotting it onto Eath's surface. Everyone's observation, whether a "hit" or a "miss", will contribute to the shape information. I wish I could be in Scotland. "♫ Oh my heart is still in Scotland, where the lassies woo the best. . . . ♪".

-- Roger Venable
October 22, 2018

credits: Hristo Pavlov, author of the Occult Watcher program, which was used to create the background maps.
Josef Ďurech and Vojtěch Sidorin for their work on the DAMIT models.
Alan Lerner, author of the lyrics to Camelot, for his naughty song, "Fie on goodness."