PRELIMINARY POLE AND SHAPE MODELS FOR THREE NEAR-EARTH ASTEROIDS

Abstract

Observations of three near-Earth asteroids (NEAs) were made between 1993 and 2016. The resulting data were used to find preliminary pole and shape models for 1863 Antinous, (5836) 1993 MF, and (154244) 2002 KL6.

CCD photometric observations of the near-Earth asteroids (NEAs) 1863 Antinous, (5836) 1993 MF, and (154244) 2002 KL6 were made by the authors at various observatories from 1997 to 2016. Additional data from Wisniewski et al. (1997) were also used. Table I gives the specific observing circumstances. We report on efforts to model pole and shape models for the three objects.

Table I.

Observing circumstances and synodic period results. Pts is the number of data points used in the analysis. The phase angle (α) is given for the first and last date (0h UT). L_PAB and B_PAB are, respectively, the phase angle bisector longitudes and latitudes for the first and last date (0h UT). The Obs column gives the observatory involved: LO = Lowell Observatory (Koehn et al. 2014); OO = Ondrejov Observatory (Pravec et al. 1999web; PDS = Palmer Divide Station (Warner 2016a, 2017, this work); WEA = Warner et al. (2016); WIS = Wisniewski et al. (1997).

Number	Name	yyyy mm/dd	Pts	Phase	LPAB	BPAB	Period	P.E.	Amp	A.E.	Obs
1863	Antinous	1999 08/07–09/14	336	43.8,13.3	4, 6	−3,+7	7.4568	0.0017	0.23	0.02	OO
		2016 02/07–02/16	450	33.3	177,182	+ 3,+l	7.453	0.005	0.18	0.02	PDS
		2016 08/06–08/10	159	42.8,40.3	6,7	−1, 0	7.471	0.005	0.33	0.02	PDS
		2016 10/03–10/06	251	7.7,8.4	7	+ 11	7.443	0.006	0.13	0.02	PDS
5836	1993 MF	1993 09/15–09/15	130	31.2	8	+ 19.2	4.96	0.01	0.53	0.02	WIS
		1997 10/23–11/04	116	22.9,16.0	67	+ 1,0	4.9543	0.0002	0.74	0.03	OO
		2016 06/05–06/09	157	53.6,55.2	304,308	+ 10	4.948	0.005	0.82	0.03	PDS
		2016 09/09–09/11	212	46.1,44.9	31	+ 11	4.953	0.005	0.88	0.02	PDS
		2016 10/14–10/18	337	19.4,15.9	40,41	+ 7,+6	4.955	0.001	0.74	0.02	PDS
154244	2002 KL6	2009 06/18–07/13	151	43.7,74.5	293,335	+12,+15	4.605	0.002	1.15	0.01	LO
		2009 09/24–09/26	104	35.4,33.3	31,32	+ 3	4.610	0.002	0.90	0.05	LO
		2009 10/16–10/22	82	12.5,6.6	35	+ 1,0	4.6081	0.0003	0.85	0.05	LO
		2016 06/10–06/27	1086	13.6,29.6	254,264	+8,+14	4.60869	0.00005	0.65	0.02	WEA
		2016 09/09–09/11	210	41.3,39.0	14	+ 7	4.609	0.005	0.98	0.02	PDS
		2016 10/14–10/19	228	3.4,1.8	23, 24	+ 2,+l	4.607	0.001	0.47	0.03	PDS

The minimum timespan between the earliest and latest data sets was about seven years (2002 KL6), while the longest timespan was nearly 23 years (1993 MF). Such long baselines of dense-in-time lightcurve data (DITD) are highly beneficial when attempting to find an accurate sidereal rotation period using the lightcurve inversion techniques developed by Kaasalainen and Torppa (2001) and Kaasalainen et al. (2001).

Sparse-in-time data (SITD) can be combined with the dense data to further enhance the inversion modeling, usually by providing data at a wide range of phase angle bisector longitudes (L_PAB; see Harris et al., 1984). Kaasalainen (2004) explored this possibility and found that if the sparse data were of sufficient quality and covered the full range (or nearly so) of longitudes, i.e., 0–360°, a reasonably accurate pole and even shape model solution could be derived. This concept was further explored by Hanus and Durech (2012) by using sparse data from the Catalina Sky Survey.

In the past few years, a number of papers (e.g., Durech et al. 2009; Hanus et al. 2011; 2013; 2016) have used both dense and sparse data to produce pole and shape models for almost 1000 asteroids. It’s worth noting that most of those papers include a very large list of coauthors who are often backyard astronomers (“amateurs”). This emphasizes the constant need of high-quality data from that large group of observers who have considerable flexibility in making observations. It’s also important to note that many of the 1000 models so far are for relatively bright asteroids, i.e., within easy reach of modest telescopes, and that – despite having good dense data sets for a given asteroid – “more is often better.”

Modeling Considerations and Restrictions

It is tempting to try to model an asteroid after obtaining a single dense lightcurve and merging in sparse data. Don’t expect to get good results, even if they appear to be so. The inversion process needs as much information as possible about how the asteroid’s lightcurve changes with different phase angles and viewing aspects, the latter judged by the phase angle bisector longitude.

As mentioned above, any sparse data points should be evenly distributed about most of the asteroid’s orbit. Just as important is that the dense lightcurves are not all within a small range of L_PAB values and that they not be diametrically opposed, i.e., about 180° separation in L_PAB. Otherwise, there is little new information about the pole or shape from one dense lightcurve over another.

Usually, dense lightcurves are needed from several apparitions at different viewing aspects before a good model can be found, and it’s particularly useful to have dense lightcurves at different phase angles during a single apparition such that the shape and/or amplitude changes significantly. Slivan (2012; 2013) discussed these requirements in detail; those papers should be mandatory reading before one starts modeling.

There is a possible exception to the multi-apparition requirement: when a near-Earth asteroid passes Earth and goes through a wide range of both phase angles and L_PAB values. In this case, especially if there are sparse data from other apparitions available, it may be possible to get a useful model. An example of this was the 2001 apparition of (5587) 1990 SB when Koff et al. (2002) obtained data over several months. Using their data alone, Warner (unpublished) was able to find a shape and pole model with a high degree of confidence and which compared favorably to one found by Kaasalaien et al. (2004) using additional dense and sparse data. It was this successful effort that initially prompted the attempts to model the three asteroids presented in this paper.

The “10 percent rule” is used to determine the quality of the period and pole solutions. In a perfect world, there would no other periods within 10% of the one with the lowest χ² value. However, most times there are several, even dozens, of points within 10% of the lowest χ² value. In these cases, the hope is for a single, well-defined minimum in the period search curve. In this case, the period with the lowest χ² value is used in the pole search. If the minimum is a broad curve, then – even if there are only a few data points below the 10% line – the sidereal period may still be too uncertain. The best solution is to get more data at future apparitions (or data from past ones) and so increase the total time span of the observations.

The same reasoning is applied in the pole search. Usually a large number of discreet poles are tested to find the one with the lowest χ² value. If there is only one solution (all others exceed the 10% rule), or there is a small number that are immediately adjacent to that solution, then the best solution based on the available data has been found.

However, the lightcurve inversion process has difficulty finding a unique ecliptic longitude when the asteroid has a low orbital inclination. Therefore, a pole search may have two clusters of discreet regions that are separated by about 180° longitude. The two regions may or may not be on the same side of the equator. This “double mirroring” is also not uncommon. In fact, it’s possible to get four solutions that mirror in both longitude and latitude.

One way to determine which pole solution is the more likely one is to plot the dense lightcurve data against each model’s curve to check which model provides the best fit. This doesn’t always work since no one model shows a significantly better fit. The only reliable way to break the ambiguity is to combine the lightcurve data with other observations such as adaptive optics, occultation profiles, and/or radar observations. The projected profiles of the model onto the sky plane at the time of the additional observations can be compared against the profiles from the other observations and so, with a little luck, resolve the ambiguities.

Another weak point of lightcurve inversion is that, when using data that are not all well-calibrated (they rarely really are), they are treated as “relative data” and so the process cannot accurately confine the Z-axis dimension. In other words, the asteroid may be “taller” or “flatter” than the model indicates. It can also happen where the model has a Z-axis that is that greater than either the X- or Y-axis dimensions. In the extreme, this would be like a pencil rotating about its longest axis instead of its shortest, such as when it is rolling off the desk. It is possible to manipulate the Z-axis in the model so that it is the shortest one and then check the resulting model curve against actual data. However, this is an arbitrary process and does not always succeed. The solution is, again, “More Data!”

Despite all these apparent obstacles, the situation is not hopeless, as confirmed by the approximately 1000 models found to-date. Like many other things, modeling takes preparation, an understanding the process and its limitations, patience, lots of practice, and more than a little luck.

Modeling the Three Candidates

All three candidates were observed at CS3-PDS at least twice during 2016 with the intent of trying to find a pole and shape model. The PDS data were combined with those from the other authors obtained from 1997 to 2016 for the analysis and from Wisniewski et al. (1997). Pravec, Kusnirak, and Ferrero sent their data directly to Warner for the analysis. The dense data for (154244) 2002 KL6 from Lowell Observatory (see Koehn et al. 2014) were downloaded from the ALCDEF web site (http://alcdef.org) and are used with permission.

Table II gives the number of dense lightcurves for each asteroid as well as the number of sparse data points obtained by the Catalina Sky Survey. The dense lightcurves were given a weight of 1.0. The CSS data were given a weight of 0.2 based on the work by Hanus et al. (2013) that evaluated the quality of sparse data from CSS and other surveys.

Table II.

N_dense is the number of dense lightcurves used in the modeling. The sparse data are considered a single lightcurve, so the N_sparse column gives the number of sparse data points.

Asteroid	Ndense	Nsparse
1863	25	111
5836	13	120
154244	39	64

Interpreting the Modeling Plots

There are several plots for each asteroid that were generated by MPO LCInvert, a program that provides a Windows interface to the C/FORTAN code available on the DAMIT web site (http://astro.troja.mff.cuni.cz/projects/asteroids3D/web.php). For each group of plots below, the left-hand plot on the first line shows the L_PAB distribution. Green circles represent the mid-point of each dense lightcurve. Red squares represent each sparse data point. The right-hand plot of the first line shows the period search χ² values versus period. The horizontal green line is at 1.1x the lowest χ² value, i.e., the 10% rule mentioned above.

On the second line, the left-hand image is the 10% pole search plot. Each small region represents a 15×15° area of the sky in ecliptic coordinates. An ideal solution would have only one dark blue region and all the others dark red (maroon). The additional image on the second line shows profiles of the asteroid model as view from over the north and south poles and in the asteroid’s equatorial plane at Z = 0° and Z = +90° rotation.

Since the inversion algorithms used here find only a convex hull, concavities appear as large flat areas on the asteroid. A good analogy is to imagine wrapping the asteroid with paper without punching holes in the paper to fill depressions. This creates flat areas in the wrapping that are covering concavities. Towards the end of this paper, we discuss some specific interpretations of these flat areas for the models that were found.

The remaining plots compare the dense data with the model curve at specific times. The red dots represent the dense data while the solid black line is the model lightcurve. The main consideration is that the dense lightcurve has the same amplitude and shape as the model. If there are multiple pole solutions, then the one that gives the best fits for all the dense lightcurves is the more likely solution – assuming it has a realistic shape.

1863 Antinous.

The main obstacles for this asteroid were the dense lightcurves being almost diametrically opposed in L_PAB and the poor distribution in L_PAB values. Even with high-quality dense lightcurves, a period search with a range of ±0.02P, about 7.3 to 7.6 h in this case, would be the norm. This would have taken several days and so the search range was limited to 7.43 to 7.49 h. This covered all the reported synodic periods. Even so, the search took more than five hours. The large number of results below the 10% line in the period search plot is a good indication that the model could be improved significantly with data from previous or later apparitions.

The pole’s ecliptic longitude seems secure and, given the positive ecliptic latitude, it’s likely that the asteroid has a prograde rotation (counter-clockwise when looking down on the North Pole).

(5836) 1993 MF.

Dense lightcurves were available from apparitions in 1993, 1997, and 2016. The 2016 data covered a significant range of L_PAB and phase angles and the sparse data were distributed over about 3/4 of the asteroid’s orbit. Given the success modeling of (5587) 1990 SB mentioned earlier, we first tried to model this NEA using only the 2016 data. When data from all apparitions were used, the pole latitude was about the same, but the favored longitude was opposite the one found from using only the 2016 data.

Initially, only the 1997 and 2016 data were available. These led to two possible solutions at (149°, −73°) and (331°, −53°). When the Wisniewski et al. data from 1993 were added, the period solution was improved by having only one solution below the 10% line.

The pole search found only one solution at (121°, −79°), thus eliminating the second pole solution and reducing the uncertainty for the first. Given the negative latitude, the asteroid’s rotation is retrograde.

(154244) 2002 KL6.

Modeling for this NEA was aided by having dense lightcurves spanning almost seven years (2009–2016) and a large range of L_PAB values. On the negative side was the lack of sparse data at L_PAB values other than within the range of those for the dense data. The former appears to have outweighed the latter, since both the period and pole solutions were found with a high level of confidence.

The elongated shape of the model is reflected in the large amplitudes of the lightcurves at all viewing aspects. As with (5836) 1993 MF, the high, negative ecliptic latitude indicates a retrograde rotation for the asteroid.

About the Derived Shapes and Pole Solutions

As mentioned previously, the lightcurve inversion process employed here does not handle concavities very well. All three asteroid models show flat areas, or straight contours, that suggest concavities. These can be relatively large craters or even a “waist” or “neck” connecting two large lobes. The model for (5836) 1993 MF may represent either a highly-bifurcated body or even a so-called “contact binary.” For (154244) 2002 KL6, one interpretation could be a small component sitting on a larger one.

Observations at high solar phase angles are generally critical for revealing the presence of concavities, although a unique non-convex solution is not possible in such a case. While the maximum solar phase angles at which the three asteroids were observed are not quite extreme, it is most useful to look at the model fits at higher phase angles. For example, in Figure 1 the model-data fit for 1863 Antinous in 2016 August shows what might be systematic residuals due to unmodeled concavities. See also Figure 3, where the fit at about α = 55° in early June for (5836) 1993 MF shows a significant deviation at the second maximum around 0.7 rotation phase. At lower phase angles in Figure 3, the model-data fits are much tighter.

Figure 1.

Modeling solutions and comparison lightcurves for 1863 Antinous.

Figure 3.

Modeling solutions and comparison lightcurves for (5836) 1993 MF.

The pole solution errors are a circle with a radius of ±10°. The sidereal periods have errors on the order of 1–2 units of the last decimal place.

YORP Considerations

The YORP effect (Yarkovsky–O’Keefe–Radzievskii–Paddack; Rubincam 2000) is the thermal re-radiation of sunlight by a body that can impart a small torque and so cause that body’s rotation to increase or decrease over time. The YORP effect increases with decreasing distance from the Sun, making NEAs the most likely candidates to find YORP acceleration after only a decade or so, as has been shown by, among others, Lowry et al. (2007), Durech et al. (2008), and Kaasalainen et al. (2008).

The change in rotation period due to YORP increases quadratically with time, meaning that a plot of sidereal period versus time would not show a linear but exponential trend. This implies that at least three data points are needed to help determine if an asteroid’s rotation rate is being influenced by YORP. Save for 1993 MF, our dense data sets were from only two apparitions. For those two, even if the modeling code allowed factoring in YORP acceleration, there would be little point in trying. Should additional data become available, either within or outside the range of dates covered by our data sets, then it may be worthwhile to revisit these two asteroids to see if there are signs of YORP acceleration.

The ALCDEF Database

The Asteroid Lightcurve Data Exchange Format (ALCDEF) database (http://alcdef.org) was used to obtain some of the data used for this paper. The web site provides access to a database of raw asteroid time-series (lightcurve) photometric data. As of 2017 Mar 29, there were almost 2.89 million observations for 11495 individual asteroids. Those doing asteroid time-series work are encouraged to upload their data for use by other researchers. Those needing raw time-series photometry can freely download data from the website. For a more complete description of the site and links to ALCDEF documentation, see Warner (2016b).

Conclusions

Trying to model an asteroid based on dense data from only one apparition is discouraged. However, it is sometimes possible to obtain a good, preliminary model if 1) there are sufficient sparse data to use in the modeling and/or 2) there are multiple dense lightcurves from the same apparition covering a sufficient range of phase angles and L_PAB values. Dense lightcurves from a single apparition that have nearly the same L_PAB will usually not provide sufficient information for the lightcurve inversion algorithms to find a reliable model.

All this adds up to say that NEAs are the only good candidates for modeling based on single apparition dense lightcurves. The viewing aspects and phase angles for most asteroids farther out, those in the main-belt and beyond, do not change sufficiently during a single apparition to derive a model beyond, at best, a modest level of confidence. These factors point out the importance of obtaining new dense lightcurves for main-belt asteroids even when the period has been well established. The most good comes when those new data are from viewing aspects and/or phase angles significantly different from previous observations.

Figure 2.

Dense lightcurve based on the 2016 Oct PDS data.

Figure 4.

Dense lightcurve based on 2016 Oct PDS data.

Figure 5.

Modeling solutions and comparison lightcurves for (154244) 2002 KL6.

Figure 6.

Dense lightcurve based on 2016 Oct PDS data.

Table III.

Pole solutions. L = ecliptic longitude; B = ecliptic latitude; P = sidereal period. The longitude and latitude errors are on the order of ±10°. The period error is approximately 2 units of the last decimal place. The a/b column gives the ratio of the two longest axes (X/Y) of the model. The a/c column gives the ratio of the longest axis (X or Y) versus the Z-axis.

Number	Name	L1	B1	P1	a/b	a/c
1863	Antinous	108	+56	7.454130	1.09	1.68
5836	1993 MF	121	−79	4.954794	1.30	1.44
154244	2002 KL6	129	−89	4.610233	1.21	2.22