Simultaneous Mass Determination for Gravitationally Coupled Asteroids

The orbit and mass determinations were made using a modified version of the CODES application (Baer 2004), whose integration force model accounts for gravitational perturbations (including first-order relativistic terms), the J₂ oblateness terms of the Sun, Earth, and Jupiter, and solar radiation pressure. In calculating gravitational perturbations, CODES uses the JPL DE430 ephemeris, which provides precise positions of the Sun, planets, and Earth's Moon for the period 1550–2650 (Folkner et al. 2014).

All calculations were based upon the 121, 612, 902 observations of the 437, 832 numbered asteroids in the Minor Planet Center's database as of 2015 June 2. Only observations of numbered asteroids were used, so as to ensure that the orbits were sufficiently well-established that residuals were dominated by observational error. Star catalog biases were removed from the optical observations using the algorithm described in Farnocchia et al. (2015).

Our first step was to create an updated observational error model. Specifically, as described in Baer et al. (2011b), we used the conventional least-squares orbit determination algorithm to calculate the orbits of the numbered minor planets, initially assuming that the observational errors were uncorrelated, with canonical uncertainties of 3 arcsec for optical observations prior to 1890, 2 arcsec for observations from 1890 to 1950, 1 arcsec for observations from 1950 to 2000, and 0.5 arcsec for observations after 2000. As a by-product of the orbit determination process, we obtained the "observed–predicted" observational residuals. These residuals were then binned, based on observatory, observation date (each bin initially being 50 days wide), apparent magnitude, location within the galactic disk, and detector technology, and used to create a statistical analysis of the demonstrated performance of each observatory, including the rms residuals in R.A. and decl. for each bin, and the correlations between closely spaced observations. In the later orbit and mass determinations, this statistical error model would allow us to assign uncertainties in R.A. and decl., and model correlations in the errors of closely spaced observations; as a result, we were able to populate the least-squares observational covariance matrix with realistic values.

Next, we used the new observational error model to recalculate the orbits of the 300 large asteroids that appeared in the JPL DE405 planetary ephemeris due to their potential perturbations of the orbits of the major planets (Standish 1998). These newly recalculated epoch state vectors were then integrated over the timespan 1800–2200, with mutual perturbations modeled using the best available published and estimated masses of the 300 asteroids. The resulting BC430 asteroid ephemeris, combined with the DE430 planetary ephemeris, helped improve our gravitational force model for subsequent calculations.

Potential mass determination encounters were identified using Opik's two-body analysis of planetary encounters, as developed by Carusi et al. (1990) and Valsecchi et al. (2003). As noted by Hilton (2002), the magnitude of the ballistic deflection angle (which is proportional to the mass of the subject asteroid and inversely proportional to the relative encounter velocity and the minimum distance) is often used to select candidate encounters. However, while finding encounters with significant deflections is clearly useful, our experience has demonstrated that relying solely upon the magnitude of the deflection angle is insufficient. Indeed, a significant deflection perpendicular to the orbit plane may not yield observable shifts in sky position near the ascending and descending nodes, while a relatively small along-track deflection may yield an easily observable shift if the observational baseline is sufficient. Therefore, we also calculated the predicted change in mean motion (see Galád 2001 for a similar approach). Combined with the elapsed time since the encounter, this allowed us to estimate the along-track deflection, and thus the predicted change in observed sky position.

Given the need to analyze the mutual encounters between over 440,000 numbered asteroids, we limited consideration to those encounters between a single subject asteroid and a single test asteroid with a relative Minimum Orbital Intersection Distance of 0.05 au or less. This threshold was a trade-off between completeness and processing time; although an exceptionally small relative velocity could yield an observable deflection at a greater distance, such an encounter would have a lengthy duration, thus making it more likely that any signal would be swamped by perturbations from other nearby asteroids. Additionally, we limited the selection to encounters that were within the period of observation of the test asteroid and that resulted in a predicted deflection angle, predicted change in inclination, or predicted change in sky position (accumulated over the test asteroid's period of observation) exceeding 0.1 arcsec, any smaller changes likely being unobservable by current technology. In total, over 31,000 candidate events were identified.

However, many subject/test asteroid pairs had multiple encounters over their shared observation timeframe; as a single mass determination solution applies to every such encounter, redundant events were eliminated. Moreover, the majority of events were due to perturbations by 1 Ceres, 4 Vesta, and 10 Hygiea; after retaining only the most promising such cases, the list was reduced to approximately 5700 candidate subject/test pairs.

Each of the remaining 5700 candidate pairs was processed by the conventional mass determination algorithm. Approximately 4400 cases were rejected based on their low significance (defined as the ratio of the calculated mass to its uncertainty), while approximately 700 others had unrealistic densities. Finally, only the solutions for 1 Ceres, 4 Vesta, and 10 Hygiea with the highest significances were retained. This yielded 242 solutions with an estimated density between 0.5 and 8.0 g per cubic centimeter and a significance greater than 2.0. (Note: We recognize there are non-trivial uncertainties in the dimensions of most main-belt asteroids. Even so, we feel these uncertainties are unlikely to justify retaining cases where the implied density exceeds 8.0 g per cubic centimeter.)

To further assess the validity of each mass solution, we conducted a variational analysis, repeating the runs with slightly different values for the masses of eight key perturbing bodies (Mars, Jupiter, Saturn, 1 Ceres, 2 Pallas, 3 Juno, 4 Vesta, and 10 Hygiea). In each case, the perturber mass was increased by the magnitude of its published uncertainty, and the resulting variant mass of the subject asteroid was compared to that from the nominal solution to determine if that mass was reliable. In a final validation test, the assumed mass of the subject asteroid used in the first iteration of the least-squares algorithm was cut in half, to assess the sensitivity of the final mass to the initially assumed value.

In all, this variational analysis yielded the 228 valid mass determinations for 30 large subject asteroids listed in Table 1 and illustrated in Figures 1–3. For each subject asteroid, we have illustrated the best-fit mass and uncertainty resulting from each separate test asteroid solution; and in cases where more than one test asteroid exists, the weighted average mass and uncertainty are calculated.

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Table 1.  Masses Measured Using the Traditional Method

Subject	Test	Mass	std dev	rms Error
Asteroid	Asteroid	(${M}_{\odot }$)	(${M}_{\odot }$)	(arcsec)	Significance
1	5303	4.73 × 10−10	1.47 × 10−12	0.452	323.2
1	79083	4.71 × 10−10	1.53 × 10−12	0.552	308.1
1	348	4.68 × 10−10	1.59 × 10−12	0.406	294.8
1	46938	4.69 × 10−10	1.64 × 10−12	0.490	285.2
1	13801	4.76 × 10−10	1.83 × 10−12	0.452	259.9
1	60592	4.76 × 10−10	1.85 × 10−12	0.477	256.9
1	85128	4.73 × 10−10	2.75 × 10−12	0.454	172.4
1	454	4.51 × 10−10	2.68 × 10−12	0.473	168.1
1	203	4.68 × 10−10	2.86 × 10−12	0.502	163.9
1	79000	4.74 × 10−10	3.33 × 10−12	0.455	142.5

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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In comparison to our previous work (Baer et al. 2011a), Figures 1–3 contain new mass estimates for eight asteroids (11 Parthenope, 27 Euterpe, 51 Neimausa, 76 Freia, 121 Hermione, 324 Bamberga, 476 Hedwig, and 532 Herculina) and significantly more precise mass estimates for six others (2 Pallas, 3 Juno, 4 Vesta, 9 Metis, 16 Psyche, and 88 Thisbe). These improvements are likely the result of several factors, including improved star catalog debiasing (Farnocchia et al. 2015), the flood of highly precise astrometric observations from modern surveys like Pan-STARRS, and the addition of almost 400,000 more numbered asteroids to the pool of potential test asteroids.

We first note that the weighted average mass for 4 Vesta ($1.322\pm 0.002\times {10}^{-10}\,{M}_{\odot }$) differs from the definitive value ($1.30284\pm 0.00001\times {10}^{-10}\,{M}_{\odot }$) obtained by the Dawn space mission (Russell et al. 2012) by approximately $10\sigma$ (calculated using this paper's uncertainty). This is clearly unacceptable, and suggests that something is amiss. Next, we point out the significant disagreement in the individual masses for 6 Hebe. Specifically, the mass estimates for test asteroids 136568 ($5.96\pm 1.10\,\times {10}^{-12}\,{M}_{\odot }$) and 1700 ($11.37\,\pm 2.21\times {10}^{-12}\,{M}_{\odot }$) differ from one another by approximately $2.2\sigma$ (based upon the weighted mass uncertainty), despite the two mass estimates having nearly identical significance. The other mass estimates continue this same pattern, clustering around two distinct values, such that the weighted average uncertainty is unrealistically smaller than the variation in the individual mass estimates themselves. Significant disagreements can also be found in the individual mass estimates for 7 Iris, 9 Metis, 11 Parthenope, 20 Massalia, 29 Amphitrite, 39 Laetitia, 52 Europa, 88 Thisbe, and 511 Davida.

A potential explanation emerges from a review of our two-body encounter modeling, which shows that the large asteroids 7 Iris and 88 Thisbe actually encountered one another eight times within the observational baseline (see Table 2). These eight encounters yielded a mass for 88 Thisbe of high significance, implying that 7 Iris was significantly perturbed by the cumulative encounters; and because 7 Iris and 88 Thisbe are of similar size, it is likely that 88 Thisbe was also significantly perturbed by these encounters.

Both 7 Iris and 88 Thisbe are included in the gravitational force model. However, their masses and epoch state vectors were determined sequentially. That is, our previously published masses were used to calculate their epoch state vectors and BC430 ephemerides; and the resulting BC430 ephemeris was then used to model the perturbations in calculating the new weighted average masses in Table 1. No attempt was made to simultaneously solve for the combination of epoch state vectors and masses that minimized the rms errors over the full observational data set. Indeed, many of the encounters that 7 Iris and 88 Thisbe experienced with their individual test asteroids lay within the same 1879–2008 timeframe as their encounters with one another; so if the trajectories of 7 Iris and 88 Thisbe were sub-optimal, the resulting test asteroid deflection geometries might also be modeled imperfectly, thus explaining the puzzlingly inconsistent mass estimates.

In general, we would describe two or more subject asteroids (like 7 Iris and 88 Thisbe) as being gravitationally coupled if their mutual perturbations alter the subsequent encounter geometries with their own test asteroids. Inconsistencies in their individual mass determinations may be a signature, suggesting the need for a simultaneous least-squares orbit/mass solution involving as many of the subject and test asteroids as feasible, in which the epoch state vectors of the subject asteroids would be included as solve-for parameters.

A review of our two-body encounter modeling also showed that 6 Hebe passed within 0.069 au of 51 Nemausa in 1981. Although 51 Nemausa is only about one-third the mass of 6 Hebe, given the long observational baseline of each asteroid, we suspected that 6 Hebe might have been sufficiently perturbed by the encounter that the geometries of subsequent test asteroid encounters may not have been modeled with adequate precision. Similar remarks apply to asteroids 29 Amphitrite, 39 Laetitia, and 52 Europa, as well as to asteroids 511 Davida and 532 Herculina.

Finally, both 4 Vesta and 11 Parthenope experienced close encounters with 17 Thetis that yielded masses of high significance, suggesting that 17 Thetis was significantly perturbed by each encounter. Specifically, Table 2 shows that 11 Parthenope and 17 Thetis passed within 0.0016 au in 1968; 17 Thetis subsequently passed within 0.0194 au of 4 Vesta in 1996 June and then passed within 0.0054 au of 11 Parthenope seven months later. Additionally, the orbits of 11 Parthenope and 17 Thetis are near 1:1 resonance, which may further enhance their mutual perturbations. Thus, the trajectories of these three large asteroids appear entangled, and given the poor agreement of 4 Vesta's weighted average mass with the Dawn space mission value, as well as the significant variation in calculated masses for 11 Parthenope, we felt that a simultaneous least-squares orbit/mass determination involving all three asteroids was likely needed.

Our mass determinations using the simultaneous algorithm are presented in Table 3, along with the corresponding weighted averages from the conventional algorithm. We calculated simultaneous mass estimates for each subject asteroid in Figures 1–3, utilizing as many test asteroids as possible in each solution. In some instances, additional solutions were run in which the subject asteroid was instead a subject/test asteroid, so that we could evaluate the extent to which including its epoch state vector as a solve-for parameter impacted its mass estimate.

Table 3.  Conventional vs. Simultaneous Mass Determinations

		Measured	Mass
Subject		Mass	Uncertainty
Asteroid	Method	(in ${M}_{\odot }$)	(in ${M}_{\odot }$)	Difference	Test Asteroids
1	Mean	4.718 × 10−10	6.66 × 10−13		5303, 79083, 348, 46938, 13801, 60592, 85128, 454,
					203, 79000, 91, 23352, 73504, 70051, 44867, 58
1	Simult.	4.724 × 10−10	6.6 × 10−13	0.69σ	5303, 79083, 348, 46938, 13801, 60592
2	Mean	1.06 × 10−10	3.26 × 10−12		106329, 582, 36741, 48343, 66300, 69508, 4971,
					16527, 52863, 18067, 5634, 5930, 3131, 381606,
					102850, 1391, 84676, 131629, 124006, 241751, 24314,
					44087, 2505, 5019, 66875
2	Simult.	1.05 × 10−10	4.25 × 10−12	0.23σ	106329, 582, 36741, 48343, 66300, 69508, 4971,
					16527, 52863
3	Mean	1.41 × 10−11	8.71 × 10−13		7201, 960, 64587, 61235, 20199, 12614, 17715,
					61232, 10383
3	Simult.	1.61 × 10−11	1.17 × 10−12	1.35σ	7201, 960, 64587, 61235, 20199, 12614, 17715,
					61232, 10383
4	Mean	1.322 × 10−10	2.36 × 10−13		17, 197, 8331, 125655, 5205, 67, 56483, 6333, 408741,
					77, 234504, 486, 234479, 157127, 33427, 112237, 3057,
					109, 6027
4	Simult.	1.291 × 10−10	6.54 × 10−13	4.40σ	197, 8331, 125655, 5205, 67, 56483
4	Simult.	1.300 × 10−10	3.46 × 10−13	5.24σ	4*, 11*, 17*
6	Mean	7.09 × 10−12	7.15 × 10−13		136568, 1700, 5295, 13152, 4497, 1150
6	Simult.	6.26 × 10−12	3.46 × 10−13	1.04σ	6*, 136568, 1700, 5295, 13152
6	Simult.	6.14 × 10−12	2.19 × 10−13	1.27σ	136568, 1700, 5295, 13152, 4497, 1150
6	Simult.	5.98 × 10−12	4.00 × 10−13	1.36σ	6*, 51*, 136568, 1700, 287
7	Mean	8.34 × 10−12	4.17 × 10−13		571, 52443, 1188, 46529, 17186, 6791, 1879, 56,
					273287, 253028, 836, 1651, 269649
7	Simult.	8.28 × 10−12	4.64 × 10−13	0.10σ	571, 52443, 1188, 46529, 17186
7	Simult.	8.38 × 10−12	6.00 × 10−13	0.04σ	7*, 88*, 571, 52443, 7629
8	Mean	4.75 × 10−12	3.63 × 10−13		967, 330
8	Simult.	5.04 × 10−12	4.82 × 10−13	0.48σ	967, 330
9	Mean	3.91 × 10−12	3.24 × 10−13		36653, 29818, 343, 7365, 8232, 94355, 29663, 20
9	Simult.	3.87 × 10−12	3.34 × 10−13	0.07σ	36653, 29818, 343, 7365, 94355, 29663
10	Mean	4.38 × 10−11	6.95 × 10−13		3946, 24433, 192839, 15187, 6143, 11215, 283315,
					75794, 111, 17109, 45496, 209, 230964, 10380,
					492, 48499, 5941, 169965, 13266, 32086, 245317,
					1287, 109507, 57493, 1259, 28872, 820, 22236, 55251,
					20905, 2619, 2824, 579
10	Simult.	4.12 × 10−11	8.13 × 10−13	2.40σ	3946, 24433, 192839, 15187, 6143, 11215
11	Mean	2.40 × 10−12	7.37 × 10−14		17, 85371, 113, 153608, 8964
11	Simult.	2.78 × 10−12	6.07 × 10−14	3.93σ	4*, 11*, 17*
11	Simult.	2.92 × 10−12	3.07 × 10−13	1.64σ	85371, 113, 153608, 8964
15	Mean	1.53 × 10−11	2.56 × 10−13		411232, 50278, 1284, 14401, 19524, 48551, 16693,
					136560, 1313, 45690
15	Simult.	1.56 × 10−11	1.47 × 10−13	1.12σ	411232, 50278, 1284, 14401
16	Mean	1.15 × 10−11	3.07 × 10−13		13206, 211012, 39054, 91495, 156553, 151878, 94,
					62211
16	Simult.	1.15 × 10−11	3.50 × 10−13	0.02σ	13206, 211012, 39054
17	Indiv.	5.60 × 10−13	4.58 × 10−14		11
17	Simult.	4.52 × 10−13	4.72 × 10−14	1.65σ	4*, 11*, 17*
19	Mean	5.53 × 10−12	2.79 × 10−13		135, 3486, 114525, 27799, 228485, 113990
19	Simult.	5.54 × 10−12	3.18 × 10−13	0.02σ	135, 3486, 114525
20	Mean	2.48 × 10−12	3.88 × 10−13		44, 60658, 49248, 253970, 3175
20	Simult.	2.69 × 10−12	4.16 × 10−13	0.36σ	44, 60658, 49248, 253970, 3175
27	Indiv.	2.17 × 10−12	6.08 × 10−13		63625
27	Simult.	1.92 × 10−12	6.99 × 10−13	0.27σ	63625
29	Mean	6.97 × 10−12	2.77 × 10−13		987, 6904, 43142, 77424, 359, 1551, 236858, 48350,
					321197, 48464
29	Simult.	7.10 × 10−12	3.07 × 10−13	0.31σ	987, 6904, 43142, 77424, 359, 1551
29	Simult.	6.77 × 10−12	3.24 × 10−13	0.48σ	29*, 39*, 52*, 124, 306, 987, 6904, 7060, 311771
39	Mean	4.58 × 10−12	1.09 × 10−12		7060, 311771
39	Simult.	3.54 × 10−12	1.33 × 10−12	0.61σ	7060, 311771
39	Simult.	4.30 × 10−12	9.30 × 10−13	0.20σ	29*, 39*, 52*, 124, 306, 987, 6904, 7060, 311771
51	Mean	2.07 × 10−12	3.05 × 10−13		287, 53824, 33850
51	Simult.	2.41 × 10−12	4.24 × 10−13	0.65σ	287, 53824
51	Simult.	2.12 × 10−12	3.79 × 10−13	0.09σ	6*, 51*, 136568, 1700, 287
52	Mean	1.53 × 10−11	7.80 × 10−13		124, 306, 1023, 993
52	Simult.	1.46 × 10−11	7.88 × 10−13	0.60σ	124, 306
52	Simult.	1.48 × 10−11	7.73 × 10−13	0.48σ	52*, 124, 306
52	Simult.	1.45 × 10−11	7.80 × 10−13	0.69σ	29*, 39*, 52*, 124, 306, 987, 6904, 7060, 311771
65	Mean	8.05 × 10−12	1.12 × 10−12		526, 70954, 1668, 226534
65	Simult.	7.52 × 10−12	9.02 × 10−13	0.37σ	65*, 526, 70954, 1668
65	Simult.	7.83 × 10−12	2.61 × 10−13	0.19σ	526, 70954, 1668, 226534
76	Indiv.	2.13 × 10−12	7.50 × 10−13		10383
76	Simult.	2.23 × 10−12	8.29 × 10−13	0.09σ	10383
88	Mean	5.24 × 10−12	9.06 × 10−14		7, 7629, 142429, 11701
88	Simult.	6.91 × 10−12	8.43 × 10−13	1.98σ	7*, 88*, 571, 52443, 7629
121	Mean	2.73 × 10−12	4.89 × 10−13		278, 5750
121	Simult.	2.78 × 10−12	5.25 × 10−13	0.07σ	121*, 278, 5750
324	Mean	6.02 × 10−12	7.72 × 10−13		15402, 31560, 310585, 1939, 24553, 158472
324	Simult.	7.00 × 10−12	4.25 × 10−13	1.11σ	324*, 15402, 31560, 310585
324	Simult.	6.54 × 10−12	2.95 × 10−13	0.62σ	324*, 15402, 310585
476	Indiv.	9.67 × 10−13	2.91 × 10−13		187901
476	Simult.	9.30 × 10−13	1.96 × 10−13	0.10σ	476*, 187901
511	Mean	2.19 × 10−11	1.15 × 10−12		1550, 532, 50529, 203, 17311, 55696, 508
511	Simult.	1.81 × 10−11	1.09 × 10−12	2.38σ	511*, 532*, 1550, 50529, 203, 191878, 17006
532	Mean	1.72 × 10−11	3.11 × 10−12		191878, 17006
532	Simult.	1.62 × 10−11	1.80 × 10−12	0.29σ	511*, 532*, 1550, 50529, 203,
					191878, 17006
704	Mean	2.21 × 10−11	1.03 × 10−12		1467, 95, 10034, 48500, 7461, 253,8624, 242001,
					43993, 340622, 101416,51293, 15856, 79476
704	Simult.	2.20 × 10−11	1.19 × 10−12	0.06σ	1467, 95, 10034, 48500, 7461, 253
804	Indiv.	2.53 × 10−12	6.21 × 10−13		1002
804	Simult.	1.89 × 10−12	6.03 × 10−13	0.74σ	804*, 1002
804	Simult.	1.90 × 10−12	6.04 × 10−13	0.73σ	1002

Note. The differences in measured masses are for the simultaneous solution relative to the mean solution for the subject asteroid. The final column lists the test asteroids in each solution; those with asterisks were subject/test asteroids, which also had their masses included as solve-for parameters.

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Most importantly, however, we also calculated simultaneous mass estimates for the gravitationally coupled cases noted in Table 2. Because our encounter analysis indicated that the subject asteroids perturbed one another, their masses and epoch state vectors were included as solve-for parameters, in addition to the epoch state vectors of the most sensitive test asteroids. As explained above, these cases were the most computationally challenging, and we were not always able to include as many test asteroids as desired.

For 14 of the 18 non-gravitationally coupled asteroids in Table 3, the simultaneous algorithm yielded masses that were within 1σ of the conventional weighted averages. Moreover, in two cases (asteroids 65 Cybele and 804 Hispania), adding the subject asteroid's epoch state vector as a solve-for parameter in the simultaneous solution yielded little apparent benefit. Indeed, in the case of 65 Cybele, adding the epoch state vector as a solve-for parameter more than tripled the mass uncertainty.

However, for 7 of the 12 asteroids in the gravitationally coupled groupings listed in Table 2 (asteroids 4 Vesta, 6 Hebe, 11 Parthenope, 17 Thetis, 88 Thisbe, and 511 Davida), the difference between the simultaneous and weighted average mass estimates was 1σ or greater. Without ground truth measurements of the actual masses of these asteroids, we are unable to conclude that the simultaneous solutions are uniformly superior. Nevertheless, as Table 4 illustrates for the 29-39-52 case, the ${\chi }^{2}$ value for each subject/test asteroid in the simultaneous solution is smaller, and the sum of ${\rm{\Delta }}{\chi }^{2}$ for the test particles is markedly lower than for the conventional solution, indicating that the simultaneous solution for these three asteroids is more consistent with the observational data.

Table 4.  ${\chi }^{2}$ Values for the Simultaneous and Conventional 29-39-52 Solutions

	${\chi }^{2}$	${\chi }^{2}$	${\rm{\Delta }}{\chi }^{2}$
Asteroid	Simultaneous	Conventional	(Sim–Con)
29	2696.35	2697.14	−0.79
39	6365.14	6366.86	−1.72
52	3521.21	3521.28	−0.06
124	4312.93	4288.24	24.69
306	3327.86	3484.94	−157.08
987	2606.41	2530.30	76.11
6904	1145.77	1143.35	2.42
7060	2059.90	2373.24	−313.34
311771	144.67	145.98	−1.31

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We also believe it is significant that the simultaneous solution for the mass of 1 Ceres ($4.724\pm 0.007\times {10}^{-10}\,{M}_{\odot }$) matches the Dawn space mission value (Preusker et al. 2016) to better than $0.01\sigma$, while the weighted conventional value is off by approximately $0.88\sigma$. In this instance, the availability of high-precision ground truth appears to demonstrate the advantage of a true least-squares solution that minimizes the rms residuals over the observations of all asteroids in the solution, as compared to a simple weighted average.

In the case of 4 Vesta, two simultaneous solutions were performed. The first nearly duplicated the conventional solution, except that 17 Thetis was excluded as a test asteroid. By contrast, the second solution focused on the gravitational coupling between 4 Vesta, 11 Parthenope, and 17 Thetis, simultaneously solving for both the epoch state vector and mass of all three asteroids. Significantly, this 4-11-17 solution yielded a mass for 4 Vesta considerably closer (0.94σ) to the Dawn space mission value than either the weighted conventional value (9.58σ) or the first simultaneous estimate (1.81σ).

To summarize, the simultaneous mass determination algorithm appears to provide significant advantages over the conventional algorithm when asteroids are gravitationally coupled, or in cases where especially high precision is required.