U.S. Naval Observatory Ephemerides of the Largest Asteroids

The U.S. Naval Observatory has produced a set of ephemerides for 15 of the largest asteroids for use in the Astronomical Almanac. The ephemerides will be available in computer readable format for public use and cover the period from 1800 through 2050. The internal uncertainty in the mean longitude at epoch ranges from 0."05 for 7 Iris through 0."22 for 65 Cybele and the uncertainty in the mean motion varies from 0."02 cen.^-1 for 4 Vesta to 0."14 cen.^-1 for 511 Davida. This compares very favorably with the internal errors for the outer planets in DE200. However, because the asteroids have relatively little mass and are subject to perturbations by other asteroids the actual uncertainties in their mean motions are likely to be a few tenths of an arc second per century, 3 Juno is a particular case in point.

As part of improving the ephemerides, new masses and densities were determined for 1 Ceres, 2 Pallas, and 4 Vesta, the three largest asteroids. These masses are: Ceres = (4.35 ± 0.05) × 10^-10 M_Sun, Pallas = (1.60 ± 0.04) × 10^-10 M_Sun, Vesta = (1.52 ± 0.09) × 10^-10 M_Sun. The mass for Ceres is smaller than most previous determinations of its mass. This smaller mass is a direct consequence of the increase in the mass determined for Pallas. The densities found are 1.98 ± 0.03 gm. cm^-3 for Ceres, 4.4 ± 0.2 gm. cm^-3 for Pallas and 3.9 ± 0.3 gm. cm^-3 for Vesta. The density for Ceres is somewhat greater than that found for the taxonomically similar 255 Mathilde, but this is probably a result of compaction by Ceres' much greater mass rather than a basic difference in composition. The similarity in the densities of Pallas and Vesta supports the conclusion of a 28-color multivariate analysis of 84 asteroids that Pallas and Vesta are taxonomically alike.

Comparison of the USNO/AE97 ephemerides of Ceres, Pallas, Juno, and Vesta with the Duncombe (1969) shows some differences that are directly attributable to the inclusion of asteroid perturbations in the physical model.

This document is designed for the reader who wants to know the details of the reduction of the data and determination of the ephemerides. A paper summarizing the process for the general reader is in preparation.

Introduction

Data

WIDE ANGLE DATA

• Hipparcos Data

RELATIVE DATA

RADAR DATA

DATA WEIGHTING

Physical Model

ASTEROID MASSES

• Asteroid Densities

Ephemerides

RESIDUALS

• Juno

COMPARISON WITH OLD EPHEMERIDES

Conclusions

References

There are many sources currently available for asteroid ephemerides. Among them is the Astronomical Almanac with ephemerides for 1 Ceres, 2 Pallas, 3 Juno, and 4 Vesta. Historically these four asteroids have been observed more than any of the others. Even in modern times few asteroids have had more attention paid to them. Ceres, Pallas, and Vesta deserve such attention because they are the three most massive asteroids, the source of significant perturbations of the planets, and are among the brightest main belt asteroids making them prime targets for photometric and spectroscopic studies aimed at understanding the composition of the asteroids.

The ephemerides currently published in the Astronomical Almanac are based on the rather old work of (Duncombe 1969). These ephemerides extend only until January 7, 2000. As a result, the U.S. Naval Observatory has produced a new set of ephemerides for use in the Astronomical Almanac.

Because of recent increase in interest about asteroids, it was decided to include more than the traditional four asteroids in the production of asteroid ephemerides. The U.S. Naval Observatory's main purpose is to improve the accuracy of navigational aids. Thus, the asteroids that USNO will be most interested in are those which will most affect our knowledge of the perturbations of the asteroids on the planets. The criteria used to select a small sample of main belt asteroids for producing ephemerides were:

1. Asteroids over 300 km in diameter. These were chosen for future studies of their perturbations of the planets.
2. Asteroids with excellent observing histories of discovered before 1850. These were chosen to explore the limits to which current asteroid ephemerides could be determined.
3. Asteroids that were the largest in their taxonomic class. These were chosen to provide useful information for the studies of the composition of the asteroids for improvement in modeling the perturbation of the asteroids on the planets.

These 15 asteroids make up the USNO/AE97 (U.S. Naval Observatory Asteroid Ephemerides of 1997). The USNO/AE97 covers the period 1800 June 21.5 (JD 2378668.0) through 2049 November 17.5 (JD 2469763.0).

The construction of the ephemerides is discussed in the following sections. Data discusses the sources of the data used to determine the ephemerides and how the data was handled. Physical Model discusses the physical model used to integrate the ephemerides. Ephemerides discusses the resulting ephemerides and how they were converted from tabular form to Chebyshev polynomials. And Residuals looks at the residuals and places limits on the accuracy of the ephemerides.

An ephemeris is only as good as the data that is used and the physical model used to generate it. Bowell et al. (1996) finds a simple relation between the uncertainty in the osculating elements depending primarily upon the length of time over which the the asteroid has been observed and the number of oppositions at which observations have been made. However, for over sampled asteroids such as Ceres, Pallas, Juno, and Vesta the uncertainty in the ephemerides can be further reduced based on the number and quality of observations. Thus an idea of the currently best available ephemeris accuracy and accuracy reachable can be inferred from Table 2.

The data used in creating these ephemerides can be divided into three categories: wide angle data, relative data, and radar data . Each of these data types are handled differently, so their sources and methods of reduction will be discussed separately.

Some aspects of the data reduction was handled using the ephemeris program PEP (Ash 1965). PEP is a high accuracy ephemeris generating program capable of of generating ephemerides using complicated physical models, comparing the results to many different observation types, adjusting designated parameters and then producing a new set of ephemerides. PEP can iterate the ephemerides until a desired level convergence in the model parameters is reached. In addition to adjusting physical model parameters, PEP can adjust such parameters as catalog corrections in right ascension and declination, electronic delay biases in delay-Doppler observations, and corrections in the location of observatories. PEP can also be used to estimate the a priori uncertainty in a set of observations for use in weighting those observations.

WIDE ANGLE DATA

This data is usually taken with a transit instrument. The measurements are not made with respect to nearby stars, but are measured by means such as time of transit or recorded using setting circles. Usually the data is referred to an equator and equinox determined from the measurements made by the instrument itself, that is a fundamental catalog, but occasionally, as in the case of the Carlsberg Meridian Circle, (Morrison et al., 1990) the observations are differentially reduced to a preexisting equator and equinox. The positions for these observations are usually given as a geocentric apparent position with respect to the equator and equinox of date.

Most asteroid observations from the nineteenth century are of the wide angle, fundamental catalog variety. The main sources of these observations are the Royal Greenwich Observatory ( Maskelyne 1811; Pond 1815-1835; Airy 1837-1883; Christie 1884-1902; Schubart 1976), l'Observatoire de Paris ( Le Verrier 1858-1867; l'Observatoire de Paris 1871; Mouchez 1880-1892; Loewy 1898-1907; Baillaud 1910-1911; Schubart 1976), the Royal Observatory, Edinburgh ( Henderson 1839-1847; Henderson & Smythe 1848-1852; Schubart 1976), the Cambridge Observatory ( Airy 1831-1836; Challis 1837-1864; Adams 1879; Schubart 1976), the Royal Observatory, Cape of Good Hope ( MacLear & Stone 1871-1872; MacLear & Gill 1900), the Sternwarte zu Kremsmunster ( Koller 1833-1839; Reslhuber 1841-1870; Strasser 1870-1880), and the U.S. Naval Observatory ( Gillis 1846; Maury 1846-1859; Gillis 1867; Yarnall et al. 1872; U.S. Naval Observatory 1906b; Harkness & Skinner 1900). Additional observations from other observatories were gathered from the Astronomische Nachrichten (1832-1900) and early observations of Ceres and Pallas made at Palermo, Milan and Seeburg provided by Schubart 1976. In all, nineteenth century wide angle observations were gathered from 39 observatories. Except for the later USNO observations ( Yarnall et al. 1872; U.S. Naval Observatory 1906b; Harkness & Skinner 1900) the data was not collected into comprehensive catalogs, but was reduced at yearly intervals.

Twentieth century wide angle observations are usually reduced fundamental system using a catalog built up over several years. Aside from Ceres, Pallas, Juno, and Vesta very few wide angle observations were made of asteroids from 1901 until 1985. There are only a few observatories to make wide angle observations. The wide angle asteroid observations for the twentieth century used were provided by the Royal Greenwich Observatory ( Blackwell et al. 1975; Buontempo et al. 1973; Christie 1903-1912; Dyson 1913-1933; Dyson & Jones 1934; Jones 1935-1953; Tucker et al. 1983), the Cape Observatory ( Stoy 1968), the U.S. Naval Observatory ( Adams et al. 1964; Adams & Scott 1968; Hughes & Scott 1982; Hughes et al. 1992; U.S. Naval Observatory 1906a; Watts & Adams 1949), the Carlsberg meridian circle ( Carlsberg Meridian Catalog 1984-1995), and the Universite de Bordeaux transit circle ( Minor Planet Center 1997). Like the nineteenth century data, the Greenwich data prior to 1940 was reduced on a yearly basis. However, the data from the U.S. Naval Observatory, the Cape Observatory, and the Royal Greenwich Observatory after 1940 were reduced to fundamental catalogs that were the result of multi-year observing programs. The Carlsberg meridian circle data was also reduced to apparent positions, but on the system of the FK5 rather than as a fundamental catalog ( Morrison et al. 1990). The Universite de Bordeaux data, taken from the Minor Planet Center, has been reduced to topocentric astrometric positions on the mean equator and equinox of J2000.0.

Hipparcos Data

Hipparcos asteroid observations were examined for inclusion in the ephemerides.

Like the Hipparcos stellar data, the asteroid data was independently reduced by two consortia, FAST and NDAC. The NDAC reduction was chosen to represent the Hipparcos data because the FAST abscissa for larger objects does not correspond to the conventional definition of the photocenter of the body (Hipparcos 1997, section 2.7.2). The FAST astrometric reduction procedure was also found to be unsuitable for objects with an apparent diameter larger than roughly 0.7 arc seconds. This is somewhat smaller than the apparent diameter of the largest asteroids at opposition.

Also, as pointed out in section 2.7.1 of Hipparcos (1997),

Each observation corresponds to a transit of the object across the instrument main grid. Because the observations are essentially one-dimensional (namely, in the direction perpendicular to the grid slits at the epoch of observation), and because solar system objects have a large daily motion, the observed direction cannot be given in the form of conventional, two-dimensional coordinates such as right ascension and declination. Instead, the observed quantity is the abscissa lambda_GC of the projected position on a reference great circle with (positive) pole P; no information is given on the object's ordinate beta_GC, except that |beta_GC| < 1°. The observed position is somewhere on an arc of a great circle perpendicular to this reference great circle, where the abscissa on the reference great circle is precisely the value given by the observation. The published quantities defining this arc are reckoned in the tangent plane of a particular point. This particular point, referred to as the reference point, is chosen to be as close as possible to the actual position given by the ephemerides. The reference point thus corresponds to the observed coordinates in the scanning direction and to the calculated coordinate in the perpendicular direction.

All positions are referred to the reference frame defined by the Hipparcos Catalog. The data for one observation consists of:

1. the astrometric coordinates (alpha₀, delta₀) of the reference point, i.e. the geocentric direction corrected for stellar aberration; and the direction w in its tangent plane of the slit motion (i.e. the trace of the reference great circle);
2. the mean epoch of the transit, reduced to the geocentre, and expressed on the TT time scale;
3. the apparent magnitude H_p in the Hipparcos photometric system.

For observations with widely spaced epochs of observation, however, the standard deviation from the mean (O - C) was much larger. The standard deviation from the mean (O - C) for all 55 observations of Vesta in Table 3 was 0."2984 in right ascension and 0."2188 in declination. In all 652 observations for Ceres, Pallas, Juno, Vesta, Hebe, Iris, Flora, Metis, Hygiea, and Psyche were examined. The standard deviation over the entire Hipparcos mission with respect to the observations within a single epoch was an average of 61 times greater in right ascension and 38 times greater in declination. This means that without accounting for the correlation between the right ascension and the declination, the Hipparcos observations have approximately the same accuracy as other wide angle optical observations. The asteroid observations for Hipparcos are clustered around epochs containing from one to 39 observations with a very high precision within the epoch. Hence, there is no assurance that the long term Hipparcos observations are normally distributed. Since the the long term accuracy is no better than that of other sources of observations, there are ample observations of the asteroids from other sources covering the same time period, and the inclusion of the Hipparcos observations would likely introduce unknown systematic errors. Thus, the Hipparcos observations were not used.

RELATIVE DATA

Relative data is that data in which the astrometric position of the asteroid is determined relative to other bodies, usually stars, in the same field. Over time the methods of taking relative data has changed radically. Since the positions of the background objects need to be known before the asteroid position can be determined, the common practice is to use positions that are all from a common epoch and corrected for proper motion of the background objects. Thus, the asteroid positions are given as topocentric positions of epoch.

Data through the late nineteenth century was made using micrometers to compare the asteroid with one or two field stars. Essentially, the position was calculated at the telescope.

Beginning in the late nineteenth century photographic plates were used to record the images of the asteroid and the background stars. These plates were later measured and a position calculated for the asteroid. Photographic plates have the added advantage of being able to collect light for extended periods of time allowing objects too dim to be viewed directly to be recorded. For this reason photographic plates have been the dominant form of recording asteroid positions through most of the twentieth century. Initially, only three or so stars were measured to determine the asteroid's position because the arithmetic needed to produce a position could become quite arduous as more background objects are used to determine the asteroid position. As computers became more commonplace, the arithmetic became less of a burden so more stars could be measured and more sophisticated reduction models could be implemented.

Modern relative positions are most often made using CCDs. A CCD produces a permanent record like photographic plates, and has the added advantage of consisting of an array of digital pixels allowing a more highly accurate determination of the center of light of the asteroid and the background objects.

The source for most of the relative observations used was the Minor Planet Center (1997). There are two advantages to using the Minor Planet Center data rather than collecting it from its original sources. First, the data is collected in a single place saving time. Second, unlike wide angle date which is reduced to apparent position of date, relative data is reduced to an epoch. The Minor Planet Center has provided the transformation from the original epoch of publication to the epoch of J2000.0. Relative observations from a total of 131 observatories were included.

The one other source of relative astronomical data was Stone (1995 - 1997) at the U.S. Naval Observatory, Flagstaff Station. These data were extremely valuable because they provided very high accuracy data at the most recent opposition of the asteroids. This provides a very solid anchor point for the modern end of the asteroid observations.

RADAR DATA

Radar data has the potential of being the most useful of all the data types because it is highly precise. A good radar observation will give the position of a body to a couple of kilometers. The largest unknown is determining the position of the center of reflection with respect to the center of mass. Fortunately, radar mapping such as done by Hudson & Ostro (1995) helps in reducing this uncertainty.

Radar data is also complementary to optical data. Rather than producing a place on the sky it produces a time delay or distance to the object and the component of the velocity along the line of sight. As a result, radar data has been shown able to produce great improvements in asteroid ephemerides for those asteroids with a rather small number of observations ( Yeomans et al. 1992, Hilton 1997b).

Until the recent upgrade of the Arecibo radio telescope, radar observations of all but the largest main belt asteroids has been impossible. To date the only time delay-Doppler observations published for the asteroids whose ephemerides are determined here are two observations of Iris by Ostro et al. (1991). This data was included in the ephemeris of Iris. Because the amount and span of other data on Iris is large and the uncertainty in the time delay was rather large (0.14 msec. and 0.08 msec) the contribution of the radar data to the ephemerides was rather small. The change in the orbital parameters was about 0.02 sigma and the reduction in the formal uncertainties in the orbital parameters was about 0.01 sigma in the final solution.

DATA WEIGHTING

The accuracy of the observations varied significantly depending on the type of observation, the equipment used, the observatory making the observation and the epoch of the observation. As a result, the weighting of the observations had to be very carefully considered.

First, the observations from each observatory were divided up into logical groups. Several criteria were used to determine what constituted a logical group. Easiest to determine were those observations that were published as part of a catalog. These data formed the bulk of the twentieth century, wide angle observations. The next check was for known changes of equipment, reduction technique, catalog used for reduction, or position of an observatory. Next determiner for a logical grouping was to examine the records of each of the observatories. If the observations from an observatory clustered around a set of epochs, those observation near each epoch were identified as a logical group. Finally, in absence of no other information, groups were devised by dividing data into groups at points where changes of equipment or techniques, such as the introduction of the impersonal micrometer and atomic time, were known to have been widely adopted.

Next, the observations were examined for relevance to the final solution. Unless the observations of an asteroid from an observatory met one of the following conditions, they were dropped from the final solution:

1. Contributed at least 0.5% of the observations to the final solution
2. Observations were made before 1847
3. Observations were made during the first observed opposition of an asteroid
4. Observations were made during the most recently observed opposition of an asteroid
5. Observations were made at an opposition that would otherwise have no observations

All of the asteroids used data from their first observed opposition through the most recent opposition given in Table 4. Except for Davida and Interamnia all of the asteroids are fit to substantially more data over substantially more oppositions than the ephemerides described in Table 2. As described in the section on Juno because there are unaccounted for perturbations on Juno the final solution for that asteroid did not contain observations from 1804 through 1838, so Juno was fit to 7318 observations in right ascension and 7174 observations in declination of a total 7428 observations covering 111 oppositions from 1839 through 1996.

The next step was to determine the least squares error in each logical grouping. Observations of different asteroids in each logical grouping were combined, and a single "catalog" least squares error was produced. An a priori error was assigned based on the age of the observation and the technique used for obtaining it. Initial ephemerides were constructed and the difference between the a priori error and the actual least squares error for each logical group was examined to determine the adjustment in the errors. Iterating with a new ephemerides calculated using the new errors was possible, but in all cases a single iteration was sufficient to determine the least square errors of the observations. For those logical group contributing 500 or more observations to the ephemerides corrections to the equinox and declination were also calculated. In all cases these corrections were so small that they did not significantly affect either the least squares error or the final ephemerides. The least squares error was then used to directly weight the observations in each logical group. Generally, except for the modern CCD observations, the relative observations were a factor of 2 worse than wide angle observations for a similar epoch.

The solar system model used to determine the ephemerides of the asteroids is the JPL ephemeris DE200 (Standish 1990). There are more recent ephemerides available such as DE403 ( Standish, et al. 1995). However, the differences between DE200 and DE403 are small. DE403 is oriented to the extragalactic reference frame which DE200 matches to a few milliarcseconds (Folkner et al. 1994). Standish, et al. show that for the inner solar system and Jupiter the difference between DE200 and DE403 is, at most, a few tenths of an arc second. For Saturn the maximum difference is an arc second. And for Uranus and Neptune the maximum difference is about 5 arc seconds. Only Pluto has a large difference in position, but it is very low mass and always at great distances from the asteroids. Hence, the differences in the perturbations caused by using DE200 over DE403 are several orders of magnitude smaller than the uncertainties in the orbits themselves. Since DE200 is still considered the standard ephemerides and the differences between DE200 and DE403 are inconsequential, DE200 was used for the solar system model.

ASTEROID MASSES

Asteroid perturbations are the largest source of incompletely modeled perturbations of the planets especially Mars and the Earth-Moon barycenter. Williams (1984) shows no less than seven asteroids capable of making periodic perturbations of more than a kilometer to Mars. The largest asteroid, Ceres, has only 0.13% the mass of Mars and is located within the asteroid belt itself. Hence it and the other asteroids are much more sensitive to the perturbations of the asteroids than are the planets. Thus the physical model for the asteroid ephemerides need to include perturbation by other asteroids. The perturbing asteroids included in the ephemeris of each asteroid are given in Table 5.

The mass for Interamnia was the mean of two values determined by Landgraff (1992). The mass for Davida is the estimate used by Viateau & Rapaport(1997). The mass for Eunomia was taken from Hilton (1997b). The masses for Juno and Psyche were estimated from their Tedesco (1989) diameters and an assumed density of 3 g cm^-1. The masses of Ceres, Pallas, and Vesta were determined contemporaneously with the ephemerides. The final masses determined in a simultaneous solution are in Table 6.

An estimate of the mass of Vesta was also made using observations of the perturbed asteroid 197 Arete. And and estimate was made of the mass of Ceres using Juno as the perturbed asteroid. However, in both cases neither perturbed asteroid resulted in a significantly different mass nor did the inclusion of the additional asteroid reduce the uncertainty in the derived mass of the perturbing asteroid.

Figure 1 shows the historic record of the determinations of the mass of Ceres in blue. Preliminary masses for these ephemerides determined using Vesta and Juno as the perturbed asteroid and the mass for Pallas used in previous Ceres mass determinations are in magenta. Preliminary masses for Ceres using Pallas, Juno, and Vesta as the perturbed asteroid and using the preliminary estimate for the mass of Pallas made while generating the ephemerides are in green, and the final mass for the mass of Ceres is in red. The final mass of Ceres is significantly smaller than most modern estimates of its mass. The one similar mass determination is that of Kuzmanoski (1995) where the author treated the encounter of 203 Pompeja with Ceres using the impulse approximation.

Figure 1. The history of mass determinations of 1 Ceres.

Asteroid            Element              Value     Uncertainty  Units

1 Ceres          mean distance        2.767837929   2 × 10-9      AU
                 eccentricity         0.07741187    2 × 10-8
                  inclination        27.143874      2 × 10-6    degrees
                ascending node       23.390569      4 × 10-6    degrees
            argument of perihelion  132.77714       1 × 10-5    degrees
                 mean anomaly       207.08208       1 × 10-5    degrees

2 Pallas         mean distance        2.773856953   2 × 10-9      AU
                 eccentricity         0.23233957    2 × 10-8
                  inclination        11.833838      2 × 10-6    degrees
                ascending node      160.858819      9 × 10-6    degrees
            argument of perihelion  323.02674       1 × 10-5    degrees
                 mean anomaly       194.831745      6 × 10-6    degrees

3 Juno           mean distance        2.669481236   1 × 10-9      AU
                 eccentricity         0.25773185    2 × 10-8
                  inclination        10.876712      2 × 10-6    degrees
                ascending node       11.70012       1 × 10-5    degrees
            argument of perihelion   46.91849       1 × 10-5    degrees
                 mean anomaly        72.053979      4 × 10-5    degrees

4 Vesta          mean distance        2.3607012339  7 × 10-10     AU
                 eccentricity         0.09035411    1 × 10-8
                  inclination        22.735663      2 × 10-6    degrees
                ascending node       18.172803      4 × 10-6    degrees
            argument of perihelion  237.07614       1 × 10-5    degrees
                 mean anomaly       138.49976       1 × 10-5    degrees

6 Hebe           mean distance        2.424774096   1 × 10-9      AU
                 eccentricity         0.20184319    3 × 10-8
                  inclination        15.512279      2 × 10-6    degrees
                ascending node       38.815444      9 × 10-6    degrees
            argument of perihelion  341.13194       1 × 10-5    degrees
                 mean anomaly       211.606827      7 × 10-6    degrees

7 Iris           mean distance        2.384906312   1 × 10-9      AU
                 eccentricity         0.23063281    2 × 10-8
                  inclination        23.084892      2 × 10-6    degrees
                ascending node      346.012362      6 × 10-6    degrees
            argument of perihelion   57.998030      8 × 10-6    degrees
                 mean anomaly       214.061641      6 × 10-6    degrees

8 Flora          mean distance        2.201299297   1 × 10-9      AU
                 eccentricity         0.15606733    3 × 10-8
                  inclination        21.982003      3 × 10-6    degrees
                ascending node       14.813338      7 × 10-6    degrees
            argument of perihelion   22.35365       1 × 10-5    degrees
                 mean anomaly         1.47253       1 × 10-5    degrees

9 Metis          mean distance        2.386443367   2 × 10-9      AU
                 eccentricity         0.12115008    4 × 10-8
                  inclination        25.935090      3 × 10-6    degrees
                ascending node       11.976521      9 × 10-6    degrees
            argument of perihelion   63.84890       2 × 10-5    degrees
                 mean anomaly       352.35224       2 × 10-5    degrees

10 Hygiea        mean distance        3.136204910   5 × 10-9      AU
                 eccentricity         0.11985212    4 × 10-8
                  inclination        24.617827      3 × 10-6    degrees
                ascending node      351.002734      8 × 10-6    degrees
            argument of perihelion  246.59315       2 × 10-5    degrees
                 mean anomaly       206.86683       2 × 10-5    degrees

15 Eunomia       mean distance        2.644308715   2 × 10-9      AU
                 eccentricity         0.18701476    4 × 10-8
                  inclination        30.019997      4 × 10-6    degrees
                ascending node      338.083503      7 × 10-6    degrees
            argument of perihelion   50.17445       1 × 10-5    degrees
                 mean anomaly       296.45252       1 × 10-5    degrees

16 Psyche        mean distance        2.921527404   6 × 10-9      AU
                 eccentricity         0.13756274    3 × 10-8
                  inclination        20.800688      3 × 10-6    degrees
                ascending node        4.29608       1 × 10-5    degrees
            argument of perihelion   15.45439       2 × 10-5    degrees
                 mean anomaly       188.69446       2 × 10-5    degrees

52 Europa        mean distance        3.099221624   9 × 10-9      AU
                 eccentricity         0.10051921    5 × 10-8
                  inclination        19.562460      4 × 10-6    degrees
                ascending node       17.54586       1 × 10-5    degrees
            argument of perihelion   94.54815       3 × 10-5    degrees
                 mean anomaly       268.51718       3 × 10-5    degrees

65 Cybele        mean distance        3.43189706    1 × 10-8      AU
                 eccentricity         0.10414568    7 × 10-8
                  inclination        20.251439      7 × 10-6    degrees
                ascending node        4.19666       2 × 10-5    degrees
            argument of perihelion  258.73969       4 × 10-5    degrees
                 mean anomaly       127.70236       4 × 10-5    degrees

511 Davida       mean distance        3.17167249    1 × 10-8      AU
                 eccentricity         0.18235227    6 × 10-8
                  inclination        23.710076      5 × 10-6    degrees
                ascending node       40.56062       1 × 10-5    degrees
            argument of perihelion   49.29935       2 × 10-5    degrees
                 mean anomaly        48.08837       1 × 10-5    degrees

704              mean distance        3.064446905   8 × 10-9      AU
Interamnia       eccentricity         0.14595013    4 × 10-8
                  inclination        31.363464      3 × 10-6    degrees
                ascending node      325.795504      8 × 10-6    degrees
            argument of perihelion   45.58639       2 × 10-5    degrees
                 mean anomaly       106.07913       2 × 10-5    degrees