In conformity with the resolution of IAU Commission 20 (New Delhi, November 1985, the Minor Planet Circulars 10193 – 10194), the so-called H, G magnitude system for minor planets is used in the EMP. A formula for the prediction of the apparent magnitude is

V = 5 log(rD) + H – 2.5 log[(1 – G)F1+ GF2],

(1)

The formula (1) predicts the observed opposition surge and the non-linear drop off in brightness at large phase angles, and is valid for 0º < b < 120º. H and G are fundamental photometric parameters for each minor planet. They are related to the previously used absolute magnitude B(1,0) and the phase coefficient. In particular, conversion from the B band of the old system to the V band of the new system is conveniently carried out using the approximate relationship H = B(1,0) – 1 mag. Further details have been given by E. Bowell, B. Hapke, D. Domingue, K. Lumme, J. Peltoniemi, and A. W. Harris in "Asteroids II" (eds. R. P. Binzel, T. Gehrels and M. S. Matthews), p. 524 – 556, Univ. of Arizona Press, Tucson, 1989).

The magnitudes in the present volume of EMP are based on the those values of H and G which were published in MPC 28103 – 28116) and in subsequent issues. The parameters for the first 3200 minor planets were determined mainly by E. F. Tedesco (Jet Propulsion Laboratory, U.S.A.). For almost all remaining minor planets the H value were determined by G. V. Williams (Harvard-Smithsonian Center for Astrophysics, U.S.A.).

Since the last year's EMP, the number of newly numbered minor planets has increased by 39891. Elements of all numbered minor planets have been determined by IAA. The sets of elements are given for the standard epoch July 31.0, 2016. They were determined as a result of orbit improving on the base of all kinds of optical observations available in the catalog of observations of the Minor Planet Center on March 5, 2015 (radar observations have not been used at this stage). The brief information on the accuracy of each set of elements is given in the Table on p. 10 – 906. It contains the following data:

1) Number of planet.

2) The number of oppositions with observations used for determination of the orbit.

3) The time interval covered by the observations used for determination of the orbit.

4) The mean-root-square error of observations, expressed in arcsec, with respect to orbits fitted by the least squares.

When estimating these data it is necessary to take into consideration the following.

1) Corrections to the initial sets of elements were determined by the least square fit of weighted conditional equations. In so doing, the observations made before 1901 were assigned weight equal to 1/16, the observations made during the time interval from 1901 to 1950 were assigned weight equal to 1/9, the observations made during the time interval from 1951 to 1995 were assigned weight equal to 1/4, and at last observations starting from 1996 were considered as having unit weight.

2) Observations of only standard accuracy were used for orbit improvement.

3) Observations in right ascension and in declination were considered as independent, so that conditional equation, e.g. in right ascension, could be excluded by "the three sigma criterion" from solution whereas equation in declination could be used or vice versa.

4) Observation is considered as used, if both conditional equations in right ascension and in declination or at least one of them are used for finding solution.

5) The mean-root-square error of observations, given in column 4 of the Table was computed by the formula

Note that weights appearing in the conditional equations are not used in the above formula.

In the Table below the brief information on the accuracy of orbital elements which are publishing in EMP-2016 is correlated with corresponding information on the accuracy of orbital elements included in minor planet orbit file maintained by Minor Planet Center. For different intervals of σ value the Table gives number of sets of elements, determined with mean errors that fall in the indicated limits.

These data show that quality of improvement of new sets is in general good. At least, 72.6 % the EMP orbits have σ < 0.5″, as compared with 73.1 % of MPC orbits. True, in the more wide interval σ < 0.8″ the MPC orbits and EMP orbits are presented approximately in equal amounts for the expense of more dense concentration of MPC orbits within interval 0.4″ ≤ σ < 0.6″.

The estimation of precision of new systems would not be complete without comparing the number of observations actually used for determining orbits. As the MPC orbits do not all were improved simultaneously with EMP orbits, only 423158 sets of elements were used for data comparison (for selected sets of elements process of improvement had started with the use of the same observations). From this number of orbits in 59208 cases the number of actually used observations was equal both for EMP and MPC orbits. In 345334 cases the number of observations used for determination of EMP orbits proves to be greater than that of used for MPC orbits and only in 18656 cases MPC orbits have been found from greater number of observations. It can be added that 5483 EMP orbits were determined on longer time interval at the expense of usage of more earlier observations.

The elements of the numbered minor planets are given with respect to the ecliptic and equinox J2000.0. The computation of osculating elements for the new standard epoch JDT 2457600.5 = 2016 July 31.0 TT was carried out by numerical integration of relativistic equations of motion in rectangular coordinates taking into account the perturbations from Mercury to Neptune and from Pluto, Ceres, Pallas, and Vesta. Coordinates and masses of perturbing planets were taken from DE 405. The perturbations from the Earth and the Moon were considered separately.

Osculating elements of minor planets are given on p. 907–6287. The first column of the Table gives the number and name or principal provisional designation of each minor planet. The second gives the absolute magnitude, H, that is, the brightness averaged over rotation for minor planets having known lightcurves and reduced to unit heliocentric and geocentric distances and to zero phase angle. The third column gives the slope parameter, G.

As noted above, the values of H and G are given in this year in accordance with the listing published in MPC 28103–28116 and in subsequent issues. In general the parameters given for the first 3200 minor planets are precisely those of the previous listing, prepared mainly by E. F. Tedesco. When H and G values were determined sufficiently precisely from photoelectric observations, frequently by least-square fitting, although in some instances G was selected, H values are given to 0.01 mag. Actual solutions for G (ranging from –0.12 to +0.60) have been made in only 110 cases. In the Table these values are given to 0.01. For by far the majority of minor planets the default value of G equal to 0.15 is used and it is denoted in the Table by X. Magnitudes based solely on photographic observations or for photoelectrically observed asteroids with large lightcurve and/or aspect variations (cf. lightcurve parameter Table, p. 6319–6360) are given to 0.1 mag. For almost all minor planets with numbers greater than 3200 the values of H were found by G. V. Williams (also to 0.1 mag and using G = 0.15) from magnitudes given on the astrometric records collected by the Minor Planet Center. For photoelectrically observed NEAs (3361), (3551), (3554), (3757) and (4015) the more precise H values from previous listing are preserved. For (4179) the H and G values suggested by D. J. Tholen (Institute for Astronomy, University of Hawaii, U.S.A.) are adopted. The new values of H determined by G. V. Williams are also adopted for 98 minor planets with numbers less than 3200. In three other discordant cases the improved H values were suggested by A. W. Harris (Jet Propulsion Laboratory, U.S.A.).

The orbital elements (mean anomaly, M, argument of perihelion, ω, longitude of ascending node, Ω, inclination, i, eccentricity, e, mean motion, μ, and semi-major axis, a) are given in the columns from 4 to 10.

Eleventh column of the Table headed TE contains last two digits of the title year of the EMP in which the orbit was first introduced and where the information on its precision can be found.

12 and 13 columns of the Table contain opposition date of minor planet in 2016 and its apparent magnitude for the fourth ephemeris date. A dash indicates that there is no opposition in 2016.

The Table of elements and opposition dates of the near-Earth asteroids (NEAs) (see p. 6288 – 6311) contains osculating elements of the numbered minor planets with perihelion distances q less or equal to 1.3 a.u., and dates of their oppositions in 2016. It is patterned after corresponding Table of elements of minor planets. The minute differences are as follows: the semi-major axis is given with less number of decimals and instead column headed TE the columns with approximate values of perihelion distance q and aphelion distance Q are given. The column headed T (Type) contains indication of minor planet type where Am stands for Amor type (a > 1 a.u., 1.017 a.u. < q ≤ 1.3 a.u.), Ap stands for Apollo type (a > 1 a.u., q ≤ 1.017 a.u.), At stands for Aten type (a < 1 a.u., Q ≥ 0.983 a.u.), and Ar stands for Atira type (Q < 0.983 a.u.). (163693) Atira is the first numbered asteroid of new type (Q < 0.983 a.u.) for which in literature one can find also other names (Apohele type, Arjuna type and so on).

The Table of elements and opposition dates of Centaurs and transneptunian objects (see p. 6312 – 6315) is constructed like that of NEAs. Column headed T (Type) contains indication of object's type where Ct stands for Centaurs (a < 30 a.u., q > 5.45 a.u.), Tn stands for transneptunian objects ("kjubiwanos" (like classical 1992 QB₁) or Plutinos), Sd stands for scattered disk objects.

The Table (p. 6316 – 6317) contains data for Jupiter-crossers, Jupiter-approachers as well as for some Mars-crossers, mainly with unusual orbits. Minor planets whose stability is protected by special mechanisms (like Trojans, Hildas and so on) are not included in the Table except in a few special cases.

The Table is constructed like two preceding ones. The header T (Type) indicates type of minor planet where Jc stands for Jupiter-crosser, Ja stands for Jupiter-approacher, Mc stands for Mars-crosser, and MT stands for Mars Trojan.

It should be noted that all objects included into the Tables of NEAs, Centaurs and transneptunian objects, and some unusual planets can be also found in the Table of elements and opposition dates of all minor planets.

The osculating elements of perturbing planets and their inverse masses used in DE 405 are given in the Table on p. 6318. They may be applied in continued numerical integration. For Ceres, Pallas, and Vesta the osculating elements correspond to those in the Table on p. 907.

On the same page values of some constants used when preparing ephemeris volume are given. Also Julian Day Number for some dates in 2015 – 2017 are given there.

The lightcurve parameters for some minor planets are given on p. 6319 – 6360. The data of the Table are based on the data published by C.-I. Lagerkvist, A. W. Harris and V. Zappalá (See: "Asteroids II machine-readable data base: March 1988 floppy disk version", National Space Science Data Center, Greenbelt MD). The Table was updated for publication in the volume for 2016 by B. Warner (Palmer Divide Observatory, Colorado Springs, U.S.A.), A. W. Harris (Space Science Institute, U.S.A.), and P. Pravec (Astronomical Institute, Academy of Sciences of Czech Republic).

The Table contains the number and name or principal provisional designation of minor planet, the period of rotational lightcurve expressed in hours, the amplitude of variation or range of amplitude observed, and a reliability code. More detailed explanations are given in the notes on p. 6359 – 6360.

     The Table on p. 6361 – 6367 includes alternative values of period and amplitude for minor planets marked in the Table "Minor planet lightcurve parameters" as having ambiguous value of period. The data for publication in the Table were supplied by B. Warner (Palmer Divide Observatory, Colorado Springs, U.S.A.), A. W. Harris (Space Science Institute, U.S.A.) and P. Pravec (Astronomical Institute, Academy of Sciences of Czech Republic).