Spacecraft Elements

Table 7: Heliocentric mean orbital elements of major interplanetary spacecraft in $HAE_{J2000}$ fitted to data provided by NSSDC. The elements are semi-major axis $a$, eccentricity $e$, mean longitude $\lambda$, longitude of periapsis $\varpi$, inclination $i$, and ascending node $\Omega$. The time parameter $y_0$ is scaled in Julian years of 365.25 days from J2000.0, periods are given in decimal Julian years from J2000.0+2000.0. The last three columns contain the precision of positions determined from the elements relative to NSSDC position data: Maximal difference in HAE distance, longitude and latitude over the period given.

Mission	Period	$a$[AU]	$e$	$\lambda[^\circ]$	$\varpi[^\circ]$	$i[^\circ]$	$\Omega[^\circ]$	$\Delta r$[AU]	$\Delta\phi[^\circ]$	$\Delta\theta[^\circ]$
Galileo	1990.4-1990.9	0.982	0.298	195.36 + 366.670$y_0$	182.17	3.39	76.51	0.014	1.2	0.10
Galileo	1991.2-1992.8	1.572	0.439	304.32 + 181.146$y_0$	-240.47	4.57	-103.37	0.036	0.7	0.09
Galileo	1993.8-1996.0	3.113	0.700	180.16 + 64.938$y_0$	-277.61	1.68	-105.39	0.014	0.8	0.09
Helios1	1977.0-1986.0	0.6472	0.5216	126.77 + 691.475$y_0$	-101.84	0.004	70.18	0.001	0.39	0.016
Helios2	1977.0-1981.0	0.6374	0.5436	147.76 + 707.453$y_0$	294.58	0.024	121.85	0.002	0.89	0.004
Pioneer10	1972.4-1973.9	3.438	0.715	291.99 + 56.479$y_0$	160.02	2.08	-17.06	0.019	0.18	0.07
Pioneer10	1974.3-2005.0	-6.942	1.727	111.81 + 19.700$y_0$	-42.02	3.14	-28.57	0.019	0.02	0.007
Pioneer11	1973.5-1974.8	3.508	0.7166	220.69 + 54.797$y_0$	195.46	3.05	16.64	0.013	0.10	0.06
Pioneer11	1975.0-1979.6	16.729	0.7767	180.91 + 5.264$y_0$	55.05	15.29	-5.24	0.012	0.25	0.20
Pioneer11	1979.7-2000.0	-8.059	2.161	127.99 + 15.668$y_0$	173.21	16.63	160.40	0.06	0.23	0.12
Ulysses	1991.1-1992.1	9.035	0.8905	143.48 + 13.272$y_0$	21.13	1.99	13.57	0.014	0.07	0.06
Ulysses	1992.2-2005.0	3.375	0.6032	256.31 + 58.073$y_0$	-22.93	79.15	-21.85	0.007	0.92	0.50
Voyager1	1978.0-1979.1	5.020	0.8009	332.66 + 31.820$y_0$	-17.71	0.93	-11.4	0.010	0.44	0.12
Voyager1	1979.2-1980.8	-4.109	2.258	302.05 + 43.088$y_0$	112.12	2.46	113.23	0.010	0.23	0.06
Voyager1	1980.9-2005.0	-3.203	3.742	332.47 + 62.642$y_0$	157.35	35.71	178.95	0.034	0.10	0.11
Voyager2	1977.9-1979.4	3.624	0.7244	65.98 + 52.225$y_0$	-20.65	0.84	-33.03	0.013	0.13	0.06
Voyager2	1979.6-1981.6	-17.345	1.2905	216.12 + 5.000$y_0$	110.80	2.58	120.05	0.017	0.10	0.06
Voyager2	1981.7-1986.0	-3.913	3.4537	324.52 + 46.379$y_0$	189.87	2.66	77.65	0.017	0.12	0.05
Voyager2	1986.1-1989.3	-2.902	6.0618	7.18 + 72.400$y_0$	-144.23	2.81	-98.07	0.034	0.12	0.12
Voyager2	1990.7-2000.0	-4.021	6.2853	256.56 + 44.661$y_0$	231.66	78.92	101.65	0.045	0.13	0.10

To determine approximate positions of spacecraft relative to each other or to planets without using positional data files it is useful to have orbital elements of spacecraft in Keplerian orbits. This excludes most near Earth missions since their orbits are not Keplerian. In Tab.7 we list orbital elements for most major interplanetary missions. We have fitted these elements to trajectory data provided by NSSDC²¹. Not much accuracy is claimed by NSSDC for the propagated trajectories of any heliospheric spacecraft. But random cross-comparison with published papers had revealed mismatches of $< 0^\circ.1$ in angles or $< 1\%$ in radial distanceHEE (R.Parthasarathy, personal communication). We used the vector method given in ch.2 of Bate et al. (1971) to calculate initial values for the elements which we then fitted to achieve the smallest maximal deviation from the position data. The deviations are listed in the last three columns of Tab.7. The spatial resolution of the NSSDC position data is only $0.1^\circ$ and the temporal resolution 1 day. This results in a poor precision of the orbital elements at perihelion specifically for the Helios mission where the spacecraft moves $8^\circ$/day. For this reason we re-calculated the Helios orbits by integration from cartesian state vectors provided by JPL and then fitted elements to the re-calculated orbits. See also the JPL Voyager home page²²for more Voyager orbital elements, and the ESA Ulysses home page²³for a discussion of Ulysses orbital elements.