Table 3 (click here) gives the postfit statistics for the observed-minus-computed residuals for the Earthbased observations grouped by data set. For each set the table indicates the type of observations and the number used versus the total number available. The statistics include the sample mean (

) of the residuals, the standard deviation (

) about the mean, and the root-mean-square (rms). The Arequipa, the 1904 Yerkes, and the USNO observations were not fit; however, statistics for them are included. To arrive at the statistics for the unused observations, we deleted all residuals greater than 30'' and applied a 3

rejection criterion to the remainder.

Set	Year	Observer/Source	Type	No.			rms	wrms	No.			rms	wrms
1	1898	Pickering	P0	7/7	-029	416	386		7/7	263	376	436
2	1899	Pickering	P0	0/3					0/3
3	1900	Pickering	P0	20/26	831	624	103		20/26	316	587	653
4	1902	Pickering	P0	2/6	-114	198	180		2/6	-315	031	316
5	1904	Pickering	P2	26/28	655	455	792		26/28	-368	742	815
6	1904	Barnard	V0	1/2	103	000	103		1/2	-390	000	390
7	1904	Perrine	P0	5/6	047	099	101	1.01	5/6	-001	058	052	0.95
8	1905	Pickering	P2	11/17	-243	121	118		11/17	-449	142	142
9	1905	Albrecht	P0	11/11	-103	094	136	1.01	11/11	152	172	224	0.99
10	1906	Perrine	P0	8/8	008	112	105	0.96	8/8	054	082	094	0.94
11	1906	Pickering	P2	6/6	406	370	528		6/6	-526	971	103
12	1906	Pickering	P2	9/9	-193	805	783		9/9	-345	571	639
13	1906	Barnard	V0	11/12	-226	425	464	1.01	12/12	-063	152	158	0.99
14	1907	Christie	P0	16/16	020	070	071	1.01	16/16	018	077	076	0.96
15	1908	Perrine	P0	2/2	191	073	198	0.99	2/2	-080	003	080	1.01
16	1908	Christie	P0	22/23	020	063	065	0.99	22/23	004	072	071	1.01
17	1909	Christie	P0	12/12	030	107	107	1.02	12/12	-004	098	094	0.99
18	1910	Christie	P0	7/7	-001	140	129	0.99	7/7	040	068	075	1.00
19	1912	Barnard	V0	12/12	168	120	204	1.02	7/7	-063	124	130	1.00
20	1913	Barnard	V0	5/5	084	066	102	0.93	3/3	-108	032	111	1.01
21	1922	VanBiesbroeck	P0	4/5	341	072	347	0.99	4/5	-145	126	182	0.98
22	1940	Nicholson	P0	1/1	-024	000	024	0.49	1/1	000	000	000	0.01
23	1942	VanBiesbroeck	P0	7/8	166	041	170	0.97	7/8	-024	025	034	0.67
24	1952	Bobone	P0	7/7	-002	102	095	1.00	7/7	002	033	031	0.61
25	1955	VanBiesbroeck	P0	8/11	109	073	129	0.96	8/11	045	048	063	0.98
26	1955	VanBiesbroeck	P0	3/3	213	056	218	0.97	3/3	079	088	107	1.02
27	1957	VanBiesbroeck	P0	8/8	034	069	073	0.97	8/8	126	070	142	0.98
28	1960	Roemer	P0	2/2	090	016	091	1.01	2/2	040	017	042	0.84
29	1968	Chernykh	P0	2/2	073	191	153	0.99	2/2	-116	051	122	0.97
30	1969	VanBiesbroeck	P0	1/3	152	000	152	0.98	1/3	038	000	038	0.77
31	1969	VanBiesbroeck	P0	1/1	-013	000	013	0.26	1/1	085	000	085	1.00
32	1975	Mulholland	P1	4/5	-026	027	035	0.99	4/5	-006	007	009	0.45
			P0	2/3	-028	081	064	0.98	2/3	-016	019	021	0.42
33	1981	Bowell	P1	8/10	108	066	124	0.99	8/10	-057	085	098	0.98
			P0	2/2	-041	057	058	0.96	2/2	-103	020	104	0.99
34	1981	Debehogne	P1	20/21	044	043	061	0.94	20/21	000	058	057	0.95
35	1982	Bowell	P1	2/2	-029	063	053	0.96	2/2	-095	114	125	1.00
36	1982	Debehogne	P1	18/21	-034	048	058	0.96	18/21	-008	060	059	0.98
37	1992	Rohde	P0	22/22	189	036	193		22/22	-026	023	034
38	1992	Whipple	P0	12/12	-010	025	026	0.52	12/12	-006	050	049	0.97
39	1993	Rohde	P0	9/9	015	026	028		9/9	-111	013	112
40	1994	Whipple	P0	2/2	-015	072	053	1.07	2/2	-040	034	047	0.93
41	1996	Whipple	C0	21/21	-037	021	042	1.06	21/21	011	029	030	0.75

Table 3: Astrometric observation residual statistics. The column entitled "Type'' gives the observation type: the letter V denotes visual, P denotes photographic, and C denotes CCD. The digit gives measurement type: 0 - absolute

and

; 1 -

and

; 2 - angular separation and

. The column entitled "No.'' gives the number of observations included in the solution/number of available observations,

denotes the sample mean,

denotes the standard deviation about the mean, rms denotes the root-mean-square, and wrms denotes the root-mean-square of the weighted residuals

The mean for each set gives an indication of systematic errors. For example, a large mean in the right ascension residuals could be attributed to an equinox offset. The standard deviation measures the scatter of the residuals about the mean and characterizes the noise level of the observations, and the root-mean-square represents the overall quality of the fit. Also included is the weighted root-mean-square (wrms), i.e., the rms of the residuals multiplied by their weights. The wrms measures the quality of the fit relative to the assumed accuracy of the observations. The weights can be recovered from the ratio of the wrms to the rms.

Overall, the orbit fits the Earthbased observations at the 1

19 level. The rms for the "old'' (pre-1940) sets is 1

48 and for the "modern'' (post-1940) sets is 0

78. Among the sets used, the poorest fit is to Barnard's 1906 micrometer measures, and the best fits are to Whipple's 1992 photographic data, Whipple's 1996 CCD data, and Mulholland's 1975 relative photographic data. Also, Nicholson's isolated 1940 observation has a surprisingly small residual. Figure 1 (click here) displays the right ascension residuals for the observations that were fit, and Fig. 2 (click here) displays the declination residuals. The figures give an indication of the time distribution of the observations and the overall quality of the fit. Considerable scatter is evident as well as some suggestion of systematic errors.

The residuals for the Voyager imaging data range in magnitude from 7 to 235 km; the rms for all eight is 104 km.

The rms for residuals of the Saturn observations from Greenwich are 0

42 and 0

28 in right ascension and declination, respectively. Table 4 (click here) gives the biases found for each observation set.

5.2. Epoch state vector

The epoch state vector obtained from the fit appears in Table 5 (click here), and the Saturnian system dynamical constants needed for the integration appear in Table 6 (click here). Two state vectors are provided: the first is a from a fit which used the complete dynamical model in post-1966 integration; the second is from a fit which used the simplified model. The simplified model was used in the pre-1966 integration in both cases. The root-mean-square of the differences between the two models over the time period 1900 to 2013 are 7.4 km in the radial direction, 50.5 km in the in-orbit direction, and 17.2 km in the out-of-plane direction.

As an aid to those wishing to reproduce the integration, Table 7 (click here) contains the state vector for the simplified model at the end of the integration.

Component	Position (km)	Velocity (km/s)
complete model
x	-12049661.9430915500	-0.5851428400610138
y	-2354538.7543550990	1.5137695265625980
z	298437.1979645121	0.7872087168834739
simplified model
x	-12049676.2666544100	-0.5851329248090125
y	-2354463.3515782810	1.5137727228222640
z	298451.8787930112	0.7872099536417393

Table 5: Barycentric state vector at Julian ephemeris date 2439440.50 (1966 Nov. 11.0) referred to the Earth mean equator and equinox of J2000

Name	Value	Units
Saturn system GM	37940629.764	km³/s²
Mimas GM	2.500	km³/s²
Enceladus GM	4.900	km³/s²
Tethys GM	41.808	km³/s²
Dione GM	73.156	km³/s²
Rhea GM	154.000	km³/s²
Titan GM	8978.200	km³/s²
Iapetus GM	106.000	km³/s²
Saturn radius	60330.0	km
Saturn J₂	162.98
Saturn J	213.74
Saturn J₄	-9.15
Saturn J₆	1.03
Saturn pole right ascension	40.58	deg
Saturn pole declination	83.54	deg

Table 6: Saturnian system dynamical constants. These are the constants currently being used in the major Saturnian satellite ephemerides, see Jacobson (1996b)

quadrupole for simplified model.

Component	Position (km)	Velocity (km/s)
x	-10039870.733667480	-1.2513317541446060
y	-6590801.243860413	1.1142229491687150
z	-2664829.368414232	0.6536617058933396

Table 7: Barycentric state vector for the simplified model at Julian ephemeris date 2414640.5 (1898 Dec. 17.0) referred to the Earth mean equator and equinox of J2000

5.3. Mean elements

Because an integrated orbit in terms of cartesian coordinates is difficult to interpret geometrically, an alternative representation in the form of mean orbital elements is often useful. Table 8 (click here) provides mean elements derived by fitting a precessing ellipse model to the integration over the period 1900 to 2013. The reference plane is the Phoebe Laplacian plane, the plane on which the orbit precesses almost uniformly. The orientation angles for the Laplacian plane pole are with respect to the Earth mean equator and equinox of J2000 system; the tilt of the plane off the Saturn equator is 26

183. The epoch mean longitude

, longitude of periapsis

, and longitude of the ascending node

, are measured from the ascending node of the Laplacian plane on the Earth mean equator of J2000. The elements may be used as replacements for those provided by Rohde & Sinclair (1992) for computing an approximate orbit. We should comment that the latter element set is referred to the ecliptic and equinox of 1950.0, hence the angular elements cannot be compared directly with ours. The root-mean-square of the differences between the integrated and the approximate orbits over the 1900 to 2013 period are 127236 km in the radial direction, 244253 km in the in-orbit direction, and 15438 km in the out-of-plane direction. These differences give an indication of the magnitude of the periodic perturbations (mostly due to the Sun) affecting the orbit.

Element	Value	Units
semi-major axis	12944346	km
eccentricity	0.16435
inclination	174.751	deg
	390.742	deg
	203.958	deg
	233.037	deg
orbital period	548.2122790	days
	1.19141	deg/yr
	0.45631	deg/yr
Laplacian plane pole	275.631	deg
right ascension
Laplacian plane pole	68.031	deg
declination

5. Processing results

5.1. Observation residuals

5.2. Epoch state vector

5.3. Mean elements