State Vectors

Table 9: Heliocentric position and velocity vectors of the Earth and the Ulysses spacecraft on Jul 31, 1994 23:59:00 UTC.

Vector	Source	System	Units	X	Y	Z
$\mathbf{r_U}_N$	NSSDC	$GEI_{B1950}$	km	-135 927 895.1	126 880 660.0	-340 567 928.0
$\mathbf{v_U}_N$	NSSDC	$GEI_{B1950}$	km/s	18.54622396	-8.287477214	2.89468231
$\mathbf{r_U}_P$	Tab.7	$HAE_{J2000}$	km	-134 998 220	-208 472 970	-362 990 550
$\mathbf{v_U}_P$	Tab.7	$HAE_{J2000}$	km/s	18.624175	-6.2275813	5.9890246
$\mathbf{r_U}_J$	Tab.7	$GEI_{J2000}$	km	-134 998 220	125 262 330	-341 329 890
$\mathbf{v_U}_J$	Tab.7	$GEI_{J2000}$	km/s	18.624175	-8.0959913	3.0176330
$\mathbf{r_U}_B$	Tab.7	$GEI_{B1950}$	km	-135 246 410	126 772 910	-340 673 310
$\mathbf{v_U}_B$	Tab.7	$GEI_{B1950}$	km/s	18.546949	-8.3037658	2.9273224
$\mathbf{r_U}_S$	SPICE	$GEI_{B1950}$	km	-135 922 227	126 877 772	-340 564 861
$\mathbf{v_U}_S$	SPICE	$GEI_{B1950}$	km/s	18.546466	-8.287645	2.894987
$\mathbf{r_{EMB}}$	Tab.4	$HAE_{J2000}$	km	94 752 993	-118 648 910	-1 355
$\mathbf{v_{EMB}}$	Tab.4	$HAE_{J2000}$	km/s	22.792	18.478	0.00025
$\mathbf{r_E}$	Tab.4	$HAE_{J2000}$	km	94 750 228	-118 652 770	-1 355
$\mathbf{v_E}$	Tab.4	$HAE_{J2000}$	km/s	22.802	18.471	0.00025
$\mathbf{r_E}_J$	Tab.4	$GEI_{J2000}$	AU	0.633 3661	-0.727 6925	-0.315 5032
$\mathbf{r_E}_{AA}$	AstrAlm	$GEI_{J2000}$	AU	0.633 3616	-0.727 6944	-0.315 5035

Note that in the version of this paper published in Planetary&Space Science, 50, 217ff, the values calculated from Tab.4&7 are calculated for Jul 31, 1994 23:59 TT, not UTC.

The position and velocity vectors (state vector, $\mathbf{r_U}_N,\mathbf{v_U}_N$ in Tab.9) of the Ulysses spacecraft which we used to determine the orbital elements in Tab.7 was provided by NSSDC for the Julian date $JD = 2449565.5000137$ (Jul 31, 1994 23:59:00 UTC) in heliocentric earth-equatorial coordinates for epoch $\epsilon_{B1950}$.

In the following we describe how to derive the state vectors for Ulysses and Earth from the orbital elements for this date and compare the values with the respective data of the JPL SPICE system. The Julian century for this date is $T_0 = -0.054195755956093$ (eqn.2). From Tab.7 we take the values for the orbital elements for Ulysses in $HAE_{J2000}$:

$$a = 3.375d, \lambda = 256^\circ.31 + 58^\circ.073\cdot T_0 \c... ... \varpi = -22^\circ.93, \Omega = 79^\circ.15, i = -21^\circ.84$$ (50)

Using eqns.26 and 30 and 1 AU$= 149 597 870$km we calculate the $HAE_{J2000}$ state vector ( $\mathbf{r_U}_P,\mathbf{v_U}_P$).

This position is in agreement with the ecliptic position available from the spacecraft Situation Center for day 213, 1994: $(\lambda_{HAE_{J2000}}=188^\circ.8, \beta_{HAE_{J2000}}=-69^\circ.4,r=2.59$ AU). To compare this vector with the NSSDC value ( $\mathbf{r_U}_N,\mathbf{v_U}_N$) we have first to transform from the ecliptic $HAE_{J2000}$ system to the equatorial $GEI_{J2000}$ system using $T(HAE_{J2000},GEI_{J2000}) = E(0,-\epsilon_0,0)$. Since $GEI_{B1950}$ refers to the orientation of the Earth equator at $\epsilon_{B1950} ({T_0}_{B1950} = -0.50000210)$ we have to calculate the precession matrix using eqn.10 :

$$P(0.0,\epsilon_{B1950}) = \left( \begin{array}{ccc} \par0.99992571 & 0.011178938 & 0.004859... ...\\ -0.0048590038 & -2.7162595\cdot 10^{-5} & 0.99998819\\ \end{array} \right)$$ (51)

Finally we derive the Ulysses state vector in $GEI_{B1950}$ ( $\mathbf{r_U}_B,\mathbf{v_U}_B$).

The distance to the original NSSDC position ( $\mathbf{r_U}_N,\mathbf{v_U}_N$) is 697950 km (0.0046 AU), the difference in velocity 36 m/s in agreement with the precision cited in Tab.7 for the orbital elements. The respective position provided by the JPL SPICE system is ( $\mathbf{r_U}_S,\mathbf{v_U}_S$), which deviates by 7062 km and 0.42 m/s from the NSSDC state vector.

Now, we calculate the $HEI_{J2000}$ state vector of the Earth at the same time. From Tab.4 we get the undisturbed orbital elements of the EMB:

$$a = 1.0000010, \lambda = -50^\circ.546769, e = 0.016710876, \... ...circ.91987, \Omega = 174^\circ.88624, i = -0^\circ.00070751475$$ (52)

To increase precision we apply the disturbance corrections by Tab.6 of Simon et al. (1994) (values available on our website) and get:

$$a = 0.99998900, \lambda = -50^\circ.549526, e = 0.016710912, ... ...\circ.91987, \Omega = 174^\circ.88624, i = -0^\circ.00070754223$$ (53)

Using eqns.26 and 30 with $\mu_E = 1/332946$(Tab.4), we get the EMB state vector in $HAE_{J2000}$ ( $\mathbf{r_{EMB}} ,\mathbf{v_{EMB}}$). Given the low precision of the Ulysses position this would already be good enough to get the geocentric Ulysses state vector but to compare with SPICE data or the Astronomical Almanac we now apply eqn.32 to get the Earth state vector in $HAE_{J2000}$ ( $\mathbf{r_E} ,\mathbf{v_E}$), where we used the Delauney argument $D=-73^\circ.746062$. Finally we transform from $HAE_{J2000}$ to $GEI_{J2000}$ using $E(0,-\epsilon_0,0)$ as above to get $\mathbf{r_E}_J$, which can be compared with the value given in section C22 of the Astronomical Almanac for 1994 ( $\mathbf{r_E}_{AA}$,which agrees with the value given by the SPICE system).