Construction of SPICE SPK files for Helios 1 and 2
--------------------------------------------------
1) Source of position data
--------------------------
High precision orbit information for the Helios 1 and 2 spacecraft (also
referred to as Helios-A and Helios-B) is neither available at JPL
(Charles H. Acton, private communication) nor at the German Space Operations
Center (GSOC) (Oliver Montenbruck, private communication). Apparently, it has
not been archived.
Low-precision position information for Helios 1 and 2, primarily intended as
support for the analysis of the Helios science data, is available from NASA's
National Space Science Data Center (NSSDC). The tabulated daily positions of
the Helios 1 and 2 spacecraft w.r.t. the Sun are available at
http://spdf.sci.gsfc.nasa.gov/pub/data/helios/helios1/traj/day/helorb1.asc
(Helios 1 trajectory)
http://spdf.sci.gsfc.nasa.gov/pub/data/helios/helios2/traj/day/helorb2.asc
(Helios 2 trajectory)
and those positions were used for the reconstruction of the trajectories.
http://spdf.sci.gsfc.nasa.gov/pub/data/helios/helios1/traj/day/helorb.txt
contains a description of the table entries.
These files contain the time and date in columns 2 to 6, the distance
Sun - Helios (in AU, 5 decimal places) in column 8, the ecliptical longitude
(in degrees, 2 decimal places) in column 11, and the ecliptical latitude
(in degrees, 3 decimal places) in column 13. The other table entries contain
heliographic longitude and latitude, the Helios - Earth distance, and the
number of rotations of the Sun referred to the launch time; these parameters
were not used for the construction of the Helios SPK files.
Position data are availabe in the period from 1974-12-10T07:54:00 to
1981-09-30T00:00:00 for Helios 1 and from 1976-01-15T06:24:00 to
1980-03-09T00:00:00 for Helios 2.
Unfortunately, the description does not mention the epoch of the ecliptic
coordinates. By comparing the coordinates of the first vector, which is a
position only 9000 km away from the Earth's center, with the ecliptical
coordinates of Earth at various epochs, the epoch of the tabulated ecliptical
coordinates can be determined to be B1950.
The tabulated ecliptical coordinates were converted to epoch J2000. All
calculations described below were performed in ecliptical coordinates of epoch
J2000. This is also the coordinate system used in the remainder of this
document, unless mentioned otherwise.
2) Reconstruction of the trajectories of Helios 1 and 2
-------------------------------------------------------
The position data described above are insufficient to directly generate SPK
kernels for the following reasons:
- low numerial precision
- incomplete state vectors (only positions, no velocities)
However, the available position data can be used to reconstruct the
trajectories of Helios 1 and 2 by finding an orbit solution that fits the
position data well. The intention behind this procedure is that the large
amount of available positions (2487 positions for Helios 1, 1516 positions for
Helios 2) compensates their low precision.
The Helios spacecraft did not perform any trajectory correction maneuvers after
the launch phase (Fritz M. Neubauer, private communication), thus their
trajectory can be idealized as a freely moving point mass in the solar system
with their trajectories being determined to a high degree of precision by the
Newtonian equations of motion.
Helios 1 and 2 are spin-stabilized spacecraft. Their attitude control system
consits of three cold gas thrusters (one for spin-up, one for spin-down, and
one for roll/pitch) (Kehr, 1977). As no pair-wise thrusters are used, attitude
maneuvers may cause small momenta. However, since the attitude of the Helios
spacecraft was kept fixed during the mission (with the spin axis being
perpendicular to the ecliptic and a nominal spin rate of 60 rpm), only few and
minor attitude correction maneuvers had to be performed after the final
attitude/spin up maneuver, thought to have a negligible effect on the
trajectories.
The most important non-gravitational perturbation is likely the solar radiation
pressure. For the Helios spacecraft, the ratio of acceleration due to solar
radiation pressure and gravitational acceleration is of the order 10^-5. Due to
the constant attitude of the Helios spacecraft, the illumination geometry did
not change throughout the mission, and the effects of the solar radiation
pressure can be accounted for by one single (unknown) parameter, the radiation
pressure coefficient, C_R (cf. chapter 3.4 of Montenbruck & Gill, 2000). The
other parameters necessary to compute the acceleration caused by the solar
radiation are the cross section A and the mass m of the spacecraft. The cross
section A was calculated from the spacecraft geometry given in Kehr (1977) as
A = 9.02 m^2, and the mass was taken from the same reference as m = 369 kg.
Uncertainties in A and m (e.g. dry mass vs. wet mass) play no significant role
as they will be absorbed in the C_R coefficient.
The following method was used to reconstruct the trajectories of Helios 1
and 2:
1. An approximation of the Helios 1 and 2 trajectories was calculated by
numerical integration of the classical equations of motion for the Sun, the
planets, and the spacecraft (which for this calculation is treated as a
massless object).
The initial conditions (state vectors) of the Sun and the planets for a
chosen starting time (see below) were obtained from the JPL DE 405
(Standish, 1998) ephemerides (using SPICE). The Earth and the Moon are
treated as a point mass (at the Earth/Moon barycenter). An initial estimate
for the state vector of Helios 1 or 2 was derived from the tabulated
positions (initial value of the velocity vector being the finite difference
of the positions of the successor minus the predecessor entry divided by the
time between both entries).
An appropriate starting time (early in the mission) for the numerical
integration was chosen such that the following conditions were met:
a) the launch phase with its possible trajectory correction maneuvers
was over,
b) the nominal spacecraft attitude and spin rate were reached, so that
for the rest of the missions only small corrections to attitude and spin
rate were necessary,
c) the distance to the Earth/Moon system was large enough so that the
point mass approximation for the Earth/Moon-system is adequate.
The exact choice of the starting time is not critical as long as the above
conditions are met.
The following table lists the launch times, the used starting times, the
number of days d elapsed between launch and starting time, and the distance
to Earth at the starting time for Helios 1 and 2:
Spacecraft launch time (UTC) starting time (UTC) d Earth dist [km]
---------------------------------------------------------------------------
Helios 1 1974-12-10T07:11:02 1975-01-01T00:00:00 21 17.4*10^6
Helios 2 1976-01-15T05:34:00 1976-02-16T00:00:00 31 26.4*10^6
For Helios 1, the final spin-up maneuver, which brought the spacecraft in
full cruise condition, occurred on 1974-12-20. The date of this maneuver
for the Helios 2 spacecraft is unknown, therefore the starting time for this
mission was delayed by 10 additional days w.r.t. to the launch time.
The numerical integration was performed using the 7th-order Runge-Kutta-
Nystrom solver for second-order differential equations described in
Montenbruck & Gill (2000), based on the coefficients derived by Dormand &
Prince (1978). All calculations were performed in extended precision (using
80-bit floating point arithmetic). The used routine does not implement step
size control.
In order to determine the optimal step size, the orbit of Venus computed
with the numerical integration described above was compared with the orbit
derived from SPICE (JPL DE 405 ephemerides). Venus was chosen for the
comparison, since it has a semi-major axis and a mean orbital velocity
comparable to those of the Helios 1 and 2 spacecraft. It turned out that
the optimal step size was slightly less than 500 s, so the value of 500 s
was used as upper step size limit. The numerical integration that takes
the simulated system from time t1 to time t2 is done in an integral number
of steps with uniform time increments delta_t. delta_t is computed such that
it is always less than (or equal to) the upper step size limit, 500 s. With
this choice, the absolute position error of the Venus orbit remains less
than 700 km for an integration over a simulated period of 6.7 years.
Using that step size, the simulation of the planetary orbits in the solar
system requires 7 CPU-seconds per simulated year on a standard PC (Intel
686-class single-core, 3.2 GHz clock rate).
The following GM values were used for the Sun and the planets:
Body GM [km^3 s^-2]
-----------------------------------
Sun 132712440018.0000
Mercury 22039.8074
Venus 324970.6932
Earth/Moon 398600.4415
Mars 42842.8707
Jupiter 126731228.6400
Saturn 37943281.8000
Uranus 5781595.0500
Neptune 6859825.0278
The following initial conditions (origin of the coordinate system at the
solar system barycenter) were used for the integration of the Helios 1
trajectory:
At starting time = 1975-01-01T00:00:00 UTC (SPICE ephemeris time
ET = -788961553.816 s):
Body x [km] y [km] z [km]
---------------------------------------------------------------------------
Sun -415652.661 -17873.453 10355.051
Mercury 35362522.896 -53244061.073 -7622106.433
Venus 72490725.741 -80797192.703 -5297523.821
Earth/Moon -26682410.797 144720430.433 18577.173
Mars -116196436.272 -196832133.518 -1259485.162
Jupiter 738194318.889 -79459833.852 -16208812.408
Saturn -362939357.441 1300993081.132 -8333898.807
Uranus -2406548603.017 -1355466209.134 26195021.054
Neptune -1560100411.553 -4254718784.892 123550601.663
Helios 1 -9612482.958 141364574.290 33423.019
Body v_x [km/s] v_y [km/s] v_z [km/s]
---------------------------------------------------------------------------
Sun 0.0012331 -0.0119122 -0.0000346
Mercury 30.7145078 29.5588730 -0.4063721
Venus 25.7677015 23.3250795 -1.1705256
Earth/Moon -29.7933096 -5.4432203 -0.0004730
Mars 21.8018393 -10.2244845 -0.7509055
Jupiter 1.2429652 13.6014084 -0.0838390
Saturn -9.8268473 -2.6185909 0.4363179
Uranus 3.2913059 -6.2517854 -0.0659556
Neptune 5.0684879 -1.8405165 -0.0788664
Helios 1 -21.1393084 -7.4599483 0.0077138
The following initial conditions were used for the integration of the
Helios 2 trajectory:
At starting time = 1976-02-16T00:00:00 UTC (SPICE ephemeris time
ET = -754315152.815 s):
Body x [km] y [km] z [km]
---------------------------------------------------------------------------
Sun -241796.200 -403454.493 5899.454
Mercury -59172709.863 -8193992.366 4782421.819
Venus -61815668.349 -89556444.319 2348509.039
Earth/Moon -107519699.665 100837740.861 10911.996
Mars -77187060.035 228120336.566 6686774.416
Jupiter 638915814.937 376650428.936 -15862684.283
Saturn -688138979.038 1169544398.464 6888591.747
Uranus -2283583117.003 -1566683537.698 23813982.826
Neptune -1383210455.487 -4314825988.801 120713186.720
Helios 2 -90385885.749 108490600.813 18803.558
Body v_x [km/s] v_y [km/s] v_z [km/s]
---------------------------------------------------------------------------
Sun 0.0085833 -0.0094532 -0.0002111
Mercury -3.7342054 -46.1972129 -3.4280783
Venus 28.5813075 -20.0741810 -1.9228840
Earth/Moon -20.9219760 -21.7852920 -0.0014383
Mars -22.0292458 -5.6861293 0.4237165
Jupiter -6.7904123 11.8650017 0.1032581
Saturn -8.8472871 -4.9232390 0.4377003
Uranus 3.8018782 -5.9337203 -0.0713989
Neptune 5.1412197 -1.6287224 -0.0849035
Helios 2 -11.8982800 -18.5631000 -0.0025700
2. Using the positions of the Sun and Helios 1 and 2 as calculated above, the
position vectors of Helios w.r.t. the Sun can be calculated and compared
with the tabulated positions. A measure for the approximation quality of the
calculated trajectory to the tabulated positions is chi^2, the sum of the
squared deviations divided by the variances:
N [ h_i ]^2
chi^2 = sum [-------- ]
i=1 [ sigma_i ]
with
h_i = ((x(t_i) - x_i)^2 + (y(t_i) - y_i)^2 + (z(t_i) - z_i)^2)^0.5
where x_i, y_i, z_i are the ecliptical Cartesian coordinates of Helios
w.r.t. the Sun obtained from the tabulated trajectory data and x(t_i),
y(t_i), z(t_i) are the ecliptical Cartesian coordinates of Helios w.r.t. the
Sun at time t_i obtained from the calculation of the trajectory.
Here it should be noted that:
a) the precision of x_i, y_i, and z_i is low due to the few significant
decimal places of the values for the Sun-Helios distance r, the
ecliptic longitude lambda, and the ecliptic latitude beta in the input
files,
b) the uncertainties in x_i, y_i and z_i significantly vary with the
distance r from the Sun.
Therefore, the normalisation of the squared deviations (h_i)^2 with the
variances (sigma_i)^2 is crucial.
sigma_i is given by:
sigma_i^2 = ((x_i - x(t_i)) delta_x/h_i)^2
+ ((y_i - y(t_i)) delta_y/h_i)^2
+ ((z_i - z(t_i)) delta_z/h_i)^2
delta_x, delta_y, and delta_z can be calculated from the the position vector
given by:
[ x ] [ cos(beta) cos(lambda) ]
[ y ] = r [ cos(beta) sin(lambda) ]
[ z ] [ sin(beta) ]
where
r is the distance from the Sun,
lambda is the ecliptical longitude (0 degrees <= lambda <= 360 degrees), and
beta is the ecliptical latitude (-90 degrees <= beta <= 90 degrees).
Hence,
(delta_x)^2 = (cos(beta) cos(lambda) delta_r)^2
+ (r sin(beta) cos(lambda) delta_beta)^2
+ (r cos(beta) sin(lambda) delta_lambda)^2
and similar expressions can be obtained for (delta y)^2 and (delta z)^2,
respectively.
delta_r is the standard deviation of r due to the limitation to 5 decimal
places,
delta_r = 0.000005 AU = 75 000 km
Similarly, delta_beta and delta_lambda are the standard deviations of
beta and lambda, due to the limitation to 2 and 3 decimal places of the
ecliptical latitude and longitude, respectively,
delta_lambda = 0.005 degrees
delta_beta = 0.0005 degrees
The standard deviation for a position in the ecliptic is:
delta_rho = ((delta_x)^2 + (delta_y)^2)^0.5
~= ((delta_r)^2 + (r delta_lambda)^2)^0.5
~= r delta_lambda (for large r)
~= 13 000 km (at 1 AU)
(the symbol "~=" denotes "approximately").
The standard deviation perpendicular to the ecliptic is:
delta_z ~= r delta_b (for large r and small beta)
~= 1300 km (at 1 AU)
The absolute accuracy of the tabulated trajectory data is highest at
perihelion of Helios (~= 4060 km) and lowest at aphelion (~= 13 200 km).
3. The so defined chi^2 is minimized by variation of 7 parameters (initial
state vector: 6 parameters; in addition, the radiation pressure coefficient
is left as a free parameter). For this purpose, the downhill simplex method
by Nelder and Mead (1965), as described in Press et al. (1992), was used.
The following state vectors (at the starting times indicated above) were
found as solutions:
Spacecraft x [km] y [km] z [km]
---------------------------------------------------------------------------
Helios 1 -9605490.174 141365781.275 34035.097
Helios 2 -90621998.461 108089412.766 22841.081
Spacecraft v_x [km/s] v_y [km/s] v_z [km/s]
---------------------------------------------------------------------------
Helios 1 -21.1396648 -7.4578390 0.0078285
Helios 2 -12.1508476 -18.4194983 0.0053951
The solutions for the radiation pressure coefficients are:
Spacecraft C_R
------------------------------------
Helios 1 -0.25231234
Helios 2 3.02079977
(N.b.: Expected values for C_R are in the range 1 < C_R < 2. C_R was not
constrained in the fitting procedure).
3) Achieved accuracy
--------------------
The following table lists the final chi^2 value, the number N of used
positions, the number M of free parameters, and the reduced chi^2,
chi^2/(N - M):
Spacecraft N M chi^2 chi^2/(N - M)
-----------------------------------------------------------
Helios 1 2465 7 10565.46 4.30
Helios 2 1494 7 2372.21 1.60
The achieved accuracy of the orbit solutions can also be judged by computing
the residuals (tabulated positions minus computed positions).
For Helios 1, the residuals in x are dominated by numerical noise (due
to the numerical uncertainty of the tabulated positions due to the small
number of significant decimal places in the input data). The noise is
modulated by the Sun - Helios 1 distance (small at perihelion, about
+/- 10 000 km at aphelion). In addition to the numerical noise, there is
a systematic component of the residuals which exceeds 20 000 km for short
times. The residuals in y are similar, except that there are two significant
spikes (which appear to have their root in the input files). The systematic
component in the y residuals is larger than in x. The residuals in z are
almost entirely due to numerical noise. It in general oscillates +/- 1000 km
around 0, the maximum deviation being less than 2500 km. The total position
error is in general less than 20 000 km, and always less than 54 000 km.
For Helios 2, the residuals in x are dominated by numerical noise, modulated
by the Sun - Helios 2 distance. The residuals in x are less than 16 000 km
throughout the entire intervall of the data. The residuals in y are similar,
and they are less than 14 000 km everywhere. However, the residuals in z
show a relatively small (< 7000 km), yet systematic structure. These systematic
residuals could also be traced back to something that appears to be an error
in the input files. There is an unexplained sudden drop by 0.004 degrees (from
0.019 to 0.015 degrees) in ecliptical latitude between 1978-10-01 and
1978-10-02 for Helios 2. The total position error is in general less than
12 000 km, and always less than 17 000 km.
The apparent problems in the input files seem to be the reason for the limited
accuracy of the provided SPK files.
4) Production of SPK kernels
----------------------------
Using the solution state vectors, the trajectories of Helios 1 and 2 were
computed by numerical integration of the equations of motion, as described in
section 2.1. The integration was performed separately as forward integration
from the epoch of the solution state vector to the end time of the available
position vectors and as backward integration from the epoch of the solution
state vector to the time of the first position vector. The Helios 1 and 2 state
vectors were written into separate output files with a step size of two hours.
In the next step, the separate forward and backward files were merged such that
the state vectors were time-ordered (covering the same time span as the
tabulated positions) and converted to osculating elements (semi-major axis,
eccentricity, inclination, longitude of ascending node, argument of pericenter,
mean anomaly).
As the last step, the program "mkspk" from NAIF was used to produce a SPK
kernel from the osculating elements.
The following set-up for mkspk was used for Helios 1:
INPUT_DATA_TYPE = 'ELEMENTS'
OUTPUT_SPK_TYPE = 5
OBJECT_ID = -301
CENTER_ID = 0
CENTER_GM = 132712440018.0
REF_FRAME_NAME = 'ECLIPJ2000'
PRODUCER_ID = 'A. Wennmacher, Univ. of Cologne, Germany'
DATA_ORDER = 'EPOCH A E INC NOD PER MEAN'
INPUT_DATA_UNITS = ('ANGLES=RADIANS' 'DISTANCES=KM')
DATA_DELIMITER = ' '
TIME_WRAPPER = '# ETSECONDS'
LINES_PER_RECORD = 1
IGNORE_FIRST_LINE = 1
POLYNOM_DEGREE = 9
LEAPSECONDS_FILE = 'naif0009.tls'
The name of the SPK kernel for the reconstructed orbit of Helios 1
is 100528R_helios1_74345_81272.bsp.
The same set-up was used for Helios 2 with the following exception:
OBJECT_ID = -302
The name of the SPK kernel for the reconstructed orbit of Helios 2
is 100607R_helios2_76016_80068.bsp.
5) Production of prediction SPK kernel for Helios 1
---------------------------------------------------
The position data in helorb2.asc and the simulated time interval cover
the entire active mission time of Helios 2, which failed fataly on
1980-03-03 due to a transmitter failure. Hence the SPK kernel
100607R_helios2_76016_80068.bsp covers the entire time span during which
Helios 2 produced science data.
The situation is different for Helios 1. The position data available in
helorb1.asc only covers the time from launch to 1981-09-30. Helios 1
failed on 1986-03-15, and so there is a considerable amount of time for
which there is no orbital information available.
In order to provide trajectory information for Helios 1 for these times,
the orbital solution found for Helios 1 as described in section 2) was used
to "predict" the orbit between 1981-09-30 and 1986-03-15.
The resulting kernel has the name 160707AP_helios1_81272_86074.bsp.
The user is advised to exercise caution when using this kernel as there is
no straightforward way to establish the precision of the trajectory data
in that kernel.
References
----------
Dormand J. R., Prince P. J., Cel. Mech. 18, 223-232 (1978)
Kehr J. (ed), Helios - Interplanetary Experience, German Space Operation Center
(GSOC), 1977
Montenbruck O., Gill E., Satellite Orbits, Springer (2000)
Nelder J. A., Mead R., Computer Journal, Vol. 7, pp. 308-313 (1965)
Press W. H., Teukolsky S. A., Vetterling W. T., Flannery B. P.,
Numerical Recipes in FORTRAN, Cambridge University Press, 1992
Standish E. M., JPL Planetary and Lunar Ephemerides, DE405/LE405,
JPL Interoffice Memorandum IOM 312.F-98-048, Aug. 26 (1998)