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Astronomical Data Analysis Software and Systems IV
ASP Conference Series, Vol. 77, 1995
R. A. Shaw, H. E. Payne, and J. J. E. Hayes, eds.
Calculating the Position and Velocity Components of HST
T. B. Ake
Astronomy Programs, Computer Sciences Corporation, Code 681/CSC,
Goddard Space Flight Center, Greenbelt, MD 20771
Abstract. The HST uses an onboard model of its orbit to perform in
real­time various control functions relating to spacecraft operations and
observation support. The flight software uses the equation of center to
solve Kepler's equation for two bodies. New coefficients for the ephemeris
are updated every other day, and are archived as FITS keywords for
each HST observation data set. Using these coefficients,an observer can
perform a variety of calibration and characterization calculations relating
to the orbital position and velocity of the telescope. We present here the
methodology for computing the HST state vectors using this information.
1. Introduction
Many precise measurements with the Hubble Space Telescope (HST ) require
knowledge of its position and velocity during the observations. Orbital parallax,
velocity aberration, Doppler shift, and light­travel time can all be significant
when converting HST observations to the geocentric system, and from there to
the barycenter of the solar system. In addition, the quality of observations are
affected by the near­earth environment in which HST operates. An observer
would be wise to understand such effects as scattered earth light, radiation
background, and geomagnetically­induced motion on the data.
One way to determine the motion of HST during an observation is through
the definitive orbit files that are archived at the Space Telescope Science Institute
(ST ScI). Every other day the Flight Dynamics Facility (FDF) at Goddard Space
Flight Center computes the position and velocity of the HST for the previous
two days based on ranging measurements of the spacecraft. This information
is forwarded to the ST ScI in the form of a list of HST state vectors for each
minute of time, with each record giving the J2000 rectangular components of
the position, in km, and velocity, in km s \Gamma1 , in the geocentric inertial coordinate
system. The observer must find and extract the appropriate file(s) from the
archive and interpolate the HST state vector data to the relevant times of the
observations.
An easier method is to use the onboard ephemeris parameters that are pro­
vided with the data sets themselves. When the definitive orbit file is generated
by the FDF, a set of ephemeris coefficients is created. The HST flight software
uses these coefficients for various spacecraft control functions. During pipeline
processing at the ST ScI, these are archived in the *.shh (non­astrometry) or
1

2
*.dbm (astrometry) header file for each observation. We summarize using this
information to compute the position and velocity of the HST .
Table 1. Onboard Ephemeris Model Parameters
FITS Keyword FITS Description Symbol
EPCHTIME epoch time of parameters (secs since 1/1/85) Ü
MEANANOM mean anomaly (radians) M 0
FDMEANAN 1st derivative coef for mean anomly (revs/sec) —
M
SDMEANAN 2nd deriv coef for mean anomaly (revs/sec/sec) ¨
M
ECCENTRY eccentricity e
SEMILREC semi­latus rectum (meters) a(1 \Gamma e 2 )
RASCASCN right ascension of ascending node (revolutions)
\Omega 0
RCASCNRV rt chge right ascension ascend node (revs/sec)
—\Omega ARGPERIG argument of perigee (revolutions) ! 0
RCARGPER rate change of argument of perigee (revs/sec) —
!
COSINCLI cosine of inclination cos i
SINEINCL sine of inclination sin i
CIRVELOC circular orbit linear velocity (meters/second) VC
TIMEFFEC time parameters took effect (secs since 1/1/85) --
2. Computations
The HST travels in a nearly circular orbit, with an altitude of about 600 km
and velocity of 7.5 km s \Gamma1 . The orbital model used onboard is based on a simple
two­body system, with perturbations to certain Keplerian elements due to the
proximity of the earth. In Table 1 we list the relevant FITS keywords and
descriptions that can be found in the header files, as well as the symbols used
in the equations below.
The steps to compute the geocentric, rectangular coordinates for the HST
using the parameters in Table 1 are as follows. First, the observer should verify
that the correct onboard ephemeris has been archived with the data. The TIM­
EFFEC keyword specifies the beginning time at which the parameters are valid
and its value should be within 2--3 days before the start of the observations.
For a time of interest, t, calculate the mean anomaly, M, from the initial
position, M 0 , at the epoch time of the parameters, Ü ,
M = M 0 + 2ú
Ÿ
—
M(t \Gamma Ü) + 1
2
¨
M(t \Gamma Ü) 2
–
: (1)
Compute the true anomaly, š, using the equation of center to solve Kepler's
equation. This is typically expressed as a series in sin nM and e. For small e,
terms only up to e 3 are needed (e.g., Smart 1965, equation V­85),
š = M+ (2e \Gamma
e 3
4 ) sin M+ 5
4 e 2 sin 2M+ 13
12 e 3 sin 3M:

3
Since trigonometric functions are computationally expensive, HST uses a differ­
ent form of this equation involving only sin M and cos M. Collecting like terms
of e n and expanding sin nM, one can show
š = M+ sin M(2e + 3e 3 cos 2 M \Gamma
4
3 e 3 sin 2 M+ 5
2 e 2 cos M): (2)
Once the true anomaly is known, then the distance from the center of the earth,
r, is
r = a(1 \Gamma e 2 )
1 + e cos š
(in meters): (3)
The main perturbation on the orbital elements due to the non­spherical
mass distribution of the earth is the regression of the ascending
node,\Omega\Gamma and
the progression of perigee, !, (Wertz 1978). The instantaneous values at t are
\Omega = 2ú
h\Omega 0 + —
\Omega\Gamma t \Gamma Ü)
i
and (4)
! = 2ú [! 0 + —
!(t \Gamma Ü )] : (5)
The geocentric HST position, in J2000 rectangular coordinates in meters, is
then
x = r
[cos\Omega cos(! + š) \Gamma cos i
sin\Omega sin(! + š)] ;
y = r
[sin\Omega cos(! + š) + cos i
cos\Omega sin(! + š)] ;
z = r sin i sin(! + š):
(6)
The corresponding equations for radial velocity can be derived by differen­
tiating those for position. Starting with x,
—
x = x
r
—
r \Gamma r(2ú —
! + —
š)
[cos\Omega sin(! + š) + cos i
sin\Omega cos(! + š)] \Gamma 2ú
—\Omega y:
To eliminate the —
r and —
š terms, we use a well­known property of elliptical orbits
that the velocity can be represented as the vector sum of two constant velocities
(Smart 1965). These are the velocity, ¯=h, perpendicular to the radius vector
and, ¯e=h, perpendicular to the semimajor axis, where ¯ = GM \Phi and h = r 2 —
š =
¯a(1 \Gamma e 2 ). Designating VC = ¯=h as the circular velocity, we have
r —
š = VC (1 + e cos š) and
—
r = eVC sin š:
The rectangular velocities can now be determined from the onboard ephemeris
parameters. Defining the auxiliary variables
a 0 = 1
r eVC sin š and
a 1 = VC (1 + e cos š) + 2ú —
!r;
we have
—
x = a 0 x \Gamma a 1
[cos\Omega sin(! + š) + cos i
sin\Omega cos(! + š)] \Gamma 2ú
—\Omega y;
—
y = a 0 y \Gamma a 1
[sin\Omega sin(! + š) \Gamma cos i
cos\Omega cos(! + š)] \Gamma 2ú
—\Omega x;
—
z = a 0 z + a 1 sin i cos(! + š):
(7)
Equations (1)--(7) yield the geocentric state vectors for HST .

4
Figure 1. Typical positional errors from an on­board ephemeris.
A HREF=''http://hoth.stsci.edu/adass­figs/aket1.eps''? Original PostScript fig­
ure (82 kB) !/A?!P?
3. Comparisons with Definitive Orbit Data
We can compare the results from the ephemeris model directly with the definitive
orbit data. In Figure 1 we show the total error in position for several days during
the first week of 1994 April. In this example the error is below 2 km for the two
days during which the ephemeris is active. After this time, the error slowly
increases, reaching about 5 km a week outside the two­day range of the model.
The error in velocity was found to be only 0.01km s \Gamma1 over the whole period
investigated. Comparisons for other weeks indicates that the positional error
can be as high as 4 km, but the velocity error is always very small, since the
orbit is nearly circular.
4. Conclusions
Results using the onboard ephemeris are accurate enough for most needs. A
positional error of 4 km translates to an uncertainty of 1 mas at 5.5 AU from
the earth, so parallax errors are small for all but nearby passing asteroids and
comets. The geographic position of HST can be determined to better than
0: ffi 03, much more accurately than needed to compute effects due to the orbital
environment. The error in velocity is much smaller than can be measured with
the HST instruments. We conclude that using the onboard ephemeris eliminates
the need to import definitive orbit data. This exemplifies the value of providing
users with self­documenting data sets so that further analyses can be performed
without resorting to additional outside information.
References
Smart, W. M. 1965, Spherical Astronomy (Cambridge, Cambridge University
Press)

5
Wertz, J. R. 1978, in Spacecraft Attitude Determination and Control (Dordrecht,
Reidel), p. 65