168 kB PostScript reprint

Astronomical Data Analysis Software and Systems IV

ASP Conference Series, Vol. 77, 1995

Book Editors: R. A. Shaw, H. E. Payne, and J. J. E. Hayes

Electronic Editor: H. E. Payne

** T. B. Ake**

Astronomy Programs, Computer Sciences Corporation, Code 681/CSC,
Goddard Space Flight Center, Greenbelt, MD 20771

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 light-travel 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 near-earth environment in which *
HST* operates.
An observer would be wise to understand such effects as scattered earth light,
radiation background, and geomagnetically-induced 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
kms, 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 provided
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` (non-astrometry) or ` *.dbm` (astrometry) header file for each
observation. We summarize using this information to compute the position and
velocity of the * HST*.

**Table:** Onboard Ephemeris Model Parameters

The * HST* travels in a nearly circular orbit, with an altitude of about 600km
and velocity of 7.5kms. The orbital model used onboard is based on a
simple two-body 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 TIMEFFEC 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, at the epoch time of the parameters, ,

Compute the true anomaly, , using the equation of center to solve Kepler's
equation. This is typically expressed as a series in and **e**.
For small **e**, terms only up to are needed (e.g., Smart 1965, equation V-85),

Since trigonometric functions are computationally expensive, * HST* uses a
different form of this equation involving only and .
Collecting like terms of and expanding , one can show

Once the true anomaly is known, then the distance from the center of the earth,
**r**, is

The main perturbation on the orbital elements due to the non-spherical mass
distribution of the earth is the regression of the ascending node, ,
and the progression of perigee, , (Wertz 1978). The instantaneous
values at **t** are

The geocentric * HST* position, in J2000 rectangular coordinates in meters, is then

The corresponding equations for radial velocity can be derived by
differentiating those for position. Starting with **x**,

To eliminate the and terms, we use a well-known property of elliptical orbits that the velocity can be represented as the vector sum of two constant velocities (Smart 1965). These are the velocity, , perpendicular to the radius vector and, , perpendicular to the semimajor axis, where and . Designating V as the circular velocity, we have

The rectangular velocities can now be determined from the onboard ephemeris parameters. Defining the auxiliary variables

we have

Equations (1)--(7) yield the geocentric state vectors for * HST*.

**Figure:** Typical positional errors from an on-board ephemeris.
Original PostScript figure (82 kB)

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 2km for the two days during which the ephemeris is active. After this time, the error slowly increases, reaching about 5km a week outside the two-day range of the model. The error in velocity was found to be only 0.01kms over the whole period investigated. Comparisons for other weeks indicates that the positional error can be as high as 4km, but the velocity error is always very small, since the orbit is nearly circular.

Results using the onboard ephemeris are accurate enough for most needs. A
positional error of 4km translates to an uncertainty of 1mas at 5.5AU 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 , 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 self-documenting data sets so
that further analyses can be performed without resorting to additional outside
information.

Wertz, J. R. 1978, in Spacecraft Attitude Determination and Control (Dordrecht, Reidel), p. 65

168 kB PostScript reprint

adass4_editors@stsci.edu