No fundamental reason has been identified for rejecting the notion of measuring the Newtonian gravitational constant G by observing an artificial binary in a near-Earth orbiting laboratory

Baker, Stephen D.

English

NASA Technical Reports Server

N8 - 15603
7f
1987
NASA/ASEE SUMMER FACULTY RESEARCH FELLOWSHIP PROGRAM
MARSHALL SPACE FLIGHT CENTER
THE UNIVERSITY OF ALABAMA IN HUNTSVILLE
SOME CONSIDERATIONS ON MEASURING THE NEWTONIAN
GRAVITATIONAL CONSTANT G IN AN ORBITING LABORATORY
Prepared by:
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Date:
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Stephen D. Baker
Professor
Rice University
Department of Physics
Space Science
Astrophysics
High Energy Astrophysics
C. A. Meegan
August II, 1987
The University of Alabama
in Huntsville
NGT-01-008-021
II
https://ntrs.nasa.gov/search.jsp?R=19880006221 2020-03-20T07:38:49+00:00Z
ABSTRACT
We have identified no fundamental reason for
rejecting the notion of measuring the Newtonian gravita-
tional constant G by observing an artificial binary in a
near-earth orbiting laboratory.
II-:l
ACKNOWLEDGEMENT
I would like to thank the scientists and staff in
the Space Science Laboratory for their hospitality and
support. My colleague, C. A. Meegan, and Branch Chief Tom
Parnell were helpful with practical details and with
generally making me feel at home. Susan Burrer and Shelby
Morris patiently made many revisions in a manuscript based
on the summer's topic. Jena Morris performed some useful
calculations for me. Finally, I appreciate the support of
the NASA/ASEE Summer Faculty Fellowship Program.
INTRODUCTION
Of the fundamental contants of nature, the Newtonian
gravitational constant G is by far the least well known
(only about one part in ten thousand, or worse). The use
of artificial binaries to determine this constant has been
regularly proposed since it became thinkable to have
orbiting laboratories. A regular objection to this method
has been that, in near earth orbit, the gravitational field
of the Earth is changing so rapidly with distance from the
earth that "tidal forces" would make the orbits of
artificial binaries unstable. This is not the case,
however; certain retrograde orbits are indeed stable and
could be used to sample the gravitational attraction
between two balls for a long time, thus permitting the
determination of G.
OBJECTIVES
The objectives of the summer's activity were to become
further familiar with the literature relevant to the
problem at hand and to draw any further oonlusions
concerning the feasibility of measuring G in a near earth
orbiting laboratory. A secondary objective was to complete
a manuscript on this topic, with Adam Falk, begun last
summer, for submission to the American Journal of Physics.
II-ii
REPORT
A common suggestion for measuring the Newtonlan
gravitational constant in an orbiting laboratory is simply
to put two balls in orbit around each other and measure the
resulting orbital elements and period of the motion of such
an artificial binary, thereby determining G. However, the
radial variation with distance of the gravitational field
of the earth is so large that "tidal forces" on the balls
in near-earth orbit can be several times greater than the
gravitational attraction between them. The presence of
these relatively strong tidal forces close to the earth has
led some writers to assume that two objects will not stably
orbit about each other and that this method of measuring G
is impossible, or at least impractical. Here is what they
say:
Farinella, Milani, and Nobili (1980)
"This means that the physical limits to the density
of [the binaries] imply that in a low Earth orbit
no stable motion of the test mass around the primary is
possible."
Avron and Livlo (1986)
"Unfortunately, such an experiment is not possible
with the Space Shuttle, the main reason being its
low altitude."
Hills (1986)
"If the semimaJor axis of the binary is
sufficiently large, the tidal field would force its
dissociation."
However, Michel Henon in a beautiful series of
papers has explored the regions of long-term stability of
many classes of orbits of point particles which are moving
under their mutual gravitation in a frame which is orbiting
about a large third body. He finds that certain retrograde
orbits are stable and states (in 1970) that:
"Contrary to the usual [and, evidently, persistent]
belief, there is no limiting [orbital] radius for the
satellites."
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Henon provides a very complete description of what is known
as Hill's Problem, where the motion of one of two two equal
mass balls is described relative to an orbiting frame of
reference in terms of dimensionless coordinates:
where the time unit is chosen to be T/2_ where T is the
orbital period of the spacecraft and the length unit is
chosen to be (m/4M) w3 A, where M is the mass of the earth
and A is the radius of the orbit of the spacecraft.
Since one must use balls of finite density (and
therefore finite size) we may investigate the tolerances on
the inital conditions of the orbit so that the balls
collide neither with each other nor with the walls of the
spacecraft. In Figure 1 we show the "launch window" for a
10 kg, B cm radius, ball started out on the neutral llne at
z - 15 cm with the velocity components shown on the axes of
the figure. The other identical ball is launched
symmetrically to maintain the center of mass at the origin.
Trajectories which result in the balls' striking each
other or the enclosure of the spacecraft are represented by
diamonds or crosses, respectively. For this example, the
enclosure is taken to be a cube of edge length 110 cm.
Those trajectories which survive for 28 orbital periods
about he earth are unmarked. Launching from points other
than on the neutral llne (the z axis) is, of course,
possible, and we find that the allowable ranges on the
initial velocities are very similar to those in Figure 1.
While the tidal forces can be significantly reduced
by going to a laboratory orbiting the earth at a large
radius, it turns out that one does not gain much in terms
of allowable ranges of initial velocities. We conclude
therefore that tidal forces do not pose any particular
problem in using an artificial binary in an experiment to
determine G.
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Are there other fundamental processes which might
make it impossible to carry out such an experiment? We can
consider several items:
I. Gradients due to components of the spacecraft. The
spacecraft in which the experiment would be performed
should be constructed to provide a minimal gradient at the
position of the orbiting balls, but it is not difficult to
do this. Since the experiment is in free fall, it is the
gradient of the gravitational field which affects the
relative motion of the balls and not the strength of the
field itself. For example, the gradient due to a I00 kg
mass a distance S m from the center of the experiment gives
a gradient only 4.0E-S times that of the earth.
2. Fluctuations in the gradient due to orbital eccentricity
and non-sphericity of the Earth. The oblateness of the
earth would create a variation of about I% in the earth's
gradient if the experiment orbits in a circular polar
orbit, and less in an equatorial orbit. Eccentricity of
the orbit would also give rise to variations in the
gradient (_bout 1% for variations of 20 km in altitude).
These effects are accurately calculable, however, and could
be accurately compensated for.
S. Electrical charging of the balls. This should not
present insurmountable problems, either. For example, to
keep the electrostatic force a million times smaller than
the gravitational force Between the Balls, used in the
example shown in Figure I, the charge on each ball would
have to be less than about one piooooulomb, not impossibly
small. Larger balls would permit larger charges.
CONCLUSION
We conclude therefore_that there is still no
fundamental reason to believe that an experiment to use an
artificial binary in a near-earth orbiting laboratory is an
unreasonable way to obtain a very good determination of the
Newtonian gravitational constant G.
If- 3
REFERENCES
P. Farinella, A. Milani, and A. M. Nobili, Astrophysics
Space Science 73(1980)417
Y. Avron and M. Livlo, Astrophysical _ournal 304(1986)L61
9. G. Hills, Astronomical _ournal 92(1986)986
M. Henon, Astronomy _ Astrophysics 30(1974)317, s41d
references therein.
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Some considerations on measuring the Newtonian gravitational constant G in an orbiting laboratory

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