Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

Re: Accuracy, etc.--astronomical measurements

I was surprised and offended by Mr. Palmer's dismissal of my contribution
yesterday on the issue of determining planetary masses. I've been a happy
camper on this list since 1992, have a Ph.D. in Astronomy, and have been
teaching physics and astronomy for 30 years. I received authoritative
bluster based on a superficial understanding of the physics and of the
observational problem. God help this man's students!

To understand the limitations on determining planetary masses, we start
with a discussion of Kepler's two body problem. Consider the motion of
masses M1 and M2 about each other under the influence of gravity. We
write the equations of motion (i.e. F = mA) for each in an outside,
inertial reference frame. The solution of these equations can be
expressed in either of two ways; the first considers the motion of M1
_relative _ to M2; the other way of looking at the solution is in terms
of the separate elliptical orbits the two bodies follow w.r.t. the center
of mass (CM). All these ellipses have the same eccentricity, and the
semimajor axis of the relative orbit, a, is related to the semimajor axes
of the two CM orbits by

a = a1 + a2

Further, the total mass of the system is M = M1 + M2. Then Kepler's third
law says

G M = (4 Pi^2) a^3 / T^2.

G is the gravitation constant, M the TOTAL mass, Pi = 3.14.., and T the
siderial period. (Sorry about the FORTRAN like notation). a and T can be
derive from (many) measurements of the positions vs. time. Knowing G, _
from Earth-bound experiment _ we can calculate the total system mass. If
the individual CM orbit semimajor axes, a1 and a2, are known, then the
ratio of the masses can be got from

M1 a1 + M2 a2 = 0,

and then the individual masses can be found. If M1 ~ M2, a prolonged
series of observations may (depending on the orbits spatial orientation
w.r.t. the observer) may reveal a "wobble" about the CM, allowing the
determination of a1 and a2. The best example of this process is the
determination of the masses of the Earth and the moon. The Earth is so
much heavier then the moon, the system CM falls below the Earth's surface,
on the line of centers. Yet the monthly wobble of the Earth can be
accurately determined - given enough observations.

On the other hand, if M1 >> M2, the tail doesn't wag the dog a detectable
amount, and the total system mass can't be split into M1 and M2. Thus the
mass of a GPS satellite can NOT be found this way. But of course you can
- and better! - weigh the thing before you blast it into orbit.

In pre-Sputnik 1955, C.W. Allen in his _Astrophysical Quantities _ gave
masses for the moon, the four Galilean satellites of Jupiter and 15 of the
larger satellites of the outer planets to 2 or 3 sig. fig. In all cases,
wobbles of the primary planet, due to the orbiting satellite, could be
observed.

Avoiding the problem of an imprecise value of G:

Now let's compare the mass of a planet, Mp, to the mass of the sun, Msun.
This requires that the planet have a satellite - or orbiting probe - of
mass Ms, with observed semimajor axis as and sidereal period Ts for motion
about the planet. The planet+satellite revolve about the sun in an orbit
with ap and Tp.

We write K III twice: once for the planet+satellite revolving about the
sun

G (Msun + Mp) = (4 Pi^2) ap^3 / Tp^2.

and for the satellite revolving about the planet

G (Mp + Ms) = ( 4 Pi^2) as^3 / Ts^2.

Dividing

(Msun + Mp) ap Ts
---------------- = ( ---)^ 3 ( --- )^2
(Mp + Ms) as Tp


Given that Msun >> Mp >> Ms, we can find, from many observations, the
value of Ms / Mp, INDEPENDENT of the value of G.

Kenneth R. Lang, in _Astrophys. Data: Planets and Stars _ , 1992, gives
the following so called "reciprocal masses" for the planets:

Msun / Mp
Merc 6023600
Ven 408525.1
Earth 332946.043
Earth + Moon 328900.555
Mars 3098710
Jup 1047.3492
Sat 3497.91
Uran 22902.94
Nep 19434
Pl 13 x 10 e 7

Lang attributes these to J. Myles Standish, Jr. (1988) at JPL. The large
number of sig. fig. quoted reflect the fact that these values depend on
the hundreds or thousands of position and time measurements needed to
establish accurate values of the orbital elements.

The values for Mercury and Venus, which lack natural satellites, must come
from orbiting or passing space probes.

Lang also quotes

G = (6.67259 +/- 0.00085) x 10 e -8 in cgs. units.

So G is essentially good to four plus sig.fig. Masses and GM values
determined from the solar mass _ in kg _ will all have comparable
precision.

I find myself at a loss about how to respond the profession of wholistic
philosophy in the last paragraph. A few points:

Determining the mass of an astronomical object is a fundamentally
different problem from plunking an object on a scale in the lab. The
physical part of the problem -- plugging into Kepler's laws -- is trivial.
The determination, with high precision, of the period and semimajor
axes is anything but trivial.

The precision of the reciprocal masses reflects the combined observations
over several centuries of observation, especially in the case of Jupiter.
Dealing with such long-term problems is characteristic of astronomy, and
is not typical of physics.

The whole business of resolution, precision, accuracy, systematic error
and statistical treatment of data goes back to the work of Gauss and other
positional astronomers in the last century.

During my time as a teacher of physics and astronomy, Planetary Physics
has emerged from astronomy, geophysics and space probes. People who
study the members of the solar system no longer reguard themselves as
astronomers and rightly so. The methods are completely different.

The one exception may be celestial mechanics. Numerical and symbolic
calculation certainly have been accelerated beyond belief by computers.
As have the complexity of the problems and the advent of resonences
and chaos. Underneath the reliance on observation and classic analytic
math looks very similar.

Beyond the heliopause (say 150 AU or about 1/2000th the way to Proxima
Cen., the nearest star) the distinctions between experimental physics and
observational astronomy remain relatively clear. Whereas physicists tend
to think of themselves as being either theorests or experimentalists,
significant numbers of astronomers carry of "numerical experiments,"
modeling the inaccessible interiors of stars, interstellar gas clouds, etc.
This is the sort of Astrophysics I've done.

============

Mr. Palmer wrote:


By the same token, the ratios of planet/Sun
mass can be found with great precision and accuracy.

That's not true, and the degree of accuracy with which those ratios
are known is much less than the accuracy to which GM anything is
known.

If that were true, then by extension one could measure the masses of
the satellites of planets and, indeed, of artificial Earth satellites
for which very accurate orbits can be determined (e.g. the Global
Positioning System). (The latter orbits are known more exactly than
are any planetary orbits.) Planetary satellite masses were not known
until experiments were done with spacecraft.

At this point, the
astronomers can go no farther--- a value of G is required to convert
from, say, solar masses to kg. But this value of G has had to be
determined by Earth-based experiment, e.g. that of Cavindish. Thus the
ratio of GM(observed) / G(Cavindish) is limited by the uncertainty in G,
and thus astronomical masses expressed in kg are relatively poorly
determined.

A further remark: a highly accurate value of GM is just what you want for
sending rockets to Mars, etc.

We sometimes say the physicists can do experiments, but astronomers must
wait for them to happen. An important distinction, I think.

That is no longer true, of course. Many scientists involved in the
exploration of space consider themselves to be astronomers.
Experimental astronomy dates back to the eighteenth century when
science at last accepted the idea that stones can fall from the sky,
and meteorites could be analyzed.

I prefer to consider all interrogation of Nature as being in one
category. I refer to this body of knowledge as empirical knowledge,
and the methods of acquisition, observation and experiment together,
as "empiry". That avoids the petty union job description battles
that ensue if one really takes that distinction to be important!

Leigh