lew@ihuxr.UUCP (Lew Mammel, Jr.) (12/09/83)
I have obtained (through the courtesy of Bill Jeffreys) a copy of the
article "On the velocity of light three centuries ago", by S.J Goldstein,Jr.
J.D. Trasco, and T.J. Ogburn III, from The Astronomical Journal, Vol 78,
no. 1, Feb 1973, pg 122.
The authors attempted a reanalysis of 40 of Roemer's orginal observations
of eclipses of Io by Jupiter. I say attempted because, sad to say, they
made a gross conceptual blunder in their fitting technique. It's very easy
to correct from the data they give in the article (assuming all their
orbital calculations are correct, and I presume they are.) I will
describe the technique, explain their error, and give my correction.
Consider the following quantities:
to[i] = times of observed events
ta[i] = calculated times of events
d[i] = calculated delay time due to light travel
tp[i] = calculated times of observations = ta[i] + d[i]
The authors adjusted the delay times to minimize the sum of (to[i] - tp[i])^2
Their blunder arose as follows: The calculated times of events were adjusted
to an "empirical" initial point. That is, the actual phase of Io wasn't
projected back over 300 years in absolute time. This would involve knowing
the orbital period of Io to better than 1 ppb. The adjustment is accomplished
by setting the sum of (to[i] - tp[i]) to zero. The blunder consisted in failing
to readjust the phase as they varied the d[i]. Hence, instead of adjusting
c to account for the VARIATION in the apparent period of Io, they were
adjusting the AVERAGE time of observation to agree with the average predicted
time of observation. This gave them the value of c right back which they
had used to adjust the initial phase of Io. I corroborated this by calculating
the error term which this gives and comparing it to the error curve in
the paper. This curve is parabolic with the approximate formula
rms = 136 + 2.5 * x^2 ( X 10^-5 days )
where x is the percent deviation of c from c0, the nominal value.
(Their graph is mislabeled "X 10^-6 days", they give .001361 days in
the article and quote it as 118 seconds, which checks. I get this precise
value from the rms deviation of their reported data also.)
According to my surmise, I calculated that the 2.5 factor on x^2 should
be given by 1/2 * (d0/100)^2/136, where d0 is the average delay time, which
the authors give as 2598 secs or 3007 * 10^-5 days. This comes to 3.3,
which compares nicely to 2.5 considering the numbers we're throwing
around here.
To get the correct result, I subtracted d0 from each d[i] to get the variation
in delay time. Calling these values d'[i], we want to minimize:
sum of ( (to[i] - tp[i]) - delta * d'[i] )^2
... where we use the authors tp[i] which include d'[i] based on c0.
This is accomplished by setting:
delta = { sum of ( (to[i]-tp[i]) * d'[i] ) } / { sum of d'[i]^2 }
The authors effectively had the additional term, delta * d0, in the sum.
This adds (delta * d0)^2 to the result, forcing delta to zero for the
minimum. I get delta = .08, meaning c = c0/1.08 .
For the estimated error, I used:
rms error / {sum of d[i]^2} = .09
So I get a value of c which is 8% low with an estimated error of 9%.
This contrasts with the authors' conclusion that the data demonstrated
a value of c to within 1/2% of the accepted value.
Of course, I may be wrong! but I really don't think so at this point.
I'm wondering if anybody has paid attention to this issue in the academic
community. Maybe I'm the nth critic to demolish this paper. I would think
that there would be some interest since Roemer's observation is cited
so often. Does anybody think I should send this to The Astronomical Journal?
After all, they published the original. Why didn't a referee spot this?
I would sincerely appreciate any comments.
Lew Mammel, Jr. ihnp4!ihuxr!lew