Requiem for Relativity

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by Jim</i>
<br />...it (the motion of the moon) has an effect on the motion of Earth's orbit and position within the solar system. The Earth's orbital speed slows and speeds up on the same time cycle as the moon although at a much lower rate. ...
<hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">

Hi Jim,

I agree, this is important. Thanks for calling my attention to it.

- Joe Keller

Dr Joe, I know you use the JPL system and that system uses the barycenter idea(I think)to generate it's positions.The idea is both Earth and the moon orbit the barycenter and thereby places the location of the Earth in the wrong place. So,(I think) the location of Earth is never where the model places it and since it is the observation platform there is a very tiny error introduced by assuming the moon is pushing the Earth around the barycenter. The true force is not about a barycenter and the true effect is not factored into the JPL generator. I think ESA astronomers(maybe they all do) use some other system to determine very tiny fuzzy stuff like you are finding in the JPL system.

My Dec. 22, 2009 post, "The Asteroid Resonance, Part 4", summarizes what is known about the axes of Davida and Laetitia. In brief:

Davida, 2007 Keck telescope determination from projected shape: (ecliptic long, lat) = (297, 21); my unweighted mean of all published determinations: (302, 22)

Laetitia, restudy of historical observations, in 2002 by Kaasalainen: (323, 35); my unweighted mean of all published: (307, 38)

On the Minor Planet Center website, are many Visual magnitudes of Monterosa to 0.01 precision, by the USNO at Flagstaff. The four oppositions, having the largest number of such observations within 30d of opposition, are those 1999 through 2003, which have 8, 11, 15 & 9. These oppositions are well-spaced around the zodiac. Assuming rotation period 5.164h per Warner, I convoluted the USNO magnitudes with sine and cosine, to find the amplitude of lightcurve variation near each opposition. Applying the original classic "amplitude phase" method which has been used for most published asteroid axis determinations, my preliminary estimate, based on the 1999 & 2003 oppositions, is that Monterosa's axis longitude is between 288 and 312.

I'll make a more refined estimate for 947 Monterosa (and eventually 1717 Arlon). Instead of assuming that the asteroid lies exactly opposite the Sun with the Sun at opposition longitude, I'll use the asteroid's heliocentric and geocentric longitude and latitude, at the mean observation time. Helmholtz' photometric principle requires that the heliocentric and geocentric coordinates be interchangeable in my formula. Let's consider extreme shape-based lightcurves (rotating broomstick) and extreme albedo-based lightcurves (rotating beachball blackened on its western hemisphere). Let the axes of illumination and observation be either 0 deg or 90 deg from the rotation axis. If either the illumination angle ("i") or observation angle ("o") is 90 deg, the lightcurve variation approaches 100%. So an improved, Helmholtzian approximate formula for the lightcurve variation, is max( sin(i)^2, sin(o)^2 ). For the Monterosa data I used, the difference between the "i" and "o" lines at the mean dates, is always less than 8 deg, and the mean dates are never more than 15d from the opposition.

For Monterosa, I'll continue arbitrarily to exclude those minorities of observations more than 30d from opposition (because of the large change in the angle of observation, and large difference between the lines of observation & illumination) or in years other than 1999-2003 (because these have at most 6 observations within 30d of opposition, hardly improve the distribution of observations around the zodiac, and risk confusion due to changing photometric standards). My axis (with the usual ambiguity of sign) will be that which optimizes the agreement between observed and predicted magnitude variation amplitudes for the previously estimated 5.16394h period.

My solution is defined as that axis, which minimizes the sum over the 4*3=12 choices of two different oppositions, of the squares of the differences, between predicted and observed, of the logs of the ratios of the amplitudes of the 516394h period magnitude variation. The axis (whether clockwise or counterclockwise cannot be determined this way) of Monterosa's rotation is thus estimated as (long, lat) = (272, -10). Other similar definitions, or reasonable weighting of the four oppositions, only alter the result by a degree or two.

For 1717 Arlon, rotation period 5.1484h, only four oppositions have USNO magnitudes recorded as described above for Monterosa. These are the four oppositions 1998 through 2003, which have 10, 10, 15 & 12 observations, resp., within 30d of opposition. Though the number of such observations is slightly larger than for Monterosa's four most-recorded oppositions, the distribution around the zodiac is poorer, but tolerable. I'll make a new post below to give my findings on Arlon when I've used the same computer program on Arlon's data, that I used for Monterosa.

The same method as the previous post, estimates Arlon's axis as (long, lat) = (250, -16), only 22 deg from Monterosa's (90 maximum due to handedness ambiguity; p = 7.4%). The most non-negligible lightspeed effect, is the Doppler effect due to Earth's moving away from the asteroid. This frequency alteration affects the convolution. A month from opposition, this amounts to 30km/s / 300,000km/s * sin(30) / 2 * (30d * 24 / 5.15h * 360 deg) = 0.022 radian; Monterosa and Arlon, according to their photometric plots on the Minor Planet Center website, have variation amplitudes no more than roughly 0.1 mag, so even the farthest from opposition, 0.022 radian, amounts to no more than roughly 0.1*0.022 = 0.0022 mag, less than half the maximum 0.005 mag rounding error, and this error, like the rounding error, would tend to cancel randomly.

For Arlon, the illumination and observation lines were, for all four mean time positions, less than 6 deg apart. The farthest of any mean observation time, from its opposition, was 11d.

Collecting the best axis determinations, I would choose the Keck telescope result for Davida because this far more powerful instrument allowed a different, simple, foolproof method. I would choose my mean of all the Laetitia determinations, because the superiority of the latest determination, in 2002, is not obvious. Here then are the results:

Davida within a few deg of (297, 21)
Laetitia within some deg of (307, 38)
Uranus rotation, Uranus' major moons orbit and rotation (257.6, 7.7)
Monterosa roughly (272, -10)
Arlon roughly (250, -16)

All five fall within 54 deg of latitude and 57 deg of longitude.

Dr Joe, Can we assume these data points(being in some sort of common pattern)indicate something important about the solar system? Can you make a determination about what it all shows us? What caused this and when?

The rotation periods of Monterosa & Arlon aren't known accurately enough to assume any relationship between the phases at the different oppositions. Nominally, Warner gives 0.001h error for Monterosa, but his determinations, both of the period and of the error, are so different from Poncy's, that likely the error is larger. The error in the period determination for faint Arlon, likely is larger than for Monterosa. Even with only 0.001h error = 1/5000 cycle for Monterosa, the 2000 rotations between oppositions give 2/5 cycle = 144deg error in the relative phase from one opposition to the next.

In the preceding posts, my convolution was with sines and cosines of period equal to the rotation period. This is appropriate for the most extreme albedo-based lightcurve, the beachball with a black western hemisphere. For the most extreme shape-based lightcurve, the broomstick, the brightness variation is a second harmonic (two peaks per rotation); for it, the convolution should be with sines and cosines of period half the rotation period. Published articles generally assume that most of the magnitude variation is shape-based and moreover that the asteroid is a triaxial ellipsoid of uniform albedo, rotating about one of its principal axes. That is, published authors assume that a second harmonic convolution is best.

When I perform the second harmonic convolution instead, I find for the axes:

Monterosa (286, 39)
Arlon (365, 28)

Thus Monterosa's axis more likely is only 20deg from Davida's and 16deg from Laetitia's.

Because I have as few as 8 data per opposition, it would be nonsense to find harmonics higher than 8th; to be conservative, I'll arbitrarily use the 1st through 4th. Instead of the amplitude of one harmonic, I'll consider the square root of the sum of these first four squared amplitudes. This gives axes

Monterosa (273, 27)
Arlon (285, -

This refinement of the technique, suggests that Monterosa differs 23deg from Davida, 30deg from Laetitia and 24deg from Uranus.

Yet another reasonable refinement, would be to use the first three even harmonics, i.e. the 2nd, 4th & 6th, because for the triaxial ellipsoid and other shapes with such symmetry, the odd harmonics give zero when convoluted with the lightcurve. This result is

Monterosa (272, 31)
Arlon (301, -42)

Besides the absolute ambiguity of (long, lat), (x, y) &lt;--&gt; (-x, -y), there is also an approximate ambiguity (x, y) &lt;--&gt; (x, -y), which is absolute if the asteroid lies perfectly on the ecliptic. That is, if (301, -42) is the best fit for Arlon, then (301, +42) must be a good fit; if (285, - is the best fit for Arlon, then (285, + must be a good fit. So, the results for Arlon are less disparate than they seem.