Review: the asteroids 511 Davida, 39 Laetitia, 947 Monterosa and 1717 Arlon comprise the set of asteroids whose rotation period clusters near the minimum. Also, they have approximately the same rotation axis (Davida and Laetitia's axes precisely determined in the literature; Monterosa and Arlon's roughly determined by me using the classical "amplitude-phase" method with 8 to 15 USNO-Flagstaff 0.01mag observations at each of four points). Their rotation period is a special time period: the moons Deimos and Phobos of Mars; and Miranda, Ariel and Umbriel of Uranus, lap each other in exact whole multiples of this same period. Particles in all but the densest one, of the rings of Uranus, lap important moons of Uranus in half-multiples, 1.5 or 2.5, of this period.

These four asteroids happen to align with each other and the Sun, in Dec. 2012. Let's define best alignment of the asteroids along a line through the Sun, as minimization of the sum of squares of sines of angles asteroid-Sun-asteroid. By this definition, the best alignment is Dec. 15.58 UT, 2012. This result and in the next paragraph, use ecliptic coordinate positions, to 0.0001deg & 0.000001AU, interpolated quadratically through the dates Dec. 1, Dec. 31 & Jan. 30.

Alternatively, let's also include Uranus as one of the asteroids. Now the result is Jan. 29.46, 2013. On this date, the mean longitude (mod 180) is 183.72 deg, and mean latitude (using appropriate signs) +4.38 deg. With or without Uranus, the standard deviation in direction, i.e. sqrt(sum(sine squared of difference angle)/(n*(n-1)/2-1))/sqrt(2), is 0.20 radian/ sqrt(2) = 8.1 deg. The Standard Error would be 8.1deg / sqrt(5) = 3.6deg.

If I choose the Keck result for Davida's axis, my average of published results for Laetitia, and my 1st-4th harmonic result for Monterosa & Arlon, together with Uranus their mean axis (long, lat) is (284, 17). This is 98.5deg from the mean Jan 29.46, 2013 direction (long, lat) = (183.72, 4.38). Also, Earth's rotation axis (at 2013.0AD) is 95.8deg from the former and 87.5deg from the latter. Within the error bars, these three are an orthogonal triple; this orthogonality recurs about every fourth time for awhile, because 6340yr is about a fourth of Earth's precession cycle. So, the event this time might be especially mild or especially severe.

In an earlier post I noted that an imaginary object with half Luna's sidereal period, would be lapped by Earth's rotation, with period exactly equal to five times Monterosa's rotation period. The Mars sidereal rotation period given by TP Snow in "The Dynamic Universe" (1983) is 24.622944h, consistent with, though more precise than, the NASA Fact Sheet. Without loss of precision, I'll use Mars' sidereal revolution period 686.980d and Luna's, 27.32166d. The apparent rotation frequency of Mars, seen by an observer in solar orbit inferior to Mars, would subtract Mars' orbital frequency. With this motivation, I find

A Support Astronomer for Hawaii's Subaru Telescope, "Ichi Tanaka" (quoted, secondhand from a messageboard, in the news website "Hawaii 24/7", article posted 9:08AM, June 29, 2011) says that the bubble was recorded by two different Mauna Kea video cameras (at the Subaru Telescope and at the Canada-France-Hawaii Telescope, known as the CFHT) and observed in real time by multiple persons at the Subaru Telescope location. Tanaka says he later communicated with astronomer Kanoa Withington, who was at the CFHT location. Tanaka says:

"It appears that the event happened not on the Summit area, but much farther away, according to the comparison of the two videos."

There is no "Ichi Tanaka" listed among the Subaru Telescope staff, but there are Support Astronomers named Makoto Tanaka and Miki Ishii.

This implies that the parallax was so small that it was not obvious to these professional astronomers. Comparison requires some care, because the center of the bubble was moving up and to the left at roughly a 30deg angle to the vertical (see, e.g. the two photos in "MailOnline", the online edition of the Daily Mail, 11:40 AM on 30th June 2011). Yet many stars were brightly seen near and through the bubble, so accurate comparison would be possible. The Subaru Telescope housing, according to their website, is at

* Altitude: 4139m (13,580ft.) * Latitude: 19d 49m 32s N * Longitude: 155d 28m 34s W

According to Wikipedia, the CFHT is at

* Altitude 4204m (13,793ft) * Latitude 19d 49m 31s N * Longitude 155d 28m 10s W

So, roughly, one telescope is 700m west of the other. If the bubble were directly northeast, then even a hasty effort, determining that the parallax was less than one degree, would correspond to a bubble distance of at least 700m/sqrt(2)*180/pi = 28km. Suppose the Daily Mail (MailOnline) photos are uncropped; this is probably so, because the same margins appear in the photo that Linda Moulton Howe put on her "Earthfiles" website, though it is a different frame. A 35mm film camera with a typical 50mm focal length lens has about 38deg field top to bottom; if the video camera is the same, then in the last Daily Mail frame, the bubble reaches ~20deg above the horizon, thus a lower bound for the height, is 10km. Tanaka says the bubble expanded to over 45deg. Strangely, the bubble's lower limb remained always near the horizon as the bubble rose and grew, so at more than 45deg apparent diameter the height would be over 24km. Also, the peak of Mauna Kea is about 27km from the nearest shore (which is northeast) as I measure on my roadmap; so, the presumed failure of the bubble to reach the shore, corroborates the presumed failure of the astronomers to notice parallax in excess of one degree.

Could a bubble in our atmosphere 24km high, or even 10km high, be so perfectly spherical? Cumulus clouds form "anvils" at the tropopause, ~30,000ft = 9km altitude. The speed of sound in a perfect gas of a given molecular weight is completely determined by its temperature: proportional to the square root of the temperature. The absolute temperature at the tropopause is typically 25% lower than at sea level; the temperature rises again, almost to the sea level value, at the stratopause, ~45km altitude (the molecular weight of the atmosphere is constant up to 100km)(Handbook of Chemistry & Physics, 68th ed., pp. F146-F147). With so much variation in the speed of sound, and refraction of sound, and presumably of shock waves also, the bubble hardly could be so spherical.

The atmosphere is halved in density roughly every 18,000ft = 5.5km. If the bubble is in our atmosphere, then the absolute humidity, i.e. the water vapor available for condensation (a phenomenon that often makes shock waves visible) is only a small fraction, at the top of the bubble, vs. at the bottom of the bubble. Yet the bubble appears almost as bright at the top as at the bottom. Likewise, if the bubble's brightness is due to spilled rocket fuel or other atmospheric contamination, the motion of the contaminant would be greatly impaired by the density of the lower atmosphere. So, the bubble is much higher than a mere 24km; it is mainly in outer space.

From the earlier of the two Daily Mail photos, it is easy to see that the bubble displays apparent (if not actual) flattening, like a rising moon, but the moon is only 1/2 deg in diameter, vs. 10 deg (if I am correct in my estimate of the frame size) for the earlier bubble in the Daily Mail article. The flattening is about right, to be due to atmospheric refraction: the 1 arcminute * tan(zenith angle) formula gives 20' at 3deg and 30' at 2deg, so the flattening of the part between 2 & 3 deg above the horizon would be about (30'-20')/1deg = 1/6, twice the flattening of Jupiter. I haven't measured it yet, but that looks about right. This flattening can happen in about the right amount, only if at 2deg above the horizon, the bubble is above, say, 1/2 the atmosphere. That gives a lower bound for the bubble height, of 5.5km*45deg/2deg = 120km.

For another rough estimate, I find that always roughly the bottom one degree of the bubble is below the horizon. (This could be determined accurately if the video width of field were known and daytime horizon photos were compared.) If the bubble is tangent to Earth's solid surface, then its distance would be 2*pi/180*7920mi/2 = 220km.

In sum, the drastic atmospheric variation, with altitude, of temperature and density, would prevent this bubble (whose height we know to be at least comparable with the tropopause, and probably with the stratopause or higher) from being so symmetrical in shape and brightness, unless the bubble is mainly in outer space, approximately tangent to Earth at all times while it grows upward. The dip of the bubble's limb below the horizon, agrees roughly with the depth of the bubble's distorted bottom portion, giving a distance of about 200km. A spherical bubble produced by a rocket's explosion, would not grow at just such a rate as to remain approximately tangent to Earth.

What is it? I'm not sure, but my best guess is, that it's "Bolon", the prophesied "ball".

"Bolon descends".

What Tortuguero Monument 6 is talking about. What the Mayan Long Count is about. An unknown astrophysical effect, which takes the form of a growing sphere tangent to Earth's surface.

could that bubble be a gravitational anomilly or do you think its electromagnetic ?... what would be the distance and diamiter of mass causing the bubble ?... does the bubble seem to be growing or moving changing in any way ?... can the bubble be tracked accros the sky ?... can you measure the frequency of light in the bubble ?... or frequency of light at the center of bubble toward any atstroids or planets in the center or center of the bubbles motion ?...

There is a video of the bubble posted in the online edition of the Honolulu Star Advertiser, a 1:30PM July 1, 2011 article by Jim Borg. The most important frames are

1. 3:39:43 top of bubble first appears 2. next frame 3:41:55 3. next frame 3:43:22.

The bubble rate of change of diameter between #2 and #1, is about the same as between #3 and #2. The difference in diameter between #2 and #1, is about 11deg, assuming the frame is 38deg high. Using 300m/s for the speed of sound, and remembering to multiply this by two because the bubble expands in all directions, the bubble height at its full 45 degree extent would be about

sin(45deg/2)/sin(11deg/2) * 300m/s * 2 * 132s = 316km, in fair agreement with the theory of the previous post.

According to the Am. Inst. of Physics Handbook, 3rd ed., 1972, Table 3d-9, p. 3-77, this value of the speed of sound is correct at 10km and again at 27km and at 69km, and never wrong by more than 10% in the range 2.5-80km.

So far, I haven't been able to gain control of any images of the bubble (due to the usual internet problems of not being able to view, save, or open files) though I've seen images on websites. I watched the CFHT video on the page of "Kanoa" (maybe Kanoa Withington of the CFHT) on vimeo.com, but apparently my attempt to save stills from it was unsuccessful, and now the video won't play on that website. All, or almost all, of the stills on the internet are from this CFHT video.

Many continuously playing Subaru videos (they have domes in the background and are not the same as the CFHT video) are on the internet, though I've not been able to magnify it enough to read the time stamps, or learn how to stop it to make a still. The Subaru video shows Jupiter to the right, a fairly easily identifiable Cassiopeia in the upper left, and identifiable (with a star map in hand) stars from Perseus, Triangulum, Aries, and Andromeda in or near the path of the bubble. I've not been able to identify the stars in any CFHT stills, however, whether colored or in a black & white version.

Another person on a messageboard remarked that the center of the bubble should not be moving, but it is. This indicates to me that the source isn't any kind of conventional explosion in the atmosphere. If a supersonic source moved while continuously exploding (e.g. a moving supersonic rocket) it would make a cone; in no projection would this cone resemble a sphere with a moving center.

Earth's rotation at 20deg latitude (i.e. Mauna Kea) is 435km/s. If the bubble were exactly NE, this projects to 435/sqrt(2) = 308km/s; leaving an unexplained vertical bubble rise of roughly 308*cot(30) = 533km/s, if the bubble's lateral westward motion is due to its stationarity as Earth rotates.

The locus of points for which their distances from two fixed points, have the same ratio, is a sphere (or in the plane, a circle). If the ratio, r, is almost one, then as r increases asymptotically to one, the center of the sphere moves away along a line to infinity, while the bottom point of the sphere (the point between the two fixed points) moves little. This geometric situation might arise from some undiscovered physical force.

The most accurate, obvious, and by far the easiest way to find the distance to the bubble, is by parallax between the two observatory's videos. Falsification of the background stars in the CFHT video would prevent this. Without knowing the distance to the bubble, the size of the bubble is unknown. Because the azimuth of the initial bubble is apparent in the videos, if the distance to the bubble were known, the point in the ocean where the bubble originated, would be known and could be searched.

This morning I realized that many true bright stars are present in the CFHT video too, though the CFHT video has deleted many stars which, judging by the Subaru video, ought to be visible. What true stars there are in the CFHT video, are disguised by the addition of many false stars, many of which are very bright and exaggerated in color.

The frauds at the CFHT didn't even bother to move the positions of the false "stars" to imitate Earth's rotation. With a ruler and magnifying glass, I checked the distance on my screen, at 03:37:29, 40:01, 43:09, 44:42 & 46:03. All five times, the distance between the lower of the two bright, light blue "stars" near the right edge of the screen, and the bright white light on the ground in the lower right corner very near the right edge, was 74 +/- 0.5 mm. The distance between the true Alpha Persei and Alpha Cas, which really is about 25deg, on the screen measured 62mm. So, in 8min34sec, that "star", which lies near 45N Decl and therefore should move upward about 8.6/4*cos(45)*cos(20) = 1.4 deg, actually moved less than 0.5mm * 25deg/62mm = 0.2 deg. On the other hand, most or all of the true stars I identify below, move in about the right direction, and in about the right amount, which due to their northern Declination, is somewhat less than 1.4*62/25 = 3.5mm.

Here is some help finding the true stars:

In each of the frames of the "Kanoa" "vimeo" video (which I am using today) that I found functioning this morning on the DiscoverMagazine/BadAstronomy site, can be seen some, or often all, of the five main stars of Cassiopeia: beta, alpha, gamma, delta & epsilon. These are seen near the top of the frame and slightly to the left of center. The bright orange "star" and the bright yellow "star" in Cassiopeia are false.

Also on this video, there is one bright orange false "star" in the righthand half of the frame, midway between the top & bottom of the frame. To the left and slightly above this "star", are some other "stars" which at first glance look like Perseus, but which really are mostly false "stars". The bright white "star" slightly above the center of the frame, is false, as is the bright white "star" slightly to the right of the center of the frame (the true alpha Per is slightly above this latter bright white false "star"). On many frames, the true, relatively dim, eta, gamma & alpha Persei are seen, in the correct relation, of position and brightness, to each other and to Cassiopeia (I used a ruler to measure distances and estimate angles on the screen).

Whatever words they might use to shirk their responsibility, professional astronomers all are paid largely through taxes and therefore all are public officials (in the Roman usage, even the assistant streetpaver is a public official: he has authority to decide the details of his work and perhaps to redirect traffic). For the public officials of the Canada France Hawaii Telescope, who by definition are affiliated with the state of Hawaii and whose observatory operates in the U.S., to lie about what is on this video, by painting in false stars, is a crime.

Now, though effectively delayed by five days (Sunday to today, Friday), by their crime, I'm going to try to determine the parallax of the bubble.

From my parallax estimate, the bubble distance at 03:42:41, probably was greater than 27km. My limiting factor was the small size of the Subaru video on my screen though I used a common kind of magnifying glass with an embedded high power lens. Another important source of error was difficulty in interpolating the slightly irregular appearance of the bubble edge.

I examined frames of the CFHT video at 03:42:48 and the Subaru video at 03:42:41. I found the distance from Delta Cas to the bubble, along the line from Gamma Cas to Delta Cas. From the rate of change of this distance in the CFHT video, I could correct the CFHT distance to the time of the Subaru video. Because the intersection point with the bubble, is 45deg up on the bubble, I must multiply the parallax by cos(45), and because the bubble is roughly NE whereas the 700m separation of CFHT from Subaru is E-W, I must multiply by cos(45) again. It happened that I found a distance of 540km, but this is not significant.

My biggest error is my measurement of the Gamma Cas to Delta Cas distance on the Subaru video; I estimate this error to be 0.3mm standard deviation, which amount would give a parallax equivalent to 27km distance.

Since the bubble was not observed by the numerous astronomical, aeronautical and other people and cameras in California, it must have been concealed by Earth's curvature there. California & Hawaii are 40deg apart; a bubble 10deg from Hawaii and 30deg from California, tall enough, (sec(30)-1)*6400km = 990km, to be just under the horizon as seen from California, would be nine times taller than needed to be seen above the horizon at Hawaii, and would appear roughly 45 deg in size at Hawaii. So, the bubble was smaller than 990km but probably bigger than 27km.

For June 22, 2011, the JPL Horizons ephemeris (airless model, sea level, apparent accounting for lightspeed) says that moonrise was exactly due east (azimuth 90), at a location 2deg41'01" latitude north of JPL's standard coordinates for the "Mauna Kea Observatory". This line of latitude is 299km N of "Mauna Kea Observatory". Also, moonrise was exactly due east, at a location 3deg00'33" longitude east of "Mauna Kea Observatory". This line of longitude is 315.5km E of "Mauna Kea Observatory" as measured along the common line of latitude. The closest approach to Mauna Kea, of the curve, on which Luna rose due east on June 22, thus was approx. NE of Mauna Kea and approx. 307/sqrt(2) = 217km distant. The precise easterly moonrise apparently potentiated the event.

Hawaii does not use Daylight Saving Time. So, the event began somewhat earlier than 03:39:43 Hawaii Standard Time, according to the time stamp on the first CFHT video frame which shows the bubble (Honolulu Star Advertiser version which I cite in my July 19 post); the Subaru video yields a similar time. This is 13:39:43 GMT (i.e. UT). To find the start time of the bubble, I measured the horizontal diameter of the bubble on four early frames (A, B, C, D) of the Honolulu Star Advertiser version of the CFHT video. Extrapolating linearly (noting that the diameter is nearly a linear function of time for the entire range of seven bubble diameters I checked), in turn through AB, AC, AD, BC & BD, gave start time estimates ranging from 03:38:31 to 03:39:01, mean 03:38:45 +/- 5.5sec SEM. This is about 1min43sec later than Marsrise, at the point on the special curve (i.e. the curve of due east moonrise) at azimuth 51 from Mauna Kea (see below).

Marsrise (i.e. elevation = 0) at Mauna Kea was 13:46 UT (JPL ephemeris, airless model, standard coordinates of Mauna Kea Observatory, 204deg31'40.1"E, 19deg49'34.0"N, elev. 4212.4m). At the coordinates of Mauna Kea but at sea level, Marsrise is 13:41 UT.

Now let's find Marsrise at sea level on the curve where Luna had risen exactly due east a few hours earlier. The midpoint between the endpoints (204deg31'40.1"E, 22deg30'35.0"N) and (207deg32'13.1"E, 19deg49'34.0"N) is approx. (206.0324E, 21.1679N). (This point is approx. 217km NE of Mauna Kea.) Marsrise at this midpoint (azimuth about 46.5 as seen from Mauna Kea) is 13:37:20 UT. At the southern endpoint (azimuth about 90 as seen from Mauna Kea) of my line segment described above, Marsrise is at 13:33:35 UT. Linear interpolation along this line, gives Marsrise at 13:37:02 UT, at the point at az. 51 from Mauna Kea.

Now I can pause and magnify the Subaru video 1000%. By looking at the earliest possible bubble frame, and using the rough estimate that the Subaru horizon camera was about the same altitude as the base of the CFHT dome buildings which appear on the Subaru camera's horizon, I can extrapolate the center of the bubble downward at roughly the 60deg slope of its travel, to find that the azimuth of rise is about halfway between the azimuths of Beta and Rho Persei.

The USNO online Celestial Navigation Data give 41.8deg azimuth for Mirfak (Alpha Persei) for the time of the frame and the latitude & longitude of the Subaru Telescope moved to sea level. From an appropriately oriented star chart I see that the midpoint of Beta & Rho Per would have about 9deg greater azimuth, i.e. 51deg.

Approx. 14sec later, according to the JPL, Marsrise occurs 0.1deg farther W and 0.1deg farther N. That is, Marsrise moves along this curve at about 0.1*sqrt(2)*111km/14s = 1120m/s. In my July 19 post I had estimated, for a bubble at about this distance, that the bubble moved to the left only about 1/8 this fast.

For the hypothesis of this post to be true, there must be yet another factor that causes a bubble to form at Marsrise at some point where Luna recently has risen due east. Otherwise, bubbles would have formed sequentially at every point on the curve where Luna rose due east.

From new observations of Comet C/2010 X1 (Elenin), the Minor Planet Center has published new orbital parameters. There has been a fundamental change; instead of a perihelion near Jupiters orbit, the comet will have an aphelion at Mercurys orbit! Of course the new comet does not belong to the class of sungrazing comets, but it will be visible in images from the coronagraph installed on the space observatory SOHO.

so DR Joe do you think this comet and the "missile" test have a connection ??? I would really like to know your thought on this comet if you know any thing about it Thanks

...do you think this comet and the "missile" test have a connection ?... MARX

My guess would be no, but I'm not ruling very much out at this point. I gather from the Wikipedia article, that the period was almost infinity when it entered the inner solar system, but the period is predicted to be perturbed to ~12,000yr for its next return.

I rechecked the coordinates of the points where, on June 22, 2011, Luna rose due east: i.e., the azimuth of Luna's center was 90 when the elevation was 0. The online JPL Horizons ephemeris gives this for an airless Earth model. That is, the angles are what would appear for an observer if Earth were airless. The effects of lightspeed (i.e., time delay, or, equivalently, aberration), precession and nutation of Earth's axis, and Earth's spheroidal shape, are included in the JPL model.

I made a small mistake in the longitude of the point east of Mauna Kea; really its E longitude is about 5' less than I said in my July 24 post. Luna rises at 90deg azimuth, along a curve that lies roughly SE to NW. I use Wikipedia's value for the "Mauna Kea observatory": long = 204:31:40.1E, lat = 19:49:34.0N. For convenience, my point A on the curve it that with the same longitude as the Mauna Kea observatory, my point B is that with the same latitude, my point C is that with latitude = 2*lat(A) - lat(B), and my point D is on the equator. By trial and error, the unspecified coordinate was found to the nearest 1", then Luna's risetime was interpolated to the nearest 1sec from the JPL times given in minutes. All points are at altitude zero above the JPL's idealized "sea level".

Point A long 204:31:40.1E lat 22:30:35.0N risetime 09:34:19UT Point B long 207:27:03.1E lat 19:49:34.0N risetime 09:22:21UT Point C long 210:25:16.1E lat 17:08:33.0N risetime 09:10:11UT Point D long 230:04:09.0E lat 0 risetime 07:49:31UT

This curve is more concave upward than a rhumb line (the longitude increment from A to B is only 99.20% that of linear interpolation between A & C) but it is close enough to a rhumb line, that quadratic interpolation, between A & C, sacrifices hardly any precision, and linear interpolation will suffice.

Now let's consider another curve on the globe. This is the curve where the elevation of Luna is arcsin(2/sqrt(6)) = 54.7356deg and the elevation of the Sun is -arcsin(1/sqrt(6)) = -24.0948deg. These, and their negatives, together with 0, are the angles, according to nonrelativistic quantum mechanics, of possible angular momentum cones when j=2, m= +/-2, +/-1, or 0 (see, inter alia, Saxon, "Elementary Quantum Mechanics", 1968, pp. 310-311). Because the celestial coordinates of the Sun and Luna change, this curve differs slightly from a line of latitude.

For linear interpolation, let's consider the corners of the rectangle whose sides are the lines of latitude and longitude through Points A & B. Let the corner below A, be called A', and the corner above B, be called B'. Let's call it Condition "L" when Luna has the desired elevation, and Condition "S" when the Sun has the desired (negative) elevation (see previous par.). From the JPL ephemeris (minute times interpolated to seconds),

Point A Condition L 13:45:04UT Condition S 13:38:16UT A' L 13:37:55 S 13:47:52 B' L 13:33:10 S 13:26:34 B L 13:26:00 S 13:36:11

By linear interpolation, I find the point on the longitude line between A & A', and the point on the longitude line between B' & B, such that Conditions L & S apply simultaneously. Again linearly, I find the intersection, of the line between these two points of simultaneity, and the line AB. The time of simultaneity on the line AB, I then find by interpolating either the L or S times at A & B. The result is 13:37:25UT.

Marsrise at this "simultaneity point" (interpolated coordinates, long 205.6996E lat 21.4339N; azimuth 34 viewed from Mauna Kea) is 13:38:12. In my July 24 post, I found by extrapolation of the bubble's linearly changing horizontal diameter, that the bubble started at 13:38:45 +/- 6sec. Using my corrected Point B, and some plane trigonometric approximations, I find that the observed "bubble start point" on the line AB, az. 51 from Mauna Kea (for determination of azimuth, see July 24 post) has long 206.132E, lat 21.043N and Marsrise 13:37:09.

In conclusion: to the accuracy of the measurements and calculations I have made, the bubble could have started at a point, on the line AB of due east moonrise, where Mars had risen about 96sec earlier. The time of this Marsrise, was about the same as the time when Conditions L & S (the special quantum mechanical angles of elevation of Luna and the Sun) prevailed at a nearby point of line AB, to which the bubble traveled during the next ~8 minutes before disappearing.

Another astronomical correlation of the Hawaii bubble

At 13:39:13 UT, June 22, 2011, according to the JPL ephemeris, Luna achieved apparent Declination (airless model, equinox of date) equal to Luna's apparent semidiameter, as observed from the North Pole. This compares to my July 24 estimate of the start time of the bubble, 13:38:45 +/- 5.5 sec. That is, the bubble started only about 28 sec before, the line from Earth's north pole to Luna's southern limb, came to lie parallel to Earth's equator. (Light travel time only affects this result by a little more than 1 sec.)

By contrast, Luna's center appeared to cross Declination Zero (apparent, of date) at 12:24:06 UT. The 28 sec discrepancy corresponds to only 28/(75*60) = 0.62% of Luna's semidiameter, i.e. 11km.

I had measured the width of the bubble, in the Canada France Hawaii Telescope observation video, at seven times, ABCDEFG. On July 24 I averaged five of the nearest linear extrapolations to zero width, namely, through AB, AC, AD, BC, and BD; the result was 13:38:45 UT +/- SEM 5.5sec.

Today I considered all 35 choices of three times, with quadratic extrapolations through each trio of points. Six of these trios gave parabolas so wild that there was no time with zero bubble width. The remaining 29 trios all gave zero times within the range [13:36:37, 13:39:28], unweighted mean 13:38:28 UT +/- SEM 7.0 sec. This is consistent with my earlier linear result: the difference is 17sec, the standard error of the difference is sqr(5.5^2+7.0^2) = 8.9sec, so the difference is 1.9 sigma from zero.

In nuclear magnetic resonance spectroscopy, one speaks of "the magic angle" of precession, which in the nonrelativistic approximation is arcsin(2/sqrt(2*3)) = 54.7356deg. According to nonrelativistic quantum mechanics, there is another, closely related angle of precession: arcsin(1/sqrt(2*3)) = 24.0948deg. Merzbacher's quantum mechanics text, among others, derives these angles. Another quantum mechanics text in the library at Iowa State Univ., happens to consider the relativistic correction to these angles, and finds that it is of order (v/c)^2, hence negligible for, say, hydrogen, but not for uranium, because for the innermost Bohr atomic electron, v/c = N/137, where N is the atomic number and 1/137 the fine structure constant. Let's suppose that the nonrelativistic (e.g. hydrogen atom) magic angle, 24.0948deg, is accurate for the Hawaii bubble.

Statistical combination of my two methods of estimating the start time of the bubble, gives 13:38:38.5 UT, only 34.5sec before Luna's southern limb reaches the plane tangent to Earth at the North Pole, at 13:39:13. At 13:39:13, Luna's center projects, according to the azimuths given by the online JPL ephemeris, onto the longitude line 125.836W. Now there are two important vectors associated with Earth: the vector Z from Earth's center to the North Pole, and the vector X, parallel to Earth's equator, from the North Pole to the southern limb of Luna.

At a point on Earth's surface, also there are two important vectors: the outward vector U which is normal to Earth's surface (more precisely, the outward normal to the approximating "reference spheroid"), and the vector V which is that point's rotational velocity (i.e. eastward parallel to the line of latitude). There are two points on Earth's surface, such that U and Z form the "magic" angle arcsin(1/sqrt(6)), and so also do V and X. For only one of these points, is Luna visible above the horizon. That point has longitude 149.931W and geographic latitude 24.095N.

Including a small correction for Earth's curvature, I find that the azimuth of this point, seen from Mauna Kea, is 50.1deg. This compares well to my estimate from the Subaru video (see earlier post), 51deg. This point, if fixed in the geocentric celestial frame, would move, relative to Earth's surface, 269m/s toward the left as projected on Mauna Kea. It is 743km from Mauna Kea. In eight minutes of time, this would be about 10 degrees of motion. I do not yet have an accurate estimate of the amount of motion in the videos, but according to the Japanese astronomer who posted online, the bubble achieved ~45deg diameter (during its >8 min duration). I did estimate from the videos that the bubble center moved upward 2x as fast as the center moved leftward, and that the bubble top moved upward 2x as fast as the bubble center moved upward; thus 40deg size after 8min implies about 10deg leftward motion in 8min. So, my predicted leftward motion of the bubble, corresponds at least roughly to the leftward motion observed.

Also it might be that the bubble, whose center apparently moved upward at about a 60deg slope, really moved upward along another "magic angle" slope, namely 45deg = arcsin(1/sqrt(1*2)), because arctan(tan(60)*cos(50.5)) = 48deg. Here cos(50.5) accounts for the projection as seen from Mauna Kea, using the azimuth angle of the great circle line of sight at the bubble.

According to the previous post, the bubble should recur at intervals of almost exactly one sidereal month. There should have been a bubble near Karachi, Pakistan, to the SW over the Arabian Sea, at 21:36 UT July 19 (02:36 July 20 Pakistan Standard Time); and near Key West, Florida, to the SW over the Gulf of Mexico, at 05:34 UT August 16 (01:34 U.S. Eastern Daylight Time).

According to the online service "Weather Underground", classic.wunderground.com/history/airport/ , Key West had clear skies and 10 mile visibility until at least 02:53 EDT on August 16, but there was a thunderstorm early that morning. The thunderstorm began at about 04:42 EDT. During the prior four hours, surface wind direction had ranged from W to SSW at speeds ranging from 3.5 to 6.9 mph. By my calculation, the bubble started 33 mi away from Key West at compass point "S by W" (slightly W of S); the bubble's center moved westward but its eastern side remained about stationary due to expansion. Soon the bubble covered about the western half of the sky at Key West, but if illuminated by Luna with more backward than forward scattering, the bubble might have been hardly visible from underneath. Thunderstorm clouds to the southwest likely partly blocked the view of the bubble, but could not have blocked it completely unless for some reason it were much smaller than the bubble seen from Hawaii. Also according to "Weather Underground", Karachi was "mostly cloudy" continuously from 18:25 July 19 to 16:30 July 20 Pakistan Standard Time.

The next bubble, according to my theory, will be 13:31:16 UT September 12, 2011 (00:31:16 Sept. 13, Okinawa Standard Time). Since the observed time in June, according to my best estimate, was 35sec earlier than given by my theory, my best estimate for the bubble start time this month, is 13:30:41 UT. The starting location will be 131.960 E long., 24.095 N lat. This is SE of Okinawa over the Pacific Ocean, about as far away as it was from Mauna Kea.

Let's all encourage Japanese astronomers to look for the next bubble! I did email the head of the Subaru observatory but got no reply; my emails to the two junior Subaru astronomers who might have been responsible for the messageboard (elsewhere) post cited in the online newspaper, "Hawaii 24/7", both bounced. If others would email the head of the Subaru Telescope observatory, there might be a response because we would not be a "lone nut"!