Requiem for Relativity

Seeing two "lobes" on Marsrock's image reminds me of the famous "ears of Saturn." When Saturn was first looked at, it was thought to have ears. The rings seen almost edge on showed up as two crescent shapes. We should expect each lobe of the nebula to have a slight blue/red shift.

In Barbarossa's Cavern There Are No Stars (Part XVII)

Part XVI's image of Barbarossa's nebula, shows the expected negative correlation (always, relative to the global mean for the region), between extinctions at points separated by vectors equal to Barbarossa's travel from 1954 (approx. date of R1) to 1991 (approx. mean date of R2 & I). (In review, Part XVI's image is constructed essentially from the average, "h", of log(#R2>R1/#R1>R2) & log(#I>R1/#R1>I), on 1deg diam disks, linearly detrended to remove the influence of the star color gradient.)

Approximately, the image maps the extinction due to Barbarossa's nebula in 1991, minus the extinction due to it in 1954 (that is, minus the extinction due to Barbarossa's nebula in 1991, evaluated at a second point 4.25deg E and 2.28deg S of the first). Because R1's spectral sensitivity curve isn't the same as R2's or I's (and my averaging and detrending only crudely correct that) what is mapped isn't exactly the finite difference h(x,y) = g(x,y) - g(x+4.25,y-2.28), where g is the extinction of Barbarossa's nebula. However, negative correlation should occur between lattice points separated by ~ (4,-2), whenever the prograde point of the pair, is near enough Barbarossa's ~1991 position, to be within Barbarossa's nebula.

A guess, that Barbarossa's nebula has radius 2deg, was confirmed roughly, by correlations of +0.65, +0.4 and ~ +0.05, between "h" values at points within 2deg of Barbarossa's 1991 position, and points, 1.4, 2.2, or 3deg, resp., away from these (in a generally retrograde direction). The lattice point pairs, one of whose members is within 2deg of Barbarossa's 1991 position, anticorrelate strongest, when the other member is 3deg N and 5deg W of Barbarossa's 1991 position; r = -0.805 and Fisher's normalized z = 3.15 ( > 3 sigma significance). The number of such pairs is n=11; I exclude separation vectors for which the number of pairs is n<8. Linear interpolation of the correlation, and of Fisher's z, between the four sets of lattice point pairs nearest (-4.25,+2.28) separation (relative to the point near Barbarossa)(n=14 or 11, total 50), gives r = -0.678 and normalized z = 2.63, p = 0.0086, 2-tailed.

If "h" is the difference in the extinction of Barbarossa's nebula at two points separated by a constant vector, the autocorrelation of "h" should be -0.5 for points such that h1 = g1-g2 and h2 = g2-g3, if g2 is as big as g1 or g3. Indeed, for the sets of lattice point pairs whose separations are the four vectors nearest (-4.25,2.28), with the pair's first member < 2deg from Barbarossa, the autocorrelation ranges from -0.613 to -0.805; when I pretend that Barbarossa orbits in the opposite direction, the autocorrelation for the analogous four sets nearest (+4.25,-2.28) separation, ranges only from -0.1 to +0.1.

Inspection of my B2/B1, R2/R1, and I/R2 plots, confirms the prograde motion of Barbarossa's inner nebula. Because autocorrelation indicates that the ordinary (inner nebula) extinction extends ~ 2deg from Barbarossa, I considered 6x6deg squares centered on RA11:18,Decl-8. For R2 vs. R1, I found the first difference in column averages of log(q) = log(#R2>R1/#R1>R2), then interpolated linearly between these to find the maximum extinction 0.45deg W of center. (Extinction of R1 & B1 is neglected because it centers 4deg W, at Barbarossa's 1954 position.) B2 vs. B1 had "0" cells prohibiting logarithms, so instead I used (#B2>B1 - #B1>B2) / (#B2>B1 + #B1>B2), weighted within each column, by number of observations. Then I followed the same procedure as for R2 vs. R1, to find maximum extinction 1.40deg W of center.

I vs. R2 are theoretically only 1deg apart, so both extinctions must be considered. Also, the I & R2 spectral windows differ much, so the star color gradient might cause a big linear trend. First, I found first differences in column averages of log(q), as for R2 vs. R1 above. Then I linearly detrended to make the last column average equal the first. If the I and R2 extinctions are the same, the midpoint should be an inflection point. The second difference changed sign in two adjacent instances; interpolation gave two nearby inflection points which I averaged to find the midpoint of I & R2 at 0.77deg E of center.

Thus, the peak extinction, estimated from the USNO-B catalog magnitudes, of B2 vs. B1, was 0.95deg W, of that of R2 vs. R1. Barbarossa's travel in that time would predict 0.5deg W. The estimated midpoint of the extinction peaks of I & R2 was 1.22deg E, of the peak of R2. Barbarossa's travel would predict 1.0/2=0.5deg E. Though the extinction travelled about twice as far as predicted, statistical errors and methodological uncertainties are big. The correct sequence of the three, is significant at p=1/6.

In Barbarossa's Cavern There Are No Stars (Part XVIII)


Dedicated to Frederick Barbarossa: "King, Knight, Hero"
(Helmut Hiller)


Barbarossa Nebula Dynamics Match Observations

The simplest dynamical estimate of the radius of Barbarossa's nebula, would be 197.7 AU * sqrt(0.0103), using the distance from Barbarossa to the sun, and the ratio of Barbarossa's mass to the sun's, to get 20.06 AU. What we want to know, is the radius of the largest possible orbit around Barbarossa. As a start, let's consider only gravity, and ignore the sun's gravitational pull on Barbarossa itself. The case of a face-on circular orbit can be solved exactly: the orbital plane is displaced toward the sun so that Barbarossa's outward pull equals the sun's inward. Barbarossa's best efficiency in this, is max(sin(theta)*cos^2(theta)) = sqrt(4/27), so the biggest possible projected radius is 20.06 * sqrt(sqrt(4/27)) = 12.44 AU.

The case of an edge-on orbit can be estimated using the circular approximation: in half a circular orbit with speed v, Barbarossa vectorially accelerates the body by approx. pi*v, but if the body is bound, again ignoring the sun's gravitational pull on Barbarossa itself, the sun accelerates the dust grain prograde by < (sqrt(2)-1)*v. The sun's efficiency in this, over half an orbit, is only 2/pi, so the biggest possible radius is 20.06 * sqrt((sqrt(2)-1)*1/2) = 9.13 AU. Thus the edge-on and face-on circular cases give approx. equal radii. Though Barbarossa's orbital motion continually converts face-on to edge-on and vice versa, this common radius could be taken as the radius of Barbarossa's nebula, if only gravity were considered.

ISU's copy of Hodge's "Interplanetary Dust", has an anonymous penciled marginal note referring to Burns, Lamy & Soter, Icarus 40:1+, 1979. I took the hint.

According to Burns, et al, Fig. 7a, p. 16, the greatest "beta", i.e., ratio of solar radiation pressure to solar gravity, 5.5, occurs for pure graphite dust of 0.08 micron size (this is for spherical particles, but shape dependence is weak according to, inter alia, Kruegel; and furthermore Burns et al give evidence that rather spherical shapes predominate anyway). So, for classical interstellar dust, i.e., carbonaceous particles of ~ 0.1 mu size, the biggest possible circular orbital radius isn't much bigger than 12/sqrt(5.5) AU (actually, the sun's gravity pulls the same on Barbarossa and on the dust grain, so the net push, equals the radiation pressure); i.e., it can subtend not much more than 1.5 deg from Barbarossa. This matches the 2deg radius found empirically in Part XVII from the autocorrelation, near Barbarossa, of the USNO-B R2 vs. R1 extinction measure (from the third, most accurate version of my image data). Or it can be found simply by looking at the images of that extinction measure, assembled by "marsrocks" above (from the first, least accurate version of my image data).

This small patch around Barbarossa, eight times the diameter of the full moon, is where backscatter of sunlight, from Barbarossa's nebula, might be detected. Farther than 1.5 deg from Barbarossa, solar radiation pressure drives away classical interstellar dust. Ultrafine, smaller-than-classical carbonaceous dust grains, which for any given extinction cause orders of magnitude less scattering, would reach 12/sqrt(1.77) AU --> 2.6 deg from Barbarossa, because (Burns et al Fig. 7a) for graphite of size < 0.004 mu (40A), "beta" has an asymptotic value of only 1.77.

The first, least accurate version of my image data (the later versions haven't yet been extended this far) shows roughly vertical strips of R2<R1 about 5.5 deg east, and R2>R1 about 12 deg west, of Barbarossa's 1987 position. These are narrow strips ~ 4 deg wide, not part of the global trend. Barbarossa's 1954 & 1987 positions are separated about 3.9 deg in RA, so if Barbarossa's outermost (gas, or low-beta non-carbonaceous dust) nebula has radius 9 deg, then the 1987 outer nebula position would cover, roughly, a vertical strip 5 to 9 deg east of Barbarossa's 1987 position (vs. ~ 5.5 observed) that wasn't covered in 1954; and the 1954 position would cover, roughly, a vertical strip 9 to 13 deg west of Barbarossa's 1987 position (vs. ~ 12 deg observed). For a 9deg radius, beta ~ 0.15 would be needed ( 12/sqr(0.15) = 31 AU --> 9.0deg ). Really, beta isn't the cause; it's the Jacobi limit, 30 AU (see Technical Detail #2, Part XIX below). So, both the Jacobi limit, and the radiation pressure limit are seen, defining an outer and inner nebula, resp.

These strips, indicating the E & W extreme portions of Barbarossa's outer nebula, give negative extinction (i.e., brightening instead of dimming). These are mainly Type M stars, so this agrees with the negative extinction found for Type K & M stars in my Harvard vs. Johnson magnitude study of the same region (statistically controlled, by samples from distant regions of equal Declination and similar galactic latitude). Reviewing all the stellar photometric studies of Barbarossa's outer nebula that I've done, I find that Type K & M stars consistently undergo paradoxical, negative V extinction there, Type F & A consistently ordinary, positive V extinction. Not only is this seen in the smooth linear variation with spectral type, in my Harvard vs. Johnson study; it's seen in my original, 4deg-resolution, study in which B2 - B1 was greatest E of Barbarossa, but R2 - R1 greatest W of Barbarossa.

I made a 1deg-resolution B2 vs. B1 study, similar to the R2 vs. R1 study the results of which are displayed in the above images by "marsrocks". Even summing results using two pairs of magnitude intervals, [16,18] & [18,20] and [17,19] & [19,21], data are sparse, and my study less thorough, but I do find positive extinction, of B2 vs. B1, out to roughly 4deg from Barbarossa's 1983 position, at least in some directions, as in the R2 vs. R1 case. Unlike the R2 vs. R1 case, the B2 vs. B1 case shows increased, not decreased, extinction of B2 vs. B1, about 9deg E of Barbarossa; and perhaps increased, not decreased, extinction of B1 vs. B2, ~ 5deg W of Barbarossa. Thus the outer nebula finding for B2 & B1, resembles the Harvard vs. Johnson finding for Type F & A stars.

My Hipparcos studies show that B-V changes (it's the reverse of ordinary extinction) in the opposite direction, to V, as though B were much less affected than V. (This holds both for Draper/Cannon Type K0, and for Draper/Cannon Types F5-B9.) This suggests that photons from stars are traded slightly up or down in energy, by a kind of stimulated emission as they traverse Barbarossa's outer nebula which has been pumped by sunlight.

Barbarossa isn't luminous like the sun, so the Poynting-Robertson and Yarkovsky effects are unimportant in Barbarossa's nebula close to Barbarossa. Barbarossa's huge angular momentum replenishes the orbital angular momentum, around the sun, of the particles gravitationally bound to Barbarossa; and the sun's radiation effects average zero on a particle's orbit around Barbarossa.

In Barbarossa's Cavern There Are No Stars (Part XIX)

Technical detail #1. I've seen the claim that the red plexiglass filter used in the Palomar Red sky survey, is about the same as a Wratten #29 filter. The Handbook of Chemistry & Physics, 44th ed., gives (with minor interpolation) 6220A as the 50% cuton for Wratten #29. The next listed, Wratten #26, has 50% cuton at 6070A. MS Bessel's Fig. 1 gives 5890A as the 50% cuton for R59F (i.e., SERC-Red). All these cutons are steep. So, if Wratten #29 is at all correct, the Palomar Red survey cuts on, 300A redder, not 300A bluer, than the SERC-Red survey. However, my data support the other claim I've seen, that Palomar Red's "xx103aE emulsion + red plexiglass filter", together, still were consistent with the "E" passsband (which in turn is about the same as "Rc" at moderate visible wavelengths, allegedly according to a Cambridge Ph.D. thesis; Rc's cuton is 285A bluer than R59F's). R2 - R1 increases toward the galactic equator (or roughly equivalently, along the local gradient of blue star fraction); this gradient in R2 - R1 is seen in the farthest corners of my pure R2>/R1> plot, as much as 20deg away from Barbarossa. So, R1 would seem more sensitive to bluer stars, than R2.

Technical detail #2. The Jacobi limit (Icarus 102:298+, 1993; or 107:304+, 1994), which uses the cube root of the mass ratio (rather than the square root as my estimate does) for Barbarossa, considering only gravity, is 29.8 AU. Empirically, however, the "most distant known satellites" (vis a vis the Jacobi limit), Jupiter's "Pasiphae and Sinope", have semimajor axis only 0.45x the Jacobi limit, corresponding to 13.4 AU for Barbarossa (Icarus 102:298+, p. 304). This agrees well with my estimate, considering only gravity, of ~ 9.13, to 12.44 AU. My estimate easily applies to the virtual negative gravity of radiation pressure, but the Jacobi limit is based on the Roche lobe, and not so easy to apply here.

Technical detail #3. Recent Earth-based photographic searches for satellites of Uranus & Neptune (Icarus 102:298, 1993; 107:304, 1994) have used the same emulsion/filter combination, IIIaJ + GG395 (passband 3950-5350A) and exposure, 60min, as the SERC-J (Blue) sky surveys. If this is best for sunlight reflected from unknown solar system objects, it's presumptively best for Barbarossa's nebula. The removal of red light reduces the red dwarf background.

In Barbarossa's Cavern There Are No Stars (Part XX)

IRAS Imaged Barbarossa's Bow Shock


Schlegel, et al, Astrophysical Journal 500:525+, 1998, Fig. 8, p. 542 ("Dust"), shows a streak of far-IR brightness, measured by the IRAS satellite in 1983, that is only 4 deg from Barbarossa's 1983.2 heliocentric position (Barbarossa was at l=265.9, b=+48.4) and extends ~ 12 deg each way. It's prograde of Barbarossa (i.e., > galactic longitude) and crosses Barbarossa's orbit at approx. 37 deg slope ("beta"). Though presumably its seeming (non-greatcircle) extension near galactic coords l=190, b=+40, is accidental (the alleged Martian canals perhaps also demonstrated the human ability to notice accidentally aligned dots) this streak is one of the most prominent curvilinear "cirrus" "dust clouds" in the sky.

With a mm ruler, I measured the positions of 8 points of this cirrus (on Schlegel's polar-projection map, Fig. chosen for their definiteness, and their roughly equal spacing with good linear interpolation between. Then I used cubic Lagrange interpolation on Schlegel's map, to find their galactic coordinates, converted to other spherical coordinates (for a nearly conformal flat map) for which the endmost points of the "cirrus" define the equator, plotted them on Keuffel & Esser graph paper using a sinusoidal projection with Barbarossa's 1983.2 position defining the prime meridian, and graphically found the best-fitting hyperbola (taking Barbarossa as a focus, using symmetry to find the slope of the directrix by eye, then successive approximations to find the eccentricity, ~ 5.5, i.e., very flat). I also made the small correction, longitude --> sin(longitude), because I'm allegedly viewing, approximately, an hyperboloid of revolution, from a finite distance; not a hyperbola. (Hausmann & Slack's Physics, Ch. XXXI, sec. 367, Fig. 355, shows a flat-nosed projectile with an hyperboloidal bow shock with e = 1.5.)

The N & S asymptotes slope 27 and 47deg, resp., to Barbarossa's orbital path. This range of slopes lies well within the theoretical range, 16.9 to 61.5deg, for initial Mach number 3.432, final Mach number > 1, and adiabatic constant k=5/3. (See Hughes & Brighton, "Fluid Dynamics", McGraw-Hill, 1967, Sec. 8.2, eqns. 8.9 & 8.10 and figs. 8-1 to 8-3, pp. 158-159; or, 2nd ed., 1991, pp. 210-211.)

The Mach number "M1" is taken to be the ratio, of Barbarossa's orbital speed in a circular orbit at 197.7 AU, to the speed of sound in rarified dissociated (i.e., atomic) hydrogen at the solar radiation Planck equilibrium temperature (this is independent of albedo, by Kirchhoff's law) at 197.7 AU. Using (HL Johnson's favored) 5800K for the sun's surface, the sun's radius, and the Stefan-Boltzmann T^4 law, gives 28.1K as the equilibrium temperature at Barbarossa. (According to Schlegel, interstellar dust is 15-19K.) The proportionality of the speed of sound, to sqrt(T/m), is so accurate that my result varied little, even when I based it on molecular hydrogen at STP (with the sqrt(T/m) correction). However, I based my calculation on the value 8.894 km/s for dissociated hydrogen at 0.1 bar and 5800K (the lowest pressure in the table; the temperature went up to 6000K, but I wanted to use HL Johnson's favored temperature of the sun)(Vargaftik, Handbook of Physical Properties of Liquids & Gases, 2nd ed., p. 34).

Corrected to 28.1K, the speed of sound in rarified dissociated hydrogen is 0.6176 km/s, giving M1 = 3.432, and a minimum possible "beta" = arccsc(M1) = 16.9deg (Hughes & Brighton, eqn. 8.. The abscissa of the intersection of the curve M2=1, with the M1=3.432 isocline (from simultaneously solving, numerically, Hughes & Brighton eqns. 8.9 & 8.10) is 61.45deg for the monatomic adiabatic exponent k=5/3 (Fig. 8-3 shows 64deg, but this seems to be for air; k = 1.404 & 1.401 for molecular nitrogen & oxygen, resp., at +15C & std. pressure).

On my R2 vs. R1 chart, I had marked degree squares with ratio < 1 and > 6; these corresponded roughly to lower and upper quartiles. Within the 3x3deg square centered on Barbarossa, the inner nebular extinction shows a northern boundary sloped at ~ 45deg.

On the B2 vs. B1 chart, I had marked squares according to whether none, one or both of the ratios used exceeded 1. Here, the average position of the arrows shows a northern boundary sloped at ~ 67deg, but data are sparse.

The average of these two charts, weighted by number of observations, gives 47deg, i.e., 45 + (67-45)/(10+1) - 27 = 20deg to Barbarossa's orbit. Hughes & Brighton's Fig. 8-3 isocline needs to have its peak shifted 2.5deg left if k=5/3; the left zero doesn't move. So, from the interpolated M1=3.43 isocline, shifted an interpolated 1deg left at the relevant position, I can read that for "beta" (the slope of the shock wave) = 37deg, "theta" (the slope of the obstruction) = 22deg, agreeing with the ~20deg observed in the extinction chart.

Schlegel's Fig. 12 maps the difference between Schlegel's infrared map, and the Burstein & Heiles, Astronomical Journal 87:1165-1189, map based on reddening of external galaxies and HI column densities. The Burstein & Heiles map shows nothing unusual in the vicinity of Barbarossa, so the "cirrus" (really, bow shock) appears on Schlegel's Fig. 12 also. (I chose Fig. 8 for my study, to stay closer to the original data.) Schlegel used DIRBE satellite data (1989-1990) to correct the (higher-resolution) IRAS data for temperature, to convert them into presumed interstellar dust column densities; Schlegel's temperature map shows nothing obviously unusual near Barbarossa.

However, even when temperature is considered, IR emission need not correlate with HI (nor presumably with reddening of external galaxies). Regarding a shell in Eridanus, "When the IR emission is compared with the HI there is little direct positional agreement. ...We are not yet convinced that we can account for apparent variations in IR brightness as resulting exclusively from temperature effects,..." (Verschuur et al, in, IAU Symposium #139, 1989; Conclusion, p. 236).

Shouldn't Barbarossa cause observable perturbations of the outer planets?