1. Introduction
This paper is only a report of a work in progress. I used a simple technique that minimizes the squared errors between Miller’s four apparent intake apex positions and my computed positions to find the true intake apex solution for the magnitude curves. I used a better technique to find both the true and the apparent exhaust apex solutions for the azimuth curves that did not depend on Miller’s apparent apex positions. This technique minimized the squared errors between computed and observed azimuth values. In the future I should use the better technique to find both sets of solutions, and I should use Miller’s actual rough observations, not the smoothed values obtained with moving averages. Better yet, I should use Miller’s original data sheets if they are available so that the precise time of each observation could be incorporated into the analysis. Ideally, Miller’s observations should be repeated with a laser light source and automated data collection at several different latitudes and for more than four epochs in a year. A modified meridian telescope might be suitable as a vertical-plane interferometer to complement Miller’s horizontal interferometer.
2. Prediction
If Miller's experiment1 were repeated in Talkeetna, Alaska, the ether velocity magnitude curve would have one sharp null every September 3 at 04:02 UT because the apparent ether intake anti-apex would be overhead and the apex would be underfoot (Fig. 1). The daily minimum would be greatest 6 months earlier or later on March 1 (Fig. 2).
If the observations were made in Fort Yukon, Alaska, two sharp nulls would occur each year on June 27 at 06:58 UT (Fig. 3) and November 19 at 00:05 UT (Fig. 4). Fort Yukon is approximately on the Arctic Circle.
The ether velocity magnitude curves are computed from the simple formula
Vmag = 11.18 sin z km/s, (1)
where Vmag is the predicted maximum observed ether velocity magnitude as a function of azimuth at the given instant of time. The velocity magnitude is actually computed from the fringe displacement in wavelengths using Eq. (3). This maximum occurs when the azimuth of the instrument is parallel to the azimuth of the apparent ether intake apex or anti-apex. The instrument is bi-directional. The coefficient of 11.18 km/s is the escape velocity from the surface of the Earth, and z is the angular zenith distance of the apparent ether intake apex or anti-apex. The derivative of the velocity magnitude function is discontinuous when the ether intake vector passes through the zenith because z cannot be negative.
This simple theory assumes the instrument measures the horizontal projection of an ether intake vector at the escape velocity from the direction of the apparent intake apex. The apparent apex is distinguished from the true apex in that the latter is fixed in direction and the former is displaced in the celestial sphere by annual and diurnal aberration. In principle, if one could stop the orbital and daily rotational motions of the Earth, the intake ether wind would appear to come from the true apex.
3. The “Reduced Velocity” Effect
There are two species of ether velocity exhibited in Miller’s Table V (p. 235). What Miller calls the “observed” velocity is on the order of 10 km/s, and what he calls the “calculated” velocity is on the order of 200 km/s. The slower velocity is calculated from the observed fringe displacement using Eq. (3). The faster velocity is inferred from the radius of the annual aberration circle, given the known orbit velocity as a function of time. Miller was unable to account for this effect.
This dual velocity manifestation is one of the obstacles to the acceptance of Miller’s findings by the orthodox physics community. We need a new model of physical reality to explain this effect, and such a model is beyond the scope of this paper. Until such a model is invented, let me refer to the projected escape velocity as being the “near” ether wind velocity and the intake and exhaust velocities as being the “far” or “cosmic” ether wind velocities. Annual aberration seems to affect only the far cosmic ether wind velocity from the intake apex (206 km/s) and the exhaust apex (57 km/s). Diurnal aberration seems to affect the far ether wind velocity from the intake apex and the near ether wind velocity to the exhaust apex. I don’t know why this should be so.
Miller’s observations are best fit by the model if the magnitude curves are presumed to be determined only by the apparent ether intake apexes, while the azimuth curves are presumed to be determined only by the apparent exhaust apexes. I have no idea as to why that should be so.
In this paper, the cosmic ether wind is assumed to operate outside an entrained ether bubble that rotates with the Earth as a “solid” body, and the near ether wind is assumed to operate inside that bubble. This means I am assuming a step in the ether wind velocity at the bubble interface. This may not be the correct physical model, but it is the basis for an ad-hoc math model that approximately reproduces the azimuth displacement. In this paper (for the azimuth analysis only; a subsequent paper will incorporate the bubble into the magnitude analysis), the radius of the entrained ether bubble varies from -1.71 earth radii in the February epoch to 9.27 earth radii in the September epoch. Since this math model is not a physical model, I won’t attempt to explain how the radius of the bubble can be negative. The negative radius is required to give a western shift in the average azimuth that Miller observed in the February epoch.
The bubble radius determines how much effect diurnal aberration will have on the apparent exhaust apex after annual aberration. With a radius of 0 there would be no diurnal aberration. At 1 earth radius, the maximum diurnal aberration of the 208 km/s intake ether wind is only about 0.1º. At 9.27 earth radii, the maximum diurnal aberration is nearly 11º. See Section 8.
The exhaust ether rays are assumed to travel upward in a helical trajectory from the Earth’s surface where the interferometer is located to the bubble interface surface in the direction of the apparent exhaust apex. The screw pitch angle of the helix is around 1.75 arcminutes per axial kilometer for all four epochs. The radius of the helix is about 1.55 earth radii (see Table V). The helical path velocity is the exhaust velocity, and the projection of that velocity on the axis of the helix is the escape velocity, which varies inversely as the square root of the radius from the center of the Earth. I have not yet applied this bubble and helix path idea to the intake apex solution.
It is useful to review how one may compute a ray velocity from its annual aberration circle radius. The aberration of light is generally observed to be an ellipse, but it is a circle for stars near the ecliptic pole. Since the true apex of the intake ether wind is only about 7º from that pole, a circle is a good approximation. The constant[2] for the aberration of light (radius of the circle or semi-major diameter of the ellipse) is 20.49552² = 0.00569320º. The product of the ray velocity and its aberration radius is a constant (when the units are km/s and degrees that constant is 1706.778), so the intake ether wind speed from the apparent apex is then given by the speed of light multiplied by the ratio of the light circle radius to the ether circle radius. From my model and Miller’s four apparent apex positions I found that the annual mean ether aberration circle radius is 8.215º.
(2)
I actually computed 207.73 km/s for the annual mean apparent ether wind speed from a least squares fit to Miller’s four apparent apex positions using my vector aberration math model and 5,844 (1.5-hour) steps in time for a year.
This annual mean apparent ether intake wind speed of 207.73 km/s should not be confused with the true cosmic ether intake wind speed of 205.6 km/s that would be observed if the Earth did not move, which is constant in magnitude and direction. The true cosmic ether wind is what the Earth’s orbital and diurnal motions aberrate to produce the apparent ether wind that is observed.
The four pairs of celestial coordinates for the apparent apex positions (one for each epoch) that define this circle (see Section 5) were determined by Miller independently from the amplitude curves and the azimuth curves and given as the observed values in his Table VI on page 236. These were the values I used in my model for the intake apex solution, and I repeat them in my Table I.
The determination of the apparent ether intake apex celestial coordinates from the magnitude curves (Miller, p. 225) does not require any particular value for the velocity. The method uses only the ratio of the minimum to the maximum velocity (or fringe displacement) over a 24-hour day and the latitude of the observatory to determine the declination of the apparent apex. The right ascension of the apparent apex is the sidereal time when the amplitude curve reaches a minimum because the apparent apex is on the meridian in that instant.
The determination of the apparent apex coordinates from the azimuth curves is also determined without regard to the ether wind speed. The right ascension requires only the sidereal times at the mean azimuth crossings. The declination can be determined from the observer’s latitude and the amplitude of the azimuth swings east and west of the mean azimuth. Miller and I disagree about the sizes of tha azimuth swings. See Section 7.
4. Interferometer Sensitivity
Even Shankland,[3] Miller’s possibly well-intentioned but evidently misguided critic, gives Miller credit for being an accurate observer (p. 169):
It is generally agreed that an experienced observer with keen eyesight, such as Miller had, can estimate the fringe position to 0.05 fringe, which may be regarded as the least count of the instrument.
However, Miller was conservative and recorded his observations in tenths of a fringe to reduce the uncertainty because each reading had to be made at a glance as each azimuth signal sounded. He followed the continuously rotating telescope eyepiece by walking in a circle. Since Miller averaged 40 readings (half-turns, the instrument is bi-directional) for each of sixteen azimuths, the effective least count is 0.1/Ö40 = 0.0158 fringe per set of azimuth readings. Since he made an average of 80 sets of observations for each epoch, the least count is 0.0158/Ö80 = 0.00177 fringe per epoch. The average minimum detectable velocity corresponding to this least count is given by
, (3)
where the numerator under the radical is the fringe displacement
in wavelengths, the denominator is the total light path in wavelengths of
5700-Angstrom light, and the coefficient of the radical is the velocity of
light. See Miller (p. 227). Thus the observed velocities are from 4 to 9
times the instrument’s sensitivity.
This formula is graphed in Miller’s Fig. 20.
5. True Cosmic Velocity and Apex Position Solution
The cosmic ether intake wind is assumed to blow at a constant velocity from a fixed position in the celestial sphere. This is what we presume one would measure if the Earth’s orbital and rotational motions could be stopped. This velocity is called the true cosmic velocity, and the position is called the true apex. If the Sun moves through a static ether matrix, then this true cosmic velocity is the negative of the Sun’s velocity through that static ether. If the Sun moves through a moving ether matrix, then this true cosmic velocity is the vector combination of the two motions. In the airplane analogy, it is the “airspeed”, the Sun’s motion is “ground speed”, and the motion of the ether matrix itself relative to the galactic reference frame is the “wind speed”. I speak of the cosmic ether wind as blowing from the source apex towards the observer because aberration of light considers light rays as being streams of photons moving towards the observer from the star.
The true cosmic velocity is analogous to the speed of light, and the true apex is analogous to a star’s true position before annual and diurnal aberration and annual parallax corrections are applied. If the source of the ether wind has a finite radial distance from the solar system, then annual parallax should be applied as well as aberration. However, the quality and quantity of Miller’s observations do not warrant a model that includes annual parallax, so it is ignored. The true cosmic ether wind velocity cannot be observed; it can only be calculated. What is observed is the apparent ether wind velocity, which is the true velocity modified by aberration.
Table I was completed before I knew that the intake and exhaust velocities were not negatives of each other. That is to say that I was assuming a static ether lattice as Miller did. Also Table I does not incorporate the concepts of the entrained ether bubble or the helical path of ether rays inside that bubble that I discovered while performing the azimuth curves analysis.
The “Reference Apex” positions observed by Miller in my
Table I were taken directly from Miller’s Table VI (p. 236) from his columns a-Obs.
(Right Ascension) and d-Obs. (Declination).
To 
convert
RA from hours to degrees, multiply by 15.
The problem solved in Table 1 is the following: given four observed apparent intake apexes at four epochs, find the true ether intake apex position and true intake velocity which when adjusted for annual and diurnal aberration produces apparent intake apexes which are nearest to all four observed apexes in a least squared error sense. The free parameters that are adjusted to minimize the rms error are the true apex RA and Dec, the true velocity, and the calendar date and time of day. The Universal Time is the decimal fraction of a day times 24 hours. One needs to calculate Earth’s velocity vector as a function of time. Aberration, either annual or diurnal, is the effect that transforms a static ray radiant into a dynamic ray radiant according to the vector equation:
, (4)
where vstatic and vdynamic are ray velocities from a source towards the observer. The true apex is the projection of -vstatic on the celestial sphere and the apparent apex is the projection of -vdynamic on the celestial sphere. The negative sign for the projection accounts for the fact that the vectors are defined as originating at the apex and pointing towards the observer while the phrase “projection onto the celestial sphere” means the radius vector from the observer to the sky. The apparent apex always moves from the true apex towards the direction of the observer’s motion.
As you can see from Table I, the minimum rms error is only 0.032º or about 1.9 arcminutes. If there were only three apexes, this would not be remarkable because a circle can be passed through any three points with zero error.
Critics may complain that I had no right to adjust the epochs. The August epoch is retarded by 15.21 days, the April epoch is advanced by 5.45 days, and the others changed by lesser amounts. Yet it is clear from my Fig. 5 and even from Miller’s Fig. 28 (p. 237) that the error between the observed and calculated epochs is mainly an error in orbit phase, since the calculated points fall on the aberration circle but are displaced in time.
I suggest that there is an unknown form of aberration at work here, and that aberration has the effect of appearing to advance or retard the phase of the orbit from the known epoch. Also, Miller’s observations at each epoch actually spanned a couple of weeks so he could get multiple coverage of the 24-hour sidereal day and still get some sleep. Therefore, the actual epoch is smeared over that time span, and the effective epoch may not be the mean epoch.
As I mentioned in the introduction, the better technique used in the azimuth and exhaust apex analysis should be applied to the magnitude curves and intake apex analysis. I suspect that when the model for the input apex solution includes the entrained ether bubble, the “unknown form of aberration” mentioned in the previous paragraph will turn out to be the “amplified” diurnal aberration.
If we force the epochs to agree with Miller’s nominal epochs, then the least squared error solution in Table II has an rms error of 0.85º.
If we use Miller’s true intake apex solution instead of my optimized values in Table II, the rms error is 1.02º as shown in Table III.
Fig. 5 plots the annual and diurnal aberration of the true intake cosmic ether wind for 5,844 (1.5-hour) steps in time over an entire year on the celestial sphere. The straight grid lines are meridians of right ascension, and the curved grid lines are circles of declination.
Miller’s observed apparent apexes are black squares with white crosses inside. The calculated apparent apexes at the optimum epochs are smaller white squares with black x’s inside from the true apex solution in Table I. The triangle points are the calculated apexes for Miller’s mean epoch dates using the Table I solution.
As a matter of curiosity, the South Pole of the Invariable Plane, which is the nadir for the total angular momentum vector for the entire solar system, is plotted as a circular symbol. November 22 is the epoch that is nearest to the south pole of the IP, and it is also plotted as a circular symbol.
Fig. 6 is an enlargement of a portion of Fig. 5 that shows how diurnal aberration of the far cosmic ether wind velocity of 208 km/s causes the apparent apex to trace small daily loops of about 0.1º radius that follow the annual aberration circle. Also see Section 8.
6. Mt. Wilson Ether Drift Velocity Magnitude Curves
In Figs. 7, 8, 9, and 10 the curves are plots the ether drift velocity magnitudes as functions of the zenith distance of the apparent anti-apex as observed from the Mount Wilson observing site at the given epoch according to the formula in Eq. (1) and using the true apex solution of Table I.
These curves are practically indistinguishable from Miller’s Fig. 26 (p. 235). This should not be surprising since both of us assume the magnitude curves are the horizontal projections of the escape velocity from the apparent ether intake apex onto the interferometer. Miller never remarked on the similarity of his “observed” velocity (Miller’s Table V) to the escape velocity from the Earth’s surface of 11.18 km/s, so it is not clear if he recognized it.
7. Mt. Wilson Azimuth Curves
Miller could not explain the effect in which the mean azimuth axis (24-hour average azimuth) is displaced generally eastward from North by as much as 55º. This azimuth displacement effect is illustrated in Miller’s Fig. 25, and Miller described it on his page 235:
In accordance with the simple theory, the direction of the cosmic motion should swing back and forth across the north and south line once in each sidereal day, because of the rotation of the earth on its axis. When the observed azimuth of motion is charted, the resulting curve of directions crosses its own axis twice in each day, as shown in Fig. 25, but this axis is variously displaced from the meridian. For the February epoch the axis is displaced by 10º to the west of north; for April the displacement is 40º east; for August 10º east, and for September 55º east.
Shankland5 cites this effect as the most important weakness in Miller’s findings (p. 169):
This azimuth anomaly has been the greatest obstacle to the acceptance of the small periodic amplitudes reported by Miller as having relevance to an aether-drift effect.
This criticism is unfair because it invokes an unproven theory to furnish the expectation. It would be more productive to seek a new comprehensive theory that fits the actual observations.
I do not have such a theory, but I did find an ad-hoc math model that yields a reasonable approximation to Miller’s Fig. 25, as you can judge from Figs. 11, 12, 13, and 14. Miller’s observations are plotted as square symbols.
I differ with Miller as to the values of the azimuth swings, especially in the February 8 epoch where Miller chooses to see -10º ±14º in his Fig. 26, and I choose to see -5º ±41º (see Fig. 11). Miller’s choice is constrained by his motionless ether model in which the ether exhaust velocity is the exact negative of the ether intake velocity, both being caused solely by the Sun’s motion through a static ether matrix. I relaxed that constraint and found that the Sun apparently moves through a moving ether matrix. This is analogous to the problem of airspeed versus ground speed for an airplane flying through a moving atmosphere.
Miller’s choice is weaker because it fails to account for the wide azimuth swings and the (mostly) eastward displacement of the mean azimuth axis, especially in the April and September epochs. My choice was forced in order to attempt to account for those swings and that displacement.
I found in Table I that the unaberrated or true ether intake velocity is 205.6 km/s from (RA, Dec) = (74.045º, -70.409º) in Mensa. I found in Table IV that the true ether exhaust velocity is 57.2 km/s towards (335.7º, +68.9º) in Cepheus. The negative sign on 57.2 in Table IV occurs when referring to ether motion towards the observer; the value is positive when described as an exhaust towards the apex. The angle between these vectors is 153.697º, which is certainly not 180º. Therefore the ether cannot be static.
This means that the Sun moves at approximately 129.06 km/s towards (89.860º, -73.165º) in Mensa and that there is a Galactic ether wind (that would be observed if the Sun’s motion could be stopped) of 78.195 km/s from (56.878º, -63.809º) in Reticulum. In the airplane analogy, the Sun appears to be flying into the wind, and the angle between the radius vector to that wind source and the Sun’s absolute “ground speed” is 14.96º.
If this is true, then the Sun does not orbit the center of the Galaxy in the Galactic plane because the angle between the solar velocity vector and the center of the Galaxy is 77.87º (not 90º), and the inclination of the solar velocity vector with respect to the Galactic plane is ±29.63º. This conclusion may raise doubt in the minds of readers who are biased by the prejudice that Newtonian gravity operates over galactic distances because the Newtonian velocity of the Sun in a circular Keplerian orbit should be on the order of 220 km/s based on the estimate of the Mass of the galaxy inside the orbit and the distance to the center of 8.5 kpc.[4]
The true intake ether wind speed (before aberration) is the vector sum of the galactic wind vector and the solar velocity vector.
Vintake = 78.20(56.878º, -63.809º) + 129.06(89.86º, -73.165º) = 205.6(74.045º, -70.409) km/s.
Likewise the true exhaust ether wind velocity is the vector difference between the solar velocity vector and the galactic wind vector.
Vexhaust = 78.20(56.878º, -63.809º) - 129.06(89.86º, -73.165º) = 57.2(335.7º, +68.9º) km/s.
I call this model ad-hoc because I had to introduce two new parameters that I don’t fully understand and can’t defend in a rigorous way: (1) a phase shift and (2) the radius for the diurnal aberration, as shown in Table IV as R/Ro. Also, I cannot explain why the magnitude curves (Figs. 7-10) seem to follow the aberration of the true intake apex and the azimuth curves (Figs. 11-14) seem to follow the aberration of the true exhaust apex.
Diurnal aberration normally occurs on the surface of the Earth, and that is what I assumed in Table I. For Table IV, I assumed that the Earth entrains a bubble of ether that rotates with the Earth as a rigid body and that diurnal aberration takes place on the bubble’s surface.
The path from that surface to the observer is assumed to be a helical path, and the twisting of the helical trajectory explains the phase shift. The axis of the helix is assumed to be parallel to the radius vector from the observer to the apparent exhaust apex but displaced from the observer’s location by the helical radius. That is, the observer is assumed to be on the periphery of the helix, not on the central axis.
At the ground terminal of this helical trajectory, the helical ether wind speed in the direction of the helical path is presumed to be the exhaust velocity of 57.2 km/s. The axial velocity from the ground to the apparent exhaust apex is presumed to be 11.18 km/s, the escape velocity from the surface of the Earth.
This may appear to be a physical model, but it is not. I hope this math model provides some clues that will lead to a physical model. A physical model would have to explain what happens at the surface of this ether bubble. One would hope that the facts that are known about the Earth’s magnetosphere could be integrated into such a physical model. Is this bubble surface identical to the magnetosphere? It would seem that further study is warranted.
At the bubble surface, the diurnal aberration is assumed to operate on an escaping ray of ether “particles” moving at the escape velocity parallel to the helix axis. If this were a physical model, one would expect the ray to be moving at the exhaust velocity, not the escape velocity.
There are two aspects to the azimuth anomaly. In addition to the displacement of the mean azimuth generally east of north, there is the problem of the large azimuth swings, especially in the February epoch. Miller chose to ignore this problem as you can see from his Fig. 27. He really had no choice with a motionless ether model. In that case the exhaust velocity must be the negative of the intake velocity, and the cosmic ether wind would be caused only by the Sun’s motion through a motionless ether.
Speaking of the February epoch, any competent theory needs to address the short-duration azimuth deflections (I call them azimuth “suckouts”) of about 20º towards north from 2 to 5 hours and again from 16 to 18 hours local civil time. A similar effect also occurs in the September epoch and to a lesser extent in the August epoch. Because these azimuth deflections seem to be coordinated with local rather than sidereal time, I suspect that the Sun somehow causes this effect. This is another topic for further study.
The slower exhaust velocity (-57.2 km/s compared to the intake velocity of 205.6 km/s) in Table IV yields a larger diameter annual aberration circle and hence a greater swing in azimuth. The effect of increasing the diurnal aberration radius above ground is to reduce the escape velocity (the assumed ray velocity for diurnal aberration) and to increase the (elevated) “observer’s” effective tangential rotation velocity, both of which increase the azimuth amplitude.
My model optimization process adjusted the true exhaust apex position and the exhaust velocity to minimize the least squared errors in the observed azimuth as a function of time of all four epochs (ignoring the data points in the azimuth suckout regions) at once, but the entrained ether bubble radius and the phase delay were adjusted separately for each epoch. The average azimuth displacement can be fine-tuned by adjusting R/Ro, the radius of the ether bubble (as a fraction of Earth’s equatorial radius) from which diurnal aberration is assumed to operate. Miller puts the average February azimuth at -10º, that is ten degrees west of north. This math model when optimized produced a February average azimuth displacement of about -5º by allowing the parameter R/Ro to be negative. I don’t know what a negative bubble radius means.
It must be noted that the ether exhaust parameter optimization process used in Table IV for the azimuth curves is quite different from the ether intake parameter optimization process used in Tables I, II, and III for the amplitude curves. The exhaust optimization used, as errors to be minimized, the differences between my computed and Miller’s observed azimuths at each time during the day for each epoch, again ignoring the azimuth suckout data.
This is really a better technique than the one I used for the intake parameter optimization for the amplitude curves because in that case the errors were the differences between Miller’s calculated apparent apex positions and my calculated apparent apex positions. If I had the time, I would recompute the intake parameters using the differences between Miller’s observed amplitude values at each time during the day and my computed amplitude values as the errors to be minimized. This problem is not solved, and I intend to continue studying it. As part of the topics to be studied further, I intend to perform that recomputation.
Table V shows the details of the presumed exhaust helix that connects the ground to the surface of the entrained ether bubble. The phase shift and the bubble radius were found in Table IV by minimizing the least squared azimuth data errors. The number of whole turns in the helix was chosen to yield a consistent if not exactly equal helix screw pitch for each epoch. The helix screw pitch in arcminutes per kilometer is
, (4)
where (R/Ro - cos z) is the axial length of the helix from the surface of the Earth to the bubble surface in earth radii, R/Ro is given in Table IV, and z is the zenith distance of the apparent exhaust apex.
The screw pitch in Table V is an approximation because I used the mean zenith distance for each epoch. The zenith distance varies over the 24-hour day from 23º to 89º in the February 8 epoch, from 47º to 78º in the April 1 epoch, from 27º to 85º in the August 1 epoch, and from 36º to 96º in the September 15 epoch. Since this is only a math model, not a physical model, the approximation is tolerated. A more rigorous analysis would need to compute the screw pitch and the helix radius at each moment in time during the 24-hour day.
I think it must be significant that a whole number of helix turns can be found for each epoch that yields about the same helix screw pitch. That mean pitch value is 1.75 arcminutes per kilometer of axial length.
The helix radius can be computed for each epoch from the total number of turns and the axial length if the helical path length is known. The helical path length can be computed from the helical speed (presumed to be the exhaust speed on the ground) and the time interval it takes a ray travel from the ground to the bubble surface. The axial length of the helix is known, and the axial velocity is presumed to be the escape velocity, so the time interval can be calculated.
The axial path velocity is the escape velocity that varies with the radius from the center of the Earth according to formula
, (5)
where R is the radius of the bubble from the center of the Earth, Ro is the equatorial radius of the Earth, and the numerator is the escape velocity at R = Ro. What we need is the average axial or escape velocity over the path from Ro to R. That average escape velocity is given by
. (6)
We assume the exhaust velocity of 57.2 km/s is the value at Ro at the surface of the Earth and that it is in a constant ratio with the escape velocity, which is 11.18 km/s at Ro. The helical velocity is the exhaust velocity, and that is presumed to be proportional to the average axial or escape velocity, so
. (7)
The average axial and helical speeds are given in Table V for each epoch. The path duration in seconds is the same for the helical path at the helical speed as it is for the axial path at the axial speed. The path duration is given by
. (8)
The helical path length, expressed in earth radii, is given by
(9)
When you unwind a helix from a cylinder you get a right triangle. Therefore, the helical path length is the hypotenuse of the right triangle whose base is the number of turns of the screw times the helix circumference and whose altitude is the axial length of the helix. Therefore, the radius of the helix is given by
, (10)
where N is the total number of turns in the helix and the sign on the radical is the sign of the helix path length. The mean value of the helix radius (ignoring the negative sign in the February epoch) is 1.55 earth radii.
8. Diurnal Aberration
The diurnal aberration of light is extremely small, 0.264
arcseconds maximum at Mt. Wilson’s latitude, because the rotational velocity is
so small compared to the speed of light.
The rotational speed of the equator is 2p(6378.137 km)/(24)(60)(60)
s = 0.4638 km/s. At Mt. Wilson’s
latitude it is (0.4638 km/s)cos(34.22º) = 0.3835 km/s. The maximum diurnal aberration is arctan(v/c),
where v is the rotational velocity and c is the ray
velocity. This is the aberration for a ray
source on the celestial equator and the meridian. Seidelmann[5]
gives the formulas for diurnal aberration.
To avoid problems with large angle offsets near the celestial pole, I
used rectangular coordinates and the geocentric velocity vector.
Table VI illustrates how the entrained bubble amplifies diurnal aberration. The last column is the maximum diurnal aberration in degrees.
For the intake ether stream, the maximum diurnal aberration is only 0.106º. The bubble ray velocity Vb is the apex velocity divided by the square root of R/Ro. The rotation velocity is 0.384 km/s times R/Ro. For the exhaust ether stream, the maximum diurnal aberration is 10.73º
9. Secular Fringe Shift
Shankland did not comment on the secular fringe shift. Miller identified it without explaining it (op. cit. p. 213).
Under ideal conditions, all the numbers in column one (and in column seventeen) [of Fig. 8] should be the same integer but actually there is always some shift of the fringe system with respect to the fiducial point. This shift is assumed to be steady, or linear, throughout the time of one turn, about twenty-five seconds, which is equivalent to assuming that the periodic displacements of the fringes take place with respect to an inclined axis.
One possible interpretation of this effect is that the secular fringe shift rate of 2.56 wavelengths per turn (in Miller’s Fig. 8) is due to a steady temperature drift, since the interferometer rotates at a steady rate of one turn in 50 seconds. If so, then Miller’s data reduction method eliminates any drift due to a steady temperature change since the triangular ramp function is subtracted from the data readings. Since Shankland’s proposed mechanism for the “random variations” he hypothesized was temperature changes, it would seem that this technique of subtracting the ramp falsifies Shankland’s criticism. If temperature drift were the cause of the secular fringe shift, we would expect to see some positive ramps and some zero ramps as well as the negative ramp shown in Miller’s Fig. 8. But Miller says that his Fig. 8 is typical of all runs (op. cit. p. 214):
The set of readings here illustrated is not exceptional; it is a fair sample as to magnitude and periodicity of the ether-drift effect. This particular displacement corresponds to an ether drift velocity of 9.3 kilometers per second. Every set of readings shows a very definite periodicity which varies both as to magnitude and phase in a systematic manner.
Another interpretation of the secular fringe shift is that it is an observation of an acceleration of the solar system with respect to the ether. If the Sun is in orbit about something, then it very definitely is being accelerated at all times, and such acceleration ought to be observed as a steady fringe shift.
The calculation of 2.56 fringes per 50-second turn was determined from Miller’s Fig. 8. The acceleration is given by
, (11)
where the coefficient of the radical is the speed of light, the numerator under the radical is the secular fringe shift during one turn and the denominator under the radical is the number of wavelengths of 2700-Angstrom light in the path. This is 92.7 times the acceleration of gravity.
If Miller’s actual data becomes available, the secular fringe shift should be computed for every data run, not just the run in Miller’s Fig. 8. If this is truly an acceleration, it should be about the same for every data run.
10. Cosmic Microwave Background Anisotropy Dipole
Fixsen et al[6] wrote regarding the cosmic background radiation anisotropy dipole, “The Doppler shift implies that the Sun’s peculiar velocity relative to the comoving frame is 371±1 km/s (95% CL) toward (galactic longitude, latitude) = (264.14º±0.15º, 48.26º±0.15º), in agreement with the microwave results from the DMR.” This translates to J2000 (RA, Dec) = (168.012º, -6.983º) in southern Leo. We can hypothesize that the frame about which the Sun orbits has itself some velocity relative to the cosmic microwave background. For convenience in notation we might call that frame the galaxy and that velocity Vgalaxy. Then we could write
Vgalaxy = VCMB - VSun = 371(168.01º, -6.98º) - 129.06(89.86º, -73.17º) = 370.8(173.80º, +12.21º)
However, suppose we hypothesize that VCMB
is the apparent ether wind from the CMB that is aberrated by the Sun’s
motion. Let’s call VCtrue
the true CMB wind vector unaberrated by the Sun’s motion. We find that VCtrue =
370.8(173.80º, +12.21º), which is the same as the old Vgalaxy
above. Now we can define a new Vgalaxy
Vgalaxy = VCtrue - VSun = 370.8(173.8º, +12.2º) - 129.1(89.9º, -73.2º) = 413.1(179.7º, +29.3º)
This is an interesting direction because it is only 12º from the North Galactic Pole. It is interesting because the Sun’s velocity vector is only 6º from the south pole of the solar system’s Invariable Plane. It makes sense that the galaxy might move in a direction not far from its own polar axis.
11. Miller’s Accomplishments
Although the quality and thoroughness of Miller’s work is clearly evident in his 1933 paper, I would like to list his accomplishments in case any reader might doubt his competency. Dayton C. Miller was a significant if somewhat forgotten figure in American science and music. His interest in astronomy dates from his childhood. As a young boy, he made a total of three refracting telescopes, each being larger and better than the preceding. In his search for more knowledge, he corresponded with the famous telescope maker of Pittsburgh, Dr. John A. Brashear. This friendship grew over many years and both enjoyed a healthy respect for each other. Miller, in one of his writing's, referred to Brashear as his “scientific father.” The following is a partial list of his non-musical accomplishments and awards[7].
1886 Ph.B. degree from Baldwin College.
1889 M.A. degree from Baldwin College.
1890 D.Sc. degree from Princeton.
1899 Appointed trustee of Baldwin-Wallace College. Chairman of the Board from
1936.
1903 Publication of LABORATORY PHYSICS.
1914 Lowell Institute lectures, SOUND ANALYSIS.
1916 Publication of THE SCIENCE OF MUSICAL SOUNDS.
1921 Became a member of the National Academy of Sciences.
1924 Honorary degree D.Sc. from Miami University.
1925 Became president of American Physical Society.
1926 Elliott Cresson medal of the Franklin Institute for fundamental
investigations in acoustics.
1926 American Association for the Advancement of Science Award; Kansas City.
1927 Honorary degree L.LD. from Western Reserve.
1927 Cleveland distinguished service medal.
1927 Los Angeles Flute Club Award; February 6th.
1927 Ambrose Swasey Chair of Physics at Case.
1927 Honorary degree D.Sc. from Dartmouth College.
1928 Wagner Free Institute of Science Lectures.
1931 Elected president of Acoustical Society of America.
1931 Elected member of the board of the Perkins Observatory, Ohio Weslyan.
1933 Honorary degree L.LD. from Baldwin-Wallace College.
1935 Publication of THE ANECDOTAL HISTORY OF THE SCIENCE OF SOUND.
1936 Honorary degree D.Eng. from Case University.
1937 Publication of SOUND WAVES: THEIR SHAPE AND SPEED.
1939 Publication of SPARKS, LIGHTNING, AND COSMIC RAYS.
1940 Honors from Case University; July.
Other memberships included: National Academy of Sciences, The National Research Council, The American Philosophical Society, The American Academy of Arts and Sciences, The Cleveland Engineering Society, American Institute of Physics, American Astronomical Society, American Mathematical Society, American Acoustical Society, American Meteorological Society, Optical Society of America, Seismological Society of America, Society for the Promotion of Engineering Education, American Musicological Society, American Guild of Organists.
12. Conclusion
Much work remains to be done. Miller’s Fig. 22 shows what appears to be noisy data that has been smoothed with moving averages. I believe that these original observations should be re-analyzed using the least-squared error technique instead of moving averages, and the cosmic model should incorporate an entrained ether bubble and consider diurnal as well as annual aberration. The reason for the scatter in Miller’s data may be largely due to lumping 15 days worth of observations onto the mean epoch. It appears that diurnal aberration is a very important factor, especially for the azimuth curves because of the entrained ether bubble, which may be the Earth’s magnetosphere.