Stellar Aberration

 

The M/M/+M® Model V2 and its Bearing estimator (Best) Equations are derivatives of the M/M/+M® Model V1.  The M/M/+M® Model V1 was developed to facilitate the measurement and control of the performance of a network of communications platforms that are moving at relativistic velocities outside our Solar System.  Figure 1 is an illustration of the M/M/+M® Model’s metrics for a Traveler at point b that is moving at Earth’s average orbital velocity (v = 29.93 km/sec).  Also, see Description_for_Figure_1.  Point b is not on the Earth’s surface or near a moon and does not encounter the spin or wobble velocity vectors associated with the Earth.  Point b is considered to be in orbit around the Sun; therefore, the M/M/+M® Models' designs do not need to account for precession or nutation of Earth's axis or the celestial plane.  There is not a design requirement for the M/M/+M® Model V2 to account for Earth's spin velocity components or changes in pole orientation due to precession or nutation because these changes are known and can be easily removed from the raw data before it is used as input to the M/M/+M® Model for the purpose of predicting the relativistic effects of a point that moves in the Earth's ecliptic plane.  Therefore, in order to simplify the discussion of relativistic stellar aberration the positional metrics convention used by astronomers that including precession, nutation, declination (d = 0^o to 90^oin the 2^nd quadrant as well as the 1^st quadrant) and right ascension (ao) will be replaced by a metrics convention for intra-galactic stellar navigation.  The Intra-Galactic Stellar Navigation (IGSN) metrics convention will define, in simple terms, the measurements taken by an Observer at point a (see figure 1) who is stationary with respects to (wrt) a solar system’s barycenter (solar system’s center of mass - a point near the center of the Sun wrt Earth’s orbital plane) as well as the measurements taken by a Traveler at point b.  The IGSN metrics system uses boundaries and bearings for topographical orientation.  In general, a topographical boundary is the border of a 2-D or 3-D geometric form and a bearing is the angular displacement of a vector from a point on a boundary wrt a designated plane or another vector.  A bearing may also be the angular displacement of a vector from a celestial body that passes through a point on a boundary wrt a designated plane or another vector.  Star vectors are aligned with points (such as points d and d' in figure 1 or points e and e' in figure 3) that are on an Observer's Spherical Inertial Frame (SIF) boundary and a Traveler's SIF boundary respectively.   

There are two spherical boundaries discussed in figure 1: the boundary of the Traveler's SIF and the boundary of the Observer's SIF.  Figure 1 is an x-y plane cross-section of the two spherical boundaries.  Their is a 2-D bi-convex kinematical form (yellow area) and a 2-D circular form (blue circle) in this view.  These two 2-D boundaries have centers at point b and they represent a single Boundary Pair.  The blue circle is the boundary seen by a traveler at point b.  A distant star that is seen through an aperture at the circular boundary at point d' by a Traveler at point b will be seen by a "stationary" Observer (in the inertial frame of the solar system's barycenter at point b) through an aperture on the bi-convex boundary at point d.  Point d' is displaced from point d along a line that is parallel to the x axis and in the direction of motion (+x direction). By definition a Boundary Pair is always in a plane that contains the velocity vector.  This plane that contains a Boundary Pair is called a Boundary Plane.  All Boundary Planes contain the velocity vector; therefore, all Boundary Planes are perpendicular to the Great Circle of the Traveler's Bi-convex Inertial Frame (BIF).  Boundary Planes are named in degrees from 0^o to 90^o in the Northern Hemisphere and from 0^o to 90^o in the Southern Hemisphere.  The method for orientation of Boundary Planes will use the Earth's orbital plane (ecliptic plane) as a reference plane for our Local Group Of Stars (LGOS) where the North Ecliptic Pole (NEP) is in the Northern Hemisphere and the South Ecliptic Pole (SEP) is in the Southern Hemisphere and where the Earth's orbital plane is the 0^o boundary (0^o North or 0^o South).  The Boundary Pair shown in figure 1 is in the 90^o North Boundary Plane that is perpendicular to the Earth's orbital plane and contains the Traveler's y axis (the y' axis) as well as the Earth's velocity vector.  Unless otherwise specified, the following discussion of stellar aberration will apply to the 90^o North Boundary Plane during event 1 when this plane contains the x axis (the velocity vector) and will apply to a q North Boundary Plane during event 2 when the vector to a star approaches being perpendicular to the velocity vector three months after event 1.  For example, a star located at q = 60^o in the 90^o North Boundary Plane will be located at q = 60^o in the 60^o North Boundary Plane during event 2 where the 60^o North Boundary Plane is at a 60^o North declination wrt the Earth's orbital plane.             

Events 1 and 2 are two key events when orbiting Travelers should record measurements from a platform with a floor in the plane of a circular orbit that is 14 x 10⁷ km from the solar system’s barycenter (same as Earth’s average distance to our Solar System’s barycenter) moving at v = 29.93 km/sec (same as Earth’s orbital velocity).  These two events involve bearing angles q and f that are measured by an Observer at point a and a Traveler at point b respectively.  Bearing angles q and f that are between 0º and 90º are declination angles measured wrt the Earth's orbital (or ecliptic) plane.  Bearing angles between 90º and 180º are supplementary to Bearing angles between  0º and 90º (i.e. a 135º bearing has the same geometry as a 45º declination wrt the Earth's ecliptic plane).  Bearing angles are NOT the same as the declination angles  (do) that are measured wrt the current orientation of the Earth's equatorial (or celestial) plane.  The bearing q will apply to two parallel vectors to the same distant star, the one that is measured by the Observer at point a and another that defines the direction to the star from point b using the coordinates of the Solar System's barycenter.  References made to the bearing q in equations and discussions of event 1 and event 2 will refer to the bearing q of the vector to the star that is re-aligned to pass through point b unless otherwise specified.  Also, discussions of event 1 and event 2 will apply to bearings where q is in the range from 0º (approx.) to 90º wrt the orbital plane during every month of a year for boundary planes in the range from 0º North (approx.) to 90º North.  Initially, annual aberration will be defined as the difference between the angle f measured by a Traveler who is in linear or orbital flight (with no spin velocity) and the angle q measured by the Observer.  That is, annual aberration is defined as f - q  for a Traveler who has only one inertial velocity component and that component is in the Earth's ecliptic plane (with no spin velocity).  This definition will pertain whenever the word aberration is used in this web page unless otherwise specified.  For example, when a star's vector is in the Earth's ecliptic plane the angle f will be zero and the angle q will be zero because both angles are declination angles with zero values wrt the orbital plane during every month of a year.  Therefore, by this consistent, unequivocal and unambiguous initial definition of annual aberration, a star in the Earth's ecliptic plane will be zero at all times during a year.  This does not mean that there is no aberration for a star in the Earth's orbital plane; it simply means that there is no aberration of the bearing f (declination wrt Earth's orbital plane) measurements taken by the Traveler for stars in the Earth's orbital plane.  Best equations for estimating aberration using bearings other than q and f will be discussed near the end of this Stellar Aberration page.  Event 1 occurs when the bearing f of the vector from point b to a given star wrt the Traveler’s platform floor (floor is in the orbital plane of point b) has a maximum deviation from the angle q (see figure 1 and discussion of star vectors in Description for figure 1).  Since a star vectors' bearings (both f and q) must be within the 90º North Boundary Plane during event 1, the Boundary Pair is in a plane that is perpendicular to the ecliptic plane as well as in a plane that is perpendicular to the Great Circle of the Traveler's BIF. All stars in the range q = 0.000---1 to q = 89.999---9º in the 90º North Boundary Plane will apply to the following discussion of event 1.  Event 2 occurs when the deviation between q and f is minimum. 

All stars in the range q = 0.000---1º to q = 89.999---9º in the q North Boundary Plane will apply for the following discussion of event 2.  Event 2 occurs when the bearing f of the vector to a star wrt the Traveler’s platform floor approaches the same value as the bearing q to the star wrt the orbital plane.  This is when the bearing vectors are in the y’,z’ plane (see “Edge view of the great circle of Traveler’s BIF - - - in the plane (x’,y’,z’) = (0, y’, z')” solid blue line in figure 1).  That is, event 2 occur when the two vectors to the star (with bearings f and q) are in a plane that is perpendicular to the velocity vector and at this time the two vector's must be merged into one vector as the kinematical form in the Observer's coordinates becomes a circular boundary and it becomes congruent with the circular boundary seen by the Traveler.  Also, it will be shown that event 2 occurs at the point where the difference between the two vectors with bearings f and q approach 0º and it will be shown that Event 2 will occur at points where the bearings f and q approach 90º (the bearings that are perpendicular to the direction of  motion).  Examples of measurements for events 1 and 2 as well as the accuracy and validity of the M/M/+M® Best Equation’s predictions will be discussed below.

The conventional astronomical notation for a bearing angle measured by a stationary Observer from the star Polaris is declination (do) where do = +89.5º (approx. current declination wrt the Earth's Equatorial Plane) that is the same topographical place as q = 66º 36' wrt the Earth's Ecliptic Plane (see figure 5 ) during every month of a year including the moments when event 1 and event 2 occur.  However, the Traveler at point b measures the declination (d) to Polaris to be d = +89.495º (or about 19 arc seconds less than 89.5^o) at event 1.  At the occurrence of event 2 the Traveler measures the bearing to Polaris to be d = +89.5º (approx.) that is the same bearing recorded for do by the Observer during event 2.  By definition, maximum aberration (f - q) = (d - do)  for any given star (e.g. about -19 arc sec. for Polaris) occurs at the time of event 1.  Also, by definition, minimum aberration = (f - q) = (d - do) = 0^o during event 2  for all star.

Figure 1 is not to scale because at v = 29.93 km/sec there is not enough separation between the angles f and q for clearly illustrating the differences in the two bearings (directions) and dimensions (distances); however, the numbered dimension and bearing callouts are precise.  The derivation of the initial M/M/+M® Best Equation for Stellar Aberration is based upon the Bi-convex Inertial Frame’s (BIF’s) spherical wave front in the Traveler’s 1^st and 4^th quadrants (i.e. the wave front moving ahead of the Traveler).  This initial M/M/+M® Best Equation for estimating q as a function of the known value for f is derived from figure 1 as follows:

tan(q) = (y_d / x’_d)
q = atan(y_d / x’_d) Where: y_d = L_o * sin(f), x_d = x’_d +  (t * v) and x_d = ((c * t)² – y_d²)^0.5
        Therefore: x’_d = x_d – (t * v) = ((c * t)² - y_d²)^0.5 – (t * v)

Finally: Stellar Aberration = f - q

The Best Equation will apply to all star vectors from q = 0^o to 89.999 - - -9^o (in the direction of motion) when a platform at point b travels from the real points a at our Solar System’s barycenter at (x, y, z) = (0,0,0) to a nearby solar system as well as to all star vectors from q = 90.000---1^o to 180^o (in the direction of motion) when the platform travels from the nearby solar system back to point (x, y, z) = (0,0,0).  However, the Best Equation will not apply at exactly q = 90^o because at the limit q = 90^o then tan(q) = n / 0 = the imaginary number infinity.  The IGSN uses a bearing angle convention with values ranging from 0^o to 180^o for the 1^st and 2^nd quadrants (+y’ or northern hemisphere) above the Traveler’s platform floor and 0^o to 180^o for the 3^rd and 4^th quadrants (-y’ or southern hemisphere) below the Traveler’s platform floor that replaces the indiscriminate declination convention currently used by astronomers with values ranging from 0^o to 90^o North for the 1^st quadrant as well as the 2^nd quadrant and uses 0^o to 90^o South for the 3^rd quadrant as well as the 4^th quadrant.  The IGSN conventions and the Best Equation will apply to star vectors for intra-galactic space Travelers as well as for Travelers who are orbiting a solar system’s barycenter.  However, in the case of an orbiting traveler, point a is an imaginary point that is stationary wrt the solar system’s barycenter at a distance of v * t behind the orbiting platform at the instant of a recorded stellar aberration event.

There are several equations in the state-of-the-art that estimate relativistic stellar aberration.  The M/M/+M® Best Equation for estimating aberration (f – q) is compared to the SRT (1905) equation developed by Einstein 1905² that can be used to estimate relativistic stellar aberration (f' - f) based upon an ellipsoid of revolution wave front where f is known rather than upon a spherical wave front.  The derivation of equation (1) following is described in section 7 of Einstein's 1905 publication "On the Electrodynamics of Moving Bodies". 

(1) According to Doppler's principle of aberration: cos(f') = (cos(f) - (v/c))/(1 - (cos(f) * (v/c)))

Therefore, the SRT (1905) estimator of stellar aberration is derived as follows:
f' = acos(cos(f) - (v/c))/(1 - (cos(f) * (v/c)))
And finally: 
Stellar Aberration =  f' - f in the SRT (1905) convention.

The SRT (1905) stellar aberration estimator will predict star locations only during event 1.  The Naval Observatory NOVAS software that is widely used by astronomers for prediction of star locations and formerly used for navigation has adapted a more accurate stellar aberration estimator that will predict aberration at any future date and not just during event 1.  This model is based upon the Astronomical Almanac (AA) 2010³ stellar aberration declination reduction basic equation (see page B28 in 2010 Astronomical Almanac) where do  is known as follows:

The declination reduction for annual aberration from a geometric geocentric place (ao, do) to an apparent geocentric place (a, d) is given by:
d =  do + (- X cos(ao) sin(do) - Y sin(ao) sin(do) + Z cos(do))/c
                where X, Y, and Z are velocity components at the geocentric place (point on Earth's surface). 
Therefore: Stellar Aberration = d - do = (- X cos(ao) sin(do) - Y sin(ao) sin(do) + Z cos(do))/c

Stellar aberration (d - do) at an orbiting point (point b) can be derived from this AA (2010) basic equation for estimating d and simplified to a function of the orbital inertial velocity X, the sidereal angle ao and the declination angle do during both events 1and 2 for a traveler in the orbit of the Earth (not at a point on Earth's surface).  If the Earth had no spin velocity, the (X, Y, Z) components of velocity can be simplified to (X, 0, 0) during events 1 & 2 because there is no Y or Z component of velocity.  The simplified AA (2010) equation for calculating stellar aberration during event 1 and event 2 from a platform with orbital inertial velocity only (no spin velocity) would become: 

Aberration (d - do) = (-X/c) cos(ao) sin(do); where X is the Traveler's inertial velocity vector wrt the Earth's equatorial plane.

        The following derivative of the AA (2010) basic equation can be applied for estimating stellar aberration (f - q) where q is known:

The declination reduction from a dynamical barycentric place (q) to an apparent place (f) wrt  the ecliptic plane is given by:
Aberration (f - q) = (- v/c) cos(ao) sin(q); where v is the Traveler's inertial velocity vector wrt the Earth's ecliptic plane.  

The Bradley 1729¹ Constant of Aberration equation (f - q) = -atan(v/c) and yields a maximum aberration limit = -0.005720166166^o = -20.592598198 arc seconds for Earth’s average orbital velocity (v =29.93 km/sec) and the velocity of light (c=299792.458 km/sec).  This constant agrees with the Falling Rain Model's prediction of the aberration for a zenith star where (f - q) = -asin((v/c) sin(f)) = -20.592598198 arc seconds and where f is measured to be 90º - asin((v/c) sin(f)).   The SRT (1905) estimate for the aberration of a zenith star during event 1 is: (f' - f) = acos(cos(f) - (v/c))/(1- (cos(f)(v/c)))) - f = +20.592598301 arc seconds.  The AA (2010) aberration estimator also returns a maximum aberration limit for an object at the zenith (do=90º) that closely approximates the Constant of Aberration (-20.592598198 arc seconds) when the Earth's axial spin is discounted, where X = v and where the sidereal angle ao cannot be defined for a zenith star.  Therefore, the AA (2010) estimate for the aberration of a zenith star during events 1 and 2 becomes: (f - q) = (- v/c) sin(q) = -20.592598267 arc seconds.  The SRT (1905} and the AA (2010) aberration estimator equations both assume that this maximum aberration limit will apply in a continuous equation and a star’s vector would pass from the 1^st quadrant to the 2^nd quadrant through the bearing angle  q = 90^o for the SRT (1905) estimator and through the declination angle do = 90^o for the AA (2010) estimator.  This limit was named the Constant of Aberration.  However, this is a misnomer; the concept as applied in the SRT (1905) estimator is a theoretical upper limit of aberration that for any given velocity of a traveling telescope occurs only for a distant star that has a declination of q = f = 90^o wrt an arbitrary "floor" that contains the telescope's velocity vector.  Furthermore, it will be demonstrated that an aberration value close to the Constant of Aberration at a declination of 90^o wrt any "floor" is irrational and that aberration actually approaches zero as a star's declination approaches 90º as predicted by the Best Model.   

Figure 2 contains a plot of the three most precise equations for estimating or predicting stellar aberration.  The plots for the SRT (1905) estimate and the Astronomical Almanac (2010) estimate are very close as they approach the value of q = 90^o in the 90^o North Boundary Plane because they both assume a value for maximum aberration (f - q) = -0.005720166^o at a bearing of q = 90^o on the y' axis.  However, the aberration at a point star on the y’ axis at a bearing of precisely q = 90^o would be a singularity and could not be observed or measured.  If the vector to a star is in the plane that is perpendicular to the direction of motion then aberration is zero for the bearing range do = 0.000---1^o to do = 89.999 - - -9 ^o.   It is an established fact that there are two times during a year when the relativistic aberration to a given star in the range do = 0.000---1^o to do = 89.999---9^o is observed to be minimum (approaches 0^o) including zenith stars where q approaches 90º and is the same place as do approaches 66º 36' except measured wrt the ecliptic plane instead of wrt the celestial plane, see figure 5 ).  This occurs every sixth month when the aberration equation returns a value approaching 0^o.  During this event (event 2) the angle do is approximately equal to the angle d, therefore, aberration (d – do) approaches 0^o.  This is a simple axiom of mathematics, when d approaches do then (d – do) approaches 0^o and when f approaches q then (f – q) approaches 0^o regardless of whether the vector to the star has a declination of 0.000---1^o, 15^o, 30^o, 45^o, 60^o, 75^o, 80^o, 85^o, or 89.999- - -9^o and regardless of whether the declinations are named q and f when measured wrt the ecliptic plane or named do and d when measured wrt the celestial plane.  This conclusion is supported by the actual observed differences in the declination angles for all stars when their vectors are perpendicular to the velocity vector.  After nutation, precession and diurnal aberration are removed from the measured declination of stars with a positive x' coordinate the data show that aberration (d – do) has a maximum negative value once a year and the aberration (d – do) data for the same star six months later show a maximum positive value when it has a negative x' coordinate.  This means that aberration (d-do) had to be zero twice a year, once when it goes from negative values to positive values and once when it goes from positive  values to negative values after nutation, precession and diurnal aberration are removed.

During a study in 1725 and 1726, James Bradley 1729¹ accumulated the most precise measurements ever taken of the star Gamma-Draconis up until the time of that study.  During November 1725, he observed that this star was moving south from his "zenith" (when his telescope was originally perpendicular to a level floor).  His "zenith" was the same location as do = 51.5^o (approx.) or at q = 75^o (approx.) wrt the Earth’s orbital plane when it was in a plane that contains the Earth's velocity vector and was perpendicular to his observatory's floor in London.  He studied the star from December 1725 through March 1726 and concluded that Gamma-Draconis continued to move south for another 20 arc seconds until March 1726 (at event 1, see figure 5 ).  He recorded this as aberration of about 40 arc seconds south during March when Gamma-Draconis was in the first quadrant in the direction of Earth’s movement (i.e. during event 1).  Three months later in June 1726 he recorded that the star had moved north about 20 arc seconds to the same declination it had been during December 1725.  Bradley's interpretation of this behavior was that the star had described a half circle with a radius of 20 arc seconds in six months and it would describe a full circle in one year.  He eventually concluded that aberration of gamma-Draconis described a circle with a radius of 20", where the aberration during March was 0", the aberration during June was 20", the aberration during September was 40" and the aberration in December was 20" (see figure 3 at Aberration_of_light ).   This conclusion was based upon his "falling rain" theory that an umbrella must be pointed forward when standing on the level surface of his observatory's "floor" that is moving in rain that is falling perpendicular to this "floor" in order to keep from getting wet.  The falling rain theory is described in figure 7A and figure 8A.  Figure 7A shows the derivation of the Constant of Aberration that is based upon the aberration of a "zenith" star where the "zenith" is defined as a distant point that is always measured to be at a 90º declination wrt  to any arbitrary "floor".  Figure 8A shows the derivation of a stellar aberration estimator based upon the falling rain theory for all celestial objects including "zenith" stars.  The falling rain theory predicts aberration values that are close to the same as the predictions of the AA (2010) model during event 1; however, as the star approaches event 2 the falling rain model's predictions are nowhere close to the AA (2010) model's predictions or empirical observations except for the prediction of aberration for a "zenith" star using the equations shown in figure 7A and 8A.  There are many other non-sequiturs in Bradley's logic; some examples are as follows.  (1) According to the falling rain theory, the aberration of a star at the "zenith" is constant all year with a value equal to the "Constant of Aberration".  Bradley assumed that this would apply to the star gamma-Draconis and it would have a constant aberration all year because it was at his "zenith" (directly overhead).  His own data contradicts this theory because he said his measurements showed that gamma-Draconis had constant aberration of 20" all year long not 20.6" as calculated by his constant of aberration equation.  (2) The velocity component perpendicular to a vector to Bradley's "zenith" was not always the same as the orbital velocity of the Earth; it is variable and depends upon the Earth's spin velocity as well as right ascension.  In short, gamma-Draconis is not a zenith star (i.e. it is not at the zenith wrt the Earth's orbital plane); therefore, according to Bradley's own falling rain theory its aberration should not be a constant value all year or describe a circle it should describe an ellipse.  (3) According to the falling rain theory aberration for gamma-Draconis should be close to ±20.6" during December and June when the star's vector is perpendicular to the Earth's orbital inertia velocity vector; but all observations since 1726 as well as Bradley's own observations prove that annual aberration for gamma-Draconis is much closer to 0" during December and June than ±20.6".  The primary problems with the falling rain theory are: (1) the predictions of aberration using the equation shown in figure 8A are nowhere close to the observed aberration of stars except when they are close to event 1, and (2)  more recent observations of the aberration of stars using computer controlled corrections for precession and nutation in right ascension and declination have shown that minimum aberration for all stars occur during event 2 when the vector to the star is perpendicular to the Earth's velocity vector (see data for Polaris below).  These observations invalidate the falling rain theory.  Therefore, for the sake of brevity the falling rain theory will not be included in further comparisons of state-of-the-art stellar aberration prediction models.  Also, the more recent observations indicate that minimum aberration for gamma-Draconis must occur when it is positioned at the location of the orange asterisk (*) shown in figure 5 during December and June, it must have aberration of -20" during March when it appears to be most distant from the North Ecliptic Pole (NEP), and it must have aberration of +20" during September when it appears to be at it's closest point to the NEP shown in figure 5.  Also, figure 3 at The stellar aberration of the star g-Draconis confirms that Bradley's own data matches the more recent observations if Bradley's aberration scale is changed to reflect the variation from aberration of 0" during December, -20" at its most distant point from the NEP during March,  0" during June, and +20" at its closest point to the NEP during September.
 
The maximum aberration for a given star depends upon its bearing angle q (see table 1 below figure 2).   The aberration estimates for the three aberration equations in table 1 for q = 75^o are within less than one millionth of one degree around the average measurement of 19.89 arc seconds.  The errors in the Astronomical Almanac (2010) and SRT (1905) estimates are over 126% wrt the Best estimate for zenith stars where q is greater than 89.999^o.  Bradley could not have actually measured the aberration of any star where q = 90^o wrt the Earth's orbital plane and when the vector to the star was precisely in the plane perpendicular to the direction of Earth’s motion as well as in the 90^o North Boundary Plane.  It is obvious that he only concluded theoretically that the maximum aberration of a star at a 90^o declination would be equal to -atan(v/c) based upon extrapolations of his observations of stars close to a 90^o declination wrt his arbitrary "floor".  Also, according to the SRT model of a kinematical form that has contracted only in its direction of motion (+x direction) there should be no deformation of the y' or z' axes that are perpendicular to the velocity vector that is the x' axis.  Therefore, there should not be any aberration in the plane containing the y' and z' axes.  This distant star with a vector that is perpendicular to the Earth's velocity vector should theoretically appear to have a center with bearings q = 90^o = f with zero aberration because the kinematic boundary that is a cross-section of the Observer's BIF has become a circle that is congruent with the circular cross-section seen by the Traveler.  However, when a star's vector is perpendicular to the Earth's velocity vector it must yield a maximum aberration value equal to the Bradley Constant of Aberration (-20.592598 arc sec.) according to the SRT (1905) and the AA (2010) aberration equations for any star regardless of its declination angle wrt the Earth's orbital plane, wrt the Earth's celestial plane, wrt Bradley's observatory floor or wrt any arbitrary plane that contains the telescope's velocity vector. 

For additional evidence that leads to the conclusion that Bradley's Constant of Aberration has not been verified, let's consider some of the unlikely predictions of the SRT (1905) and AA (2010) models for a star at q = 90^o wrt the Earth's orbital plane.  According to the SRT (1905) and AA (2010) aberration equations, a star precisely at q = 90^o would have a constant aberration value of -20.592598 arc sec. at all times during a year because this star's vector is always perpendicular to the orbiting observer's velocity vector.  This prediction of an aberration value = -20.592598 arc sec. for a star when its vector is perpendicular to a Traveler's direction of motion including a star at q = 90^o has no credibility due to the following incriminating circumstances.  (1) A star or distant celestial object precisely at the NEP (q = 90^o North wrt the Earth's orbital plane) has never been observed and widely reported to have annual aberration as great as the constant of aberration at all times during a year.  (2)  A distant celestial object at precisely q = 90^o North wrt the Earth's orbital plane (d₂₀₀₀ = +66^o 33' wrt the Earth's ecliptic plane) would have been much too dim for Bradley to have seen through his telescope.  It is extremely unlikely that Bradley could have seen any of the zenith objects with maximum aberration significantly below -20" such as the Cat's Eye Nebula (do = +66^o 38' with a 2.5' radius ) or the galaxy NGC 6552 (do = +66^o 36' with size 1'x 0.7' ) which are the brightest objects within 5' of the NEP (d₂₀₀₀ = +66^o 33') because these objects have a magnitude of 14 or dimmer which is too dim to have been observed using the telescope that was available to Bradley.  The aberration for a distant object at q = 90^o wrt the Earth's orbital plane has either not been authoritatively confirmed or has not been widely reported by anyone including Bradley.  (3)  When a star has a vector that is in the ecliptic plane, the bearing q = 0º and the bearing f = 0º at all times during a year because these bearings are defined to be the slope of a star's vector wrt the ecliptic plane.  Therefore, if a star is in the ecliptic plane its aberration (f - q) must be 0º at all times during a year including the times when the star's vector is perpendicular to the direction of motion of an observer at a point that is orbiting in the ecliptic plane.  (4) The empirical telescopic data has not recorded aberration for objects within 5' of the NEP.  The official historical record (that has passed the bias of peer reviewers) has only extrapolated the aberration of an object at the NEP based upon data from stars with declinations outside of this within 5' of the NEP range.  If aberration is maximum at the NEP (q = 90º North wrt the Earth's orbital plane) or the SEP (q = 90º South wrt the Earth's orbital plane) during event 2 because the Earth's velocity vector is perpendicular to the vector to the star, then why hasn't someone provided actual observational data from a telescope that confirms that aberration for such an object is maximum (i.e. approx. -20") during event 2?  Also, why hasn't someone provided actual observational data from a telescope that confirms that aberration for stars such as Gamma-Draconis (at q = 75^o) has been observed to be maximum (i.e. approx. -20") during event 2 when Earth's velocity vector is precisely perpendicular to the vector from those objects?   (5)  Finally, both the SRT (1905) and the AA (2010) aberration estimators agree that maximum aberration has the constant of aberration value when q = 90^o (see figure5).  However, these two theories are not compatible in any other respect because the AA (2010) model is based upon a scenario where aberration is a function of the velocity of the traveling telescope on Earth as well as the velocity of light while the SRT (1905) model is based upon a relationship between a spherical dynamical form and an ellipsoidal kinematical form that has changed the slope of the traveling telescope due to the traveling telescope's velocity.  Isn't it peculiar that these two estimators of stellar aberration cannot agree closely on the aberration of any star viewed from an orbiting body except upon the aberration of a celestial body at the NEP and then only if the Traveler  happens to be orbiting at precisely the same velocity as the Earth's orbital velocity?

If the foregoing unlikely predictions and incriminating circumstances are not enough to shake your faith in the validity and reliability of the SRT (1905) and AA (2010) models, let's consider the empirical data collected by actual telescopic observations which unequivocally proves that the Best Model (M/M/+M® Model V2 software that applies the Best Equations) is a more reliable predictor of stellar aberration than the SRT (1905) and AA (2010) models.  (1)  The empirical data proves that the aberration for Gamma-Draconis, Polaris and all other stars between q = 0^o and q =  90^o  has been observed to be near zero when the Earth's velocity vector is approximately perpendicular to the vector to the stars in this range because the Earth's velocity vector was perpendicular to the vector to the stars in this range.  (2) Table 3 shows that the Best Model's predictions of annual aberration precisely agree with telescopic observations of Polaris from August 1, 2006 through October 31, 2006.  When the effect of Precession (P) and Nutation (N) are added to do we get  do with P & N.  Annual aberration (dº - doº) is derived by subtracting do with P & N from the raw observations that are do with P, N, and Aberration (A).  The four values for observed annual aberration (dº - doº) in table 3 are precisely the same as the annual aberration values (f - q) estimated by the Best Model.  These observational data confirm that annual aberration is near zero when the vector to the star is approximately perpendicular to the Earth's velocity vector.  (3) Table 4 contains Bradley's annual aberration calculations based upon his own observational data for gamma-Draconis (Bradley's observations were taken from figure 3 at Aberration of light- Wikipedia ).  Bradley's aberration measurements in table 4 have been translated from declinations used by Bradley that were wrt his observatory floor to declinations wrt the Earth's ecliptic plane that are used by the Best Model to calculate annual aberration values (f - q).  After Bradley's observational data are translated to bearing aberration values (f - q), his observations confirm that aberration is near zero when the vector to the star is approximately perpendicular to the Earth's velocity vector.  These data also confirm that aberration is maximum when the vector to the star is at the plane that is perpendicular to the Earth's ecliptic plane and contains the Earth's velocity vector.   

Isn't it obvious that the Polaris and gamma-Draconis observational data in tables 3 and 4 confirm that aberration is zero approximately 91.3 sidereal days after the occurrence of event 1?  Based upon the Polaris and gamma-Draconis observational data, isn't it more logical to conclude that the aberration to any star is zero when the Earth's velocity vector is precisely perpendicular to the star's vector?   Then, doesn't it follow that maximum aberration does not occur when the Earth's velocity vector is precisely perpendicular to the star's vector?  Also, doesn't the Polaris and gamma-Draconis observational data confirm that maximum aberration does occur when the vector to the star is at the plane that is perpendicular to the Earth's ecliptic plane and contains the Earth's velocity vector.  The SRT (1905) and AA (2010) aberration equations are based upon the theory that two events (events 1 and 2) must be separated by three months on the Traveler’s calendar where aberration during event 1 is maximum and aberration during event 2 is 0" for all objects except an object that is precisely at the NEP or SEP where aberration has a constant value of approx. -20".  According to this theory the aberration of a point at the NEP or SEP (q = 90º) must be the same value during every month of the year; therefore, f = approx. 89º 40" (90º + approx. -20") during each month of the year.  This is a non-sequitur because aberration (f - q) has been measured to be 0" twice a year for all stars that have actually been observed through a telescope.  Therefore, if aberration is constant during all months of a year and f = 90º twice a year because q = 90º then aberration must be 0" during each month of the year instead of approx. -20".   Table 1 (below figure 2) shows that the AA (2010) and the SRT (1905) estimates of maximum aberration (20.592598 arc sec.) occurs at q = 90º.  However, it has been demonstrated that this is a non-sequitur with a burden of empirical proof to provide evidence that an object at q = 90º is an exception to the rule that stellar aberration is measured to be 0" twice a year during event 2.  Table 1 shows that the true maximum aberration limit (-20.550811 arc sec.) occurs at q = 87.89^o according to the Best Model and that aberration approaches 0" during event 2 when  q =approaches 90º.  Telescopic observations have verified that aberration for all stars actually measured have an aberration of 0" during event 2.  The theory that aberration is approx. -20" during every month of the year for a point precisely at the NEP or SEP has never been verified by telescopic observation.

The Bearing estimator (Best) stellar aberration model is based upon Einstein’s principle that aberration is caused by the relationships between a kinematical form and a dynamical form.  The Best Model was a natural derivative of the M/M/+M® Model V1 and was fully developed as a predictive model in the M/M/+M® Model V2R0 software (R0 is a Beta Test Version).  The Best Model has a much more simple and easy to understand geometric form than the geometric form described in the SRT (1905) paper.  Also, the connection between the geometric form and the Best equations for calculating aberration can be easily visualized and understood by anyone with a basic knowledge of Euclidean geometry.  No “understanding” of obscure concepts such as the concept of simultaneity (that is reputed to separate the men from the boys) will be required in order to understand the Best stellar aberration model.  The Best stellar aberration model was based upon the geometric form described in figure 3.  Figure 3 describes the three boundary plane types:  the 0º Boundary Plane (yellow area), the 90º Boundary Plane (light grey area) and a 75º Boundary Plane (dark grey area).  Maximum aberration (Event 1) occurs when the vectors (qa  and qa) from a star is in the 90º North (or South) Boundary Plane where a is the right ascension angle described on the Earth’s orbital plane (yellow area) in figure 3 and where a is always 0º for a 90º Boundary Plane.  Minimum aberration (Event 2) occurs when the vectors (q₉₀ and f₉₀) from a star are in the nº North (or South) Boundary Plane where n has the same value as q.  For example, figure 3 shows the 75º North Boundary Plane (dark grey area) that contains the Boundary Pair for a star at q = 75^o during event 2.  Points ea and ea’ can be thought of as being apertures on the Observer’s and Traveler’s SIF through which the light from a star is seen by the Observer and Traveler respectively.  The Observer sees the star at q = 75^o through point e₀ and the Traveler sees the star through point e₀’ during maximum aberration (event 1).   During minimum aberration (event 2), the Observer sees the star at q = 75^o through point e₉₀ and the Traveler sees the star through point e₉₀’ (e₉₀ and e₉₀’ are at the same physical location during event 2).  Therefore, the declination wrt the Earth’s ecliptic plane to points e₉₀ and e₉₀’ (i.e. the Bearings angles q₉₀ and f₉₀) are both equal to 75^o and the annual aberration (q₉₀ - f₉₀ ) is zero.  

The M/M/+M® Model V2 Best equation: tan(q) = (y_d / x’_d)  will determine q when the value of f during event 1 is given (see figure 1) where q in figure 1 is the same as q₀ in figure 3 and where f in figure 1 is the same as f₀ in figure 3.  The V2 model uses points e and e' shown in figure 3 that are points on the Observer's and Travelers SIF's respectively just as points d and d' shown in figure 1 are points on the Observer's and Travelers SIF's respectively.  The relationships between these four points are discussed in video 2 at the Video Tutorials page.  Another Best equation has been derived that will determine f when the value of q during event 1 is given.  The values for q₉₀ and f₉₀ during event 2 are the same (i.e. q₉₀ = f₉₀ during event 2); therefore, when one value is known the other value is also known.  The M/M/+M® V2 Best Model will also calculate annual aberration at any sidereal day.time-of-day offsets (from the event 1 day.time-of-day) for any platform with a given orbital speed and orbital cycle (days / year) and will alternatively calculate the aberration at any sidereal angle for a platform in extended linear flight.  The V2 Best Model calculates sidereal values based upon the geometry shown in figure 3 and figure 4.  Although the geometry in these two figures may appear to be complex, the model software is very user friendly; it will be easy to learn how to get the M/M/+M® V2R0 model to do all of the stellar aberration calculations for any star location during event 1 and event 2.  The Best Model V2R0 Beta Test release is simplified for training and testing purposes and prediction of star locations for sidereal angles up to a =89.9999999º can be accomplished by simply pressing the Execute control button after setting one target value such as either of the three aberration values (f-q, a'-a, or ebe' ), a sidereal day value, a right ascension value (a' or a) or an observed declination bearing (f).  Table 4 contains a column with Best Model estimates of Sidereal Aberration (a' - a) for each of Bradley's Sidereal day observations of gamma-Draconis where a and a' are angular displacements to projections of the Boundary Pair and points e and e' respectively onto the 0º Boundary Plane.  Table 4 also contains a column with Best Model estimates of Real Aberration of the angle ebe' for each of Bradley's sidereal day observations of gamma-Draconis where ebe' is the "real" angular displacement between the two vectors eb and e'b in the Boundary Plane for each sidereal day.  The angle ebe' is equal to the aberration value e' - e where e' and e are the bearings (or slopes) of the lines e'b and eb wrt the Great Circle of the Traveler's BIF respectively (see figure_1 and figure 4).  The Sidereal Aberration values (a'-a) and the Real Aberration values (e'-e) are automatically calculated with each Bearing Aberration (f - q) calculation.

Intra-galactic space travel can be expected to involve much greater velocities than the slow orbital velocity of 29.93 km/sec of the Earth.  For example, at a velocity of c * cos(30^o), the velocity where the Traveler’s clock ticks an one-half the rate of the Observer’s clock,  the error in the AA (2010) estimate and the SRT (1905) estimate for maximum aberration for Gamma-Draconis at q = 75^o would have an error of over 32% (see table 2  below figure 2) wrt the Best Model.  The AA (2010) estimate for aberration of Gamma-Draconis at q = 75^o (do = 51.5^o) (approx.) would have an error over 83%.  Finally, the developer’s of the AA (2010) and SRT (1905) equations apparently went to a lot of trouble to calibrate their equations so that they would match Bradley’s Constant of Aberration value (f - q) = -20.592598 arc seconds down to the nearest one millionth of one arc second for the Earth’s orbital velocity v = 29.93 km/second (see the comparison of Bradley’s constant with the SRT (1905) estimate at q = 90^o and the bottom of table 1).  However, these two equations grossly over estimate the value of Bradley’s Constant of Aberration at v = c * cos(30^o).  The AA (2010) estimate exceeds Bradley’s Constant by 21.34% and the SRT (1905) estimate exceeds Bradley’s Constant by 46.72% (see the comparison of Bradley’s constant with the SRT (1905) estimate at q = 90^o and the bottom of table 2).  The maximum aberration for any bearing at v = c * cos(q) occurs at q = 67.53^o where aberration (f - q) = 37.762407^o according to the Best Equation (see table 2).  Ironically, the M/M/+M® Best estimate for maximum aberration is much closer to Bradley’s prediction of maximum aberration (Bradley’s Aberration Constant) than the maximum aberration estimates of the AA (2010) or the SRT (1905) equations at v = c * cos(q) despite the apparent effort that was made to calibrated these equations so that they would precisely agree with Bradley’s Aberration Constant at Earth’s orbital velocity v = 29.93 km/second.

In Conclusion, the M/M/+M® Best Model is a precise predictor of relativistic stellar aberration and does not have the validity and reliability problems of equations that agree with Bradley’s Aberration Constant at Earth’s orbital velocity.  The Best Model and the IGSN metrics convention will greatly simplify the development of a Stellar Navigation and / or Galactic Positioning System (GPS) for space travelers and will dramatically improve precision at velocities greater than Earth’s Orbital velocity.  GPS has made the onboard NOVAS program obsolete for Navigation and Global Positioning on Earth; but an onboard system will be needed for space travel outside our Solar System.  The M/M/+M® Best Model that has been described can be adapted in algorithms that account for the relativistic aberration of vectors to stars that appear in Travelers’ 1^st and 4^th quadrants  (i.e. ahead of the Travelers) as they move in the +x direction away from our Solar System’s barycenter at (x,y,z) = (0,0,0).  This same Model where q is expressed as a function of f can be adapted in algorithms to account for the relativistic aberration of vectors to stars that appear in Travelers’ 2nd and 3rd quadrants (i.e. ahead of the Travelers) as they move in the -x direction on their way back to our Solar System’s barycenter at (x,y,z) = (0,0,0).  The Best Model as used in this article to determine q as a function of f applies only to star vectors that are ahead of Travelers in the direction of their motion and is not a continuous equation that applies to star vectors that are behind the Travelers’ direction of motion.  Reciprocal forms of the Best Model that determine f as a function of q can be adapted in algorithms to account for the relativistic aberration of vectors to stars that appear in Travelers’ 2^nd and 3^rd  quadrants (i.e. behind the Travelers) as they move in the +x direction away from our Solar System’s barycenter as well as to account for the relativistic aberration of vectors to stars that appear in the Travelers’ 1^st and 4^th quadrants (i.e. behind the Travelers) as they move in the –x direction to return to our Solar System’s barycenter at (x,y,z) = (0,0,0).

Please send questions or comments to ken-more@ken-more.com.  

For additional information on the M/M/+M® Model V2R0 and the M/M/+M® Best Model V2R0 currently available for trainning and testing purposes please send an e-mail to ken-more@ken-more.com .


1 Bradley, J. 1729, An account of a new discovered motion of the fixed stars, Phil. Trans. Royal Society 35:637.

2 Einstein, A. 1905, On the Electrodynamics of Moving Bodies, revised and translated in The Principle of Relativity, Dover, NY, 1923, pp. 35-65.

3 Astronomical Almanac 2010, Washington: U.S. Government Printing Office, p. B28.