P10 Acceleration: Light Speed Doesn't Extrapolate

From: Ralph Sansbury (r9ns_at_bestweb.net)
Date: 07/16/04 

Date: 16 Jul 2004 13:50:37 -0700

Light speed delay is assumed to extrapolate to the most distant stars
and galaxies but for distances beyond the GPSS satellites at 11,000
miles etc, the evidence is not as clearcut. (See Appendix)
   The recently observed anomalous acceleration of Pioneer 10
provides the first clearcut evidence that light speed delay does not
extrapolate beyond one minute. That is, the predicted Doppler shifted
frequencies of a radar frequency sent to the spacecraft and returned
to earth two light times later were used to adjust successive
Newtonian calculated positions and velocities of Pioneer 10 as it
moved away from the Earth.
     When the transmission and receptions earth site motions hours
apart, used to compute the Doppler shifted return frequency, are
replaced by earth site motions 1 minute apart, the anomalous
acceleration disappears
   No longer do the observed frequencies increase slightly but
systematically with respect to the frequencies implied by the
relative motions of the earth and the spacecraft. Thus it is no
longer necessary to assume an anomalous acceleration of the craft to
the sun to keep the predicted frequencies equal to the observed.

 The method is as follows:

    First,referring to the NASA Horizons ephemeris, we project the
Madrid earthsite velocity wrt sun,V=(v1(t),v2(t),v3(t)),a vector
starting at Madrid at a specific time( eg t=21:24 Oct 7 1987) onto the
line between Madrid and the craft position assuming the nearly
instantantaneous light delay model, at this same time. The coordinates
of the craft positions however are based on the above estimation
procedure and earth site motions assuming the conventional light
delay model.
   The velocity coordinates of the earth site wrt sun are
v1(t)=(x(t)-x(t-1))/60sec., etc.
(in this example the earthsite velocity is V=30.028km/sec and the
projected velocity on the line from the earth site to the ephemeris
craft position at this same time is W= 25.43728km per second toward
the craft.. The projected velocity of the craft onto this same line is
12.841164 (from 13.06)away from the earthsite. Thus the difference,
12.5801242 is the total uplink velocity and twice this is the total
total uplink plus downlink velocity which is 25.16022929.
   The ratio of the projected earth site part of the total is
25/30=10/12 whereas the projected craft part of the total is 12/13
which is a smaller fraction. If we change the position of the craft by
changing these angles of projection implied by a change in the angle
of projection of the total total uplink plus downlink velocites and
assume tentatively a slight change in the velocity of the craft, we
can make the ephemeris position of the craft and its velocity give
results that match more closely the received radar tracking data-at
least for this minute.
    After repeating this procedure a few times we find no further
changes are needed to sustain a close match minute by minute, and we
can have confirmation of the trajectory determined in this way.
   
1) We take as the the angle of projection arcos(25.16023/30.02854)The
angle of projection is arcos(0.837877)= 33.083 deg.

2) change the magnitude of the projected earthsite velocity at the
Madrid earthsite by trial and error in the spread*** to 24.8392593 (
arccos(24.8392593/30.02854)= 34.1890 deg; or 1.106 degrees more than
initially assumed)
   This means that if the craft position is at a slightly larger angle
of projection,the motion of the earthsite to the craft would be
reduced enough so that when the craft velocity away from the earth
assumed to be the same as in the old position, is subtracted, the net
velocity of the earth to the craft is smaller and enough smaller to
make the predicted frequency match the observed frequency to within
.004Hz.
   We have ignored the effect of the implied craft position change on
the craft velocity to Madrid but we can assume tentatively that the
craft velocity wrt the sun is slightly greater so as to compensate for
this effect. Of course we could also assume an even larger velocity
of the craft away from the earth and a smaller increase in the angle
of
projection of the earthsite velocity onto the line between the
earthsite and the new craft position. If this assumption produces a
trajectory that requires even less adjustment than our first
assumption we can change this later.
   
   In this example the positions of the earth sites given by the
ephemeris(NASA Horizons Telnet, observer table) are 55 seconds later
than the times for the frequencies in the tracking data. Thus the
change in position of the Madrid earthsite wrt the Sun from 21:23 to
21:24 divided by 60 seconds and associated with the spread*** time,
21:24, represents the average velocity during this minute in the CT
time system but in the GMT time system this is the average velocity
from 21:22 to 21:23 which produces the received frequency in the
tracking data recorded at 21:23 etc. So we compare the spread***
predicted frequency for 21:24 with the observed received frequency for
21:23 (or a linear interpolation of the value for 21:23:05)etc.
    .
3)To determine a new better fitting trajectory, we must change the
coordinates of the craft position wrt the earthsite near Madrid, from
x,y,z to a point on the spherical arc 1.106 degrees(or fewer if we
assume a larger craft velocity away from the earth) further along the
great circle arc in the direction from x1,x2,x3 to xyz. Note x1 = r
times v1/(v1^2+v2^2+v3^2)^1/2, etc. where r denotes the distance from
Madrid at this time to the craft at the same time. These two points
and their diametric opposites a distance 2r apart define a sphere of
radius,r. Our problem then is to find the coordinates of third point
on the same great circle of this sphere 1.106 degrees in a clockwise
direction from x,y,z:
                                           
(x1,y1,z1) = 1.839 357 595, 5.303 190 978, 2.630 169 494
  (x2,y2,z2) = -1.583 053 321, 5.525 134 415, 2.367 217 205
   (x3,y3,z3) = ?,?,?
    
If we draw a line parallel to the X axis through point, y=5.3,
x=1.8,z=0, and a second line from this point to x=-1.6 , y= 5.5,z=0
and a third line from this point to the point x=-1.6,y=5.3,z=0, we
have formed a right triangle where the angle at the first point
between the first two lines is
theta=arcos((y2-y1)/((y2-y1)^2+(x2-x1)^2)^1/2
   Let*s form a new axis line, X*, coinciding with the old line,
y=5.3 so that the first point above in the X*YZ coordinate system
becomes, (1.8,0,0) and the second point becomes (-1.6,(5.5-5.3),0) .
If we rotate the X* and Y axes to get X**,Y*, kept perpendicular with
respect to each other, about the Z axis through this angle,
theta=1.106deg., the Y* axis so formed coincides with the line
between the first and second point. The coordinates of the first
point become (x1**,0,0) where x1**=(x1)cos(theta) and the coordinates
of the second point become (x2**,0,0) where x2**
=((1.6+1.8)/cos(theta) - (1.8)cos(theta))
    Now allow z to be z1 and z2, so(x1,y1,z1 ) becomes
(x1**,0,z1)=(a,b) and (x2,y2,z2) becomes (x2**,0,z2)=(c,d)
   The points (a,b) and (c,d) are on a circle of radius r and (c,d)
is a point in a counter clockwise direction from (a,b) through an arc
of 33.083 degrees while (e,f) is in a clockwise direction from (a,b)
through an arc of 1.106deg=theta..
   The chord distance between these points is s=2((r)(sin(theta/2))
   The angle between the chord and the tangent at the point (a,b) is
(½)intercepted arc =theta/2. This angle is equal to the angle between
a line drawn from (a,b) perpendicular to a point on the line drawn
from (0,0) to (e,f). Thus
(s)(sin(theta/2))= (e-a) and (s)(cos(theta/2))= (b-f)
   Having thus found (e,f) we must determine what (e,f)= (x3**,0,z3)
is in the original system of coordinates.
   Draw a line parallel to the original X axis through the point,(x=0,
y=5.3)and then draw a line from the point (y=5.5,x=-1.6) to the point,
(x=1.8,y=5.3) and extend this line sloping downward about a quarter
of an inch and label this endpoint
(e,f)=(x3**,0,z3). We see immediately that (x3**)(sin(theta))=5.3-y3
where y3 is the value of the Y coordinate of (e,f) in the original
system and (x3**)(cos(theta))+1.6=x3 is the value of the X coordinate
of (e,f) in the original system.
  The value of the Z coordinate can be determined from the equation,
z3^2 = r^2 - x3^2-y3^2.

4)Given this new position of the craft wrt sun we can determine the
acceleration of the craft with respect to and due to the sun and
project the velocity produced by this acceleration onto the line
between this new position and the earth site at this same time and add
this velocity to the assumed previous velocity magnitude along the
previous craft-earthsite line so as to produce a new position in this
new trajectory.

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