Deriving Equations for Deflection, Splay and Critical Distances
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Knowing the amount of deflection of BTTPs at the sun’s photosphere enables
us to calculate the deflection caused by any object, at any distance, and this
lets us calculate gravitational critical distances.
The deflection of BTTPs at the sun’s photosphere is, as we
found in Article 8, α = 0.87555”, or α = (0.87555/(60 x 60))π/180 = 4.24479 x 10-6 radians. Since the deflection is in proportion to the mass of the deflecting object, and, as discussed
previously, inversely proportional to the distance from the object, we can derive a general equation for deflection of:
α = Mass in Solar Masses x 4.24479 x 10-6 radians/Distance
in solar radii
This equation would be more useful if it were in terms of
light years, and since there are 13,598,852.68 solar radii in a light
year, we can rewrite the equation as:
α = Mass in Solar Masses x ((4.24479 x 10-6)/(13,598,852.68))/Distance in ly, or:
α = msuns x 3.121432925 x 10-13/dly
radians . . . (22)
As we mentioned in the gravitation section, over
considerable distances there is a splay
in the direction of BPs moving forward in time due to the curvature of the
Universe, and when this splay exceeds the deflection of BTTPs, then the
gravitational force becomes repulsive. The angle of deflection, α, when
two objects interact, is the deflection on just one side of the interaction, so
the corresponding splay that needs to exceed the deflection to turn the gravitational
intensity factor, I, negative, is just one half of the total splay angle
between two objects. At one radian separation, 46 billion light years, there is one radian
(57.29578°) of total splay, so the splay on each side is half this, or half a radian.
Based on this we can calculate the splay (σ), as follows, from distance in
light years (dly):
σ = 0.5 x dly/4.6000 x 1010 =
dly x 1.0870 x 10-11 radians
This splay equation is based on the assumption that the Universe is exactly
spherical. If there are areas that are flattened out, and areas that have a greater
curvature to make up for the flatter areas, then the splay will change in those
areas. To account for these, I will introduce a “flatness factor”, “f”, which
equals 1 at the exactly spherical curvature, and increases or decreases with
the effective radius of curvature, so that, for instance, if the curvature is
flattened out half way to being flat, where its radius of curvature is doubled,
then f = 2. The splay will reduce in proportion to how much “f” increases, so
the general equation becomes:
σ = dly x 1.0870 x 10-11/f radians . . . . . (23)
What is most useful to know about deflection and splay is
the critical distance for any two objects, when gravitational attraction turns
to repulsion. This is the distance where the deflection caused by the larger object equals
the splay due to the distance — that is when σ = α. In terms of Equations 22
and 23, this is when:
dly x 1.0870 x 10-11/f = msuns
x 3.1214 x 10-13/dly
dc = 1.1368 x (mf)˝ . .
. . . . . (25)
For f = 2,500, the critical distance for our sun would be close
to 8.5 light years. A star one fourth the mass of our sun would then have a critical
distance of about 4.25 light years, and one four times the mass of our sun would have
a critical distance of about 17 light years. Critical distances of this order would ensure
stars in globular clusters, in the central bulges of galaxies, and even in open clusters,
would attract each other. Stars in the spiral arms of our galaxy and other spiral galaxies,
on the other hand, would be far enough apart that their gravitational attraction would be
significantly diminished or they would slightly repel each other, making it less likely
they would have the kind of close interactions that are common in clusters, helping to give planetary
systems around suitable single stars in the spiral arms the stability needed for life to evolve.
So, it seems like space in the neighborhood of
galaxies has a “flat” (Euclidian) geometry to a fairly high degree of flattening, probably about f = 2,500.
What, then, could be the cause of this flattening? This will be discussed in the next article in this series: “Structure of the Universe — Local Flattenings.”
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