Scale of Curvature

Shawn Halayka

shalayka@gmail.com

May 27, 2007

1) Abstract

The relativistic precession rate of two co-orbiting bodies is approximated by taking into account the change in the curvature of energy at all scales within the system.

As orbit velocity increases, the curvature of energy at the quantum scale is reduced due to Special Relativistic time dilation (unification of energy direction). At the fictitious orbit velocity of v = c, the binding energy of the body is fully directed forward, leading to a precession rate of 2pi orbit plane rotations per orbit.

In a binary system of near-equal body mass, the reduction of gravitational curvature at the orbit scale due to eccentricity also leads to precession, but at the larger system scale.

2) Precession as the Scale of Curvature

The mass of the Sun and Mercury are:

M = 1.98892e30.

m = 3.302e23.

It is considered that the mass of Mercury makes an insignificant contribution to the system's gravitational field in comparison to the mass of the Sun. As a result, the Sun attains no appreciable orbit radius, velocity or eccentricity.

Oppositely, Mercury's orbit apoapsis and periapsis distances from the gravitational barycenter (taken to be the centre of Sun) are:

a = 69817079e3,

p = 46001272e3.

From these, the average orbit radius and orbit eccentricity are calculated as:

r = (a + p) / 2 = 57909175500,

e = 1 – p/r = 0.20563.

The circular orbit velocity v for r is:

v = sqrt(GM/r) = 47873.54.

The maximum and minimum of the two masses are:

MMax = M,

MMin = m.

And their respective eccentricities:

eMax = e = 0,

eMin = e = 0.20563.

The ratio between the maximum and minimum masses, and its inverse are:

MRatio = MMin / MMax,

MInvRatio = 1 – MRatio.

Given that eccentricity is the dissolution of curvature at the orbit scale, the stable mass of the Sun all but counteracts the system-wide effects of Mercury's orbit eccentricity. System scale eccentricity, as calculated by weighted sum e' is:

e' = eMax*MInvRatio + eMin*MRatio.

Given these parameters, precession occurs at a rate of:

n = [2pi[1 – cos(asin(v_sun / c))] + 2pi[1 – cos(asin(v / c))]] / cos^2(asin(e')),

n = [0 + 2pi[1 – cos(asin(v / c))]] / 1.

The orbit circumference C and orbit time t for r are:

C = 2pi*r,

t = C/v,

resulting in a precession rate in arc seconds of a degree per century of:

del = n / t (360*60*60)(100*365*24*60*60),

del = 43.08.

As Mercury orbits around the Sun, it is constantly accelerated by gravitation into taking a curved path through space-time. Special Relativistic time dilation takes effect, reducing the amount of quantum curvature within the planet. Redirected as such, the energy produces an increase in space-time rotation at the larger orbit scale.

For the fictitious circular orbit v = c, quantum curvature is completely reduced. The space in which the orbiting body travels rotates enough per orbit so that the total orbit velocity would appear to be 2pic + c.

In an eccentric orbit like Mercury's (11), velocity varies due to non-constant orbit radius (3, 4), but the amount of precession (20) remains equal to the precession that occurs in a circular orbit of constant radius (5) and velocity (7).

Like the reduction in the quantum curvature of Mercury due to velocity, the reduction in orbit path curvature due to eccentricity translates to a rotation of space-time at the system scale. The Sun's mass at near rest effectively holds the system fast from rotating due to Mercury's eccentric path through space.

In the case of binary star PSR 1913+16, both bodies are of near equal mass, orbit radius, velocity and eccentricity. Using averaged values for simplification, the combined orbit and system scale precession in degrees per year is:

M = 1.414*1.98892e30,

a = 1576800e3,

p = 373300e3,

e' = 1*e + 0*e,

del = n / t (360)(365*24*60*60),

del = 4.42.