Perihelion precession of Mercury from hyperbolic anisotropy
- From: "Just G. Waller" <wallermax@xxxxxxxxxxx>
- Date: Tue, 27 Feb 2007 22:53:38 +0000 (UTC)
The hyperbolic anisotropy of vacuum can address
the anomalous precession of any orbiting body, by:
dw = W (Exp(V^2/c^2)- 1),
where
dw is the precession angular velocity,
W is the angular velocity,
V is the magnitude of difference vector of escape velocities
between the orbiting body and observer, and
c = 2.99e+5 km/s is speed of light in the vacuum.
For a quick computation on planet Mercury observed from the Earth,
we only need these data:
GM, standard gravitational parameter of the Sun,
R_m, Mercury average distance to the Sun,
R_e, the Earth average distance to the Sun,
c speed of light in vacuum,
and the favourable alignment: Earth and Mercury distance vectors to
the Sun being orthogonal.
The solar system escape velocity from Mercury, under newtonian laws,
is
V_m = sqrt(2GM/R_m)
The escape velocity from the Earth is
V_e = sqrt(2GM/R_e).
So, it yields a magnitude for the vector difference of
V = sqrt(V_m^2 + V_e^2)=
= sqrt(2GM(1/R_m + 1/R_e)),
because we've assumed each escape velocity is defined in the
direction of its distance vector to the Sun, and they are
orthogonal.
The average angular velocity of Mercury is
W_m = sqrt(GM/R_m^3)
Finally, the precession is
dw = W_m (Exp(V^2/c^2) - 1),
dw = sqrt(GM/R_m^3) (Exp(2GM(1/R_m + 1/R_e)/c^2) - 1)
For average distance of Mercury to the Sun
R_m = 55.0 e+6 km ,
and average distance of the Earth
R_e = 147.0 e+6 km, it yields
dw = 43.12 arcsec per century,
which is a pretty good prediction!.
The factor z = Exp(V^2/c^2) - 1 is actually a shift,
exactly it is a gravitational shift. But, for a body
in an outer orbit with respect to the Earth it would be
z = Exp(-V^2/c^2) - 1
.
So, in this case it is z < 0, and rather than a perihelion precession
we would measure a perihelion delay. The observation of precession
(z > 0)implies a Doppler red-shift for any signal coming from the
body to us, and the observation of an orbit delay implies
a Doppler blue-shift.
The core of this gravitational anisotropy resides in the
difference vector V. For inertial systems, the shift z is
expressed as z = Exp(v/c) - 1, where v is the relative speed
between source and observer. See my last post on this issue at
http://groups.google.com.gi/group/sci.physics.research/browse_thread/thread/48cd5e1ba31c5b28/?hl=en#
For an observer at large distance, such that the gravitational
potential can be regarded as zero, it yields
z = Exp(2GM/rc^2) - 1,
where r is the distance of the body to the center of the system.
Let us compare this solution with the relativistic one for
gravitational
red-shifts, z_s, of non-rotating, uncharged masses which are
spherically
symmetric,
z_s = 1/sqrt(1 - (2GM/rc^2)) - 1
For sake of simplicity, call x = 2GM/rc^2
When we express z and z_s in their respective expansion series,
we see that they match until the second order approximation,
z = x + x^2/2 + x^3/6 + .. + x^n/n! + ...
z_s = x + 3x^2/8 + 5x^3/16 + ...
we see the discrepancy in the second term, it is 4/3. A good
approximation is achieved for low 2GM/r with respect to c^2.
.
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