Mercury’s Orbital Precession
Attributable To The Sun’s Spin
Roots of Gravitomagnetic Theory. (Article on this website.)
THE FEYNMAN LECTURES ON PHYSICS, volume 2, Section 14-5.
In Newton’s gravitational
force law, no distinction is made between a non-spinning central body and a
spinning central body. In most cases the central body mass can be modeled as a point
mass. Given a massive central body whose center is at rest in an inertial frame,
a satellite’s orbit can be analytically
found to be a closed ellipse.
That is, the orbit continually repeats itself, and has one point (called the
perigee) where the distance to the central body is less than any distance at
other points in the orbital path. Fig.1 depicts the orbit of Mercury around the
Sun which (owing to its much greater mass) is assumed to be at rest at the
origin of the x-y plot.
That is, the orbit continually repeats itself, and has one point (called the perigee) where the distance to the central body is less than any distance at other points in the orbital path. Fig.1 depicts the orbit of Mercury around the Sun which (owing to its much greater mass) is assumed to be at rest at the origin of the x-y plot.
Orbit of Mercury in the Sun's Gravitational Field
In Gravitomagnetic Theory
(GMT) Newton’s gravitation law again applies to a central body (whether the
body spins or not). Quite as the electric
field of a sphere of charge (whose center is at rest) is inverse square
regardless of whether or not the sphere spins, so is the gravitational field of a sphere of mass inverse square.
However, analogously to a
spinning charge, GMT specifies that a spinning
mass has an Imaginary gravitomagnetic field (dubbed the O field) quite as a spinning charge has a Real B
field. And application of the GMT version of the Lorentz force law indicates
that a satellite moving through said field experiences a gravitomagnetic
force, in addition to the omnipresent gravitational force. Whereas the
gravitational force on Mercury is always toward the Sun, however, the gravitomagnetic force
is away from the Sun.
Invoking Newton 2, the resulting satellite path in the case of a spinning Sun can be computed (see Appendix A). And it is found that the analytical closed elliptic orbit, attributable to gravity alone, is very slightly different when the central body spins. However, this difference is not great enough to be seen in a plot like Fig. 1. But the difference can be greatly exaggerated by increasing the gravitomagnetic force to be say .1 of the gravitational force. Fig 2. plots the computed Mercury path around a spinning Sun whose gravitomagnetic field has been exaggerated. Note that there is no perihelion. Of course in reality the opening out of the orbit is very much less obvious.
Under the Influence of a Much Enhanced Gravitomagnetic Force
#Python 3.3.4 (v3.3.4:7ff62415e426, Feb 10 2014, 18:12:08) [MSC v.1600 32 bit (Intel)] on win32
#Type "copyright", "credits" or "license()" for more information.
References and suggested background reading:
(1) Roots of Gravitomagnetic Theory. (Article on this website.)
(2) THE FEYNMAN LECTURES ON PHYSICS, volume 2, Section 14-5.
Please Note: "GMT" is short for "Gravitomagnetic Theory".
This program computes the orbit of Mercury as it travels
around the Sun, with the Sun spinning and, according to GMT, engendering a gravitomagnetic field
in addition to the omnipresent gravitational field. The center of the sun is assumed to
be fixed at the origin of a rectangular coordinate system in inertial space. When the sun spins CCW,
its angular velocity points in the positive z-direction.
In GMT the spinning sun is theorized to
engender a dipolar gravitomagnetic field that (a) points in the positive z-direction at internal
points of the equatorial plane, and (b) points in the negative z-direction at points in the
external equatorial plane. The orbit of Mercury
is also assumed to lie in the xy plane and, from the perspective of the positive z axis, also to be
CCW. Mercury is assumed to be acted upon by a gravitational force and
by an opposing gravitomagnetic force. Analytically the orbit is a perfect ellipse
when only gravity acts. However, with the small (and opposing) graviyomagnetic force also acting,
the result should hypothetically be an open ellipse.
It is expected that the computed result
may not be precisely what is observed owing to certain realities. First the spinning Sun
is modeled as a spinning square line mass, and not as a spinning, oblate sphere of mass
with a complicated internal density and with zones that rotate at different rates.
Secondly, the Sun's spin equatorial plane does not actually coincide precisely with the plane
of Mercury's orbit. (Mercury's orbital plane is tilted slightly, relative to the
Sun's spin equatorial plane.) Finally there may be
small numeric computer errors.
Feynman(Reference 2) models a square electric current loop as a square line charge q,
with sides a, and line charge density lambda=q/4a.
He assumes that this line charge circulates CCW in the xy-plane with a current,
I=q/tau, where tau is the time for q to make one trip around the loop. I.e. I=q*omega/(2*pi).
The magnetic moment for this model is mu=I*a**2, or mu=(q*omega*a**2/(2*pi).
For a point r>a in the loop's plane
he finds that B points in the -z direction and has the
We model the spinning Sun as a square line mass M, where M=MassOfSun,
and with side a equal to R, the radius of the Sun, and with a line mass density of lambda=M/(4*R).
We assume that this line mass circulates CCW with a current I=lamda*v, where v=omega*2*R/pi is
the line mass speed around the loop.
At points in the equatorial plane and at greater distances
r>R our model indicates that Oz=-mu*G/(c**2*r**3)=-M*OmegaOfSun*R**2*G/(4*c**2*r**3).
#Constants for Sun.
#Constants for Mercury.
DT=OrbitalPeriod/N #Time increment betweem orbital step computations
G=6.67408e-11 #Gravitational constant.
#First determine what OzAdjust must be in order for the gravitomagnetic force on Mercury
#to be 1/10 the gravitational force.
F=-G*MassOfSun*MercuryMass/Dis**2 #F is gravitational force. Masses are Imaginay; hence force is negative or attractive.
Oz=-MassOfSun*RadiusOfSun**2*G*OmegaOfSun/(4*SpeedOfLight**2*Dis**3) #Oz is Imaginary.
GMF=MercuryMass*PerihelionSpeed*Oz #GMF is the gravitomagnetic force. Note use of left hand rule. GMF opposes F.
print("OzAdjust=",OzAdjust) #For reference.
#Now compute orbit with sun spinning.
#Do trial orbit to determine numeric error.
print("computing. Please wait.")
FX=gX*MercuryMass #Gravitational force.
#Oz=0. Execute this statement if computing gravity alone (non-spinning Sun).
TotalFX=FX+GMFX #Points toward Sun, but less than force when Sun does not spin.
#Set up for t=DT loop.
TotalFX=FX+GMFX #Points toward Sun.
#Now do rest of orbit.
for index in range(2,2*N-1):
TotalFX=FX+GMFX #Points toward Sun, but less than force when Suin does not spin.
#Execute following 3 statements for plotting of x-axis.
for index in range(0,2*N-1):
#Execute following 3 statements for plotting of y-axis.
for index in range(0,2*N-1):
print("End Of Program")