Celestial Mechanics

• The Sun's mass is 1.99*10³⁰ kg and the Earth's mass is 5.98*10²⁴ kg.  This makes the Sun more than 330,000 times as massive as the Earth, so the motion of the Sun as a result of the gravitational pull of the Earth will be very small and considered negligible.  Therefore, any motion of the sun would result in an equal motion in the frame of reference for the Earth's orbit and would cancel out.
• The Earth's orbit has an eccentricity of 0.017, also very small, so I will describe it as being circular.
• The linear velocity of the Earth through space with respect to the Sun is 29.8 km/s which is 0.01% of the speed of light, so relativistic effects will be ignored.
• For the time being, I will work with a universe containing only the Sun and Earth.  Under these conditions, the only perterbation to the Earth's orbit will be variations in the mass of the Sun.  As the Sun consumes itself its mass continuously decreases which results in an Earth orbit that slowly spirals outward.  The period of time I will be using will be on the order of a few years or less, so this effect will be ignored as well.
• Another source of perterbation would be the distribution of the mass of the Sun, but since the distance between the Earth and Sun is 214 times the radius of the Sun (6.96*10⁸ m) I will assume that it can be considered a point source.
• The "primary" law of physics, F = ma, adequately models reality at the celestial scale.
• To be complete, I also assume that time and space are linear and that space is frictionless.

[ the following sections are currently under review ]

Now that I have stated all that I will ignore, let me go on to that which cannot be ignored.  For any given initial velocity, position,  and direction of the Earth with respect to the Sun, its orbit is completely defined.  In other words, there is one and only one orbit that will result from the given initial conditions.  The general polar equation that describes the motion of a body under the influence of a central force is^[1]:

1     G * M
- = --------- * (1 + e * cos(theta))
r   (r₀ * v₀)²
 

     r₀ * v₀²
e = --------- - 1
      G * M

where G is the universal gravitational constant (6.67*10^-11 m³ / s²kg), M is the mass of the body causing the central force, and e is the eccentricity of the orbit which can be derived as shown from the initial conditions stated above.  The values r₀ and v₀ are those that characterize the perigee of the orbit, assuming that the perigee is placed where theta = 0.  From this follows directly that for any angular position around the central body, there is one and only one radius that satisfies the equation.  There are three conic sections that are described depending on the eccentrity e:

e > 1  hyperbola (two values of theta for which r goes infinite)
e = 1  parabola  ( one value of theta for which r goes infinite)
e < 1  ellipse   ( no values of theta for which r goes infinite)

This next diagram shows how the Earth's orbit changes when its initial distance is held constant while its velocity with respect to the Sun at its perigee increases:
 
 

Note that when e >= 1 the escape velocity has been exceeded and the Earth would not return.  When e < 1 this is the case where the escape velocity has not been exceeded, hence the perpetual elliptical orbit..  When e = 0 the radius is constant, hence the orbit is circular.  (A circle is just the special case of an ellipse where there is zero distance between the two focii)  From this point forward I will use the term 'orbit' to describe an elliptical orbit.

This leaves us in a precarious position when trying to explain how the Earth's orbit as it is today came to be.  If the Earth was ever at a different position than one in its present orbit, it could not achieve its present orbit.  If the Earth was ever moving at a different velocity than its present velocity, it would not be in its present orbit.  If the Earth did not start out in its present orbit, it would not be in and could not end up in its present orbit.  Let me restate that. If the Earth did not start out in its present orbit, it would not be in and could not end up in its present orbit.  Remember, this conclusion is the result of the unequivocal laws governing planetary motion and my previously stated assumptions.  This is sure to raise an eyebrow or two among those in the scientific community who have not already studied this subject.

There are three scenarios that come to mind to explain the origin of the Earth's orbit.  The first is that the Earth in some primordial state was "freely" roaming through space until it came into proximity with the Sun at which point it was captured and started to orbit.  For this to produce an orbit at all the Earth's velocity must be below the escape velocity for its distance from the Sun.  Assuming this is the case, it simply could not produce a circular orbit of the present radius for if the Earth started at some much larger distance from the Sun than it is now, it would have no choice but to return to that point.

The second scenario is that the Earth was "spawned" from the Sun somehow by a blob of the Sun's mass which happened to escape and condense into a mass of molten rock and started to orbit that way.  In that case, again, it either exceeded escape velocity or it didn't.  If it did, there's no orbit.  If it didn't, an orbit that originates at the surface is destined to return to the surface.  Once the Earth was separated from the Sun and at that point only under the influence of the Sun's gravity, its orbit would inevitably bring it right back to the Sun and it would never complete its first orbit.

The last scenario is that the Sun and Earth formed in place from a spinning disc of matter, with the matter that made the Earth traveling at a velocity with respect to the Sun that was just right according to the Earth's present orbit.  This scenario falls in line with my conclusion.  To allow for this to happen once takes a certain amount of faith and/or luck.  To allow for this to happen nine times, once for each planet, and then again 33 more times to account for all the planets' satellites, at the same time allowing for the retrograde rotation of Venus and Uranus and the greater than 0.2 eccentricities of the orbits of Mercury and Pluto, takes faith on the order of that which the most religious among us have in a creator.  In fact, while this wouldn't prove the existence of a creator, it certainly points in the direction of intelligent design.  However, I will admit that this scenario deserves a closer look as it is the one I know the least about.  After all, why a spinning disk?  Why not a spinning sphere?  How could a spinning disk appear in the first place?  Perhaps if the masses and motions of each body were carefully choreographed...

Now let me examine the assumptions I made that led up to this conclusion.  First, I assumed that the pull of the Earth on the Sun was negligible.  If this was not the case, the equations describing the motions of the Sun and Earth would be more complex than the one above, but this by no means would make it any easier for the Earth's present orbit to arise.  I then assumed that the Earth's orbit is circular.  This is arbitrary.  Even the slighlty elliptical orbit it has, or any orbit for that matter, is still governed by the equation above.  That's the reason the Earth's distance from a Sun of today's mass could not deviate from its present distance without there being a corresponding difference in the shape or size of the orbit it defines.
Next I ignored relativistic effects, which accounts for the linearity of space and time.  For the Earth to be moving fast enough for relativistic effects to be taken into account, it would take some miraculous event to put in its present orbit at its present velocity.  Lastly, as far as the frictionlessness of space and the constancy of the Sun's mass are concerned, the few years or less of time that are required to make the necessary observations would make these factors insignificant.

Finally, I assumed a universe containing only the Sun and Earth.  This obviously is not the case so now I will now attempt to include all nine planets in my analysis.  The following table lists the mass and mean orbital distances (as measured center to center) of the 10 major bodies in our solar system^[2]:
 

From this data, one can calculate the acceleration due to gravity of each body due to every other body.  This acceleration would be greatest when the bodies are closest to one another, and the case where all 10 bodies are co-linear is a close if not the best approximation of this.  Keep in mind that the mass of the body being accelerated has no effect.  That's why feathers and hammers fall at the same rate on the moon.  Using the following equation, the set of accelerations are shown with the body in the column doing the pulling and the body in the row being pulled:

               G * M
acceleration = -----
                 r²

In the following table, all the accelerations are normalized to the smallest, the pull on Pluto by Mercury (6.43*10^-13 m/s²):
 

	Sun	Mercury	Venus	Earth	Mars	Jupiter	Saturn	Uranus	Neptune	Pluto
Sun		1.02E+04	4.33E+04	2.76E+04	1.29E+03	3.26E+05	2.89E+04	1.10E+03	5.28E+02	1.78E+00
Mercury	6.16E+10		2.01E+05	7.31E+04	2.32E+03	3.80E+05	3.14E+04	1.15E+03	5.41E+02	1.82E+00
Venus	1.77E+10	1.36E+04		3.52E+05	4.65E+03	4.39E+05	3.38E+04	1.19E+03	5.54E+02	1.85E+00
Earth	9.17E+09	4.02E+03	2.86E+05		1.10E+04	5.00E+05	3.60E+04	1.22E+03	5.65E+02	1.88E+00
Mars	3.97E+09	1.18E+03	3.51E+04	1.02E+05		6.52E+05	4.09E+04	1.30E+03	5.85E+02	1.93E+00
Jupiter	3.41E+08	6.58E+01	1.13E+03	1.57E+03	2.22E+02		1.39E+05	2.07E+03	7.71E+02	2.36E+00
Saturn	1.01E+08	1.81E+01	2.89E+02	3.79E+02	4.64E+01	4.64E+05		4.37E+03	1.13E+03	3.10E+00
Uranus	2.51E+07	4.32E+00	6.62E+01	8.38E+01	9.60E+00	4.50E+04	2.85E+04		4.02E+03	6.76E+00
Neptune	1.02E+07	1.73E+00	2.62E+01	3.28E+01	3.67E+00	1.42E+04	6.26E+03	3.41E+03		3.16E+01
Pluto	5.93E+06	1.00E+00	1.51E+01	1.88E+01	2.08E+00	7.51E+03	2.95E+03	9.86E+02	5.45E+03

Armed with this collection of data, we can see that the Sun's closest competitor to its pull on the Earth is Jupiter, but it is "outweighed" by a factor of 18,380, so it has little influence.  We can see that the Sun's pull on the planets exceeds the pull between any two planets.  The closest is Jupiter's pull on Mars, and that is 9 times weaker than the Sun's pull on Pluto, and 6089 times weaker than the Sun's pull on Mars.  It would appear that even the influence of the other planets is not sufficient to explain a spontaneous generation of Earth's orbit much less our entire solar system.

Since my immediate area of concern is the Earth, let me also include the effects of the Moon.  The Moon's mass is 7.36*10²² kg and its mean distance from the Earth is 3.80*10⁸ m.  The resulting acceleration of the Earth is 3.40*10^-5 m/s².  This is more than enough to significantly alter the Earth's motion.  However, the effect is only local.  The center of mass of the Earth/Moon system is now what orbits the Sun as described by the above equations.  Now, the orbit of the Moon is not as "perfect" as the planetary orbits.  Since the Earth is not perfectly spherical and its surface water tends to bulge toward the Moon, its orbit is not perfectly elliptical.  Another source of perterbation is the Sun.  The Earth's pull on the Moon accelerates it at a rate of 2.76*10^-3 m/s², while the acceleration due to the Sun is more than twice as much at 5.90*10^-3 m/s²!  This makes the Moon's orbit with respect to the Earth dependent upon the Earth's position with respect to the Sun, and makes its orbit more complex than a simple ellipse.  While you may or may not find all of this interesting, what I previously stated about the Earth also applies to the Moon.  The Moon's orbit could only be as it is by starting out as it is.

When bringing other bodies into consideration, one must also take into account the possibility of collisions.  While the nine known planets could not have collided with each other and have their present orbits, it would have been possible for bodies that are not in today's solar system to have caused collisions and significantly effect the orbit of the planet it collided with.  However, with the masses and velocities involved, annihalation is a more likely outcome.

More to come...

1. Vector Mechanics for Engineers: Statics and Dynamics.  Fourth Edition. Ferdinand P. Beer and E. Russell Johnston, Jr.  McGraw-Hill, Inc.  1984
2. Physics: Parts 1&2.  Third Edition.  David Halliday and Robert Resnick. John Wiley & Sons, Inc. 1978