Two Geophysical Coincidences

    

Which of the following two coincidences is more "impressive"?

  (1)    gT = c          (The acceleration of gravity at the earth's
                          surface multiplied by one period of the 
                          earth's orbit equals the speed of light).

  (2)  (D/d)s = (D/d)m   (The diameter over the distance for the sun 
                          equals the same ratio for the moon.)

Of course, these "equalities" are only approximate.  Numerically we 
have roughly  

            gT                          (D/d)s               
           ----   = 1.0315            ----------  =  0.9604
            c                           (D/d)m

I suppose coincidence (2) has been historically more impressive, since 
the astonishing precision of the match is displayed so vividly during 
solar eclipes.  In contrast, it's hard to think of any physically
perceivable consequences of coincidence (1).  On the other hand, the
appearance of the physical constant c in (1) seems quite remarkable.

An interesting related question is whether such coincidences are, in 
effect, compounded by the fact that they apply to (and only to) our 
own planet Earth, which is distinguished by several other seemingly 
unique properties, not least of which is its being the only site (so 
far as we know) of the spontaneous emergence of life.

Of course, it's exceedingly well known that trying to judge the 
significance of events after they have occurred is a very tricky
undertaking.  ("Something improbable is bound to happen.")  
Nevertheless, the apprehension of "coincidences" is one of the 
foundations (maybe THE foundation) of rational thought (as well 
as much irrational thought.)

In any case, on the subject of coincidence (1), note that the orbital 
periods and surface gravities of the nine planets are listed in the 
table below.

             T          g
          Time to    Surface
          complete   gravity     T*g          T*g
          one orbit  (Earth     (year-        ---
          (years)     gravs)     gravs)        c
         ----------  -------    -------     -------
Mercury    0.241      0.380      0.09158     0.0943
Venus      0.615      0.900      0.55350     0.5701
Earth      1.000      1.000      1.00000     1.0300
Mars       1.881      0.380      0.71478     0.7362
Jupiter   11.860      2.640     31.31040    32.2497
Saturn    29.460      1.130     33.28980    34.2876
Uranus    84.010      0.890     74.76890    77.0119
Neptune  164.790      1.130    186.21270   191.7983
Pluto    248.500      0.050     12.25000    12.6175

The product T*g has units of velocity, so we can express it in 
dimensionless form if we divide it by some standard velocity, such
as c (the speed of light).  The right hand column lists these 
dimensionless values for the nine planets.  As can be seen, Tg/c 
ranges from about 1/10 up to nearly 200.  

For a planet of mass m and radius r in a roughly circular orbit of
radius R around a star of mass M, the surface gravity is about
g = Gm/r^2  (where G is Newton's gravitational constant) and the 
period of revolution is about T = 2pi r^(3/2) / sqrt(GM).  Therefore,
the product gT for this hypothetical orbiting planet is
                            ____
                           / G     / R \2
          gT   =  2pi m   / ---   ( --- )
                        \/  M R    \ r /

For example, the mean distance from the Earth to the Sun is about 
R = 1.49E+11 meters, the Earth's mass is about m = 5.98E+24 kg, the 
Earth's radius is about r = 6.37E+06 meters, and the Sun's mass is 
about M = 2.0E+30 kg.  Newton's constant is 6.67E-11 Nm^2/kg^2, so 
we have gT approximately 3x10^8 m/sec, confirming that gT/c is 
about 1 for the Earth.

Are there any physical or biological reasons for us to expect to 
find ourselves on a planet for which gT/c is close to 1?  Would there
be any special obstacles to the development of life on a planet 
orbiting a star if the value of gT/c was as great as, say, 100, or 
as small as 1/100?

Another interesting coincidence (of a different sort) is the fact 
that the period of a circular orbit around a gravitating body is 
identical to the period of a tableau pendulum at the same altitude.
Consider a flat table at a height R from the center of a gravitating
body, and suppose a frictionless puck is placed on this table, as
illustrated below.
                 
In the small-theta approximation we have cos(theta) ~ 1 and 
sin(theta) ~ theta, so the equation of horizontal motion for the
puck is  GM/r2 theta = - r theta" where M is the mass of the 
gravitating body.  Thus we have

               d^2 theta      GM
               ---------  +  ----- theta  =  0
                 dt^2         r^3

which has the solution theta = A cos(wt) where
                         ______
                        / GM
                 w =   / -----
                     \/   r^3

This is identical to the period of a circular orbit of radius r, so
we could substitute the period of a tableau pendulum at the Earth's
distance from the Sun in place of the Earth's orbital period in the
first geophysical coincidence noted above.