js2j@mhuxt.UUCP (sonntag) (09/25/85)
Ted Holden has been arguing for weeks in net.origins about how
huge dinosaurs could never have existed in our current gravitational
field, due to square-cube types of problems. His postulated cause for
the reduced felt-accelleration is that during the extreme past the Earth
orbited Saturn, producing extremely large tidal effects, (on a 'locked'
system so that one side of the earth faces Saturn all of the time.)
I worked up the equations describing the situation, in the hopes that
they'll indicate that the earth would have to be grazing Saturn's atmosphere
(or inside it) for the free-fall accelleration at the ends of the earth
to be much lower. I don't have handy a table full of the constants I'd
need to evaluate the results, however... perhaps someone out there would
be kind enough to evaluate my results.
delta_a = G * M * r1 (1/r0^3 - 1/r1^3)
Where: G is the gravitational constant (as in F=(G*m1*m2)/r^2)
r1 is the radius from the center of saturn to the inner side of
the earth. (r1=r0 - radius of earth)
r0 is the radius from the center of saturn to the center of earth.
M is the mass of Saturn
delta_a is the changed in free-fall accelleration at the inner and
outer ends of the earth. A delta_a of -5 meters would
correspond to a 50% reduction in weight.
Derivation:
delta_a = a - a
centripetal Saturn's gravity
= (2*pi*r1)^2 / (T^2 * r1) - G * M / r1^2 (where T=orbit time)
Due to the fact that the earth is in orbit, we know that delta_a=0 when
evaluated with r0 instead of r1. This allows us to solve for T^2:
T^2 = (4 * pi^2 * r0^3) / (G * M)
Substituting this into the origional equation and rearranging, we get
the result quoted above. One last simplification we can make:
(1/X^3 - 1/(X - del)^3 = 3 * del * X^-4 (for large X/del)
Using this in the original equation:
delta_a = 3 * G * M * (radius of earth) * r0^-3
This is simple enough that I hope that *someone* out there can find
the orbital radius required to cause say a 50% reduction in weight. Please
compare this result to the radius of Saturn. If it turns out to be
less than the radius of Saturn, do you think Ted will be quiet?
--
Jeff Sonntag
ihnp4!mhuxt!js2j
Silly quote: "There are a few off-the-wall extremists, who are shunned
by us moderates." - Don Blackmatt@oddjob.UUCP (Matt Crawford) (09/27/85)
Oooh, excellent point, Jeff! I did the derivation another (simpler) way and got out an Earth-Saturn distance of 4 Saturn radii, which made my approximations invalid, so I did it again solving the quartic (with help from macsyma) and got an Earth-Saturn distance of 49.3 megameters. The radius of Saturn is about 60.3 megameters, so the Earth would have to be down in the clouds somewhere to lower our surface gravity at the sub-Saturn point by 500 cm/sec^2. Of course this won't stop the Velikovskians. They're probably all Capricorns. :-) _____________________________________________________ Matt University crawford@anl-mcs.arpa Crawford of Chicago ihnp4!oddjob!matt
kurtzman@uscvax.UUCP (Stephen Kurtzman) (09/30/85)
In article <974@oddjob.UUCP> matt@oddjob.UUCP (Matt Crawford) writes: >Oooh, excellent point, Jeff! I did the derivation another (simpler) >way and got out an Earth-Saturn distance of 4 Saturn radii, which >made my approximations invalid, so I did it again solving the quartic >(with help from macsyma) and got an Earth-Saturn distance of 49.3 >megameters. The radius of Saturn is about 60.3 megameters, so the >Earth would have to be down in the clouds somewhere to lower our >surface gravity at the sub-Saturn point by 500 cm/sec^2. > >Of course this won't stop the Velikovskians. They're probably all >Capricorns. :-) >_____________________________________________________ >Matt University crawford@anl-mcs.arpa >Crawford of Chicago ihnp4!oddjob!matt Since the fact that the earth orbited Saturn cannot be controverted, this of course implies that the atmosphere of Saturn is breathable! :-)