If the exponent is not exactly 2 (or -1) then there will be *no* elliptic
orbits. Rather, in general, the orbit will be an open figure that does
*not* retrace itself with each revolution, but yet remains in an annular
band between the fixed perihelion and fixed aphelion distances. If for a
given orbital angular momentum the energy is as low as possible
(consistent with this angular momentum) then the perihelion and aphelion
distances merge and the orbital band narrows to a circle. Just how wide
the annular orbital band is depends on how much extra orbital energy the
system has above this minimal value. The location and structure of the
orbit (including just how "habitable" it is) is determined once the
initial conditions (e.g. the energy, the angular momentum, etc.) for the
system are specified.
If you were to posit a universe that had a Kepler/Coulomb exponent
something other than 2, why do you worry about whether the Earth
would stay in a "habitable zone" or not? I'd be more worried about
whether there would even be stars & planets or even any kind of
aggregated matter, radiation, etc. at all or not. The answer to such
concerns would depend on the details of just what the newly proposed
laws of nature would happen to be. You cannot just change one
parameter (e.g. the exponent of attractive force laws) and pretend that
the rest of the theory can logically hold together. Because such
parameters tend to be determined by the mathematics of physical theory at
the deepest levels, you are not free to change them without changing
those deep-level theories in a *very* drastic way. You can't say what
the universe would be like until you have actually carefully specified
the new deepest-level theory and then worked out all of its consequences
at the more superficial phenomenological levels.
> Secondly, in response to this,
<snip>
>[snip]
>
>I should also have added, that one must show why those deeper physical laws
>took the values they did so that they can then give gravity the values it
>needs. One can possibly construct an infinitely regressing sequence of
>anthropic questions.
I don't know if the sequence would be infinite or not, but such anthropic
arguments would tend to revolve about just how and why the deepest-level
theory/ies take/s the form/s it/they do/es. All the other phenomenlogical
behavior of physical reality would be settled once the questions of the
deep-level theory/ies are settled. Parameters like central force
exponents, the details of the resonance levels of nuclei, atomic structure
and all of chemistry are determined by the deep-level theory. Treating
a host of phenomenological level parameters as somehow independently
freely adjustable in anthropic arguments is, IMO, quite illegitimate.
>>>So why shouldn't a randomly chosen universe have a gravitational force with
>>>an exponent of 10?
>>
>>Because that then would not be a gravitational-type force. It would be
>>something else. Gravitation has to do the effects and the means by which
>>matter curves spacetime and spacetime's curved geometry influences the
>>behavior of the matter in it. The effects of curved spacetime do not and
>>cannot produce an exponent of 10.
>
>But that is the point of the anthropic argument. even assuming the many
>universe hypothesis of Everett, where all conditions are found in one of
>the universes, in a truly randomly chosen universe the exponent could be 10
>and life would be impossible.
There are very many ways for life (as we know it) to be impossible by
making a slight adjustment of just a relatively few fundamental
constants (e.g. number of dimensions of space & time, numbers of
generations of elementary particles, coupling constants and mixing angles
for the fundamental interactions, the masses of the elementary particles,
etc.) whose values determine all the rest of the phenomenology of the
physical world. Presumably, once (or if) an acceptable Theory of
Everything is found, it will require that these (formerly fundamental)
parameters be determined to have the values they do by the logic of
mathematical necessity (and will even be in principal calculable from that
theory even if the calculation ends up being too intractable to actually
carry out). But even granting the existence of such a mathematically
acceptable and consistent Theory of Everything--even if it is found that
such a theory is logically unique--there are *still* deep metaphysical
questions that science is would be incapable of addressing. Questions
such as why there is something rather than nothing, why the Theory of
Everything happens to be realized in nature, just what/Who enforces the
theory so that things do happen according to how the theory describes
them so logically well, why the workings of nature should obey
mathematics and logic anyway, etc., etc.. Even the deepest level theory
of physics is *still* descriptive of nature, not prescriptive. Science,
math & logic cannot explain *why* the description contained in the theory
actually happens in nature. It at best they can say that *given* that
the deep level theory actually describes nature, then things must work
out according to the logic of that theory. But this is essentially
tautological and begging the question at its heart.
>>Regarding the business of attempting to calculate the a priori
>>probabilities for the various constants of nature, I agree with Howard's
>>comments on that score.
>
>Then do you discount the anthropic principle? Barrow and Tipler do not
>calculate probabilities, but the entire anthropic argument is that it is in
>some emotional sense, difficult to belief that all the constants could be
>correct for life to exist. If this isn't an implicit probability
>calculation I don't know what is.
I don't know about whether or not it is an "implicit probability
calculation" but I, for one, *do* profoundly get the feeling (from
natural theology) that the universe is somehow made *for us*, or at
least that we are meant to be here. I don't know if natural theology
can go much (or any) further than this though.
David Bowman
dbowman@georgetowncollege.edu