From: jmc@SAIL.Stanford.EDU (John McCarthy)
Subject: averting doom
Reply-To: jmc@cs.Stanford.EDU
Organization: Society for Celestial Mechanical Engineering
Date: 29 Dec 92 21:10:51

from a U.P. story

	     WASHINGTON (UPI) -- Life on Earth as we know it will
     come to an end in 1,500 million years and the planet will
     look more like its dusty, volcanic sister Venus in 2,500
     million years, scientists said Wednesday.
	     But mirrors or shades potentially could shield Earth
     from increasing heat from the sun and delay the catastrophic
     consequences, they said. A collision with a comet or other
     major change in the atmosphere could speed up the end of
     life.
	     Ken Caldeira and James Kasting of Pennsylvania State
     University calculated the doomsday estimates using computer
     models of temperature and atmosphere changes and projections
     of the sun's increasing heat.
	     As the sun continues to brighten and warm over time,
     the amount of carbon dioxide in the atmosphere drops --
     eventually to a point too low for plants to survive, the
     scientists said in the British journal Nature.
.
.
.
	     ``After 2 1/2 billion years, all of Earth's water
     would have been lost to space. That's a pretty strong
     barrier to life,'' said Caldeira, an Earth systems
     scientist.
	     Tyler Volk, an applied science professor at New York
     University, suggested there may be ways to delay or prevent
     the eventual demise of life.
	     ``Our descendants or descendant species would not
     have to run from the devolution...they could fight,'' said
     Volk. ``Shades in space or mirrors on the Earth that keep
     out a small fraction of the elevated future (heat from the
     sun) would be an option.''
	     Other possible solutions include constructing closed
     environments such as the Biosphere 2 project in Arizona, in
     which cycling of carbon dioxide, water and other essential
     matter would be controlled.  Establishing controlled
     Earth-like environments in space also could be considered,
     Volk said.

(Is it really true that mainly doom-seekers become Earth systems
scientists, and mainly problem-solvers become applied science
professors?)


The most straightforward way to avoid the death of the earth from the
overheating predicted by Caldeira and Kasting is to move the earth
farther from the sun on whatever schedule seems appropriate to our
descendants.

Here's the best way to do it that I have been able to think of.  It is
along the same line as what Thomas Clarke offhandedly suggested in a
reply to my original post.  The present post contains only a
qualitative discussion with a few numbers taken from calculations I
made for a slightly different project - moving Mars closer to the sun
in order to improve its climate.  I need to work on the formulas and
the numbers some more before giving them.

The method involves no new science and only predictable improvement
in present technology.

Our object is to transfer energy from the orbit of Venus to the
orbit of the earth so that Venus will move closer to the sun and
the earth farther away.  Jupiter could also be used.  Unfortunately,
it seems that the matter is a bit more complicated than this, because
the process must not only conserve energy, but it must also conserve
angular momentum of the earth and Venus about the sun.  Maybe this
can be done at the cost of giving Venus a more eccentric orbit, but
maybe it requires a third planet.  (This is analogous to processes
in atomic physics that require a third body in order to satisfy
all the conservation laws.)

The problem is to arrange for a coupling between the orbits of the
earth and Venus and possibly another planet as well.

Our tool for doing this is what I shall call a *tame asteroid*.
A tame asteroid is one that has repeated encounters with planets.
A small deflection of the asteroid's orbit before an encounter
is magnified by the encounter.  The asteroid is always controlled
so that it never stops having encounters.  The deflections
(delta-v s) are accomplished as many encounters in advance as the
noise in the system will permit.  

[This process is analogous to the trajectory of the spacecraft
Galileo, but apparently the plan with Galileo is to give up the close
encounters when it enters the Jovian system.  It JPL were to keep
control of it by a sequence of encounters, it could be returned to the
vicinity of the earth after spending sufficient time in the Jovian
system.  I have no idea what use this might be.]

The asteroid, say Ceres, has repeated encounters with the earth
and Venus.  It passes in front of the earth and behind Venus on
each encounter.  Thus it adds energy to the earth's orbit and takes
it from Venus's orbit.

When I was thinking about moving Mars, I made some calculations
involving the masses of Ceres, Mars and Venus and the ratio between
the escape velocity from Mars and the delta-v s needed to move
Mars to the more salubrious distance of the earth's orbit.
I got a figure of at least 330 encounters taking at least about
two years each.  This assumed that each encounter with Mars
transferred the maximum possible amount of energy.  Therefore, the
computation is optimistic by some small factor, say 5.

Since the earth is 9 times as massive as Mars, about 9 times as
many encounters would be required.  At least at first, the encounters
would take less time, because the earth and Venus are closer to the
sun.

The problem with conserving angular momentum is one I only encountered
recently, so I haven't figured out what additional encounters might
be needed.


Keeping a Tame Asteroid Tame

Ceres has a mass of 10^21 kilograms, so it would be important to
make its deflections as much in advance as possible.  I assume that
the gravitational fields of the bodies involved will have been
measured accurately long before the project is attempted, i.e. the
high order harmonics of the gravitational potentials will be accurately
known.  What I don't know is this: What is the largest source of noise, i.e.
unpredictable deflections, in the system?  I speculate that it is
weather in the sun causing unpredictable fluctuations in the sun's
gravitational field.  I don't know if this is right, and I hope someone
else can shed light on how large they are likely to be.


Taming Ceres

I confess I haven't yet computed how large a delta-v must be imparted
to Ceres to tame it.  Making its orbit eccentric enough to cross that
of Mars would suffice, because it could be arranged to encounter Mars
in an appropriate way to tame it.  Besides doing this with a single
delta-v, it would probably be cheaper to put Ceres into a suitable
resonance with Jupiter, so that Jupiter would do the work of making
the orbit of Ceres eccentric enough to tame it.  I don't know how
to do this kind of computation.

The most straightforward way to impart a delta-v to Ceres is to install
a large number of nuclear reactors on it, and use the energy to expel
fragments of Ceres in a desired direction and at an appropriate velocity.
If maximal energy efficiency is wanted, i.e. to impart the maximum
momentum per unit of energy used, then the appropriate velocity is
a simple optimization problem, and it turns out to be  sqrt(2) vec,
where  vec  is the escape velocity from Ceres.  The actual velocity
with which the matter leaves the Ceres gravitational field is then
vec.

The number of reactors needed is large.  A million reactors each
of 1000MWE, would give Ceres a delta-v of 1 km/sec in 1,000 years -
if I did the arithmetic correctly.  If our descendants installed
one reactor every thousand years, they would install a million in
a billion years, and that is time enough to avert the prophesied
doom.  A non-trivial fraction of the mass of Ceres would be consumed
if energy is optimized and if 1 km/sec is needed.  Our descendants
might decide to use a higher exhaust velocity.  More likely, it
will turn out that a considerably smaller delta-v will do with the
aid of Jupiter.  Our descendants might decide to use a smaller
asteroid.


The Broken Kepler Approximation

The full computations required to determine all these trajectories
will be quite large (though within the capacity of present computers).
Moreover, the required information about the gravitational fields
won't be available for a long time.  Here is an approximation that
can give a qualitative picture and an estimate of the time required
to move earth out a given distance or to move Mars in (a shorter time
project).

Make the following assumptions:

1. The planets and the asteroid are point masses.

2. The system is planar.

3. The planets have masses small compared to the that of the sun.

4. The asteroid has a mass small compared to that of any planet
involved.

5. The encounters between the asteroid and a planet are elastic
collisions of point masses.  However, we assume that the possible
angles of the collisions are limited by the radii of the planets.

6. Any individual collision has a small effect on the trajectory
of the planet.

7. The asteroid departs from each collision with a velocity that
ensures a subsequent collision with the same or a different planet.

Making these assumptions leads to the following conclusions.

1. The trajectory of the asteroid is a sequence of segments of
Keplerian ellipses about the sun.  That's why we call the above
set of assumptions the broken Kepler approximation.

2. The segments are separated by elastic collisions with the planets
conserving energy and momentum.

3. For each collision there is a discrete set of deflections that
ensure subsequent collisions.  They form a sort of spectrum.

4. Computing the next collision does not require the solution of
differential equations.  Instead one has transcendental equations
to solve analogous to Kepler's equation (the one used to compute
the position of a planet as a function of time).  However, it
looks like the transcendental equation will involve two unknown
parameters instead of the one that appears in Kepler's equation.

It would be nice to have a program that would compute broken
Kepler trajectories and display them for our contemplation.


I hope I have convinced you that our distant descendants can
survive the warming up of the sun until it becomes an actual
nova.

By the way, it seems to me that if the above idea is sound, it settles
the question of the stability of the solar system - in the negative.
Very likely an asteroid could be tamed over a sufficiently long time
with as small an expenditure of delta-v as might be desired.  Once
tamed it could be used with infinitesimal external force to expel a
planet from the system.  This tells us that the current trajectory of
the solar system is arbitarily close to one in which a planet is
expelled.  Of course, the probability that a planet actually would be
expelled by this mechanism in some particular finite time is extremely
low, because maintaining the required sequence of encounters requires
an improbable precision in the initial conditions.  I suppose a lower
bound on the probability could be computed and from it an expected
upper bound on the gravitational lifetime of the solar system could be
obtained.

Criticism and comments are welcome.  For a certain reason, I even
welcome comments, however uninformed, to the effect that the whole
idea is preposterous.  I prefer such comments to be postings rather
than email.

What would be most welcome is a collaborator on a paper that could
be published in _Nature_.
--
John McCarthy, Computer Science Department, Stanford, CA 94305
*
He who refuses to do arithmetic is doomed to talk nonsense.

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