% This preamble comes from /u/jmc/let/latex-96.tex which is copied into
% latex sources by the macro latex-include-96 which is itself called
% by the macro mklatex-96 executed in /u/jmc/let/files.
% Citations for use with latex2html are in /u/jmc/1/citations.tex
% Main citation list is /u/jmc/1/biblio.bib

\documentclass[12pt]{article}
%\documentclass[a4paper,12pt]{article} % if a4 is wanted
% In case we want to use latex2html.
\usepackage{html}
% for author date style
\usepackage{/u/jmc/1/cslibib} % Fri Oct 15 1999 Tom gave me
                              % this version that avoids a bug.
\usepackage{varioref}
\usepackage{amssymb}

%\usepackage{color}
% usage:  \textcolor{cyan}{ Reasoning about action}
% available colors: black, white red, green, blue, yellow, cyan and
% magenta

% for 3400 display, compile on Powerbook 3400
%\setlength{\textwidth}{8in}
%\setlength{\textheight}{5in}
% configs on Oztex, select A4 landscape

% \usepackage amstex % in case we want full mathematics
% \usepackage{chess} % /usr/local/lib/texmf/doc/chess/ boards-2e.ltx
% /usr/local/lib/texmf/tex/inputs/chess.sty
% \usepackage{a4wide}  % auxiliary to chess

%\usepackage[dvips]{epsfig} % UNCOMMENT IF FIGURES ARE WANTED
% usage: \epsfig{file=blocks1.eps}
%*****************************

% BEGIN stuff for times and dates
\newcount\hours
\newcount\minutes
\newcount\temp
\newtoks\ampm

% set the real time of day

\def\setdaytime{%
   \temp\time 
   \divide\temp 60
   \hours\temp 
   \multiply\temp -60
   \minutes\time 
   \advance\minutes \temp 
   \ifnum\hours =12 \ampm={p.m.} \else
   \ifnum\hours >12 \advance\hours -12 \ampm={p.m.} \else \ampm={a.m.} \fi \fi
}

\setdaytime

\def\theMonth{\relax
  \ifcase\month\or
    Jan\or Feb\or Mar\or Apr\or May\or Jun\or
    Jul\or Aug\or Sep\or Oct\or Nov\or Dec\fi}

\def\theTime{\the\hours:\ifnum \minutes < 10 0\fi \the\minutes\ \the\ampm}

\def\jmcdate{ \number\year\space\theMonth\space\number\day}
% END stuff for times and dates

%
\begin{document}
% % for author date style
%\bibliographystyle{cslibib}
\bibliographystyle{alpha}
%
\title{CHAOS AND MOVING MARS TO A BETTER CLIMATE}
\author{
{\Large\bf John McCarthy} \\
Computer Science Department \\
Stanford University \\
Stanford, CA 94305 \\ 
{\tt jmc@cs.stanford.edu} \\
{\tt http://www-formal.stanford.edu/jmc/}}
\date{\jmcdate ,\ \theTime}
\maketitle

\begin{abstract}
In a chaotic dynamical system, small deviations from given initial
conditions produces large changes over time.  This suggests that
chaotic systems are controllable by making small changes initially.
The solar system is a somewhat chaotic dynamical system, so maybe
humanity can control its evolution to produce desired effects.

The specific idea of this article is to move the planet Mars to an
orbit at the same distance from the sun as the earth's orbit.  This
would make Mars warmer, which might make it more habitable.

The scheme is to use a \emph{tame asteroid} that makes several
thousand encounters with Mars, Venus and Jupiter to exchange energy
and angular momentum among these planets, thus moving Mars to the
desired orbit.  Conservation of both energy and angular momentum
requires that two other planets besides Mars be used.

A \emph{tame asteroid} is one that makes repeated encounters with
planets that magnify small perturbations of its orbit.  The object is
to achieve the goal of moving Mars with minimal total $\Delta v$,
i.e. minimal rocketry.

Moving Mars will take some tens of thousands of years, but not
millions of years.\footnote{This article is an elaboration of a web
  page (http://www-formal.stanford.edu/jmc/future/mars.html) put up
  for my students in a class in Technological Opportunities for
  Humanity at Stanford University.  It may change.}


\end{abstract}

%\tableofcontents %UNCOMMENT IF TABLE OF CONTENTS WANTED

% begin here


\section{Chaos and controllability}
\label{sec:intro}

Consider a system governed by systems of differential equations that
evolves chaotically.  Thus small variations in initial conditions are
amplified to produce large changes in later states.  Suppose humans
can make small purposeful changes.  Then perhaps the later states can
be controlled.  Suppose some initial conditions for a system are
specified, and we can compute the future history.  Small deviations
from the initial conditions grow exponentially with time.  Thus making 
a large change with a very small expenditure of energy requires a long 
time, but the time grows only logarithmically with the inverse of the
energy expended.  We can ask what states are reachable by small
changes in initial conditions.

Take the solar system as an example.  There are certain integrals of
the motion of the solar system as a whole, e.g. energy and angular
momentum.  It is clear that we cannot change the total energy of the
system by more than the energy we have available to expend.  However,
we can in principle change the energy of one planet at the expense of
energy of some others.

\footnote{This conclusion depends on regarding the solar system as
  isolated.  If we consider a large system involving Alpha Centauri
  and our own solar system, we can imagine increasing our solar
  system's energy by taking energy from the Alpha Centauri system.  It
  seems apparent that this would take a very long time.  If it were
  determined to take more than (say) $10^{100}$ years, then we could
  regard the energy of our own system as essentially unchangeable.
  Actually it might take only tens or hundreds of thousands of years.}

Angular momentum is also conserved, but it isn't obvious to me what
the trade-off is between expending energy to change angular momentum.
Taking a mass $m$ a distance $d$ from the sun allows making a change of
angular momentum of $m\Delta v d$.  If we want the changed angular
momentum to be effective in the inner solar system, then $\Delta v$
must not be so large that the mass escapes.  The time for the mass to
return to the inner solar system is proportional to $d^{3 \over 2}$.

As a concrete problem, consider moving Mars to an orbit at the same
distance from the sun as the earth, keeping it on the other side of
the sun from the earth to eliminate gravitational interaction with the 
earth.  This would make Mars warmer, which would facilitate human
settlement.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{A tame asteroid}
\label{sec:tame}

The idea is to use an asteroid to transfer energy and angular momentum 
among Mars, Jupiter and Venus in a way that makes Mars move closer to
the sun.  We would expect Venus and Jupiter to move further away, but
not much. We'll see how the mathematics turns out.  We hope to do this 
with only incidental $\Delta v$ applied to the asteroid by rockets.

The largest asteroid is Ceres, which has a mass about ${1 \over
  1,000}$th that of Mars.  The most favorable encounter of Ceres with
Mars is when Ceres comes as close to Mars as possible and has its
velocity reversed in the Mars-centered co-ordinate system.  The
$\Delta v$ of Ceres is then approximately the escape velocity from
Mars. [We'll give the details later].  The $\Delta v$ given to Mars is
then ${1 \over 1,000}$ of this escape velocity.  Actually, the
encounters will not usually be so favorable, so there will be some
efficiency factor less than 1.0.  The escape velocity from Mars is
about 5,100 meters per second, so a $\Delta v$ for Mars of 24 km/sec
would require at least 4,800 encounters.  If we imagine that the
orbital period of an asteroid within the orbit of Jupiter is about 10
years, then we are talking about some multiple of 48,000 years.  Very
likely, our descendants will become interested in starting such a
project only after they have had a technological civilization for some
tens of thousands of years.

Ceres is the largest asteroid, but it looks like would be very
expensive to get Ceres out of its very stable orbit.  Therefore, we
propose using an asteroid from the Kuiper belt which would be much
more easy to manipulate at first.  (This idea belongs to
\cite{Kory01}.)  The biggest of these isn't known yet, but we can
imagine it to be $1 \over 10$th the mass of Ceres.

We want a tame asteroid.

Asteroids large enough to have significant gravitational effects are
expensive to move much with feasible rockets.  Therefore, we want an
asteroid orbit that passes by many planets.  Passing close to a planet 
should amplify any perturbation of the orbit.  The earlier the
perturbation can be made the greater the amplification.  Note that you 
get no amplification out of a full Keplerian ellipse about the sun,
because the ellipical orbit amounts to a focussing effect.  A small
perturbation now will cause a slightly changed ellipse that will
repeat itself unless there is a further perturbation.  

The one effect that does grow with time in a perturbed elliptical
orbit is the time change $\Delta T$ of arrival at a given point.  If
the period is changed, then the perturbation will grow linearly with
time.  This effect can be used to adjust the phase of the next
encounter of the asteroid with the orbit of the planet.  It looks like
by waiting long enough we can achieve the next encounter with an
arbitrarily small $\Delta v$ of the asteroid.

The first mathematical question is whether there can be a tame
asteroid.  Ideally its orbit would, if unperturbed, graze planets for
the indefinite future.  Let's call this a tame orbit.  The second
question is whether an early enough small perturbation could make the
asteroid switch to a different tame orbit.  If so, the third question
is what is the set of tame orbits reachable from an initial orbit.  As 
a first approximation to these questions we regard the asteroid as
having infinitesimal mass, so the planets themselves are not
perturbed.

The next main question arises when the finite mass of the asteroid is
taken into account.  What class of planetary configurations can be
reached with an arbitrarily small tame asteroid?   Can these
configurations be reached with an arbitrarily small $\Delta v$ given
enough time?

The specific question is whether an asteroid from the Kuiper belt can
be tamed and used to move Mars to an orbit on the opposite side of the 
sun at the earth's distance from the sun.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Conservation of Energy and Angular Momentum}
\label{sec:conservation}

Since the asteroid is small compared to planets, and our goal is to
directly apply very little total $\Delta v$ for the asteroid, the main 
effect is exchange of energy and angular momentum among the three
planets. 

Before trying to design orbits for the tame asteroid, we compute the
changes in the orbits of Jupiter and Venus required to move Mars to
one AU from the sun.  We assume that energy and angular momentum are
conserved, i.e. that the asteroid itself overall contributes nothing,
because of its small size.

Here are some equations in which we assume that the planets have
circular orbits.  We derive relations between the energy
$\mathcal{E}$, the angular momentum $h$ and the distance of a planet 
from the sun.  We use the speed $v$ of the planet in its orbit as an
intermediate variable.

We start with equations for a single planet in a circular orbit.

\begin{equation}
  \label{eq:c1}
  \mathcal{E} = {1 \over 2}mv^2 - {m\mu_{\odot} \over r}
\end{equation}

\begin{equation}
  \label{eq:c2}
  h = mvr, \mbox{assuming circularity}
\end{equation}

\begin{equation}
  \label{eq:c3}
  {v^2 \over r} = {\mu_{\odot} \over r^2}, \mbox{ and so}
\end{equation}
%
\begin{equation}
  \label{eq:c4}
  v = \sqrt{\mu_{\odot} \over r}
\end{equation}

Circularity also gives
\begin{equation}
  \label{c5}
  \mathcal{E} = {1 \over 2}m{\mu_{\odot} \over r} - m{\mu_{\odot} \over r} = -{m\mu_{\odot}
    \over 2r}
\end{equation}

\begin{equation}
  \label{eq:c6}
  h = mvr = m\sqrt{\mu_{\odot} r}, \mbox{ and so}
\end{equation}

\begin{equation}
  \label{eq:c7}
  r = {h^2 \over m^2\mu_{\odot}}
\end{equation}
and finally

\begin{equation}
  \label{eq:c8}
  \mathcal{E} = - {m\mu_{\odot} \over 2r} = -{m\mu_{\odot} \over 2{h^2 \over m^2\mu_{\odot}}}
= -{m^3\mu_{\odot}^2 \over 2 h^2}.
\end{equation}

Having expressed $\mathcal{E}$ in terms of $h$, we are ready to write
the conservation laws for the total energy $\mathcal{E}$ and the total 
angular momentum $h$.  We use subscripts for the quantities associated 
with the three planets.

Conservation of energy for the three planets, e.g. Mars, Venus and
Jupiter gives for the new configuration

\begin{equation}
  \label{eq:c9}
  \mathcal{E}_1 + \mathcal{E}_2 + \mathcal{E}_3 = \mathcal{E},
\end{equation}
where $\mathcal{E}$ is the original total energy.

Conservation of angular momentum gives for the new configuration
\begin{equation}
  \label{eq:c10}
  h_1 + h_2 + h_3 = h,
\end{equation}
where $h$ is the original total angular momentum.

We assume that planet 1, e.g. Mars, is to be moved to a desired
circular orbit, and therefore we know $\mathcal{E}_1$ and $h_1$.
 We need to solve for $\mathcal{E}_2$, $\mathcal{E}_3$, $h_2$, and
$h_3$.  We use (\ref{eq:c8}) to express the energies in terms of the
angular momenta, so (\ref{eq:c9}) then becomes

\begin{equation}
  \label{eq:c11}
  {m_1^3\mu_{\odot}^2 \over 2 h_1^2} + {m_2^3\mu_{\odot}^2 \over 2 h_2^2} +
{m_3^3\mu_{\odot}^2 \over 2 h_3^2} = \mathcal{E}.
\end{equation}

(\ref{eq:c10}) and (\ref{eq:c11}) must be solved for $h_2$ and $h_3$,
which will then allow determining $r_2$ and $r_3$ by substituting in
(\ref{eq:c7}).

These equations have the form

\begin{equation}
  \label{eq:12}
  \begin{array}[l]{l}
x + y = a \\
{c \over x^2} + {d \over y^2} = e.
  \end{array}
\end{equation}

Substituting $y = a - x$ gives
\begin{equation}
  \label{eq:13}
{c \over x^2} + {d \over (a - x)^2} = e.  
\end{equation}

Solving these equations for moving Mars to 1.0 AU with the aid of
Venus and Jupiter yields

new Venus distance  = 1.99886e+09 compared to 1.07700e+11

new Jupiter distance = 7.79062e+11 compared to >7.78000e+11

Venus comes out distressingly close to the sun, whose radius is
6.96000e+08.  Oh well, nobody we know lives on Venus.

All distances are in meters.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{A scheme with minimal total $\Delta v$}
\label{sec:minimal}

Consider an asteroid that starts very far out.  The further the better
as concerns energy spent on adjusting the orbit of the asteroid, but
the further out the longer the whole process will take.  The asteroid makes
one encounter with a planet per trip into the inner solar system.  On
successive trips it encounters Mars, Venus and Jupiter but not
necessarily in a fixed order.  The encounters are on the correct side
to get the sign of the $\Delta v$ of the planet correct.  The distance
of the encounter from the planet is large enough so that the asteroid
will come back out again after the encounter.  The successive
encounters arranged so that energy and angular momentum are
transferred in the correct proportions in two senses.  The first sense 
is that the asteroid should always have its perihelion inside the
orbit of Venus and its aphelion way out.  The second sense is that
energy and angular momentum are exchanged in the correct ratio among
the three planets.  Because of conservation of energy and angular
momentum, these two conditions will agree.

When the asteroid is very far out very little $\Delta v$ is required
to adjust the phase of its next planetary encounter.  In the limit of
the asteroid going to infinity, zero $\Delta v$ is required.  Because
there is only one encounter per trip, only one condition has to be
satisfied.  The adjustment can take into account the inclination of
the orbit of the planet.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Differential equations}
\label{sec:diff}

It is interesting to see how $r_2$ and $r_3$ vary as $r_1$ is varied
but keeping the total energy and the total angular momentum of the
three planets constant.  As usual, we preserve circular orbits.

Conservation of energy gives
\begin{eqnarray}
  \label{eq:diff1}
  0 &=& d\mathcal{E} \nonumber \\
    &=& d\mathcal{E}_1 + d\mathcal{E}_2 + d\mathcal{E}_3 \nonumber \\
    &=& d({\mu_{\odot} m_1 \over 2r_1}) + d({\mu_{\odot} m_2 \over 2r_2})
 + d({\mu_{\odot} m_3 \over 2r_3}) \nonumber  \\
    &=& {\mu_{\odot} \over 2}(m_1r_1^{-2}dr_1 + m_2r_2^{-2}dr_2 +
    m_3r_3^{-2})dr_3). \nonumber
\end{eqnarray}
%
Thus 
\begin{equation}
  \label{eq:diff2}
  m_2r_2^{-2}dr_2 + m_3r_3^{-2})dr_3 = -m_1r_1^{-2}dr_1.
\end{equation}

Conservation of angular momentum gives
\begin{eqnarray}
  0 &=& dh \nonumber \\
    &=& dh_1 + dh_2 + dh_3 \nonumber \\
    &=& d(m_1{\mu_{\odot} r_1}^{1 \over 2}) + d(m_2{\mu_{\odot} r_2}^{1 \over 2})
 + d(m_3{\mu_{\odot} r_3}^{1 \over 2} \nonumber \\
    &=& {1 \over 2}\mu_{\odot}^{1 \over 2}(m_1r_1^{-{3 \over 2}}dr_1
                      + m_2r_2^{-{3 \over 2}}dr_2
                      + m_3r_3^{-{3 \over 2}}dr_3) \nonumber
\end{eqnarray}
%
Thus
\begin{equation}
  \label{eq:diff3}
  m_2r_2^{-{3 \over 2}}dr_2
                      + m_3r_3^{-{3 \over 2}}dr_3) =
-m_1r_1^{-{3 \over 2}}dr_1.
\end{equation}
%
We can solve (\ref{eq:diff2}) and (\ref{eq:diff3}) for $dr_2$ and
$dr_3$, giving

\begin{equation}
  \label{eq:diff4}
  dr_2 = - {m_1 \over m_2}{{r_1^2 - r_1^{-{3 \over 2}}r_3^{-{1 \over
          2}}}
\over {r_2^{-2} - r_2^{-{3 \over 2}}r_3^{-{1 \over 2}}}}dr_1.
\end{equation}

Since the 2 and the 3 subscripts play symmetrical roles, we have

\begin{equation}
  \label{eq:diff5}
  dr_3 = - {m_1 \over m_3}{{r_1^2 - r_1^{-{3 \over 2}}r_2^{-{1 \over
          2}}}
\over {r_3^{-2} - r_3^{-{3 \over 2}}r_2^{-{1 \over 2}}}}dr_1.
\end{equation}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Remarks and Acknowledgments}
\label{sec:ack}

Consider the possibility that introducing an arbitrarily small tame
asteroid could throw a planet out of the solar system.  This would be
a different kind of instability than those that have been so far
considered.  It would be of mathematical interest to prove that a
system consisting of a sun and three planets could be disrupted by a
single arbitrarily small tame asteroid.

Notes: 

1. Another way would use multiple asteroids from the Kuiper belt or even
the Oort cloud.  Maybe no Jupiter or Venus, and each asteroid makes a
single pass and is kicked out of the solar system.  Many can be en
route at once.  I suspect several times the mass of Mars will be
required.  specifically in the ratio of the escape velocity from a
solar orbit at Mars distance from the sun to the Mars escape velocity.
A single Jupiter pass may improve the numbers.

2. It looks like a lot can be worked out by considering single
collisions.  For example, suppose an asteroid from far out has an
elastic collision with Mars.  It will take energy and angular momentum 
from Mars.  I suppose that if it were to collide again with Mars it
would tend to give back what it took.  How does this modify if we have 
an intermediate collision with Jupiter or Venus?  In the end we need
both, but let's see what happens with one.

3. It may be a lot easier to move Mars than the above considerations
suggest.  The solar system is a lot more chaotic, even in the short
term than was thought 10 years ago.  I found \cite{Portercvit05} very
informative about exotic orbits as applied to spacecraft and as
observed and calculated for the short period comet Oterma.  Whether
this would help move Mars would require difficult mathematics and
computations, at least by present standards.  Our descendants my find
the matter obvious.


Tom Costello commented usefully on an early version of this article.
I have learned from \cite{Kory01}, and its ideas have triggered
further thinking, as has discussion with Don Korycansky and Greg
McLaughlin.  R. William Gosper used Macsyma to solve the quartic
equations.


\bibliography{/u/jmc/1/biblio}
% Its full name is /u/jmc/1/biblio.bib
%\bibliography{biblio} % Tom worked some magic.
% for use with Alpha on Powerbook 3400
\vfill
\begin{latexonly}
{\tiny\rm\noindent /@steam.stanford.edu:/u/jmc/w01/mars.tex: begun Wed Feb  7 16:55:34 2001, latexed
\today\ at \theTime}
\end{latexonly}
\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Related files:


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Leftover material:



