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C00002 00002	THE FEASIBILITY OF INTERSTELLAR TRAVEL
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THE FEASIBILITY OF INTERSTELLAR TRAVEL


	Optimists have proposed many schemes for interstellar travel,
usually aimed at reaching nearer stars within a human lifetime, but
these schemes usually involve extrapolations of present science.
Pessimists, finding flaws in these schemes have often concluded that
interstellar travel is forever infeasible.  The purpose of this article
is to show that interstellar travel is entirely feasible based on
present science and quite small extrapolations of present technology,
provided travel times of several hundred to several thousand years
are accepted.  Naturally, no-one is going to start journeys that
take a thousand years within the next few hundred years unless fleeing
a danger, because a faster method of travel may be discovered
that will permit an earlier arrival with a later start.
However, since interstellar travel in a few thousand years is
feasible with even our present technology and
our solar system will support life
for several billion years, it is hard to imagine that journeys of many
thousand years will not be undertaken over the course of this time
unless all of mankind comes under a dictatorship that forbids it.

	The technology presumed is using a nuclear fission reactor
to generate electricity which is used to expel a working fluid
at an exhaust velocity optimized over time for the journey being
undertaken.  If the exhaust velocity is chosen too low, the mass
ratio of the rocket system is unacceptable, and if the exhaust
velocity is too high, very little thrust can be obtained given
plausible assumptions about the power that can be handled.  In
many cases, the exhaust velocity used should vary during the journey.

	We shall derive formulas for the time required to  accomplish
an interstellar journey of distance  s using the above technology in an
optimal way.  We shall assume that  the  performance  of  the  system
(reactor  +  rocket)  is characacterized by a number  p  equal to the
power the  system  can  handle  per  unit  mass  of  apparatus.   The
plausible  values  of   p   are  between  one  watt/kilogram and 1000
watts/kilogram.  Since the time  T  turns out  proportional  to   p-1/3 ,
this  will  only  mean  a  factor  of  ten  in  the  time required to
accomplish a given journey.

	We introduce symbols as follows:

	s = length of journey

	T = time of journey

	t is a time variable

	M = initial mass of the system

	m0 = final mass of the system

	m is a mass variable

	α = M/m0 = the mass ratio

	w is an exhaust velocity variable

	a(t) is the acceleration

	p = power available for unit mass of system

	P is a power variable.

	Conservation of momoentum and conservation of energy give the
following equations:

         .
1)	-mw = ma
                 .  2
2)	P = -1/2 m w .

                                                                   .
	Solving for  w  in 1), substituting in 2) and solving for  m 
gives

3)


	We  now  distinguish  two cases: in a single stage rocket, we
have

4)

expressing  the  fact that the power available is proportional to the
final mass of the system.

	In a continuously staged rocket, we have

4')	P = pm,

expressing the fact that the power available is proportional to the
current mass.  (We suppose that every so often a nuclear reactor or
a rocket is taken out of service, vaporized and expelled as working
fluid).

	In the two cases, we get the equations


5)


and


5')


	Taking  into  account the initial and final masses we get the
following results by integration:


6)


and


6')


	Using  α =M/m   and setting for the two cases

7)	q = p(1 - 1/α)

and

7') 	q = p log α,

we get the following equation valid in both cases:


8)


	Assuming that the journey begins and ends at rest we have


9)


The final distance is given by


10)	s =


from which


11)	s =


follows by integration by parts.

	Our  goal is now to determine the acceleration profile  a(t) 
satisfying equations 8),9) and 11) so that  T   is  minimized  for  a
given  s.  Before doing this, however, we shall treat the simple case
in which we use a constant magnitude acceleration reversed in sign at
the midpoint of the journey.  This assumptions gives from 8) and 11)


12)

and

13)	s =


	Solving for  T  and  a  gives


14)

and


15)	a =

	We shall now labor mightily to optimize  a(t), but the  eager
reader  is  warned  that  this  only  changes  the  co-efficient 2 in
equation 14) to 1.817 which might not be considered worth either  the
mathematics or the engineering.  Well, onward!

	Instead of holding  s  fixed and minimizing  T, we  take  the
equivalent   but  simpler  problem  of  maximizing   s   holding   T 
constant and maintaining the validity of 9).
Introducing Lagrange multipliers, we must hold stationary the integral

	16)

subject to the conditions 8) and 9) for arbitrary variations of a(t).
This gives

	17)

which must hold for arbitrary variations  a(t).  Therefore

18)

	Combining  18) with 8), 9), and 11) gives (if we have finally
gotten the algebra right)


19)


and


20)	T =


	Thus  if we optimize acceleration and use continuous staging,
we get


21)	T =


	As  a  numerical  example,  we let  s = 10   meters ~ 100 light years,
p = 1000 watts/kilogram, and α = e  ~ 3000.  We then get

	T = .9  10   sec = 3000 years.

Remarks:  The  zero  in  acceleration  at the midpoint means that the
calculation is invalid at that time because with constant power, that
corresponds to infinite exhaust velocity.  The actual optimum profile
involves emitting only the waste products of the nuclear reactor near
the midpoint.

	Since  the  time  is  proportional  to  the  2/3 power of the
distance, clearly the formula is incorrect for  long  distances.   It
becomes  incorrect when it recommends exhaust velocities greater that
corresponding to emitting only the  waste  products  of  the  nuclear
reaction.
We can calculate this condition most readily for the case of constant
acceleration and continuous staging.  Solving equations 1) and 2) for
w  gives


22)	w =


which leads finally to


23)	w =


	As a numerical example, choose  α = e  and  p = 1000 again,
and solve for s getting


24)	s =


in MKS units.  The limiting case of a fission reactor using all
fissionable material and expelling only the fission products as
reaction mass and using 200,000,000 electron volts as the energy
of fission gives  s = 1700 light years, i.e. the equations of
this article are valid for journeys shorter than this.  Longer
journeys are energy limited, and the formulas customary for
chemical rockets apply.  If controlled fusion is feasible as
a source of energy and gives more energy per gram of fuel, then
the formulas apply out to correspondingly longer distances.

	The conclusion is that the galaxy could be occupied by
humanity in a time small compared to the time during which our
solar system will support life.


THE SOCIOLOGY OF INTERSTELLAR TRAVEL


	Most discussions of interstellar travel assume a unified
society that decides whether to undertake interstellar travel
taking into account its feasibility and the values of the society.
This is certainly one of the possibilities but there are others
including at least the following:

	1. Humanity continues to be divided into sovereign nations
up into the period in which interstellar travel becomes feasible.
Rival ideologies may lead to competitive expansion into the solar
system and beyond it.

	2. A political group or a religion which is unable to gain
control of a country may sponsor emigration from earth.

	3. Very small groups down to 20 people may pioneer.

	4. The loser in a war or other power struggle may escape.
In particular, the loser in a war to unify the earth may escape.
In my opinion, either side in World War II, feeling doomed to
lose, would have launched an escape expedition if it could.  I
have in mind one in which many people co-operate to let a few
escape in order to preserve an ideology or a leader or a way of
life.

	Quite likely either capitalism or communism, if doomed to
defeat, would launch esacpe expeditions.

	Naturally, it would depend on circumstances whether those
holding power on earth would tolerate interstellar escape or try
to prevent it.

	The time when interstellar travel becomes feasible depends on
the motivation of the travellers.  It is hard to imagine that a
secure group or nation would launch an expedition taking hundreds or
thousands of years as long as waiting a fraction of that time for
more advanced technology would ensure an easier journey and an earlier
arrival.  On the other hand, a group that expected to lose its ability
to launch an escape expedition, might do so as soon as possible.

	How soon is possible?  If we assume that the Shuttle is produced
on schedule and that Shuttles can be bought or Shuttle launches rented,
then a group could launch an expedition before the year 2000 at a cost
between a few hundred million and a billion dollars.  This would be an
absolutely austere expedition as we shall show later.  There will be many
individual fortunes capable of affording this cost, and it is quite likely
that some of them will be politically insecure enough to be motivated to
escape, although as long as escape to other countries on earth is feasible,
there may not be a motivation for interstellar or interplanetary escape.
There are many mass movements and religions that can also raise that kind
of money, and almost any government would be capable of it.

	Perhaps one could conclude that if the world remains sufficiently
pluralistic so that no major way of life is threatened with extinction,
interstellar travel will be postponed until it is clear that waiting
longer to start will not lead to an earlier arrival.  Otherwise some groups
will escape unless they are overwhelmed.  Emigration to interplanetary
space is an intermediate possibility that would probably be preferred
by pioneers and utopian colonists unless the governments on earth refused
to allow independent pioneering.


THE SOCIOLOGY OF THE INTERSTELLAR EXPEDITION


	Suspended animation would obviously make long interstellar expeditions
easier, but there is as yet no scientific guarantee that this is possible, and
in keeping with our commitment to base this paper entirely on present science,
we will not assume it.
Therefore, we will assume a multi-generational expedition whose individuals
don't live longer that we do today.

	Since the society of such an expedition would be quite different from
our present urban culture, one may worry about whether the expedition could
maintain its purpose or even survive.  A number of science fiction stories
have imagined the society relapsing into savagery or a terrorist dictatorship.
The closest analogy in human existence would seem to be isolated tribes on
small islands.  We know that there exist many such tribes which have apparently
maintained cultural isolation in groups of a few tens to a few hundreds for
centuries.  In particular, they have maintained isolation long enough for
their languages to drift to the point where they are incomprehensible to all
their neighbors.  Other isolated tribes seem to have maintained language stability.

	An interstellar expedition has both advantages and disadvantages compared
to these tribes.  First, it has written records and can use computer aided
instruction.  Second, if it has friends on earth, it can maintain radio
contact with them even at interstellar distances with the delays imposed
by the speed of light and with limitations on information transmission rate.
Its disadvantage is that it must maintain a more complex technological
capability than any isolated tribes with the possible exception of the
Eskimoes, who had quite a complicated technology.  Probably ten adult men
can maintain enough capability to maintain a nuclear rocket including fuel
separation and electronics and computers and the life support system.  I
can't prove it couldn't be done by one man with good reference books.
A lot depends on whether the group can be initially selected for intelligence
and energy and the extent to which these qualities can be maintained, i.e.
the mean to which the initially selected group will revert must be rather
high.

	While some human groups of a few tens may have survived 1000 years
of inbreeding, a sperm bank is obviously desirable to maintain genetic
diversity.