Date: Tue, 21 Nov 1989 17:14-EST
From: space-tech-request@cs.cmu.edu
To: "~/st/lists/stdigest"
Subject: Space-tech Digest #41
Contents:
Marc Ringuette Space Elevators
Michel van Roozendaal Re: Space Elevators
Marc Ringuette Re: Space Elevators
Marc Ringuette Re: Space Elevators
Michel van Roozendaal Re: Space Elevators
Marc Ringuette Re: Space Elevators
Christopher Neufeld Re: Space Elevators
Hans Moravec Re: Space Elevators
Marc Ringuette Re: Space Elevators
Marc Ringuette Re: Space Elevators
Christopher Neufeld Re: Space Elevators
------------------------------------------------------------
Date: Mon, 20 Nov 1989 11:53-EST
From: Marc.Ringuette@DAISY.LEARNING.CS.CMU.EDU
To: space-tech@cs.cmu.edu
Subject: Space Elevators
Here's some mail I've recently been exchanging with some folks about space
elevators and tethers. It's pretty long (600-odd lines).
But first, I'll explain for those of you who haven't seen these ideas:
Space elevator:
Attach a cable to the earth, pay it out past the geosynchronous
point, and attach a counterweight. It hangs there, and you can
winch yourself into orbit up the cable.
The cable has to be strong; stronger than current materials. But
with a tapered cable (thickest in the middle where the most tension is)
it's within a factor of 10-50 of current materials.
Rotating tethers (pinwheels)
Put a tether in low earth orbit, and spin it so that the end closest
to the earth has zero relative velocity with the surface, and the far
end has twice orbital velocity. It looks like the spoke of a wheel
rolling around the earth in low orbit.
If you grab the low end when it approaches the surface, and let go
at the top, you've just been given twice orbital velocity, and will
zoom out into the solar system.
The strength requirements aren't as high; they're probably within
the strength of some current materials. However, there are plenty
of othe problems.
Non-rotating tethers in low orbit
If you want to de-orbit a shuttle from a space station, you can
attach a cable and pay it out, so that the shuttle hangs closer to
the earth. As you pay out the cable, gravity creates tension; the
station is sped up and the shuttle slowed down. The net effect is
that the station is boosted into a higher orbit, and the shuttle needs
less fuel to de-orbit.
This is a practical idea, and is being tested right now.
------------------------------
[ This message was from sci.space. --Marc ]
From: ESC1759@esoc.profs (Michel van Roozendaal ECD)
Subject: Re: Space Elevator
Date: 14 Nov 89 01:52:57 GMT
Space Elevators (or: Space Tethers...)
Your article is based on the initial concept proposed by Tsiolkowsky
back in 1895; he calls is the "Equatorial Tower". I quote:
Upon ascending such a tower gravity decreases gradually; at an altitude of
34,00 werst (Russian pre SI; 1 werst=1067 meters) gravity is totally
eliminated. For that reason at a still higher altitude it is displayed
with a force directed away from the critical point; the direction is
reversed so that mans head faces the Earth.
An other Russian involved in similar work is Y.N. Arsutanov; in 1959
he proposed the Space Funicular. Other people with similar proposals are
Isaacs (1966; the sky hook), Colombo, Grossi etc (Tethered Satellite System,fro
m 1974)
In 1979 Arthur C. Clarke wrote "The fountains of paradise", in which
Dr. Morgan builds an orbital tower on the island Taprobane (=Ceylon).
The story is placed in 2142.
In all kinds of variations one can use the original concept. Most of these
concepts are referred to as to TETHERS. I will not list them all here,
because I found I never read long stories on this net myself.
One of the applications which is think is particularly nice is the
deorbit of the Space Shuttle from a Space Station using a long cable
(=tether). Rather than braking the shuttle using rocket engines,
and thus consuming propellant, we connect the station with the shuttle
on a tether. The will be given an initial displacement (Station above the
shuttle), due to gravity gradient forces they will separate; i.e. the
shuttle will go down, and the station will go up. Gravity Gradient:
In the centre there is an equilibrium between the centrifugal and the
gravity force. Closer to the Earth, the gravity force is larger, farther
away, the centrifugal force is higher.
Okay, due to the fact that both vehicles have the same initial velocity,
but are now in different orbits on different altitudes, the station
will lag behind the shuttle. Trick: the gravity gradient forces are
pointing towards/away from the centre of the Earth, they will accelerate
the station, and slow down the shuttle in such a way the shuttle and the
station are both on the same local vertical. In this process angular
momentum was transferred from the shuttle to the station. We can now
disconnect the shuttle from the tether, and start reentry, having saved
a considerable amount of fuel.
Dynamic behaviour is a big problem; can we control the (in the case above
50 km-) long Tether?? This should be demonstrated with the TSS (Tethered
Satellite System) mission in march 1991 (STS 46 with Eureca).
Coming back on an other issue you raised; the material which we can use
to build a tower/elevator/tether.
In view of our applications a nice property in order to compare materials
is the free breaking length in a 1-G environment. This is the length
of a cable of a certain material when it will collapse under its
own weight, without tapper/1G. Remember; the free breaking length
of any material is infinite when we apply tapper.
Lets list a few materials:
Steel 37 km
Glass 76 "
Nylon 92 "
Carbon 171 "
Aramide 193 "
HP-PE 336 "
So we talk about strength per density; we need in space a strong,
light weight material.
The best candidate is the High Performance Poly Ethylene fibre. This
material is very simple in structure, but all (No; many of) the molecules are
aligned in the direction of the load.
The only HP-PE fibre soon in production is DYNEEMA SK-60 from the Dutch
firm DSM-Stamicarbon. (Pilot plant in the Netherlands)
An aramid fibre will be used in the TSS mission (in this case Kevlar).
This material is as strong but weighs more:
Strength Density
Aramid Fibre 2 GPa/density 1.44 kg/dm3
HP-PE 3 " 0.97 "
Drawbacks are for a HP-PE; exposure to atomic oxygen/high temperatures.
Other hazards for all possible materials and applications: Debris and
micrometeoroids, which can cut the tether.
That's it for the moment; I realise this posting is to long, sorry
for that.
I did my thesis work on tethers, so if somenbody wants extra info
(and you think I might be able to help you), ask
me. I can also send long lists with references.
One example: Guidebook for analysis of Tether applications, Joseph A. Carrol,
Martin Marietta corp, march 1985. (Contract RH4-394049O)
^-----------------------------------------^----------------------------------^
^ Michel van Roozendaal ^ EARN: ESC1759 at ESOC ^
^ c.o. European Space Operations Centre ^ ^
^ Robert Bosch strasse 5 ^ valid till 20 december 1989 ^
^ 6100 Darmstadt FRG ^ ^
^ tel. (0)6151-886376 ^ ^
^ (0)6151-595725 ^ ^
-------------------------------------------------------------------------------
-> ESA does very little work on TETHERS, they certainly don't pay me for it <-
*******************************************************************************
------------------------------
Date: Tue, 14 Nov 1989 23:15-EST
From: Marc.Ringuette@DAISY.LEARNING.CS.CMU.EDU
To: ESC1759@esoc.bitnet@vma.cc.cmu.edu (Michel van Roozendaal ECD)
Subject: Re: Space Elevator
Michel,
I have a couple of questions about using tethers for orbital maneuvers -
- Has anybody done any simulations on dynamic behavior of tethers? Is
some sort of active control a decent option?
- Is any short of debris shielding practical? (for instance, have a thin
protective layer which vaporizes debris before contact)
- Presumably you've heard about rotating tethers (pinwheels) in low orbit.
They raise a couple of questions.
- Do you know how accurately we can rendezvous with the lower end?
Can you see any way to have the rendezvous be with an aircraft or
other craft in the very high atmosphere?
- What's a good technique to reboost?
- What do you do with the payload once it's going twice orbital velocity?
If you have interesting answers to any of these, that would be great.
-- Marc Ringuette // CMU CS Dept, Pittsburgh // Internet: mnr@cs.cmu.edu
------------------------------
Date: Wed, 15 Nov 1989 20:12-EST
From: Marc.Ringuette@DAISY.LEARNING.CS.CMU.EDU
To: ESC1759@esoc.bitnet@vma.cc.cmu.edu (Michel van Roozendaal ECD)
Subject: Re: Space Elevator
I just came up with two ideas about how to handle the problem that for a
pinwheel, the payload comes out with twice orbital velocity and zooms out of
orbit.
1. Climb up to the center of the pinwheel, then let go.
2. Ride it to the top, then hook onto another tether that's flying
by, and use it to slow you down. It actually slows you down
_too much_, so you have to use a third to speed up some again.
-- Marc Ringuette // CMU CS Dept, Pittsburgh // Internet: mnr@cs.cmu.edu
------------------------------
Date: Thu, 16 Nov 89 10:50:59 SET
To: Marc.Ringuette@daisy.learning.cs.cmu.edu
From: ESC1759%ESOC.BITNET@VMA.CC.CMU.EDU (Michel van Roozendaal)
Comment: File TETHER SCRIPT A1
Hi Marc,
Thank you for your reply, I'll try to give you some info. Let me first
explain my background, so you can place what I say in the right context.
At did my M.Sc. thesis work on tethers (in fact the ir. or ingenieur
degree, as it is called in the Netherlands) at the aerospace
department of the Delft University of Technology. I was virtually working
on my own, supported only by some people at ESTEC. (the technology
centre of ESA in Noordwijk). I visited the second international
conference on tethers in Venice, october 1987. Since I started
working here at ESOC (operations centre of ESA in Darmstadt) I have
not worked on tethers anymore (I do rendezvous at the moment). So my
info is a bit old. (I completed my studies in May 1988)
My address here remains valid till 20 December 1989, after which date
I will move to France (Fontainebleau), where I probably can get a new
EARN address. So if could put me on your 'Space-Tech' list, that would be
great; I'll also be interested in back issues etc.
Your questions:
1. Tether Dynamics: as you know one of the key issues for tether
applications. So there are MANY simulations done. Mainly in preparation
for the TSS missions. Still people are very concerned whether the tether
can be stabilised. For TSS-1 we got a 20 km tether, with (as I remember
well) a maximum tension (20 km tether deployed) of about 40 N. That's
very little; the subsatellite will use additional thrusters.
But about simulations: I myself ran a simulation on a PC (using
simulation software; ISIM) which used a simplistic Langragian
model, without airdrag but with an optimal control strategy.
It suggested that it should be possible to control the dynamic
behaviour of a tether.
In the real world, different groups are building dynamic models,
which are supposed to be as accurate as possible. I guy named
Gordon Gullahorn seems to be busy to compare these models, and
to compare their behaviour in different cases. (i.e. feeding these
models with the same input.) He works at the Harvard Smithsonian
Centre for Astrophysics, Cambridge, Mass. One of the models I
know is TETHSIM, built by Aeritalia.
Conclusion: TSS-1 will show whether we can trust these models.
2. Tether shielding; I did some work on possible tether materials
and coatings etc. I never heard about a shielding which could
vaporise the debris, and I don't think it is possible. This because
the velocity difference between a particle in orbit and the tether
will be very large. Impactvelocities above the speed of sound of tether
will become explosive. (impact > about 9 km/sec )
Normally it is assumed that if the edge of an incoming particle comes
in the centre of the tether, the tether will be cut.
This is very limiting; a classical example of a single point failure.
For the moment I found that micrometeoroids are more dangerous then
debris, mainly because very small dust particles can become fatal for the
tether.
If you apply a jacket, you will reduce the population of potentially
fatal particles, because they are unable to penetrate deep enough in the
tether. But the mass of the tether increases very rapidly.
I found: for 2.5 mm diameter at 500 km: 0.09 cuts/year
The TSS tethers will use teflon coating, which is a very weak material,
but which will consume a high percentage of the weight of the tether.
I think the safest solution should be to use a structure of the
form of a ladder; with two legs, and cross-connections every so
many meters. This limits the chance of fatal cut of the tether.
Problem is how to retrieve this complex structure.
Conventional Tether: Fatal hit
------------------ * ---------------------------------------
Ladder Tether
Start:
------------------------------------------------------------
| | | | | |
------------------------------------------------------------
Non fatal hits:
------- * ------------------------------ * -------- * ------
| | | | | |
------------------ * ---------------------------------------
Fatal hits:
------------------------- * --------------------------------
| | | | | |
------------------------------- * --------------------------
The scenario to save the Shuttle when during a tethered deorbit the
cable is cut is to use scavenging tanks. The fuel 'scavenged' during
the tethered deorbit is pumped into tanks, wich are mounted at the
lower (shuttle) end of the tether. Suppose the tether would be cut
during the deorbit, the shuttle has enough fuel onboard to complete
reentry. Once it is on the desired low altitude, the fuel in the tanks
will be retrieved, and stored at the station.
3. Rotating tethers; (I don't know too much about these; I can
recommend a paper by Jerome Pearson which gives an overview:
Space Elevators and space tethers; rotating structures in space,
presented in Venice 87 This paper contains many references)
Accuracy: a matter of time available, and the velocity difference between
the chaser and the target. Use differential GPS/Video cameras.
Paper: Edmondo Turci (aeritalia) Transportation of payloads from
sub-orbital trajectory to the Space Station using long tethers. This
is about docking a payload launched from a B-747 with the lower
end of a non-rotating tether connected with the station.
Phasing seems a problem to; at the moment you pick up payload-1 you have
to launch payload-2. Otherwise (for heavy payloads) you are out of
balance. But the first time you use the tether you can not avoid this.
In general, I think active control is required, what will consume
quite a lot of propellant.
I think the aim is to launch the payload into some sort of orbit once
it is at the far end (i.e. away from the Earth) of the tether. Again,
phasing?
What one should work out is a rotating tether, with a certain length,
and the middle in a certain orbit, so that the velocity/orbit after
release is reasonable.
Anyway; I am not really familiar with the concept, so I wont pollute
the system with my uneducated ideas. (In fact, I am a bit sceptical..)
I just received your mailing concerning your answer to Chris Neufeld:
Where is the advantage of the fact the tether is rotating? What do
you do with it? I mean, the advantage of the rotation is that
you use this to get the mass up. If you crawl up, you have to propel
yourself?
If you're interested, I'll briefly explain two funny things I came
across during my thesis work.
1. After having attended a presentation of D.A. Arnold concerning
dumbbell tethers (= 0----0 ) having positive orbital energy. He
found this for one case as a sort of unexpected funny result.
At the same time I was doing simulations with very long tethers, and
I had many sort of similar results. Anyway, I sort of proofed (I am
bad in mathematics) that in a general case with a central attracting
body, two masses connected on a massless tether in a circular body,
there are orbits possible with positive orbital energy. (Remember;
closed orbit, Energy < 0, Parabola = 0 , Hyperbola > 0 ). Since the
lower mass must be orbiting above the surface of the central body,
the tether must be very long. If you assume a point mass as central
body, any length of the tether is possible.
Standard case , showing orbital energy of a mass in circular orbit.
^ | Radius ->
| |------------------------------------------------------
E | *****************
| *********
| ****
| **
| *
| *
| *
Case with two masses connected on a tether, in a stable orbit.
^ |*
| |*
E |*
|*
| * Radius ->
|-*-----------------------------------------------------------------
| * *****************************
| * *****
| * *
| * *
| * *
| * *
| **
There are many funny things involved here; e.g. when you deploy a tether,
the CG (Or whatever you want to call it) of the two-mass system
does *NOT* stay in the same orbit, but will lower. (Under the assumption
of conservation of angular momentum.)
2. A also proposed what I call the "Celastial Funicular", which is
a variation on all the proposal for lifts etc. A used the original
idea of building one or two large towers on the moon. You can
consider the moon in the same way as the space station; a large satellite
in orbit around the Earth. Off course, due to the high mass of the moon,
the system is more complex; the Lagrange points etc. My idea was to build
a big tower on the point centre of the 'Light' side of the moon,
pointing towards the Earth. we can now discriminate two parts of the
tower; 1. Moon-L1 and 2. L1-Earth. On part one the net force acting
on the tower is towards the moon, On part two the net force acts
towards the Earth. The trick is: if you have masses in baskets, which
are picked up on the moon, and released Far beyond L1, the system is
rotating on itself. The masses on the second part of the tower propel
the masses on the first part.
* = mass L/7=baskets
|* 7 7 7 7 7 *
|=============L1================= EARTH
| L* L* L* L* L*
/ <-- <- -> --> --->
Force
Moon
The system is rotating counter-clockwise. The empty baskets go back
to the moon, and will be loaded with new masses.
Maybe you can do something with it; I hope I can make things clear.
Sorry, I have to write everything in a hurry because I got some
real work waiting here on my desk. Do ask me about the things I did not
make clear enough. If you want I can send you hardcopies of the papers
I mentioned, and something I wrote about the things I discussed here.
Are you in some way involved in tetherwork, or is it just a hobby?
Curious the hear from you, regards;
Michel van Roozendaal
c.o. European Space Operations Centre
Robert Bosch Strasse 5
6100 Darmstadt FRG
(0)6151-886376
(0)6151-595725
EARN: ESC1759 at ESOC
------------------------------
Date: Sat, 18 Nov 1989 02:14-EST
From: Marc.Ringuette@DAISY.LEARNING.CS.CMU.EDU
To: ESC1759%ESOC.BITNET@VMA.CC.CMU.EDU
Subject: Re: Tethers
Hi, and thanks for the ideas.
Why rotating tethers? Mainly to match velocities. How are we to get payloads
from a point on Earth out into LEO? If our tether is in LEO, it's whizzing
by at 7.8 km/s, so we can't just grab on. But if the tether is rotating in
such a way that the low end has zero velocity, we can just latch on. This
benefit is separate from any benefit you might get from using the tether to
'throw' the payload.
Your moon-through-L1 conveyor is nifty; I get the idea. However, it is likely
to be an overly complex solution to the problem of getting stuff off of the
moon.
-- Marc Ringuette // CMU CS Dept, Pittsburgh // Internet: mnr@cs.cmu.edu
------------------------------
Date: Wed, 15 Nov 89 21:33:14 EST
From: Christopher Neufeld
To: Marc.Ringuette@DAISY.LEARNING.CS.CMU.EDU
Subject: Re: More about a spacial lift
Hi Marc,
You write:
>
> I have a couple of nontrivial questions about rolling skyhooks for Earth.
> I wonder if you've seen any answers that might work.
>
> - Is any short of debris shielding practical? (for instance, have a thin
> protective layer which vaporizes debris before contact)
>
It is certainly practical. The mass of a shield increases as the radius
of the tether, while the mass the tether can support increases as the
square of the radius. The impact of a shield which doesn't support the load
can thereby be reduced to an arbitrarily small amount. It may not be
necessary to shield the tether. Grit might sandblast the thing slowly, but
the tether has a small cross-section and moves in the same direction as
most of the grit, so the problem may not be so bad. Bigger things can't be
blocked by a conceivable shield.
We may be able to get rid of grit with mirrors :-). I think my mailer
has been having problems, because I've been unable to get answers from
Steve about using one of his sails as a mirror.
> - Do you know how accurately we can rendezvous with the lower end?
> Can you see any way to have the rendezvous be with an aircraft or
> other craft in the very high atmosphere?
>
From Moravec's article in _The Endless Frontier_: the intercept should
be about 50km up. Safety considerations keep it from being much lower,
since the tether bobs up and down with each payload. At 50km, air-breathing
craft are probably unlikely, but a completely reusable aircraft with
hydrogen/LOX engines which could meet the tether should be quite feasible
with current technology.
> - What's a good technique to reboost?
>
Presumably the tether is built non-rotating. Some mechanism would have
to spin it up when it's completed. This might be used to spin up the tether
also. The technique I favor for spinning it up (or down) is this: put a
ballast mass on the two arms of the tether. On the tether which is going up
as seen from the earth, pull the mass in. On the one going down, send the
mass to the tip of the arm. The earth exerts a torque on the tether,
spinning it up. The energy to spin it up comes from the motors which pull
the mass to the centre of the tether on the up stroke. To increase the
orbital velocity a thruster at the hub will probably be necessary.
See also the description below for a lunar tether.
> - What do you do with the payload once it's going twice orbital velocity?
> It's great for planetary probes, but how can you get into Earth orbit?
>
You can release the payload at any point in the stroke of the tether.
If you want to go into a high earth orbit, let go of the rope partway
through the stroke, then do an apogee burn to circularize the orbit.
> I wrote:
> > - What do you do with the payload once it's going twice orbital velocity?
> > It's great for planetary probes, but how can you get into Earth orbit?
>
> How about this idea: Once you've latched on to the end of the tether,
> climb up to the center, then let go. You're in LEO.
>
Yes, though unfortunately you're also sharing the orbit with the
tether, and space is a bit dangerous around there. Better is to let go at
another part of the tether, and make a thruster burn to separate the orbit
from that of the tether.
> This "winch yourself to the center" plan has the problem that the payload
> spends a longer time being accelerated. If the payload is a person and the
> acceleration is, say, 5g, it would be nicer if you could accelerate only for
> the 5 minutes it takes to get halfway round, not the longer time it takes
> to traverse a thousand km of cable to get to the center.
>
Taking from Moravec's article again: a tether made of this hypothetical
graphite whisker could be built with a taper of ten, a mass ratio of fifty,
a length roughly equal to that of a transatlantic telephone cable (but
smaller in cross section and lighter). Orbital period is two hours, and the
tether is vertical every twenty minutes. At pickup, the tip undergoes a
acceleration of 1.4g with negligible horizontal velocity. This makes
rendezvous tricky, but not impossible. Anyway, sustained accelerations of
2.4g or less shouldn't be a problem.
By the way, a lunar tether could be made of Kevlar with a taper of 4
and a mass ratio of 13. The spin-up/down scheme for a lunar tether is nice.
A mass launcher can do both by firing at the right time and having the
tether catch the payload. The mass driver can be built with accelerations
of 1000g making it virtually useless as a cargo transport and unthinkable
as a passenger vehicle, but it would be able to spin up the tethers which
could then do the actual lifting. This strikes me as a rather elegant
solution to the problem.
> Another choice is to ride the tether to the top, then hook onto another
> tether that's flying by, which slows you back down. I'm just working it
> out, but I think you need 4 stages to reach LEO given this sort of setup.
>
The cable covers a few thousand kilometres of height for Moravec's
terrestrial non-synchronous skyhook. It should be possible to step off at
the height of LEO and circularize the orbit even with a low thrust thing
like the IUS. Granted, it would be nice to do the whole thing using only
tethers, but because classical gravitational orbits around a monopole
source are closed you'd be hard pressed to keep the orbits of the tether
and the payload from intersecting at some later date. This could have
unfortunate consequences.
> -- Marc Ringuette // CMU CS Dept, Pittsburgh // Internet: mnr@cs.cmu.edu
>
By the way, I checked up again on lunar synchronous skyhooks. They are
possible, using Lagrange points instead of geosynchronous orbit, but the
cable would be 300000km long through L1 and 550000km long through L2. Mass
ratios would be about 100 and 200 respectively with currently available
bulk materials. I think a small non-synchronous tether is more likely than
a bridge which goes 3/4 of the way from the moon to the earth.
As a final note, have you ever seen Charles Sheffield's scheme for
anchoring an earth-synchronous skyhook? There's something which would
terrify the Christics!
Christopher
------------------------------
Date: Wed, 15 Nov 1989 21:42-EST
From: Hans.Moravec@ROVER.RI.CMU.EDU
To: Marc.Ringuette@DAISY.LEARNING.CS.CMU.EDU
Subject: Re: Space Elevator
A pinwheel boost you to somewhat over twice circular orbital velocity
if you hold on for one half turn. Since escape is sqrt2 times circular,
that throws you far (to Saturn orbit with an "optimum" size earth
version). But, if you let go sooner, you can be left in an elliptical
orbit with any apogee you choose, that can be circularized later at
your leisure.
Btw, a synchronous lunar skyhook can be built through the L1 and L2
points, pointing directly towards earth, or directly away from it.
The midpoints on both sides are about 60,000 km from the lunar surface.
--Hans
------------------------------
Date: Tue, 21 Nov 1989 16:44-EST
From: Marc.Ringuette@DAISY.LEARNING.CS.CMU.EDU
To: space-tech@cs.cmu.edu
Subject: Re: Space Elevators
There seems to have been some confusion on the necessary lengths for
the lunar skyhooks. The 60,000 number sounds correct for the L1 side,
since the Earth is pulling on the tether. On the L2 side, it seems to
me that it should be much farther out, however.
--Marc
------------------------------
Date: Tue, 21 Nov 1989 16:44-EST
From: Marc.Ringuette@DAISY.LEARNING.CS.CMU.EDU
To: space-tech@cs.cmu.edu
Subject: Re: Space Elevators
OK, the actual poop on lengths needed for lunar space elevators is like this.
If we assume the cable needs to go out through L1 or L2 and just a little bit
farther, and be terminated by a large counterweight, then the lengths are just
the distances to L1 and L2 plus a little:
L1 - 58500 km from the moon's center
L2 - 63900 km from the moon's center
The bigger numbers, on the order of 300,000 km, come from assuming a tapered
cable all the way out, terminated by a payload-sized counterweight.
--Marc
------------------------------
[ Here are the relevant calculations from Chris Neufeld, which he sent
to me... -- Marc ]
Date: Tue, 21 Nov 89 15:57:29 EST
From: Christopher Neufeld
To: Marc.Ringuette@DAISY.LEARNING.CS.CMU.EDU
Subject: Length of lagrange stabilized synchronous lunar tethers
Marc,
Here are my calculations on the cable lengths needed.
******
Variables:
G = gravitational constant, 6.67e-11 N m^2/kg^2
M1 = mass of the moon, 7.35e22 kg
R1 = radius of moon, 1.738e6 m
M2 = mass of earth, 5.977e24 kg
r = mean earth-moon distance, 3.84e8 m
w = angular frequency w = 2pi/T, T = sidereal month. w = 2.6617e-6 /s
x,y = distances measured from the centre of the moon, in metres
Y = Tensile strength of the cable in N/m^2
p = Density of the cable in kg/m^3
A particle on the earth-moon line feels an effective acceleration from
the masses and from the centrifugal pseudo-force. First I will consider the
inner (first) Lagrange point.
acc(x) = (G M1 / x^2) + w^2 (r - x) - (G M2 / (r - x)^2)
Setting this equal to zero gives the distance from the centre of the moon
to the L1 point. d1 = 5.820e7 m.
Now, obtain an integral equation for the area of the tether as a function
of distance from the centre. At a given height, the cable has to support
the apparent weight of the cable below it plus a constant amount for the
payload. Notation: int^y means a definite integral from some lower limit
to y.
The force which the cable can exert is Y * Area.
Y * A(y) = int^y { p * A(x) acc(x) } dx + Const.
Differentiating both sides with respect to y, we get a very simple first
order differential equation.
A'(y) = p / Y * A(y) * acc(y)
The solution of this is trivial
A(y) = Const. * exp {p/Y * int^y acc(x) dx }
And int^y acc(x) dx is explicitly:
-G M1 / y - w^2 (r - y) / 2 - G M2 / (r - y)
Evaluating the area function at y=radius of moon gives the anchor point
area. A(anchor) = # * exp( -4.38e6 p/Y) where p/Y is measured in units
of s^2/m^2. Doing the same thing at L1 gives A(L1) = # * exp( -1.68e6 p/Y)
where the constant in front (#) is the same in both cases.
The taper is the quotient of these two numbers,
taper = exp(2.7e6 p/Y)
Taking values of p/Y inferred from the table of hanging lengths included in
the article by Michel van Roozendaal on this list, we get the following
tapers:
Nylon 20
Carbon 5
Aramide 4.2
HP-PE 2.3
Now, to find the balance length of the cable (length for which is will be
stationary without a ballast and without an anchor), we find another value
of y which gives the same area as the anchor. Newton's method of successive
approximations gives a length of 292000 km, or about 3/4 of the way from
the moon to the earth.
CAVEAT: The possibility of rounding errors in plugging in the numbers
limits the accuracy of all the number here. Don't expect better than 5%
accuracy.
***
This calculation can be repeated for the L2 point. To within the
precision I used (3 digits on the input parameters), all numbers are the
same, save length, which is 678000 km, or more than 1.5 times the
earth-moon distance.
--
Christopher Neufeld....Just a graduate student | "Space probe may be
cneufeld@pro-generic.cts.com | Doomsday machine!"
neufeld@helios.physics.utoronto.ca | -Toronto Star article
"Don't edit reality for the sake of simplicity" | on Galileo 19/11/89
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End of Space-tech Digest #41
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