Many of us, in a fanciful, Sci-Fi moment, have no doubt thought of the idea of simply extending an elevator up into space as a means by which we could reach orbit. Nearly as many of us dismissed the concept almost as quickly as it came, recognizing that the materials required to create such an elevator would have to be impossibly strong. Now, although it is true that creating a space elevator on earth would truly be a challenge, the idea is not completely useless; it is possible to create such systems on smaller bodies (large asteroids or small moons), and more over, the physics of space elevators is highly interesting. Elevators, if used properly, allow for extreme energy conservation in space exploration, a great advantage when most of the systems in current use are extremely inefficient and wasteful. More over, space elevators allow energy to be drawn from the rotational energy of planetoids, thereby providing access to vast quantities of energy (as will be seen, vast is an understatement) for powering interplanetary travel. As such, despite of difficulties of creating such systems, a detailed understanding and analysis of space elevators is of significant value. Now, the modern concept of a space elevator is that of having a satellite beyond geosynchronous orbit which is tethered to the earth by a cable, up and down which an elevator would travel.
Geosynchronous orbit is special since this is the distance at which an object orbiting the earth would have to travel around the earth at the same rate the earth rotates, the point at which a geosynchronous object’s centripetal force exactly cancels out its gravitational force. If an object is within geosynchronous orbit and is moving around the earth with the same period (time per orbit) as the period of rotation of the earth then its gravitational force would exceed its centripetal force and pull it down. However if such an object were beyond geosynchronous orbit then its centripetal force would exceed its gravitational force and it would be pulled outwards. The design of a space elevator is a balancing act. All the components of an elevator rotate around the earth with the same period as the earth’s rotation since they are physically connected to the surface of the earth, and so, all the components of the elevator below geosynchronous orbit, most of the cable and the elevator, would be pulled down while all the components above this orbit, some of the cable and the satellite, would be pulled upward. Since the forces on a cable used in a space elevator would certainly be immense it is necessary that no unessential forces be acting on the system, and as such the forces above geosynchronous orbit would have to be balanced against those below. To do this the distance beyond geosynchronous orbit, and mass of the satellite, must only exert enough force to support the cable, the elevator at its heaviest plus a small additional force on the earth to stabilize the system. The point at which these forces converge is the point on the cable at geosynchronous orbit. Up or down from this point, the amount of cable exerting force reduces and hence the stress on the cable reduces. It is at the point of geosynchronous orbit that the cable has to be capable of taking the greatest stress. Now, there are two basic designs for a space elevator, the straight cable and the gradiated cable types. The straight cable design involves tethering the beyond geosynchronous orbit satellite to the ground using a cable of uniform thickness and hence strength along its length. Since the greatest stress on a cable is at geosynchronous orbit such a cable’s strength at every point along its length would have to be stronger than the pressure at geosynchronous orbit. The disadvantage of this design is that the cable would be thicker and hence heavier than absolutely necessary. The advantage is that any point on the cable could withstand the pressure at geosynchronous orbit, and so the cable could be designed to move.
Since the cables could move, the force could be distributed over two strands of cable, each with an elevator attached and, by use of pulleys, by moving one cable and hence elevator up the other could be moved down. As such, it would be possible to place electric motors, generators and electrical storage on the ground, allowing the system to be lighter and leaving more room in the elevators for cargo. To determine the minimum cable strength necessary to support an elevator of this design it is necessary to recognize that the minimum requirement for a space elevator is that the beyond geosynchronous satellite must support its own cable, that everything necessary beyond geosynchronous orbit must support everything necessary below geosynchronous orbit. Since the weight of the elevators themselves plus the additional stabilizing force exerted on the earth could be reduced to fit whatever additional strength remains in the cable after that used to support itself, it is only necessary that the cable strength be greater than that necessary for self-support. Now, the amount of weight do to the cable pulling down, which the beyond geosynchronous satellite must support through pressure on the cable is the sum of the forces on the cable from the surface to geosynchronous orbit, at which the force goes to zero. The force at any given point on the cable is the force of gravity plus the centripetal force. The force on the cable do to gravity per unit length is
where
where the velocity v is the distance a segment travels around the earth, the circumference of its orbit 2pr, divided by the time it travels that distance, this time being the period of rotation p of the earth, 23 hours and 56 minutes (4 additional minutes per day are generated by the earth’s rotation around the sun to produce a total of 24 hours). Therefore the centripetal force can be rewritten as
Putting these together gives the total force on a segment
The sum of this force over the length of the cable up to geosynchronous orbit gives
where
Now,
using these formulas the minimum cable strength per cable density,
In order to give a relation between these necessary minimum cable strengths and the strengths of available materials
As can be seen from these numbers, commonly available materials for producing cables prove to be significantly weaker than those necessary to support the pressures of this type of space elevator. The second type of space elevator uses the gradiated cable. As the cable approaches the earth the amount of cable and hence the necessary thickness of the cable reduces. By reducing the thickness and hence mass of the cable so that it is only as massive as necessary it is possible to reduce the force on the cable to its minimum and hence be able to use cables significantly weaker than those necessary for a straight cable elevator. The disadvantage of the gradiated cable is that the cable is only the right thickness to support the elevator at a particular height and so such a cable could not be moved. As a result the elevators for such a cable would have to be designed to climb under their own power. Consequently, the elevator would have to contain the mechanisms to power its climb as well as the mechanisms to draw the energy out of its decent. A straight cable elevator could be reduced to being little more than a cargo hold, and life support if crew were to be transported, whereas a gradiated cable elevator would also have to contain electric motors, generators and energy storage. As a result, the difference between a straight and gradiated cable space elevator is whether the weight is put in the cable or in the elevator. It has been suggested that energy could be transmitted from the ground to a cable-climbing elevator, however this possibility destroys the greatest advantage of a space elevator, the extreme level of efficiency that could potentially be achieved if such a system were designed properly.
Now, in order to calculate
the force on a gradiated cable the forces acting on each segment of the
cable must be summed over the distance from the earth to geosynchronous
orbit. This is the same
action done previously with the straight cable except that, in this
case, the mass per unit length
In
this case the force F cannot be derived directly, rather it is necessary
to solve for the formula
where
R is the distance, measured from the center of the earth, of the point
on the cable of interest. The
solution to this formula is
Although
this formula was derived for the special case
Now, despite the difficulty of finding a cable material strong
enough to be able to create a space elevator, the benefits that an
elevator would have over other means of entering space makes developing
such a material well worth it. There are three main benefits of a space elevator system.
The first and simplest benefit is the potential efficiency of
such a system. This
efficiency comes in two forms. Firstly,
in the case of rocket based systems of reaching orbit the great speed
the vehicle must travel produces extreme amounts of air resistance,
thereby drawing large amounts of energy out of the system.
A space elevator, on the other hand, could travel as slowly as
desired. On the space leg
of the trip to geosynchronous orbit high speed would be desirable so as
to shorten the journey, however during the short atmospheric leg, slow
speeds would reduce the lose of energy to air resistance, or even render
this effect negligible. The
second form of efficiency is in the ability to use more efficient
sources of propulsion. In
the case of the straight cable design of elevator this advantage would
reach its greatest extent. Since
a straight cable system would act by moving the cables, the engines that
produce this effect could be located on the ground.
As such, such engines would have no size or weight restrictions,
and could therefore be brought to their maximum practical efficiencies.
The second main benefit of
space elevators is their energy conservation ability. In a standard elevator this takes the form of the
elevator-counter balance system, in which the energy lost as the
elevator rises is countered by the energy gained by the counterweight as
it drops. Although this
system is highly advantageous for elevators that travel short distances,
for a space elevator the weight of the elevator and counter balance
change greatly over the distance traveled.
For
instance, for the two elevator, each of the same mass, straight cable
system shown above, an elevator starting at the ground has the full
force of gravity acting on it while the counter elevator at
geosynchronous orbit has no net force, the force of gravity being
countered by the centripetal force.
As the elevator climbs the force times distance and hence the
energy needed to raise it is substantial while the force times distance,
energy, produced by the counter elevator is zero.
This need to exert substantial energy to raise the one elevator
while receiving little return energy from the downwards elevator
continues until the elevators meet halfway. At the halfway mark the elevators experience equal force and
hence equal energy and so are perfectly counterbalancing each other.
As the downwards elevator travels beyond the halfway point it
experiences greater force than the upwards elevator and so, instead of
having to put energy into the system, energy can be generated from the
system. In fact, the amount
of energy that could be drawn from the elevators after they pass is
equal to that that would have to be exerted on the elevator system to
reach that point. Therefore,
although a space elevator could not be effectively counterbalanced, as
can be done with a standard elevator, energy put into the elevator
system by electric motor during the first half of the trip could be
recaptured by using the energy generated during the second half to power
an electric generator. In
fact, so long as the amount of mass being transported down the elevator
is equal, on average, to the amount transported up, the energy being
used up moving mass up would be equal to that being generated when
moving mass down and so, the elevator would consume a net zero energy
(actually, small inefficiencies in the mechanisms would prevent this
system from reaching absolute zero consumption of energy).
So, the elevator and all its mechanisms, in the case of the
gradiated cable system including the onboard mechanisms to generate,
recapture and store energy, since they travel up and down during every
trip, would consume no overall energy.
With a space elevator the mass of the elevator itself would not
factor into the energy consumed, only the mass of the cargo would use up
energy, and even then, only if the cargo is traveling one way.
For instance, if a satellite were transported into orbit, since
the elevator would carry the mass of the satellite up but would be empty
on the way down, only the energy cost of transporting the satellite up
would need to be expended. Now,
this would be fairly expensive, however, in the case of a rocket the
energy cost of transporting the rocket and part of the cost of
transporting its fuel into space, not to mention the energy lost to air
resistance, would also have to be paid. As such, using an elevator to transport cargo into space is
immensely cheaper than using rockets.
However, say a space elevator were used to regularly rotate crews
and transport supplies. In
this case, for every crew member transported up, another would be
transported down and likewise, with supplies, the mass of the supplies
going up would be countered by the mass of the wastes that would have to
be transported down. As
such, the use of an elevator for crew rotation, as well as space
tourism, where as many people are returning as are embarking, and for
supplying space facilities would be particularly advantageous in that
such systems have the potential of consuming nearly no energy, and hence
of being exceedingly cheap to maintain.
Now, this benefit of cheap
transport to and from orbit alone is a huge benefit of space elevators,
however the energy conservative nature of this system of transport can
be extended to interplanetary travel.
It has been suggested that space elevators could be used, not
only to move cargo into space, but also to shoot cargo out of orbit.
The suggestion is that a space elevator would simply need to be
extend beyond geosynchronous orbit until the centripetal velocity on the
cargo becomes great enough to allow it, once released, to brake orbit.
To determine this height exactly it must be recognized that, in
order to escape the earth’s gravity, the energy imparted by traveling
up the elevator must equal or exceed the energy necessary for escaping
the gravitational field. The
total energy imparted by the elevator is the sum of the change in
kinetic energy do to increasing centripetal velocity and the change in
potential energy do to change in height within a gravitational field.
The
energy necessary to escape the gravitational field of the earth is
likewise the change in kinetic energy, this time from the centripetal
velocity at the surface of the earth to the zero velocity at infinite
distance, plus the potential energy of traveling an infinite distance
out of the gravitational field.
Equating
these two formulas gives us
Solving
in terms of
Calculating
the necessary cable length,
These
cable lengths are only about 30% greater than those necessary to reach
geosynchronous orbit. However,
in order to counterbalance the cable below geosynchronous orbit further
mass would still be required above this distance, and so producing a
space elevator that could shoot cargo out of orbit would require no
greater strength of cable than producing an elevator to put cargo into
orbit.
Now, in order to explain the
potential of space elevators for interplanetary travel, for the sake of
argument we will imagine two identical planets. Besides sharing the same mass, diameter and period of
rotation, these planets will also be stationary with respect to each
other, no external gravitational fields will be acting on them and the
planes passing through their equators will be one and the same.
Now, if a space elevator were placed at the equator of both these
planets, with the cable being extended high enough to allow cargo to
leave orbit, a tethership could be grappled to the top of one of these
elevators. Assuming that
the technical issues of timing and trajectory were dealt with the
tethership could release from the space elevator of one planet (a),
cross the distance at velocity (b) and grapple onto the second elevator
(c).
If
the timing and trajectory were perfect this action would consume no
energy and, once the elevator swings around the other side of the planet
the tethership could be released to travel back to the original planet.
In this way a ship could travel back and forth between two
planets perpetually at no energy cost, however this alone would be
insufficient to allow cheap cargo transport from one planet to another.
In order to complete this process, three steps are necessary.
Firstly a cargo would have to be transported from one planet to
the top of its space elevator. In
order to do this at no energy cost there would have to be an equal mass
at the top of the elevator to be transported down.
Secondly the cargo would have to be transported via tethership to
the second planet. Finally
the cargo would have to be transported down the second planet’s
elevator to the surface. The
problem with this procedure is that, in order to repeat the process
another mass would have to be present at the top of the first elevator.
However, when the cargo travels down the second elevator it would
be possible to use the resulting generated energy to transport an equal
mass on the second planet up at no energy cost.
Further, since the tethership would have to return to the first
plant to repeat the process, nothing would be lost by transporting this
mass in the process.
So,
to sum up this process, ballast masses of equal mass to the cargo to be
transferred would be placed at the top of the two elevators.
The cargo would then be transported up the first elevator while
ballast would be transported down producing a net zero energy
consumption (a). The cargo
would then be transported by the tethership to the second planet and
then exchanged for ballast which would then be transported back to the
first planet to replenish the ballast at the top of the first elevator
(b). Finally, as the cargo is transported down the second elevator
the energy this produces would be used to transport any odd mass up (c).
In this way, the ballast at the top of the second elevator would
be replenished at no energy cost. Since
there would be ballast at the top of both elevators after this process
was completed (d), the process is repeatable.
This method shows that it is theoretically possible to transport
large quantities of cargo indefinitely and at no energy cost between two
planets; that it is merely the inefficiency of rocket propulsion that
makes space travel expensive.
Now, although this example
shows that it is possible to transport material between planets energy
free, this situation is so precise that it is not applicable to any
realistic situation. Therefore
it is of some interest to show how it is that this system of transport
could be applied to typical planetary systems.
In the case of variation in mass, rotational period and external
gravitational effects, variation of the distance up an elevator at which
a tethership is grappled for one planet relative to that for another and
the trajectory along which a tethership is released is sufficient to
allow a system to maintain its efficiency.
Relative motion of planets towards or away from each other is a
bit more problematic. When
two planets are moving away the tethership would arrive at its
destination with less relative velocity than when it was launched and as
such, each transit would require an addition of energy to compensate for
this. On the other hand, in
the case of two planets moving towards each other the tethership would
arrive with greater velocity relative to the destination planet.
Diagram
a) shows that for each transit, one at time t=a and another, after being
swung around the planet at t=b, for planets moving towards each other,
the tethership would gain the difference in the velocities of the
planets while b) shows that, for planets moving away from each other the
tethership would lose the difference in velocity each transit.
Now, although travel between planets with relative velocities
towards or away from each other does not conserve the energy of the
system, this is not a problem. Within
any contained system of planets, such planets must move towards each
other as often as apart. As
such, there would be plenty of opportunity within such a system to
balance the energies lost by transit between receding planets with
energy gained between approaching planets.
Now, this is fine for a set of planets within an enclosed space
without significant gravitational interactions, however, for an orbital
system it is not possible to release a tethership in any direction.
The speed of the earth around the sun is quite substantial, so
much so that in order to use a tether system to bring a ship to a
complete stop, to cancel the velocity it is traveling along with the
earth at, the tether required would have to be over eleven times longer
than the distance required to reach geosynchronous orbit.
So, generally, if an object were to be released from any
reasonable sized space elevator, rather than travelling backwards
towards another planet it would instead reduce in its orbital velocity
around the sun and hence fall towards the sun into an elliptical orbit.
However, it is the properties of an elliptical orbit that present
the solution to energy conservation in the orbital situation.
Do to conservation of energy an object leaving the earth at a
particular velocity into an elliptical orbit would be travelling at the
same velocity when its path again crosses earth’s orbit.
So if an object were released from earth into any elliptical
orbit, if that orbit would cause the object to meet up with earth at a
later date then the object could be recaptured by earth with no net loss
of energy. Likewise, if an elliptical path were to intersect with mars
such an object could be captured by mars and then, at a later date,
released from mars at the same velocity, and hence energy, to enter an
equivalent elliptical orbit. In
this way a tethership could be released into a mars intersecting
elliptical orbit. At mars
the ship would be captured at its reduced speed, having lost kinetic
energy and hence speed moving away from the sun.
Later, the tethership could be released at the same velocity it
arrived at mars with, into an equivalent elliptical orbit that would
intersect earth. As the
ship approaches earth, it would gain speed as its distance from the sun
reduces until, as it reaches earth it would reacquire the speed it had
upon leaving. In this way
the tethership could be recaptured at earth with the same energy it left
with, with no net lose of energy during the trip.
Now, the solutions to these problems has been relatively simple, requiring just height, trajectory and timing adjustments, however the situation where two planets fail to share the same plane passing through their equators requires a more technical solution. An equatorial located space elevator can only release or receive a tethership along its equatorial plane and as such, in order to apply the tethership system to normal circumstances, mechanisms would have to be put in place to deal with this limitation. One solution is simply to move the elevator north or south of the equator. If the elevator were located north of the equator then a component of the gravitational force on the tethership would be to the south and as such, when released, the tethership would be deflected slightly south of the equatorial plane. However, if the tethership’s target were to be located at a significant angle to this plane this method’s effect would be insufficient. The alternative is to use a tethership-satellite system. In this system a tethership, instead of being attached through a space elevator to the earth, would be attached instead to a heavy satellite. This pair would then be made to rotate about each other such that the satellite would move at precisely twice the velocity around their center of mass than their center of mass would orbit the earth. Now, cargo could be transferred from orbit to the tethership by using an elevator which would pick up the cargo at the point along the cable at which their center of mass was located and move this cargo along the cable to the ship. So long as the mass moved to the tethership was kept equal to that moved from the ship to orbit, i.e. cargo is exchanged for equal ballast and vice versa, then the system would maintain its rotational velocity and no energy would need to be expended to keep things going. Once the cargo was loaded and the tethership-satellite system reached the correct location the tethership could be released. The tethership, being lighter and hence traveling at much greater velocity than the satellite, would leave orbit, however the satellite, travelling at twice the orbital speed, would reverse its orbit and proceeds to travel in the same orbit and with the same velocity as the system but in the opposite direction. Now, as the tethership approached the target planet a second tethersatellite, likewise in a reverse orbit, could be set up to receive the ship.
By
ensuring precise timing and trajectory the tethership could grapple to
the cable of the satellite such that the combined velocity, the velocity
of the pairs center of mass, is equal and opposite to that of the
satellite. In this way the
orbit of the satellite would reverse and the tethership would enter
orbit. Once the
tethership’s cargo was unloaded, either by replacing it with new cargo
or with ballast, the procedure could be repeated in the opposite
direction. This procedure
is completely conservative and as such could theoretically be used to
transport cargo between planets indefinitely at no energy cost.
Since this method does not rely on ground based space elevators
the orbits do not have to be equatorial and so orbits that are in line
with the orbital plane of the planets could be used.
Therefore this system could be used to transfer cargo between
planets regardless of the orientation of the their equatorial planes.
Now, as can be seen from the diagram above, it is not necessarily
the case that the tethersatellite would enter exactly the same but
reverse orbit as that of the tethership-tethersatellite system.
If the tethership were released while it was at its furthest from
the planet as the ship and satellite rotate about each other, then the
tethersatellite would be released into a lower than geosynchronous orbit
whereas if the tethership was released while at its closest then the
tethersatellite would enter into a slightly higher orbit.
However, it is only necessary that the orbit the tethersatellite
enters be stable and circular so that satellite would be at the same
altitude and velocity when it comes time to grapple another tethership.
If the tethersatellite must be released at a slightly lower
altitude, then to achieve a circular orbital path it would only be
necessary to set up the configuration of the tethered ship-satellite
system such that the velocity of rotation of the tethersatellite around
the system’s center of mass would be slightly greater than twice the
velocity of the system around its planet.
In this way, when the tethership separates from the
tethersatellite the satellite would enter its slightly lower orbit with
the slightly greater velocity necessary to keep this lower orbit
circular. Now, space elevators would allow cargo to be
transported into geosynchronous orbit while tethership-satellite systems
would allow for transport between planets into ecliptical orbits. However, in order to complete this system it is necessary to
transfer cargo between geosynchronous and ecliptical orbits.
Now, although switching an objects location within a given orbit
costs relatively little energy, the amount of energy approaching zero as
the time it takes to change location approaches infinity, for an object
to change to another orbital path always takes a considerable amount of
energy. However, an energy
conservative tether-system exists to solve this problem as well.
Take a cargo in 24-hour ecliptical orbit around the earth,
perhaps unloaded from a tethership-satellite system.
A counter mass equal to the cargo could be placed in the same
orbital path as this cargo but moving in the opposite direction such
that the two pass each other at the point where the ecliptical orbit
meets the geosynchronous orbit.
As
the cargo and counter-mass pass each other they could grapple together
with a tether and would then proceed to spin about each other.
Once the cargo’s direction of velocity shifts to that for
geosynchronous orbit the two would release, the cargo moving in
geosynchronous orbit and the counter-mass in counter geosynchronous
orbit. After the cargo is
exchanged for ballast the counter-mass’s counter geosynchronous motion
could be used to return this ballast to ecliptical orbit by repeating
the process. There is one
potential problem with this solution, that when the two masses are
linked they have a net zero orbital velocity and hence would begin to
fall, gaining downwards velocity. However,
by doing this the masses, once released, would enter a slightly
elliptical orbital path and by the nature of elliptical orbits, which is
proven but too far off topic to present here, if an elliptical orbital
path is generated by a velocity being added to a circular orbital path
towards the source of gravity then once the object moves to the exact
opposite side of this source of gravity it would possess the same
distance and a velocity equal to the original circular orbital velocity
and an added outwards velocity equal to the inwards velocity it gained
previously. Therefore, if
the masses gain a downwards velocity by being coupled together this
would later translate into an equal upwards velocity when they meet on
the opposite side of the earth which would then act to counter the
downwards velocity they would gain during this second linking.
As such, so long as the two masses couple together equally
between the two points at which the geosynchronous and 24 hour
ecliptical orbits cross, no energy would be lost through this practice.
All these systems, when taken together, show that it is possible
under real world situations to transfer cargo between planets at
negligible energy cost.
Now, the energy conservative
nature of tether systems, as shown, leads to impressive results, however
these results are completely dependent on being able to find an equal
mass (ballast) at the cargo’s destination.
Unfortunately, when transporting cargo into orbit one way, such
as when putting up a satellite or space station, this advantage no
longer applies. However, it is under such situations that the third energy
advantage of space elevators comes to the fore. To show this property of elevators it is necessary to
determine the energy an object would possess upon reaching a particular
height up an elevator and relate this to the energy that the elevator
would impart into such an object. The
energy an object would possess, the amount the elevator would have to
have imparted into it to bring it from the ground to a particular height
and speed, is the change in its potential energy plus the change in its
kinetic energy.
Now,
in order to calculate the energy that would be imparted into an object
by the elevator, the force the elevator would exert on this object to
lift it, this force being equal and opposite to that the object would
exert do the centripetal motion and gravitational force, has to be
integrated over the distance up the elevator the object travels.
where
the force
F, being the negative of the centripetal-gravitational force
acting on the object at
r, is given by
Putting
these together gives the result
These
two results (
To get the total amount of energy drawn by
the elevator from the earth’s rotation it is necessary to take the
difference of the imparted and expended energies.
Now,
there are three questions of interest that can be drawn from this
formula, what percentage of the energy required to reach geosynchronous
orbit
Applying
this formula to earth, mars and the moon gives
This
shows that using an earth based space elevator to reach geosynchronous
orbit would result in 16% of the energy imparted into the cargo coming
from the earth’s rotational energy, which gives a system efficiency of
119%, greater than any direct energy propulsion system could ever hope
to achieve, being as such systems are always limited to a maximum of
100% efficiency. Now, the
point at which all the energy needed to raise a cargo comes from the
rotational energy of the earth is the point
Calculating
this formula for earth, mars and the moon to find the distance
As
can be seen, an earth bound space elevator would have to have a cable
length four times greater than the minimum cable length (about three
times the practical cable length) in order to draw all its energy from
the earth. This in an
interesting result since it is also exactly the same distance as is
necessary for a particular design of space elevator that has been
suggested. The basic design for a space elevator is the
minimum practical cable length design in which the cable would be run
from the earth’s surface to just past geosynchronous orbit with a
counter mass being used beyond this orbit to hold the cable up.
Alternatively, an elevator could be created, straight or
gradiated, with no such counterweight, instead with the cable simply
continuing up until the centripetal force on the cable itself is great
enough to counterbalance the cable below geosynchronous orbit.
In order to calculate the height of such a cable, it is necessary
to use the formula for the force at geosynchronous orbit for a straight
cable and the mass per unit length formula for a gradiated cable.
In the case of the straight cable the force at geosynchronous
orbit do to the cable below this orbit is given by
while
the counter force from the cable above geosynchronous orbit is given by
In
order for the cable to be in equilibrium both these forces must be equal
in magnitude so
This
formula is identical to the previous formula for calculating
Now,
that the finite straight cable that travels from the surface to
This
is the case when
This
is equivalent to the previous formula for a self-supporting straight
cable and therefore shows that all the infinitesimal cables are
themselves balanced and so adding them to a balanced straight cable
would not disturb the balance of the system.
Therefore a self-supporting gradiated cable would be precisely
the length required to draw all its net operating energy off the
earth’s rotation. Now, these results are of great significance to the practical application of space elevators.& |
.
,
necessary for creating a straight cable space elevator on all the
likeliest places such an elevator would be built, earth, mars and the
moon, can be calculated.
.
,
the force can be taken as F=-(Tensile strength/Density)/
,
regardless of what the cable is made of.
)
as a function of height for earth, mars and the moon gives
,
distance up elevator measured from center of earth
,
for earth, mars and the moon gives
and
)
show that the energy that the object gains is the potential plus the
kinetic energy while the amount of energy the elevator expends to give
the object this energy is only the potential energy minus a value equal
to the gain in kinetic energy.
(x)
