Accurate Geo-registration by Ground-to-Aerial Image Matching
Energy Given to Mass By Lifting From Ground Up Into GEO
Transcript of Energy Given to Mass By Lifting From Ground Up Into GEO
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Calculating the energy given to mass by lifting if from the ground and putting it into
Geostationary Earth Orbit
By Jim Cline
How much of the energy that is consumed by rocket launch vehicles going to GEO is actually
given to the lifted payload itself?
Here we find, through three different paths of reasoning, that it is 15.7 KWhr/kg, 7.3 KWhr per
pound mass,
That is, about 73 cents of electricity per pound lifted up from ground into GEO.
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Much greater efficiency than rockets provide, is needed to enable large scale projects such as
SPS in GEO. Yet the energy given to mass by putting it into GEO is quite small, so there is
the potential for economically building SPS in GEO.
Energy is the ability to do work, and it is indeed work to lift something from the ground all the
way up into the Geostationary Earth Orbit, GEO. 22,300 miles above the Earth's equator. Just
how much work it is, however, limits what is worthwhile to lift up there. Current transportation
modes are extraordinarily wasteful of energy, requiring far more work expended in the rumblehustle to space than is actually useful energy spent. but it is the best we have gotten so far;
can we now look toward doing much better? It is very possible, as is shown on these pages.
So let's look at how much work is involved that is actually supplied to the payload; that is the
useful part; all else is very expensive wasted fluff. We will find that the limiting values of
energy are quite low, enabling quite a new range of possible projects to be achieved up there,
given a new and sufficiently efficient transportation mode if we choose to do so (such as
explored in many of these pages); some new large scale projects begin to look feasible then,
that could enormously help our civilization along with the planetary ecosystem we live within.
There are several aspects involved. One is the actual energy added to the payload mass by
having been lifted up there, and that is a very specific value. Other aspects involved concern
the actual kind of grunt work done to lift it up there, and current techniques are extremely
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basically is a simple calculation: multiply the weight of the object and multiply it by the height it
is lifted. Note that the weight of something is its mass multiplied by the force of gravity existing
at its location. So we assume here that the force of gravity does not significantly vary over the
height being raised, limited to, say, the height of existing tallest buildings.
So if we lift something off the ground which weighs (in English units) one pound, to a position
which is one foot higher, the work done on the object is one foot-pound: Energy equals
Weight times height lifted:
E = W x h
Shifting from English units of measurement to the simpler International System of Units of
measurement
However, when one gets into more complex calculations, it has been found that the use of a
system of units known as the International System of Units (SI) is clearer and simpler in the
long run. instead of a "foot", it uses a meter (about 3 feet) as its basic unit of length. And for its
unit of weight ... well, to make things clearer overall, SI separates out the mass of the object
from the force of gravity at its location at the moment, to give it more adjustable values for useat different locations, more universally usable. In SI units, the "kilogram" is the basic amount
of mass; and the gravitational acceleration (which, when multiplied by the physical mass of
something gives the weight of that something where it is at) on the ground (on the surface of
the Earth) is 980 cm per second per second ("cm" stands for a hundredth of a meter, or
"centimeter"). Note that the notion of "time" is embedded into the notion of "acceleration",
which could be thought of as how fast something is speeded up per unit of time... therefore it
may be of interest that the notion of "time" is embedded into the notion of the "weight" of
something.
The force of gravity becomes less the higher up we go, so the higher we go the less work it
takes to go even higher
The value of gravitational acceleration varies with the height of the object. The further one
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gets from the center of the mass of the earth, the lower the gravitational acceleration one
experiences. And this force of acceleration decreases rapidly, inversely proportional to the
square of the distance from the center of the Earth, as the fixed gravitational pull is spread out
in all directions from the planet ever more thinly to include ever larger encompassed spherical
surface high above the planet.
This means that the higher one gets above the ground, the easier it is to lift the object higher
yet. And the amount drops off as the square of the distance from the center of the Earth.
(Note that referring to the "square" of something simply means the length of line moved
sideways through a distance equal to its own same length, thus covers an area which is its
"square".)
And it means that the equation we used above (E = W x h), to determine the energy given to
an object by lifting it off the ground, is only effectively valid for use near the ground, since the
value of gravitational acceleration used to determine the object's Weight "W" changes when
we get much higher above the ground. There are slightly more sophisticated ways to calculate
the work, energy, needed to lift payload up through a varying strength gravitational field.
Arthur Clarke's simplified calculation for the energy given to a payload's mass when lifted from
the ground up out of the gravity well
Arthur C. Clarke many decades ago pointed out that the theoretical amount of energy needed
to lift a mass up from the earth's surface out into far distant space is mathematicly equivalent
to the energy needed to lift that mass up one planetary radius' altitude within a constant
gravitational acceleration equal to that located at the planet's surface. (Ref "The Exploration of
Space" by Arthur C. Clarke. Harper edition.) This makes it easy to make a calculation of the
theoretical energy needed to do one thing: without vehicular overhead costs considered, lift a
mass up from the ground and go to a theoretically infinite distance from the planet, not
considering any other gravitational fields.
Note that the work, or energy, required to lift something is its weight ("weight" is the force of its
mass times the gravitational accelleration where it's at) times the height through which it is
lifted, the force exerted upward being equal in magnitude to its weight.
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Energy, Work, Work = weight x height change
Clarke's calculation shows that, theoretically, the work required to lift something from ground
to a point far distant above the Earth, is mathematically equivalent to that required to lift it
from the ground to one planetary radius, as if the acceleration of gravity were to remain
constant throughout the lift. Since the radius of the Earth is about 4000 miles, the work is
equivalent to raising it 4000 miles up as if with a constant weight of that which it had on the
surface. So for one pound of stuff, raised to a distance of one planetary radius under a
constant surface gravitational field strength:
Work = 1 lbf x 4000 miles x 5280 ft / mile
Work = 2.1E7 ft lbf
Since 1 ft lbf equals 3.77E-7 KwHr,
Work = 2.1 x 10E7 ft lbf x 3.77E-7 KwHr,
Work = 7.9 KwHr
Stated otherwise, the energy cost to accelerate 1 pound of mass to escape velocity = 7.9
KwHr, nothing else considered.
Note that, at current American electric power costs of ten cents per KHr, this is 79 cents.
(And yes, of course there is a lot more expense to be considered for the process of lifting
payload to space!)
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Another way to make this calculation is to use the gravitational equation.
Calculating theoretical work (energy) required to go from the ground up to essentially beyond
the influence of the planet's gravitational field:
Given:
* Gravitational constant, G = 6.67E-11 m3 / Kg Sec2
* Earth mass = 5.983E24 Kg
* R0 is the Earth equatorial radius = 6.378E6 meters
* Destination R = infinity
* 1 joule = 1 Kg m2 / Sec2 = 2.78E-7 KwHr
Then, calculating the work (energy) from equatorial Earth surface to essentially beyond the
influence of the planet's gravitational field:
where
R0 = radius of equatorial Earth surface = 6.378E6 m
R = infinity
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W = GMm ((1 / R0) - (1 / Rinfinity))
W = (6.67E-11)(5.983E24)((1 / 6.378E6) - (1 / infinity))
W = (4E14)(1.57E-7) = 6.27E7 Joules / Kg to escape = 17.4 KwHr / Kg; @2.2 Kg / lbm = 7.9
KwHr per pound to GEO
Thus calculating both ways yields the same amount of electrical energy to escape the
gravitational influence of earth: 7.9 KwHr per pound mass. Note that this is the same amount
of energy which would be given up by the one pound mass if it returns to rest again on the
equator.
At a cost of electric power of 10 cents per KwHr, 7.9 KwHr costs $0.79, which thus is the basic
added-energy cost to move a pound of payload electrically from the ground to far distant
space from earth. Any closer destination would cost less.
So how much would it cost to move payload electrically only to GEO, the earth-synchronousClarke Belt orbit? Let's calculate it two ways, first using the gravitational equation again, then
using Clarke's simplification, a bit expanded. These calculations only calculate the enerrgy
needed to raise payload to an altitude, and the energy needed to give the payload orbital
velocity to stay in orbit needs to be added to find the total cost.
First, calculating the lesser theoretical energy cost of going from Earth's equator into
geosynchronous Earth orbit, using the gravitational equation.
Calculating theoretical work (energy) required to go from the ground up to GEO, the Clarke
Belt:
Given:
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* Gravitational constant, G = 6.67E-11 m3 / Kg Sec2
* g0 = 9.807 meters / sec2
* Earth mass = 5.983E24 Kg
* Earth equatorial radius = 6.378E6 meters
* RGEO = 4.23E7 meters (22,300 mi above the equator)
* 1 joule = 1 Kg m2 / Sec2 = 2.78E-7 KwHr
Then, calculating the work (energy) from equatorial Earth surface to Geosynchronous Earth
Orbit (the Clarke Belt):
where R = radius of GEO altitude = 4.23E7 meters
R0 = radius of equatorial Earth surface = 6.378E6 m
W = GMm ((1 / R0) - (1 / RGEO))
W = (6.67E-11)(5.983E24)((1 / 6.378E6) - (1 / 4.23E7))
W = (4E14)(1.33E-7) = 5.31E7 Joules / Kg to GEO = 14.76 KwHr / Kg; @2.2 Kg / lbm = 6.71
KwHr per pound to GEO
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That is the theoretical energy needed to lift one pound mass up to GEO; to stay there it also
needs to be given orbital velocity:
Calculating orbital velocity at GEO, using the gravitational acceleration there as the
centrepetal force's "a" as in F = m * a:
Calculating the gravitational acceleration at GEO altitude, gGEO:
gGEO = (GMm) / (RGEO)2 = (6.67E-11)(5.983E24) / (4.23E7)2
gGEO= 0.224 m / Sec2
Since gGEO = a = V2 / RGEO, where RGEO = 4.23E7 meters [this is the distance of GEO
from center of the Earth]
Then V2 = a * R = (gGEO)(RGEO)
Therefore V = (a * R)1/2 = (0.224 * 4.23E7)1/2
= 3.078E3 m / S = orbital velocity in GEO
Another way to calculate the orbital velocity in GEO is to calculate the circumference of the
orbit then divide it be number of seconds in a day, one orbit:
VGEO = (2 * pi * 4.226E7 Meters) / 8.64E4 seconds = 3.073E3 meters per second, about the
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same.
The Earth's equator, probably the starting point for payload to GEO, rolls along at 1,000 mph,
or 4.47E2 m / S
So the delta V from equator to GEO, component at right angle to Earth's radius, is (3.073E3 -
4.47E2) m / S = 2626 m / S
The 1/2 mV 2 kinetic energy to be added from equator to GEO, horizontal component, then is
3.45E6 Joules per Kg; becoming an additional 0.96 KwHr/ Kg to orbit at GEO; at 2.2 lbm / Kg,
that is 0.436 KwHr per pound mass to GEO to provide the orbital velocity component after
getting up there.
So the total electrical energy to place one pound mass into geosynchronous earth orbit is the
6.71 KwHr to lift the mass up there plus another 0.44 KwHr to accelerate it to orbital velocity
at that altitude, or a total of about 7.15 KWh to emplace one pound mass from earthsurface
into GEO, Earth synchronous orbit, the Clarke Belt.
At $0.10 per KwHr average cost of electrical power in the US currently, that equals $0.72 per
pound from equatorial Earth surface up to GEO.
We can also use the Clarke equivalency, slightly modified, to calculate the energy needed to
lift from the ground to GEO:
We can also find the same result by using the same Clarke's simplification equivalency that
we used earlier, to calculate the energy from GEO to far distant space, subtract this from the
energy to lift from the ground to far distant space, thus the energy needed to go from the
ground up to GEO. We already calculated above that it takes 7.9 KwHr of energy to escape
the earth's gravitational well. And we calculated above that the gravitational acelleration ith
the altitude of GEO is
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Copyright 2008 James E. D. Cline