Monday, January 08, 2007

PUSH TOWARD INTERSTELLAR

CONTENTS
The interplanetary performance envelope (particle exhaust velocity of 10%-50% c) will need considerable "pushing" for interstellar flights to happen. Thus, our Thought Experiment (TE) proposes a performance envelope for interstellar travel via accelerator propulsion.
TE assumes an interstellar exhaust velocity (VExh) of at least 86.6% light speed, .866 c. Thus, as exhaust velocity approaches every closer to c; spaceship range increases. When we achieve this; interstellar flights will become feasible. However, for interstellar flights to become practical, we must overcome significant problems.
---Reliability becomes ever more important for interstellar voyages.  Precise quantities of near light, exhaust particles must exit vessel for every second of every day.
---Flight duration expands by several orders of magnitude.  Exhaust particles must maintain near light speeds for years while interplanetary need only days of much slower exhaust speeds.
EXPAND ENVELOPE
from INTERPLANETARY
particle accelerator
... will transform several grams/sec of water  to ions and expel them at speeds from 10% to 50%c.
This will produce equivalence and quick flights to nearby planets (days to weeks).
to INTERSTELLAR
particle accelerator
... will transform several kilograms/sec of water  to ions and expel them at growth factors from 2 to 11 (i.e. corresponding particle exhaust velocities of 86.6%c to 99.6%c).  This will produce g-force acceleration for lengthy durations from 100 days to over a year..
Review some assumptions:
I. Assume onboard particle accelerator....
...can consume reasonable quantities of fuel particles, original fuel flow per second (ffsec), and accelerate them into a constant flow of exhaust particles (ffExh) .
For this initial example, further assume particles' exhaust speed  (VExh) is 86.6% c.  One can express exhaust velocity as a product of decimal component (dc) and light speed, c. 
EXAMPLE: VExh  =  dc × c = .866 c
Thus, dc = .866 = VExh / c
Such notation facilitates calculation of relativistic mass growth via the
Lorentz Transform
mr = mo

(1-dc2)
Particle Size
Lorentz Transform quantifies the relativistic growth due to particle speed. Thus, fuel flow per second (ffsec) is the original mass at rest, and exhaust fuel flow (ffExh) is the relativistic mass at an accelerated velocity.
Notional fuel quantity increases per following table.
ffExh = n × ffsec
Original
Fuel Flow
Exhaust
Velocity
Growth
Factor 
Exhaust
Fuel Flow
ffsecVExh =dc×c nffExh
1 kg .8667c 2.0 2.0 kg
Given1

 (1-dc2)
n×ffsec
This example shows a one kilogram of fuel growing 100% to 2.00 kg if individual particles accelerate to 86.666% light speed.  Subsequent examples will consider other quantities at increased exhaust speeds.
II. Assume g-force ...
Example -2: Assume: VExh = 94.3% c = dc × c
Thus, dc = .943
Original
Fuel Flow
Exhaust
Speed
Growth
Factor
Ship
Mass
ffsecVExh = dc×cnMShip
1 kg.943 c383,623 mT
2 kg.943 c3173,246 mT
GivenGiven1

(1-dc2)
(n2-1)×30.57×106ffsec 
dc=VExh/c
n=1/(1-dc2)
n2=1/(1-dc2)
1-dc2=1 / n2
dc2=1 - (1 / n2) = (n2 - 1) / n2
dc=(n2 - 1) / n
...comes from momentum exchange. One second of fuel flow's small mass times enormous speed equals spaceship's huge mass times 9.8065 m/sec velocity increase for one second (acceleration due to near Earth gravity, g).
MShip × g = ffExh × VExh
Both right side terms can be re-expressed. Exhaust fuel mass (ffExh) can be rewritten as growth factor, n, times fuel flow per second (n×ffsec).  Particle exhaust velocity (VExh) can be rewritten as decimal component times light speed (dc×c).
MShip × g = (n×ffsec) × (dc×c)
Rewrite equation as shown below. Note that left panel shows decimal component, dc, defined in terms of n, growth factor. or dc = (n2 - 1) / n.
MShip = (n×ffsec) × ((n2 - 1) / n×(c/g)
c = 299,792,458 m/sec  g= 9.80665 m/sec2 
c/g = 30,570,323 sec
The two constants, light speed (c) and acceleration due to gravity (g) can combine for a third constant (c/g), 30.57 mega-sec. The two "n"s cancel out and the implicit "/sec" of ffsec  cancels out the sec from c/g.
MShip = (n2 - 1) × 30.57 mega-ffsec 
At the same exhaust speed, the greater the fuel mass, the greater the ship's initial mass which increases 9.8065 m/sec for every second of powered flight (aka "g-force").
III. Assume g-force ship's gross weight decreases...
... due to a fairly consistent fuel consumption.
EXAMPLE: If ship's g-force propulsion system consumes 1 kg/sec anytime during a particular day; then, one could further assume that day's consumption as about 86,400 kg.  We don't know exact amount of consumption per second for any given ship; however, we can designate that value as an variable, ffsec.
(One day = 24 hours × 3,600 sec /hour = 86,400 seconds.)
ffday =day × ffsec =86,400 × ffsec 
Daily % decrease in ship's gross weight can be approximated by the daily decrease divided by ship's GW for that day.
∇= 
ffday

GW
Day × ffsec×g

ffsec × (n2-1) × c
Day × g

 (n2-1)×c
c = 
299,792,458 m/sec
g = 
9.80665 m/sec2
Day = 
86,400 sec
Day × g /c  = .002826 = .2826%
∇ = .2826% / (n2-1)
Conclusion: Inspection shows daily decrease depends on particle size, n (multiple of original ffo), which depends on particle speed, d (decimal portion of c, light speed).
THUS,  vessel's range depends
---not on actual quantity of ffsec
---nor actual ship's gross weight, GW.
Consider ffSec and ship's GW.  Example:  ffSec=1.0 kgRECALL: 1,000 kilograms (kg) = 1 metric Tonne (mT)
Particle
Exhaust
Speed
Exhaust
Fuel
Flow
Ship's
Gross
Weight
 Daily
Decrease
VExh= dc×cffExh=n×ffsecGW
.866 c2.00 kg52,949 mT0.16 %
.943 c3.00 kg86,465 mT0.10 %
.968 c4.00 kg118,397 mT0.07 %
Givenffsec

(1-dc2)
ffsec×(n2-1)×c

g
Day × ffsec

GW
Don't need ffSec and ship's GW; rewrite as follows:
Relativistic
Growth
Factor
Decimal
Component
Light Speed
 Daily
Decrease
ndc
2.8660.16 %
3.9430.10 %
4.9680.07 %
Given(n2-1)

n
.2826%

(n2-1)
IV. Assume Range: Minimum of 100 days G-force acceleration.
EXAMPLE:  Independently vary particle's growth factor from n=2 to n=7.  Assume initial fuel load to be 50% of ship's Gross Weight (%TOGW=50%); thus, continuous flow of high speed exhaust particles will decrease ship's weight over many days until a minimum gross weight of half of ship's initial GW.
Efficiency, (E), will likely improve as humanity learns to design better g-force propulsion systems. Assume inefficiency (E') decreases with growth factor, n; perhaps, E'=.5×.9n-2.  Subsequently, Efficiency (E) will increase (1-E'); efficiency factor (ε = 1/E) will decrease.
Daily exhaust flow,∇, is the amount of charged particles needed to achieve g-force momentum. 
Daily consumption rate.  ε∇, is the amount of charged particles needed to ensure required exhaust flow. It accounts for inevitable inefficiencies. AXIOMATIC: ε∇ always exceeds ∇.
Percent Take Off Gross Weight (%TOGW) is the portion of ship's initial mass allocated for fuel.
APPROX. RANGE: Propulsion Time, tp
Let Percent Take Off Gross Weight (%TOGW) = 50%
Relativistic
Growth
Factor
Decimal
Component
Light Speed
Vessel's
Propulsion
Exhaust Rate
Forecast
Efficiency
Factor
RANGE:
Propulsion
Time
ffExh=n×ffsecVExh= dc ×c=ffDay/GWεtp
2.8660.163%2.000212 days
3.9430.100%1.818381 days
4.9680.073%1.681565 days
5.9800.058%1.574763 days
6.9860.048%1.488975 days
7.9900.041%1.4191,197 days
Given(n2-1)

n
.2826%

(n2-1)
1

1 -(.5×.9n-2)
log(1-%TOGW)

log(1-ε∇)
CONCLUSION. As particle exhaust speed increases, vessel's range increases.
Model ship's range with daily decrease, ∇, in ship's mass to reflect fuel consumption. Also consider efficiency factor, ε, and Percent Take Off Gross Weight, %TOGW.
EXAMPLE:  Independently vary ship's %TOGW (portion of ship's mass dedicated to fuel) from 40% to 60%. Assume particle's exhaust speed as constant 99.0% c; then, previous work leads us to determine following values.  Decimal component (dc) is .990; growth factor (n) is 7; and daily exhaust flow, , is 0.041% Ship's GW per day.
Efficiency factor (ε) is inverse of efficiency (1/E). For interstellar performance, Thought Experiment arbitrarily assumes an efficiency model: ε = 1/E = 1/(1-.5×.9n-2). (NOTE: There's no way of knowing the actual efficiency of future g-force propulsion systems, but we're sure their efficiency will improve.)
Daily exhaust flow,∇. Previous work approximates this value by dividing .2826% by the term, (n²-1).
Daily consumption rate.  ε∇, product of efficiency factor and daily exhaust flow. ε ensures sufficient quantity for daily exhaust by consuming more sure than needed.  Design flaws and peripheral needs will compel a consumption rate greater than exhaust flow.
Percent Take Off Gross Weight (%TOGW).  For any given particle exhaust speed, range (R) increases with %TOGW. (See following table.)
Ship's
Fuel
Exhaust
Speed
Growth
Factor
Exhaust
Rate
Consume
Rate
RANGE:
Prop. Time
%TOGWdcnε∇tp
40.00%0.86620.163%/day.326%/day156.3 days
50.00%0.86620.163%/day.326%/day212.1 days
60.00%0.86620.163%/day.326%/day280.3 days
GivenGiven1

(1-dc2)
.2826%

(n2-1)


1-(.5×.9n-2)
log(1-%TOGW)

log(1-ε∇)
Logarithm helps transform %TOGW into available propulsion time.

CONCLUSION. As fuel load (%TOGW) increases, range increases.
V. Assume interstellar flight profile ....
...such that ship accelerates for 100 days, cruises at constant speed for several years, and then decelerates for last 100 days of voyage.
REASON: To maintain Earth-like gravity (g-force) for the pax and crew throughout the multi-year voyage, the vessel must use two methods: 1) propulsion via a high speed ion stream and 2) centrifugal force by spin. Initially, ship uses ion stream to accelerate at g (g = 9.80665 m/sec² = 0.489 AU/day²) for a feasible duration (at least 100 days). Even after a year of such g-force, ship will be far short of midway between any two stars. However, TE assumes fuel limits require first few star ships to stop propulsion after only 100 days; thus, it must then cruise at constant velocity (approx .25 c) until it reaches the deceleration point 100 days prior to destination.  During cruise, it must change into a habitat (roughly cylindrical shape) and spin about its longitudinal axis at an exact angular velocity to produce centrifugal g-force at inside of outer hull.  At deceleration point, ship will again emit ion stream to slow vessel down to orbiting velocity at destination.
Growth FactorAt relativistic speeds, particle grows by multiple, n.
Particle VelocityEach multiple, n, maps to a specific, relativistic speed (vExh).
Exhaust FlowTo achieve g-force acceleration, ship must consistently expel numerous high speed particles which collectively decrements ship's GW, expressed as a daily percentage.
Consume RateDue to inevitable inefficiencies, consumption rate  (ε∇) must exceed exhaust flow as determined by an Efficiency Factor (ε).
Prop. TimeTotal of 200 days. TE assumes initial interstellar vessels will g-force accelerate for 100 days, cruise for years; then, g-force decelerate for 100 days; thus, total propulsion time would be 200 days. 
Accel. TimeDivide propulsion time (tp) by 2; thus, 200 days/2 = 100 days.
Max-VelocityTE assumes 100 days of g-force produces cruise velocity of about .25c derived from exponential (t) as shown. Vt = c(1-(1-Δ)t)   For more....
Accel DistanceTE assumes 100 days of g-force travels about .0353 LY, derived from natural logarithm. d = c×t + V/ [ln(1-Δ)].  For more....
Interstellar G-force
LY = 9,460,730,472,580.8 km = 63,241 AU
Year = 365.25 daysc = 173.145 AU/Day = 299,792,458 m/sec
Gro.
Fact.
Part.
Vel.
Exh.
Flow
Effic.
Factor
Cons.
Rate
200 Day
Cons.
nvExhεε∇%TOGW
2.866 c.163%

day
2.00 .326%

day
47.99%
3.943 c,100%/Day1.818..182%/Day30.49%
4.968 c.073%/Day1.681..123%/Day21.76% 
5.980 c.058%/Day1.574..091%/Day16.61%
6.986 c.048%/Day1.488..071%/Day 13.26%
7.990 c.041%/Day1.419.058%/Day10.93% 
8992 c.036%/Day1.362.048%/Day9.24%
9994 c.032%/Day1.314..042%/Day7.97%
10995 c.028%/Day1.274.036%/Day6.98%
11996 c.026%/Day1.240.032%/Day6.2%
Given(n2-1)

n
.2826%

(n2-1)
1

1 -(.5×.9n-2)
ε × ∇100%-(1-ε∇)200
CONCLUSION:For a given propulsion time (i.e., 100 days of g-force accel + 100 days of g-force decel), fuel rqmt decreases as exhaust particle velocity increases.
To go interstellar, greatly expand interplanetary profile.

FINAL NOTE-1:
Send Robots First. To pave the way for human travelers, precede a human expeditionary force with robotic explorers. With only Artificial Intelligence (AI) occupants, a vessel could forego all the items required to support humans; no food, no air, no entertainment, no social life.  Most notable, there is no need to simulate Earth gravity; thus, propulsion could be as great or as small as needed.
Communications. One would expect constant streams of data between Earth and this vessel, but interactions must delay due to light speed limitation.  (i.e., if a vessel is one light year away; signal will take at least a year to reach the vessel, and another year for the response to reach original sender.)
Autonomy will be compelled by communication circumstances.  Thus, a robotic "pathfinder" vessel will need to be extremely sophisticated, because such systems will prove essential. With a plethora of stellar systems to choose from, robots must explore all prospective destinations to provide data for humanity to choose best prospects of which planets to visit first.
FINAL NOTE-2:
Even an optimistic flight profile to the nearest star system, Alpha Centauri (AC), would take several years.
Thus, a 50%TOGW fuel load could easily be completely consumed; this leads to following flight plan options:
1) One-way mission with no chance of returning to Earth.  Vessel occupants would resign themselves and their descendants to orbit AC indefinitely in their spaceship.
2) In situ materials.  Gather AC indigenous comets for fuel for return trip.  This is one of many areas where the robotic pathfinder would prove its worth; they could survey the system for comet availability.
3) Very high exhaust speeds.  Particle speeds in excess of .999c would result in growth factors of eleven and higher.  Thus, a 50%TOGW fuel load could accommodate four fuel burns of one year each.  Thus, a vessel could g-force for one year for initial acceleration to AC, cruise for a few years,  g-force for final year to decelerate just prior to AC.  For return trip, vessel could again accelerate for a year, cruise for a while; finally, decelerate just prior to Sol, Earth's sun.
CONCLUSION
Push the particle envelope, push exhaust speed closer to c.

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