Monday, January 08, 2007

PUSH TO 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 that interplanetary g-force propulsion particles will need an exhaust velocity (VExh) from 10% light speed, c; we should learn to do it for much higher exhaust velocities. Thus, as exhaust velocity approaches c; spaceship range increases. When we achieve these technologies; interplanetary flights will become much more routine; however, significant problems remain for interstellar flights.
---Performance problems become much more difficult for interstellar voyages.  Exhaust particles must maintain consistent speeds greater than .999c. Also, there are greater reliability concerns ; interstellar flight duration will be years versus days for interplanetary.
---However, TE also assumes that human ingenuity will rise to the occasion and solve all the engineering problems involved with the much greater interstellar performance envelope.
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 7 to 11 (i.e. corresponding particle exhaust velocities of 99.0%c to 99.6%c).  This will produce g-force acceleration for lengthy duration of a year or more.
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 86.67% c.  One can express exhaust velocity as a product of decimal component (dc) and light speed, c. 
EXAMPLE: VExh  =  dc × c = .867 c
Thus, dc = .867 = 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
.8667 c
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 86.7% 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: By inspection we can see the daily decrease depends on particle size, n (multiple of original ffo), which depends on particle speed, d (decimal portion of c, light speed). It does not depend 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 ship's range...
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.
.... can be modeled with daily decrease, ∇, in ship's mass to reflect fuel consumption.  Model must 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%0.99070.041%/day.058%/day882 days
50.00%0.99070.041%/day.058%/day1,197 days
60.00%0.99070.041%/day.058%/day1,583 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 about a year, cruises at constant speed for several years, and then decelerates for same duration as the acceleration.
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 = 0489 AU/day²) for a feasible duration (perhaps one year). Even after year of such g-force, ship will be far short of midway between two stars. Thus, the ship must then stop propulsion and 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.  Finally, the ship must once again turn on the ion stream for propulsion during last year of voyage; ship must then decelerate from the very high cruise speed to orbit about the destination.
Growth FactorAt relativistic speeds, particle grows by multiple, n.
Particle VelocityEach multiple, n, maps to a relativistic speed.
Exhaust FlowDaily mass (% ship's GW) of high speed particles for g-force.
Consume RateDaily mass consumed to maintain exhaust flow.
Prop. TimeUse consume rate (ε∇) and %TOGW in logs to determine.
Accel. TimeDivide propulsion time (tp) by  four.
Max-VelocityFormula uses exponential (t) as shown.
Accel DistanceUse natural log of daily, light speed remainder [ln(1-Δ)].
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
Cons.
Rate
RangeAccel
Time
Ship's
Max Vel.
Accel
Dist
nvExhε∇tptAccVMaxdAcc
2.866 c.163%/Day.326%/Day212 day53 day24 AU/dy655 AU
3.943 c,100%/Day.182%/Day381 day95 day41 AU/dy2,038 AU
4.968 c.073%/Day.123%/Day565 day141 day57 AU/dy4,295 AU
5.980 c.058%/Day.091%/Day763 day191 day72 AU/dy7,510 AU
6.986 c.048%/Day.071%/Day975 day244 day86 AU/dy11,709 AU
7.990 c.041%/Day.058%/Day1,197 day299 day99 AU/dy16,870 AU
8992 c.036%/Day.048%/Day1,429 day357 day110 AU/dy22,939 AU
9994 c.032%/Day.042%/Day1,669 day417 day120 AU/dy29,842 AU
10995 c.028%/Day.036%/Day1,915 day479 day128 AU/dy37,490 AU
11996 c.026%/Day.032%/Day2,166 day542 day136 AU/dy45,793 AU
See previous
tables.
tp

4
c[1-(1-Δ)t]
c×t+VMax

ln(1-Δ)
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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