Tuesday, February 10, 2009

PUSH PARTICLES TO INTERPLANETARY

CONTENTS
Our thought experiment proposes performance envelope for interplanetary travel via accelerator propulsion.  This envelope must greatly exceed the closest current technology we now have to "accelerator propulsion", ion propulsion. Ion propulsion transforms a gas to ions and discharges them to propel a craft.  It electrically accelerates the ions to 30 km/sec (note that Earth's orbital escape velocity is 11 km/sec). As the xenon ions exit the spacecraft at this comparatively high speed, the ions push the spacecraft in the opposite direction.
Making the leap from ion drives to onboard particle accelerators will be an enormous technological challenge. While current ion propulsion greatly reduces travel time to nearby planetary destinations, it has serious limitations.  The travel time is still lengthy, i.e., several months to Mars at near zero g conditions. However, Thought Experiment (TE) proposes a sufficient flow of high speed (10% to 50% c, light speed) exhaust particles to accelerate vessel at rate, g.  Such performance would enable Einstein's "equivalence" (simulating gravity), and quick flights to other planets will inevitably become routine.


EXPAND ENVELOPE
from current ion propulsion ... which exhausts milligrams/sec of xenon ions at about 30 km/sec to accelerate a relatively small vessel much slower than g. Thus, flight time to Mars reduces by a few months.  Vessel occupants will experience "micro-g" conditions.
to onboard particle accelerator...  which will transform several grams/sec of water to ions and expel them at speeds from 10% to 50%c (about 30 to 150 million meters per second). Thus, flights to nearby planets will be relatively quick (days to weeks). Furthermore, such propulsion can produce equivalence (occupants feel same gravity as if static on Earth's surface); thus, the term "g-force".
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 10% c.  One can express exhaust velocity as a product of decimal component (dc) and light speed, c. 
EXAMPLE: VExh  =  dc × c = .1 c
Thus, dc = .1 = 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 gm
10% c
1.005
1.005 gm
Given 1

 (1-dc2)
n×ffsec
This table shows relativistic mass growth.  EXAMPLE: If an exhaust particle accelerates to 10% light speed; then, one gram of fuel will grow 0.5% to 1.005 gm.  Subsequent examples consider other quantities at increased exhaust speeds.
II. Assume g-force ...
Example -2: Assume: VExh = 20% c = dc × c
Thus, dc = .2
Original
Fuel Flow
Exhaust
Speed
Growth
Factor
Ship
Mass
ffsecVExh = dc×cnMShip
1 gm.2 c1.02066.24 mT
2 gm.2 c1.020612.48 mT
GivenGiven1

(1-dc2)
(n2-1)×30.57×106ffsec 
Rewrite equation as shown: 
MShip = (n × ffsec) × ((n- 1) / n × (c/g)
Note decimal component, dc, defined in terms of n, growth factor. 
or dc = (n- 1) / n
c = 299,792,458 m/sec  g= 9.80665 m/sec2
  c/g = 30,570,323 sec = 30.57 mega-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 = (n- 1) × 30.57 mega-ffsec 
At the same exhaust speed, the greater the fuel mass, the greater the ship's initial mass that can be increased another 9.8065 m/sec for every second of powered flight (aka "g-force").
...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 (happens to equal acceleration due to near Earth gravity, g).
MShip × g = ffExh × VExh
dc=VExh

c
Define decimal component
of decimal light speed.
n=1

(1 - dc2)
AXIOMATIC:  'n' = Growth factor.
As if LT's mo = 1
n2=1

(1 - dc2)
Square both sides.
1-dc2=1

 n2
Rearrange.
dc2=
1  -1

n2
Rearrange.
dc2=(n- 1)

n2
Rearrange.
dc=(n- 1)

 n
Square root both sides.
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)

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 gm/sec anytime during a particular day; then, one could further assume that day's consumption as about 86,400 gm.  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 gm
RECALL: 1 million grams (106 gm) =1 metric Tonne (mT)
Particle
Exhaust
Speed
Exhaust
Fuel
Flow
Ship's
Gross
Weight
 Daily
Decrease
VExh= dc×cffExh=n×ffsecGW
0.1 c
1.005 gm
3.072 mT
2.83 %
0.2 c
1.021 gm
6.240 mT
1.41 %
0.3 c
1.048 gm
9.613 mT
0.94 %
Givenffsec

(1-dc2)
ffsec×(n2-1)×c

g
Day × ffsec

GW
Don't need ffSec and ship's GW; rewrite as follows:
Decimal
Component
Light Speed
Relativistic
Growth
Factor
 Daily
Decrease
VExh= dc×cn
0.1
1.005
2.83 %
0.2
1.021
1.41 %
0.3
1.048
0.94 %
Given1

(1-dc2)
.2826%

(n2-1)
IV. Assume ship's range...
EXAMPLE:  Independently vary particle's exhaust speed from 10%c to 60% c.  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 a range of days until a minimum gross weight of half of ship's initial GW.
Efficiency factor, ε, enables us to account for inevitable inefficiencies in the propulsion system. Artificially set efficiency factor at 2 (ε = 2.0); thus, TE assumes consumption rate is two times the exhaust flow. 
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%
Let efficiency factor (ε) = 2.0
Decimal
Component
Light Speed
Relativistic
Growth
Factor
Vessel's
Propulsion
Exhaust Rate
RANGE:
Propulsion
Time
VExh= dc×cffExh=n×ffsec=ffDay/GWtp
.11.0052.81%11.9 days
.21.0211.41%24.2 days
.31.0480.94%36.4 days
.41.0910.71%48.7 days
.51.1550.57%60.9 days
.61.2500.47%73.2 days
Given1

(1-dc2)
.2826%

(n2-1)
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 56.5% c; then, previous work leads us to determine following values.  Decimal component (dc) is .565; growth factor (n) is 1.211; and daily exhaust flow, , is 0.5% Ship's GW per day.
Efficiency factor (ε) is inverse of efficiency (1/E). For interplanetary performance, Thought Experiment arbitrarily assumes an efficiency of 50%, E=.5; thus, ε = 1/E = 2. (NOTE: There's no way of knowing the actual efficiency of the future g-force propulsion system, but we're sure it's less than 100%.)
Daily exhaust flow,∇. Previous work approximates this value by 86,400×ffsec divided by ship's current Gross Weight (GW); this reduces to 0.2826% divided by (n²-1).
Daily consumption rate.  ε∇, product of efficiency factor and daily exhaust flow. ε ensures sufficient quantity for daily exhaust.  Design flaws and peripheral needs compel a consumption rate greater than exhaust flow.
Percent Take Off Gross Weight (%TOGW).  For any given particle exhaust speed, range increases as %TOGW increases.  (NOTE: Range is also known as propulsion time (tp). See following table.)
Ship's
Fuel
Exhaust
Speed
Growth
Factor
Exhaust
Rate
Consume
Rate
RANGE:
Prop. Time
%TOGWdcnε∇tp
40.00%0.5651.2110.50%/day1.00%/day50.82 days
50.00%0.5651.2110.50%/day1.00%/day68.96 days
60.00%0.5651.2110.50%/day1.00%/day91.17 days
GivenGiven1

(1-dc²)
.2826%

(n2-1)
ε×∇log(1-%TOGW)

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



CONCLUSION. As fuel load (%TOGW) increases, range increases.
V. Assume interplanetary flight profile ....
...such that ship accelerates for half the distance/time and then decelerates for remaining half.  Thus, a g-force vessel can use above profile to achieve quickest time for a direct flight from departure to destination. If the ship accelerates further then mid-way; then, ship must decelerate at greater than g-force to achieve orbital speed at destination.
REASON: Maintaining near Earth gravity for the pax and crew is a good thing; thus, ship needs to maintain acceleration rate, g = 9.8 m/sec2, for as much of the flight as poss.
Exhaust Vel.Assume interplanetary envelope up to 50%c for exhaust particles.
Growth..Fact.Relativistic mass growth affects propulsion performance.
Exhaust FlowDaily mass (% ship's GW) of high speed particles for g-force.
RangeUse consume rate (ε∇) to determine max days of propulsion time.
Accel. TimeFor max acceleration time, divide propulsion time (tp) by two.
Mid-VelocityAchieve max velocity at midway.  Compute via traditional Newtonian formula.
Accel Dist.Use traditional Newtonian method to compute distance to max midway.
Total Dist.AXIOMATIC: If Accel dist. is halfway; then, double to determine max total dist.
Interplanetary G-force
Original fuel flow: ffsec = 1.0 kg/sec
g = 9.80665 m/sec2 = .489AU/day2
Exh.
Vel.
Growth
Factor
Exh.
Flow
RangeAccel
Time
Mid.
Vel.
Accel
Dist
Total
Dist
dcntptAccVMaxdAccdTtl
.101.0052.81%/Day12.0 day6.0 day2.93 AU/dy8.8 AU 17.5 AU
.201.0211.38%/Day24.7 day12.3 day6.04 AU/dy37.2 AU74.5 AU
.301.0480.90%/Day38.2 day19.1 day9.34 AU/dy89.3 AU 178.6 AU
.401.0910.65%/Day53.2 day26.6 day13.0 AU/dy172.8 AU 345.7 AU
.501.1550.49%/Day70.5 day35.2 day17.2 AU/dy303.4 AU 606.9 AU
Given.1

√(1-dc2)
.2826%

(n2-1)
log(1-%TOGW)

log(1-ε∇)
tp

2
g×tAcc g×tAcc2

2
2×dAc
To g-force to interplanetary, expand ion drive to accelerator drive.
FINAL NOTE-1:Send AI First. To pave the way for human travelers, precede a human expeditionary force with Artificial Intelligence (AI) devices. AI vessels 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.  If a vessel is one AU away; signal will take at least 8 min to reach the vessel, and another 8 min for the response to reach original sender. For 10 AU, signal will take about 83 minutes between vessels and 166 mins for a two way response.  Thus, two way communications will become extremely difficult; immediate responses will become impossible.
Autonomy will be compelled by such communications.  Thus, a robotic "pathfinder" vessel will need to be extremely sophisticated, because such systems will prove essential.
FINAL NOTE-2:Even an pessimistic g-force profile to the Kuiper Belt (beyond the planets) takes weeks.
However, the Solar System extends well beyond Kuiper; some say it extends a full LY from Sol out to the Oort Cloud.
Particle exhaust speeds of 0.5 c will fall far short of getting us that far; we'll need an expanded envelope to truly explore entire Solar System not to mention visit neighbor stellar systems.
CONCLUSION:
Push the particle envelope, accelerate ions much closer to light speed.



VOLUME I: ASTEROIDAL
VOLUME II: INTERPLANETARY
VOLUME III: INTERSTELLAR



SLIDESHOW 

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