PUSH TO INTERSTELLAR
TE assumes that interplanetary gforce propulsion particles will need an exhaust velocity (V_{Exh}) 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 gforce acceleration for lengthy duration of a year or more. 
...can consume reasonable quantities of fuel particles, original fuel flow per second (ff_{sec}), and accelerate them into a constant flow of exhaust particles (ff_{Exh}) .
For this initial example, further assume particles' exhaust speed (V_{Exh}) is 86.67% c. One can express exhaust velocity as a product of decimal component (d_{c}) and light speed, c.
EXAMPLE: V_{Exh} = d_{c }× c = .867 c
Thus, d_{c} = .867 = V_{Exh }/ c
m_{r} =  m_{o} √(1d_{c}^{2}) 

Particle Size
Lorentz Transform quantifies the relativistic growth due to particle speed. Thus, fuel flow per second (ff_{sec}) is the original mass at rest, and exhaust fuel flow (ff_{Exh}) is the relativistic mass at an accelerated velocity.
Notional fuel quantity increases per following table.
ff_{Exh} = n × ff_{sec}
Original Fuel Flow  Exhaust Velocity  Growth Factor  Exhaust Fuel Flow 

ff_{sec}  V_{Exh} =d_{c}×c  n  ff_{Exh} 
1 kg

.8667 c

2.0

2.0 kg

Given  1 √(1d_{c}^{2})  n×ff_{sec} 
Original Fuel Flow  Exhaust Speed  Growth Factor  Ship Mass 

ff_{sec}  V_{Exh} = d_{c}×c  n  M_{Ship} 
1 kg  .943 c  3  83,623 mT 
2 kg  .943 c  3  173,246 mT 
Given  Given  1 √(1d_{c}^{2})  √(n^{2}1)×30.57×10^{6}ff_{sec} 
d_{c}  =  V_{Exh}/c 

n  =  1/√(1d_{c}^{2}) 
n^{2}  =  1/(1d_{c}^{2}) 
1d_{c}^{2}  =  1 / n^{2} 
d_{c}^{2}  =  1  (1 / n^{2}) = (n^{2 } 1) / n^{2} 
d_{c}  =  √(n^{2 } 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).
M_{Ship} × g = ff_{Exh} × V_{Exh} 

Both right side terms can be reexpressed. Exhaust fuel mass (ff_{Exh}) can be rewritten as growth factor, n, times fuel flow per second (n×ff_{sec}). Particle exhaust velocity (V_{Exh}) can be rewritten as decimal component times light speed (d_{c}×c).
M_{Ship} × g = (n×ff_{sec}) × (d_{c}×c) 

Rewrite equation as shown below. Note that left panel shows decimal component, d_{c}, defined in terms of n, growth factor. or d_{c} = √(n^{2 } 1) / n.
M_{Ship} = (n×ff_{sec}) × (√(n^{2 } 1) / n×(c/g) 

c = 299,792,458 m/sec g= 9.80665 m/sec^{2}
c/g = 30,570,323 sec
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 megasec. The two "n"s cancel out and the implicit "/sec" of ff_{sec }cancels out the sec from c/g.
M_{Ship} = √(n^{2 } 1) × 30.57 megaff_{sec} 

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 "gforce").
EXAMPLE: If ship's gforce 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, ff_{sec}.
(One day = 24 hours × 3,600 sec /hour = 86,400 seconds.)
ff_{day} =day × ff_{sec} =86,400 × ff_{sec} 



c =

299,792,458 m/sec


g =

9.80665 m/sec^{2}

Day =

86,400 sec

Day × g /c =  .002826 = .2826% 
Particle Exhaust Speed  Exhaust Fuel Flow  Ship's Gross Weight  Daily Decrease 

V_{Exh}= d_{c}×c  ff_{Exh}=n×ff_{sec}  GW  ∇ 
.866 c  2.00 kg  52,949 mT  0.16 % 
.943 c  3.00 kg  86,465 mT  0.10 % 
.968 c  4.00 kg  118,397 mT  0.07 % 
Given  ff_{sec} √(1d_{c}^{2})  ff_{sec}×√(n^{2}1)×c g  Day × ff_{sec} GW 
Relativistic Growth Factor  Decimal Component Light Speed  Daily Decrease 

n  d_{c}  ∇ 
2  .866  0.16 % 
3  .943  0.10 % 
4  .968  0.07 % 
Given  √(n^{2}1) n  .2826% √(n^{2}1) 
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 gforce propulsion systems. Assume inefficiency (E') decreases with growth factor, n; perhaps, E'=.5×.9^{n2}. Subsequently, Efficiency (E) will increase (1E'); efficiency factor (ε = 1/E) will decrease.
Daily exhaust flow,∇, is the amount of charged particles needed to achieve gforce 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.
Efficiency, (E), will likely improve as humanity learns to design better gforce propulsion systems. Assume inefficiency (E') decreases with growth factor, n; perhaps, E'=.5×.9^{n2}. Subsequently, Efficiency (E) will increase (1E'); efficiency factor (ε = 1/E) will decrease.
Daily exhaust flow,∇, is the amount of charged particles needed to achieve gforce 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.
Relativistic Growth Factor  Decimal Component Light Speed  Vessel's Propulsion Exhaust Rate  Forecast Efficiency Factor  RANGE: Propulsion Time 

ff_{Exh}=n×ff_{sec}  _{VExh= dc ×c}  ∇=ff_{Day}/GW  ε  t_{p} 
2  .866  0.163%  2.000  212 days 
3  .943  0.100%  1.818  381 days 
4  .968  0.073%  1.681  565 days 
5  .980  0.058%  1.574  763 days 
6  .986  0.048%  1.488  975 days 
7  .990  0.041%  1.419  1,197 days 
Given  √(n^{2}1) n  .2826% √(n^{2}1)  1 1 (5×.9^{n2})  log(1%TOGW) log(1ε∇) 
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 (d_{c}) 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×.9^{n2}). (NOTE: There's no way of knowing the actual efficiency of future gforce propulsion systems, but we're sure their efficiency will improve.)
Daily exhaust flow,∇. Previous work approximates this value by dividing .2826% by the term, √(n²1).
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 

%TOGW  d_{c}  n  ∇  ε∇  t_{p} 
40.00%  0.990  7  0.041%/day  .058%/day  882 days 
50.00%  0.990  7  0.041%/day  .058%/day  1,197 days 
60.00%  0.990  7  0.041%/day  .058%/day  1,583 days 
Given  Given  1 √(1d_{c}^{2})  .2826% √(n^{2}1) 
∇
1 (5×.9^{n2}) 
log(1%TOGW)
log(1ε∇) 
Logarithm helps transform %TOGW into available propulsion time.
CONCLUSION. As fuel load (%TOGW) increases, range increases.
V. Assume interstellar flight profile ....
REASON: To maintain Earthlike gravity (gforce) for the pax and crew throughout the multiyear 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 a year of such gforce, ship will be far short of midway between two stars. However, fuel limits require the ship to stop propulsion after first year; thus, it must then cruise at constant velocity until it reaches the deceleration point one year 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 gforce at inside of outer hull. At deceleration point, ship will again emit ion stream to slow vessel down to orbiting velocity at destination.
Growth Factor  At relativistic speeds, particle grows by multiple, n. 
Particle Velocity  Each multiple, n, maps to a relativistic speed. 
Exhaust Flow  Daily mass (% ship's GW) of high speed particles for gforce. 
Consume Rate  Daily mass consumed to maintain exhaust flow. 
Prop. Time  Use consume rate (ε∇) and %TOGW in logs to determine. 
Accel. Time  Divide propulsion time (t_{p}) by four. 
MaxVelocity  Formula uses exponential (t) as shown. 
Accel Distance  Use natural log of daily, light speed remainder [ln(1Δ)]. 
Gro. Fact.  Part. Vel.  Exh. Flow  Cons. Rate  Range  Accel Time  Ship's Max Vel.  Accel Dist  

n  v_{Exh}  ∇  ε∇  _{tp}  t_{Acc}  V_{Max}  d_{Acc}  
2  .866 c  .163%/Day  .326%/Day  212 day  53 day  24 AU/dy  655 AU  
3  .943 c  ,100%/Day  .182%/Day  381 day  95 day  41 AU/dy  2,038 AU  
4  .968 c  .073%/Day  .123%/Day  565 day  141 day  57 AU/dy  4,295 AU  
5  .980 c  .058%/Day  .091%/Day  763 day  191 day  72 AU/dy  7,510 AU  
6  .986 c  .048%/Day  .071%/Day  975 day  244 day  86 AU/dy  11,709 AU  
7  .990 c  .041%/Day  .058%/Day  1,197 day  299 day  99 AU/dy  16,870 AU  
8  992 c  .036%/Day  .048%/Day  1,429 day  357 day  110 AU/dy  22,939 AU  
9  994 c  .032%/Day  .042%/Day  1,669 day  417 day  120 AU/dy  29,842 AU  
10  995 c  .028%/Day  .036%/Day  1,915 day  479 day  128 AU/dy  37,490 AU  
11  996 c  .026%/Day  .032%/Day  2,166 day  542 day  136 AU/dy  45,793 AU  
See previous tables.  t_{p} 4  c[1(1ε∇)^{t}] 

FINAL NOTE1:

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 NOTE2:

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) Oneway 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 gforce for one year for initial acceleration to AC, cruise for a few years, gforce 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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