Gforce Interstellar Range: Ship Mass Half Life
recognizes that even with particle accelerator propulsion systems, interstellar travel will take years (versus days for interplanetary). Thus, the interstellar portion of our thought experiment uses better computation methods to determine gforce ranges.
Interstellar Profile Assumptions
Constraint: gforce needed to maintain Earthlike conditions and best environment for human travellers.
Need gforce acceleration to attain sufficient portion of light speed to make interstellar flight duration reasonable. (few years).
Need gforce deceleration to slow down.
However, gforce propulsion requires fuel, and interstellar voyages take so long that a starship would easily consume well over 100% of its mass in fuel during the multiyear voyage.
Thus, interstellar flights also need cruise phase constant velocity. No fuel consumption, no gforce; thus, regain gforce through longitudinal spin, and crew/pax lives on inside of outer hull.
Fuel Consumption Assumptions
 For interplanetary flights, TE assumed that half the ship's mass could easily suffice(maybe water). For interstellar flights, TE might need to consider higher percentages of vessel mass to be available as fuel. More fuel increases potential propulsion time, t_{p} , which can increase the durations of both acceleration/deceleration phases to further decrease the cruise phase duration:
 cruise distance will be slightly shorter
 cruise velocity will be slightly greater
 Higher performing particle accelerators could accelerate fuel particles to even greater exhaust speeds. The greater momentum would result in an even better energy exchange between exhaust particles and ship (less fuel could propel a bigger ship).startfuel
Ship's propulsion system accelerates a predetermined mass of ionized particles (plasma) to exit vehicle at relativistic speeds. Since these ions come from designated, onboard material, this material fuels the spaceship much like gasoline fuels our automobiles. For both vehicles, when we add fuel, the total gross weight (GW) increases; when fuel transforms into energy to produce motion, fuel is consumed, and GW decreases.
The fuel is consumed at a certain rate; thus, ship's gross weight (GW) decreases at this same rate. If we know the amount of fuel and the consumption rate; then, we can calculate the time required to consume this amount, the ship's range.
Additional Assumptions
 Artificially assume 100% efficiency for fuel flow conversions for convenience. 100% Efficiency is very unlikely; however, for convenience:
TE artificially assumes every single particle travels completely through the many kms of waveguides and exits spacecraft exhaust as intended.
TE further assumes no additional energy is required for other energy requirements, such as electromagnetic fields to guide the particles.
Thus, TE artificially assumes 100% efficiency for convenience. Future work will adjust for less then perfect efficiency.  Marginal Fuel Requirements  it makes sense to carry "spare fuel"; it's always a good policy to plan the mission to end with some fuel remaining. Since this fuel is not planned for consumption, it might be better to consider marginal fuel as part of ship's infrastructure. Thus, thought experiment disregards spare fuel for computing range of spacecraft.
 Assume continuous gforce acceleration due to constant particle exhaust speed.
 TABLE1 varies fuel loads given just one particle exhaust speed (v_{Exh} = .99c) to determine Vessel Ranges based on fuel load.
 TABLE2 varies particle exhaust speeds given just one fuel load (F_{0} = 50% GW_{0}) to determine potential Vessel Ranges based on particle exhaust speed.
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Set particle exhaust speed such that, v_{Exh} = .99 c ; thus, d_{c} = .99
Thus, Lorentz Transform (LT) shows that exhaust particles
will increase over 7 times greater due to relativity. ff_{Exh} = m_{r} ff_{sec} = 7.09 ff_{sec}
Given g=9.8065 m/sec^{2} and c=299,792,458 m/sec, previous work leads to:
Thus, this example defines ∇, Daily Difference in Ship's Gross Weight: ∇ = %TOGW_{Day} = .04%/day 
Determine Propulsion Time, t_{Prop}BACKGROUND: Thought Experiment (TE) assumes onboard particle accelerator increases particle speed to relativistic velocities and expels a predetermined mass of high speed particles out of ship's exhaust; resultant momentum propels ship for a velocity increase equal to gforce accelerationRecall following points.ff_{sec }Fuel flow per second determines fuel consumption; thus, ever decreasing gross weight of gforce vessel. For every second of flight, our notional spaceship consumes a quantity of "at rest" fuel which we'll call fuel flow per sec (ff_{sec}). This is the fuel's mass prior to conversion to plasma (ions, particles with charges). Fuel consumption is reckoned in terms of fuel at rest "ff_{sec}" which is at zero velocity in relation to its host vessel. "At rest" fuel gives the daily difference in gross weight (∇, a certain gross weight percentage decrease per day).
ff_{Exh }Fuel exhaust mass provides propulsion for gforce vessel. For every second of flight, a particle accelerator uses a system of magnetic and electrical fields to accelerate these charged particles to large fraction of light speed. Relativism and a century of particle accelerator experience both tell us that ff_{sec} will grow as particles accelerate to near light speed; we call this new quantity ff_{Exh}, fuel flow of exhaust particles. Collective exhaust particle momentum (near light speed velocity times relativistic mass) enables spaceship's propulsion. To account for relativistic growth, TE uses relativistic mass growth factor, m_{r}; this is computed with Lorentz Transform (LT). To state this exhaust rate, TE borrows the symbol, ∇ ("nabla"), and uses it to represent percentage of ship's gross weight consumed each day, the "daily difference".
then, gforce fuel consumption =∇ ∇= .04% GW/day If ship's initial gross weight (GW_{0}) is 100 metric Tonnes (mT); then, ship will consume .04 mT of fuel on first day of powered flight. F_{Day} = ∇ * GW_{0} = .04% * 100mT = .04 mT GW_{1} = GW_{0}  F_{Day} = 100mT  .04mT = 99.96 mT 
1st day of trip, GW_{1} = GW_{0}  first day's fuel consumption = 99.96 tons. 2nd day, ship consumes .o4% of the remainder which will be less then .04 mT. On day, t, fuel consumption will be even less and so on as shown in following table. To daily decrement 1% of ship's gross weight (GW_{t}), multiply original quantity (GW_{o}), by (1∇) as shown.


Exponentials can readily approximate fuel consumption. Any inexpensive scientific calculator has an exponent function; thus, we can readily calculate numerous guesses as to how many days is required to reduce ship's original gross weight (GW_{0}) by 50%.
Consider that as fuel is consumed, GW decreases; thus, a lesser GW needs less fuel to propel it, and this cycle continues throughout powered flight.
we eventually determine t integer which produces a value much closer to 50%.
There might be a more graceful way to determine this range. Logarithms A more convenient method might use logarithms. A quick review follows: log10^{a} = a For example, log(100) = log(10^{2}) = 2 What if a^{t} = b, where a and b are known, but t is unknown? Exponential characteristics allow us to accomplish following:
For present case, use following values:
Thus, logarithms can quickly approximate range for different Fuel Loads.
 
Term "half life" means a duration for which a specified shrinking substance decreases by 50%. Shrinking could be due to radioactivity like thorium or uranium; or it could be due to fuel consumption like automobiles, aircraft or gforce spacecraft. For example, radioactive thorium might decrease by 50% in 60 seconds.This could be described by M_{tho}(1Δ)^{t} = .5 * M_{tho} where Δ is % difference in mass per second and the t is time in seconds. Divide both sides by M_{tho} and set t = 60 sec; we get: (1Δ)^{60} = .5 Percent Initial Gross Weight (%GW_{0}) is the amount of fuel required for entire mission. This value is expressed as a percentage of the vessels' initial gross weight, GW_{0}. %GW_{0}= F_{0}/GW_{0} 
RelativisticGrowth  Decimal c  DailyDiff.
 Prop. Time  Accel. Time  Max.Velocity  Accel Dist  Total Dist  

m_{r}  d_{c}  ∇  t_{P}  t_{Acc}  V_{Max}  d_{Acc}  d_{P}  
%/day  Days  Days  AU/day  AU  LY  
3  0.943  0.10%  695 


 .40 LY  
4  0.968  0.07%  952 


 .72 LY  
5  0.980  0.06%  1,204 


 1.08 LY  
6  0.986  0.05%  1,454 


 1.48 LY  
7  0.990  0.041%  1,703 


 1.96 LY  
8  0.992  0.036%  1,951 


 2.48 LY  
IV  √(m_{r}^{2}1) m_{r}  0.282%/dy √(m_{r}^{2}1)  log(%GW_{Fin}) log(1∇) %GW_{Fin}=.5=50%  t_{Prop} 4  c(1(1Δ)^{tAcc}) Δ = .2826% 
LY = 63,241 AU  4 * d_{Acc} 
 m_{r} is relativity growth factor computed from Lorentz Transform.
 d_{c} is decimal component of particle exhaust speed when expressed as decimal light speed.
 ∇, "daily difference", is the amount of fuel needed for each day's propulsion, expressed as a percentage of ship's gross weight (%GW).
 Δ , "daily delta", is percentage of light speed achieved after first day of gforce flight. Δ= 86,400 g / c = .2826%
 %GW_{Fin }is the final gross weight of the spacecraft after the flight's required fuel (%TOGW) is consumed. %GW_{Fin }is expressed as a percent of orginal ship's mass and can be computed: 100% %TOGW
 t_{P} is total powered flight time; this is also known as the "range".
 t_{Acc} is acceleration time.
 v_{Max}is velocity achieved after gforce acceleration for t_{Acc} days. c = 173.15 AU/day.
 d_{Acc} is acceleration distance. 1 light year (LY) = 63,241 AU.
 d_{P} is total distance for powered portion of flight for both destination and return flights. This includes distances for acceleration and deceleration (but not cruise).Powered Destination Distance = d_{Acc} + d_{Dec}Powered Return Distance = d_{Acc} + d_{Dec}TE assumes same propulsion distances for both destination and return flights. For consistent gforce, TE also assumes acceleration distance = deceleration distance; thus:d_{P} =Destination (d_{Acc} + d_{Dec} )+ Return (d_{Acc} + d_{Dec} )= 4 * d_{Acc}
 TE assumes ever improving technology will lead to higher performance, higher efficiency gforce propulsion systems such that particle exhaust speed will keep increasings.
 Ever increasing particle speed and system efficiencies will lead to ever shrinking ∇ s, daily increment of %GW needed to maintain gforce.
 Leads to greater vessel range; i.e., available propulsion time, t_{P}.
 Half Life Method
Radioactive Half Life GW Remainder As duration increases, radioactivity decreases remaining mass of certain elements. As number of days of gforce flight increase, fuel consumption decreases ship's mass. ...easily computed with logarithms. ...easily computed with logarithms.
 Portion of t_{P} actually used for propulsion; that portion divided by 4 to determine acceleration time, t_{Acc }.
 Recall t_{Acc} is gforce duration from beginning of flight to start of cruise phase. The longer the initial gforce duration, the higher velocity achieved, the shorter the cruise phase.
 At completion of curise phase, an equal duration is required to decelerate to destination.
 These three phases compose the departure leg (acceleration, cruise, deceleration); only the acceleration and deceleration phases required gforce power.
 Similarly, return leg requires same three phases, and TE assumes their durations equal corresponding departure phase.
 Thus, there are four gforce durations, each of them = initial t_{Acc}.
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