Interstellar: Temporary Mat'l
Upcoming tasks:
262011: Spreadsheet/table with following:
Distance / T_{Acc} / V_{Acc}
LY / Monotonic increase /Monotonic increase
Distance is interstellar
T_{Acc} increases from base of 360 days
Arbitrary increase of 1% for every LY.
Subsequent increase in V_{Acc} will decrease cruise time.
From Volume I, recall following benefits of constant gforce acceleration:
 Much quicker travel times.
 Simulated gravity.
Further recall that gforce acceleration requires following flight profile.
 Accelerate to the midpoint.
 Reverse direction of spacecraft so that exhaust propellant particles now point to destination.
 Decelerate remaining half of journey to destination.
Interstellar distance vastly increases scope of this profile. Consider following items:
1. Accelerating to Midpoint. Interplanetary distances are just a few AUs; thus, gforce acceleration will take just a few days. However, midpoint to the nearest star, AC, is over 2 LYs which is over 120,000 AUs; this takes years.
2. Velocity at Midpoint approaches c, light speed. Numerous published works in both theory and experiment continue to confirm that nothing can travel at a speed greater then c; but continuing to accelerate can take you closer and closer. Thus, relativistic effects can become quite pronounced.
3. Theoretical Best Time would be acheived at light speed, c. Of course, we know spacecraft's final velocity, V_{fin} can ever approach but never achieve 100% c; thus, we can use c as an intuitive upper limit, which means 4.3 years with no time for accel/decel.
Even this theoretical best time is a long time to accelerate a constant flow of ions out of our spacecraft. Can this be done? This is discussed further in next chapter, Practical Range. This chapter, Practical Profile, discusses several trip profiles which might prove practical for interstellar flight??
Orbital:

Interplanetary:

Interstellar:
 



 
Allows easy comparison with escape/orbital velocities of dept/dest planetary systems. (i.e., Earth's escape velocity, e= 11 km/sec.)  Since thot exp. considers most interplanetary flights to take just days, g= 0.489 AU/day^{2} can be a very convenient value of g, and AUs per day is easily calculated.  Determines %c achieved by first day's gforce acceleration. Also enables discussion in considering relativity to compute %c velocities for subsequent days of acceleration.(See below.) 
We must transition from the convenient values used in first volume, because the much longer interstellar voyages cannot afford to give away value gained through more precise values.
Transition from "convenient values" used in Vol I, Interplanetary 

g
≈
10 m/sec^{2}
≈
864 km/sec /day
≈
0.5 AU/day^{2 }
≈
0.3%c /day
≈
g
to more precise values in this Vol II, Interstellar.
g
=
9.80665 m/sec^{2}
=
847.3 km/sec /day
=
0.489 AU/day^{2}
=
0.282%c /day
=
g
Transition from Newton's Formulas to Einstein's.
Newton formulas work well for the special case of extremely slow motion which is the portion of the speed spectrum where we live our lives. Thus, Newton formulas are "good enough" for the interplanetary portion of our thought experiment because our accelerating spacecraft needs only days to reach nearby planets and thus never achieves enough speed compared to light speed. However, we need the more general Einsteinian formulas for instellar travel. Key example: While traveling to Mars, our gforce spacecraft might accelerate for two days before starting the deceleration phase. After one day of gforce acceleration, spacecraft has achieved .282% c; Newton's formula, acceleration times time, would give us two days times acceleration or 2 days times 0.282%c/day or .564%c, barely one half of one percent c, still a relatively slow speed even tho much greater then our current experential base. On the other hand, flights to midway from Sol to AC would take at least two years; Newton's formula ( g * t) would give us 731 days * .282%c/day = 206% c, twice the speed of light. Of course, this is impossible.
However, this volume considers interstellar travel. Our particle accelerator, propulsion system spacecraft will take years to perform this travel; this is sufficient time for constant gforce acceleration to achieve significant fractions of light speed. Thus, this volume must use more general formulas. These are discussed in following tables.
Transition to interstellar.
Recall again Einstein's Special Relativity axiom that all observers should measure same value for c regardless of observer's velocity. Thus, our thought experiment might expand to include a series of observers who are able to accelerate for a certain time then maintain the attained velocity indefinitely. Therefore, observer A might remain static at the starting point; but all other observers accelerate at gforce. Observer B might accelerate at gforce for one day, then maintains constant velocity while observers C through Z continue accelerating. Observer C would stop accelerating after for two days, then stop, and so on. Eventually, we have a series of observers separated by one day's gforce acceleration from the previous one and the subsequent one. After all observers have stopped accelerating, relative velocities between them becomes static.
All observers (A through Z) would consistently measure light speed, c, as 299,792,458 meters per second at all times. However, they would measure each other's speed differently. Thought experiment conjectures that Observer A would measure B's velocity as .282%c after the end of first day. However, B continues to observe light speed as 299,792,458 meters per second, and thought experiment further proposes that B will measure Observer C's velocity as .282%c after end of second day. What value does A measure for Observer C's velocity after end of day two??
To answer that question, define a term, R, as remainder of light speed, c, not yet attained by a specified observer. For example, A would observe B's remainder, R, to be c (1  .00282) for B after first day of gforce acceleration. Of course, B would observe same R = c (.99718) for C after second day of gforce acceleration.
All observers (A through Z) would consistently measure light speed, c, as 299,792,458 meters per second at all times. However, they would measure each other's speed differently. Thought experiment conjectures that Observer A would measure B's velocity as .282%c after the end of first day. However, B continues to observe light speed as 299,792,458 meters per second, and thought experiment further proposes that B will measure Observer C's velocity as .282%c after end of second day. What value does A measure for Observer C's velocity after end of day two??
To answer that question, define a term, R, as remainder of light speed, c, not yet attained by a specified observer. For example, A would observe B's remainder, R, to be c (1  .00282) for B after first day of gforce acceleration. Of course, B would observe same R = c (.99718) for C after second day of gforce acceleration.


=
c (1.00282)(1.00282)
=c (.99718)^{2}
Recall Observer C accelerates for 2 days, then constant velocity.
This leads us to final equation for R:
R_{i}
=
c (.99718)^{i}
=c (1  Δ)^{i}
"i" indicates number of days traveled at gforce acceleration.
Transition our thought experiment's attention from notional observers to notional spacecraft after one, two, three, . . . i days of gforce acceleration. To determine spacecraft's velocity attained after i days of acceleration, use following equation:
v_{i} = c  R_{i}
v_{i} = c[1 (1 Δ)^{i}]
v_{i} = c[1 (1 Δ)^{i}]
Exponential formulas transition from Newton to Einstein
Conclusion two part interplanetary profile won't work for interstellar.
To achieve practical gforce acceleration, we must overcome some limitations. These include the following:
1st, TECHNOLOGY. We have yet to design, build and operate a spaceship with a particle accelerator as the propulsion system; thus, the most compelling limit will be the transition from notional to actual. Of the numerous design tasks, an extremely difficult one will involves enormous energy to bring exhaust particles to near light speeds. Once solved, the increased mass and speed of the exhaust fuel flow (ff_{Exh}) will product an enormous momentum to propel the spaceship at gforce acceleration throughout powered flight. Throughout these volumes, several tables show that the higher the particle exhaust speed, the lower the quantity of original fuel mass (fuel flow per sec, ff_{sec}) needed to propel the ship at gforce. This means that less fuel needs to exit the spaceship for same distance traveled. HOWEVER, this increased momentum comes at a cost; much more energy is required by spaceship to accelerate the particles to these near light speeds.
2nd, FINITE RANGE. Above gforce requires fuel consumption, and our space vehicle will eventually deplete the fuel supply. Further considerations:
%TOGW Limit is ½ ship’s mass. Percentage of ship’s TakeOff Gross Weight (TOGW is also known as initial ship’s mass) must be used for fuel with the rest needed for infrastructure and payload. Thot exp assumes 50%, but it could range from 10% to 90%. 
Return flight mandates a reserve. Cautious planning leads us to assume no refueling at destination or anywhere else along the flight path. Thought experiment assumes another 50% reduction to plan for return flight. 
Efficiency factor needs a margin. It's safe to assume that our initial flight design will not be perfect. Even if it was, we still have to consider energy needs for life support, and many other auxilliary energy requirements. 
Above factors lead us to determe total propulsion time (t_{p}) from effectively converting 1/8th of the ship’s TOGW to kinetic energy. A further range limitation comes from the flight profile required to maintain gforce throughout the voyage. 
Flight Profile. Accelerate for ½ t_{p}. Another range limitation comes from thot exp’s self imposed flight profile where the ship accelerates to half distance, then decelerates for remaining half. We’ve chosen this profile as best way to maintain gforce throughout the trip. We could possibly improve the range by accelerating perhaps 90% of the distance then decelerating at a much greater force for remaining 10% of distance. However, this much greater force would greatly exceed Earth like gravity and thot exp assumes this to be undesireable. 
3rd. STELLAR DISTANCES. Our finite range seems to keep getting whacked by above considerations; however, the near light speed of exhaust particles impact so much specific impulse that interplanetary space travel is easily accomplished with considerable margin. Consider above interplanetary table, the most demanding range (Kuiper Belt) is easily covered by the least capable row in the interstellar table (where n= 2).
On the other hand, margin is definitely not there for interstellar travel. Consider the most generous table in the interstellar table (exhaust particle traveling so fast that relativistic effects grow it 11 times mass of same particles at relative rest with the spacecraft). Conservatively, resulting momentum gives spacecraft a range of 329 days of acceleration to take us to 1/3 of one LY; another 329 days of deceleration would take us to .64 LY, the entire distinace is not even close to the Oort Cloud, edge of our Solar System.
On the other hand, margin is definitely not there for interstellar travel. Consider the most generous table in the interstellar table (exhaust particle traveling so fast that relativistic effects grow it 11 times mass of same particles at relative rest with the spacecraft). Conservatively, resulting momentum gives spacecraft a range of 329 days of acceleration to take us to 1/3 of one LY; another 329 days of deceleration would take us to .64 LY, the entire distinace is not even close to the Oort Cloud, edge of our Solar System.
Thus far, our notional spacecraft hasn't made any significant progress toward our nearest stellar neighber; yet, our notional spacecraft has already exhausted available fuel. Thus, we must now use reserve fuel to return back to departure point, while still far short of our goal.
4th. RELATIVISTIC PARADOX. Consider an extreme scenario from one of Robert Heinlein's novels, which elaborates on the "twin paradox". Two identical twins grow up together then separate, where one twin accelerates to near light speed, then travels on an interstellar journey for a few years. The traveling twin ages a few years and eventually returns to Earth as still a young man. Upon his return, he's amazed to discover that his meager savings has made him very wealthy and his twin has become an old man; so old, that he marries his twin's greatgranddaughter. This fanciful story illustrates that relativistic effects could become very pronounced; thus, this thought experiment assumes that near light speed travel will be accomplished in a carefully controlled manner. For example, initial trips to Alpha Centauri might attain a cruise velocity of perhaps .5c, observing results, then incrementing subsequent trip's cruise speed to perhaps .51c, and so on.

For initial flights from Earth to other planets, flight planners must assume that spacecraft needs entire fuel load for all four phases. Perhaps after humans have gained considerable spaceflight experience, we might learn to gather fuel from other places in the Solar System. Til then, we'll have to assume the only fuel available is what we start off with.
Getting there??
Seesaw velocity curve vs. flattop velocity curve.
Problem: maintain near Earth gravity conditions for entire interstellar voyage.
One way: alternately accelerate then decelerate for fairly long periods. This constant force could maintain near Earth gravity conditions as long as the ship could maintain powered flight. Note following items:
Thus, trip will be a series of acceldecel cycles.
Arbitrarity pick convenient numbers to make up this cycle; thus, we'll accelerate for 300 days:
v_{Fin} = g * t = 150 AU/dy = .866 c
Note: size increase and time dilation effects make this choice particularly interesting.
v_{Ave} = g * t / 2 = 75 AU/dy = .433 c
d_{Accel} = t * v_{Ave} = g * t^{2}/2 = 300 days * 75 AU/dy = 22,500 AU = .3565 LY
To maintain near Earth gravity throughout deceleration part of cycle,
Above profile might get us to Oort Cloud, it won't get us to even the nearest stellar system (happens to be Alpha Centauri, AC).
Building upon above model, three possibilities come to mind.
1. Cyclus Gigantus .
We must adjust our thought experiment. Additive method clearly does not work because after 350 days of gforce acceleration, it shows final velocity exceeding light speed, an Einsteinian impossibility.
Thus, we hypothesize another method based on following two assumptions which relate spacecraft velocity to c to the observer’s position and velocity.
First assumption, onboard observers always see light speed, c, as 300,000 km/sec faster (then their current spacecraft speed).
· On board propulsive force still accelerates them; thus, they still feel gforce to simulate near Earth gravity.
Thus, we hypothesize another method based on following two assumptions which relate spacecraft velocity to c to the observer’s position and velocity.
First assumption, onboard observers always see light speed, c, as 300,000 km/sec faster (then their current spacecraft speed).
· On board propulsive force still accelerates them; thus, they still feel gforce to simulate near Earth gravity.
· Distance to destination continues to decrease.
· Their velocity ever increases in relation to Earth bound observers; thus, their ship (including passengers and crew) undergo relativistic changes
· Their velocity ever increases in relation to Earth bound observers; thus, their ship (including passengers and crew) undergo relativistic changes
· · ship’s mass increases.
· · time decreases.
These changes also affect the ship bound observers; thus, they don’t notice them.
Second assumption, Earth bound observers closely monitor the spacecraft as it proceeds toward its stellar destination. They also see c as 300,000 km/sec faster (in relation to their position/speed).
· They see static distance for the destination,
· They see spacecraft distance increase in a non linear fashion.
· They eventually determine the daily increase in spacecraft velocity to be 0.288% of the difference between c and spacecraft's speed for previous day.
Second assumption, Earth bound observers closely monitor the spacecraft as it proceeds toward its stellar destination. They also see c as 300,000 km/sec faster (in relation to their position/speed).
· They see static distance for the destination,
· They see spacecraft distance increase in a non linear fashion.
· They eventually determine the daily increase in spacecraft velocity to be 0.288% of the difference between c and spacecraft's speed for previous day.
Recall from Vol. 1, that most interplanetary flights require less than 10 days for gforce acceleration. Thus, interplanetary flight profiles all consisted of accelerating to midway, then deceleration to destination.
PEAK PATTERN. Extend interplanetary pattern to interstellar. Accelerate to midpoint, then decelerate to dest.Earthbound observer would measure following times for gforce spacecraft.
1 year. Spacecraft .38 LY dist . traveled, .64c velocity.
2 yeas. 1.15 LY, 87% c.
3.1 years, reached midway from Sol to AC, 2.15 LY, .95c. Ship rotates 180° so that exhaust particles reverse direction and point toward AC, destination. Maintaining symmetry.
1.1 years from midpoint or 4.2 years into voyage, 3.15 LY, 87% c.
5.2 years slowed back down to 64%c, and total traveled dist = 3.92 LY.
6.2 years, voyage ends for total distance of 4.3 LY and slowed down to very close to zero velocity.
Problems with technology and relativity.
Tech. Previous work with momentum equivalence equation indicates that to maintain gforce throughout the entire voyage would require enormous performance from the particle accelerator used as the propulsion system. For example, if exhaust particle speed stays at .866c (particle size grows to twice original size, it's going so fast), initial estimates for total propulsion time would be a max of 601 days because at that duration, propulsion system would have consumed 100% of ship's mass. To make the flight from Sol to AC (as shown in accompanying diagram with fuel reserves to allow for reasonable infrastructure, payload, and sufficient fuel reserves to allow for peripheral power requirements, return flight and expected inefficiencies, thot exp. projects that n would have to be at least 22 (exit fuel particle = 22 x entrance fuel particle.) Thot exp. assumes problematic to maintain consistent, reliable, controllable fuel particle exhaust speed such that
ff_{Exh} = 22 x ff_{sec}
.Relativistic effects on ship and contents (which includes lots of crew and pax) is unknown and should be approached cautiously.
Scale. Finally, the diagram displays best possible case of interstellar travel because it's to the nearest stellar neighbor. Other stellar destinations would require even greater scale.
SAWTOOTH PATTERN. Another profile might be a “sawtooth” repetition. Create a cycle of acceleration/deceleration and til arrival at destination star. This maintains gforce throughout flight and still significantly reduces travel time compared with vehicles traveling a much slower, constant velocity (recall examples: Voyager and Pioneer spacecraft).
For example, let's arbitrarily pick 325 days as the number of days to accelerate then observe the results.
Vfin = .6c
Dist Acc = .31 LY. = Dist Dec
Dist cycle = .62 LY = Dist Dec + Dist Acc
Total cycles = DistA.C. / Distcycle = 4.3 AC / Dist cycle = 7 cycles
NOTE: For convenience, chose arbitrary time/distance such that cycles from Sol to AC would be an integer number.
NOTE: While thought experiment presumes impracticality of this profile, one can conjure of up scenarios where spacecraft might need to decelerate in midst of trip. For example, "Snowballs from Oort" describes method where midflight refueling could happen.
Thus, the flight profile would become:
Accelerate at g; constant velocity travel, c; decelerate at g.
Corresponding distances:.516 LY. ; 2.97 LY ; .516 LY.
Corresponding travel times:1.0 year; 3.0 year; 1.0 year.
Total travel time = 5.0 years.
FLATTOP PATTERN. Accelerate, cruise, decelerate.
Key Concepts
In our thought experiment, a spaceship accelerates at gforce to simulate gravity and attain enormous speeds. Since our spaceship will eventually orbit some destination, it must slow down at the same gforce for the same amount of time. Thus, our spaceship is constrained to accelerate to midpoint of voyage, then decelerate for remaining half of the flight.Given above scenario, thought experiment defines key concepts, Δ, and R.Δ  Daily difference in spaceship's velocity is due to acceleration caused by constant gforce. We approximate this percentage value by using our more precise values to close approximate speed attained after just one day of gforce acceleration. Velocities attained after subsequent days are determined as shown below (to be added later)R – Remainder of c not yet attained.
We now know that a massless photon travels at velocity, c; but our gforce spacecraft can never attain that velocity regardless of how many days accelerated.
Unfortunately, Newtonian physics presents a paradox. As previously described, Newton’s laws of motion has a gforce spaceship accelerating at about .489 AU/day^{2} (= g = 9.80665 m/sec^{2}). Since light speed, c, equals 173.15 AU/day, Newtonian physics has our notional spaceship traveling at light speed in 355.5 days (=c/g) which we now know is impossible. Thereafter, Newtonian physics has ship exceeding c, which is even more impossible.
Clearly, we need another way to describe our spaceship’s motion. Perhaps Calculus will show us a way.
Einstein stated that c is always observed as a constant value regardless of the velocity of the observer. Thus, multiple observers at different velocities will all observe photons traveling at same speed, c= 173.15 AU/day. All the observers might observe each other traveling in far ranging speeds, but two facts stay steadfast:
1. No one is observed at or over c, light speed.
2. All observers consistently measure photons at the same velocity, 173.15 AU/day.
1. No one is observed at or over c, light speed.
2. All observers consistently measure photons at the same velocity, 173.15 AU/day.
Consider two observers: one is on the notional spacecraft, and the other observer stayed behind at the ground support system on planet, Earth. After one day of gforce travel, Earth bound observer (O_{E}) measures spacecraft’s velocity at 0.2826% (= .489 AU/day / 173.145 AU/day) of light speed. The one day observer (O_{1}) observes Earth receding at .002826c, but he continues to measure c as 173.15 AU/day; this is the same value for c as determined by O_{E}.
O_{E} 
observes O_{1}'s speed as 0.2826%c

O_{1}
 
O_{E}

observes Ship's speed as everincreasing

Ship
 
O_{E}  observes c as 173.15 AU/Day===>  c 
Therefore, we determine that after first day of constant gforce acceleration, Earth bound observer will measure our spacecraft to have attained approximately 0.2823% of light speed. Thus, we now need to determine likely measurements for succeeding days.
Thought experiments are wonderful things! Not only are they extremely low cost models, but they’re not constrained by life support considerations. For example, Einstein’s famous thought experiment was about the accelerating elevator accelerating through deep space, and it didn’t need to carry an array of equipment for the occupant, not even food or air. Similarly, our thought experiment is going to discharge observer, O_{1}, after exactly one of gforce acceleration when O_{1} will maintain velocity attained by spaceship after exactly one day. In like manner, spaceship will accelerate for a second day and discharge O_{2} to maintain greater velocity achieved after two days of gforce acceleration. After these two days, consider objects as shown in following diagram.
O_{E} 
observes Ship's speed as ever increasing========>

Ship
 
O_{1}

observes Ship's speed as ever increasing=====>
 Ship  
O_{1}

observes O_{2}'s speed as 0.2823%c ===>

O_{2}
 
O_{E} 
observes O_{1}'s speed as 0.2823%c ===>

O_{1}
 
O_{1}

observes c as 173.15 AU/Day ===>

c
 
O_{E}

observes c as 173.15 AU/Day ===========================>

c
 
_{}

c = 299,792,458 m/sec = 173.15 AU/Day ; thus, 0.2823%c = 0.49 AU/day

O_{1} is discharged exactly at end of day one and serves as permanent marker of spaceship's velocity at that time.
Similarly, O_{2} is discharged exactly at end of day two to show spaceship's velocity at that time.
Why do we say that O_{E} and O_{1} both observe same initial velocity increase of their successor? It seems counterintuitive for these speeds to be the same, but consider following items:
 Einstein says that all observers observe light speed as c, regardless of observers' respective velocities.
 Both O_{E} and O_{1} initially observe a relative ship's initial velocity as zero.
 O_{E} certainly observed ship beginning of flight, start of day one when relative velocity was zero. He then observed ship accelerate at g for rest of the day.
 O_{1} was discharged by ship at start of day one. For an instance upon discharge, he shared ship's velocity; thus, he initially observed ship at relative velocity, zero, exactly at start of day two, when he began his observation. Thereafter, he observed ship accelerating at g.
 O_{2} was discharged by ship at start of day two. Being at constant velocities, O_{1} and O_{2} both serve as permanent markers of gforce spaceship velocities achieved after days one and two respectively.
 Given above conditions, O_{E} and O_{1} have to measure same velocity increase for their respective successors.
 On the other hand, O_{E} observes ever increasing velocity for the gforce spaceship; thus, he must measure different velocities for O_{1} and O_{2}. Thus, O_{E} and O_{1} must observe different values for O_{2}.
 This brings us to the question:
What does O_{E} observe as O_{2}, the ship's second daily velocity?
To help O_{E} predict gforce spaceship's velocities for succeeding days; let's define terms: delta and remainder.
Delta Concept 
What if our starship picked different duration to discharge observers: hourly, every min, every sec?
Every second!!!!!!!!!!!!!!
Everyday, a string of 86,400 observers would each measure the starship's speed for every second throughout the entire day. If every single observer observed exactly same relative value to its successor, this value would prove to be remarkably consistent, and it would certainly be a more precise value then then the Daily Diff, Δ, mentioned above.
Δ_{Sec }(%c/sec) Thought experiment's gforce spaceship accelerates at rate, g = 9.80665 m/sec^{2}; thus, increasing velocity 9.80665 m/sec after one second. Thus, each of our 86,400 observers must observe its successor's velocity at 9.80665 m/sec greater then it's own velocity. To express this relative velocity increase as %c:
Δ_{Sec }=  g * 1 sec c  =  9.80665 m/sec 299,792,458 m/sec  


g =  9.80665 m sec^{2}  =  .489 AU day^{2}  


g * day
c
=
.489 AU/day
173.145 AU/day
Δ =  .00282c  =  .282% c 

This presents a quandary. First: if first day of gforce acceleration brings the spacecraft to .282% light speed, does 2nd day double it to .564% c??
Second: if so, can we continue to add/multiply days of gforce acceleration times 0.282%?
Of course, we immediately realize the answer to second question must be "No"; otherwise, in about 355 days, this method exceeds light speed, an Einsteinian impossibility.
Thus, we eventually realize the answer to first question must also be "No". The model which describes gforce velocity must apply universally. It cannot apply to gforce spacecraft for 354 days then stop applying on day 355. It must apply for all durations; even for small durations such as two days and less. Thus, 2nd day velocity cannot double first day's velocity.
Perhaps we can further leverage the Delta Concept to resolve this quandary.
O_{3}

observes c as 173.15 AU/Day===>

c
 
Ship 
observes c as 173.15 AU/Day===>

c
 
O_{2}

observes c as 173.15 AU/Day======>

c
 
O_{1}

observes c as 173.15 AU/Day=========>

c
 
O_{E} 
observes c as 173.15 AU/Day============>

c

_{}
 
_{}
 
_{}
 
Ship
 
O_{2}  <==O_{3}observes O_{2}'s speed as Δ. 
O_{3}
 
O_{1}  <==O_{2}observes O_{1}'s speed as Δ.  O_{2}  
O_{E}  <==O_{1}observes O_{E}'s speed as Δ.  O_{1} 
_{}

_{}
 
_{}
 
_{}
 
Ship
 
O_{2}  O_{2}observes O_{3}'s speed as +Δ.==> 
O_{3}
 
O_{1}  O_{1}observes O_{2}'s speed as +Δ.==>  O_{2}  
O_{E}  O_{E}observes O_{1}'s speed as +Δ.==>  O_{1} 
_{}

_{}
 
_{}
 
_{}
 
O_{E}  O_{E} observes ship's speed as ever increasing ======> 
Ship
 
O_{E}  O_{E}observes O_{3}'s speed as +Δ + ??======> 
O_{3}
 
O_{E}  O_{E}observes O_{2}'s speed as +Δ + ?====>  O_{2}  
O_{E}  O_{E }observes O_{1}'s speed as +Δ.==>  O_{1} 
_{}

Which brings to the current question: If we use Newton's Laws to estimate initial day's velocity due to gforce acceleration, how can we more accurately estimate 2nd day's velocity, 3rd day, succeeding days??
(Note: Recall that Newton's Laws are most accurate for initial day, as time goes on, they become less accurate. In an extreme case, if we continued Newton's Laws to describe gforce spaceship's velocity for an entire year, our ship's velocity would exceed light speed, an impossibility.)
Remainder concept. R = (1  Δ) . From the viewpoint of the Earthly observer, spacecraft reaches 0.283 % c after one day of gforce acceleration. Conversely, the spacecraft has 99.717% c left to accelerate before reaching light speed. This remainder can be expressed as:
R = 1  Δ = 100% c  0.283% c = .99717 c = 172.766 AU/day
Above table clearly indicates that if we assume same value for each observer's relative velocity for initial daily velocity increase; then, it follows that each observer would also observe same value for relative velocity of initial daily remainder of light speed.
We make the intuitive leap (i.e. assumption) that O_{E} will interpret succeeding R values as shown below.
We assume second day's remainder value to be 99.717% of R_{1}
R_{2} = .99717 * R_{1} = R^{2} = c (1  Δ)^{2}
It follows that R_{t} = R^{t} = c (1  Δ)^{t} . For example, we can readily compute the remainder for the 100th day of gforce acceleration: R_{100} = (1  Δ)^{100} which is readily available from any calculator with the x^y function. Recall that we assume a consistent Δ which always equals 0.2826% of previous day's attained velocity.
R_{100} = c (1  .002826)^{100} = c (.99717)^{100} = c * 75.32% = 0.7532 c
Of course, velocity is easily computed from any remainder value by subtracting it from 100%.
v_{100} = c (1  R^{100}) = c (1  (1  Δ)^{100}) = c (1  0.7532 ) c = .24678 c
Recall thought experiment assumes c = 173.15 AU/day
v_{100} = .24678 * 173.15 AU/day = 42.73 AU/day

Ship
O_{E}
O_{E}observes O_{3}'s speed as c(1R^{3}) = 1.47 AU/day==>
O_{3}
O_{E}
O_{E }observes O_{2}'s speed as c(1R^{2}) = .978 AU/day==>
O_{2}
O_{E}observes O_{1}'s speed as c(1R^{1}) = .490 AU/day==>
O_{1}
_{}
Earthbound Observer measures monotonic increase of ship's velocity.
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