Introduce-Flight profile (transient)

Table-4. Exhaust Particle Size Impacts Fuel Burn

Relativistic Mass Increase
ffExh= ffsec/SQRT(1-v2Exh/c2) vExh (per cent c) ffExh (ffsec multiples) (% c) (* ffsec) 50 1.15 60 1.25 70 1.40 80 1.67 90 2.29 IV DV	Apply Lorentz Transform to determine how much exhaust particle grows from original (ffsec). Top row example: When particle exhaust speed is 50% light speed, then exhaust particle mass is 1.15 times original mass. ffsec. Bottom row example (vExh = 90% c): If ffsec = 1.0 kgm, then ffExh= 2.29 times original mass (ffsec).
Particle Size Maps to Particle Speed
We get a more pleasing picture if we list ffExh as integer multiples of ffsec. Thus, we restate relationship between exhaust particle's speed vs. mass. Above table independently increases mass in ffsec multiples, then determine required velocity. For example, for mass to double, particle speed must be 86.6% c. For mass to grow 6 times, speed must be 98.6% light speed.	vExh= c * SQRT(1-ff2sec/ff2Exh) vExh (per cent c) ffExh (ffsec * n) (% c) n 86.6 2 94.3 3 96.8 4 98.0 5 98.6 6 DV IV
Size & Speed to Ship Mass
ffExh= n * ffsec ff2sec/ ff2Exh = 1/n2 vExh (per cent c) ffExh (n * ffsec ) MShip (mega-ffsec ) (% c) n (106 * ffsec) 86.6 2 51.96 94.3 3 84.85 96.8 4 116.19 98.0 5 146.97 98.6 6 177.48 99.0 7 207.85 c * SQRT(1-1/n2) ffExh/ffsec vExh * ffExh / g	New ship size (Mship) column lists values as "mega-ffsec". Thus, if fuel flow values are in grams, then Mship would be metric Tonnes. Further, if ffsec values are in kilograms, then Mshipvalues would be in kilo metric Tonnes (kmT)... vExh= c * SQRT(1-ff2sec/ff2Exh). Let ffExh = n ffsec where n is integer multiple of ffsec , then vExh= c * SQRT(1-ff2sec/(n2 * ff2sec)) vExh= c * SQRT(1-1/n2) ; thus, restate MShip as MShip = c * SQRT(1-1/n2) * n ffsec / g NOTE: n * SQRT(1 - 1/n2) = SQRT(n2 -1) MShip = c * SQRT(n2 - 1) ffsec / g
Determine Daily Diff, Δ
The daily portion of ship's mass that must convert to kinetic energy is a very small percentage. Furthermore, this percentage gets even smaller as exhaust particle speed increases. Note that we kept exhaust particle size as the IV. TakeOff Gross Weight = Fuel + Infrastructure + Payload %TOGW = Percent of Ship mass required for fuel %TOGWTtl = ffTtl/MShip ffDay= Sec per Day *ffsec = 86,400 * ffsec %TOGWDay = ffDay/MShip %TOGWDay = 86,400 * ffsec /(c * SQRT(n2 - 1) ffsec / g). %TOGWDay = 86,400 * g /(c * SQRT(n2 - 1)) Let Sec per Day = 86,400 secs/day c= 300,000,000 m/sec g = 10 m/sec2 Then quantity, Sec per day * g /c = .00288/day Δ = 0.288% /SQRT(n2 - 1)	Δ = %TOGWDay = ffDay/MShip ffExh (n * ffsec ) MShip (mega-ffsec ) %TOGWDay (Δ) n (106 * ffsec) (%MShip/Dy) 2 51.96 0.17% 3 84.85 0.10% 4 116.19 0.07% 5 146.97 0.06% 6 177.48 0.05% 7 207.85 0.04% ffExh/ffsec c*SQRT(n2-1)*ffsec g 0.288% SQRT(n2-1)

Legacy:Table-8. Typical Profile - Max Practical

Total profile: accel to midway; decel to dest; reverse on return leg.

g= 0.5 AU day2		Departure Leg				Return Leg
ffExh (multi-ffsec )	Practical Range (RgePrac)	Phase I Time (tHalf)	Phase II Time	One Way Time (t1-way)		Phase III Time	Phase IV Time	Dest Dist. (d1-way)
* ffsec	Days	Days	Days	Days		Days	Days	AU
2	208	52	52	104		52	52	1,353
3	340	85	85	170		85	85	3,613
4	455	114	113	228		113	113	6,778
5	589	147	147	295		147	147	10,848
6	712	178	178	356		178	178	15,823
7	833	208	208	417		208	208	21,704
8	955	239	239	477		239	239	28,489
9	1,076	269	269	538		269	269	36,180
10	1,197	299	299	599		299	299	44,775
11	1,318	329	329	659		329	329	54,276
IV	DV	RgePrac/4	RgePrac/4	=tPh-I+tPh-II		RgePrac/4	RgePrac/4	g * t21-way 4

2nd draft: Table-8. Typical Profile - Max Practical

Total profile: accel to midway; decel to dest; reverse on return leg.

g= 0.5 AU day2		Departure Leg						Return Leg
ffExh	Practical Range	Phase I		Phase II		One Way		Phase III		Phase IV				Two Way
n*ffsec	(RgePrac)	t	d	t	d	t1-way	d1-way	d	t	d	t	(t2-way)
n	Days	Days	AUs	Days	AUs	Days	AUs	Days	AUs	Days	AUs	Days	AU
2	208	52	676	52	676	104	1,353	52	676	52	676
3	340	85	676	85	676	170	3,613	85	676	85	676
4	455	113	676	113	676	226	6,778	113	676	113	676
5	589	147	676	147	676	295	10,848	147	676	147	676
6	712	178	676	178	676	356	15,823	178	676	178	676
7	833	208	676	208	676	417	21,704	208	676	208	676
8	955	239	676	239	676	477	28,489	239	676	239	676
9	1,076	269	676	269	676	538	36,180	269	676	269	676
10	1,197	299	676	299	676	599	44,775	299	676	299	676
11	1,318	329	676	329	676	659	54,276	329	676	329	676
IV	DV	RgePrac/4				RgePrac/4	=tPh-I+tPh-II

TABLE-7. Vary Ship Size.

Fuel per second			Ship propulsion		Per day
Original Mass	Exhaust Velocity	Exhaust Mass	Ship Velocity	Ship Mass	Daily %TOGW
ffsec	vExh	ffExh	g	MShip	%TOGWDay
kgm/sec	%c	kgm/sec	m/s2	mT	% MShip
0.48	86.6	0.96	10	25,000	0.17%
0.96	86.6	1.92	10	50,000	0.17%
1.73	86.6	2.88	10	75,000	0.17%
1.92	86.6	3.84	10	100,000	0.17%
ffExh 2	Given	MShip * %TOGWDay 86,400 sec/day	Con.	Given	Given

Since the Earth-like gravity condition requires extremely stable incremental velocity increase throughout the flight, any increased momentum from increased exhaust mass-velocity must go to increase amount of mass being propelled; this is size of spaceship, MShip(= ffExh* vExh / g).

Discussion when planning a mission, it's likely that planners will specify payload requirements or perhaps payload capacity and fuel flow will be adjusted to satisfy requirements. Thus, it makes sense to make table such that one can determine required fuel requirements to satisfy mission.

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TABLE-8. Rate Gives Range

Travel to Planets in terms of %TOGW

Given Ship Size, MShip Original Fuel Flow Exhaust Velocity Exhaust Mass Ship Vel/sec Ship Mass Daily %TOGW ffsec vExh ffExh g MShip %TOGWDay kgm/sec %c kgm/sec m/s2 mT % Ship's Mass 1.92 86.6 3.88 10 100,000 0.17% ffDay = ffsec* 86,400 s/dy Given ffExh = 2 * ffsec Observed IV ffDay MShip

Dest. Dist. Time Fuel Total %TOGW Max Velocity 1 way 2-way 2-way 2-way midpoint AU Days mT % Ship Size km/sec NEO 1.0 5.7 960 0.94% 1,223 Mars 2.5 8.9 1,488 1.49% 1,932 Asteroid Belt 4.0 11.3 1,882 1.88% 2,444 Jupiter 6.2 14.1 2,342 2.34% 3,042 Saturn 10.5 18.3 3,048 3.05% 3,957 Uranus 20.2 25.4 4,228 4.23% 5,492 Neptune 31.1 31.5 5,246 5.25% 6,817 Kuiper Belt 40.0 35.8 5,944 5.95% 7,728 Observed 4SQRT(d/g) MShip * %TOGWTtl t * %TOGWDay tHalf*864km/s

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Due to Lorentz Transform (LT), mass of exhaust particle (ff_Exh) is double that of original (ff_sec) in this specific case when exhaust particle's speed is 86.6% light speed.

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After one day of g-force acceleration, spacecraft achieves 864 km/sec.
Compute: v_Final = t * g ;
where t = time in seconds and
g = 10 meters/sec per sec = 10 m/sec²
v_Final = 86,400 secs * 10 m/s² = 864,000 m/s = 864 km/sec
This is an extremely large value compared with velocity required to escape Earth's gravity, e = 11.5 km/sec. Of course, a vessel wishing to orbit Earth would have to decrease its velocity to less then e.

For above destinations, max velocity happens a midway between dept and dest. In a previous chapter, we have determined that two way travel time (go and return) for a flight profile where ship must accelerate to midway, then decelerate from midway to destination for one way. Thus max velocity is acheived at midway. To compute time to midway: t_Half = SQRT(d/g)

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TABLE-9. Maximum Range

%TOGW yields Max Days of Powered Flight

Disregard Fuel Flow & Ship Size Exhaust Velocity Ship Vel/sec Daily %TOGW vExh g %TOGWDay %c m/s2 % Ship's Mass 86.6 10 0.17% Given Observed g * SQRT(1-v2Exh/c2) * 86,400 secs/day vExh

Dist. Time %TOGW Max Velocity 1 way t2-way 2-way midpoint AU Days % Ship Size km/sec 27 29.4 5% 6,350 108 58.8 10% 12,182 243 88.2 15% 19,051 433 117.8 20% 25,436 675 147.0 25% 31,752 tHalf = SQRT(d/g) 2,702 294.1 50% 63,526 t1-way= 2 * SQRT(d/g) 6,083 441.2 75% 95,300 t2-way= 4 * SQRT(d/g) 10,812 588.2 100% 12,7051 g=0.5 AU/day2 g*t22-way 16 %TOGW %TOGW Day IV tHalf*864km/s

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1)	ffExh	=	ffsec SQRT(1-v2Exh/c2)
2)	ffsec	=	ffExh * SQRT(1-v2Exh/c2)
3)	ffday	=	ffsec*86,400 sec/day
4)	ffday	=	ffExh * SQRT(1-v2Exh/c2)*86,400 sec/day
5)	%TOGWDay	=	ffday MShip
6)	%TOGWDay	=	ffExh * SQRT(1-v2Exh/c2)*86,400 sec/day MShip
7)	ffExh	=	g * MShip vExh
8)	%TOGWDay	=	g * MShip * SQRT(1-v2Exh/c2)*86,400 s/dy vExh..................*...................MShip
9)	%TOGWDay	=	g*SQRT(1 - v2Exh/c2)*86,400 s/dy vExh

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PERSPECTIVE from bottom row of values: After we've expended 100% of the ship's mass, we accomplished following profile of go and return spaceflight.

PHASE-I. ACCELERATE TO HALFWAY. We have departed from vicinity of Earth and accelerated at g for half the one way distance (5,406 AUs = 10,812AUs/2). At this distance, constant g-force acceleration for 147 days causes our notional ship to attain 127,051 kms/sec which 42% light speed.

PHASE-II. DECELERATE TO DESTINATION. Our ship then decelerates for 147 days and another 5,406 AUs. Then, our velocity decreases to essentially zero, and we've completed the initial one way distance, 10,812 AUs. One LY = 63,115 AU; so, this takes us far less then 1/4 distance to Oort Cloud (appox 1 LY from Sol).

PHASE- III. ACCELERATE BACK TO HALFWAY. With 50% theoretical mass left, notional spaceship then reverses course for return trip and accelerates for 147 days and about 5,400 AUs toward Earth.

PHASE-IV. DECELERATE BACK TO START POINT. Finally, the spaceship decelerates for 147 days and remaining 5,406 AUs for a safe orbital velocity in Earth's vicinity.

1) BAD NEWS!! After we've accomplished the tremendous, remarkable feat of accelerating enormous amount of mass to 86.6% speed of light for a 2-way trip of almost 600 straight days, we've still barely moved out of the center of our Solar System.

1a) Even Badder News!!!! We can't use 100% of ship's mass as fuel. Let's arbitrarily pick a max of 50% as a reasonable limit for subsequent discussion.

2) GOOD NEWS!! Range of 50% TOGW is better then we might think. For example, if daily %TOGW is 1% and our ship weighs 100 tons, then first day, ship will consume 1 ton of fuel. A crude approx, 50 tons of fuel, thus, range is 50 days. However, range is really better then this.

The %TOGW_Day is really only good for first day. After the first day, we must really use percent Gross Weight (%GW) because the GW decreases as the fuel is consumed.

2nd day of trip, GW = %TOGW - first day's fuel = 99 tons. 2nd day, ship consumes 1% of 99 tons which is slightly less then 1 ton,

3rd day, fuel consumption will be even less and so on as shown in following table.

Days	Assoc'd Calculation		Remaining GW
0	None		100 mT
1	100mT * .99		99 mT
2	99 mT * .99		98.81 mT
3	100mT * .99 * .99 * .99		97.03 mT
4	100mT * .994		96.06 mT
5	100mT * .995		95.10 mT
...	... ... ... ... ...		... mT
n	100mT * .99n		100*.99n mT

A more convenient way to approximate fuel consumption might involve the simple use of exponentials. For this example, we can use an inexpensive scientific calculator (any exponent function) and readily calculate following:

.99⁵⁰ = .6050; too high.

.99¹⁰⁰ = .3660; too low.

.99⁶⁰ = .5472; too high.

.99⁷⁰ = .4948; too low, but close.

.99⁶⁹ = .4998; about right.

	0.998350	=	0.9184
	0.9983100	=	0.8435
	0.9983200	=	0.7112
	0.9983400	=	0.5063

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.Similiar, exercise for fuel consumption rate of .17% /day which we calculate from an exhaust particle speed of .866c and we get table to right.

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Perhaps a more convenient method might involve logarithms.

Recall log10¹⁰ = x

For example, log10¹⁰⁰ = 2.

What if a^x = b, where a and b are known, but x is unknown.

Due to exponent characteristics, x = log b / log a;

in our case, .9983^x = .5000;

then x = log .5000 / log .9983 = 407.4.

Thus, exponentials and logarithms give a more accurate approximation of how many days 50% TOGW will take us if we're expending .17% GW per day.

EVEN BETTER NEWS!! It's very likely that we'll eventually design propulsion systems to accelerate particles much faster then .866c which means a much less fuel consumption rate and consequently much greater range.