Thursday, April 17, 2008

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

IVDV

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

DVIV

Size & Speed to Ship Mass

ffExh= n * ffsecff2sec/ 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/ffsecvExh * 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 LegReturn Leg
ffExh
(multi-ffsec )
Practical
Range
(RgePrac)
Phase I Time
(tHalf)
Phase II Time One Way
Time
(t1-way)
Phase III TimePhase 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
IVDVRgePrac/4RgePrac/4=tPh-I+tPh-IIRgePrac/4RgePrac/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 LegReturn Leg
ffExh
Practical
Range
Phase I Phase II One Way
Phase III Phase IV Two Way
n*ffsec (RgePrac)tdtdt1-wayd1-waydtdt(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
IVDVRgePrac/4RgePrac/4=tPh-I+tPh-II








TABLE-7. Vary Ship Size.

Fuel per secondShip propulsionPer day
Original
Mass
Exhaust
Velocity
Exhaust
Mass
Ship
Velocity
Ship
Mass
Daily
%TOGW
ffsecvExhffExhgMShip%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. GivenGiven



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
ffsecvExhffExhgMShip%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 * ffsecObservedIV
 ffDay
MShip
Dest. Dist. Time Fuel

Total
%TOGW

Max
Velocity
1 way 2-way 2-way 2-waymidpoint
AU Days mT % Ship Sizekm/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 (ffExh) is double that of original (ffsec) 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: vFinal = t * g ;
where t = time in seconds and
g = 10 meters/sec per sec = 10 m/sec2
vFinal = 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: tHalf = 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
vExhg%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-waymidpoint
AU Days % Ship Sizekm/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
IVtHalf*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 %TOGWDay 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.

DaysAssoc'd Calculation

Remaining
GW

0None

100 mT

1100mT * .99

99 mT

299 mT * .99

98.81 mT

3100mT * .99 * .99 * .99

97.03 mT

4100mT * .994

96.06 mT

5100mT * .995

95.10 mT

...... ... ... ... ...

... mT

n100mT * .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
.

.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.

.

Perhaps a more convenient method might involve logarithms.

Recall log1010 = x

For example, log10100 = 2.

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

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

in our case, .9983x = .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.

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