Saturday, November 10, 2007

Quick Look Tables

To summarize previous chapters, g-force acceleration can take us to planets in days; even the most distant destinations within the Kuiper Belt will take at most weeks.



Interplanetary Flight Profile
PHASE I. Departing Earth,our notional spaceship constantly accelerates at g (10 m/s2) to simulate gravity and travel much quicker then constant velocity vehicles. After only 3.16 days, it reaches midway (2.5 AU) of our 5 AU journey.
PHASE II. Midway between Earth and Jupiter (i.e., d/2 = 2.5 AU), spaceship velocity exceeds 2,700 km/sec. To decrease this speed and still simulate gravity,spaceship reverses fuel exhaust vector to decelerate at g for remaining 3.16 days of travel.
g = 0.5 AU/dy2
g = 10 m/sec2
Distance Acceleration Time Max Velocity Deceleration Time Accel + Decel
to Earth Dept to Midway (at midway)Midway to Dest Travel Time
(AU) (days) (km / sec) (days) days)
NEO

1

1.41

1,218

1.41

2.83

Mars

2

2.00

1,728

2.00

4.00

Jupiter

5

3.16

2,730

3.16

6.32

Saturn

10.0

4.47

3.864

4.47

8.94

Uranus

20.0

6.32

5,464

6.32

12.64

Neptune

30.0

7.75

6.693

7.75

15.50

Kuiper Belt

45

9.49

8,200

9.49

18.97

Earth's escape velocity,
e = 11 km/sec
d
tACCEL = d
g
g = 864km/sec / day
VMax = tACCEL*g
tDECEL = d
g
tTTL = 2 *d
g

Traditional Momentum Exchange (Newtonian)


vfuel : 10%c to 90%c

300 * mfuel * vfuel/ Vship = Mship

mfuel vfuel Vship Mship
gram dec. c m/s mT

1

0.1

10

3

1

0.2

10

6

1

0.3

10

9

1

0.4

10

12

1

0.5

10

15

1

0.6

10

18

1

0.7

10

21

1

0.8

10

24

1

0.9

10

27

Const. IV Const. DV
Mship: 3 to 27 metric Tonnes (mTs) {traditional momentum model}

k * mfuel * vfuel/ Vship = Mship

Definitions
  • mfuel : mass of fuel consumed for each second of flight (grams)
  • vfuel : velocity of fuel particles as they exit spacecraft (decimal c, light speed). Assume c = 300,000,000 m/s.
  • Vship : velocity increase of spacecraft during one second of flight (m/sec)
  • Mship : mass of spacecraft which can be propelled by momentum of fuel particles (metric Tonnes)
  • k : conversion constant to account for mix of units in both velocity and mass. In this case, k= 300.





Relativistic Momentum Exchange (Einsteinian)


vfuel : 10%c to 90%c

300 * mf-LT * vfuel/ Vship = Mship

mfuel =1.0 gmc=300,000,000 m/sVship= 10 m/s
mf-LT vfuel Mship
gram dec. c mT

1.00

0.1

3

1.02

0.2

6

1.05

0.3

9

1.09

0.4

13

1.16

0.5

17

1.25

0.6

22

1.40

0.7

29

1.67

0.8

40

2.29

0.9

62

mfuel/(1-v2fuel)1/2 IV DV
Mship: 3 to 62 metric Tonnes (mTs) {relativistic momentum model}

mf-LT= mfuel/(1-v2fuel)1/2



k * mf-LT * vfuel/ Vship = Mship

Definitions
  • mfuel : mass of fuel consumed for each second of flight (grams). Assume a constant 1.0 grams for this table.
  • mf-LT : Uses Lorentz Transform (LT) to determine increased mass of fuel after being accelerated to vfuel, a significant fraction of light speed.
  • vfuel : velocity of fuel particles as they exit spacecraft (decimal c, light speed). Round c to 300,000,000 m/s.
  • Vship : velocity increase of spacecraft during one second of flight (m/sec) . Assume a constant 10 m/s.
  • Mship : mass of spacecraft which can be propelled by momentum of fuel particles (metric Tonnes)
  • k : conversion constant to account for mix of units in both velocity and mass. In this case, k= 300.








Daily Fuel Flow

vfuel : 10%c to 90%c

ffday / Mship=%TOGW / day

mfuel = 1.0 gmvfuel Vship= 10 m/s ffday=86.4 kg
mf-LT vfuel Mship %TOGW /day
gram decimal c mT % / day
1.00 0.1 3 2.87%
1.02 0.2 6 1.41%
1.05 0.3 9 0.92%
1.09 0.4 13 0.66%
1.16 0.5 17 0.50%
1.25 0.6 22 0.38%
1.40 0.7 29 0.29%
1.67 0.8 40 0.22%
2.29 0.9 62 0.14%
mfuel/(1-v2fuel)1/2 IV DV ffday/Mship
%TOGW: 2.8% to 0.14% (less is better!!)

mf-LT= mfuel/(1-v2fuel)1/2



k * mf-LT * vfuel/ Vship = Mship

Definitions
  • mfuel : mass of fuel consumed for each second of flight (grams) Remains a constant 1.0 grams for this table.
  • ffday: amount of fuel consumed per day of flight. Since 1 day = 86,400 secs, this amount is 86,400 times fuel per second. Since we're assuming 1.0 gram per sec, ffday turns out to be 86.4 kgm for this table.
  • mf-LT : increased mass of fuel after being accelerated to vfuel, a significant fraction of light speed.
  • vfuel : velocity of fuel particles as they exit spacecraft (decimal c, light speed). Assume c = 300,000,000 m/s.
  • Vship : velocity increase of spacecraft during one second of flight (m/sec)
  • Mship : mass of spacecraft which can be propelled by momentum of fuel particles (metric Tonnes)
  • %TOGW: Percent Take off Gross Weight is the amount of ship's mass needed to convert to energy (i.e., must be in form of fuel) to propel the ship for the entire trip.
  • %TOGW/day: Amount of ship's mass needed to convert to energy to propel ship for that day; hence, %TOGW = ffday / Mship .
  • k : conversion constant to account for mix of units in both velocity and mass. In this case, k= 300.







Ranges and Efficiencies


vfuel : 10%c to 90%c

Theoretical Range: Rtheo=100% /%TOGW / Day

Feasible Range: Rfeas=50% /%TOGW / Day

Practical Range: Rprac=25% /%TOGW / Day

Example: k * mfuel * vfuel / Vship = Mship
k = 300 mfuel=1.0 gm vfuel= .1c Vship= 10 m/s Mship= 3 mT
c=300,000,000 m/s ffday =86,400 * mfuel
vfuel %TOGW / day RtheoRfeasRprac
dec. c % / day Days Days Days
0.1 2.87% 35 17 9
0.2 1.41% 71 35 18
0.3 0.92% 109 54 27
0.4 0.66% 152 76 38
0.5 0.50% 200 100 50
0.6 0.38% 263 132 66
0.7 0.29% 345 172 86
0.8 0.22% 455 227 114
0.9 0.14% 714 357 179
IV (increments) ffday/Mship 100%/%TOGW / day 50%/%TOGW / day 25%/%TOGW / day
Rprac: 9 to 179 days (more is definitely better!!!)

mf-LT= mfuel/(1-v2fuel)1/2



k * mf-LT * vfuel/ Vship = Mship

Definitions

  • mfuel : mass of fuel consumed for each second of flight (grams) Remains a constant 1.0 grams for this table.
  • mf-LT : increased mass of fuel after being accelerated to vfuel, a significant fraction of light speed.
  • vfuel : velocity of fuel particles as they exit spacecraft (decimal c, light speed). Assume c = 300,000,000 m/s.
  • Vship : velocity increase of spacecraft during one second of flight (m/sec)
  • Mship : mass of spacecraft which can be propelled by momentum of fuel particles (metric Tonnes)
  • k: conversion constant. In this case, k= 300.
  • ff day: amount of fuel consumed per day of flight.
  • %TOGW/day: Amount of ship's mass needed to convert to energy to propel ship for that day; hence, %TOGW / day = ffday / Mship
  • Rtheo : Theoretical Range. Number of days of propulsion available if we convert entire mass of spaceship into energy. While impractical for spacecraft with payloads and reuseable infrastructure, it might be an option for refueling missions where most mass was fuel.
  • Rfeas : Feasible Range. Most spaceships have design limits. For simplicity, we'll assume that our notional spacecraft has fuel tanks such that half its mass can be in fuel.
  • Rprac : Practical Range. By nature, human designed devices tend to start out with large inefficiencies and continually improve. Furthermore, even perfectly designed propulsion systems will have inherent sources of inefficiency. Onboard energy will be required for many nonpropulsion functions: life support, navigation, communication, auxiliary power for the propulsion system (i.e., accelerator magnets will need a lot of energy even with superconductors). For simplicity, we'll assume that only only half the fuel goes for propulsion. If fuel is 50% of ship's mass, then 25% of total mass can apply toward propulsion.








Ranges and Distances


vfuel : 10%c to 90%c

Maximum Distance: Dmax= g * R2prac/2

Middle Distance: Dmid= g * (0.5 * Rprac)2/2

Practical Distance: Dprac=2 * Dmid

Example: k * mfuel * vfuel / Vship = Mship
k = 300 mfuel=1.0 gm vfuel= .1c Vship= 10 m/s Mship= 3 mT
c=300,000,000 m/s ffday = 86,400 * mfuel g = .5 AU/day2 t = Rprac t= 0.5 * Rprac
vfuel %TOGW / day Rprac Dmax Dmid Dprac
dec. c % / day Days AUs AUs AUs
0.1 2.87% 8.71 19 5 9
0.2 1.41% 17.73 79 20 39
0.3 0.92% 27.17 185 46 92
0.4 0.66% 37.88 359 90 179
0.5 0.50% 50.00 625 156 312
0.6 0.38% 65.79 1,082 271 541
0.7 0.29% 86.21 1,858 464 929
0.8 0.22% 113.64 3,228 807 1,614
0.9 0.14% 178.57 7,972 1,993 3,986
IV (increments) ffday/Mship 25%/%TOGW / day g * t2 / 2 g * t2 / 2 2 * Dmid
(Range as Practical Distance) Dprac: 9 to 3,986 AUs

mf-LT= mfuel/(1-v2fuel)1/2



k * mf-LT * vfuel/ Vship = Mship

Definitions

  • mfuel : mass of fuel consumed for each second of flight (grams) Remains a constant 1.0 grams for this table.
  • mf-LT : increased mass of fuel after being accelerated to vfuel, a significant fraction of light speed.
  • vfuel : velocity of fuel particles as they exit spacecraft (decimal c, light speed). Assume c = 300,000,000 m/s.
  • Vship : velocity increase of spacecraft during one second of flight (m/sec)
  • Mship : mass of spacecraft which can be propelled by momentum of fuel particles (metric Tonnes)
  • k: conversion constant. In this case, k= 300.
  • ff day: amount of fuel consumed per day of flight.
  • %TOGW/day: Amount of ship's mass needed to convert to energy to propel ship for that day; hence, %TOGW / day = ffday / Mship
  • Rprac : Practical Range. Even perfectly designed propulsion systems have inherent sources of inefficiency. If fuel is 50% of ship's mass, then 25% of total mass can apply toward propulsion. This gives us a practical limit of total time of powered spaceflight (in days).
  • g, acceleration due to near Earth gravity, is 9.8 m/sec/sec which we commonly round to 10 m/s2. Instead of meters and seconds, let's use roughly equivalent term with AUs and days; this turns out to be 0.5 AU/day2.
  • Dmax : Maximum acheivable distance. To determine distance flown during constant acceleration, use well known formula d = 0.5 g * t2. If spaceship accelerates for the whole flight; then time, t, equals practical range, Rprac.
  • Dmid : Middle distance. Since constant acceleration will produce some very high speeds, spacecraft must SLOWDOWN!!!! prior to destination. It makes great sense to do this at midway between departure and destination. Using this constraint to determine half of the max practical distance; change time, t, to one half of practical range: Rprac/2.
  • Dprac : Practical distance. Since Dmid is half the possible distance from practical range, we can determine total practical distance by doubling Dmid. Thus, a practical flight profile would involve accelerating to midpoint then decelerating to destination.





Ranges and Planets


vfuel : 10%c to 90%c

Destination CoHabitat Mars Jupiter Saturn Uranus Neptune
tmid=(2*(Dmid)/g)1/2 tmid=SQRT(d/g)

d = Typical Dist. (AUs)

1 2 5 10 20 30
taccel = tmid = tdecel t = taccel + tdecel

t= 2 * tmid (days)

2.83 4.00 6.32 8.94 12.65 15.49
Dmid = d/2
c=300,000 km/sec 
AU=150,000,000km

t2-way= 2 * t (days)

5.66 8.00 12.65 17.89 25.30 30.98
vfuel %TOGW / day Rprac Total Two Way %TOGW Needed To Travel
dec. c % / day Days %TOGW %TOGW %TOGW %TOGW %TOGW %TOGW
0.1 2.87% 8.71 16.24% 22.96% OUT OF RANGE OUT OF RANGE OUT OF RANGE OUT OF RANGE
0.2 1.41% 17.73 7.98% 11.28% 17.84% OUT OF RANGE OUT OF RANGE OUT OF RANGE
0.3 0.92% 27.17 5.20% 7.36% 11.64% 16.46% 23.27% OUT OF RANGE
0.4 0.66% 37.88 3.73% 5.28% 8.35% 11.81% 16.70% 20.45%
0.5 0.50% 50.00 2.83% 4.00% 6.32% 8.94% 12.65% 15.49%
0.6 0.38% 65.79 2.15% 3.04% 4.81% 6.80% 9.61% 11.77%
0.7 0.29% 86.21 1.64% 2.32% 3.67% 5.19% 7.34% 8.99%
0.8 0.22% 113.64 1.24% 1.76% 2.78% 3.94% 5.57% 6.82%
0.9 0.14% 178.57 0.79% 1.12% 1.77% 2.50% 3.54% 4.34%
IV (increments) ffday
Mship
25%

%TOGW/day
= t2-way* (%TOGW/day)
25% is max practical limit.
(Clearly, as fuel particle's exhaust speed, vfuel, increases, the spaceship's two way practical range correspondingly increases. While a vfuel of one tenth light speed, c, might get us to Mars, we need nearly four tenths c to reach Neptune.)

mf-LT= mfuel/(1-v2fuel)1/2



k * mf-LT * vfuel/ Vship = Mship

Definitions

  • mfuel : mass of fuel consumed for each second of flight.
  • vfuel : velocity of fuel particles as they exit spacecraft (decimal c, light speed). Assume c = 300,000,000 m/s.
  • mf-LT.: relativistically increased mass of mfuel after being accelerated to vfuel .
  • Vship : velocity increase of spacecraft during one second of flight (m/sec)
  • Mship : mass of spacecraft which can be propelled by momentum of fractional light speed, fuel particles.
  • k: conversion constant. In this case, k= 300.
  • ff day = (mfuel * 86,400): daily quanity of fuel.
  • %TOGW/day: Amount of ship's mass needed to convert to energy to propel ship for that day; hence, %TOGW / day = ffday / Mship
  • Rprac : Practical Range. Even perfectly designed propulsion systems have inherent sources of inefficiency. If fuel is 50% of ship's mass, then 25% of total mass can apply toward propulsion.
  • g, acceleration due near Earth gravity, equals 9.8 m/sec/sec which we've rounded to 10 m/s2. Instead of meters and seconds, we've opted to use roughly equivalent term using AUs and days; this turns out to be 0.5 AU/day2.
  • CoHabitat : (Co-Orbiting Habitat) New concept which will require elaboration in another place. Expected points:
    • Definition: Large (O'Neill Habitat-3 size) habitat made of asteroid materials which occupies Terran orbit and either leads or lags Earth by 60o.
    • PURPOSE: Receive and process asteroidal materials for Earth but mainly other habitats. (Much safer for Earth, to have this done 1 AU away.)
    • To visualize cohabitat's placement in Terran orbit, consider that Earth, Sol, and the habitat will be at the three verticies of an equilateral triangle where all sides = 1 AU.
    • To construct diagram, copy circle which represents Terran orbit about Sol, and center it about Earth. Intersections between Terran orbit about Sol and the Terran centered circle will be the two places 60o away from Earth.
    • OUT OF RANGE :
    • d
    • dmid
    • tmid
    • taccel
    • tdecel
    • t
    • t2-way








    Kuiper Belt and Beyond


    vfuel : 10%c to 90%c

    Destination Kuiper Belt and Beyond >>>>> >>>>>
    tmid=(2*(Dmid)/g)1/2 tmid=SQRT(d/g) d = Typical Dist. (AUs) 50 100 150 200 250 300
    taccel = tmid = tdecel t = taccel + tdecel t= 2 * tmid(days) 20.00 28.28 34.64 40.00 44.72 48.99
    Dmid = d/2 c=300,000 km/sec

    AU=150,000,000km
    t2-way= 2 * t (days) 40.00 56.57 69.28 80.00 89.44 97.98
    vfuel %TOGW / day Rprac Total Two Way %TOGW Needed To Travel
    dec. c % / day Days %TOGW %TOGW %TOGW %TOGW %TOGW %TOGW
    0.1 2.87% 8.71 OUT OF RANGE OUT OF RANGE OUT OF RANGE OUT OF RANGE OUT OF RANGE OUT OF RANGE
    0.2 1.41% 17.73 OUT OF RANGE OUT OF RANGE OUT OF RANGE OUT OF RANGE OUT OF RANGE OUT OF RANGE
    0.3 0.92% 27.17 OUT OF RANGE OUT OF RANGE OUT OF RANGE OUT OF RANGE OUT OF RANGE OUT OF RANGE
    0.4 0.66% 37.88 OUT OF RANGE OUT OF RANGE OUT OF RANGE OUT OF RANGE OUT OF RANGE OUT OF RANGE
    0.5 0.50% 50.00 20.00% OUT OF RANGE OUT OF RANGE OUT OF RANGE OUT OF RANGE OUT OF RANGE
    0.6 0.38% 65.79 15.20% 21.50% OUT OF RANGE OUT OF RANGE OUT OF RANGE OUT OF RANGE
    0.7 0.29% 86.21 11.60% 16.40% 20.09% 23.20% OUT OF RANGE OUT OF RANGE
    0.8 0.22% 113.64 8.80% 12.45% 15.24% 17.60% 19.68% 21.56%
    0.9 0.14% 178.57 5.60% 7.92% 9.70% 11.20% 12.52% 13.72%
    IV (increments)
    ffday 
    Mship
    25% 
    %TOGW/day
    = t2-way* (%TOGW/day)
    25% is max practical limit.
    (As two way travel time, t2-way, increases to much further distances in our Solar System, we see even more limitations. We now see that we have to investigate increasing fuel particle's exhaust speed, vfuel, beyond 90% c.)

    mf-LT= mfuel/(1-v2fuel)1/2



    k * mf-LT * vfuel/ Vship = Mship

    Definitions

    • mfuel : mass of fuel consumed for each second of flight.
    • vfuel : velocity of fuel particles as they exit spacecraft (decimal c, light speed). Assume c = 300,000,000 m/s.
    • mf-LT.: relativistically increased mass of mfuel after being accelerated to vfuel.
    • Vship : velocity increase of spacecraft during one second of flight (m/sec)
    • Mship : mass of spacecraft which can be propelled by momentum of fractional light speed, fuel particles.
    • k: conversion constant. In this case, k= 300.
    • ff day= (mfuel * 86,400): daily quanity of fuel.
    • %TOGW/day: Amount of ship's mass needed to convert to energy to propel ship for that day; hence, %TOGW / day = ffday / Mship
    • Rprac : Practical Range. Even perfectly designed propulsion systems have inherent sources of inefficiency. If fuel is 50% of ship's mass, then 25% of total mass can apply toward propulsion.
    • g, acceleration due near Earth gravity, equals 9.8 m/sec/sec which we've rounded to 10 m/s2. Instead of meters and seconds, we've opted to use roughly equivalent term using AUs and days; this turns out to be 0.5 AU/day2. .
    • CoHabitat :

    • OUT OF RANGE : This arbitrary range limit is based on two considerations:

    1) Fuel supply is finite and must be a portion of a ship which must devote some mass to infrastructure and payload.

      2) Efficiency in converting fuel to propulsion directed energy will always be less then 100% for vessels which need energy for other things (i.e., life support, communication, auxiliary power for propulsion system, etc.)

      For convenience, we've chosen fuel to be 50% of ship's mass and efficiency to also be 50%. These optimistic estimates combine for a total of 25% of ship's mass to convert to propulsion energy. We could be much more optimistic and use 90% for each to give us 81% of ship's mass for propulsion. On the other hand, we could be much less and estimate 20% of ship's mass for fuel with a 25% efficiency; this would give us an effective 5% of ship's mass for propulsion. Of course, the possible combinations are numerous and will need actual experimentation for better estimates which will be eventually confirmed with actual operational experience.

      • d
      • dmid
      • tmid
      • taccel
      • tdecel
      • t
      • t2-way

      0 Comments:

      Post a Comment

      Links to this post:

      Create a Link

      << Home