Saturday, May 19, 2007

Vol 1. Tables, A Summary

Table 1. Momentum Exchange.

Recall momentum exchange: M * V = m * v

Apply to fast moving fuel particles and a slow moving spaceship: Mship * Vship = mfuel * vfuel

Rearrange to solve for spaceship size: Mship = mfuel * vfuel / Vship


Per above text, give terms following values:

  • mfuel = 1.0 gram
  • Vship = 10 meters / second
  • vfuel = one tenth light speed = 30,000,000 m/s
  • Mship = (1.0 gm * 30,000,000 m/s) / (10 m/s)

Basic Momentum
Mship = mfuel * vfuel / Vship

Mship

mfuel

Vship

vfuel

gm

gm

m/sec

m/s

3,000,000

1.0

10

30,000,000





Conversion Constant: 300/c
Mship = (300/c)(mfuel * vfuel / Vship)

Mship

mfuel

Vship

vfuel

mT

gm

m/sec

dec. c

3

1.0

10

.1 c

6

1.0

10

.2 c

1,000,000 gms = 1,000 kgms = 1.0 metric Tonnes = 1 mT
Thus, Mship = 3,000,000 gm = 3.0 mT

300,000,000 meters /sec = light speed = c
Thus, Vfuel = 30,000,000 m/s = .1c

It’s more convenient to express Mship in mT and vfuel in decimal c.

To do this, introduce a conversion constant, k1, as follows:
Mship = k1 * mfuel * vfuel / Vship
where k1 = 300/c



Reconsider: mfuel * vfuel = Mship * Vship. Divide both sides of original equation by one sec. (mfuel * vfuel)/1sec = (Mship * Vship)/1sec

Rearrange: (mfuel /1sec) * vfuel = Mship * (Vship/1sec)

Recall that we set Vship equal to 10 m/s; thus, Vship/ 1sec = 10m/s/s = g, acceleration due to gravity near Earth’s surface.

We introduce new term, ff (fuel flow), which we define as mfuel/1sec.

Thus, we’ll set ff equal to 1.0 gm/sec.
ff * vfuel = Mship * g

To continue expressing vfuel as decimal c and Mship in metric Tonnes, we'll continue use of conversion constant, k1 = 300/c.

This gives equation at top of table: Mship = 300/c (ff * vfuel / g)


g

ff

vfuel

Mship

con

con

IV

DV

m/s/s

gm/sec

dec. c

mT

10

1

0.1

3

10

1

0.2

6

10

1

0.3

9

For perspective, note that Vfuel of .1 c means the particles are exiting at one tenth the speed of light, 30,000,000 m/s.

Momentum exchange allows one gram of high speed particles to propel a 3 ton vessel for an additional 10 m/s.

CYCLOTRON IN SPACESHIP
Perhaps a particle accelerator can propel a spaceship by emitting a stream of very high speed particles.

(For more "Momentum" details.)



Table-2. Lorentz Transform.

To quantify mass increases due to relativistic (near light speed) velocities of exhaust particles, use the Lorentz Transform (LT). This mass increase also contributes to the momentum exchange which propels the ship.

  • Original fuel flow (ff). Convert one gram of matter to plasma every second; accelerate all those particles to near light speed; and expel them from the spacecraft. (See T-1. Momentum Exchange.)


  • Relativistic fuel flow (ffLT). Use LT to adjust original fuel flow to account for relativistic mass increase. (See following table.)


At very high speed, particles will increase mass due to relativity.
Confirmed by numerous scientific observations, Lorentz Transform (LT) describes how mass increases with speed. Original mass (mo), or mass at rest, can be considered the input mass input as plasma into the heart of the cyclotron. We'll consider mo to be a constant one gram (1 gm); thus, we can observe effects of high speed


gffffLTvfuelMship
conconDVIVDV
m/s/sgm/secgm/secdec. cmT
1011.0050.13.02
1011.0210.26.12
1011.0480.39.43
1011.0910.413.09
*ffFuel flow is consumed at non-relativistic rate
**ffLTFuel flow is expelled at relativistic rate which affects momentum and propulsion.

Transition from mr equation to ffLT equation


Terms in mr equation

  • mo, original mass
  • c, light speed, is a constant.
  • v, velocity achieved by particles at time of exhaust from spaceship.
  • mr, relativistic mass, which increases as v increases.

Rearrange Some Expressions

  • Express v as decimal c, then
    mr = mo /SQRT(1-v2/ c2)
    reduces to
    mr = mo/SQRT(1-v2ratio).


  • Divide both sides by one second.
    mr/1sec = (mo/1sec) /SQRT(1-v2 fuel).


Terms in mr equation

  • mo/sec becomes ff (fuel flow), grams per second. This is used to determine fuel consumption.
  • mr/sec becomes ffLT, fuel flow's mass increased by relativistic velocity. This is used to determine momentum exchange.


(For more LT details.)



Table-3. Percent Take Off Gross Weight (%TOGW).


      Daily Fuel Flow (ffDay)
      gffffLTvfuelMshipffday
      conconDVIVDV86.4 ff
      m/s/sgm/secgm/secdec. cmTkgm/day
      1011.0050.13.0286.4
      1011.0210.26.1286.4
      1011.0480.39.4386.4
      1011.0910.413.0986.4
      1011.1550.517.3286.4
      While ff units are grams per second, ffday are kgm/day.

      Note that ship size (Mship) grows as output particle speed (vfuel) increases; however, daily fuel flow (ffday) stays constant.

      Determining fuel flow per day (ffday) is straightforward. We arbitrarily chose fuel consumption to be 1.0 gram per second. Since there are 86,400 secs/day, daily fuel flow must be 86,400 times this amount. (86,400 sec/day =60 secs/min * 60 mins/hr * 24 hrs/day).

      86,400 grams per day can be expressed as 86.4 kgm/day.

      For convenience, express ffDay in kilograms. We can then use conversion constant = 86.4; and we now see that 86.4 kgms can theoretically propel a multiton spaceship for an entire day of spaceflight.

      It's very likely that our actual fuel flow will never be exactly 1.0 gram per second; however, we can always convert gm/sec to kgm/day by multiplying by 86.4. Conversely, we'll be able to do reverse conversion of kgm/day to gm/sec by dividing by 86.4.

      Finally, fuel quantity consumed differs from fuel quantity in momentum exchange which propels the ship. The propulsion particles are energized to significant fractions of light speed, c; thus, these particles have increased mass which is quantified by the term, ffLT. Consumed particles (changed to plasma ions) are at rest with relation to spaceship; thus, weight decreases by original mass consumed, mo, NOT mass expelled, ffLT.




      Small Part of Ship's Mass Converts to Energy

      %TOGWday=ffday
      Mship


      1 kgm= 1,000 gms

      1 mT = 1,000 kgms
      = 1,000,000 gms

      g

      ff

      ffLT

      vfuel

      Mship

      ffday

      %TOGWday

      con

      con

      DV

      IV

      DV

      86.4 ff

      DV

      m/s/s

      gm/sec

      gm/sec

      dec. c

      mT

      kgm/day

      %/day

      10

      1

      1.005

      0.1

      3.02

      86.4

      2.87%

      10

      1

      1.021

      0.2

      6.12

      86.4

      1.41%

      10

      1

      1.048

      0.3

      9.43

      86.4

      0.92%

      10

      1

      1.091

      0.4

      13.09

      86.4

      0.66%

      10

      1

      1.155

      0.5

      17.32

      86.4

      0.50%

      10

      1

      1.250

      0.6

      22.50

      86.4

      0.38%

      When we've determined how much mass is required to move a space ship, we can convert this to percentage of spacecraft mass per day.

      As propulsion particle speed, vfuel, independently varies in increments of .1 c, we can observe how this increases the dependent variable, Mship.

      1 gram at .1c, one tenth light speed, propels 3.02 tons of vessel for one second. Doubling this gram's exhaust speed to .2c approximately doubles vessel size (6.12 tons). The slight differential over exactly double is due to relativistic mass increase.

      Converting the ratio of 86.4 kgm by ship size gives the percent value of %TOGW per day. This tells how much of ship's mass must convert to energy to maintain accelerated space flight. Note this percentage decreases as propulsion particle speed increases.

      CONCLUSION: Faster is Better. The greater the propulsion particle speed, the more mass that can be used as payload vs. fuel. Paradoxically, this means we must inject energy into the particles inside the ship; so, where do we get the energy to do that???

      Note: Determine equation for %TOGW which strictly depends on vfuel and not directly on Mship



      Table-4. Spaceship Range (in days of flight).
      When we've determined %TOGW, we can then determine how many days of flight can be sustained. This range will depend on vfuel, speed of propulsion particles.

      Hold Ship Size Constant

      g

      *ff

      **ffLT

      vfuel

      Mship

      ***ffday

      %TOGWday

      ....m/s/s....

      ..kgm/sec..

      kgm/sec

      ..dec..c..

      .......mT.......

      mT/day

      %/day

      10

      33.2

      33.333

      0.1

      100,000

      2,866

      2.87%

      10

      16.3

      16.667

      0.2

      100,000

      1,411

      1.41%

      10

      10.6

      11.111

      0.3

      100,000

      916

      0.92%

      10

      7.6

      8.333

      0.4

      100,000

      660

      0.66%

      10

      5.8

      6.667

      0.5

      100,000

      499

      0.50%

      10

      4.4

      5.556

      0.6

      100,000

      384

      0.38%

      10

      3.4

      4.762

      0.7

      100,000

      294

      0.29%

      con

      IV

      Size of typical aircraft carrier.

      • Determine equation for %TOGW to strictly depend on vfuel.
      • With %TOGW, determine how many days of flight can be sustained.
      • Thus, range will depend on vfuel, speed of propulsion particles.

      Academically, it makes senses to see how much ship can be be propelled by one gram of particles ejected at very high speeds.

      Above tables show that one gram at .1 c can propel 3 mT, one gram at .2 c can propel slightly more then 6 mT, and so on. That’s an impressive fuel/mass ratio, but note following two things. 3 metric Tonnes is about the mass of my Ford Explorer; our spaceship will have to be much bigger. If we increased fuel consumption to 1 kgm per second, that could propel 3,000 mT to accelerate another 10 m/s during one second. 3,000 mT is much better then just 3 mT, but it still may not be enough to launch a scientific expedition.

      Furthermore, using fuel flow to determine ship size is really putting cart before the horse. It would be more appropriate to use ship size determine required fuel flow. Therefore, let’s readjust terms so that we keep ship size constant (though substantial) and determine required fuel flows needed at incremental propulsion particle speeds. Thus, we’ll arbitrarily pick a nice round number, 100,000 mT as the mass of our ship, determine required fuel flow as we uniformly increase particle propulsion speed as shown by the table

            Table-4. Spaceship Range (in days of flight).

            g

            ff

            vfuel

            Mship

            ffday

            %TOGWday

            ***Range

            con

            DV

            IV

            con

            DV

            DV

            DV

            m/s/s

            kgm/sec

            dec c

            mT

            mT/day

            %/day

            days burn

            10

            33.2

            0.1

            100,000

            2,866

            2.87%

            8.7

            10

            16.3

            0.2

            100,000

            1,411

            1.41%

            17.7

            10

            10.6

            0.3

            100,000

            916

            0.92%

            27.3

            10

            7.6

            0.4

            100,000

            660

            0.66%

            37.9

            10

            5.8

            0.5

            100,000

            499

            0.50%

            50.1

            10

            4.4

            0.6

            100,000

            384

            0.38%

            65.1

            10

            3.4

            0.7

            100,000

            294

            0.29%

            85.1

            10

            2.5

            0.8

            100,000

            216

            0.22%

            115.7

            ***Assumptions about range of spaceship
            CapfuelCapacity of fuel likely limited to one half mass of Mship
            EfffuelEfficiency of fuel is 50% due to auxilliary energy requirements.
            DisregardExponential function which considers declining fuel consumption due to decreasing mass of Mship.



          Table-5. Acceleration Basics

          Basic Acceleration
          t g vfin vave d
          (sec) 10m/s/s m/s m/s m

          1

          10

          10

          5

          5

          2

          10

          20

          10

          20

          86,400

          10

          864,000

          432,000

          37,324,800,000

          Incr. Con t * g vfin / 2 t * vave

          Force of gravity will accelerate a free falling object (near Earth's surface) 10 m/s for every second of free fall. After one second, object is falling 10 m/s. After two seconds, object is falling 20 m/s. If spaceship self propels at force of gravity for one entire day (86,400 seconds), it will have achieved final velocity of 864,000 m/s.

          If spaceship (or falling object) starts with initial velocity of 0 m/s (zero) and maintains constant acceleration throughout duration, then average velocity equals one half of final velocity (vave = .5 vfin). Thus, after one day of g-force accelerated flight, our space ship has averaged 432,000 m/s.

          Distance traveled equals time times average velocity (d = t * vave). Thus, our spaceship has traveled a very large distance of 432,000 m/s times 86,400 seconds.






          If units of meters and seconds give us huge numbers inconvenient for long durations of acceleration, let's try days and AUs.


          Seconds to days. Simple arithmetic: 60 sec/min * 60 min/hr *24 hr/day = 86,400 sec/day.

          Meters to AUs. Accelerating for one second at g will move an object 5 meters, accelerating for one day at g will move an object how far?? Basic acceleration calculations give us about 37.5 billion meters (= 37.5 million kms) distance for one day of g-force acceleration.

          Since 1 AU equals 150 million kms, one day acceleration distance equals 1/4 AU.


          Convert to Daily
          86,400 seconds = 1 day
          37,324,800,000 m = .25 AU
          t g vfin vave d
          (day) AU/dy/dy AU/dy AU/dy AU

          1

          0.5

          0.5

          0.25

          0.25

          Given vfin / t 2 * vave d / t Incr.
          10 m/s/s = g = 0.5 AU/day/day
          d = 0.5 * g * t**2

          • d = t * vave
          • d = t * vfin / 2
          • d = t * g * t / 2
          • d = 0.5 g * t**2

          We now have values t and d in days and AUs for one day's g-force acceleration. What about correspondings values for Vave, Vfin, and g?

          Vave = distance/time = d/t = .25 AU /1 day = .25 AU/dy

          Vfin = twice average speed = 2 * Vave = 2 *

          .25 AU/dy = .5 AU/dy

          g = velocity increase per duration = Vfin/t = (.5 AU/dy) / (1 day) = .5 AU/dy/dy






          Using the constant, g (= .5 AU/dy/dy), we can determine all inflight speeds, times, and distances with the variables: time and distance (t & d).


          Thus, we've chosen to independently vary t in one day increments and observe resulting distances after g-force acceleration from an initial speed of zero.

          d = 0.5 * g * t**2



          On the other hand, we can rearrange this equation to make distance the Independent Variable (IV) and time the Dependent Variable (DV).

          t = SQRT(2*d/g)


          Daily Acceleration

          tgvfinvaved
          (day)0.5 AU/dy/dyAU/dyAU/dyAU

          1

          0.5

          0.5

          0.25

          0.25

          2

          0.5

          1

          0.5

          1

          3

          0.5

          1.5

          0.75

          2.25

          4

          0.5

          2

          1

          4

          Incr.Cont * gt*g / 20.5g*t**2






          Table-6. Enroute to Mars: Distances and Flight Times.

          Recall our thot experiment has spaceships which use particle accelerators to constantly accelerate throughout their spaceflights anywhere in our Solar System.

          Our notional spaceships can easily reach the orbit of Mars within a few days.

          The table shows several approximate distances which vary from .5 AU (absolute closest with Mars and Earth on same side of Sun) to 2.5 AU (Mars and Earth separated by Sun). (See orbit diagram.)

          However, constant g-force acceleration achieves very high velocities, and the spacecraft must slow down before it reaches Mars so that it can start orbiting. On the other hand, we want to maintain g-force for as much of the flight as possible so the occupants can enjoy as close to Earth like conditions as possible.

          Useful flights must adopt a flight profile as follows:
          1. Accelerate from rest to the midpoint between departure and destination. Occupants feel simulated gravity due to force from constant acceleration; this enables comfort and optimal productivity.
          2. Decelerate from midway to destination by using same g-force in opposite direction. I conjecture this would be achieved by turning the space vessel 180 degrees and pointing stream of accelerated particles toward the destination.

          Examining table, we can see that only one half day of g-force acceleration would bring the ship to 432 km/sec, much higher then Mars's escape velocity of 5 km/sec (ship would have to be slight slower to orbit around Mars) .

          Using acceleration/deceleration profile, we can then determine flight times for various distances.

          Possible Acceleration/Deceleration profiles: Earth to Mars
          EarthMarsg = 10 m/s/s = 0.5 AU/day/day
          mean distance from Sol11.52After 1 day, Vfin = 864 km/sec
          escape velocity (km/sec)11.185.02g = 0.5AU/dy / dy = 864km/sec / day
          Profile to Midpoint (Acceleration)Final V at MidpointProfile fm Midpoint (Deceleration)Totals
          Dist (AU)Time (day)Velocity (km/sec)Dist (AU)Time (day)Total Dist (AU)Total Time (Day)
          d1t1VMidd2t2DT
          0.250.504320.250.500.51.00
          0.500.716110.500.711.01.41
          0.750.877480.750.871.51.73
          1.001.008641.001.002.02.00
          1.251.129661.251.122.52.24
          D / 2SQRT(2*d1/g)864 * t1d2 = d1SQRT(2*d2/g)Givent1 + t2


          1. Even the shortest flight (.5 AU when Mars and Earth are at their closest) for a constant g-force accelerating spacecraft achieves an enormous speed at the midpoint of the flight. Since spacecraft needs the force from constant acceleration to simulate Earth gravity, then the spacecraft must reverse direction of propulsion flow of particles to apply same force in opposite direction. This would continue gravity conditions for second half of flight and would decelerate the spacecraft.

          2. For example, a vessel would have to start from Earth at least as fast as the indicated escape velocity, 11.2 km/sec. After accelerating for one day, the ship will have achieved 864 km/sec, 77 times greater. To operate in the Martian planetary system, the spacecraft must reduce speed to about 5 km/sec, less then 1% of VMid . For convenience, next table will approximate departure and destination velocities as zero.

          3. Finally, we can simply the table by changing relevant equations as follows:

          D/2 = d1 = d2

          T = t1 + t2 = SQRT(2d1/g) + SQRT(2d2/g)

          T = SQRT(2 * D/2 /g) + SQRT(2 * D/2 /g) = 2 * SQRT(D/g)






          Table-7. To the Planets.


          Solar System DestinationMarsJupiterSaturnUranusNeptuneKuiper Belt
          Typical Distance from Earth (AU)1510203040
          Flight Time (Days) = 2 * SQRT(D/g)2.836.328.9412.6515.4917.89






          Solar System DestinationMarsJupiterSaturnUranusNeptuneKuiper Belt
          Typical Distance from Earth (AU)1510203040
          2 way Flight Time (Days) = 4 * SQRT(D/g) = t5.6612.6517.8925.3030.9835.78
          %TOGWDay**Range (Days)vfuel (dec. c)%TOGW%TOGW%TOGW%TOGW%TOGW%TOGW
          2.87%8.70.1016.21%36.25%51.26%72.49%88.79%102.52%
          1.41%17.70.207.98%17.85%25.24%35.69%43.72%50.48%
          0.92%27.30.305.18%11.58%16.38%23.17%28.37%32.76%
          0.66%37.90.403.73%8.35%11.80%16.69%20.45%23.61%
          0.50%50.10.502.82%6.31%8.92%12.62%15.46%17.85%
          0.38%65.10.602.17%4.86%6.87%9.71%11.90%13.74%

          Increment

          g = 0.5 AU/dy/dy**Range Limit-1: Assume fuel is limited to one half mass of spaceship. This limits %TOGW to 50%.
          **Range Limit-2: Assume fuel efficiency of 50%. This further limits %TOGW to 25%.





































            SIMPLIFY FOR CONVENIENCE


            Consistent Units

            Recall Newton's momentum exchange equation, total quantity of momentum (mass times velocity) remains the same for a closed system of finite quantity of particles.


            _____M * V=m * v_____

            Use this equation to describe propulsion process for a spaceship. Left side is a huge spacecraft increasing its speed slightly, and other side is a small quantity of fuel particles traveling at very high velocity. (Disregard relativity for now.)

            Mship * Vship= mfuel * vfuel

            Use same dimensions to ensure consistency. For example, use grams for both mass terms and m/s for both velocity terms. Rearrange:

            ____Mship = mfuel* vfuel / Vship


              Specify that fuel velocity as one tenth light speed, c. Light speed is 299,792,458 m/s; thus,

              ____vfuel = 29,979,246 meters/sec

              Specify ship velocity as the velocity gained due to gravity for one second. Acceleration due to near Earth gravity, g, is 9.80665 meters /second2 (m/s2); thus, we want:

              ___Vship = 9.80665 m/s

              We've arbitrarily chosen value of the fuel's mass to be one gram.

              _______mfuel = 1.0 gram

              Mship = mfuel* vfuel / Vship
              Mship = 1.0 gm * 29,979,246 m/s / 9.80665 m/s
              Mship = 3,057,032.32 gm


                Above equation is simple, and exact values of c and g do add some precision to the calculation. However, we're going to sacrifice a little bit of precision for a whole lot of convenience. Let's do a couple of things: approximate values and convert units.

                Why Approximate?

                Approximate values are much easier to work with.

                300 million m/s is commonly used for c, speed of light; so, we'll use that value.

                Similarly, 10 m/s2 is also commonly used for g, gravity near Earth; so, we'll use that as well.

                Reaccomplishing above equation with new values:

                Mship = mfuel * vfuel/ Vship

                Mship = 1.0 gm * 30,000,000 m/s / 10 m/s

                Mship = 3,000,000 gm

                NOTE: ratio of .1 c to g for one second (29,979,246 / 9.80665 )is closely approximated by 30,000,000 / 10 because 29.98 million is just under 30 million and 9.8 is just under 10.

                Even with approximation, 3 million gms and 30 million m/s both have a lot of zeros. For further convenience, use conversion constants to mix units.


                Why Mix Units?

                For even more convenience, let's mix the units (i.e., use inconsistent dimensions). For example, I'm happy with using grams for fuel mass (mfuel), but I'd much rather use metric Tonnes (mTs) for ship mass (Mship). I.e, 3 mTs is easier to work with then 3,000,000 grams. In like manner, I'm happy with Vship being 10 m/s, but for vfuelI'd much rather use .1c vs. 30,000,000 m/s.

                For below equations, introduce terms and definitions:

                • shipgm designates ship mass (Mship) in grams.
                • shipmT designates ship mass (Mship) in metric Tonnes.
                • fuelm/s designates fuel particle velocity (vfuel) in meters per sec.
                • fueldec_c - fuel particle velocity (vfuel) in decimal c.

                By definition: 1 mT = 1,000,000 gms
                which leads to: 1 gm = 1 mT / 1,000,000

                Thus, shipgm (Mship in grams) is equivalent to shipmT, (Mship in metric Tonnes) if one factors in a conversion constant of k1 = 1,000,000.

                shipgm = (1,000,000) shipmT = k1 shipmT.

                Similarly, c = 300,000,000 m/s; and 1 m/s = ( c/300,000,000)

                fuelm/s = k2 fueldec_c = (300,000,000) fueldec_c.

                Conclusion, to convert fuel particle velocity from m/s to decimal c, apply a conversion constant of k2 = 300,000,000.

                Using consistent units, let's review:

                • Mship = mfuel * vfuel/ Vship

                To clearly indicate that we're using only units of grams and meters/sec, let's make suitable subsitutions for Mship and vfuel.

                • shipgm = mfuel * fuelm/s / Vship
                • shipgm = 1.0 gm * 30,000,000 m/s /(10 m/s)
                • shipgm= 3,000,000 gm

                As discussed above, we want to mix units; so, let's substitute equivalent expression for shipgm and fuelm/s.

                • k1 shipmT = mfuel * k2 fueldec_c / Vship
                • shipmT = (k2/k1) mfuel * fueldec_c / Vship
                • k2/k1 = 300,000,000/1,000,000 = 300 = k3
                • shipmT = k3 * mfuel * fueldec_c / Vship
                • shipmT = 300 * 1.0 gm * .1c / 10 m/s = 3.0 mT

                Thus, given following set of dimensions:

                • mfuel as grams (gm)
                • vfuel as decimal light speed (c)
                • Mship as metric Tonnes (mT)
                • Vship as meters per second (m/s)

                k3=300 is a valid conversion constant and following equation is valid:



                Mship = k3 * mfuel * vfuel / Vship
                Mship = 300 * mfuel * vfuel / Vship

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