Thursday, March 11, 2010

MORE SNOWBALLS

If one resupply vessel is great,
multiple resupplies must be even greater!

SNOWBALL FROM OORT introduces the concept of a 7G cargo vessel launching from Sol and overtaking a previously launched 1G vessel, much slower with passengers (pax).

SNOWBALL FROM OORT shows how a cargo vessel can overtake the prior pax vessel exactly one year into its cruise phase.

MORE SNOWBALLS extends this profile with two variations.

First, suppose that mission planners decide to up the game and send an annual resupply vessel throughout the pax vessel's cruise period.

Second, consider a variable 1G vessel profile with different cruise velocities achieved by different durations of 1G acceleration (explained further below).
Towed vs. Throwed
Harvesting the "snowballs" i.e., comets:
Before we "throw" the snowballs on an interstellar profile, we might want to "tow" them back to Sol's inner orbit for processing and additional cargo..
TOWED. Economic feasibility will compel vessels to tow many comets back to Earth from Oort Cloud.  Many missions will require routine travel between Earth and Oort Cloud (system upgrades, crew changes, etc.), vessels may as well bring raw cargo on these "retrograde" trips.
1G Acceleration Phase
Ship starts off trip by accelerating at g-force for 365¼ days and 23,841.5 AU, to a velocity of .6443c (64.43% light speed). Occupants will feel a gravity equivalence.
1G Cruise Phase
After 1 year of 1G propulsion, vessel maintains constant velocity of 111.5 AU/day (64.43% light speed). Artificial gravity continues due to carefully controlled longitudinal spin. Earth observers can measure ship's cruise progress for each year of flight.
Throwed Snowballs
These "snowballs" could resupply interstellar spaceships enroute to their destination.

Unmanned, they would not be limited by the need to simulate near Earth gravity; so, they could accelerate well past 1 g to quickly attain much greater velocity.

They could intercept the 1G "passenger" (pax) vehicle and travel in tandem.  During the remainder of the cruise period, the "snowball" could be harvested by the pax vessel occupants in a thoughtful, well planned manner.

Upon approaching stellar destination, the two vessels could  decelerate in tandem, and both could achieve an orbit at their new stellar home.
Annual Resupply Vessels
Mission stakeholders could plan a resupply flight for each year of the Pax vessel's cruise phase.

Linked table (7G Dec) uses numerical methods to determine duration and distance (48¼ days and 6,484 AUs) required to decelerate from 7G cruise speed (149.9 AU/day) to 1G cruise speed (111.8 AU/day).
Interception Point 7G Decel Start Point
time dist time dist
1 Yr.23,842 AUn/a n/a
2 Yr.64,567 AU682.25 days58,083 AU
3 Yr.105,293 AU1,047.50 days98,809 AU
4 Yr.146,018 AU1,412.75 days139,534 AU
5 Yr.186,744 AU1,778.00 days180,260 AU
Given See Prev. Tablet- 48¼ days d - 6,484 AU

To determine each 7G Deceleration Start Point, 
simply subtract indicated values from corresponding Interception Point.
Insets show 7G deceleration phase (48¼ days and 6,484 AUs) for several successive years of Pax vessel's cruise phase.When the 7G vessel starts deceleration, it is 1,104 AUs (= 6,484AU - 5,380AU) behind the 1G vessel.  Throughout the 48¼ days, this gap gradually decreases till they rendezvous at the end of this duration. Thus, 7G vessel precisely matches 1G vessel's cruise velocity at each Interception Point.
7G Cruise End Points: 682.25 days; ⓑ 1,047.5 days;1,412.75 days.7G Cruise Start Points: 360.2 days;  ② 453.7 days;  ③ 547.3 days.
7G Cruise End Point 7G Cruise Start Point
time (t) dist (d) time dist
682.25 days58,083 AU360.2 days9,802 AU
1,047.50 days98,809 AU453.7 days9,802 AU
1,412.75 days139,534 AU547.3 days9,802 AU
1,778.00 days180,260 AU640.9 days9,802 AU
Previous Table
t -d - 9,802 AU

149.9 AU/day
Given

"7G Cruise End Point" coincides with the
 "7G Deceleration Start Point".
After 100 days of 7G acceleration, Resupply vessel travels 9,802 AUs.
Determine start time for each example by subtracting 100 days.
Thus, first start time, A = -100 days = 260.2 days.
Similarly, B = 353.7 days, and C = 447.3 days after Pax vessel launch.
After 100 days of 7G acceleration, each Resupply vessel starts cruise phase with constant velocity of 149.9 AU/day. (Points: ①  ②  ③)
At end of each cruise phase (Points: ⓐ  ⓑ  ⓒ ), Resupply vessel must decelerate for 48¼ days to match speed of Pax vessel.
DESIRED END STATE:  Resupply (7G) vehicle matches velocity of Pax vessel exactly upon interception. For rest of flight, vessels fly in tandem.  CONCLUSION:  Such maneuvers can happen multiple times during the flight.

Achieve Different 1G Cruise Speeds
by adjusting duration of 1G accelerations for Passenger vessel.
Faster Cruise Speeds come from longer accelerations.
EXAMPLE: 1G propulsion for 1.1 year (401.7 days) takes vessel to a velocity of 117.9 AU/day over a distance of 28,072 AU.

BASELINE: By comparison, 1G propulsion for 1.0 year takes vessel to 111.8 AU/day and 23,876 AUs.

Slower Cruise Speeds come from shorter accelerations.
EXAMPLE:  1G propulsion for .9 year (328.7days) takes vessel to a 105.1 AUs/day over 19,913 AUs.
Acceleration: 1 × g
Time (t)
Velocity (Vt)Distance (dt)
YearsDaysPer
Cent c
AU per
day
Light
Year
Astro-
Units
Faster1.1 Y 401.7 d67.63%c117.1AU/d   0.458 LY28,972AU
Baseline1.0 Y365¼ d64.43%c111.9AU/d   0.377 LY23,842AU
Slower0.9 Y328.7 d 60.70%c105.1AU/d   0.315 LY19,913AU
Given
Vt=c ×(1-(1-Δ)t)
Δ = .2826%
dt =c × t+Vt

ln(R)
R = 1-Δ = 0.997174
For convenience, arbitrarily define BASELINE 1G acceleration profile with a duration of 1 year; thus, start acceleration at day 0 and end after 365.25 days.  Resultant Velocity becomes vessel's cruise velocity.  See dashed line on above diagram.
NOTE: To readily compare different cruise phases (following frames), arrange acceleration start times for all cruises to start at a common time at 365.25 days.

For faster cruise, increase baseline acceleration duration by 10% from 365.25 days to 401.7.  Thus, start faster acceleration at -36.45 days (before day zero, BASELINE start.)

For slower cruise, decrease baseline acceleration duration by 10% from 365.25 days to 328.7 days.  Thus, start slower acceleration at +36.55 days (after day zero, BASELINE start.)
EXAMPLE:  Consider a slower than baseline trip to our nearest stellar neighbor, Alpha Centauri, 4.13 Light Years (LYs) from Sol. During a typical interstellar flight profile; a spaceship could
  • Accelerate at 1G from Sol for 0.9 years to attain reasonable cruise speed (.60c) in a distance of .31 Light Year (LY).
  • Decelerate at 1G for 0.9 year (same duration as acceleration) starting .31 LY prior to destination.
  • Between acceleration and deceleration, cruise for several years at constant velocity, 60% Light Speed, c.
  • Cruise distance (dCRU) is total dist minus Accel dist minus Decel dist:
    dCRU = 4.3 LY - .31 LY - .31 LY = 3.68 LY
  • Earth bound entity would observe cruise duration as 6.13 years (3.68 LY ÷ .60 LY/yr).
For each year (as observed from Earth) of each vessel's cruise phase, one could plan for each 1G vessel to intercept a 7G resupply vessel.
PREPLANNED ANNUAL INTERCEPTS
Cruise Start
(after 1 yr Baseline 1G)
Cruise Distances
(at annual interviews)
Dist. (d0)Vel. (V)2 yr3 yr4yr
Faster
28,972 AU
117.9 AU/d
72,036AU

115,098AU

158,161 AU
Baseline
23,842 AU
111.8 AU/d
64,678AU

 105,511.9 AU

146,347AU
Slower
19,913 AU
105.1 AU/d
58,301AU

96,689AU

135,076AU
SeePrev. Tabled0 + V × (t-1)
First three years are shown in above table and associated diagram (to the right.)
Different 1G Vessel Velocities Require Diff. 7G Decelerations
NOTE:  See 7G deceleration values from following link: (7G Dec)
MATCH SPEED OF 1G's FASTER CRUISE (117.9 AU/day):
To match this velocity, 7G vessel must decelerate from its cruise speed (149.9 AU/day) for 43 days and 5,843.2 AUs.
BASELINE CRUISE speed is arbitrarily chosen as vessel velocity (111.8 AU/day) attained after 1 year of 1G propulsion:  Thus,  7G vessel must decelerate for 48.25 days and 6,444.9 AUs.
MATCH SPEED OF 1G's SLOWER CRUISE (105.1 AUs/day):
7G vessel must decelerate for 53.4 days and 6,974.1 AUs.
Consider Selected Annual Intercepts
Determine "start point" of 7G Decel.
Annual
Intercept
7G Deceleration
Interval
7G Deceleration
 Start Point
time
(tAnn)
distance
(dAnn)
time
(tInt)
dist
(dInt)
timedist
Faster2 Yr 
72,036 AU
43
days
5,843.2
AU
687.5
days
66,192.8
AU
Baseline3 Yr
 105,511.9 AU
48.25
days
6,444.9
AU 
1,047.5
days
99,067
AU
Slower4 Yr
135,076 AU
53.4
days
 6,974.1
AU
1,407.6
days
133,668.4
AU
Previous
Tables
7G Dec tAnn - tInt
 Assume measurements accomplished by Earth bound observers.

After slight 7G deceleration interval, resupply vessel matches velocity of pax vessel at exact time/distance (Selected examples include points:   , .) Thereafter, both vessels travel in tandem.
End point of 7G cruise segment;
coincides with start point of 7G Deceleration (Previous Table).
7G Cruise Segments
Cruise End PointCruise Start Point
Time (t1)Dist. (d1)Time (t0)Dist. (d0)
Faster687.5 days66,193 AU  312 days9,841 AU
Baseline1,047.5 days99,067 AU  452 days 9,841 AU
Slower1,407.6 days 133,668 AU  582 days 9,841 AU
See Prev. Tablet0 = t1 - (d1 - d0)/V
Cruise start point of 7G cruise segment;
coincides with endpoint of 7G Acceleration.
Jan Hendrik Oort
"GREAT OAK OF ASTRONOMY"
(28 April 1900 - 5 November 1992)  
Subrahmanyan Chandrasekhar:
"The great oak of Astronomy has been felled, and we are lost without its shadow".
Jan Hendrik Oort was a Dutch astronomer; best known for Oort Cloud of comets which bears his name. Oort Cloud is the de facto boundary of our Solar System.
In 1935, Dr. Oort became professor at the observatory of the University of Leiden; Ejnar Hertzsprung was the director. (Dr. Hertzsprung is well known for the Hertzsprung-Russell diagram, an essential tool in the study of stars.) Fascinated by radio waves from the universe, Dr. Oort pioneered radio astronomy with an old radar antenna confiscated from the Germans after WWII. Radio interferometry was suggested by Oort well prior to their experimental tests by others (Ryle in Cambridge and Pawsey in Sydney).   In the 1950s, he raised funds for a new radio telescope in Dwingeloo, in the east part of the Netherlands, to research the center of the galaxy.
In 1950, Dr. Oort observed most comet origins to be within the Solar System.  HYPOTHESIS:  SOL IS SURROUNDED BY BILLIONS OF COMETS. While very few enter the inner Solar System, and even fewer are observed; an analysis of these few indicate the existence of the Oort Cloud.
Summarize 7G Resupply Solution
1) Determine Annual Intercepts of 1G Cruises. Plan for resupply vessels to join passenger vessels at specific time/distance points.  For convenience, arbitrarily choose these points at one year intervals (as measured by Earth observers).

2) Determine Start Point of 7G Deceleration.  7G vessel must decrease its cruise velocity (from 149.9 AU/day) to precisely match velocity of its target pax vessel (1G cruise velocity of about 110 AU/Day.)

3) Determine Start Point of 7G Cruise. Upon completion of 7G acceleration (100 days), Resupply Vessel starts the cruise portion of its journey at 149.9 AU/day.

4) Determine Start Point 7G Accelerations at 100 days prior to corresponding end-times.  (Recall these end-times coincide with start-points of 7G cruise segments.)
COMPARE INTERSTELLAR
 FUEL CONSUMPTIONS:
Consider different 1G profiles due to
different 1G acceleration durations
Following three examples consider 1G acceleration durations of .9 year, 1.0 year and finally 1.1 years. 
HYPOTHESIS: Increased durations increase fuel consumptions.
Oort's other discoveries include:
  • In 1924, Oort discovered the galactic halo, a group of stars orbiting the Milky Way but outside the main disk.
  • In 1927, he calculated that the center of the Milky Way was 5,900 parsecs (19,200 light years) from the Earth in the direction of the constellation Sagittarius
  • Mass of Milky Way exceeds 100 billion Solar masses. 
  • Light from Crab Nebula is produced by synchrotron emission.
1G Vessel: Least 1G Duration (0.9 Yr)
 Daily Difference (Δ)
Previous work estimates that under certain conditions, a vessel consumes .0468% of its gross weight (GW or "mass") for one day of 1G-force propulsion.  Thus,
Δ = .000468 GW
Thus, compute daily remainder:
RΔ = (1 - Δ)GW = 0.999532 GW
Similarly, assume 7G daily diff is seven times greater; thus, for 7G propulsion:
7×Δ = .00328 GW
R= (1-7×Δ)GW = .99672 GW.
Flight
Phase
Passenger VesselResupply Cargo Vessel
n×G
time (t)
Phase Rem.
Cum. Rem.
Cum. Fuel
Consumed
n×G
time (t)
Phase Rem.
Cum. Rem.
Cum. Fuel
Consumed
Accel1×G
329days
RΦ=.8573GW
RC=.8573GW0
F=.1427GW0 
7×G
100 days
RΦ=.7118GW
RC=.7118GW0
F=.2882GW0 
Cruise
RΦ   (1-.000468)t

F = 1 - RC = 1 - P(RΦ)
RΦ  = (1-.00328)t
Decel
7×G
53.4days
RΦ=.8342GW
RC=.5938GW0
 F=.4062GW0 
Cruise
F = 1 - RC = 1 - P(RΦ)
Decel
1×G
329days
RΦ=.8573GW
RC=.735GW0
F=.2651GW0 
1×G
329days
RΦ=.8573GW
RC=.509 GW0
F=.491GW0
1G Vessel: Baseline Duration (1 Yr)
Flight
Phase
Passenger Vessel
Phase Remainder (RΦ)
1G profile has two propulsion phases: 1) 1G acceleration to cruise speed; 2) 1G deceleration to destination.
7G profile has three propulsion phases: 1) 7G accel to cruise; 2) 7G decel to match 1G velocity; 3) finally, 7G profile's third phase is the same 1G deceleration propulsion as used in the 1G profile.
Phase Remainder (RΦ) is vessel mass remaining after a vessel completes a  propulsion phase.  For each 1G propulsion, compute RΦ as follows:
RΦ = (1-Δ)t= (.999532)t
For each 7G propulsion, compute RΦ as follows:
RΦ = (1-7×Δ)t= (.99672)t
Resupply Cargo Vessel
n×G
time (t)
Phase Rem.
Cum. Rem.
Cum. Fuel
Consumed
n×G
time (t)
Phase Rem.
Cum. Rem.
Cum. Fuel
Consumed
Accel1×G
365¼days
RΦ=.8377GW
RC=.8377GW0
F=.1623GW0 
7×G
100 days
RΦ=.7118GW
RC=.7118GW0
F=.2882GW0 
Cruise
RΦ = (1-.000468)t

F = 1 - RC = 1 - P(RΦ)
RΦ  = (1-.00339)t
Decel
7×G
48¼days
RΦ=.8540GW
RC=.608GW0
 F=.392GW0 
Cruise
F = 1 - RC = 1 - P(RΦ)
Decel
1×G
365¼days
RΦ=.8377GW
RC=.7017GW0
F=.2983GW0 
1×G
365¼days
RΦ=.8377GW
RC=.5092GW0
F=.491GW0  
1G Vessel: Greatest Duration (1.1 Yr)
Flight
Phase
Passenger VesselResupply Cargo Vessel
Cumulative Fuel Consumption
increases as cumulative remainder (RC) decreases.  RC is product of previous phases:
RC = P(RΦ)
For 1 G profile:
RC= R1G-Acc × R1G-Dec
For 7 G profile:
RC=R7G-Acc×R7G-Dec ×R1G-Dec

For all profiles:
F = (1-RC) GW0
n×G
time (t)
Phase Rem.
Cum. Rem.
Cum. Fuel
Consumed
n×G
time (t)
Phase Rem.
Cum. Rem.
Cum. Fuel
Consumed
Accel1×G
401days
RΦ=.8289GW
RC=.8289GW0
F=.1711GW0 
7×G
100 days
RΦ=.7118GW
RC=.7118GW0
F=.2882GW0 
Cruise
RΦ = (1-.000468)t

F = 1 - RC = 1 - P(RΦ)
RΦ = (1-.00339)t
Decel
7×G
43 days
RΦ=.8641GW
RC=.6150GW0
 F=.385GW0 
Cruise
F = 1 - RC = 1 - P(RΦ)
Decel
1×G
401days
RΦ=.8289GW
RC=.6870GW0
F=.3130GW0 
1×G
365¼days
RΦ=.8289GW
RC=.5098GW0
F=.490GW0  
 CONCLUSION: More Propulsion Time Requires More Fuel.
AXIOM: Most interstellar voyages require same energy for deceleration as for acceleration.
For 1G profile, following three variations assume: deceleration energy = acceleration energy
VariationAcceleration
Time (tAcc)
Deceleration
Time (tDec)
Propulsion
Time (t)
1G Fuel
1) Slower
0.9 year328.73 days656.46 days26.49% GW0
2) Baseline
1.0 year 365.25 days730.50 days28.96% GW0
3) Faster
1.1 year 401.78 days803.56 days 31.35% GW0
GivenGiven365.25 days/yr × tAcctAcc+ tDec (1 - (1 - Δ)t) GW0
CONCLUDE:  1G fuel consumed increases as propulsion time increases.
For all 3 1G variations, resupply vessels use 100 days of 7G acceleration
to high speed cruise (149.9 AU/day). 
Thus for all 3 variations:
t7G-Acc = 100 days
f7G-Acc = (1 - (1 - 7×Δ)100) GW0 = .2882 GW0
7G Propulsions1G PropulsionsTotal
Variation
(Cruise Speed)
7G Decel.
Time(t7GD)
7G Prop
Time (t7G)
7G Fuel
Consumed (f7G)
1G Decel.
Time (t1GD)
1G Decel
Fuel (f1GD)
Ttl Resupply
Fuel (fResup)
1) Slower     (105.1AU/day)
53.4 days153.4 days39.55%328.73 days14.26% GW053.81% GW0
2) Baseline (111.8 AU/day)
48¼ days148¼ days38.52% 365.25 days15.72% GW054.24% GW0
3) Faster    (117.9 AU/day)
43 days143.0 days37.45%401.78 days17.14% GW054.59% GW0
GivenNumerical
Methods
(t7GD + 100) dy(1 - (1 - 7×Δ)t) GW0See Prev.
Table
(1 - (1 - Δ)t) GW0(f7G + f1GD) GW0
CONCLUDE:  Fuel consumed still increases as 1G propulsion time increases.



VOLUME I: ASTEROIDAL
VOLUME II: INTERPLANETARY
VOLUME III: INTERSTELLAR




1 Comments:

Blogger glenn robinson said...

Jim - Congratulations on the excellence of your presentation! Your methodical approach, giving each factor due consideration, has the makings of a textbook, but the exposition would suit a wider readership as well. Your idea regarding use of a particle accelerator as a drive falls naturally out of your material, but is not the first thing a space enthusiast would consider.
My own interest is in bootstrapping a space economy in the inner solar system based on use of asteroid materials and sunlight. Best regards and thanks again for the well-directed effort!

7:36 AM  

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