Thursday, January 11, 2007

SNOWBALL FROM OORT

RESUPPLY MISSIONS. During the several years that interstellar vessels must "cruise" between stars, new provisions from the Oort Cloud could be put to good use
Oort Cloud (OC) is a huge, spherical body of comets orbiting Sol from 0.3 light-year (LY) to beyond an entire LY.  It is the most likely source of most long-period comets. For more on Oort Cloud.
As a natural border of Sol's system, Oort Cloud contains "trillions of comets" which could be harvested to benefit humankind.

 It is a fair distance from Earth and Sol, human's current environment. At 1 LY (63,241 AU), it will take a typical g-force spaceship profile over two years to travel from Earth. Once there, Sol’s gravity still controls the orbit, but the extreme distance will cause Sol to appear as one of a myriad of stars.
1-G Acceleration
Exterior View:
Refashioned asteroid

Interior View:
Mostly empty space to accommodate human habitation.
Fuel stored between inner and outer hulls.
Acceleration: 1 × g
Time (t)Velocity (Vt)Distance (dt)
percent cLight Year
365¼ days  64.43%c0.377 LY
Given
Vt=c ×( 1 - (1-Δ)t)
Einsteinian
Velocities
dt =c × t+Vt

ln(1-Δ )
Einsteinian
Distances
7-G Acceleration Assume:  Embedded Infrastructure.
a. System of robots/nanobots to process heterogeneous mix of water, methane, ammonia, etc.
b. Self contained propulsion system can turn of/off as required to eventually match speed of target, the 1G pax vessel.
c. Infrastructure Services Include:  Navigation, tracking and communications
Assume:  Resupply Vessel Starts Near Sol.
While source material comes from far ranging comets, assume they are all dispatched to the Sol area for processing before starting their 100 day acceleration.
Acceleration: 7 × g
Time (t)Velocity (Vt)Distance (dt)
percent cLight Year
100 days86.44%c0.155 LY
Given
Vt = c ×( 1 - (1-Δ)t)
Einsteinian
Velocities
dt =c × t+Vt

ln(1-Δ)
Einsteinian
Distances
A very few comets from Sol's Oort Cloud could be refashioned as "snowballs" to resupply interstellar spaceships with many metric Tonnes (mT) of ices.

While manned vessels must accelerate at 1-G to simulate Earth gravity for comfort of crew and passengers, unmanned snowballs would not be bound by this requirement. Thus, these resupply vessels could accelerate well past 1g.  For example, 7g acceleration enables a vessel to reach higher speeds and greater distances much quicker than a 1g-force vessel. Interstellar ships must fly long voyages with no hopes for enroute resupply from other sources; thus, Oort Cloud Objects (OCOs) could provide following materials:
  • Fuel from water and other "ices" which can be superheated and ionized into particles.
  • Life support from water and perhaps liquid oxygen.
  • Building materials from "rocky" materials as well as exports from inner Solar System.
To derive max benefit from resupply vessels, they should optimize their flight times with minimal waste of energy  and quickest flight time.

After 1 year (365 days) of 1-g acceleration, TE's interstellar vessel crew decides to stop propulsion due to fuel concerns.



Vessel will maintain constant velocity of 64.4% light speed for several years.

Vessel's distance can be described by following equation;



d1G = A365  + V1G × t = .377 LY + .644c × t
After 1-g vessel accelerates for 265.25 days, 7-g vessel initiates 100 days of 7-g acceleration. Thus, both vessels start their respective constant velocities around the time the 1-g vessel completes the one year acceleration.




7-g vessel maintains constant .866c for a certain duration.



Vessel's distance can be described as follows:


d7G = A100  + V7G × t = .155 LY + .866c × t
AT CONSTANT VELOCITY, vessels intercept as shown.
For first year, both 1-g passenger vessel and the resupply 7-g vessel go through their respective acceleration profiles as described above. Throughout 2nd year, both vessels cruise on the same path to the same destination star at their respective constant velocities.  The 1-g pax vessel is .377 LY (23,841.5 AU) from Sol with velocity of .644 c (= 111.5 AU/day = 193,066 km/sec). The 7-g resupply vessel is much closer to Sol .155 LY (9,802 AU) from Sol with much greater velocity, .866 c (= 149.9 AU/day = 259,620 km/sec); thus, the resupply vessel will inevitably overtake the pax vessel.
Compute point of interception via common solution to above linear equations:
.377 LY + .644c×t=.155 LY + .866c×t
t = 1.0 years
Unfortunately, there is a huge velocity differential between the two vessels
 .866c -.644c = .222c = 38.44 AU/day  = 66,554 km/sec
For the two vessels to rendezvous, their velocity must be equal at the point of interception; thus, the resupply vessel must decelerate prior to meeting pax vessel.
Deceleration requires following adjustments to Resupply Vessel Profile:
Linear programming method starts with desired end point; then, work backward to determine required start point.
7G's Deceleration Endpoint
must match 1G vessel's velocity at pre-determined intercept.
7G resupply vessel must decelerate from 149.9 AU to match 1G pax vessel's velocity (111.8 AU/day) exactly at an intercept point. Arbitrarily chose an intercept point of 2.0 years and 64,824 AU after 1G vessel launch.
7G Decel.
time
Velocity7G Decel
distance
1G Trip
Distance
1G Total
Trip Time
0  days149.93 AU/dy0.0  AU 59,416 AU 682.25 days
1  day 149.46 AU/dy149.4  AU 59,565  AU683.25 days
...............
48 days112.1 AU/dy6,444.9 AU64,796 AU730.25 days
7G VESSEL MATCHES CRUISE VELOCITY OF 1G VESSEL
48¼ days111.8 AU/dy6,472.8 AU64,824 AU730.5 days
Given
(1-Rt) × c
dt - dt-1
c × t + Vt

ln(R)
 tA + tcr + tD
Numerical methods help us back off 48¼ days from deceleration endpoint to the deceleration start.  For more details, click: 7G Dec
ADJUST ENDPOINTS OF 7G CRUISE

ⓐ 360.2 Days.  Change 7G cruise start from 365.25 days. Cruise velocity remains 149.9 AU/day; thus, the 7G acceleration duration must also remain 100 days. To accommodate the shifted acceleration 100 day duration (see below), 7-g cruise distance has new ordinate intercept (YO):
dAU = -44,986 AU + tDy×149.9 AU/day

365.25 Days (remains the same). Stop 1G acceleration; start 1G cruise; velocity = 111.5 AU/day as described above.

ⓒ  400 days: Resupply vessel lags pax vessel by 11,982 AU.

ⓓ  500 days: Due to much greater speed, resupply vessel shortens the lag to 8,142 AU.

ⓔ  600 days: The lag is further shortened to 4,302 AU.

ⓕ  684 days: When resupply vessel reaches 1,076 AU behind the pax vessel, it prepares to decelerate. Cruise segment ends at point where deceleration starts.
7G Resupply Vessel Shifts Acceleration Period.

① t = 0.0 days. Initiate acceleration of 1-g vessel for passengers ("pax").

② t = 260.2 days. 7G resupply vessel launches and accelerates. This 5 day adjustment proves necessary to eventually intercept the pax vessel in two years past point .

③ t = 360.2 days.  7-g vessel stops acceleration and starts constant velocity, a linear function. For more details, click: 7G Acc

④ t = 365.25 days.  1-g vessel stops acceleration and starts constant velocity. For more details, click: 1G Acc
Resupply Mission Ends
 With 1 G Deceleration

Traditional 1G Profile: Pax vessel initially accelerates at 1G for one year to attain velocity of 64.43c over a distance of .38 LY.  After an indeterminate cruise duration (most likely several years), pax vessel must decelerate for same time/distance just prior to destination.

Resupply 1G Profile starts the same as traditional profile.  However, it differs when pax vessel is joined by resupply vessel which partially decelerates from much higher velocity to match pax vessel's cruise speed.  Finally, the pax and resupply vessel decelerate in tandem just prior to destination. For more, click: 1G Dec

COMPARE INTERSTELLAR FUEL CONSUMPTIONS: 1G vs. 7G

Accel.a = 1 × ga = 7 × g
Daily Diff.ε∇ = .0468% GWε∇ = .339% GW
Daily Rem.(1-ε∇) = 99.9532% GW(1-ε∇) = 99.61% GW
Time (t)Velocity (Vt)Distance (dt)Fuel (ft)Velocity (Vt)Distance (dt) Fuel (ft)
percent cLight Yearpercent GW % light speedLight Year%Gross Wt.
1 day0.28%c0.000 003 9 LY 0.048% GW 1.98%c0.000 03LY 0.339% GW
48¼ days12.8 %c0.0087LY2.35%GW62.0%c0.0475LY14.6% GW
100 days 24.65%c0.035 LY4.73%GW86.44%c0.155LY28.82%GW
200 days 43.22%c0.129 LY9.24%GWPracticality Prevents Powered Progress
Beyond 100 Days at 7G.
300 days 57.22%c0.268 LY13.54%GW
365¼ days 64.43%c0.377 LY16.23 %GW
Given
Vt =173.145 AU

day
× ( 1 - (1-Δ)t )
Einsteinian Velocities
dt =173.145 AU

day
× t+Vt

ln(1-Δ)
Einsteinian Distance
ft = 1 - (1-ε∇)t
Fuel Consumed
For much faster cargo delivery, consider 7g acceleration.
Assume fuel exhaust particles have velocity, VExh = 99.2%c, with a corresponding growth factor, n = 7.
Thought Experiment proposes Einsteinian motions and exponential fuel consumption.
1G Acc7G Acc 7G Dec1G Dec
Flight
Phase
Passenger VesselResupply Cargo Vessel
n×G
time
Incr. Rem
Cum. Rem
Fueln×G
time
Inc. Rem
Cum. Rem
Fuel
Consump.
Accel1×G
365¼days
RI=.8377GW
RC=.8377GW0
F=.1623GW0 
7×G
100 days
RI=.7118GW
RC=.7118GW0
F=.2882GW0 
Cruise
RIncr. = (1-.000468)t

RCum = P(Ri)
R = (1-.00339)t
Decel
7×G
48¼days
RI=.8540GW
RC=.608GW0
 F=.392GW0 
Cruise
RCum = P(Ri)
Decel
1×G
365¼days
RI=.8377GW
RC=.7017GW0
F=.2983GW0 
1×G
365¼days
RI=.8377GW
RC=.5092GW0
F=.491GW0  
CONCLUSION:
7G Total Fuel Consumption
is 49.1% of vessel's
initial Gross Weight (GW0)
after following power profile:
1) 100 days 7G acceleration to high speed cruise.
2) 48¼ days 7G deceleration to rendezvous with 1G Pax vessel.
3) 365¼ days 1G deceleration in tandem with Pax vessel to orbit at destination.

For more, see More Snowalls.



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



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