Saturday, May 13, 2017

VOLUME 0: ELEVATIONAL Table of Contents

VOLUME 0: ELEVATIONAL
VOLUME I: ASTEROIDAL
VOLUME II: INTERPLANETARY
VOLUME III: INTERSTELLAR
Long term deployment 
of space borne enterprises
 requires reliable raising
of Earth bound resources
 to Earth bound orbits
without risky rockets.
bl
Volume 0: ELEVATIONAL
1.NEED TO ELEVATE. Humanity's space presence will require frequent, reliable elevation of Earth resources into orbit.  BACKGROUND: Space Elevator's theoretical "tether" concept could be a start.
2.UP-LINK TETHER: Climbers Ascend Tether to GEO Node. Associated "strain" might break even the strongest cable at anytime. Thus, a physical cable requires an instant backup capability.
3.DEDICATED DOWN-LINK: Climbers Descend Tether to Marine Anchor.  During normal ops, dedicated downlink tether would provide downward bound service for climbers to return to Earth. 
4.OUT-LINK TETHER: Beyond GEO to Apex Anchor. (AA) . Thought Experiment (TE) considers innovative ways to leverage Space Elevator (SE) beyond Geosynchronous Equatorial Orbit (GEO).
5.LIFEBOATS: Ride the Virtual Tether.  For a reliable, rapid return to Earth's surface, ride the virtual tether, an Earth based, high speed, particle beam to enable "lifeboats" to land at zero velocity.
6.BASE STATION:  Forget Down-Link's physical tether; instead, exclusively use lifeboats to travel Down-Link Virtual Tether back to Base Station with a reliable particle beam capability.
7.SEVERED TETHER: Some scenarios end badly.  EXAMPLES: 1) Taut tether might "snap" and recoil. 2) A broken tether might wrap around the Earth.  3) ... perhaps even a "ring of fire" around the globe. 
8.ELEVATIONAL ALTERNATIVES  Besides the Space Elevator (SE), there are several other non-rocket proposals to lift a payload from Earth's surface to Earth's orbit.  Also, see: SKYHOOK.
9.BEAM ME UP! Thought Experiment (TE) proposes another non-rocket method to reach GEO.  Replace Up-Link's physical tether with a virtual tether. Also, see STREAMING ION BEAMS.
10.HIGH STAGE ONE: Leverage Lofstrom's Launch Loop concept to avoid wind and weather in lower atmosphere and get a head start on trip to GEO.  
11.CONSTRUCTING SPACE ELEVATOR. Using physical tether to construct subsequent tethers might make sense. HOWEVER, virtual tether makes more sense; no rocket risk nor risk of sever.
12.SPACE TUG: Earth Orbits. TE's Space Tug could move vehicles between tethers; but it would be best used in moving Space Habitats among different terrestrial orbits. Also, see "Ion Thrusters" and "Specific Impulse".
13.BASE OFF EQUATOR?  Can a tether anchor off the Equator??  Not much.  HOWEVER, a Base Station can locate well away from equator and still project an ion beam toward the GEO Node.
14.ELEVATIONAL CHALLENGES MAIN CHALLENGE: Quickly move pax/cargo from Earth's Equator to GEO without the absurdly long cable (might extend 1/4 distance to the Moon!!!!).
15.BEAM ME OUT!!! Expand range of orbital destinations from strictly GEO to well beyond. Even escape Earth's gravity to travel throughout the Solar System.
VOLUME 0: ELEVATIONAL
VOLUME I: ASTEROIDAL
VOLUME II: INTERPLANETARY
VOLUME III: INTERSTELLAR

posted by JimODell at 4:05 AM 0 comments

Saturday, August 30, 2014

VOLUME I: ASTEROIDAL Table of Contents

Over the next century,
mankind's dwellings will include habitats
throughout the Solar System.
  • HABITATS:  Some human colonies will live in large, spinning cylinders constructed from "in situ" material from asteroids and comets.
  • EARTH LIKE GRAVITY: Spinning cylinders will simulate g-force via centrifugal force along longitudinal axis.
  • ENERGY SOURCE: Large, adjacent mirrors will reflect sunlight into habitats for plentiful energy.
  • FRESH FOOD: Essential ingredients from Mother Earth will enable robust agricultures.
  • SPENDING MONEY: Harvesting of space bound materials will enable a robust economy among the many habitats as well as Mother Earth
  • POPULATIONS: Smaller habitats will easily house 10,000 people; larger habitats will provide comfortable quarters for well over 100,000.
Volume I: ASTEROIDAL
1.HABITATS CAN ORBIT. Transform selected asteroids into orbiting habitats; i.e., "orbiters". SEE APOLLO. NOTE: Background material: Orbit Elements, Inclination, and Omega, Orbit "Rolls"
2.HABITATS CAN TRANSPORT.  As transporters, habitats can move large human populations throughout the Solar System; for example, to Mars orbit via a simple "semi-orbit", described in an associated TABLE.
3. HABITATS CAN CYCLE BACK TO EARTH.  Refashion selected Near Earth Asteroids (NEAs) as resonant cyclers which rendezvous with Earth every 2 years. See associated: TABLE I and TABLE II.
4.HABITATS CAN CYCLE TO MARS:  TE coins term "Marsonance"; design Habitat's orbit to  routinely rendezvous with Mars due to well designed orbital "resonance". (See TABLE.)
5.HABITATS CAN IMPORT RESOURCES FROM EARTH. To properly provision initial habitats, use "space elevator" to export many materials from Mother Earth to her children. 
6.HABITATS CAN TERRAFORM. For Earth-like environment, habitat transform processes include:
1)  import Earth soil for minerals and microbes; 2) spin for gravity; 3) mirrors reflect sunlight for energy, etc. 
7.HABITATS CAN COMMUNICATE: Refashioned habitats can communicate all through the Solar System by deploying specialized comm capsules (i.e. COMMUNICATORS).
8. SPACE TUG: Solar Orbits. Another TE brainstorm could help Space Habitats transition to different orbits around the Sun. Also, see "EP Thrusters" and "Specific Impulse".
9.NEW MOON: LAUNCHING ALPHA.(Leverage New Moon for a slightly smaller, slightly quicker Solar orbit.) Alpha could be a huge habitat, 60° ahead of Earth.  To build such a habitat, use Luna as a launch platform. See Alpha Launch Tables.
10.FULL MOON AND BEYOND: LAUNCH OMEGA.  (Leverage Full Moon for a slightly larger, slightly slower orbit around Sol.) Omega could be a huge habitat at Lagrange Point 5 (L5), 60° behind Earth. At 1 AU from Earth. See Omega Launch Tables
11.EARTH's ELLIPTICITY Helps a Lot!!! Earth's orbit is near circular, but it still has enough "eccentricity" for a habitat to more quickly attain a parking sport of 60° from Earth.
12.LEVERAGING LAMBDA for Cyclers. Synergize Habitats Alpha (α) and Omega (Ω) with 2 year cycler orbits.  Requires careful placement of lambda (ℓ), semi-latis rectum.  See 2 YEAR CYCLER TABLE.
13.HABITATS CAN MINE MINERALS from the Asteroid Belt, a plentiful source of materials required to construct and maintain habitats in space.
14.HABITATS CAN EXTRACT ELEMENTS from the Gas Giants, a plentiful source of essential elements required to power habitats in deep space beyond the reach of effective sunlight.
15.HABITATS CAN MIGRATE Initial habitats will orbit and cycle, but g-force propulsion will enable much quicker migration to the planets in days and to nearby stars in years. 
VOLUME O: ELEVATIONAL
VOLUME I: ASTEROIDAL
VOLUME II: INTERPLANETARY
VOLUME III: INTERSTELLAR

posted by JimODell at 5:16 AM 0 comments

VOLUME II: INTERPLANETARY Table of Contents

Interplanetary flights will become routine
(much like airline travel today)
because travel time will become reasonable.
Furthermore, spacecraft will maintain comfortable Earthlike conditions (gravity, atmosphere, comfortable billets, entertainment, etc) throughout the flight.
Consider Einstein's thought experiment about an accelerating elevator. If the elevator accelerates at same rate as free falling objects near Earth's surface, then occupants will feel same g-force as if they're static on Earth's surface. (Einstein called this "equivalence".)
Instead of Einstein's elevator, our thought experiment notionalizes a high performance spaceship to accelerate at rate, g, to produce gravity like force (g-force). A g-force trip to nearby planets will take days or weeks (orbiting vehicles now take years). We speculate that g-force propulsion can be accomplished with well known science and slight advances in current technology.
Volume II: INTERPLANETARY
1. ACCELERATE FOR A DAY: Consider Einstein's famous "Thought Experiment" which proposes an equivalence between being static on Earth's surface with accelerating at 1 g through space.
2.ACCELERATE TO THE PLANETS in days; G-force acceleration greatly increases ship's velocity. However, the ship must SLOWDOWN throughout the second half of the flight.
3.MOMENTUM MAKES IT HAPPEN: High speed exhaust particles approach light speed; thus, their relativistic mass gain further increases momentum to further enhance propulsion.
4.MASS TO MOTION: On board particle accelerators can drive exhaust particles with enormous momentum to drive spaceships. Express fuel consumption as % initial gross weight per day.
5. PUSH PARTICLES TO INTERPLANETARY . Particle exhaust speeds from .1c to .5c will transport people, cargo and habitats to nearby planets inside the Kuiper Belt (KB).
6.TERRESTRIAL TUG UPGRADES to interplanetary.  Prime example of a "traveling tug" is Ion Thruster technology as  developed by Dr. Chang-Diaz's VASIMR team at Ad Astra Corp.
7.PROFILE TO THE PLANETS: G-force can propel vessel via straight line to destination. However, flight profile must be carefully planned and executed. 
8.G-FORCE TO MARS: Water can get us to our red neighbor quickly. Thought experiment assumes water as the source of plasma particles for the propulsion system of the g-force spaceship.
9.FINITE FUEL: Consider "g-force" propulsion, specified quantity of charged particles exit aft at near light speeds. At a given consumption rate, logarithms can approximate vessel's range.
10.ACCELERATORS IN SPACE: Let particle accelerators propel g-force vessels.  Space environment provides some benefits.
11.G-FORCE ELEVATORS: BEAM ME UP quickly move pax/cargo from Earth's Equator to Geographic Equatorial Orbit (GEO) Node without the absurdly long tether.
12.TO URANUS:Why go?? Helium-3, an isotope of Helium, can produce cheaper, cleaner fuel for onboard power needs for space faring enterprises. He-3 is plentiful on the Gas Giants; with the lowest escape velocity, mining HE-3 from Uranus might prove fruitful.
13.KEEPERS FROM KUIPER: Space communities will eventually depend on raw materials from space. Kuiper Belt has plenty.
14.SUPER G-FORCE: Can interplanetary cargo travel quicker than passengers at g-force?
YES!!!!  Less time requires more fuel, but it may be worth it.
15.EXTRAPLANETARY: Prep for the Stars. Transition to interstellar. Eventually, interplanetary flights will become routine. When they do, the practicality of interstellar flights will become imminent.
VOLUME 0: ELEVATIONAL
VOLUME I: ASTEROIDAL
VOLUME II: INTERPLANETARY
VOLUME III: INTERSTELLAR

posted by JimODell at 5:12 AM 0 comments

Thursday, August 28, 2014

VOLUME III: INTERSTELLAR Table of Contents

With traditional propulsion, 
flight to nearby stars take long centuries.
FORTUNATELY, g-force reduces flight time to a few years.
Unfortunately, fuel can severely limit duration/distance for interstellar g-force voyages; though, fuel is not a problem for interplanetary flights.  G-force vessels can easily carry sufficient fuel to accelerate at constant g-force throughout a trip to Mars, which would take a few days and only a few percent of the ship's mass for fuel.  However, interstellar vessels would easily consume well over 100% of its weight in fuel during the multi-year voyage.
Thus, interstellar g-force ships will conduct their voyages via 
three phases:
  1. PHASE I. Accelerate at g-force for about a year to a significant percentage of light speed.
  2. PHASE II. Cruise for a few years at this speed to save fuel (simulate gravity via longitudinal spin).
  3. PHASE III. Decelerate at g-force back to near zero velocity for orbital operations at the destination stellar system.

Volume III: INTERSTELLAR
1.INTERSTELLAR SCENARIOS: Thought Experiment (TE) has grouped some common scenarios as: 
1) Theoretical 2) Feasible 3) Practical.  Subsequent chapters focus on the practical.
2.PUSH TOWARD INTERSTELLAR: Interplanetary performance envelope will need considerable "pushing" for interstellar flights. Particle exhaust speeds must be at least 86.6% light speed (.866 c).
3.ACCELERATE FOR A YEAR: Compare spaceship's g-force speeds with c, light speed. See associated 1G TABLE: Accelerate for 1 Year.
4.DETERMINE DUE DISTANCE:  Use exponential to determine g-force interstellar speeds;
from integral calculus, use integral to determine distance duly traveled.
5.TO NEIGHBOR STARS: Between an initial, year long, g-force acceleration and the voyage's final year of g-force deceleration, interstellar flights need a multi-year cruise phase to conserve fuel.
6.PRACTICALITY: LIMITED RANGE: G-force acceleration requires mass/energy conversion. Since spaceship has limited mass; it has limited range. Inevitable inefficiencies limit range even further.
7.DYNAMIC EFFICIENCY FACTOR: Make the feasible range more practical with a dynamic efficiency factor which inversely correlates with vessel performance (particle velocity as it exits vessel's exhaust).
8.FUSION WILL WARM US: Fusion reactors will probably provide the power (though not the propulsion) for humans to live well and prosper during multi-year journeys to the stars. 

9. INTERSTELLAR SUPER G:  Unmanned AI vessels might use greater than g-force propulsion to resupply manned, interstellar vessels between the stars.  See associated 7G TABLE: Accelerate for 100 Days.
10.SNOWBALL FROM OORT: From Oort's comets, construct ice encased vessel to travel at interstellar super G throughout our stellar neighborhood.  See associated 7G TABLE: Decelerate for 48¼ days.
11.ANNUAL SNOWBALLS:  Interstellar travelers might need multiple resupplies throughout their voyage.  Thus, TE proposes more ways to throw more snowballs. See 1G TABLE: Decelerate for 1 Year.
12.ENHANCED TIME DILATION: With annual snowballs, g-force vessels could conceivably accelerate to 86.6%c; then, passengers would age ½ as fast as Earth observers. (NOTE: Without "snowballs," practical cruise speed is 64.4%c; thus, STANDARD TIME DILATION TO AC.)
13.INTERSTELLAR COMMUNICATIONS: Maintaining huge data flows over extreme distances will likely involve well planned placement of AI controlled, interstellar beacons.  IMPORTANT, these necessary, autonomous devices must actively avoid collisions, especially with the very vessels they support.
14.INTERSTELLAR LIGHTHOUSES: Like Earth's coastal lighthouses, neighboring stars can help us avoid enroute hazards. Singularities (aka "Black Holes") are especially insidious.
15.HUBS: Sol's closest stellar neighbors can help humanity travel to even further stars. Vessels can stop there to replenish resources before traveling on. Also consider:  OCTANTS: TE groups neighboring stellar systems (perhaps 51 within 15 LYs) into 8 octants and BEARINGS.
VOLUME 0: ELEVATIONAL
VOLUME I: ASTEROIDAL
VOLUME II: INTERPLANETARY
VOLUME III: INTERSTELLAR

posted by JimODell at 4:11 PM 0 comments

Sunday, July 20, 2014

7G TABLE: Decelerate for 48 ¼ Days

CONTENT
INTRODUCTION
MUST SLOWDOWN!!!
7G ACCEL FOR 100 DAYS
DECELERATE
INTERSTELLAR VESSELS
CRUISE AT CONSTANT VELOCITY
CRUCIAL EVENTS
XXXXXXXX
INTERSECTION POINT
START 7G DECEL
SUMMARY
7G DECEL DAYS 0-6
7G DECEL DAYS 7-12
7G DECEL DAYS 13-18
7G DECEL DAYS 19-24
7G DECEL DAYS 25-30
7G DECEL DAYS 31-36
7G DECEL DAYS 37-42
7G DECEL DAYS 43-48¼
7G DECEL DAYS 49-54

INTRODUCTION

To effectively accomplish an interstellar resupply mission, an Artificial Intelligent (AI) vessel could conceptually use 7G propulsion to quickly intercept the primary 1G vessel. After 100 days of 7G propulsion, the resupply vessel reaches an enormous velocity of 149.9 AU/day (.866 c).  THEREFORE,  the superfast, resupply vessel must slow down to match cruise velocity of the pax vessel. This brainstorm chapter proposes daily progress throughout a conceptual, deceleration duration of 48¼ days.
RESUPPLY VESSEL MUST SLOWDOWN!!!
7-G PROPULSION FOR 100 DAYS
Table, 7G Acceleration for 100 Days, describes daily progress of vessel accelerating from zero velocity at 7g.
After 100 days of 7g acceleration, a vessel travels total distance of 9,842 AUs (.155 Light Year, LY). It gained velocity of about 150 AU/day (0.866 c=86.6% light speed).
Assume 7G vessel cruises at this velocity until the optimal time/distance to start decelerating at 7g.
DECELERATE FOR 48¼ DAYS
Decel.
Time(t) 		Spot
Velocity (Vt)		Spot
Distance (dt)		Daily
Dist. (dΔ)		7G
Dist.(d7G) 		Consumed
Fuel (F)
0 days149.93 AU/dy86.59% c9,811.1 AU0.1551 LY0.0 AU9,811.1 AU27.974% GW0
1 days149.46 AU/dy86.32% c9,661.7 AU0.1528 LY149.4 AU9,960.5 AU28.210% GW0
2 days148.98 AU/dy86.04% c9,512.8 AU0.1504 LY148.9 AU10,109.4 AU28.445% GW0
3 days148.48 AU/dy85.76% c9,364.4 AU0.1481 LY148.4 AU10,257.9 AU28.679% GW0
4 days147.98 AU/dy85.46% c9,216.5 AU0.1457 LY147.9 AU10,405.8 AU28.913% GW0
5 days147.46 AU/dy85.17% c9,069.1 AU0.1434 LY147.4 AU10,553.2 AU29.146% GW0
6 days146.94 AU/dy84.86% c8,922.2 AU0.1411 LY146.9 AU10,700.1 AU29.378% GW0
Given
(1 - (1-Δ)100-t) × c
c×(100-t)+ Vt
ln(1-Δ)
dt-1 - dt dΔ + ΣdΔ 1-(1-ε∇)100+t
For success, the 7G resupply vessel must slow down to match the 1G vessel's cruise velocity (.644c = 111.5 AU/day).
This table describes daily progress throughout the 48¼ days of 7G deceleration.
INTERSTELLAR VESSELS: 1G PAX/CARGO & 7-G RESUPPLY (SNOWBALL)
Interstellar 1G Vessel
starts its one year of acceleration at day 0.
After this year, vessel stops acceleration and cruises at 111.5 Astronomical Units per day  (.644 c).
One year of 1G acceleration has taken vessel to 23,841 AUs,  .377 Light Year (LY).
Interstellar 7G Snowball
In this example, 7G snowball (traditional habitat encased in large volume of ice) starts its 100 day acceleration at day 260 of 1G vessel’s acceleration.
On day 360, 7G vessel stops propulsion, and cruises at 149.9 AUs per day (.865c).
100 days of 7G acceleration takes vessel to 9,842 AU, (= .155 LY).
AFTER 1G ACCELERATION, PAX VESSEL CRUISES AT CONSTANT VELOCITY
Decel.
Time(t) 		Spot
Velocity (Vt)		Spot
Distance (dt)		Daily
Dist. (dΔ)		7G
Dist.(d7G) 		Consumed
Fuel (F)
7 days146.40 AU/dy84.56% c8,775.8 AU0.1388 LY146.4 AU10,846.5 AU29.609% GW0
8 days145.86 AU/dy84.24% c8,630.0 AU0.1365 LY145.8 AU10,992.3 AU29.840% GW0
9 days145.30 AU/dy83.92% c8,484.7 AU0.1342 LY145.3 AU11,137.6 AU30.070% GW0
10 days144.73 AU/dy83.59% c8,340.0 AU0.1319 LY144.7 AU11,282.3 AU30.299% GW0
11 days144.15 AU/dy83.26% c8,195.8 AU0.1296 LY144.1 AU11,426.4 AU30.527% GW0
12 days143.56 AU/dy82.91% c8,052.3 AU0.1273 LY143.6 AU11,570.0 AU30.755% GW0
Given
(1 - (1-Δ)100-t) × c
c×(100-t)+ Vt
ln(1-Δ)
dt-1 - dt dΔ + ΣdΔ 1-(1-ε∇)100+t
CRUCIAL EVENTS FOR SUCCESSFUL RENDEZVOUS
Points ⓐ thru ⓕ map out following events:
ⓐ After 360 days, resupply vessel ends 100 days of 7G propulsion and starts cruise at 149.9 AU/day.
ⓑ After 365¼ days of 1G acceleration, primary vessel cruises at 111.5 AU/day.
ⓒ After 400 days (about 35 days of constant velocity), resupply vessel is about 12,000 AU behind primary vessel.
ⓓ After 500 days (about 135 days of constant velocity),, resupply vessel closes the gap to slightly over 8,000 AU.
ⓔ  600 days (100 more days of constant velocity), gap closes even more to just over 4,000 AU.
ⓕ  684 days, gap shrinks to about 1,000 AU in prep for 7G vessel’s deceleration duration of 48¼ days needed to match speed of primary vessel.
Describe primary vessel’s cruise distance by linear equation:
d = 111.5 t - 16,884 AU
Resupply vessel’s cruise distance:
d= 149.9 t - 44,152 AU
By inspection, we expect a pending intercept of these two cruise tracks about one LY and two years from start of primary vessel’s track. HOWEVER, the resupply vessel track must become nonlinear at about 684 days for 48¼ day deceleration maneuver.
7G DECELERATION DURING DAYS 13 THRU 18
Decel.
Time(t) 		Spot
Velocity (Vt)		Spot
Distance (dt)		Daily
Dist. (dΔ)		7G
Dist.(d7G) 		Total
Fuel (F)
13 days142.96 AU/dy82.57% c7,909.3 AU0.1251 LY143.0 AU11,712.9 AU30.982% GW0
14 days142.34 AU/dy82.21% c7,767.0 AU0.1228 LY142.3 AU11,855.3 AU31.208% GW0
15 days141.72 AU/dy81.85% c7,625.3 AU0.1206 LY141.7 AU11,997.0 AU31.433% GW0
16 days141.08 AU/dy81.48% c7,484.2 AU0.1183 LY141.1 AU12,138.1 AU31.658% GW0
17 days140.42 AU/dy81.10% c7,343.7 AU0.1161 LY140.4 AU12,278.6 AU31.881% GW0
18 days139.76 AU/dy80.72% c7,203.9 AU0.1139 LY139.8 AU12,418.3 AU32.105% GW0
Given
(1 - (1-Δ)100-t) × c
c×(100-t)+ Vt
ln(1-Δ)
dt-1 - dt dΔ + ΣdΔ 1-(1-ε∇)100+t
7G DECELERATION DURING DAYS 19 THRU 24
Decel.
Time(t) 		Spot
Velocity (Vt)		Spot
Distance (dt)		Daily
Dist. (dΔ)		7G
Dist.(d7G) 		Consumed
Fuel (F)
19 days139.08 AU/dy80.32% c7,064.8 AU0.1117 LY139.1 AU12,557.5 AU32.327% GW0
20 days138.38 AU/dy79.92% c6,926.4 AU0.1095 LY138.4 AU12,695.9 AU32.549% GW0
21 days137.67 AU/dy79.51% c6,788.7 AU0.1073 LY137.7 AU12,833.6 AU32.770% GW0
22 days136.95 AU/dy79.10% c6,651.7 AU0.1052 LY137.0 AU12,970.6 AU32.990% GW0
23 days136.21 AU/dy78.67% c6,515.4 AU0.1030 LY136.3 AU13,106.9 AU33.210% GW0
24 days135.46 AU/dy78.24% c6,379.9 AU0.1009 LY135.5 AU13,242.4 AU33.428% GW0
Given
(1 - (1-Δ)100-t) × c
c×(100-t)+ Vt
ln(1-Δ)
dt-1 - dt dΔ + ΣdΔ 1-(1-ε∇)100+t
DETERMINE INTERSECTION POINT
Assume 1G pax vessel accelerates at 1G for one year; then, cruises at constant velocity. Thus, determine distance traveled per following linear equation:
d1G = -.267 LY + .644 c × t
Assume resupply vessel starts 7G acceleration at 265.25 days after mission begins (1G vessel starts).  100 days later (365.25 days into mission), 7G vessels stops acceleration and starts cruise. Thus, determine total distance traveled via following linear equation:  
 d7G = -.711 LY + .866 c × t
To determine intersection point, set equations equal as shown:     
d1G = -.267 LY + .644c × t = -.711 LY + .866c × t  = d7G
For convenience, substitute: LY = 63,241 AU and c = 173.15 AU/day:     
-16,885 AU + 111.5 AU/day × t = -44,964 AU + 149.9 AU/day × t
thus, solve for: tInt = 731.2 days and dInt= 64,644 AUs.  
These linear equations show two cruising vessels will meet at time of 731.2 days (2 years) since day zero and distance equals 64,644 AU (about 1 LY) from 1G pax vessel's starting point. However, the two vessels cannot gracefully rendezvous because they still have a huge velocity differential of about .22 c.  Following section proposes a graceful deceleration method.
INITIATE 7G DECELERATION
After 684 days of mission time, resupply vessel flight profile used 7G acceleration and high speed cruise to reach a position 58,340 AU from Sol.
At this point, 7G vessel is only 1,076 AU behind the 1G vessel.
The 7G vessel must start the 48¼ days of  7G deceleration required to slow down from 149.9 AU/day to 111.5 AU/day, cruise velocity of the primary vessel.

For functional rendezvous, both the primary mission vessel and the resupply vessel must match velocities  at the same position at the same time.

In this example, both vessels simultaneously reach 64,796 AU (about one LY) along the track from Sol to neighboring star.

Decel.
Time(t) 		Spot
Velocity (Vt)		Spot
Distance (dt)		Daily
Dist. (dΔ)		7G
Dist.(d7G) 		Consumed
Fuel (F)
25 days134.70 AU/dy77.79% c6,245.1 AU0.0988 LY134.8 AU13,377.2 AU33.646% GW0
26 days133.91 AU/dy77.34% c6,111.1 AU0.0966 LY134.0 AU13,511.2 AU33.864% GW0
27 days133.12 AU/dy76.88% c5,977.9 AU0.0945 LY133.2 AU13,644.4 AU34.080% GW0
28 days132.30 AU/dy76.41% c5,845.5 AU0.0924 LY132.4 AU13,776.8 AU34.296% GW0
29 days131.47 AU/dy75.93% c5,713.9 AU0.0904 LY131.6 AU13,908.4 AU34.512% GW0
30 days130.62 AU/dy75.44% c5,583.1 AU0.0883 LY130.7 AU14,039.1 AU34.726% GW0
Given
(1 - (1-Δ)100-t) × c
c×(100-t)+ Vt
ln(1-Δ)
dt-1 - dt dΔ + ΣdΔ 1-(1-ε∇)100+t
Decel.
Time(t) 		Spot
Velocity (Vt)		Spot
Distance (dt)		Daily
Dist. (dΔ)		7G
Dist.(d7G) 		Total
Fuel (F)
31 days129.76 AU/dy74.94% c5,453.2 AU0.0862 LY129.9 AU14,169.0 AU34.940% GW0
32 days128.88 AU/dy74.43% c5,324.2 AU0.0842 LY129.0 AU14,298.1 AU35.153% GW0
33 days127.98 AU/dy73.91% c5,196.1 AU0.0822 LY128.1 AU14,426.2 AU35.366% GW0
34 days127.06 AU/dy73.38% c5,068.9 AU0.0802 LY127.2 AU14,553.4 AU35.577% GW0
35 days126.13 AU/dy72.84% c4,942.6 AU0.0782 LY126.3 AU14,679.7 AU35.788% GW0
36 days125.17 AU/dy72.29% c4,817.3 AU0.0762 LY125.3 AU14,805.0 AU35.999% GW0
Given
(1 - (1-Δ)100-t) × c
c×(100-t)+ Vt
ln(1-Δ)
dt-1 - dt dΔ + ΣdΔ 1-(1-ε∇)100+t
Decel.
Time(t) 		Spot
Velocity (Vt)		Spot
Distance (dt)		Daily
Dist. (dΔ)		7G
Dist.(d7G) 		Consumed
Fuel (F)
37 days124.20 AU/dy71.73% c4,692.9 AU0.0742 LY124.4 AU14,929.4 AU36.208% GW0
38 days123.20 AU/dy71.15% c4,569.5 AU0.0723 LY123.4 AU15,052.8 AU36.417% GW0
39 days122.19 AU/dy70.57% c4,447.1 AU0.0703 LY122.4 AU15,175.2 AU36.626% GW0
40 days121.15 AU/dy69.97% c4,325.7 AU0.0684 LY121.4 AU15,296.6 AU36.833% GW0
41 days120.10 AU/dy69.36% c4,205.4 AU0.0665 LY120.3 AU15,416.9 AU37.040% GW0
42 days119.02 AU/dy68.74% c4,086.1 AU0.0646 LY119.3 AU15,536.1 AU37.246% GW0
Given
(1 - (1-Δ)100-t) × c
c×(100-t)+ Vt
ln(1-Δ)
dt-1 - dt dΔ + ΣdΔ 1-(1-ε∇)100+t
Decel.
Time(t) 		Spot
Velocity (Vt)		Spot
Distance (dt)		Daily
Dist. (dΔ)		7G
Dist.(d7G) 		Total
Fuel (F)
43 days117.92 AU/dy68.11% c3,968.0 AU0.0627 LY118.2 AU15,654.3 AU37.452%GW0
44 days116.80 AU/dy67.46% c3,850.9 AU0.0609LY117.1 AU15,771.4 AU37.657%GW0
45 days115.66 AU/dy66.80%c3,735.0 AU0.0591 LY115.9 AU15,887.3 AU37.861%GW0
46 days114.49 AU/dy66.12% c3,620.2 AU0.0572 LY114.8 AU16,002.1 AU38.065%GW0
47 days113.30 AU/dy65.44% c3,506.6 AU0.0554 LY113.6 AU16,115.7 AU38.268%GW0
48 days112.09 AU/dy64.74% c3,394.2 AU0.0537 LY112.4 AU16,228.0AU38.470%GW0
Resupply Vessel Matches Cruise Velocity of Baseline "Pax" Vessel (1.0 yr of 1G Accel.)
at exact time/distance of the intercept.
48¼days111.78 AU/dy 64.56% c3,366.3 AU 0.0532 LY27.9 AU16,256.0 AU38.520%GW0
Given
(1 - (1-Δ)100-t) × c
c×(100-t)+ Vt
ln(1-Δ)
dt-1 - dt dΔ + ΣdΔ 1-(1-ε∇)100+t
Decel.
Time(t) 		Spot
Velocity (Vt)		Spot
Distance (dt)		Daily
Dist. (dΔ)		7G
Dist.(d7G) 		Consumed
Fuel (F)
49 daysxxx.xx AU/dyxx.xx% cx,xxx.x AUx.xxxx LYxxx.x AUxx,xxx.x AUxx.xxx%GW0
50 daysxxx.xx AU/dyxx.xx% cx,xxx.x AUx.xxxx LYxxx.x AUxx,xxx.x AUxx.xxx%GW0
51 daysxxx.xx AU/dyxx.xx% cx,xxx.x AUx.xxxx LYxxx.x AUxx,xxx.x AUxx.xxx%GW0
52 days106.98 AU/dy61.79% c2,966.6 AU0.2636 LY107.3 AU16,667.7 AU39.272%GW0
53 days105.64 AU/dy61.01% c2,860.5AU0.2652 LY106.0 AU16,773.7 AU39.471%GW0
Resupply Vessel Matches Cruise Velocity of Slower "Pax" Vessel (0.9 yr of 1G Accel.)
at exact time/distance of the intercept.     
53.4 days105.10 AU/dy60.70% c2,818.4 AU0.2659 LY42.0 AU16,815.7 AU39.551%GW0
54 days104.27 AU/dy60.22% c2,755.7AU0.2662 LY62.6 AU16,836.3 AU39.669%GW0
Given
(1 - (1-Δ)100-t) × c
c×(100-t)+ Vt
ln(1-Δ)
dt-1 - dt dΔ + ΣdΔ 1-(1-ε∇)100+t

posted by JimODell at 2:01 PM 0 comments