Monday, October 09, 2006

HUBS

SELECT A HUB FOR EACH OF THE 8 OCTANTS.

OCTANTS:TE's way of displaying our neighboring stellar systems. Thus, far humanity has discovered about 51 such systems within 15 LYs of Sol.BEARINGS can help vessels precisely track remaining distance. This will prove essential for decelerating at exact, required distance to destination.
Consider notional (X, Y, Z) coordinates in Octant I
O = (0, 0, 0) Origin
P1 = (1, 2, 2); P2 = (2, 4, 4); P3 = (3, 6, 2)

Pythagorean Theorem
Compute 3-D distances from origin in following manner:
D = √(X2 + Y2 + Z2)
D0,1 = √(12 + 22 + 22) = √(1 + 4 + 4) = √(9) = 3
D0,2=√(22+42+42)=6 ; D0,3=√(32+62+22)=7

Furthermore,quickly determine 3-D distances between any two points by further leveraging P-Theorem on the coordinate differences.
EXAMPLE: compute 3-D distances from P1 in following manner.
Da,b = √[(Xb-Xa)2 + (Yb-Ya)2 + (Zb-Za)2]
to P2: D1,2 = √[(2-1)2 + (4-2)2 + (4-2)2] =√[1+4 +4]=3
to P3: D1,3= √[(3-1)2 + (6-2)2 + (2-2)2] =√[4+16 +0]=4.47

Determine Distances Between Adjacent Stars

Observed Astrometrics
Consider Groombridge 34 (G..34) and Teegarden's Star (T..St),
Sol's neighbors in Octant One.
StellarRADecDist.
Systemαδd
G..344.6°44°11.62 LY
T..St43°17°12.51 LY
Right Ascension (RA=α) and Declination (Dec=δ)
are readily obtained in  decimal degrees.
Observed distance are traditionally obtained for nearby stars
by carefully measuring brightness and parallax.

Compute 3-D Coordinates
Seq.StardCEPxyz
i =0Sol0  LY0  LY0  LY0  LY
i =1G..348.36 LY8.3 LY0.7 LY8.1 LY
i =2 T..St11.6 LY8.7 LY8.2 LY3.6 LY
Givend×cos(δ)dCEP×cos(α)dCEP×sin(α)sin(δ)
Use trigonometric functions 
to convert astrometrics
to three dimension coordinates.
Compute Leg Distances
Determine  two legs of voyage to interstellar destination.
EXAMPLE: Leg-1 distance from Sol to Hub, G..34;
and Leg-2 distance is from hub to destination, T-Star .
Seq.StardLegΔdXΔdYΔdZ 
i =0Sol0  LY0  LY0  LY0  LY
i =1G..3411.62 LY8.3 LY0.7 LY8.1 LY
i =2 T..St8.76 LY-0.4 LY-7.5 LY-4.5 LY
GivenSee belowXi - Xi-1Yi - Yi-1Zi - Zi-1
Use Pythogorean Theorem to obtain leg distances..
 dleg = [(ΔdX)2 + (ΔdY)2 + (ΔdZ)2]
EXAMPLE: dleg2 = [(0.4)2 + (7.5)2 + (-4.5)2] = 8.76 LY 

Total Travel Times from Sol to Hub to Stars

Consider Hub Concept Instead of direct to all stars, use a well situated star as stop over "hub" for its neighbors. Thus, interstellar voyages could transit this hub enroute to other destinations.
Likely criteria:
1) PROXIMITY:  Closeness to Sol reduces flight time.
2) WELL SITUATED:  Position among other stars is very useful.
3) WELL PROVISIONED:  In situ materials (comets and asteroids) could resupply transit vessels.
Convert interstellar distances (LYs) to time (Yrs)
Previous TE work assumes following model
for insterstellar g-force travel times (Yrs).
Choose to G-force Accelerate for 1 Year:
 tAcc = 1 year; distance = dAcc= .38 LY;
achieve cruise velocity; vCru 6443c
G-force Decelerate for same duration as acceleration:
tDec = 1 year; distance = dDec= .38 LY
Cruise at Constant Velocity with no Propulsion:
Distance:dCru = dW - dAcc - dDec = 7.8 LY-38 LY- 38 LY =7.04LY
Time:
tCru =  dCru / vCru = 7.04 LY / (.644c) = 10.93 years
Travel Time
tTtl =  tAcc +  tCru  + tDec = 1.0 yr + 10.93 yr + 1.0 yr = 12.93 yrs
Stopover Flights
EXAMPLE: Consider Wolf 359 (W359) as Hub;for spokes, try Ross 129 (R129) and Lalande 21185 (L185)
DeptLeg-1HubLeg-2DestTtl-Time
DistTimeDistTime
Sol7.8 LY12.9 YrsW359 3.8 LY6.7YrsR12819.6Yrs
Sol7.8 LY12.9 YrsW359 4.1 LY7.2YrsL18520.1Yrs




One long voyage can divide into two shorter flights; use a well placed star as a hub between Sol and subsequent destinations. Thus, a  stopover flight profile might prove more tolerable and probably much more useful than direct flights to the more distant neighbors. Stopover flights make even more sense if the "hub" has plentiful "in situ" resources (comets/asteroids) to build and provision other star ships. 
Put it all together:
From each octant, select most suitable star as a hub.

Express traditional observed coordinates:
Right ascension (α) in degrees (°)
Declination (δ) in degrees (°)
Distance.in Light Years (LY)

Transform into three dimensional (3D) Cartesian Coordinates.
X, Y and Z in LY.
Assume Common Ecliptic Plane (CEP) as base for X-Y plane.

Determine typical travel time:
given cruise time of .6443 c
with one year acceleration (accomplished during initial .37 LY from Sol)
with 1 year deceleration (accomplished during final distance of .37 LY prior to destination).
IIIIIIIVVVIVIIVIIIAll 8 Octants
OCTANTSExample
Hub
RADecDist3-D Coord.Time
αδdxyzt
OriginSoln/an/a00000
OCT-IGroombridge344.6°44°11.628.30.78.118.9
OCT-IIWolf 359164°7.78-7.42.10.912.9
OCT-IIIBarnard's Star270°5.96-0.1-5.90.510.1
OCT-IVRoss 248356°44°10.327.34-0.67.216.9
OCT-VLuyten 726-8 25°-18°8.797.53.5-2.714.4
OCT-VISirius101°-17°8.58-1.68.2-2.614.4
OCT-VIIα Centauri217°-63°4.37-1.6-1.4-3.87.6
OCT-VIIIRoss 154283°-23°9.71.9-8.7-3.815.9
OBSERVED:  α;  δ;  ddCEP  = d × cos(δ)
x = dCEP×cos(α)y = dCEP×sin(α)z = d×sin(δ)
INTERSTELLAR PROFILE TRAVEL TIME:
t = 1 yr + (d-.76 LY)/.644c + 1 yr
SUMMARY: HUBS
Hub stars are selected to leverage as much as possible the three hub criteria of 
    1) proximity to Sol, 
    2) situation to other stars, 
    3) provision of on site resources.

Hub's stellar names are placed to conveniently designate parent octants; they are not placed to show exact location within each octant.

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