Friday, June 03, 2011


Divide the heavens into 8 octants.

Stellar System
RADecDist3-D Coord.Time
OCT-IIWolf 359164°7.78-
OCT-IIIBarnard's Star270°5.96-0.1-5.90.510.1
OCT-IVRoss 248356°44°10.327.34-
OCT-VLuyten 726-8 25°-18°8.797.53.5-2.714.4
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)
t = 1 yr + (d-.76 LY)/.644c + 1 yr


Upper octants originate at the four quadrants on the Z=0 plane and extend upward.

Quadrants and Octants. The X,Y axes of a two-dimensional Cartesian system divide the plane into four infinite regions, called quadrants. Each quadrant is bounded by two perpendicular, linear rays which originate at the Origin (0,0) where X = 0 and Y = 0.  Similarly, a three-dimensional Cartesian system uses the X, Y and Z axes to divide space into eight regions, octants. In Thought Experiment (TE) model, the CEP is at the Z=0 plane.  Celestial equatorial coordinates are based on the location of stars relative to Earth's equator projected out to an infinite distance. This view describes the sky as seen from the Solar System, and modern star maps almost exclusively use equatorial coordinates. TE assumes CEP to be centered at Sol, but fixed relative to distant stars.

Lower octants originate at the same four quadrants on the CEP plane but extend downward.
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.
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
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
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 .
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
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)
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. 
Figure shows a small sample of nearby stars in eight octants centered on Sol, our sun. The octants are arbitrarily aligned such that the X axis aligns with the Point of Aries, with Right Ascension (RA) at 0 hours or 000&amp°, and the Y axis aligns with RA at 90°.

This random sample only a few of the stars within 15 LYs.  For example, it only shows one of the suggested hubs (i.e., Groombridge 34) in below figure.
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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