A Thought Experiment: HUBS

CONTENT
8 OCTANTS
X, Y, Z COORDs
DISTANCES TWEEN STARS
TRAVEL TIMES
SELECT HUB
SUMMARY

SELECT A HUB FOR EACH
OF THE 8 OCTANTS.

Thus far, humanity has discovered about

51 stellar systems within 15 LYs of Sol.

I	II	III	IV	V	VI	VII	VIII	All 8 Octants

DETERMINE CARTESIAN (X, Y, Z) COORDINATES

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)
D_0,1 = √(1² + 2² + 2²) = √(1 + 4 + 4) = √(9) = 3
D_0,2=√(2²+4²+4²)=6 ; D_0,3=√(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.
D_a,b = √[(X_b-X_a)² + (Y_b-Y_a)² + (Z_b-Z_a)²]
to P₂: D_1,2 = √[(2-1)² + (4-2)² + (4-2)²] =√[1+4 +4]=3
to P₃: D_1,3= √[(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.

Stellar	RA	Dec	Dist.
System	α	δ	d
G..34	4.6°	44°	11.62 LY
T..St	43°	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.	Star	d_CEP	x	y	z
i =0	Sol	0 LY	0 LY	0 LY	0 LY
i =1	G..34	8.36 LY	8.3 LY	0.7 LY	8.1 LY
i =2	T..St	11.6 LY	8.7 LY	8.2 LY	3.6 LY
Given		d×cos(δ)	d_CEP×cos(α)	d_CEP×sin(α)	d×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.	Star	d_Leg	Δd_X	Δd_Y	Δd_Z
i =0	Sol	0 LY	0 LY	0 LY	0 LY
i =1	G..34	11.62 LY	8.3 LY	0.7 LY	8.1 LY
i =2	T..St	8.76 LY	-0.4 LY	-7.5 LY	-4.5 LY
Given		See below	X_i - X_i-1	Y_i - Y_i-1	Z_i - Z_i-1

Use Pythogorean Theorem to obtain leg distances..
d_leg = √[(Δd_X)² + (Δd_Y)² + (Δd_Z)²]

EXAMPLE: d_leg2 = √[(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:
t_Acc= 1 year; distance = d_Acc= .38 LY;
achieve cruise velocity; v_Cru= 6443c

G-force Decelerate for same duration as acceleration:
t_Dec = 1 year; distance = d_Dec= .38 LY

Cruise at Constant Velocity with no Propulsion:

Distance:	d_Cru = d_W - d_Acc - d_Dec = 7.8 LY-38 LY- 38 LY =7.04LY
Time:	t_Cru = d_Cru / v_Cru= 7.04 LY / (.644c) = 10.93 years

Travel Time
t_Ttl= t_Acc+ t_Cru + t_Dec= 1.0 yr + 10.93 yr + 1.0 yr = 12.93 yrs

Stopover Flights

**EXAMPLE: Consider Wolf 359 (W359) as Hub;**
Dept	Leg-1		Hub	Leg-2		Dest	Ttl-Time
Dept	Dist	Time	Hub	Dist	Time	Dest	Ttl-Time
Sol	7.8 LY	12.9 Yrs	W359	3.8 LY	6.7 Yrs	R128	19.6 Yrs
Sol	7.8 LY	12.9 Yrs	W359	4.1 LY	7.2 Yrs	L185	20.1 Yrs

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.

SELECT OCTANT'S MOST SUITABLE STAR AS HUB

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 decimal 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.

Use typical Interstellar Profile to determine travel time:

given cruise velocity of 0.6443 c,
with one year acceleration (accomplished during initial 0.37 LY from Sol),
with 1 year deceleration (accomplished during final distance of 0.37 LY prior to destination),
BEARINGS to nearby stars help determine when to transition from cruise to deceleration.

I	II	III	IV	V	VI	VII	VIII	All 8 Octants

OCTANTS

Example
Hub

dec

Dist

3-D Coord.

Time

n/a

4.6°

44°

11.62

8.3

0.7

8.1

18.9

OCT-II

Wolf 359

164°

7°

7.78

-7.4

2.1

0.9

12.9

OCT-III

Barnard's Star

270°

5°

5.96

-0.1

-5.9

0.5

10.1

OCT-IV

Ross 248

356°

44°

10.32

7.34

-0.6

7.2

16.9

OCT-V

Luyten 726-8

25°

-18°

8.79

7.5

3.5

-2.7

14.4

OCT-VI

Sirius

101°

-17°

8.58

-1.6

8.2

-2.6

14.4

OCT-VII

α Centauri

217°

-63°

4.37

-1.6

-1.4

-3.8

7.6

OCT-VIII

Ross 154

283°

-23°

9.7

1.9

-8.7

-3.8

15.9

OBSERVED: α; δ; dd_CEP = d × cos(δ)

x = d_CEP×cos(α)	y = d_CEP×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.