The course directional values are either:
Note: Chapter 6 helps define many, but not all, possible SGC navigation problems. Further examples could be made into a SGC navigational reference. I will now only attempt to lay the groundwork in understanding and resolving SGC navigational problems.
For interstellar flights of say, less than 20 light years, it would seem that a direct sighting of the destination star and some backup triangulation would be sufficient. However, as interstellar trips become longer and more complex, it will be useful to be able to factor out such inherent errors in the Apparent Stellar Distance, ASD, of the triangulation stars as viewed from Earth compared to their positions as viewed from a starship in mid-mission.
Since we are dealing with a multitude of confusing points and
apparent distances, it would be wise at this time, to go over some Point
Abbreviations comparing points and distances as seen from 0,0,0 versus
as seen from a mid-mission point M:
Point Abbreviations:Real and apparent SGC point positions and distances defined
(Star or Object) ' (XYZT or type of measurment as in RSD, ASD, etc.) (type of coordinate in lowercase) ' (viewpoint in lowercase) ' (CET)
|Points and positions as viewed from point 0,0,0||Points and positions as viewed from point M|
|Special rule for 0,0,0:
||Special rules for view from point M or from any other point:
|Apparent Stellar Distances (ASD)|
|The distance from 0,0,0 to the ASP of star S will be referred to as ASD or S'ASD.||The distance from point M to the ASP of star S
will be referred to as S'ASD'm.
(The differential X, Y, Z and T values between point M and star S are transient values and are used only to find the ASD, the RSD or course headings. To avoid additional confusion, these values will be kept as variables; S'X1'm or simply X1, as needed.)
|Apparent Stellar Positions (ASP)|
|S'X, S'Y, S'Z and S'T are the ASP of star S as viewed from 0,0,0. (If no other conflicting coordinates are referenced, the S' may be dropped, but only when the coordinates are viewed from 0,0,0).||S'X'm, S'Y'm, S'Z'm and S'T'm are the ASP of star S, as viewed from point M.|
|The galactic Right Ascension to the ASP of star S, as viewed from 0,0,0, is Gr.||The galactic Right Ascension to the ASP of star S, as viewed from point M, is S'Gr'm.|
|The galactic declination to the ASP of a star the as viewed from 0,0,0 is Gd.||The galactic declination to the ASP of star S, as viewed from point M, is S'Gd'm.|
|Real Stellar Distances (RSD)|
|The distance from 0,0,0 to the RSP of star S will be referred to as RSD or S'RSD.||The distance from point M to the RSP of star S will be referred to as S'RSD'm. (See above note in the ASD section regarding differential values.)|
|Real Stellar Positions (RSP)|
|S'Xr, S'Yr, S'Zr and S'Tr are the real coordinates of star S as viewed from 0,0,0 or from point M. Real coordinates do not change depending on viewpoint.|
|0,0,0,T are the real current coordinates of 0,0,0 and the Current Earth Time (CET) is the T coordinate.||X'm, Y'm, Z'm and T'm are the real coordinates of point M.|
|The real galactic Right Ascension to the RSP of star S, as viewed from 0,0,0, is Grr, or S'Grr.||The real galactic Right Ascension to the RSP of star S, as viewed from point M is S'Grr'm.|
|The real galactic declination to the RSP of a star, S, as viewed from 0,0,0, is Grd, or S'Grd.||The real galactic declination to the RSP of star S, as viewed from point M, is S'Grd'm.|
Let us return to the example of our navigator, Celeste,
and her mission to Sirius. Before she embarks on the Sirius mission she
could easily resolve some navigational problems of distance. With her complete
chart and list of Annual Shifts, she would know the real coordinates of
Sirius as well as the apparent coordinates. Knowing, also, that her average
planned velocity for the mission is .36c, it becomes a simple matter
to find the real distance to Sirius using the good ole pythagorithian right
angle hypotenuse rule in 3D.
Where Xr, Yr, Zr are the real coordinates for Sirius, S'RSD'm is the Real Stellar Distance from 0,0,0.
Using this RSD distance, Celeste may also easily
find the time t in Earth years, it would take to travel this
far based on her average velocity of travel, AVT, of .36c
(.36 the speed of light).
Unfortunately, it is not as easy as that to resolve a distance required to travel to a moving star, because one must consider that the star is moving towards or away from 0,0,0. Simply knowing how far it is at any given time, will not tell you how far the ship will need to travel to meet the star. The point of rendezvous of a starship with the star will be called NPI the Nearest Point of Intersection. (This issue will be resolved later in an example in 6.5 below.)
Let's look closer at a particular mission in order
to understand the navigational problems regarding distance. Suppose Celeste
is now almost half way into the mission, 4 light years from Earth but still
5 light years from Sirius. How can one find the remaining real distance
to Sirius from a mid-mission point? It is easily done using SGC
with only a one line formula!
Where Xr, Yr and Zr are the real coordinates of
Sirius, a known value, and
X'm, Y'm and Z'm are the
ship's X, Y and Z coordinates, a measured value. RSD
is then the real distance from the ship at mid-mission point M to
Note: This formula turns out to particularly important if Celeste has strayed from a direct course. This would result in a situation where she may be 4 light years from Earth but more than 5 light years from Sirius. In that case, formula 6.1.4 would set the record straight and allow for accurate calculations to be made.
However, if Celeste wishes to do a visual sighting for Sirius, she must work with the ASD, Apparent Stellar Distance to Sirius and not it's RSD, Real Stellar Distance. (She would obviously also need to know the ASP Apparent Stellar Position as seen from her ship, i.e. Sirius's apparent galactic declination and galactic Right Ascension. This question is covered in 6.2. Section 6.1 is only about distances, let's stay for now, with only the distance question.)
The light arriving from Sirius is as old as how many light years Celeste is away from Sirius. When we check the above table 6.1.1, we find that this distance is referred to as S'ASD'm. So, for Celeste to find how far away Sirius appears to be, she must know this value.
Where the current Earth time is Feb. 10, 2011. The mission route is shown as a solid black line from E0 to S3. The starship is currently at point M. RSD values, the real distances, are shown as dotted lines; the RSD0 from 0,0,0 at E0 to Sirius at S0 at the beginning of the mission is 9 light years. The RSD3 value from E3 to S3 at the end of the mission is 9 light years plus or minus a distance caused by Sirius's Radial Motion. (In reality there would only be a minus 3 minutes 30 light seconds over the 25 year mission.) The ASD, light seen from point M, are shown as solid dark blue lines. At point M, light arriving from the 0,0,0 is 4 years old and from Sirius is 5 years old. All T coordinates are given for each point as dates. (It is a coincidence that the T coordinate for point M and S1 are the same. The 0,0,0, coordinates, our Sun, actually does move slowly towards X +. So, showing the various E coordinates along this line is correct.)
Celeste started the mission on January 1, 2000 at point E0 and travelling at an AVT (Average Velocity of Travel) of .36c she would reach 4 light years from Earth at point M, on February 10, 2011 CET (Current Earth Time) at point M. When Celeste makes some observations at point M she finds that some odd things have happened.
If Celeste wishes to find the correct distance for S'ASD'm, point M to S1, she could do the following step by step process:
Where XAS, YAS and ZAS are the known SGC Annual Shifts of Sirius and S'ASD'm is the Apparent Stellar Distance of Sirius as viewed from the ship at point M.
In fact, even if Celeste had done a little sight seeing and found that she was now 4 light years from Earth but still, say, 7 light years from Sirius, formulae 6.1.4 and 6.1.6 would still work fine because the Real Stellar Distance, RSD, would be greater and this would automatically adjust the solution in formulae 6.1.6.
Note: There is no conflict between the SGC coordinates that Celeste sees of Sirius and the coordinates that her stay at home sister, Candice, sees from Earth as long as Celeste remembers the SGC rule regarding the T coordinate. To arrive at her own mid-mission SGC T coordinate, she must add the distance back to the Earth (at 0,0,0) .
Once this concept is understood, even more complicated distance problems
are made easy. It is then possible, for example,
to find the Real Stellar Distance between any two stars in the SGC
coordinate system. This is just a variation of formula
Where RSD2 represents the distance between a star, P with real coordinates P'Xr, P'Yr, P'Zr and another star, Q, with real coordinates Q'Xr, Q'Yr and Q'Zr.
Going one step further, it is even possible to find the Apparent Stellar Distance as viewed from Celeste's ship at a mid-mission point to a star other than Sirius. The value we seek shall be called ASD4.
Use a variation of formula 6.1.4
to find the Real Stellar Distance between the starship at mid-mission point
and star W. The Real Stellar Distance between these points shall
be called RSD4. Once the RSD4 is found, the coordinates of
star W's Apparent Stellar Position relative to the starship at point
Where X'm, Y'm and Z'm are the SGC coordinates of the ship at a point M, RSD4 is the real distance between the starship at point M and star W. W'Xr, W'Yr and W'Zr are star W's real coordinates (given by the SGC star list) while W'X'm, W'Y'm and W'Z'm are the apparent coordinates of star W as viewed from point M.
Using these new apparent positions, W'X'm, W'Y'm and W'Z'm, it is possible to run the last part of formulae 6.1.6 again and get our end goal value for ASD4, the apparent stellar distance from star W to the starship at point M.
In summary, it can be seen that such a comprehensive SGC chart and list of Annual Shifts can be a great aid to finding interstellar distances, both real and apparent.
SGC values transferred to Galactic Right Ascension, Gr
|For Coordinates||To find Gr|
|X+, Y-||X/T = cos b, 90° - b = Gr
| Y/T | = cos a = Gr
|X+, Y+||X/T = cos b, 90° + b = Gr
Y/T = cos a, 180° - a = Gr
|X-, Y+||| X/T | = cos b, 270° - b =
Y/T = cos a, 180° + a = Gr
|X-, Y-||| X/T | = cos b, 270° + b =
| Y/T | = cos a, 360° - a = Gr
|X=0, Y=0||Gr = 0°|
|X=0, Y-||Gr = 90°|
|X=-, Y=0||Gr = 180°|
|X=0, Y+||Gr = 270°|
Translating Gr to the new SGC value, galactic heading, Gh.
Gh = [0 to <4]
Gh = Gr ÷ 90
SGC values transferred to Galactic declination, Gd
|For Coordinates||To find Gd|
|Z+||Z/T = sin y, y = Gd|
|Z = 0||Gd = 0°|
|Z-||| Z/T | = sin y, Gd = 0° - y|
|Z = T||Gd = 90°|
|Z = - T||Gd = -90°|
Translating Gd to the new SGC value galatic altitiude, Ga.
Ga = [1 to -1]
Ga = Gd ÷ 90
Return to Table of Contents | The Appendix
Last updated: October 3, 2002 |