In astronomy, we generally give stars 3D spherical coordinates. They are the standard right ascension, declination and distance coordinates of the stars. These standard stellar coordinates are cataloged for many stars (Epoch J2000 is used here).
From the stellar spherical coordinates (α,δ,R), we compute the corresponding rectangular (X,Y,Z) coordinates from which the direct distance (D) between two stars can then be computed.
For any given star, let:
α (alpha) = Right Ascension (0 to 360 degrees) = Astronomical longitude
δ (delta) = Declination (0 to ±90 degrees) = Astronomical latitude
R = Distance in light years
Given the spherical (α,δ,R) coordinates of a star, the corresponding
rectangular (X,Y,Z) coordinates may be computed from:
Given the spherical (α,δ,R) coordinates for two stars, distinguished
by subscripts, to compute the distance between the stars, we first
compute the corresponding rectangular (X,Y,Z) coordinates:
For the first star:
For the second star:
We now take the differences between the computed (XYZ) coordinates:
Finally, the distance between the two stars
may be computed from:
NUMERICAL EXAMPLE:
In this example we will compute the distance between the bright stars Sirius (= 1st Star) and Vega (= 2nd Star) and compute how bright each star appears to be from the perspective of the other.
This is the computational data pertaining to the stars used in the problem:
J2000.0 CATALOG DATA FOR SIRIUS AND VEGA AS VIEWED FROM EARTH/SUN
-------- --------------- ---------------- ------- -------
1ST STAR RIGHT ASCENSION DECLINATION DIST LY VIS MAG
Sirius 06h 45m 08.9s -16° 42' 58" 8.6 -1.46
101.2870833 deg -16.7161111 deg
-------- --------------- ---------------- ------- -------
2ND STAR RIGHT ASCENSION DECLINATION DIST LY VIS MAG
Vega 18h 36m 56.3s +38° 47' 01" 25.3 +0.03
279.2345833 deg +38.7836111 deg
Performing the distance computation using the given data:(Angles expressed in degrees)
DISTANCE BETWEEN THE STARS SIRIUS AND VEGA Rectangular Coordinates of Sirius: x1 = 8.6 * Cos(101.2870833) * Cos(-16.7161111) = -1.6121048578 y1 = 8.6 * Sin(101.2870833) * Cos(-16.7161111) = 8.0772729746 z1 = 8.6 * Sin(-16.7161111) = -2.4736166278 Rectangular Coordinates of Vega: x2 = 25.3 * Cos(279.2345833) * Cos(38.7836111) = 3.1648925282 y2 = 25.3 * Sin(279.2345833) * Cos(38.7836111) = -19.4661817544 z2 = 25.3 * Sin(+38.7836111) = 15.8474358553 Rectangular Coordinate Differences: dx = (x2 - x1) = 4.7769973860 dy = (y2 - y1) = -27.5434547290 zx = (z2 - z1) = 18.3210524831 The distance (D) between the stars Sirius and Vega may then be found from: D = SqRoot(dx*dx + dy*dy + dz*dz) = 33.4233835290 LY So, D = 33.42 light years
APPARENT BRIGHTNESS
Once we know the distance (D) between two stars, we may then determine how bright each star would appear to be (ideally) as viewed from the other's perspective, as opposed to their apparent brightness (or magnitudes) as viewed from Earth.
In this case, the general stellar magnitude vs. distance
formula may be expressed as:
Where:
m = Visual magnitude of star as viewed from distance d
M = Visual magnitude of star as viewed from distance D
d and m respectively refer to the distance to and visual magnitude of one of the stars as viewed from Earth.
D is the distance between the two stars previously computed above.
M is the visual magnitude of one of the stars at distance D.
The known values are (d, m, D) from which we compute M for each star.
NUMERICAL EXAMPLE:
For Sirius:
m = +0.03 = Magnitude of Vega as viewed from Earth d = 25.3 = Distance to Vega from Earth in light years D = 33.4 = Distance between Sirius and Vega in light years M = Magnitude of Vega as viewed from Sirius M = 5*Log10(33.4/25.3) + 0.03 = 0.603129728185 + 0.03 = 0.633129728185 From Sirius, the magnitude of Vega is +0.63 For Vega: m = -1.46 = Magnitude of Sirius as viewed from Earth d = 8.6 = Distance to Sirius from Earth in light years D = 33.4 = Distance between Sirius and Vega in light years M = Magnitude of Sirius as viewed from Vega M = 5*Log10(33.4/8.6) + (-1.46) = 2.94624007784 - 1.46 = 1.48624007784 From Vega, the magnitude of Sirius is +1.49
-