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Have a look at Fig. 6:
If b is small compared to c, we can assume that
the two longer sides of the triangle, c, have the
same length as the centre line.
With the usual equations for a right-angled triangle
we find:
sin(β/2) = (b/2)/c
We can use the small-angle approximation sin x
= x, if we are dealing with very small angles
(but only when the angle is measured in
radians). This approximation may seem less justified,
but it can be mathematically proven to
be very good for small angles.
Try this approximation yourself by calculating sin(1°), sin(1’), sin (1’’). Note that you have first to convert the angles from degrees to radians.
Now you have a simple relationship between b, c, and β without the trigonometric function: β/2 = (b/2)/c c = b/β
1 arcminute = 1' = 1/60 of a degree = 2.9089 × 10-4
1 arcsecond = 1'' = 1/3600 of a degree = 4.848 × 10–6 radians
1 milliarcsecond (mas) = 1/1000 arcsecond
Speed of light (c) = 2.997 × 108 m/s
1 parsec (pc) = 3.086 × 1013 km = 3.26 light-years
1 kiloparsec (kpc) = 1000 parsec
1 Megaparsec (Mpc) = 106 parsec
1 nanometer (nm) = 10–9 m