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Scale and Distance
For two points P and Q on the sky at the same distance
from Earth, the distance S between these two points can
be calculated if the distance D from Earth and the angular
distance on the sky A are
known. For angular distances less than 10 degrees, the
following formula is good to within one percent:
S = .0175 D x
A(degrees) (1)
IMPORTANT NOTES ON THE LIMITATIONS
OF THIS FORMULA: This formula applies if the two
points are at approximately the same distance from
Earth, for example points on the two edges of the moon,
or a star cluster, or a galaxy, or two stars at the
same distance. The stars in the constellations are in
general NOT at the same distance.
For distances greater than about 500 million light
years, the effects of the expanding universe must be
taken into account, so the simple formula (1)
does not apply. See
http://www.astro.ucla.edu/~wright/cosmo_02.htm for
a detailed discussion of these effects.
Example
1. The moon is about 0.51 degree across,
and is about 384,000 kilometers from Earth. Using
Equation (1), we calculate that the
diameter of the moon is approximately:
S =( 0.0175) x (384,000) x
(0.51) = 3430 kilometers.
For more distant sources the angular distance is
usually measured in arc minutes or arc seconds. One
degree contains 60 arc minutes and 3,600 arc seconds,
so Equation (1) is modified as
follows:
For angular distance in arc
minutes,
S = .00029 D x
A(minutes) (2)
For angular distance in arc
seconds,
S = .0000048 D x A(arc
seconds) (3)
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Enter the distance in kilometers or light
years and the angular size in degrees to
compute the linear scale in kilometers or
light years. This calculator is good for
angles less than 10 degrees, and distances
less than 500 million light years.
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Example
2. If two stars in a star cluster 800 light years
away are 1 arc minute apart, then using Equation (2) their
separation in light years is calculated to be:
S = (.00029) x (800) x 1 = .23 light
years
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Enter the distance in kilometers or light
years and the angular size in arc minutes
to compute the linear scale in kilometers
or light years. This calculator is good for
angles less than 10 degrees, and distances
less than 500 million light years.
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Example
3. If a bright star cluster in a galaxy 100
million light years away is 20 arc seconds away from the center
of that galaxy, then according to Equation (3), the distance of the star cluster
from the center of the galaxy is:
S = (.0000048) (100,000,000)x (20) =
9,600 light years.
Note that in the above formulas, S and D
must always be in the same units. That is, if S is in
kilometers, D is also in kilometers, etc.
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Enter the distance in kilometers or light
years and the angular size in arc seconds
to compute the linear scale in kilometers
or light years. This calculator is good for
angles less than 10 degrees, and distances
less than 500 million light years.
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In case you wonder where the numbers in the conversion formulas
come from:
The natural unit for measuring angles is the radian, which is
derived from the formula for the circumference C of a circle: C = 2πR, where R is the radius. If R = 1, then C = 2π, so the radian is defined such that a circle has 2π
radians. It also has 360 degrees, so:
1 radian = 360/2π = 360/6.28 = 57.3 degrees
so
1 degree = 0.0175 radian
1 arcmin = 0.00029 radian
1 arcsec = 0.0000048 radian
See also: Scales and Angular Measurement
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