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Technically Speaking: Angular Diameter and Linear Diameter
When an angle is small, the equations of trigonometry
simplify and we obtain small angle formulas. A very
important small angle formula in astronomy relates the angular diameter of an object
on the celestial sphere to its distance from us
and its linear diameter. It can be expressed
as (see the adjacent figure)
a = 206,265 (d / r )
where a is the angular diameter in seconds of arc,
d is the linear diameter, and r is the distance to the object. (The
equation in this form gives the angle in arc seconds because the factor 206,235 is the
conversion from radians to arc seconds.)
For example, the Sun has a linear diameter of about 1.4 million km and is on
average about 150,000,000 kilometers from us. Its angular diameter is thus
a = 206,265 x (1,400,000 km / 150,000,000 km) = 1925" of arc
which is a little over 1/2 degree.
Alternatively,
the above equation can be solved for any one of the three variables given the value of
the other two. For example,
the Moon
also has an angular diameter of about 1/2 degree
but is at a distance of only 384,000 km. We can solve the above equation for d to
conclude that the Moon must be about 3500 km in diameter.
That we can observe such spectacular
total solar eclipses from Earth thus results from a remarkable
coincidence: even though the Sun and Moon are at very different distances and have
very different linear diameters,
they subtend almost exactly the same angular diameter on the celestial sphere.
This Java
calculator can be used to solve the above equation automatically.
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