Small-Angle Formula

angularvslinearfinal-1

In astronomy, the sizes of objects in the sky are often given in terms of their angular diameter as seen from Earth, rather than their actual sizes. аFor a given observer, the distances D, d, and angleаθ in radians (as portrayed in the picture above) form a right triangle with the trigonometric relationship:

tandD

Since these angular diameters are often small, we can use the small angle approximation which will give us:

tanthetatheta

Sinceаθ is in units of radians, and we often need to convert our measurements into units of degrees. аOne radian is 180/π ≈ 57.3 degrees. аSo we can rewrite our small angle approximation as:

smallangledegrees



When dealing with astronomically distant objects, where angle sizes are extremely small, it is often more practical to present our angles in terms of arcseconds, which is 1/3600th of one degree. аSince one radian equals 3600⋄180/π ≈ 206265 arcseconds, we can again rewrite the small angle formula as:

angularsize

Since it is easy to measure the angular size of astronomical objects, we often use this to solve for other unknowns, such as the distance or the diameter of a celestial body. аIf two objects are roughly the same distance from the observer, you can also use the formula to find the distance between the two objects.


Example:

On August 27, 2003, Mars made the closest approach to Earth in recorded history due to a near synchronization of Earth being at aphelion (furthest orbital point from the sun) and Mars being at perihelion (closest orbital point from the sun). The distance between the planets that day was a mere 55.8 million km. аObservations of Mars were quite easy to do that night, since the angular diameter of the planet was observed to be about 25.1 arcseconds. аGiven this information, find the diameter of the red planet.


Solution:

θ = 25.1"

D = 55.8 million km = 55.8 x 106 km = 5.58 x 1010аm

angularsizeexample

So the planet's diameter is about 6790 km, which is a little more than half the diameter of Earth.

ай 2014 University of Iowa, Dawg