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1.16 Geometry

Practical astronomy often deals with the relation between distance (D), physical size (d), and angular size(Θ), given by Θ ∝ d / D. We can make an approximation that will allow us to turn this proportionality into an equation. Let us consider angles that are small enough that the diameter of an object at a certain distance (d) is nearly equal to the length of an arc of a circle at that distance (s). The approximation s ∝ d is valid as long as the angle Θ is not too large. In practice, the approximation is good if Θ < 10°, which is true for the angular size of almost all astronomical objects.

The ratio of the arc length s to the circumference of the circle is the same as the ratio of the angle Θ (measured in degrees) to the number of degrees in a complete circle. In equation form:

s / 2πr = Θ / 360

Multiplying both side of the equation by 2πr, we get:

s = 2πrΘ / 360

Now we can make three substitutions. We use our approximation to substitute d for s, we substitute the distance D for the radius of the circle, and we note that the numerical factor 2π/360 can be written as 57.3:

d = Θ D / 57.3

The small angle equation relates distance, physical size and angular size and it is widely used in astronomy. If we can determine any two out of d, D, and Θ, the small angle equation will allow us to calculate the third quantity. You will also see the equation written in the form where Θ is measured in units of arc seconds. Since there are 3600 arc seconds in a degree, the denominator becomes 57.3 x 3600 and the small angle equation is d = Θ D / 206,265.

Other useful geometric formulas — involving the areas and volumes of spheres, cylinders, and cubes — are given below:

• Area of a rectangle of sides a and b, A = ab

• Volume of a cube of side a, V = a3

• Volume of rectangular solid of sides a,b,c, V = abc

A circle with labeled circumference and diameter. Click here for original source URL.

• Circumference of a circle of radius r, D = 2πr

• Area of a circle of radius r, A = πr2

• Surface area of a sphere of radius r, A = 4πr2

• Volume of a sphere of radius r, V = 4/3 πr3

• Volume of a cylinder of radius r, height h, V = πr2h

Unusual shapes are rarely encountered in astronomy. Positions on the sky are best described in terms of angles marked off on a circle, and astronomical objects like planets and stars are almost always perfect spheres. This symmetry is of course caused by gravity.