Have you got your workbook with you Have you got your workbook with you? Learning Astronomy by Doing astronomy Activity 1 – Mathematical and Scientific Methods step 2 – Small Angle Formula Let’s get started!

For angles smaller than 0 For angles smaller than 0.5 radian (about 30°), the sine of the angle, the tangent of the angle, and the angle are all approximately equal to one another. That is, sin 𝚯 = tan 𝚯 ≈ 𝚯, when the angle is expressed in radians. 𝚯 is the Greek letter theta, and is just another way to express a variable to which we can assign a value. What is a “radian”? A radian is another way to express the size of an angle. It is generally more convenient to use than a “degree”; but, we need to remember to have our calculators in the “rad” mode and NOT the “deg” mode.

“Wait. ” you say. “I don’t know any trigonometry “Wait!” you say. “I don’t know any trigonometry!” Good news: When we use the small angle formula, we don’t need to know any trigonometry at all. We just need to know how to manipulate a simple equation to solve for one of the variables. In this activity workbook, we use the small angle formula almost exclusively when trying to figure out either the actual size of an object (if we know its distance) or get a good estimate of its distance (if we know its size).

Definitions of angular distance and angular size The angular size of the Moon is ½ degree, or 0.5 degree, or: 0.5 deg ÷57.3 𝑑𝑒𝑔 𝑟𝑎𝑑 =0.009 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 https://dept.astro.lsa.umich.edu/ugactivities/Labs/coords/angdistsize.jpg

We can almost always, in astronomy for objects extremely far away, measure its “angular size” 𝜃. 𝜃= 𝑠 𝑑 s 𝜃 d If we know 𝜃and the actual size of an object, and want the distance, we use: 𝑑= 𝑠 𝜃 . If we know 𝜃 and have an estimate of the distance, we can get the actual size. We use: 𝑠=𝑑 × 𝜃.

𝚯 = s/d. This equation is known as the small-angle formula, where is the angular size in radians, s is the actual (linear) size of the object being measured, and d is the distance to the object. The units of s and d must match. For example, if s is in meters, d must also be in meters. If s is in light-years, then d must also be in light-years. If we know the distance to an object and can measure its angular size, we can calculate its actual size. If we know the actual size and measure the object’s angular diameter, we can determine its distance. This is a very important formula in astronomy.

Actual Diameter (s - kilometers) Distance (d - kilometers) Your turn: Angular Diameter (𝚯 - radians) Actual Diameter (s - kilometers) Distance (d - kilometers) 0.01 3,480   0.009 1.5 × 108 3,474 3.84 × 105 𝑑=𝑠/𝜃 𝑠=𝑑×𝜃 𝜃= 𝑠 𝑑 5. Use the small-angle formula to fill in the blank places in the preceding table.