MEASURING DISTANCE TO THE STARS How do astronomers measure the distance to the stars? Measuring Tape? Radar? Obviously, you cannot use a tape measure Bouncing radar off the surfaces of stars would not work because: (1) stars are glowing balls of hot gas and have no solid surface to reflect the radar beam back (2) the radar signal would take years to just reach the nearest stars. MEASURING DISTANCE TO THE STARS

A favorite way to measure great distances is a technique used for thousands of years: look at something from two different vantage points and determine its distance using trigonometry. The object appears to shift positions compared to the far off background when you look at it from two different vantage points. The angular shift, called the parallax, is one angle of a triangle and the distance between the two vantage points is one side of the triangle. Basic trigonometric relations between the lengths of the sides of a triangle and its angles are used to calculate the lengths of all of the sides of the triangle. PARALLAX

The size of the parallax angle p is proportional to the size of the baseline. If the parallax angle is too small to measure because the object is so far away, then the surveyors have to increase their distance from each other Ordinarily, you would use tangent or sine, but if the angle is small enough, you find a very simple relation between the parallax angle p, baseline B, and the distance d: p = (206,265 × B)/d PARALLAX CONTINUED

EARTH’S ORBIT AS BASELINE Trigonometric parallax is used to measure the distances of the nearby stars. The stars are so far away that observing a star from opposite sides of the Earth would produce a parallax angle much, much too small to detect. As large a baseline as possible must be used. The largest one that can be easily used is the orbit of the Earth. In this case the baseline = the distance between the Earth and the Sun---an astronomical unit (AU) or 149.6 million kilometers! A picture of a nearby star is taken against the background of stars from opposite sides of the Earth's orbit (six months apart). The parallax angle p is one-half of the total angular shift. EARTH’S ORBIT AS BASELINE

1 parsec = 206,265 astronomical units = 3.26 light years The distances to the stars in astronomical units are huge, so a more convenient unit of distance called a parsec is used (abbreviated with ``pc''). A parsec is the distance of a star that has a parallax of one arc second using a baseline of 1 astronomical unit. 1 parsec = 206,265 astronomical units = 3.26 light years FYI: The nearest star is about 1.3 parsecs from the solar system. Using a parsec for the distance unit and an arc second for the angle, our simple angle formula above becomes extremely simple for measurements from Earth: p = 1/d THE PARSEC

The angles involved are very small, typically less than 1 second or arc! (Remember that 1 arc_second = 1/3600 of a degree). To determine the distance to a star we can approximate the equation given in the previous section with the small angle approximation: d = r / p where d is the distance to the star, p is the parallax angle expressed in radians (see diagram), and r is the baseline, in this case 1 Astronomical Unit (A.U.) -- the radius of the Earth's orbit. Since there are 206,265 arc-seconds per radian, the formula can be re-written as: d (in AU) = 206,265 / p with p measured in arc-seconds . Or If we define the distance of one parsec as 206,265 AU, we get: d (in parsecs) = 1 / p This is the distance unit astronomers use most frequently, and it is equivalent to 3.26 light-years. ONE MORE TIME

PARALLAX ONLY GOOD FOR NEARBY STARS Parallax angles as small as 1/50 arc second can be measured from the surface of the Earth. This means distances from the ground can be determined for stars that are up to 50 parsecs away. If a star is further away than that, its parallax angle p is too small to measure and you have to use more indirect methods to determine its distance. Stars are about a parsec apart from each other on average, so the method of trigonometric parallax works for just a few thousand nearby stars. PARALLAX ONLY GOOD FOR NEARBY STARS

DISTANCE-INVERSE SQUARE LAW When the direct method of trigonometric parallax does not work for a star because it is too far away, an indirect method called the Inverse Square Law of Light Brightness is used. This method uses the fact that a given star will grow dimmer in a predictable way as the distance between you and the star increases. If you know how much energy the star emits, then you can derive how far away it must be to appear as dim as it does. A star's apparent brightness (its flux) decreases with the square of the distance. The flux is the amount of energy reaching each square centimeter of a detector every second. Energy from any light source radiates out in a radial direction so concentric spheres (centered on the light source) have the same amount of energy pass through them every second. As light moves outward it spreads out to pass through each square centimeter of those spheres. DISTANCE-INVERSE SQUARE LAW

The same total amount of energy must pass through each sphere surface. Since a sphere has a surface area of 4𝜋 𝑟 2 : the flux on sphere-1 = (the flux on sphere #2) × [(sphere #2's radius)/(sphere #1's radius)]2.