**Sidebar: How can we measure the size of a star, its radius?**

**Step 1: estimate the distance to the star** (“estimate” = “measure” – all measurements are formally estimates, with a finite precision and finite accuracy). There’s detail at http://hypertextbook.com/facts/KathrynTam.shtml. Basically, it’s a direct geometric measurement that’s been done for many nearby stars, as well as to planets in our own solar system.

The lines from the Earth toward the star and continuing beyond are lines of sight of a telescope on Earth. The star (or other celestial body – a planet, for example) is sighted from Earth against the background of far more distant objects in the sky. This is done while the Earth is near one extreme of its orbit, in the direction perpendicular to the direction to the star. (It’s not practical to view the star at a grazing angle through the atmosphere, so that the diagram is an exaggeration; one views the star at higher elevations.) Then, much later, the star is sighted again. It appears to have moved relative to the distant objects. First it appeared near the distant object *A*, at least in projection, and later, near distant object *B*. The angular distances between *A* and *B* are readily measured; they are the same as the angle *θ *(theta) in the sketch. By geometry, *d* equals the orbital shift, *s*, multiplied by the sine of this angle. Since the angle is extremely small, this sin(*θ)* equals *θ * itself, when *θ*is expressed in radians (one radian is an angle subtended by an arc that is the same length as the radius of the circle; there are then 2 pi radians in a circle, with one radian being 360 degrees divided by 2 pi, or about 57.29 degrees). For Proxima Centauri, the angle is tiny but it is measurable with excellent telescopes, though not amateur telescopes. Working backwards (not having measured the angle myself), I find that it is, again in radians, (2 Earth orbital radii)/(4.25 light-years). Using 1 Earth radius as 149 million km and figuring 1 light-year as

We can then divide, to get

An arc-second is 1/60^{th} of an arc-minute. An arc-minute is 1/60^{th} of a degree. Small as this angle is, it is measurable. A great telescope can measure angles down to about 0.1 arc-second on Earth. An array of telescopes spread out over great distance can compare signals that show the same pattern of waves (a technique called interferometry) and can measure 0.001 arc-second…in a long dedicated view of one area of the sky.

**Step 2: more geometry: **To get the actual physical size (diameter or radius) of the star, we can again use geometry. For nearby stars that are large enough, one can actually see both edges of the star, which form an even smaller angle between them. For Proxima Centauri, the relevant distance is now the diameter of the star rather than the distance across the Earth’s orbit. Since this diameter is about 100,000 km, the angle is 0.005 arc-seconds. Measuring this requires a telescope array. Proxima Centauri’s size was measured in 2002 by two astronomers, Pierre Kervella and Frederic Thenevin.

**Alternative: **The most common way of measuring the angular size of a star takes advantage of the wave nature of light. Light is comprised of oscillating electric and magnetic fields. Over time and over distance, the fields vary from positive to negative in a (generally) simple pattern, a sine wave. Then waves from different sources, such as different areas on the same star, meet at a point, they add their field strengths. If one wave is going positive and the other is also going positive, the added fields are stronger. If they are out of phase, one going positive and the other negative, the sum of the two fields is less than either one. In time and in space, then, there are patterns of positive and negative reinforcement, or interference. The theory is explained, with some advanced math, in several places – a book on optics (Introduction to Modern Optics, GR Fowles, Dover, 1975), a website maintained by the Astronomy Café, and another website from the University of Denver, among others. The first use of this optical theory was to measure the diameter of the giant red star, Betelgeuse, with a complicated optical setup on the telescope on Mount Wilson, near my alma mater, Caltech. Betelgeuse was found to have a diameter 280 times as large as our Sun; it would extend beyond the orbit of Mars, were it our central star! Simpler setups now work effectively, using optical theory developed by Robert Hanbury-Brown and Richard Twiss.