The red supergiant star Betelgeuse in the constellation Orion has \(T_{\mathrm{eff}} \approx 3500 \mathrm{~K}\) and a diameter of \(0.045^{\prime \prime}\). Assuming that it is \(140 \mathrm{pc}\) from us, show that its radius \(R \approx 700 R_{\odot}\), and that its luminosity \(L \approx 10^{5} L_{\odot}\).

Red supergiant stars are one of the largest types of stars in the universe. These stars are massive, often more than 10 times the mass of the Sun, and have expanded to hundreds of times their initial size. Because of this expansion, their outer layers are cool and emit light that appears red, hence the name 'red supergiant.' Betelgeuse, in the constellation Orion, is a prime example of a red supergiant star, showcasing a massive and cool surface with an effective temperature of around 3500 K.

Calculating the radius of stars, such as Betelgeuse, involves understanding its angular diameter and converting it to the actual radius. Given an angular diameter in arcseconds, we first convert it to radians using the formula:\[ \delta_d = \frac{\text{diameter in arcseconds}}{206265} \]Next, we find the angular radius by dividing the angular diameter by 2:\[ \delta = \frac{\delta_d}{2} \]Using the small-angle formula, we can then calculate the actual radius of the star. The small-angle formula states that the radius in AU can be calculated through:\[ R = d_{pc} \times \tan(\delta) \]For small angles, the tangent can be approximated by the angle itself in radians, simplifying the process. Finally, converting the radius from AU to solar radii, given 1 AU is approximately 215 solar radii.

A star's luminosity is a measure of how much light it emits, and it is crucial for understanding a star's energy output. The formula for luminosity is:\[ L = 4\pi R^2 \sigma T_{\mathrm{eff}}^4 \]where:\* \( R \) is the radius of the star\* \( \sigma \) is the Stefan-Boltzmann constant (approximately \(5.67 \times 10^{-8} W m^{-2} K^{-4}\))\* \( T_{\mathrm{eff}} \) is the effective temperature of the star. To compare the luminosity of Betelgeuse to the Sun, we use the relative formula:\[ \frac{L}{L_{\odot}} = \left( \frac{R}{R_{\odot}} \right)^2 \left( \frac{T_{\mathrm{eff}}}{T_{\odot}} \right)^4 \]Given Betelgeuse's radius is about 700 times that of the Sun, and its effective temperature is lower, yet with this large radius, its luminosity is found to be approximately 100,000 times the Sun's.

Understanding the conversion of angular diameter is essential in stellar astrophysics. Angular diameter is the apparent size of an object as seen from a given point, here represented in arcseconds. Betelgeuse has an angular diameter of \(0.045^{\prime \prime}\). To convert this to radians, we use the fact that 1 radian equals 206265 arcseconds. Therefore,\[ \delta_d = \frac{0.045^{\prime \prime}}{206265} \text{ radians} \]This conversion allows us to later find the actual size of the star when factoring in its distance from Earth, making it a crucial step in stellar size calculations.

The small-angle formula is pivotal for calculating astronomical distances and sizes. For very small angles in radians, the formula states:\[ R = d \times \delta \]where:\* \( R \) is the actual radius of the object\* \( d \) is the distance to the object\* \( \delta \) is the angular diameter in radians. For angles as small as those found in many astrophysical observations, \( \tan(\delta) \approx \delta \) greatly simplifies calculations. This formula allows us to determine the actual radius of Betelgeuse (in AU) from its angular radius and distance. By applying this understanding, we then convert from AU to more familiar units like solar radii.