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Chapter 20: Problem 36

The central star in a newly formed planetary nebula has a luminosity of \(1000 \mathrm{~L}_{\odot}\) and a surface temperature of \(100,000 \mathrm{~K}\). What is the star's radius? Give your answer as a multiple of the Sun's radius.

Short Answer

Expert verified
The radius of the central star in the newly formed planetary nebula expressed as a multiple of the Sun's radius.

Step by step solution

01

Understand the luminosity formula

Recall the formula for stellar luminosity: \( L = 4\pi R^2 \sigma T^4 \) where \( L \) is the luminosity, \( R \) is the star's radius, \( \sigma \) is the Stefan-Boltzmann constant (\(5.67×10^{-8} \: W\: m^{-2} K^{-4}\)), and \( T \) is the star's surface temperature.
02

Rearrange the formula to solve for R

Because we need to determine the radius, we must rearrange the formula before proceeding: \( R = \sqrt{L / (4\pi\sigma T^4)} \) .
03

Substitute the given values into the formula

Substitute the given values into the rearranged formula: \( R = \sqrt{1000 L_{\odot} / (4\pi\sigma (100,000\,K)^4)} \). Because \(L_{\odot}\) is the luminosity of the sun, remember to convert it into the basic unit of watts (the SI unit) where \( L_{\odot} = 3.828×10^{26} \,W \) before making the substitution.
04

Calculate the radius in meters

Evaluate the expression inside the square root first, then take the square root to get the radius in meters.
05

Convert the radius to multiples of the Sun's radius

With the radius in meters, we now convert to multiples of the Sun's radius (\( R_{\odot}\)) to get the final answer. Using the conversion factor \( R_{\odot} = 696340 \,km \), or equivalently \( R_{\odot} = 696340000 \,m \), divide the obtained radius in meters from the earlier step by the Sun's radius.

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Most popular questions from this chapter

. Use a telescope to observe the remarkable triple star 40 Eridani, whose coordinates are R.A. \(=4^{\mathrm{h}} 15.3^{\mathrm{m}}\) and Decl. \(=-7^{\circ} 39^{\prime}\). The primary, a 4.4-magnitude yellowish star like the Sun, has a 9.6-magnitude white dwarf companion, the most easily seen white dwarf in the sky. On a clear, dark night with a moderately large telescope, you should also see that the white dwarf has an 11 th-magnitude companion, which completes this most interesting trio.

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