How massive would a main-sequence star need to be, to be 100 times brighter than the Sun?

Name: Problem 6, Chapter 58: Pathways to Astronomy
Uploaded: 2020-11-09T16:57:57-08:00
Duration: 5 min 18 s
Channel: Matthew Miranda
Description: How massive would a main-sequence star need to be, to be 100 times brighter than the Sun?

How massive would a main-sequence star need to be, to be 100...

Key Concepts

Main-Sequence Stars

Main-sequence stars are those stars that are in a phase of stable hydrogen burning in their cores. Their properties, including luminosity, temperature, and lifetime, are closely linked to their masses. The mass of a main-sequence star not only determines where it is located on the Hertzsprung-Russell diagram but also how quickly it burns through its nuclear fuel, making the understanding of the mass-luminosity relationship particularly important in studying these stars.

Stellar Mass–Luminosity Relationship

The stellar mass–luminosity relationship is a fundamental concept in astrophysics that describes how a star's brightness (luminosity) increases steeply with its mass. For main-sequence stars, this relationship is typically expressed by a power law (L ? M^?), with the exponent ? usually ranging between 3 and 4 for stars similar to the Sun. This concept is essential to understanding how much more massive a star must be to achieve a luminosity significantly greater than that of the Sun.

Transcript

00:01 Question number 18 wants us to categorize a g2 star with 100 times the luminosity of the sun.

00:08 Categorizing a star based on its spectral class in luminosity requires the use of the hr diagram.

00:16 As you can see on the version of the diagram here, we have four different luminosity classes of stars categorized on it.

00:25 All we need to do is find a point on the diagram that intersects with, the spectral class g and a luminosity equal to 100 times that of the sun.

00:35 In doing so, we find that stars in the spectral class g with a luminosity equal to 10 to the 2 times, or 100 times that of the sun, are mainly those that are in the giants as far as luminosity class goes.

00:52 Now we're asked to determine how the radius of this star we just plotted compares with that of the sun.

00:59 To find that out, we need to use this formula for luminosity.

01:03 L is equal to 4 pi r squared times sigma times t to the fourth.

01:12 R is the star's radius, sigma is the stefan boltzmann constant, and t is the star's temperature.

01:21 We need to rearrange this formula so that we're solving for radius.

01:25 So let's get r squared alone on one side, and now our expression becomes r squared is equal to l over 4 pi times sigma times t to the fourth.

01:40 We're going to have two versions of this equation, one involving the radius of the sun, and another involving the radius of the star we've just plotted.

01:50 We'll have r solar squared is equal to l solar, the sun's luminosity, divided by 4 pi sigma times the sun's temperature to the fourth power.

02:05 And then we'll have the radius of the star squared is equal to that star's luminosity, l star, divided by 4 pi times sigma times the star's temperature to the 4 pi times the star's temperature to the 4.

02:20 Power...

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