Alnitak is the brightest O-class star in the night sky, located at the eastern end of Orion's be

Alnitak is the brightest O-class star in the night sky,...

Key Concepts

Stellar Parameter Estimation

Stellar parameter estimation involves determining fundamental properties of stars such as luminosity, temperature, and radius using observational data and theoretical relations. By placing stars on the H-R diagram and applying laws like the Stefan-Boltzmann law, astronomers can infer physical characteristics and compare different types of stars.

Stefan-Boltzmann Law

The Stefan-Boltzmann law states that the total energy radiated per unit surface area of a black body is directly proportional to the fourth power of its temperature. In astrophysics, this law is critical for relating a star's luminosity, temperature, and radius, which allows for the estimation of a star’s size when its luminosity and temperature are known.

Stellar Classification

Stellar classification categorizes stars based on their spectral characteristics, which correlate with their temperature, color, and intrinsic luminosity. Spectral types, such as O-class for very hot, luminous stars and M-class for cooler, less luminous stars, give insight into a star's temperature, mass, and stage in its life cycle.

Hertzsprung-Russell Diagram

The Hertzsprung-Russell (H-R) diagram is a fundamental tool in astrophysics that plots stars according to their luminosity (intrinsic brightness) and surface temperature (or spectral class). This diagram reveals patterns such as the main sequence, giant branch, and white dwarf regions, which are used to understand stellar evolution and the physical properties of stars.

Transcript

00:01 Question 17 asks us to find the relative sizes of two stars whose peak wavelengths are as follows.

00:09 Peak wavelength for the first star is at 500 nanometers, and the peak wavelength for the second star is at 700 nanometers.

00:23 The other bits of information that we're given is that their apparent brightnesses, are at a ratio of l1 over l2 is equal to 0 .091, and that their distances from us are equal to one another.

00:48 We can use the information about apparent brightness and distance to help us get our answer.

00:55 The formula for apparent brightness is little l is equal to big l, the l, the lumines, divided by four times pi times the distance squared.

01:08 Doing a little bit of rearrangement on this equation, we can find that the distance to the star squared can be set equal to its luminosity over four times pi times its apparent brightness.

01:25 Since we know that both stars are equidistant from our vantage point, we can say that the luminosity of the first star over 4 times pi times its apparent brightness is equal to the luminosity of the second star over 4 times pi times its apparent brightness.

01:49 We're also given a bit of information about the ratios of their apparent brightness to each other.

01:55 We can use that to help us with our equation.

01:59 First, let's cancel out the four pies, as that's just a constant.

02:03 That brings us down to luminosity of the first star over its apparent brightness is equal to the luminosity of the second over its apparent brightness.

02:15 Now we can rearrange this equation.

02:17 Just multiply both sides by little l sub 1 over big l sub 2, and we get the luminosity of the first star over the luminosity of the second is equal.

02:29 To little l sub 1 over little l sub 2.

02:33 Well, we know what little l sub 1 over little l sub 2 is.

02:38 So now we can set luminosities to that number.

02:41 So 0 .091 not only is equal to the apparent brightness of the first star over the apparent brightness of the second, but it's also equal to the luminosity of the first star over the luminosity of the second.

02:56 So we're getting a little bit further along.

02:59 But what we want to know is the relative sizes of the two stars.

03:03 How do we find that? well, we can use the stefan boltzmann equation.

03:08 The stefan boltzman equation relates the power put out by a star with its area and temperature as follows.

03:15 It says that p, the power, is proportional to the star's surface area times its temperature to the fourth power.

03:22 The power put up by a star can also be thought of as its luminosity.

03:26 So basically, p is equal to l.

03:29 Given that logic, we can say that 0 .091 is not only equal to big l sub 1 over big l sub 2, but also p sub 1 over p sub 2.

03:41 Going a bit further, we can say that p sub 1 over p sub 2 can be equated with surface area of the first star times its temperature to the fourth power over surface area of the second star.

03:59 Times its temperature to the fourth power.

04:02 Now the surface area of a sphere, we're assuming both of these stars are spherical, is equal to four times pi times the radius squared...

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