D.1b - Luminosity and Brightness

The Dimming of Betelgeuse

ESO telescope sees surface of dim BetelgeuseUsing ESO's Very Large Telescope (VLT), astronomers have captured the unprecedented dimming of Betelgeuse, a red supergiant star in the constellation of Orion. The stunning new images of the star's surface show not only the fading red supergiant but also how its apparent shape is changing.

Guest Star Luminosity

Make some comparisons of the 5 'stars' in the room, what are some measurements that you would want to take regarding the 'stars'?

05 Feb 2024:

Stars 1 and 4 are a distance of 3.2 meters apart. Using your luminosity data, where would you have to position yourself so that the two stars had the same brightness?

Your Luminosity is Blinding!

Luminosity v. Brightness: (Option D)

In astronomy (option D), luminosity is the total amount of energy emitted per unit of time by a star, galaxy, or other astronomical object.
The total power radiated by a star in all directions is known as its luminosity and the SI unit for luminosity is watts ( W ). When you compare this to the power received by an observer on the Earth, you can see that the two quantities are quite different. The power received per unit is known as the star’s brightness and this is measured in watts per metre squared (W/m^2).
List of Luminous Stars with various scales of measuring luminosity.
Radiating Area: This will be essential in determining the size of a star!

The luminosity of a star is the total power radiated by the star and it depends on the surface area of the star and its absolute temperature. This is often called the Stefan-Boltzmann law and is given by the following equation;

Using the simulation below, compare the luminosity based on Temperature and Size of star.

Luminosity of Betelgeuse

Determining the luminosity of Betelgeuse.

Emissivity / Power Radiated: (Topic 8)

- The radiation emitted by such a body at constant temperature is called black-body radiation.
- The Stefan-Boltzmann law states that the power of radiation emitted by a black body per unit area is proportional to the fourth power of its temperature.
Most objects are not black bodies. They radiate a fraction of the power per unit area compared to a black body at the same temperature. The value of this fraction depends on the object and is called the object’s emissivity (e).
Emissivity = power per unit area radiated by the object / power per unit area radiated by a black body at the same temperature.
The equation for the power radiated by an object with emissivity e can be given by the diagram in the previous section (Black-body radiation).

The most luminous stars are hundreds of thousands times more luminous than the sun; the least luminous shine with only a few hundred thousandths of the sun’s power.

From the various 'stars' in the classroom, compare the luminosity of your assigned pair of stars.

Brightness - Luminosity in action (Option D)

Intensity of Sun's Radiation (Topic 8)

If two stars were at the same distance from Earth, the one that had the greatest luminosity would also have the greatest brightness. However, because stars are at different distances from the Earth, their brightness will depend on the luminosity as well as the distance from Earth. The brightness (b) of a star will decrease with distance according to the inverse square law.

Determining the radius of Sirius

Using the luminosities of the stars in the classroom, determine the brightness of star to your given distance.

We also calculated the intensity of the Sun's radiation in W m-2 using:

From these two equations we should see:

Intensity (I) = brightness (b)
Power (P) = Luminosity (L)
4𝜋d^2 = Area

TOK: Can the IB get it act together and use the same eq's?

Relative brightness of Altair

How hot are you little star? - Wien's Law

The surface temperature of a black body could also be found if the wavelength of the maximum intensity of radiation on a black body curve is known. The effect of temperature on the black body curve can be investigated using the simulation below. You will need FLASH.

Wien's Law - Option D

Wien's Law - Topic 8

Please add any comments you would like to leave for the IB below:

How Big are you little Star? - Putting it all together

The Magnitude Scale...or Hipparchus' curse

How do we define the apparent brightness of stars in the night sky? The Greek astronomer Hipparchus cataloged the stars in the night sky, defining their brightness in terms of magnitudes (m), where the brightest stars were first magnitude (m=1) and the faintest stars visible to the naked eye were sixth magnitude (m=6).

First confusing point: Smaller magnitudes are brighter!

The magnitude scale was originally defined by eye, but the eye is a notoriously non-linear detector, especially at low light levels. So a star that is two magnitudes fainter than another is not twice as faint, but actually about 6 times fainter (6.31 to be exact).

Second confusing point: Magnitude is a logarithmic scale!

A difference of one magnitude between two stars means a constant ratio of brightness. In other words, the brightness ratio between a 5th magnitude star and a 6th magnitude star is the same as the brightness ratio between a 1st magnitude star and a 2nd magnitude star. Are we confused yet?

So, do we toss out this confusing, archaic measure of brightness?? Absolutely not!

We refine it, and precisely define the scale such that a difference of 5 magnitudes is equal to a factor of 100 in brightness.

So what is the brightness ratio which corresponds to 1 magnitude difference?

So a 1st magnitude star is 2.512 times brighter than a 2nd magnitude star, and 2.5122=6.31 times brighter than a 3rd magnitude star, and 2.5123=15.9 times brighter than a 4th magnitude star, 2.5124=39.8 times brighter than a 5th magnitude star, and 2.5125=100 times brighter than a 6th magnitude star.

When Hipparchus first invented his magnitude scale, he intended each grade of magnitude to be about twice the brightness of the following grade. In other words, a first magnitude star was twice as bright as a second magnitude star. A star with apparent magnitude +3 was 8 (2x2x2) times brighter than a star with apparent magnitude +6.

In 1856, an astronomer named Sir Norman Robert Pogson formalized the system by defining a typical first magnitude star as a star that is 100 times as bright as a typical sixth magnitude star. In other words, it would take 100 stars of magnitude +6 to provide as much light energy as we receive from a single star of magnitude +1. So in the modern system, a magnitude difference of 1 corresponds to a factor of 2.512 in brightness, because

2.512 x 2.512 x 2.512 x 2.512 x 2.512 = (2.512)5 = 100

A fourth magnitude star is 2.512 times as bright as a fifth magnitude star, and a second magnitude star is (2.512)4 = 39.82 times brighter than a sixth magnitude star.

How to Measure the Radius of a Star

It is important to keep the end in mind when starting a calculation such as this, the radius is a measurement of distance measured in meters.

TWO ASSUMPTIONS:

Radius of unknown star will be compared to our own SUN.
Considering the formulas that we have for this unit and their meaning should provide a path to a final answer. Through some analysis the most appropriate equation that would describe the radius of a star is the equation for LUMINOSITY:

If we can determine the LUMINOSITY and TEMPERATURE of the star, we can then find the radius.

Determine Surface Area of a Sphere

Procedure overview:

1. Determine Relative Brightness:

Compared to our sun.
You can confirm this brightness ratio using the magnitude calculator found here.

2. Determine Distance to Star:

Using parallax angle.
This will give the distance in Parsec - Convert to A.U. 1 Parsec = 206625 A.U.

3. Determine Relative Luminosity:

If the difference in brightness and distance are known, we can compare Luminosities
We want the Luminosity of the Sun compared to the Luminosity of the Star.
Distance can also be in A.U.'s (as long as they are consistent).

4. Determine Radius:

Based on the Luminosities, you can then solve for the radius of the star based on the Radiating Area (surface area of the star).

Calculations in Practice: Determine the relative size of the star Betelgeuse.

Guide
Why Betelgeuse?
- Wikipedia Page
- It's in my favorite constellation
- Immortalised in Dave Mathews Song: Black and Bluebirds
- ABOUT TO GO SUPER NOVA!!!!

Black and BlueBird- Dave Matthews Band- DMB from Come Tomorrow

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