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Blackbody radiation for AQA A-level Physics

Blackbody radiation

This page covers the following topics:

1. Brightness and apparent magnitude
2. Black bodies
3. Absolute magnitude
4. Stellar spectral classes
5. Stefan's law
6. The Hertzsprung-Russel diagram
7. Wien's displacement
8. Supernovae
9. Inverse square law
10. Black holes

Apparent magnitude measures the brightness of a star as it appears from the Earth. The Hipparcos scale is a logarithmic scale that was originally used for classifying stars based on their apparent magnitude. The scale has a minimum value of 1, which represents the brightest stars, and a maximum value of 6, which represents the dimmest stars. The Hipparcos scale can be used to relate brightness and apparent magnitude. When the apparent magnitude increases by 1, the brightness increases by a factor of 2.51.

Brightness and apparent magnitude

A black body radiator is a perfect emitter and absorber of all wavelengths of radiation. Stars can be approximated as black bodies. It is called a black body because it would appear black when any wavelength of light is shone on it, since it would absorb all of it. The radiation emitted by black bodies is over a continuous range of wavelengths with a peak wavelength that has a higher maximum intensity. The peak wavelength of a black body is determined by its temperature. The hotter the black body, the shorter the peak wavelength of radiation will be.

Black bodies

An astronomical unit (AU) is the average distance between the Earth's and the Sun's centres. A parsec is the distance at which 1 AU subtends an angle of 1 arc second. A light year is the distance which an electromangetic wave travels in a year in a vacuum. Absolute magnitude of an object is the apparent magnitude if the object was at a distance of 10 parsecs from the observer. Absolute and apparent magnitude can be related by the following formula: m โˆ’ M = 5log(d/10), where m is the apparent magnitude, M is the absoltue magnitude and d is the distance of the object in parsecs.

Absolute magnitude

The strength of the absorption lines of stars can be used to classify them into spectral classes. The energy of the particles of a star depends on the temperature, and consequently the absorption lines will depend on the temperature of the star.

Stellar spectral classes

The power of a black body is directly proportional to the surface area of the object and its absolute temperature. This is given by Stefan's Law: P = ฯƒATโด, where ฯƒ is Stefan's constant (5.76 ร— 10โปโธ mโปยฒKโปโด), A is the surface area and T is the absolute temperature.

Stefan's law

The Hertzsprung-Russel diagram is a diagram which classifies stars. It has luminosity as compared to the Sun on the y-axis and temperature in Kelvin on the x axis. The dimmer stars are found at the bottom, whereas the brighter ones are found at the top. The hotter planets are found on the left, whereas the cooler ones are on the right.

The Hertzsprung-Russel diagram

Wien's displacement Law can be used to find the temperature of a black body being observed using its peak wavelength. Wien's displacement Law states that the temperature of the black body is inversely proportional to the wavelength which gives peak intensity. This is given by the formula: ฮปmaxT = 2.9 ร— 10โปยณ mK. The peak wavelength of a black body decreases when its temperature is increased. The intensity of a black body increases at any wavelength when its temperature is increased.

Wien's displacement

When a star runs out of fuel, fusion will stop and the core will suddenly collapse inwards and become rigid. This is because it can no longer be forced any closer toghether. The outer layers of the star will fall into its core and rebound off of it. This creates a huge shockwave/explosion known as a supernova. As this shockwave passes through other surrounding material, more fusion reactions occur and elements heavier than iron are flung into space. A supernova's absolute magnitude is rapidly increasing. All types of supernovas occur at the same critical mass, therefore the peak absolute magnitude achieved in each is approximately the same. Supernovas produce light curves, with time usually being measured from the peak, as given in the diagram.

Supernovae

The inverse square law states that the intensity of a star is inversely propotional to the distance from the observer to the star. This is given by the following formula: I = P/4ฯ€dยฒ. The denominator of this formula is the expression for the surface area of a sphere with radius d. This is because the light emitted by a star is approximated to be equal in all directions from the source point, thus giving the shape of the sphere.

Inverse square law

When the core of a giant star collapses, gravity forces the neutrons together to create a singularity called a black hole. Black holes have immense gravitational pull, to which even light is susceptible. Escape velocity is the the velocity an object would need to be at to move out of the gravitational pull of a black hole. The point at which the escape velocity becomes greater than the speed of light is called the event horizon. The radius of the event horizon is given by the Schwarzchild radius.

Black holes

1

Use the Hertzsprung-Russel diagram to compare the brightness main sequence stars and supergiants.

Most supergiants are significiantly brighter than main sequence stars, with only a few of them having about the same luminosity.

Use the Hertzsprung-Russel diagram to compare the brightness main sequence stars and supergiants.

2

Describe Stefan's Law.

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Describe Stefan's Law.

3

Calculate the absolute temperature of a black body which has a peak wavelength of 8 ร— 10โปโถ m.

T = 2.9 ร— 10โปยณ mK/(8 ร— 10โปโถ m) =362.5 K.

Calculate the absolute temperature of a black body which has a peak wavelength of 8 ร— 10โปโถ m.

4

Explain what a value of 6 on the Hipparcos scale represents.

A value of 6 means that the star looks the dimmest it possibly could from Earth.

Explain what a value of 6 on the Hipparcos scale represents.

5

Describe the radiation emitted by black bodies.

The radiation emitted by black bodies is over a continuous range of wavelengths with a peak wavelength that has a higher maximum intensity.

Describe the radiation emitted by black bodies.

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