IB Physics

Option F - Astrophsyics

Stellar Radiation

luminosity / black bodies / wien's law and stefan-boltzmann law / stella spectra

Luminosity

The total power radiated by a star in all directions is known as its luminosity and, as suc, 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 are is known as the star's brightness and this is measured in watts per metre squared (W m-2).

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 luminosity of a star will decrease with distance according to the inverse square law.

The relationship between luminosity (L), brightness (b) and distance (d) is given by the following formula:

It is , therefore, possible for two stars with very different luminosities to have the same brightness as viewed from Earth. This is because the more luminous star must be further away.

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Black Body Radiation

A black body is an object which absorbs 100% radiation that hits it so it does not reflect any radiation and therefore appears to be black. It is also a perfect emitter of radiation, as at a particular temperature, it would emit the maximum amount of energy for that particular temperature. This is known as the black body radiation. The maximum wavelength emitted is infinite and it emits a definite amount of energy for each wavelength for a particular temperature. All objects emit radiation above absolute zero. Black bodies are usually solid but stars are exceptions because the gas particles in stars are very dense that they are able to absorb most of the radiant energy.

This is a black body curve for a temperature of 5000K. This graph shows the black body radiating energy at every wavelength. As seen from the graph, the peak wavelength, the point where it emits maximum energy, is around 5 x 10-7 m. Sometimes the y-axis is labelled as 'intensity'. To be absollutely correct, it should be called the 'intensity function'. In this way the area under the graph between two wavelengths gives the intensity of radiation emitted. Said another way, the area under the graph represents the total power radiated between those two wavelengths.

Black body radiation curves at different temperatures.

As the temperature increases, the peak wavelength emitted by the black body decreases. As temperature increases, the peak wavelength moves towards the visible part of the spectrum. Since the graphs do not touch the x-axis, black bodies emit at every wavelength which means that some radiation is emitted even at low temperatures and at any temperature above absolute zero, a black body will emit some visible light. Also, as temperature increase, the total energy emitted increases as the area under the curve increases.

Since stars act as though they were black bodies, there radiation spectrum can be used to determine their surface temperature. This is the main application of black body radiation curves in astronomy. More on this in the next section.

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Wien's Law and the Stefan-Boltzmann Law

WIen's Law

Wiens’s Law tells us that the temperature of a hot object determines the peak wavelength in its spectrum.  Said another way, the temperature of a black body (or the surface temperature of a star) is inversely proportional to the peak wavelength given by the black body radiation spectrum for that object. Stars at a very high temperature have a smaller peak wavelength than stars at a lower temperature.  The peak wavelength emitted by a star is related to its temperature by Wien’s Displacement Law.

Where:

λmax is the wavelength in the spectrum of the star which corresponds to the peak intensity function on the radiation spectrum

T is the temperature in Kelvin.

and the constant is 2.90 x 10-3 Km

Hotter objects emit most of their radiation at shorter wavelengths; hence they will appear to be bluer while cooler objects emit most of their radiation at longer wavelengths; hence they will appear to be redder. Wien's law is very useful to astronomers because by measuring the spectrum of radiation being emitting from a star, the star's surface temperature can be determined.

Stefan-Boltzmann Law

The Stefan-Boltzmann law connects the luminosity of a star with its size and temperature. The total power radiated by the star (luminosity) is directly proportional to its surface area and to the fourth power of its temperature.

Where σ is the Stefan-Boltzmann constant : 5.67 x 10-8 W m-2 K-4

A is the surface area of the star in metres squared (m-2)

T is the temperature in Kelvin (K)

For the full derivation of this formula please see the following website: http://en.wikipedia.org/wiki/Stefan-Boltzmann_law

Make sure that if you are using these formulae that your temperature is in Kelvin and the wavelengths are in metres (and not nanometres).

In terms of stars, A is the surface area of the star, The radius of the star, r, is linked to its surface area using the equation A = 4πr2 . This means that if we know the luminosity of a star and its temperature ((from Wien's Law), we can work out its radius or size.

Practise Questions

  1. Two stars have equal radii and the temperature of the first star is twice that of the second.  What is the value of the ratio of the luminosity of the first to that of the second?

    1:16

  2. The brightest star in Orion is Betelgeuse.  This star has a surface temperature of 3250K and a distance of 500 pc from earth.  Second brightest is Rigel.  It ahs a surface temperature of 11,600K and is a distance of 400 pc from earth.  Assuming they are the same size, find the ratio of apparent brightness.

    253.6:1

  3. The sun has a surface temperature of 5800 K, what is the peak wavelength in the solar spectrum?

    5.00 x 10-7 m

  4. The star Betelgeuse has a surface temperature of 3250 K, what is the peak wavelength and What colour would it be?

    8.90 x 10-7 m, Red giant

  5. Sirius B is a white dwarf, which is smaller than earth, has a very high density of a temperature of 30,000 K, what is its peak wavelength?

    9.67 x 10-8 m

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Stella Spectra

The discovery of spectral lines was made by Fraunhofer who, in the early 1800's, magnified the Sun's spectrum and discovered dark lines which could be identified with particular elements (based on spectra in the laboratories).

Spectrum from the Sun

Laboratory spectra generally look like the inverse of the Sun's spectrum: instead of dark lines, it has bright lines superimposed on a faint continuum. But the bright lines show up at the same wavelengths as the dark lines, and can be used to identify elements in stars.

Absorption spectrum of helium

Another wave of conceiving absorption spectrum

There are three types of spectra:

  • Continuous spectrum - a solid or liquid body radiates an uninterrupted, smooth spectrum (called a Planck curve).

  • Emission spectrum - a radiating gas produces a spectrum of discrete, bright spectral lines (like helium above).

  • Absorption spectrum - a continuous spectrum that passes through a cool gas and has specific spectral lines removed (inverse of an emission spectrum). This is what most stellar spectra look like (like the sun above).

    The missing wavelengths in a star's absorption spectra correspond to the absorption specturm of a number of elements in the star. THe absorption is taking place in the outer layers of the star, which means we have a way of telling what elements exist in the star - at least in its outer layers. The spectra of stars are also used to classify them into spectral classes.

Ways of obtaining different types of spectra

Spectral Classes

Stars are divided into a series of spectral types based on the appearance of their absorption spectra. Some stars have a strong signature of hydrogen, others have weak hydrogen lines, but strong lines of calcium and magnesium.

There are 7 basic spectral classes: O, B, A, F, G, K and M. Note that the spectra classes are also divisions of temperature such that O stars are hot, M stars are cool. Between the classes there were 10 subdivisions numbered 0 to 9. For example, our Sun is a G2 star. Sirius, a hot blue star, is type B3. The syllabus, however, alludes to the notion that you only have knowledge of the principle spectral classes (OBAFGKM).

Why do some stars have strong lines of hydrogen, others strong lines of calcium? The answer is not composition (all stars are 95% hydrogen) but rather surface temperature. As temperature increases, electrons are kicked up to higher levels by collisions with other atoms. Large atoms have more kinetic energy, and their electrons are excited first, followed by lower mass atoms.

Relative strengths of absorption lines for different spectral classes

If the collision is strong enough (high temperatures) then the electron is knocked off the atom and we say the atom is ionized. So as we go from low temperatures in stars (a few thousand K) we see heavy atoms, like calcium and magnesium, in the stellar spectra. For stars with higher temperatures, we see lines from lighter atoms, such as hydrogen. The heavier atoms are all ionized by this point and have no electrons to produce absorption lines.

Absoprtion spectra for different types of stars classified according to spectral class

Follow the link below to complete a tutorial on determining which elements are present in a star from its spectra.

http://www.learner.org/teacherslab/science/light/color/spectra/spectra_1.html

The characteristics of the different spectral classes are summarised in the table below.

Class Effective Surface Temperature/K Colour
O
28 000 - 50 000
blue
B
9900 - 28 000
blue-white
A
7400 - 9900
white
F
6400 - 7400
yellow-white
G
4900 - 6000
yellow
K
3500-4900
orange
M
2000-3500
orange-red

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