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Stellar Radiation
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luminosity / black bodies / wien's law and stefan-boltzmann law / stella spectra The total power radiated by a star in all directions is known as its luminosity and, as such, 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 area on the receiving end 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 brightness 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:
In essence, as
the light from the star spreads out in all directions it is distributed
over an area equal to the surface area of a sphere of radius d. 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 may be located further away. 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, and at a particular temperature, it would
emit the maximum amount of energy for that particular
temperature. The whole spectrum of light emitted by such a body
is known as black body radiation. Although the
gas making up a star would normally be considered transparent, and thus
a poor absorber of light, the layers of gas are effectively so deep
that they absorb 100% of the light striking the star. None is
transmitted through the star, and none is
reflected. Further contributing to this effect is the fact that
the gas is ionized, which scatters the light randomly and leads to
multiple opportunities for the light to be absorbed before escaping.
Since a star
behaves like a perfect absorber of light, it also behaves like a
perfect emitter.
This is a
black body curve for a temperature of 5800K. 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 absolutely 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. Notice that as the temperature increases, the peak wavelength emitted by the black body decreases, meaning the star is emitting light of a higher average frequency. The peak shifts into the red, then orange, then yellow, and so on into the ultraviolet and ultimately the X-ray end 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 increases, the total energy emitted increases as the area under the curve increases. Since stars act as though they were black bodies, their radiation spectra 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. 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. Practice Questions
When hot ionized gas is viewed against a dark background, one observes line spectra corresponding to the emission lines of the elements on the gas. You may have observed these lines in your chemistry class using a diffraction grating. However, since
a good emitter is a good absorber, this means that a cold gas will
absorb at exactly the same frequencies it would emit if it was hot. 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 way of conceiving absorption spectrum There are three types of spectra:
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, which by tradition are called O, B, A, F, G, K and M. (A common mnemonic for remembering the classes is "Oh be a fine girl(guy) and kiss me.") 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
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.
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