Inverse Square Law, Blackbody Radiation The Inverse Square Law for Radiation The amount of energy emitted in one second by a source of light is called its luminosity and is ... The flux from the blackbody’s surface ( F) is directly proportional to th
Blackbody Radiation A blackbody is a surface that • completely absorbs all incident radiation • emits radiation at the maximum possible monochromatic
Wien's Displacement Law When the temperature of a blackbody radiator increases, the overall radiated energy increases and the peak of the radiation …
What is a ‘black body’,and what is the Stefan ... Derivation of Stefan’s constant σ in P = σT4 from the Planck formula for the spectrum of blackbody radiation. 2.
Impact of Both Series and Shunt Resistance; Effect of Temperature; Effect of ... the variation in the emitted blackbody spectrum as a function of temperature. ...
Doing Physics with Matlab Thermal Physics Blackbody Radiation 3 and 2 1 3 2 1 2 21 exp 1 ff Af ff b hf P R df df c hf kT [W.m-2] (7) The equations (6) and (7) give the Stefan-Boltzmann law, equation (2) when the bandwidths extend from 0 to .
Blackbody Radiation References 1) R.A. Serway, R.J. Beichner: Physics for Scientists and Engineers with Modern Physics, ... resulting in the characteristic blackbody curve. The intensity of the light bulb is reduced, reducing the temperature, and the
Blackbody radiation derivation of Planck‘s ... A black body is an object that absorbs ... - It accurately describes the spectrum of thermal radiation from ...
o Blackbody Radiation o Atomic Line Spectra ... • This spectrum is know as blackbody spectrum.
“Blackbody Curve” - a graph of an object’s energy output versus wavelength. The WAVELENGTH at which the PEAK of this curve occurs tells us about the object’s TEMPERATURE and COLOR. UV IR put Wavelength 11. Hotter objects emit light that PEAKS at shor
Blackbody Radiation Blackbody = something that absorbs all electromagnetic radiation incident on it. A blackbodyy does not necessarilyy look black. Its color depends on its temperature. The Sun and other stars behave approximately like blackbodies. The amount of electromagnetic radiation, with a given wavelength, emitted by a square meter t off a blackbody’s bl kb d ’ surface f iin one second d iis given i bby th the Pl Planckk blackbody bl kb d radiation law. It is displayed as a graph on the following slide. The temperature scale used by physicists and astronomers is the absolute Kelvin scale. The temperature at which a body can emit no energy is absolute zero. One Kelvin = 1 Celsius degree = 1.8 Fahrenheit degree. The freezing point of water is about 273 K = 32oF.
flux in watts per square m meter
Comparison of the Solar (T = 5800 K) Continuous Spectrum with that of a 3500 K Star
wavelength in nanometers
The 3500 K star is brightest at a wavelength around 828 nm. This is infrared radiation, so we don’t see it. Since the visible part of this star’s spectrum is strongest at red and orange wavelengths, it has a faint red - orange color.
flux in watts per square meter
Planck’s Law for Blackbody Radiation For a star with T = 7500 K
For a star with T = 5800 K The visible spectrum (about 400 to 700 nm) is superimposed on the graph). wavelength in nanometers
The peak of the graph for the star with a surface temperature of 7500 K is at a wavelength of 386 nm (violet). In the visible light from this star, there would be a bit more blue and violet than the other colors. The star would have a faint blue color. Our sun has a surface temperature of 5800 K. The wavelength at which it is brightest is 500 nm (green). It has a faint yellow color.
Wien’s Law and the Stefan-Boltzman Law 7000 K
Notice that (a) the higher the temperature, the shorter the wavelength at which the star is brightest and (b) the higher the temperature the greater the area under the graph.
(a) is Wien’s law. Stated more precisely, it
says that the wavelength λmax at which the blackbody is brightest is inversely proportional to its absolute temperature T. If T is in Kelvins and λ is in nm, then λ=
2.898 × 106 T
(b) is the Stefan-Boltzman Law: The energy radiated in one second by one square meter of the blackbody’s surface (E) is directly proportional to the fourth power of its temperature. If E is in watts per square meter and T is in Kelvins, 4
E = σT 4
E A ⎛ TA ⎞ =⎜ ⎟ . E B ⎝ TB ⎠
σ is a constant whose value we won’t need because we’ll always use the second form of the Stefan-Boltzman law.
Example 1: The surface temperature of the Sun is about 5800 K, and the surface temperature of Sirius is about 11,000 K. Compare the surface flux (the electromagnetic energy emitted by 1 m2 of surface in 1 second) of the star Sirius with the surface flux of the Sun.
A = Sirius and B = Sun FA ⎛ 11000 ⎞ =⎜ ⎟ FB ⎝ 5800 ⎠
TA = 11,000 K
FA 4 = (1.897 ) FB
TB = 5800 K
FA = 12.9 FB
Sirius is about 13 times brighter than the sun.
Example 2: At what wavelength does Sirius emit the most electromagnetic energy? λ=
2.898 × 106 . T
T = 11,000 K
2.898 × 106 . 11000
λ = 263 nm
This is a bit shorter than violet, so Sirius is brightest at ultraviolet wavelengths. The visible wavelengths at which it is brightest are violet and blue, so Sirius has a pale blue color.
Atoms • • • • • • •
A neutral atom consists of a nucleus, made of protons and neutrons, and enough electrons to neutralize the positive charge of the protons. The diameter of the nucleus is about 1 femtometer (1 fm = 10-15m). m) The “cloud” of electrons, on the other hand, occupies a volume with a diameter of about 0.1 nm (1 nm = 10-9 m). Since mp = 1836 me and mn = 1839 me, most of the mass of an atom is concentrated in its nucleus. An ion is an atom that has either a deficit or a surplus of electrons. The process of removing an electron from an atom is called “ionization”. The number of protons in the nucleus determines the chemical identity of the atom.
The Bohr Model of the Hydrogen Atom
The electron in a hydrogen atom moves in a circular orbit. The radius can only have one of a set of discrete values, rn (n = 1, 2, 3, …). n = 1, 1 the th lowest l t possible ibl orbit, bit is i called ll d the th ground state of the atom. The states above the ground state are called excited states.
n=1 The electron can transfer to a higher orbit if it absorbs a photon of energy equal to the energy difference between the two orbits. When an electron transfers to a lower orbit, it emits a photon with energy equal to the energy difference between the two orbits. A transfer from one orbit to another is called a transition.
An energy level diagram is a graphical representation of the energies that are available to the electron in the atom. The following is an energy level diagram for the hydrogen atom. Paschen Series - infrared
n=5 n=4 n=3
Balmer Series - visible n=2
α β γ
Lyman Series - ultraviolet
Any atom can be described in terms of its energy levels. He, for example, has twice as many protons as H. Its energy levels are closer together, and the ionization potential for removing one electron from a neutral He atom in its ground state is 24.5 eV.
The higher energy levels are increasingly close together and blend into the continuum at an energy of 13.6 eV above the ground state. When a hydrogen atom absorbs a photon with energy greater than or equal to 13.6 eV, the electron is removed from the atom. 13.6 eV is the ionization potential of the hydrogen atom. λ Hα = 656.3nm λ Hβ = 486.1nm λ Hχ = 434.0nm
Kirchoff’s Rules I. A solid, liquid, or dense gas has a continuous spectrum. II. A hot, low-density gas has an emission (bright line) spectrum. III. When light with a continuous spectrum passes through a cool gas, the result is an absorption spectrum. The dominant component of a star’s spectrum is a set of absorption lines. The visible lines in the absorption spectrum of the Sun are shown below.
Dependence of Stellar Spectra on Temperature (Spectral Classification)
Hydrogen Balmer Line Strength
Approximate Surface Temperature
Singly ionized helium lines either in emission or absorption. Strong ultraviolet continuum.
11,000 - 25,000
Neutral helium lines in absorption.
7,500 - 11,000
Hydrogen lines at maximum strength for A0 stars, decreasing thereafter.
6,000 - 7,500
Metallic lines become noticeable.
5,000 - 6,000
Solar-type spectra. Absorption lines of neutral metallic atoms and ions (e.g. once-ionized calcium) grow in strength.
3,500 - 5,000
Metallic lines dominate. Weak blue continuum.
Molecular bands of titanium oxide noticeable.
> 25,000 K
The Luminosity Effect
From “An Atlas of Stellar Spectra” by Morgan, Keenan, and Kellman
• • • •
In a dense stellar atmosphere, the atoms collide more frequently than in a low density stellar atmosphere. The collisions broaden the absorption lines in the star’s spectrum. spectrum Giant stars are larger and less dense than main sequence stars, so their absorption lines are narrower than those of main sequence stars. The spectra shown above illustrate this effect. Notice that the absorption lines in the spectra of the giant (13 Mono ) and supergiant (HR 1040) stars are narrower than the absorption lines in the main sequence star α Lyrae. This enables us to use the star’s spectrum to determine its luminosity as well as its temperature.
T = 50,000 K
T = 27,000 K
T = 11,000 K
T = 7200 K
T = 6000 K
T = 5100 K
T = 3700 K
Composition of the Sun Element Hydrogen
% by Number of Atoms 91 0 91.0
% by Mass 70 9 70.9
The Doppler Effect Vr
Vr V Vt
Doppler Simulation V Vr < 0
Vr λ observed − λ lab = c λ lab
Vr = radial velocity
Vr > 0
Vt = tangential velocity
Example: The laboratory wavelength of the blue-green line in the hydrogen spectrum is 486.1 nm. If the wavelength of this line in the spectrum of a star is 485.9 nm, what is the d ve velocity oc y of o thee star? s radial
λ lab = 486.1 nm, Vr = c
λ observed − λ lab λ lab
λ observed = 485.9 nm,
c = 3.00 × 108 m / s
m ⎞ 485.9 − 486.1 ⎛ = ⎜ 3.00 × 108 ⎟ = −1.2 × 105 m / s = −1.2 × 102 km / s s ⎠ 486.1 ⎝