Color/Temperature Relation

•Blackbody Radiation

•Planck’s law and Astrophysics

•Monochromatic Luminosity and Flux

•Bolometric Magnitude

•Filters, measured flux

•The Color Index

Color/Temperature Relation

Betelguese(3100-3900K)

Rigel (8000-13,000K)

What does the color of a celestial object tell us?

Blackbody Radiation

• Any object with temperature above absolute zero 0K emits light of all wavelengths with varying degrees of efficiency.

• An Ideal Emitter is an object that absorbs all of the light energy incident upon it and re-radiates this energy with a characteristic spectrum.Because an Ideal Emitter reflects no light it is known as a blackbody.

• Wien’s Law: Relationship between wavelength of Peak Emission max and temperature T.

• Stefan-Boltzmann equation: (Sun example)

Blackbody Radiation Spectrum

Blackbody

L:Luminosity A:area T:Temperature

http://hyperphysics.phy-astr.gsu.edu/Hbase/hframe.html

http://hyperphysics.phy-astr.gsu.edu/Hbase/hframe.html

Blackbody Radiation

http://www.mhhe.com/physsci/astronomy/applets/Blackbody/frame.html

Blackbody Radiation

• http://hyperphysics.phy-astr.gsu.edu/Hbase/bbcon.html#c1

Classical Spectral Energy DensityRayleigh-Jeans/Wien’s

• Energy per unit volume per unit frequency within the blackbody cavity u(f,T)

• Power radiated J(f,T) is related to u(f,T) by J(f,T)=u(f,T)c/4 (cavity radiation is isotropic and unpolarized)

• How to calculate u(f,T)?– Wien’s Exponential Law (loosely

based on Maxwell’s velocity distribution for molecules)

– Rayleigh-Jeans Law

• Consider EM radiation to be in equilibrium with cavity walls

Not Quite Right

Ultraviolet Catasrophe

Planck’s Law for Blackbody Radiation

•Planck used a mathematical “sleight of hand

” to solve the ultraviolet catastrophe.

•The energy of a charged oscillator of frequency f is limited to discrete values of Energy nhf.•During emission or absorption of light the change in energy of an oscillator is

hf.•The mean energy at high frequencies tends to zero because the first allowed oscillator energy is so large compared to the average thermal energy available kBT that there is almost zero probability that this state is occupied.

•Planck seemed to be an “Unwilling

revolutionary”.He viewed this “quantization” merely as a calculation trick…Einstein viewed it differently…light itself was quantized.

Planck’s Law and Astrophysics

Power radiated per unit wavelength per unit area per unit time per steradian: (units are W m -2 m-1 sr -1 )

Power radiated per unit frequency per unit area per unit time per steradian: (units are W m -2 s sr -1 )

In spherical coordinates the amount of radiant energy per unit time having wavelengths between anddemitted by a blackbody radiator of temperature T and surface area dA into a solid angle is given by:

In terms of B:

Consider a model star consisting of a spherical blackbody of radius R and temperature T. Assuming that each patch dA emits isotropically over the outward hemisphere, the energy per second having wavelengths between anddemitted by the star is:

Angular integration yields a factor of , the integration of dA over the surface of the sphere yield 4R2.

Note that:

Monochromatic Luminosity and FluxMonochromatic Luminosity

Monochromatic Flux received at a distance r from the model star is:

Fd is the number of Joules of starlight energy with wavelengths between anddthat arrive per second per one square meter of detector aimed at the model star, assuming that no light has been absorbed or scattered during its journey from the star to the detector. Earth’s atmosphere absorbs some starlight, but this can be corrected. The values of these quantities usually quoted for stars have been corrected and would correspond to what would be

measured above Earth’s atmosphere.

Why do we keep the wavelength dependence?

Filters!!! Sf(

Color/Temperature Relation Color Indices

• A star’s absolute color magnitudes can be determined from the apprarent color magnitudes by using eqn 3.6 if the distance is known.

• U-B color index:difference between its ultraviolet and blue magnitudes.

• V-B color index:difference between

its blue and visual magnitudes.

Stellar magnitudes decrease with increasing brightness. A star with a smaller B-V index is bluer than a star with a larger value of B-V!!!!

Because a star’s color index is a difference in magnitudes it is independent of the star’s distance

The Color IndexUVB Wavelength Filters

• Bolometric Magnitude: measured over all wavelengths.

• UBV wavelength filters: The color of a star may be precisely determined by using filters that transmit light only through certain narrow wavelength bands:

– U, the star’s ultraviolet magnitude. Measured through filter centered at 365nm and effective bandwidth of 68nm.

– B,the star’s blue magnitude. Measured through filter centered at 440nm and effective bandwidth of 98nm.

– V,the star’s visual magnitude. Measured through filter centered at 550nm and effective bandwidth of 89nm

• U,B,and V are apparent magnitudes

Sensitivity Function S()

Filter Response

http://astro.unl.edu/naap/blackbody/animations/blackbody.html

Apparent Magnitude and Radiant Flux

If one assumes that B is slowly varying across the bandwidth of the filter S can be approximated by a step function S=1 inside the filter’s bandwidth and S=0 otherwise.The integrals for U,V and B can be approximated by the value of the Planck function B at the center of the filter bandwidth,multiplied by that bandwidth. Therefore for the filters listed on p 75 of the text, we have

Look at example 3.6.2 T=42000K U-B=-1.19 and B-V=-0.33

The constants CU,CB and CV differ and are chosen such that the star Vega(T=9600K, use applet) has a magnitude of zero as seen through each filter. This does not imply that Vega would be equally bright when viewed through them.

Bolometric Magnitude/Correction

Monochromatic Flux received at a distance r from the model star is:

Bolometric Magnitude mbol is a measurement of total brightness integrated over all wavelengths.

S

Cbol is a constant that allows the magnitude scale to be defined such that the bolometric correction BC is negative for all stars while still being as close to zero as possible. (However some supergiant stars have positive bolometric corrections!!) (see problem 3.17)

Bolometric correction BC=mbol-V=Mbol-MV is the difference between the bolometric and visual magnitudes.

Color Index

• A star’s absolute color magnitudes can be determined from the apparent color magnitudes by using the following equation if the distance is known.(example 3.2.2)

• U-B color index:difference between its ultraviolet and blue magnitudes.

• B-V color index:difference between its blue and visual magnitudes.

Stellar magnitudes decrease with increasing brightness. A star with a smaller B-V index is bluer than a star with a larger value of B-V!!!!

Related to “color”/surface temperature of star.

Because a star’s color index is a difference in magnitudes it is independent of the star’s distance

Spectral Type, Color and Effective Temperaturefor Main-Sequence Stars

Spectral Type B-V Te(K)

O5 -0.45 35,000

B0 -0.31 21,000

B5 -0.17 13,500

A0 0.00 9,700

A5 0.16 8,100

F0 0.30 7,200

F5 0.45 6,500

G0 0.57 6,000

G5 0.70 5,400

K0 0.84 4,700

K5 1.11 4,000

M0 1.24 3,300

M5 1.61 2,600

From Frank Shu, An Introduction to Astronomy(1982), Adapted from C.W. Allen, Astrophysical QuantitiesNote that this table does not quite agree with our text!!!!

Spectral Type, Color and Effective Temperaturefor Main-Sequence Stars

Spectral Type, Color and Effective Temperaturefor Main-Sequence Stars (continued)

Color-Color Diagram• Relation between the U-B and B-V

color indices for main sequence stars.

• Would be a straight line if stars were true black bodies. Some light is absorbed as it travels through a star’s atmosphere. The absorption being wavelength dependent alters the distribution of radiation from that of a blackbody.

• Best agreement to Blackbody radiation for very hot stars….

Interstellar Reddening

One also needs to correct color indices for interstellar reddening. As the light propagates through interstellar dust, the blue light is scattered preferentially making objects appear to be redder than they actually are…

Hertzsprung-Russell Diagram

Spectroscopic Parallax

•Identify star as Main-Sequence•Obtain Te from B-V measurement•Use H-R diagram to read off absolute luminosity•Measure apparent luminosity•Calculate distance modulus to obtain distance to star

Rayleigh-Jeans LawLord Rayleigh reasoned that the following assumptions could be used:

Electromagnetic Waves in Cavity

•Standing Waves

•Considered at temperature T, each mode an equal amount of energy kT

•Analogous to 1-d Harmonic Oscillators

Energy Density

•Product of number of standing waves(oscillators)in the frequency range f - f+df and the oscillators average energy

From Boltzmann’s distribution law the average oscillator energy is kBT. The probability of finding an individual system (oscillator) with energy above E0 with an ensemble of sytems at temperature T is:

For a discrete set of energies the varage would be:

In the classical case considered by Rayleigh, an oscillator can have any energy in a continuous range from 0 to infinity. Leading to:

Density of Modes

• Calculation of the number of modes in a cavity

– Consider cubical cavity of side L

– Consider standing waves that have frequencies between f and f+df

– Use Maxwell’s Equations and boundary condition that the wave vanishes at the boundaries.Each component of the electric field satisfies an equation of the type:

– Solutions of the form:

Where

•Number of standing waves between k and k+dk is:

•Accounting for 2 polarizations

•Using

•We obtain the number of modes between f and f+df

•And the number of modes between and +d

Rayleigh-Jeans Law• Putting everything together we obtain

the Rayleigh-Jean’s expression for

spectral density.

Ultraviolet Catasrophe

Derivation of Planck’s Law

• Planck solved the “catasrophe” by “quantizing” the energy of the standing waves in the cavity to obtain a mean energy:

• Solving this by using

• And the steps

•We obtain the mean oscillator energy:

•Multiplying by the density of modes

•We obtain the spectral energy distribution

•Stefan-Boltzmann’s Law can be obtained from Planck’s Law by simply integrating the spectral density function over all wavelengths

•Subsituting and evaluating

•We obtain:

Derivation of Stefan-Boltzmann Law

The Stefan-Boltzmann constant is a derived constant depending on kB, h and c !!!!!

Derivation of Wien’s Law

http://en.wikipedia.org/wiki/Wien%27s_displacement_law#Derivation_from_Planck.27s_Law