Chapter 3

Stars: Radiation

Nick Devereux 2006 Revised 2007

Blackbody Radiation

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The Sun (and other Stars) radiate like Blackbodies

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The Planck Function

I = 2 h

W m-2 Hz-1 sterad-1

c2 (ehkT - 1) h is Planck’s constant 6.626 x 10-34 J s k is Boltzmanns constant 1.380 x 10-23 J/K c is the speed of light 2.998 x 108 m/s T is the temperature in K is the frequency in Hz and Iis the Specific Intensity

Where

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Ivs. I

It is important to know which type of plot you are looking at Ior I.

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Transferring from Ito I

Id= I d(equivalent

energy)

Since c = c/ Thus, d-c d Nick Devereux 2006

Then,

IIdd I = - 2 h c

c2 (ehkT - 1)

I = - 2 h c2 W m-2 m-1 sterad-1

(eh kT - 1)

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Wiens Law

Differentiating Ileads to Wien’s Law,

max T = 2.898 x 10-3

Which yields the peak wavelength, max (m).for a blackbody of temperature, T.

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Blackbody Facts

Blackbody curves never cross, so there is no degeneracy. The ratio of intensities at any pair of wavelengths uniquely determines the Blackbody temperature, T.

Since stars radiate approximately as blackbodies, their brightness depends not only on their distance, but alsotheir temperature and the wavelength you observe them at.

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Temperature Determination

To measure the temperature of a star, we measure it’sbrightness through two filters. The ratio of the brightnessat the two different wavelengths determines the temperature.

The measurement is independent of how far away the staris because distance reduces the brightness at all wavelengthsby the same amount.

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Filters and U,B,V Photometry

Filters transmit light over a narrow range of wavelengths

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The Color of a Star is Related to it’s Temperature

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Color IndexA quantitative measure of the color of a star is providedby it’s color index, defined as the difference of magnitudes at two different wavelengths.

mB – mV = 2.5 log {fV/fB} + c

The constant sets the zero point of the system, definedby the star Vega which is a zero magnitude star.Magnitudes for all other stars are measured with respectto Vega.

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Dealing with the constantIn the basic magnitude equation, there is a constant, c, whichI can now tell you is equivalent to

mo = -2.5 log (the flux of the zero magnitude star Vega).

So, for a star of magnitude m* we can write

m* - mo = 2.5 log {fo/f*}

Note: There is no constant !In this equation mo = 0 of course because it is the magnitude ofa zero magnitude star. However, the flux of the zero magnitude star, fo is not zero, as you can see on the next slide. Nick Devereux 2006

Zero Magnitude Fluxes

Filter (m) F ( W/cm2 m) F (W/m2 Hz)

U 0.36 4.35 x 10-12 1.88 x 10-23

B 0.44 7.20 x 10-12 4.44 x 10-23

V 0.55 3.92 x 10-12 3.81 x 10-23

1 Jansky (Jy) = 1 x 10-26 W/m2 Hz

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Calculating Fluxes

Now you know what the fluxes are for a zero magnitude star, fo,you can convert the magnitudes for any object in the sky(stars, galaxies, etc) into real fluxes with units of Wm-2 Hz-1, at any wavelength using this equation!

m* = 2.5 log {fo/f*}

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Vega ( also known as -Lyr)

Vega has a temperature ~ 10,000 K, so it is a hot star.Vega is the zero magnitude star, it’s magnitude isdefined to be zero at all wavelengths. Be aware - This does not mean that the flux is zero at all wavelengths!!

Magnitudes for all other stars are measured with respect to Vega, so stars cooler than Vega have B-V > 0, andstars warmer than Vega have B-V < 0.

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Color and Temperature

The B-V color is directly related to the temperature.

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Bolometric Magnitudes ( MBol )When we measure Mv for a star, we are measuring only the smallpart of it’s total radiation transmitted in the V filter. To get theBolometric magnitude, MBol which is a measure of the stars total output over all wavelengths, we make use of a Bolometric Correction (BC). So that, MBol = Mv + BC

The BC depends on the temperature of the star because Mv includesdifferent fractions of MBol depending on the temperature (see Appendix E).

Question: The BC is a minimum for 6700K – Why?

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The Sun

The Sun has a BC = -0.07 mag and a bolometric magnitude,Mbol(sun) = +4.75 mag, and an effective temperature = 5800K.

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Spectral Types

There is a system for classifying stars that involves letters ofthe alphabet; O,B,A,F,G,K,M. These letters order stars byTemperature, with O being the hottest, and M the coldest.

Our Sun is a G type star.

Vega is an A type star.

The letter sequence is subdivided by numbers 0 to 5, with0 being the hottest. So a BO star is hotter than a B5 star.

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Luminosity Classes

Stars are also subdivided on the basis of their evolutionarystatus, identified by the Roman numerals I,II,III,IV and V. There will be more about this later.

Stars spend most of their lives on the main sequence, luminosity class V.

The Sun is a GOV.

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Stellar Luminosity

The Stellar Luminosity is obtained by integrating the Planck function over all wavelengths, and eliminating the remaining units (m-2 sterad–1), by multiplying by 4π D2, the spherical volume over which the star radiates, and the ,the solid angle the star subtends, to obtain

L = 4π R2 W

Where R is the radius of the star, T is the stellar temperature,and is the Stefan-Boltzmann constant = 5.67 x 10-8 W m-2 K-4

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Relating Bolometric Magnitude to Luminosity

The bolometric magnitudes for any object, Mbol* , may be comparedwith that measured for the Sun, Mbol, to determine the luminosityof the object, L* in terms of the luminosity of the Sun, L○.

Mbol - Mbol* = 2.5 log{ L* / L }

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Where we are going …..You now know how to measurethe luminosity and temperatureof stars.

Next, we need to find their masses.

Once we have done that we can plota graph like the one on the left.

Stars populate a narrow range in thisdiagram with the more massive ones having higher T and L.

Understanding the reason for this trend will lead us to an understanding of the physical nature of stars.

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