Chapter 3 Stars: Radiation  Nick Devereux 2006 Revised 2007

Blackbody Radiation  Nick Devereux 2006

The Sun (and other Stars) radiate like Blackbodies  Nick Devereux 2006

The Planck Function I = 2 h   W m -2 Hz -1 sterad -1  c 2 (e h kT - 1) h is Planck’s constant x J s k is Boltzmanns constant x J/K c is the speed of light x 10 8 m/s T is the temperature in K is the frequency in Hz and I is the Specific Intensity Where  Nick Devereux 2006

I vs. I It is important to know which type of plot you are looking at I or I.  Nick Devereux 2006

Transferring from I to I I d = I d (equivalent energy) Since c =  c/ Thus, d -c d    Nick Devereux 2006

Then, I I d  d I = -2 h  c  c 2 (e h kT - 1)  I = -2 h c 2 W m -2 m -1 sterad -1   (e h kT - 1)  Nick Devereux 2006

Wiens Law Differentiating I leads to Wien’s Law,  max T = x  Which yields the peak wavelength, max (m).  for a blackbody of temperature, T.  Nick Devereux 2006

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 also their temperature and the wavelength you observe them at.  Nick Devereux 2006

Temperature Determination To measure the temperature of a star, we measure it’s brightness through two filters. The ratio of the brightness at the two different wavelengths determines the temperature. The measurement is independent of how far away the star is because distance reduces the brightness at all wavelengths by the same amount.  Nick Devereux 2006

Filters and U,B,V Photometry Filters transmit light over a narrow range of wavelengths  Nick Devereux 2006

The Color of a Star is Related to it’s Temperature  Nick Devereux 2006

Color Index A quantitative measure of the color of a star is provided by it’s color index, defined as the difference of magnitudes at two different wavelengths. m B – m V = 2.5 log {f V /f B } + c The constant sets the zero point of the system, defined by the star Vega which is a zero magnitude star. Magnitudes for all other stars are measured with respect to Vega.  Nick Devereux 2006

Dealing with the constant In the basic magnitude equation, there is a constant, c, which I can now tell you is equivalent to m o = -2.5 log (the flux of the zero magnitude star Vega). So, for a star of magnitude m * we can write m * - m o = 2.5 log {f o /f * } Note: There is no constant ! In this equation m o = 0 of course because it is the magnitude of a zero magnitude star. However, the flux of the zero magnitude star, f o is not zero, as you can see on the next slide.  Nick Devereux 2006

Zero Magnitude Fluxes Filter (  m) F ( W/cm 2  m) F (W/m 2 Hz) U x x B x x V x x Jansky (Jy) = 1 x W/m 2 Hz  Nick Devereux 2006

Calculating Fluxes Now you know what the fluxes are for a zero magnitude star, f o, 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 {f o /f * }  Nick Devereux 2006

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 is defined 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, and stars warmer than Vega have B-V < 0.  Nick Devereux 2006

Color and Temperature The B-V color is directly related to the temperature.  Nick Devereux 2006

Bolometric Magnitudes ( M Bol ) When we measure M v for a star, we are measuring only the small part of it’s total radiation transmitted in the V filter. To get the Bolometric magnitude, M Bol which is a measure of the stars total output over all wavelengths, we make use of a Bolometric Correction (BC). So that, M Bol = M v + BC The BC depends on the temperature of the star because M v includesdifferent fractions of M Bol depending on the temperature (see Appendix E). Question: The BC is a minimum for 6700K – Why?  Nick Devereux 2006

The Sun The Sun has a BC = mag and a bolometric magnitude, M bol(sun) = mag, and an effective temperature = 5800K.  Nick Devereux 2006

Spectral Types There is a system for classifying stars that involves letters of the alphabet; O,B,A,F,G,K,M. These letters order stars by Temperature, 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, with 0 being the hottest. So a BO star is hotter than a B5 star.  Nick Devereux 2006

Luminosity Classes Stars are also subdivided on the basis of their evolutionary status, 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.  Nick Devereux 2006

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 D 2, the spherical volume over which the star radiates, and the , the solid angle the star subtends, to obtain L = 4π R 2   W Where R is the radius of the star, T is the stellar temperature, and  is the Stefan-Boltzmann constant = 5.67 x W m -2 K -4  Nick Devereux 2006

Relating Bolometric Magnitude to Luminosity The bolometric magnitudes for any object, M bol*, may be compared with that measured for the Sun, M bol , to determine the luminosity of the object, L * in terms of the luminosity of the Sun, L ○. M bol  - M bol* = 2.5 log{ L * / L  }  Nick Devereux 2006

Where we are going ….. You now know how to measure the luminosity and temperature of stars. Next, we need to find their masses. Once we have done that we can plot a graph like the one on the left. Stars populate a narrow range in this diagram 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.  Nick Devereux 2006