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Transcript
```Chapter 3
 Nick Devereux 2006
Revised 2007
 Nick Devereux 2006
The Sun (and other Stars) radiate
like Blackbodies
 Nick Devereux 2006
The Planck Function
I =
2 h 3
c2 (eh/kT - 1)
Where
 Nick Devereux 2006
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 I is the Specific Intensity
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
2
 Nick Devereux 2006
Then,
I = Id
d
I = -
I =  Nick Devereux 2006
2h3
c
c2 (eh/kT - 1)  2
2 h c2
5 (eh /kT - 1)
Wiens Law
Differentiating I leads to Wien’s Law,
max T = 2.898 x 10-3
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.
mB – mV = 2.5 log {fV/fB} + 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
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 of
a 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)
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
F
( W/cm2 m)
1 Jansky (Jy) = 1 x 10-26 W/m2 Hz
 Nick Devereux 2006
F (W/m2 Hz)
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*}
 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 ( MBol )
When we measure Mv 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, 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?
 Nick Devereux 2006
The Sun
The Sun has a BC = -0.07 mag and a bolometric magnitude,
Mbol(sun) = +4.75 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.
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 D2, the
spherical volume over which the star radiates, and the ,
the solid angle the star subtends, to obtain
L = 4π R2 T4
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
 Nick Devereux 2006
Relating Bolometric Magnitude
to Luminosity
The bolometric magnitudes for any object, Mbol* , may be compared
with that measured for the Sun, Mbol, to determine the luminosity
of the object, L* in terms of the luminosity of the Sun, L○.
Mbol - Mbol* = 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
```
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