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Blackbody Radiation

Nick Devereux 2006

The Sun (and other Stars) radiate like Blackbodies

Nick Devereux 2006

The Planck Function

I = 2 h 3W m-2 Hz-1 sterad-1

c2 (eh/kT - 1)

Where

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

Nick Devereux 2006

Ivs. I

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

Nick Devereux 2006

Transferring from Ito I

- Id= Id(equivalent energy)
- Since c =
- = c/
Thus, d = -c d

2

Nick Devereux 2006

I = Id

d

I = - 2 h 3c

c2 (eh/kT - 1) 2

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

5 (eh /kT - 1)

Nick Devereux 2006

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.

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) 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

Nick Devereux 2006

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

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.

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 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

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.

Where we are going ….. Nick Devereux 2006

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