A. THE BASICS

a) SPEED OF AN ELECTROMAGNETIC WAVE:

In a vacuum, all electromagnetic waves moves at the speed of light, 300,000 km sec^-1:

c   =   λ × ν

 
b) ENERGY OF AN ELECTROMAGNETIC WAVE:

EM radiation can also be thought of as consisting of discrete "bits" or quanta of energy, a consequence of the wave - particle duality of mass/energy. The energy of a photon can be expressed in terms of its wave parameters:

E   =   h × ν

where h is "Planck's constant", which has a value of 6.63 x 10^-27 ergs s^-1

 
c) ORIENTATION OF THE WAVE:

EM waves consist of ocillating electric and magnetic fields. The orientation of an EM wave, also known as the wave polarization is determined by the direction of the electric field vector:

 
B)   BLACKBODY RADIATION

a) DEFINITION:

A blackbody radiator will eventually reach thermal equalibrium with its surroundings, and will re-emit energy at the same rate that energy is absorbed. The spectrum of emitted energy is described in a very specific way:



where B_λ is the intensity of radiation emitted at a wavelength of λ. Intensity is defined as energy per unit time, having a wavelength between λ and λ+Δλ, emitted by a blackbody radiator of temperature T and surface area dA into a solid angle dΩ = sinθ dθ d&phi:, which in cgi units is expressed in these units:

[ergs s^-1 Å^-1 cm^-2 sr^-1]

Note that B_λ can be converted into B_ν by the following transformation:



b)   CHARACTERISTICS OF THE BB CURVE:

i)   KIRCHOFF'S 1ST LAW:   ®   Every BB emits some amount of radiation at every wavelength.

ii)   ENERGY DISTRIBUTION DIFFERENCES:   ®   At every wavelength, a hotter BB emits more energy than a cooler one.

iii)   MAXIMUM ENERGY OUTPUT:   ®   For a BB at a given temperature, there exists a specific wavelength, λ _peak, at which the BB radiates a maximum amount of energy. A temperature increases , the peak wavelength decreases.



 
c)   WIEN'S LAW

λpeak   =   0.290 / T   [cm]

where l_peak is expressed cm , and T is expressed in Kelvins . Wien's Law is derived from the Planck equation, by setting dI / dλ = 0.

 
d)   STEFAN-BOLTZMANN LAW  



These radiation laws allow us to derive the total energy emitted by an object with a uniform temperature. We sum the energy per second emitted from each square centimeter of a star's surface, over all wavelengths to derive the total flux :

F   =   σ4

where σ is the Stephan-Boltzmann constant having the value 5.672 x 10^-5 ergs K^-4 cm^-2 s^-1

Now, sum this emitted energy over the entire surface of the star to get the star's luminosity:

L   =   total energy / time   =   4 π R2 σ T4

EXAMPLE:   Given two stars, one appearing blue, and the other appearing red . Both stars have the same apparent magnitude and are located at the same distance from Earth.
What can you conclude about their relative sizes?

Red star   ⇒   lower temperature (per Wein's Law)   ⇒   lower flux (per the S-B Law)

Blue star   ⇒   higher temperature ⇒   higher flux

Since both stars are observed to have the same brightness AND are located at the same distance, each must have the same true luminosity.

If     L_red   =   L_blue     then     4πR²_red F_red   =   4πR²_blue F_blue  

therefore:

R_red   >   r_blue,   if   F_blue   >   F_red

 
e) LOOKING AT THE EXTREMES:

i) WEIN APPROXIMATION


ii) RAYLEIGH-JEANS APPROXIMATION



f) WHAT'S NOT A BLACKBODY ?
• A blue shirt (reflects short wavelengths, absorbs all others)
• A window (transmits all light, without absorbing or reflecting it)
• A white piece of paper (reflects all wavelengths)
Stars are fairly good approximations to BB's:

 
C)   INTENSITY and FLUX

a) INTENSITY:

Defined as the amount of energy per unit time, per unit area on the surface of source, within a particular direction. Units are: ergs cm^-2 Hz^-1 sr^-1 s^-1]. Intensity is also known as "surface brightness":


Based on the above definition for intensity, the amount of energy with frequency between ν and ν + dν that passes through an area dA on the surface of the source, into a solid angle dΩ = sinθdθdφ, in time dt, is:

dEn = In cosq dA dt dW dn

 
b) LUMINOSITY (L):

where A is the area on the surface of the source (star), and the "4p" comes from an integration over all solid angles.

 
c) FLUX (F)

Defined as the energy at a specific frequency (or wavelength) passing through an area per unit time. Units are: ergs cm^-2 Hz^-1 s^-1

First, let's look at the equation for monochromatic flux, which means flux of a particular wavelength (or frequency):


The total flux is simply the sum of all the monochromatic fluxes (from the blue to the red ends of the spectrum):

where the 4pR² is the surface area of the star, assuming F is the flux at the surface of the star, or the area of the detector, if F is the flux of the star at the distance of the telescope + detector.

 
d) INVERSE-SQUARE LAW

Note that flux follows the inverse square law, but intensity does NOT. As an explanation, consider the following:


Consider a square-shaped object that is some distance D from the observer. This object subtends an area of 9 squares in the figure (the green area). The detected energy (flux) from the object is "smeared out" over these 9 squares, and we could compute some average energy per square from the above "image" by dividing the total energy in all 9 squares by "9":
surface brightness   =   F / (9 squares)
Now let's take the same object and place it a distance that is three times further than before. That object will subtend an area of only one square (see yellow square in figure), being so much further away. The energy (flux) from this object is also weaker by a factor of 3²=9, compared to the total energy detected from it when it was closer, since the flux falls off with the square of the distance. Even though the energy from that object is weaker by a factor of 9, it is concentrated into just that one square, as compared to being smeared out over 9 squares, as it was before:
surface brightness   =   (F / 9) / (1 square)   =   F / 9
which is the same as before. So the total amount of energy per square does not change, since the inverse-square law of the flux is exactly cancelled by the fact that the object subtends a smaller angular size with distance.
Surface brightness (or "intensity") is invariant with distance, and does NOT follow an inverse-square law.