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Bibliography
· · · · · Rybicki and Lightman: "Radiation Processes in Astrophysics" Longair: "High Energy Astrophysics" Rohlfs and Wilson: "Tools of Radio Astronomy" Dyson and Williams: "The Physics of the Interstellar Medium" Shu: "The Physics of Astrophysics I: Radiation"


Radiation Processes
We can measure the following quantities: · The energy in the radiation as a function of a) position on the sky b) frequency · The radiation's polarisation. From these measurements we can hope to determine · Physical parameters of the source (e.g. temperature, composition, size) · The radiation mechanism. · The physical state of the matter. Need to understand the difference between often used terms: luminosity, flux, flux density, specific intensity and specific energy density.


Luminosity (L)
· · · · A source's luminosity is the power that it emits in radiation. SI units are W. Also used are erg s-1 10-7 W Have to define a frequency range

e.g. the total (or bolometric) luminosity of a lightbulb is 50 W the luminosity in the visible range is only 1 W · For a spherical blackbody source L = 4R2T4

E = Р L dt


Flux ()
· The flux received The flux source at
=
L 4d
2

of a source is the power per unit area. from a uniform spherical distance d is given by

·

SI units are Wm-2.

· e.g. Flux from a light bulb at a distance of 10 m is 0.04 Wm-2

E=

РР

dt dA


Flux Density (F)
· Flux density is the power received per unit area per unit frequency. Flux is the integral of flux density with respect to frequency.

=

= 2 -

Р



2

1

F d
1

is the bandwidth

· Also called specific flux - `specific' refers to Hz-1. · NB Astronomers often say `flux' when they mean `flux density' · SI units are Wm-2Hz
-1 -26

filter

· Also used are Jy 10

Wm-2Hz

-1

E=

РРРF

d dA dt


Specific Intensity, Surface Brightness (I)
· Specific intensity (or surface brightness) is the power received (or emitted) per unit area per unit frequency per unit solid angle. ч F at surface of a spherical source = I · · Specific intensity is independent of distance. Also define the mean intensity J
J =
1 4

F = Р I cos d



Р

4

I d

· An isotropic radiation field is one where I is independent of angle, so J = I . · Units are Wm-2Hz-1sr
-1

filter

E=

Р РРР

I cos d d dA dt


Calculating F and I for a blackbody
· The specific intensity of a black-body is given by the Planck function, B


Ж for a uniform brightness source В Ь A derivation of B is given in the appendix.
· In the Rayleigh-Jeans region where h << kT
I = B = 2kT 2

2h 3 1 Ф F I = B = 2 Г= c (exp h - 1) Х kT

e.g. A black-body source of angular radius 1 arcsec and temperature 2700 K is observed by a telescope with a beam FWHM = 1 arcmin at 1GHz. I (source) = 3.02 x 10
-17

Wm-2Hz-1sr-1. (We are in the Rayleigh-Jeans region)
-27

F (source) = I (source) x (source) = 2.2 x 10 ((source) = (1/3600 x /180)2 )

Wm-2Hz-1 = 0.22 Jy.


Determining
· In general the integration for calculating the received flux density from the specific intensity using

F = Р I cos d
· · is not trivial since it must take into account the telescope's response function, also called the beam. If the source is much smaller than the telescope's beam then this makes a negligible difference and F = I (source) (as in the previous example). Another simple case is if the source has constant specific intensity inside the beam; the integration then gives I (beam) where (beam) is the effective beam solid angle. The effective beam solid angle for a Gaussian beam of FWHM 2 FWHM is (beam) = FWHM 4 ln 2 Extending the previous example: the telescope also picks up radiation from the 2.7 K Cosmic Microwave Background . I (CMB) = 3.02 x 10
-20

Wm-2Hz-1sr-1. (We are in the Rayleigh-Jeans region)
-27

F(CMB) = I (CMB) x (beam) = 2.9 x 10 ((beam) = (1/60 x /180)2 x 1.13)

Wm-2Hz-1 = 0.29 Jy.


Brightness Temperature (TB)
· Radio astronomers often use the concept of brightness temperature for sources that do not have a black-body spectrum. It is the temperature that a black-body source would have in order to be the same brightness. In general the brightness temperature is a function of observing frequency, and does not correspond to an actual physical temperature.
I 2 TB ( ) = 2k

e.g. At 350 GHz the atmosphere emits with a surface brightness of 2x10 Therefore it has a brightness temperature of 52 K. At 30 GHz the surface brightness of the atmosphere is 2x10 corresponds to a brightness temperature of 7 K.
-18

-15

Wm-2Hz-1sr-1.

Wm-2Hz-1sr-1. This


Radiation Energy Density (U)
· u() is the specific energy density i.e. the energy in the radiation field per unit frequency per unit volume per unit solid angle. The volume of field incident on the target is c cos dA dt. Therefore

dE =

Р РРР

u ()c cos d d dA dt
I c

ч u () =
·

u is u() integrated over all angles; it has units Jm-3Hz-1

· U is u integrated over all frequencies i.e. the total energy density in EM fields; it has units Jm-3. · So for an isotropic field

u = Р u () d =

4J c



U = Р u d =

4 c

Р

J d


Radiation Pressure (P)
· For photons of mtm p, E = pc. Since p = p cos, then the momentum per unit area per unit time per unit frequency, p of a radiation field is

p =

1 c

Р

I v cos2 d

· For isotropic radiation (i.e. J = I) perfectly reflected by a wall (so the total momentum change is twice the incident momentum), and remembering P = dF/dA, F = dp/dt then
P=

РР
2

2 c

I cos2 d d

P = 1U 3
(d = 2 sin d )