Astrometry and Cosmology
For reference purposes, the earth’s surface at sea level may be represented by revolving an ellipse of eccentricity, e, and major axis, ae, about the polar axis. The flattening factor, f, or ellipticity, is related to the eccentricity, e, by the equation
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n=(y+l)/(y+j).
If the magnetic field is randomly distributed and N(E) is given by Eq. (1-174) then I(v) is given by Eq. (1-175) with a new constant of order unity substituted for ct.(y). In this case, however, the degree of polarization is zero. Measurements of the degree of circular polarization, n o, of the synchrotron radiation from an ensemble of ultrarelativistic electrons (E~me2) may be used to determine the magnetic field intensity, H, of the radiating source (LEGG and WESTFOLD, 1968; MELROSE, 1971). For a monoenergetic distribution of electrons, the degree of circular polarization is given by n o = 20 cot(1 F(v/vJ
J/3
(vHSin(1)1/2{(~)3/2 Kl/3(~) v
Vc
Vc
(v)
ql((1) +[ 2 + - tan (1J(V)-1/2[V - K Z/3 cp((1) Vc Vc Vc
(1-177) -
-1 F (v)J} , 2 Vc
37
Synchrotron Radiation from an Ensemble of Particles
where v is the frequency of radiation, the gyrofrequency vH =eHj(2nmc), the critical frequency, vc ' is given by Eq. (1-154), K is the modified Bessel function, the angle 0 = rJ. -I/; where rJ. is the pitch angle and I/; is the angle between the direction of observation and the velocity vector of the electron, the proportion of electrons ha ving val ues of 8 within the range 8 to 8 + d 0 in the solid angle dQ=2nsinOdO is cp(O)dQ, the first derivative of cp(8) with respect to 0 is cp'(8) and is unity for an isotropic distribution, and the function F(x) is given by 00
F(x) = x SK S /3('1)d'1.
(1-178)
x
For an ensemble of electrons with apower law energy distribution of index, as given by Eq. (1-174), the degree of circular polarization is given by
n
=
o
I"~
(3"+8) T -i-T -1 ( 3"+4) ~ ~ cotO(VH sinO)1/2 12 12
V3
y(y+!)
TC~~1)TCi~;7)
v
(1-179)
CP'(O)] , x [ y+2+tanO-cp(O) for y>~. Values of the gamma functions, T, are tabulated by LEGG and WESTFOLD (1968) for values of}' in the range 0.4-9.0. The degree of circular polarization given by Eqs. (1-177) and (1-179) is the ratio of the fourth Stokes parameter, V, to the first Stokes parameter, I. MELROSE (1971) has extended these formulae to include the effects of re absorption and differential Faraday rotation on the degree of circular polarization. Both effects can cause the circular polarization to become slightly smaller and to reverse in sign. The polarization of gyrosynchrotron radiation in a magnetoactive plasma is given by RAMATY (1969). From Eq. (1-175) we see that the synchrotron radiation from electrons with apower law distribution of index -y has apower law frequency spectrum proportional to v', where the spectral index, rJ., is given by
(y -1)
rJ.=---
2
.
(1-180)
Most radio sources are observed to have apower law frequency spectrum, and some examples are shown in Fig. 4. Spectral indices for radio galaxies and quasistellar sources with known redshifts are included in Table 31. It follows from Eqs. (1-164) and (1-174) that the total electron energy, U e , is given by (1-181)
38
Continuum Radiation
100000
30000
10 000
r:'-
3000
::l~
.....: IN
I-JI VI'"
."'=' 'E
1000
§ ...-u x
'VI
~~ .!; mOJ
300
N
u:
j:;o'o .~ OJ
