Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://star.arm.ac.uk/preprints/2010/571d_OnlineI.pdf Äàòà èçìåíåíèÿ: Wed Sep 26 14:09:39 2012 Äàòà èíäåêñèðîâàíèÿ: Tue Oct 2 05:48:15 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: ðåëÿòèâèñòñêîå äâèæåíèå

Online do cumentation. Part I:
Built-in electron distributions
As has been said in the main text, although the algorithm itself is valid for a general case, the analytical built-in electron distribution functions G(E , µ) have the factorized form G(E , µ) = u(E )g (µ), (1)

that is, they can be written as a product of the energy (or momentum) distribution function u(E ) and the angular distribution function g (µ). Currently, to cover a representatively broad range of possibilities needed for practical applications including contribution from thermal and nonthermal electrons, nine types of energy distributions and five types of angular distributions are implemented. The index of the energy distribution is specified by the parameter ParmIn[17] in a call to the gyrosynchrotron code (see online documentation, Part II), and the index of the angular distribution is specified by the parameter ParmIn[19]; any combination of the energy and angular distribution is possible. However, in the library libGS Std HomSrc CEH the selected anisotropy is applied throughout any assumed distribution including the Maxwellian component if present, while in the library libGS Std HomSrc C the Maxwellian component (in THM, TNT, TNP, and TNG distributions) is always adopted isotropic. The distribution parameters (whose number and meaning depend on the type of distribution) are specified by different elements of the array ParmIn. These parameters will be described below. For the factorized electron distribution (1), the functions u(E ) and g (µ) can be normalized independently. We assume that the distribution functions satisfy the following normalization conditions
Emax 1

2
Emin

u(E ) dE = ne ,
-1

g (µ) dµ = 1,

(2)

where ne is the number density of electrons having the energy between Emin and Emax . Sample emission spectra in Figures 1­9 were calculated for the following source parameters: · visible source area S = 10 · source depth L = 10
10 20

cm2 ;

cm;

· magnetic field B = 180 G; · thermal plasma density n0 = 3 â 109 cm-3 , unless different is specified (for the thermal, thermal/nonthermal, and kappa distributions); · thermal plasma temperature T0 = 3 â 107 K, unless different is specified (for the thermal, thermal/nonthermal, and kappa distributions); · viewing angle = 45 , unless different is specified (for the anisotropic pitch-angle distributions).

1


Figure 1: Thermal electron distribution (for n0 = 3 â 109 cm-3 and different electron temperatures) and single power-law electron distribution over kinetic energy (for nb = 3 â 107 cm-3 , Emin = 0.1 MeV, Emax = 10 MeV, and different power-law indices ). For the thermal distributions, the emission spectra at high frequencies are dominated by free-free emission.

Distributions over energy
Sample emission spectra for different energy distributions (in Figures 1­7) were calculated under the assumption that the pitch-angle distribution is isotropic (ISO). Thermal distribution (THM; index 2) Relativistic thermal distribution is given by the expression u( ) d = where is the K2 is In n0 2 - 1 exp - 2 K2 (1/) d , (3)

n0 is the number density of the thermal electrons, is the Lorentz-factor, = kB T0 /(mc2 ) normalized thermal energy for the temperature T0 , kB is the Boltzmann constant, and the MacDonald function of the second order. our gyrosynchrotron codes, the parameters of this distribution are specified as:

· ParmIn[2] = T0 [K]; · ParmIn[11] = n0 [cm-3 ]; · ParmIn[17] = 2. Note that the above parameters, T0 and n0 , are also used beyond the case of thermal distribution. For other energy distributions, n0 and T0 are considered as the background plasma density and temperature, respectively. They are used to calculate the dispersion parameters of the electromagnetic waves (n0 ) and the free-free contribution (both n0 and T0 ). However, the gyrosynchrotron contribution of the thermal electrons is calculated only if one explicitly chooses the thermal distribution as the "main" electron distribution by setting ParmIn[17] = 2 (or, more broadly, the class of thermal/nonthermal distributions, see below); otherwise, this contribution is neglected in GS computation. Examples of the thermal distributions and the corresponding gyrosynchrotron emission spectra are shown in Figure 1. 2


Figure 2: Double power-law electron distribution (for nb = 3 â 107 cm-3 , Emin = 0.1 MeV, Ebreak = 1 MeV, Emax = 10 MeV, 1 = 4, and different high-energy power-law indices 2 ). Single power-law distribution (for the same particle number density and = 4) is given for reference. Single p ower-law distribution over kinetic energy (PLW; index 3) Power-law distributions of the nonthermal electrons over kinetic energy E = mc2 ( - 1) are widely used for interpretation of solar radio and hard X-ray emissions. These distributions are given by the expression u(E ) dE = AE
-

dE for E

min


max

,

(4)

and 0 otherwise. The normalization constant A equals A= -1 nb , 1- 1- 2 Emin - Emax (5)

where nb is the number density of the nonthermal electrons. The logarithmic normalization for = 1 is not implemented, however, one can arbitrarily approach this case taking very close but slightly different from 1. In our gyrosynchrotron codes, the parameters of this distribution are specified as: · ParmIn[6] = E · ParmIn[7] = E
min max

[MeV]; [MeV];

· ParmIn[9] = ; · ParmIn[12] = nb [cm-3 ]; · ParmIn[17] = 3. Examples of the single power-law distributions and the corresponding gyrosynchrotron emission spectra are shown in Figure 1. Double p ower-law distribution over energy (DPL; index 4) In this case the electron spectrum consists of two parts (high-energy and low-energy), where both the high-energy and low-energy parts are described by power laws, but with different 3


indices. These distributions (double power-law or broken power-law) can be described by the following expression: u(E ) dE = dE A1 E A2 E
- -
1 2

, for E , for E

min

< E Ebreak , break E < Emax ,

(6)

-1 -2 and 0 outside the range from Emin to Emax . In the above expression, A1 Ebreak = A2 Ebreak (to make the function continuous), 1 = 1, and 2 = 1. In the library libGS Std HomSrc CEH the normalization factor is given by

A-1 = 1

2 nb

E

1-1 min

1-1 - Ebreak +E 1 - 1

2 -1 break

1-2 Ebreak - E 2 - 1

1-2 max

,

(7)

i.e., nb is the number density of nonthermal electrons between Emin and Emax , and A2 is found using the above continuity condition. In the library libGS Std HomSrc C, normalization (5) is instead used for the purpose of easier comparison between the DPL and PLW results. In our gyrosynchrotron codes, the parameters of this distribution are specified as: · ParmIn[6] = E · ParmIn[7] = E · ParmIn[8] = E
min max

[MeV]; [MeV]; [MeV];

break

· ParmIn[9] = 1 ; · ParmIn[10] = 2 ; · ParmIn[12] = nb [cm-3 ]; · ParmIn[17] = 4. Examples of the double power-law distributions and the corresponding gyrosynchrotron emission spectra are shown in Figure 2. Thermal/nonthermal distribution over energy (TNT; index 5) This distribution ensures a smooth transition from nonthermal to the thermal distribution at low energies by means of expression u(E ) dE = dE uTHM (E ), for E Ecr , AE - , for Ecr E < E
max

,

(8)

and 0 for E > Emax . In the above expression, uTHM (E ) is the thermal distribution function - (3), A = uTHM (Ecr )Ecr to make the function continuous, the matching point Ecr satisfies the condition Ecr < Emax , and > 1. In our codes, the matching point Ecr is defined as the energy corresponding to the momentum pcr p2 (9) p2 = THM , cr where pTHM is the mean thermal momentum corresponding to the energy kB T0 , and the parameter specifies location of the turning point (the distribution becomes purely thermal when < p2 /p2 (Emax )). cr For small , number density of the nonthermal electrons (with E > Ecr ) is much less than that of the thermal electrons. Therefore we assume that the normalization condition remains approximately the same as for the thermal distribution (3), and the total electron number density ne n0 . However, it should be noted that actually the total electron density slightly exceeds n0 . In our gyrosynchrotron codes, the parameters of this distribution are specified as: 4


Figure 3: Thermal/nonthermal electron distribution over kinetic energy (for n0 = 3 â 109 cm-3 , T0 = 3 â 107 K, = 4, and different matching parameters ). For = 0.05 and = 0.03, the emission spectra at high frequencies are dominated by free-free emission. For = 0.03, the emission spectrum (shown by dotted line) is nearly the same as for the purely thermal distribution with T0 = 3 â 107 K (shown by green dashed line in Figure 1), because the contribution of nonthermal particles is negligible in this case. Red dashed line represents the nonthermal "tail" of the thermal/nonthermal distribution; for = 0.1, this "tail" behaves as the single power-law distribution with nb = 106 cm-3 , Emin = 0.03 MeV, Emax = 10 MeV, and = 4. · ParmIn[2] = T0 [K]; · ParmIn[3] = ; · ParmIn[7] = E
max

[MeV];

· ParmIn[9] = ; · ParmIn[11] = n0 [cm-3 ]; · ParmIn[17] = 5. Number density of the nonthermal electrons nb is not specified explicitly, while it is calculated consistently using n0 , T0 , , , and Emax . Examples of the thermal/nonthermal distributions over energy and the corresponding gyrosynchrotron emission spectra are shown in Figure 3. Kappa distribution (KAP; index 6) Another way of describing the smooth transition from the thermal distribution to a nonthermal tail is a so-called Kappa distribution, which is widely used to quantify particle distributions in the interplanetary plasma. It is convenient to express the Kappa distribution in terms of the Lorentz-factor : u( ) d = A
3/2



2 - 1
+1

-1 1+ ( - 3/2)

d for E < E

max

,

(10)

5


Figure 4: Kappa distribution (for ne = 3 â 109 cm-3 , T0 = 3 â 107 K, and different values of the parameter ). Thermal distribution (for the same particle number density and temperature) and single power-law distribution (for nb = 3 â 107 cm-3 , Emin = 0.1 MeV, Emax = 10 MeV, and = 4) are given for reference. For the thermal distribution and kappa distributions with = 6 and = 20, the emission spectra at high frequencies are dominated by free-free emission. and 0 otherwise. In the above expression, = kB T0 /(mc2 ) is the normalized thermal energy for the temperature T0 , and is the distribution parameter. The normalization factor A is calculated numerically by using normalization condition (2). Kappa distribution becomes purely thermal distribution when . In our gyrosynchrotron codes, the parameters of this distribution are specified as: · ParmIn[2] = T0 [K]; · ParmIn[4] = ; · ParmIn[7] = E
max

[MeV];

· ParmIn[11] = ne [cm-3 ]; · ParmIn[17] = 6. Note that for kappa distribution, there is no unique demarkation between the thermal and nonthermal particles, so the total number density is specified. Examples of the kappa distributions and the corresponding gyrosynchrotron emission spectra are shown in Figure 4. Power-law distribution over momentum (PLP; index 7) Power-law distribution of the nonthermal electrons over the absolute value of momentum is given by the expression u(p) dp = Ap
-

dp for pmin < p < p

max

,

(11)

and 0 otherwise. The normalization constant A equals A= nb -3 , 3- 2 pmin - p3- max (12)

where nb is the number density of nonthermal electrons, pmin = p(Emin ), and pmax = p(Emax ), the case of = 3 is not implemented. In our gyrosynchrotron codes, the parameters of this distribution are specified as: 6


Figure 5: Power-law electron distribution over momentum (for nb = 3 â 107 cm-3 , Emin = 0.1 MeV, Emax = 10 MeV, and different power-law indices p ). Single power-law distribution (for the same particle number density and = 4) is given for reference.

Figure 6: Power-law electron distribution over Lorentz factor (for nb = 3 â 107 cm-3 , Emin = 0.1 MeV, Emax = 10 MeV, and different power-law indices ). Single power-law distribution (for the same particle number density and = 4) is given for reference. · ParmIn[6] = E · ParmIn[7] = E
min max

[MeV]; [MeV];

· ParmIn[9] = ; · ParmIn[12] = nb [cm-3 ]; · ParmIn[17] = 7. Note that this distribution is not a power-law when expressed via the electron energy. And, vice versa, power-law distribution over energy becomes non-power-law when expressed via the electron momentum. Examples of the power-law distributions over momentum and the corresponding gyrosynchrotron emission spectra are shown in Figure 5.

7


Power-law distribution over Lorentz factor (PLG; index 8) Power-law distribution of the nonthermal electrons over Lorentz factor is given by the expression u( ) d = A
-

d for

min

<<

max

,

(13)

and 0 otherwise. The normalization constant A equals A= nb -1 1- 2 min -
1- max

,

(14)

where nb is the number density of nonthermal electrons, min = (Emin ), and max = (Emax ), the case of = 1 is not implemented. In our gyrosynchrotron codes, the parameters of this distribution are specified as: · ParmIn[6] = E · ParmIn[7] = E
min max

[MeV]; [MeV];

· ParmIn[9] = ; · ParmIn[12] = nb [cm-3 ]; · ParmIn[17] = 8. Again, this distribution is different from the power-law distributions over energy or momentum. Examples of the power-law distributions over Lorentz factor and the corresponding gyrosynchrotron emission spectra are shown in Figure 6. Thermal/nonthermal distribution over momentum (TNP; index 9) This distribution is similar to the thermal/nonthermal distribution over energy (index 5) with the only difference that the nonthermal part (at E > Ecr ) is described by the power-law distribution over the absolute value of momentum, that is u(p) dp = dp and 0 for expressed point and In our uTHM (p), for p < pcr , Ap- , for pcr p < p , (15)

max

p > pmax . In the above expression, uTHM (p) is the thermal distribution function (3) via momentum, pcr is given by Eq. (9), pmax = p(Emax ), and location of the matching the matching conditions are the same as for the TNT distribution. gyrosynchrotron codes, the parameters of this distribution are specified as:

· ParmIn[2] = T0 [K]; · ParmIn[3] = ; · ParmIn[7] = E
max

[MeV];

· ParmIn[9] = ; · ParmIn[11] = n0 [cm-3 ]; · ParmIn[17] = 9. An example of the thermal/nonthermal distribution over momentum and the corresponding gyrosynchrotron emission spectrum are shown in Figure 7. 8


Figure 7: Different thermal/nonthermal electron distributions (for n0 = 3 â 109 cm-3 , T0 = 3 â 107 K, = 0.1). All the distributions have different numbers of fast electrons above Ecr . Thermal/nonthermal distribution over Lorentz factor (TNG; index 10) This distribution is similar to the thermal/nonthermal distribution over energy (index 5) with the only difference that the nonthermal part (at E > Ecr ) is described by the power-law distribution over the Lorentz factor, that is u( ) d = d uTHM ( ), for < cr , A - , for cr < , (16)

max

and 0 for > max . In the above expression, uTHM ( ) is the thermal distribution function (3) expressed via Lorentz factor, cr = (pcr ), max = (Emax ), and location of the matching point and the matching conditions are the same as for the TNT distribution with index 5. In our gyrosynchrotron codes, the parameters of this distribution are specified as: · ParmIn[2] = T0 [K]; · ParmIn[3] = ; · ParmIn[7] = E
max

[MeV];

· ParmIn[9] = ; · ParmIn[11] = n0 [cm-3 ]; · ParmIn[17] = 10. An example of the thermal/nonthermal distribution over Lorentz factor and the corresponding gyrosynchrotron emission spectrum are shown in Figure 7. If the energy distribution index differs from the above values (2-10) then the single power-law distribution over energy (index 3) will be used.

9


Distributions over pitch-angle
Sample emission spectra for the different pitch-angle distributions (in Figures 8­9) were calculated under the assumption that the energy distribution is a single power-law (PLW) with nb = 3 â 107 cm-3 , Emin = 0.1 MeV, Emax = 10 MeV, and = 4. Isotropic distribution (ISO; index 1 or 0) In this case, the electron distribution does not depend on pitch-angle, that is 1 g (µ) = const = . 2 In our gyrosynchrotron codes, the parameters of this distribution are specified as: · ParmIn[19] = 1. For the library libGS Std HomSrc CEH: the use of indices 0 and 1 yields equivalent results, given all other parameters are identical. For the library libGS Std HomSrc C: the use of index 1 yields the result computed according to the new continuous code, while the use of index 0 activates the original (less accurate but the fastest) Petrosian-Klein code. Exp onential loss-cone distribution (ELC; index 2) Symmetric loss-cone distribution with exponential boundary is given by the expression for |µ| < µc , 1, |µ| - µ c g (µ) = A , for |µ| µc , exp - µ (17)

(18)

where µc = cos c > 0 is the loss-cone boundary, and the parameter µ determines the sharpness of the loss-cone boundary. The normalization factor A is given by A-1 = 2 µc + µ - µ exp µc - 1 µ . (19)

In our gyrosynchrotron codes, the parameters of this distribution are specified as: · ParmIn[19] = 2; · ParmIn[20] = c [degrees]; · ParmIn[22] = µ. An example of the exponential loss-cone distribution and the corresponding gyrosynchrotron emission spectrum are shown in Figure 8. Gaussian loss-cone distribution (GLC; index 3) Symmetric loss-cone distribution with gaussian boundary is given by the expression for |µ| < µc , 1, (|µ| - µc )2 g (µ) = A , for |µ| µc , exp - µ2

(20)

10


Figure 8: Exponential and gaussian loss-cone distributions (for c = 75 and µ = 0.3). The emission spectra are calculated for the propagation angle = 30 . where µc = cos c > 0 is the loss-cone boundary, and the ness of the loss-cone boundary. The normalization factor 1 -1 µ erf A = 2 µc + 2 parameter µ determines the sharpA is given by -µ µ
c

,

(21)

where erf is the error function. In our gyrosynchrotron codes, the parameters of this distribution are specified as: · ParmIn[19] = 3; · ParmIn[20] = c [degrees]; · ParmIn[22] = µ. An example of the gaussian loss-cone distribution and the corresponding gyrosynchrotron emission spectrum are shown in Figure 8. Gaussian distribution (GAU; index 4) Gaussian distribution is given by the expression g (µ) = A exp - (µ - µ0 )2 , µ2 (22)

where µ0 = cos 0 is the beam direction, and µ is the beam angular width. The above expression represents the beam along the field line for µ0 = ±1, the transverse beam for µ0 = 0, and an oblique beam (or a hollow-beam) otherwise. The normalization factor A is given by 1 - µ0 1 + µ0 -1 µ erf + erf . (23) A= 2 µ µ In our gyrosynchrotron codes, the parameters of this distribution are specified as: · ParmIn[19] = 4; · ParmIn[21] = 0 [degrees]; 11


Figure 9: Gaussian distribution (for 0 = 60 and µ = 0.5) and supergaussian distribution (for the same 0 and µ, and different values of the parameter a4 ). The emission spectra are calculated for the propagation angle = 60 . · ParmIn[22] = µ. If 0 = /2, this distribution coincides with GLC distribution with c = /2, otherwise they are different from each other. An example of the gaussian distribution and the corresponding gyrosynchrotron emission spectrum are shown in Figure 9. "Sup ergaussian" distribution (SGA; index 5) This distribution is very similar to the GAU distribution near its maximum (µ0 ) but decreases more rapidly at some angular distance from µ0 . Such a shape is achieved by adding a term with fourth degree of (µ - µ0 ) to the argument of exponent in (22), that is g (µ) = A exp - (µ - µ0 )2 + a4 (µ - µ0 )4 , µ2 (24)

where µ0 = cos 0 is the beam direction, and the beam angular width and shape near the maximum are determined by the parameters µ and a4 . The normalization factor A is calculated numerically by using normalization condition (2). In our gyrosynchrotron codes, the parameters of this distribution are specified as: · ParmIn[19] = 5; · ParmIn[21] = 0 [degrees]; · ParmIn[22] = µ; · ParmIn[23] = a4 . Examples of the "supergaussian" distribution and the corresponding gyrosynchrotron emission spectra are shown in Figure 9. If the angular distribution index differs from the above values (0­5) then the isotropic distribution (index 1) will be used; however, values above 100 should be avoided as they are reserved for future use and testing. 12