Accurate Methods for
Determining the
Parameters of Radio Pulse
Propagation Medium
A.I. Shevtsovaa
a
Department of Astrophysics, Institute of Radio Astronomy of NAS of
Ukraine Krasnoznamennaya str.4, Kharkov 61002, Ukraine
E-Mail: ashevtsova@rian.kharkov.ua
Supervisor: O.M. Ulyanov a
E-Mail: oulyanov@rian.kharkov.ua
MAIN GOAL:
To investigate the properties of the propagation
medium of the pulsar radio emission using its
polarization characteristics
PLAN :
-
Formulation of the direct problem;
Modeling the polarized signal from the source;
Modeling the propagation medium;
Simulation of the process of signal propagation in
the medium and the process of the signal registration
in the laboratory frame;
- Estimation of the propagation medium parameters;
- Interpretation of the results.
2
Simulation of the Pulse Signal with the Elliptical
Polarization
⃗
Ė 0x (t , i , z) = Ai⋅e −i ( k( ω )⃗z−2 π f
i
i
t+ φ i )
{
−i ( ⃗
k (ω i )⃗z−2 π f i t − π / 2+ φ i )
Ė 0y (t ,i , z) = Bi⋅e
Ė 0z (t , i , z ) = 0
Ė x (ψ , i , z ) = G 1 (ψ ,a , σ G1 )⋅(1−G 2 (ψ , a , σ G2 ))⋅Ė 0x (ψ , i , z )+Nx i (ψ )
Ė y (ψ , i , z) = G1 (ψ , a , σ G1)⋅Ė 0y (ψ , i , z)+Ny i (ψ )
Ė z (ψ ,i , z ) = Nz i (ψ )
RLC
[
][
{
B
A
ε=
]
Ė x (ψ ,i , z )
cos( χ ( ψ )) −sin( χ (ψ )) 0 Ė x (ψ ,i , z )
RLC
Ė y (ψ ,i , z ) = sin( χ (ψ )) cos( χ (ψ )) 0 ⋅ Ė y (ψ , i , z )
0
0
1 Ė z (ψ , i , z)
Ė RLC
(ψ ,i , z )
z
1-G2
B
A
ε
G1
χ (ψ ) = arctg
RLC
RLC
(ψ )
( −a)
m
3
Layered Model of the Propagation
Medium
⃗ rec
Ė
I
RLC
Dn, Rn
n
s
⃗ rec =∏ D R ⃗
Ė
Ė
i i
i
Di
Ri
}
RLC
I
...
⃗s
Ė
D1, R1
(
⃗ s=
Ė
n
∏ D i Ri
i
−1
)
⃗ rec
⋅Ė
The Matrix of the Dispersion and Rotation Phase Shifts
(Dispersion Delay and Rotation Measure Matrix)
- is the cross-section «radius of the light cylinder»
- is the cross-section «free space under Ionosphere»
4
The Propagation Medium Model
Equation of the
Wave Propagation:
[ ]
Ė Ix (ω )
Ė RLC
(ω )
x
*
Ė Iy (ω ) = D × R × Ė RLC
(ω )
y
Ė Iz ( ω )
Ė RLC
(ω )
z
I
⃗
⃗ RLC
Ė = [ K̇ ]⋅Ė
Eikonal Equation
∇ φ Ο , Χ (ω )=n Ο , Χ (ω ) ⃗
k (ω )
φ Ο , Χ (ω , ψ ) ≈ ω
D=
[
−i φ
e
DM
L
DM
RM
− φ
( ω ,ψ ) ∓ φ O , X ( ω , ψ )
c
(DM , ω )
0
0
0
−i φ DM ( DM , ω )
e
0
[
Matrix of the
Faraday Rotation R=
[
1 1
1
R =
−i i
√2 0 0
*
]
e
0
0
1
]
RM
0
e
iφ
RM
X
(RM , ω )
0
[
2
ωp
n Χ ( ω )= 1−
ω ( ω +ω H )
Matrix of the
Dispersion
Delay
i φ O (RM , ω )
0
0
√
√
2
ωp
n Ο ( ω )= 1−
ω ( ω −ω H )
0
1 i
1
0 ×R×
1 −i
√2 0 0
√2
0
0
√2
]
0
0
1
]
The model
value of the
Rotation Measure
Transition from linear to circular
coordinate system and back
5
Taking into Account the Reflective Surface
[
]
[ ]
I
Ė REC
(
ω
,
ψ
)
Ė
x'
x (ω , ψ )
REC
I
Ė y ' (ω , ψ ) = S eff ×COS ×(UN + REF )× Ė y (ω , ψ )
I
Ė REC
(
ω
,
ψ
)
Ė
z'
z ( ω ,ψ )
[
−ρ̇ p ( ε̇ , σ , ω ) e−i ϕ
REF =
0
0
0
ρ̇ s (ε̇ , σ , ω )e−i ϕ
0
0
0
ρ̇ p ( ε̇ , σ , ω ) e−i ϕ
]
REF is the Fresnel
Coefficient Matrix
2h
cos( α ) is the phase shift upon reflection
c
h is the altitude of the dipole phase center above the ground surface
ϕ =ω
[ ]
1 0 0
UN = 0 1 0
0 0 1
is the unit matrix, describes the presence of an incident
wave
The direction cosine matrix, which determines
the projection of the transformed components of
the E vector to the new axis:
[
cos(∡ x ' x) cos (∡ x ' y) cos(∡ x ' z )
COS = cos (∡ y ' x) cos(∡ y ' y ) cos(∡ y ' z )
cos(∡ z ' x ) cos(∡ z ' y ) cos(∡ z ' z)
]
S eff =
[
√
A
S eff (α ,θ , ω )
0
S dH
eff ( α ,θ , ω )
0
0
√
0
B
S eff (α ,θ , ω )
0
dH
S eff ( α ,θ , ω )
0
√
C
S eff ( α , θ , ω )
dV
eff
S ( α ,θ , ω )
]
6
Polarization Parameters in the Different Sections of
the Propagation Medium
RLC
I
REC
7
8
Faraday Rotation and
RM Estimation
2
( )
ψ −ψ Χ
2π c
ψ ( RM , ω )= Ο
=RM λ 2=RM
2
ω
RM = π2
c
[
f c2 ( f c+ Δ f F )
(
2
2
2
f c+ Δ f F ) − f c
]
3
f
c
≈ π2
c 2Δ f F
∆fF
3
fc
Δ f F ≈ π2
c 2 RM
RM = -11.7 rad/m2,
fc= 23.7 MHz,
Δf = 524 kHz.
RMEst= -11.62 rad/m2
Errors of the method:
δRM ≤ 0.5 ·100/nmax%
Errors in this case:
1.79%.
Δ ψ =π
Errors in the
determination of
RM
fc=23.7 MHz
ψ
˙ (t ,ψ ) = FFT [∣E˙ x (ω ,ψ )∣2 ]
Pw
The Most Precise Definition of the
RM and Polarization Parameters of
Radio Emission in the Observer
Frame of Reference
Ellipticity
Position Angle Trend
RM Estimations
Errors of the methods
9
The Largest Radio Telescope at Decameter
Wavelengths UTR-2
UTR-2
ΔF: 8–33 MHz,
Δλ: 9.1 - 37,5 m
10
The Data Analysis
Polarization Parameters of the Single Pulses of PSR J0814+7429
(PSR B0809+74), (fc=23.7 MHz, Δf=1.53 MHz )
Δ RM rad / m =RM Est −RM ATNF
2
δ RM =0.5⋅100 / nmax
δ RM rad / m =δ RM ⋅RM ATNF / 100
2
11
The Accurate Results of Data Analysis of Radio Emission. 12
Single Pulse of the PSR B0809+74 and New Estimations of
the RM and Polarization Degrees
L = √ Q 2 ( ω , ψ )+U 2 (ω , ψ )
PSR J0953+0755 (PSR B0950+08)
UTR-2 Observations 22 February 2013 ; Δf = 18-28 MHz.
t
13
Revised Values of the Rotation Measure and
Position Angle for PSR B0950+08
14
Visible value of the Rotation Measure for PSR
B0950+08 in the Different Frequency Ranges
* - M.P. van Haarlem, M.W. Wise, A.W. Gunts, G. Heald, et al., LOFAR: The Low Frequency Array.
arXiv: 1305.3550v2 [astroph.IM] 19 May 2013
** - Johnston, S., Hobbs, G., Vigeland, S., Kramer, M., Weisberg, J. M. & Lyne, A. G., 2005. Evidence for alignment of the rotation and velocity vectors in pulsars. MNRAS, 364, 1397-1412.
15
Conclusions :
- It is shown that the use of the eikonal equation
can describe the propagation medium via commuting
matrices, one of which describes the dispersion delay
at different frequencies, and the second - the rotation
of the polarization plane.
- New more accurate method for RM estimation at
decameter wavelength is proposed. The most
accurate value of the Rotation Measure for PSRs
B0809+74 and B0950+08 are detected.
- Probably that the theory that emission at different
frequencies is generated at different distances from
pulsar surface is confirmed.
15
Thank You !
The End
The Relative Orientation of the Magnetic Field in the
Magnetospheres of the Pulsars and in the Interstellar Medium
The value of the angles β between the magnetic axis B and the axis of rotation
Ω were taken from the book Malov IF, Radio Pulsars - Moscow: Nauka, 2004
15
The Largest Radio Telescopes in the
Decameter Wavelengths UTR-2 and URAN-2
UTR-2
URAN-2
A1
Variations of the RM in the Earth's Ionosphere
IRI 2007
day
IGRF 2011
night
Coordinates of RT
UTR-2:
49° 38' 17.6" N.Long.
36° 56' 28.7" E.Lat.
Estimations for Night :
∆DM = 0.97 ·10-6 , pc/cm3
∆RM= 0.29 rad/m2
Estimations for Day:
∆DM = 4,3 ·10-6 , pc/cm3
∆RM= 1.34 rad/m2
A2
Determination of the Signal Polarization Parameters
Polarization Tensor:
[
] (
I xx ( ω , ψ ) I xy (ω , ψ )
c ⟨ Ė x ( ω , ψ )⋅Ė x ( ω ,ψ )⟩ ⟨ Ė x ( ω , ψ )⋅E˙ y (ω , ψ )⟩
J (ω ,ψ )=
=
I yx ( ω ,ψ ) I yy ( ω , ψ ) 4 π ⟨ Ė y ( ω ,ψ )⋅E˙ x (ω ,ψ )⟩ ⟨ Ė y ( ω ,ψ )⋅E˙ y ( ω , ψ )⟩
)
Stokes Parameters:
I (ω ,ψ ) = I xx (ω ,ψ )+ I yy ( ω , ψ )
Q( ω ,ψ ) = I xx ( ω ,ψ )−I yy ( ω , ψ )
U ( ω , ψ ) = I xy ( ω , ψ )+I yx (ω , ψ )
V ( ω ,ψ ) = i (I yx ( ω ,ψ )−I xy (ω ,ψ ))
I ( ω ,ψ ) = I rr ( ω , ψ )+I l l ( ω , ψ )
Q( ω , ψ ) = I rl (ω ,ψ )+ I lr ( ω , ψ )
U (ω , ψ ) = −i ( I rl (ω , ψ )− I lr ( ω , ψ ))
V ( ω , ψ ) = I rr (ω , ψ )− I l l ( ω , ψ )
Parameters of the Polarization Ellipse:
Relative Polarization Parameters
( polarization degrees):
1
arg (Q(ω , ψ ) + iU (ω , ψ ))
2
V (ω , ψ )
1
ξ (ω , ψ ) = arcsin[
]
2
2
2
2
(Q
(
ω
,
ψ
)+U
(
ω
,
ψ
)+V
(
ω
,
ψ
))
√
χ (ω , ψ ) =
P c (ω ,ψ ) =
P l ( ω ,ψ ) =
ε (ω , ψ ) = tan( ξ (ω , ψ ))
P t ( ω ,ψ ) =
A3
√P
2
t
2
V (ω , ψ )
I ( ω ,ψ )
√Q 2 (ω , ψ )+U 2(ω ,ψ )
I ( ω ,ψ )
√ Q 2 (ω , ψ )+U 2 (ω , ψ )+V 2 (ω , ψ )
I (ω , ψ )
2
2
(ω , ψ )+P l (ω , ψ )+P c (ω , ψ )+P d (ω , ψ ) = 1
Evaluations of Variations of the Propagation
Medium Parameters
A4
A5
PSR B1133+16
A6
A7
PSR В0950+08
O.M. Ulyanov, V.V. Zakharenko Energy of Anomalously
Intense Pulsar Pulses at Decameter Wavelengths //
Astronomy Reports, 2012, Vol. 56, No. 6, pp. 417–429.
A8
Fresnel Reflection Coefficients
for «S» and «P» Polarizations
ε̇ sin (α )− √ ε̇ −cos 2 (α )
ρ̇ p (ε̇ , σ , ω ) =
ε̇ sin (α )+ √ ε̇ −cos 2 (α )
ρ̇ s (ε̇ , σ , ω ) =
sin (α )−√ε̇ −(cos 2 ( α ))
sin (α )+√ε̇ −(cos 2 ( α ))
Complex dielectric ground permittivity
ε̇ = Re [ ε̇ ] − i 60 σ λ (ω )
α — is the elevation angle
Ϭ - is the ground conductivity
√
2
4π e Ne
ω p=
me
ωH=
e∣ ⃗
B∣
me c
- is the plasma cyclic
frequency
- is the gyrotropic
cyclic frequency
Propagation the Ordinary (O) and Extraordinary (X) Waves
at Different Direction of the External Magnetic Fields
A9
Transformation Matrix from Linear to Circular Basis
[
1 i
0
Τ lc=
1 −i 0
√2 0 0 √2
1
[ ][
]
[
1 1 0
Τ cl =
−i i 0
√2 0 0 √2
1
] [
]
] [ ]
s
E˙ x rec (ω )
e −i α (DM , ω )
0
0
e−i φ (RM , ω )
0
0
Ė x ( ω )
−1
0
e−iα ( DM , ω ) 0 Τ lc
0
e−i φ ( RM , ω ) 0 ⋅Τ lc⋅ E˙ sy ( ω )
E˙ y rec ( ω ) =
0
0
1
0
0
1
0
Ė z rec ( ω )
Ο
Χ
A 10
Methods of Estimation of the Position Angle
The method of "Stokes parameters"
1
U
χ res = arctg ( )
2
Q
1
χ res= arg ( Q+iU )
2
t,sec
The method of "spectral response"
1
χ res =− arg ( Ṗ n ( nmax ) )
2
The method of "Jones vector"
t,sec
J̇ x , y = Ė y / E˙ x
χ res=−arctg
ℜ J̇ x , y
1−∣ J̇ x , y∣2
t,sec
A 12
Faraday Rotation and RM Estimation
2
ψ Ο −ψ Χ
2π c
2
ψ ( RM , ω )=
=RM λ =RM
2
ω
2
2
Δ ψ =π
1
1
2
( )
Δ ψ ( RM , f )=c RM
RM = π2
c
[
∆fF
fc=23.7 MHz
[( ) (
fc
−
2
f c +Δ f
f c ( f c+ Δ f F )
2
2
2
( f c+ Δ f F ) − f c
]
)]
Δ f =Δ f F
3
fc
≈ π2
c 2Δ f F
∆fF
3
f
c
Δ f F ≈ π2
c 2 RM
ψ
fc=30 MHz
ψ
A 13
Criterion a Quasi-Longitudinal
Propagation
M O,X
2 √ u⋅cos(∡ ⃗
k⃗
BI )
=
2
2
4
2
u⋅sin (∡ ⃗
k⃗
B I ) ± √ u ⋅sin (∡ ⃗k ⃗
B I ) + 4 u⋅cos (∡ ⃗
k⃗
BI )
Zheleznyakov V. V. 1996, Radiation in
Astrophysical Plasmas, Astrophysics and
Space Science Library, vol. 204, Kluwer
Academic Publishers, Dordrecht
(published in Russian in 1997)
ωH 2
e∣⃗
B∣
u=( ω ) =
me c ω
( )
Ellipticity for normal modes in the range of angles
between the magnetic field and the propagation
direction from 0 to π for frequency f = 23.7 MHz
(red curve) and f = 111 MHz (blue curve).
⟨∣ ⃗
B ISM∣⟩ ≈ 1 μ G
u ISM = 1.39⋅10−14
∣B⃗I∣=50419⋅10
−9
M O = −1/ M X
f = 23.7 МГц
∣⃗
B PSR∣ ~ 0.1 T= 1000 G
u PSR = 13950.64
T≈ 0.5 G
∣B⃗Iv∣=46311.6⋅10 T≈ 0.46 G
u I = 3.49⋅10−3
2
−9
A-14
Review of the Previous Works.
Komesaroff
1970
Radhacrishnan
and Cooke 1969
1.
Radhacrishnan V., Cooke D.J.,
Magnetic poles and the polarization
structure of pulsar radiation, 1969 ,
The Astrophysical Letters, Vol.3, pp
225-229
Goldreich and
Julian 1969
2.
Goldreich P. and Gullian W.H.,
Pulsar Electrodynamics, 1969, The
Astrophysical Journal, Vol. 157,
p.869-880
3.
Komesaroff M.M., Posible
Mechanism for the Pulsar Radio
Emission, 1970, Nature, V.225, pp.
612-614
A-15
Comparison of the Actual and
Model Spectra of the Individual
Pulses PSR B0950 +08
The Spectrum of the Polarization
Response of the Real Signal.
Spectrum of the response of model
elliptically polarized signal obtained using
the polarization parameters estimated at the
laboratory frame.
The Difference Spectrum of Real and
Model Signals
A-16
Thank You Indeed !
The End