Numerical Code for Fitting Radial Emission Profile of a Shell Supernova Remnant

0.49 MB·English

by  Bojan Arbutina

Publ. Astron. Obs. Belgrade No. 91 (2011), 1 - 4 Poster
NUMERICAL CODE FOR FITTING RADIAL EMISSION
2
PROFILE OF A SHELL SUPERNOVA REMNANT
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0
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B. ARBUTINAand S. OPSENICA

n
Department of Astronomy, Faculty of Mathematics, University of
a
Belgrade, Studentski trg 16, 11000 Belgrade, Serbia
J
E–mail [email protected]
4 E–mail [email protected]

E] Abstract. Expressions for surface brightness distribution and for flux density have been
theoretically derived in the case of two simple models of a shell supernova remnant. The
H
models are: a homogenous optically thin emitting shell with constant emissivity and a
. synchrotronshell sourcewith radial magnetic field. InteractiveDataLanguage (IDL) codes
h
forfittingtheoreticallyderivedemission profilesassumingthesetwomodelstomeanprofiles
p
of shell supernovaremnantsobtained from radio observations havebeen written.
-
o
r
t
s 1. MODELS OF EMISSION OF SHELL SUPERNOVA REMNANTS
a
[ 1. 1. HOMOGENOUS OPTICALLY THIN EMITTING SHELL WITH CONSTANT EMISSIV-

1 ITY
v
In this paper, we investigated two models of emission of shell supernova remnants
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(SNRs). If we consider homogenous emitting shell with emissivity ε = const, for
5 ν
8 specific intensity, if the medium is optically thin, we have
0
1. Iν = ενds= εεν((rr2′′+−rr1′′+)),+εν(r1′−−r2′−), 0R−<∆sinθsi<nθR−d∆R, (1)
0 Z ( ν 2+− 2− d ≤ ≤ d
2
′
1 where ds=dr . Cosine theorems (see Fig. 1)
:
v (R ∆)2 =d2+r′2 2dr′ cosθ, (2)
i − 1 − 1
X R2 =d2+r′2 2dr′ cosθ, (3)
r 2 − 2
a give us
r′ =dcosθ (R ∆)2 d2sin2θ, (4)
1± ± − −
q
r′ =dcosθ R2 d2sin2θ. (5)
2± ± −
Finally, we have p
C sin2θ sin2θ sin2θ sin2θ , 0<θ <θ
ν 2 1 1
I = − − − (6)
ν
( Cν(cid:16)psin2θ2 sin2θ, p (cid:17) θ1 θ θ2,
− ≤ ≤
p
1
S. OPSENICA and B. ARBUTINA
Figure 1: Radiation from an optically thin homogenous shell with thickness ∆ and
′ ′
radius R, at the distance d from the observer. r and r are the distances from
1 2
intersection of the line of sight for a given θ with the inner and outer radius of the
shell, respectively.
where θ = arcsinR−∆, θ = arcsinR and C = 2ε d. From the last equation one
1 d 2 d ν ν
can see that I0 = 2ε Rδ and Imax = 2ε R δ(2 δ), where δ = ∆/R. Brightness
ν ν ν ν −
distribution i.e. specific intensity in units 2ε d givenby equation (6), for δ =∆/R=
ν
p
0.1 and R/d=0.01, can be seen in Fig. 2.
For the total flux density we have
2π θs
S = I cosθsinθdθdϕ
ν ν
Z0 Z0
θ1
= 4πε R2 d2sin2θ (R ∆)2 d2sin2θ cosθsinθdθ
ν
− − − −
Z0 (cid:16)p q (cid:17)
θ2
+ 4πε R2 d2sin2θcosθsinθdθ. (7)
ν
−
Zθ1
p
After integration we obtain the expected result
4π R 3 R ∆ 3 ε V V L
ν ν ν
S = ε d − = = E = , (8)
ν 3 ν d − d d2 4πd2 4πd2
h(cid:16) (cid:17) (cid:16) (cid:17) i
where the shell volume is V = 4πfR3, f = 1 (1 δ)3 is the volume filling factor,
3 − −
=4πε is total volume emissivity (ε is emissivity per unit solid angle) and L is
ν ν ν ν
E
luminosity.
1. 2. SYNCHROTRONSHELLSOURCEWITH RADIALMAGNETICFIELD
If we have a synchrotron shell source with radial magnetic field, emission coefficient
is ε (Bsinθ′)α+1ν−α i.e.
ν
∝ ε =ε˜ (sinθ′)α+1. (9)
ν ν
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RADIAL EMISSION PROFILE OF A SHELL SUPERNOVA REMNANT

0.005
0.004
0.003
I

2 d
0.002
0.001
0.000
0.000 0.002 0.004 0.006 0.008 0.010 0.012
sin
Figure 2: Brightness distribution for an optically thin homogenous shell-like source
with δ =∆/R=0.1 and R/d=0.01.
Sine theorem (see Fig. 1) gives us
′
r sinΘ
′
= , Θ=θ θ, (10)
d sinθ′ −
i.e.
′ ′
r =d(cosθ sinθ cotθ ) (11)
−
and
dθ′
′
ds=dr =dsinθ (12)
sin2θ′
Intensity is then
I = ε ds=ε˜ dsinθ (sinθ′)α−1dθ′ (13)
ν ν ν
Z Z
i.e.
2C sinθ µ2− 1 µ2 α−22 dµ, 0<θ <θ
I = ν µ1− − 1 (14)
ν  Cνsinθ µRµ22+− 1(cid:0)−µ2 (cid:1)α−22 dµ, θ1 ≤θ ≤θ2,
where µ=cosθ′, µ1,2± =∓√siRn2siθn1θ,21(cid:0)−,2sin2θ a(cid:1)nd Cν =ε˜νd.
RatherthandirectintegrationwewillfindfluxdensitythroughS = Lν = EνV ,
ν 4πd2 4πd2
where
2π π π
= ε dω′ = ε˜ (sinθ′)α+1sinθ′dθ′dϕ=2πε˜ (sinθ′)α+2dθ′ (15)
ν ν ν ν
E
Z4π Z0 Z0 Z0
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S. OPSENICA and B. ARBUTINA

0.004
0.003
I
0.002
2 d
0.001
0.000
0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014
sin
Figure 3: Radial profile with δ = ∆/R = 0.1 and R/d = 0.01 convolved with a
Bessel function with HPBW=5 10−4 rad.
×
i.e. =2π√πΓ(α+23)ε˜ andtheshellvolumeisasbeforeV = 4πfR3,f =1 (1 δ)3.
Eν Γ(α+4) ν 3 − −
2
2. IDL CODES
2. 1. DIRECT PROBLEM: SIMULATION OF OBSERVATIONS OF SHELL SUPERNOVA
REMNANTSWITHRADIOTELESCOPE
If we observe a shell SNR with a radio telescope, picture we get is a convolution of
real intensity of radiation of the SNR and power pattern of the telescope, so we get
”convolved” intensity (Fig 3). When simulating this convolution numerically, one
must choose an expressionfor real intensity and an expressionfor power pattern of a
radio telescope. In our case, expressions for intensities are (6) and (14), according to
the models. ForpowerpatternP (θ),twopossiblecaseshavebeenchosen: Gaussian
n
approximation and approximation with Bessel function of the first kind. Each pat-
tern has defined half power beam width (HPBW). Usually one takes HPBW=1.02λ
D
from Bessel function approximation (D is diameter of radio telescope and λ is wave-
length, see Rohlfs and Wilson 1996, Uroˇsevi´c and Milogradov-Turin 2007). Because
of technical limitations, we must consider that power pattern takes zero value for
angles greater than some critical angle θ . In the case of Gaussian approximation of
c
power pattern, θ of 5 sigmas (σ = HPBW/(2√ln2)), while for the approximation
c
with Bessel function θ of 8 HPBW has been chosen.
c
Expression that is used for numerical simulation of convolution of intensity of
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RADIAL EMISSION PROFILE OF A SHELL SUPERNOVA REMNANT
radio emission from a SNR and power pattern of a radio telescope is:
′
I (θ)P (θ )sinθdθdϕ
Iconv(θ )= intersection ν n . (16)
ν 0 R R 2π 0θcPn(θ)sinθdθ
′ R
Angleθ isrelatedtootherangularparametersthroughfollowingrelationofspherical
trigonometry:
′
cosθ =cosθcosθ +sinθsinθ cosϕ. (17)
0 0
Region of double integration in numerator of the expression (16) is the intersection
between regions where two convolving functions I (θ) and P (θ) are defined. That
ν n
double integration is performed by the IDL function INT 2D. Integration in denomi-
nator of the expression (16) is performed by the IDL function QROMB.Integrations in
theexpression(14)areperformedbytheIDLfunctionsQROMBandINT TABULATED,as
well as by ”handwritten” function that calculates definite integrals using rectangular
method.
In the case of first model (with constant emissivity), user of the program enters
parameters C , θ , θ , as well as parameter of antenna HPBW, and the program
ν 1 2
performsaconvolution. Inthecaseofsecondmodel(withradialmagneticfield),user
also enters an additional parameter of object - spectral index α.
2. 2. INDIRECT PROBLEM: FITTING MODEL TO OBSERVED PROFILE OF A SUPER-
NOVAREMNANT
Indirectproblemisthefollowing: userentersobservedradialemissionprofileofashell
SNRintheformofatable,aswellasthe parameterHPBW,andthe programshould
find the best values for parameters C , θ and θ in the case of first model, or C ,
ν 1 2 ν
θ , θ and α in the case of second model, by fitting the chosen model to the entered
1 2
data. This is performed by the iterative IDL procedure CURVEFIT. To perform this
procedure, user has to estimate initial values for the parameters. That can be done
using observed radial profile and the expression (6) or (14). Initial value for spectral
index α can be taken to be 0.5. This parameter is, however, better to be held fixed
(assuming that it is known from spectra). In addition to finding the best values of
the parameters, the program also calculates their errors (i.e. standard deviations).
Finally, the program calculates the flux density of an SNR.
The program has been tested with artificially generated data. Results of the
application of the program to the real data will be given elsewhere.
Acknowledgements. Duringtheworkonthispapertheauthorswerefinancially
supportedbytheMinistryofEducationandScienceoftheRepublicofSerbiathrough
the projects: 176004 ’Stellar physics’ and 176005 ’Emission nebulae: structure and
evolution’.
References
Rohlfs, K., Wilson, T. L.: 1996, Tools of Radio Astronomy, Springer, Berlin.
Uroˇsevi´c, D., Milogradov-Turin, J.: 2007, Teorijske osnove radio-astronomije, Matematiˇcki
fakultet, Beograd.
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