Discovery of Kolmogorov-like Magnetic Energy Spectrum in Tycho's Supernova Remnant by Two-point Correlations of Synchrotron Intensity
Jiro Shimoda Takuya Akahori A. Lazarian Tsuyoshi Inoue Yutaka Fujita


firstpage–lastpage
Semi-regular Dubuc-Deslauriers wavelet tight frames
    Alberto Viscardi
    
===================================================





The spectral slope of the magnetic energy in supernova remnants (SNRs) can be obtained by
analysis of spatial two-point correlation functions of synchrotron intensities.
This method has been originally developed for the analysis of magnetic field structure
in diffuse interstellar medium and applied when the geometry of the emission region is simple and known.
In this paper, applying this correlation analysis to Tycho's SNR,
for which the synchrotron emission region is known to be a spherical shell,
we find that the magnetic energy spectrum shows the Kolmogorov-like scaling.
Our results can be explained by turbulence developed downstream of the shock front via
Richtmyer-Meshkov instability or
an amplification of upstream magnetic field induced by cosmic rays.
They could be discriminated by future observations with
a sub arcsecond resolution such as Square Kilometer Array.





ISM: supernova remnants—acceleration of particles—turbulence—magnetic fields—(magnetohydrodynamics) MHD—shock waves








§ INTRODUCTION

Supernova remnant (SNR) shock waves are believed to be an accelerator of Galactic cosmic-rays (CRs)
with energies at least up to 10^15.5  eV (called “knee energy”).
However, there is no firm evidence that the SNRs are accelerators of the knee-energy CRs.
In the standard scenario, the CR particles are accelerated 
through the “diffusive shock acceleration” (DSA) mechanism <cit.>
which is accompanied by simultaneous generation of magnetic-field disturbances
at the vicinity of the shocks <cit.>.
In the DSA mechanism, CR particles are scattered
through interactions with the field disturbances
to go back and forth between upstream and downstream of the shock,
suffering the shock heating repeatedly and increasing their energy.
If the mean free path of the accelerated particles is sufficiently larger than
a radius of the SNR,
they escape from the SNR shock and the acceleration is finished.
Thus, the maximum energy of the accelerated particles depends on their diffusion coefficient.
If we consider a monochromatic field disturbance with scale length l,
a CR particle with gyroradius r_g≃ l interacts resonantly with
the monochromatic field disturbance
resulting in a pitch-angle scattering <cit.>.
The strength of the scattering depends on an energy density of the magnetic field disturbance.
Because the field disturbance
usually has a continuous energy spectrum due to a turbulent cascade,
the CR particles with different energies resonate with the field disturbances at different scale lengths.
Hence,
the diffusion coefficient
and the maximum energy are related to the spectrum.
For the field disturbances with an energy spectrum proportional to l^m,
the diffusion coefficient of the particles with energy E can be written as



    κ(E)∼κ̃(E_0)(E/E_0)^1-m,



where E_0 is the energy of the CR particle giving the representative diffusion coefficient
κ̃(E_0)=cr_g(E_0)/3 <cit.>.
c is the speed of light.
<cit.> evaluated the maximum energy of CR protons as a function of the spectral slope m
for a number of SNRs
(Cas A, Kepler, Tycho, SN 1006 and G347.3-0.5) by deriving the magnetic-field strength
from the thickness of non-thermal X-ray filaments.
For Tycho's SNR, if m∼2/3 (corresponding to Kolmogorov-like turbulence),
the maximum energy of CR protons reaches around the knee energy.
[γ-ray emissions from Tycho's SNR may originate in high energy CR protons
with an energy at least ∼10^14  eV
but there is no consensus on the maximum energy of CR protons because it depends on emission models
 <cit.>.]
While the slope of the magnetic energy spectrum should be related to the maximum energy of CRs,
it has not been well determined observationally.
Moreover, there is a controversy over the shape of the magnetic energy spectrum
realized in the SNRs
such as a single power-law <cit.>,
a broken power-law <cit.>
and a spectrum containing several discrete peaks <cit.>.
The uncertainty of the spectrum prevents us from determining the maximum energy based on
the standard scenario.
Therefore, it is important to obtain the spectral slope observationally to reveal
acceleration process of CRs at SNRs.
It is widely recognised for interstellar medium (ISM)
that the two-point correlation function of synchrotron intensities reflects
statistical nature of turbulent magnetic field including the spectral slope of magnetic energy
 <cit.>.
This correlation analysis, however, suffers from the uncertainty of the geometry of the emission region
because of the projection effect.
In other words, the correlation function reflects not only the structure of the magnetic field
but also the geometry of the emission region.
Therefore, the geometry must be determined separately to obtain the spectral slope
of magnetic energy.
Fortunately, the emission region of young SNRs is often a spherical shell <cit.>.
Thus, if we select arbitrary points on a concentric circle of an SNR image,
the depth of the emission region along the line of sight through the points is constant.
This means that we can exclude the geometrical effect from the correlation function.
As a result, the derived correlation function depends only on the spectral slope of magnetic energy.
In this paper, we explore the spectral slope of magnetic energy in Tycho's SNR
by applying the correlation analysis.
This paper is constructed as follows.
In section <ref>, we briefly explain our analysing method and
the application for Tycho's SNR.
The results are shown in section <ref>,
and we discuss the origin of magnetic field structure in Tycho's SNR in section <ref>.








§ ANALYSIS OF MAGNETIC FIELD CORRELATION


To extract magnetic field correlation,
we consider the second-order correlation function of
the synchrotron intensity per frequency I_ν
on the circle S centred on the SNR centre.
The function is given by



    C^(2)_I_ν,S(λ)    =   ∫_S I_ν(X)I_ν(X') d^2X/∫_Sd^2X
       ≡   ⟨ I_ν(X)I_ν(X+λ) ⟩_X,S,




where X=(x,y) is the two-dimensional sky position and
λ=X'-X is the position vector of two separated
sky positions X and X'=X+λ
 (see Appendix <ref> for detail).
Here we select X and X'=X+λ from the region



    S(X,R) = {X | (R-δ)^2≤ f(x,y)≤ (R+δ)^2
    },



with f(x,y)=(x-x_c)^2+(y-y_c)^2,
where (x_c,y_c) is the centre of the SNR,
R is the radius of the circle we are interested in and
δ≪ R is the width.
Note that although the correlation function C^(2)_I_ν,S(λ) is defined in a two-dimensional space,
it is mostly represented as one-dimensional function owing to the condition δ≪ R,
that is, the domain of definition of C^(2)_I_ν,S(λ)
is much elongated in the azimuthal direction.
For Kolmogorov-like turbulent field, the one-dimensional correlation function
shows the scaling relation of λ^2/3 <cit.>.
We study Tycho's SNR using the correlation function.
We analyse a 1.4  GHz image, which is published in <cit.>,
obtained by Very Large Array: project VLA/13A-426 (PI J. W. Hewitt).
Figure <ref> shows the image.
The synthesized beam size (angular resolution) is 1.92  arcsec and the image pixel size is 0.4  arcsec.
The image noise level is σ_ Tycho=5.3×10^-5  Jy beam^-1.
Supposing that the distance
to Tycho's SNR is d=4  kpc <cit.>,
1  arcsec≈0.02  pc(d/4  kpc).
We calculate the intensity centroid from pixels with I_ν≥3σ_ Tycho at the rim of SNR,
and set the centre of concentric circles at the centroid:
(R.A.,Dec.)=(0^ h25^ m19^ s.1,+64^∘08'23”.0).
It mostly agrees with the geometrical centre derived from the X-ray image <cit.>:
 (R.A.,Dec.)=(0^ h25^ m19^ s.9,+64^∘08'18”.2).
We analyse eight concentric circles with radii,
R=1.00R_ SNR (white),
0.97R_ SNR (red),
0.94R_ SNR (purple),
0.91R_ SNR (green),
0.88R_ SNR (blue),
0.85R_ SNR (light blue)
0.82R_ SNR (yellow)
and 0.79R_ SNR (orange),
where R_ SNR = 632  pixels≈ 253  arcsec≈ 5  pc is the SNR radius.
The width of each circle is set to be δ=0.0091R_ SNR.
Errors on the correlation function are evaluated from the lower and upper limits of C^(2)_I_ν,S.
There are pixels having weak signals of I_ν<3σ_ Tycho.
We regard such weak signals as noise.
Meanwhile,
if we assign a pseudo-signal of 3σ_ Tycho to those pixels,
we obtain the maximum value of the correlation function.
We regard it as the upper limit.
Similarly, if we assign zero to those pixels,
we obtain the minimum value as the lower limit.
We regard these limits as the errors of C^(2)_I_ν,S.








§ RESULTS







Figure <ref> shows the second-order correlation functions C^(2)_I_ν,S.
The line width indicates the errors evaluated from the upper and lower limits of C^(2)_I_ν,S.
In order to clarify the correlation of fluctuating component,
we display |C^(2)_I_ν,S(λ)-C^(2)_I_ν,S(λ_ min)|,
where λ_ min≈1.4 arcsec is the minimum separation distance between X and X'.
For the outer circles with radii R≥0.90,
C^(2)_I_ν,S follows a power law and has a positive slope
at relatively small scales (λ/R  0.5),
i.e. larger-scale magnetic field disturbances are predominant.
The slope is close to the Kolmogorov scaling λ^2/3.
Such a single power-law with the Kolmogorov scaling is predicted by <cit.> (henceforth, GS95) as
the developed magnetohydrodynamics (MHD) turbulence.
On the other hand, for the inner circles with radii R≤0.85,
C^(2)_I_ν,S are somewhat flatter than the outer ones at relatively large scales λ/R0.2,
although a Kolmogorov-like scaling is implied in the small scales λ/R0.2.
It may indicate that the nature of
field disturbances varies at R 0.9 R_ SNR.
Actually,
by the combination of measurements of X-ray imaging and spectroscopy,
<cit.>
found that the contact discontinuity,
at which Rayleigh-Taylor instability (RTI) works,
is located at R≈0.9 R_ SNR.





§ DISCUSSION


The correlation functions for R 0.9 R_ SNR imply the developed GS95 turbulence that is trans-Alfvénic.
If this is the case, the velocity dispersion of the largest eddy, u_ inj,
should be close to the Alfvén velocity, C_ A,
and it gives the Alfvén Mach number of turbulence, M_ A,turb≡ u_ inj/C_ A≈ 1.
For comparison, if we consider the SNR shock velocity 
<cit.>
and the magnetic field strength in the ISM <cit.>,
we obtain a high Alfvén Mach number of ∼500 for the shock.
Thus, in comparison with the shock, our results imply
smaller gas velocity and/or larger magnetic-field strength at R 0.9 R_ SNR.
The non-thermal X-ray filaments with the thickness ∼0.01 R_ SNR seen in Tycho's SNR imply
a significant cooling of high energy CR electrons,
suggesting the presence of a strong (likely amplified) magnetic-field <cit.>.
We discuss how the condition M_ A,turb≃1 is satisfied at the vicinity of the shock.
Multidimensional MHD simulations <cit.> showed that
the SNR shock is rippled owing to the interaction with
density fluctuations pre-existing in the ISM <cit.>.
Because of the shock rippling,
the velocity component tangential to the shock surface is generated downstream,
yielding the velocity dispersion just behind
the shock <cit.>.
Using three-dimensional MHD simulations,
<cit.> showed that the strength of the downstream velocity dispersion Δ u
can be expressed by using growth velocity of the Richtmyer-Meshkov instability (RMI) u_ RMI:



    Δ u≃ u_ RMI≃ A ⟨ u_ sh⟩,



where A=(Δρ/⟨ρ⟩)/(1+Δρ/⟨ρ⟩) is the Atwood number
and ⟨ u_ sh⟩ is the mean shock velocity.
Δρ and ⟨ρ⟩ are the dispersion of upstream density fluctuation and the mean upstream density, respectively.
They assumed a weak magnetic field in the upstream region,
M_ A=⟨ u_ sh⟩/C_ A,1≃100,
where C_ A,1 is the upstream Alfvén speed,
and obtained super-Alfvénic turbulence (i.e. M_ A,turb>1) behind the shock (though the result depends on Δρ, see <cit.> for details).
The downstream magnetic field is amplified by the turbulent dynamo process induced by the RMI-driven super-Alfvénic turbulence.
The amplified field is able to explain the orientations of observed magnetic fields
in young SNRs <cit.>.
The field amplification becomes significant at a distance



    d_ RMI≃ l_Δρ/(r_cA),



from the shock front <cit.>,
where l_Δρ is the scale length of upstream density fluctuations
and r_c is the shock compression ratio.
<cit.> examined
the ambient density of Tycho's SNR from infrared dust emissions
and found
order-of-magnitude variations in density at the scale length of
l_Δρ∼ R_ SNR.
Such density variations are also inferred by observations of the expansion rate of Tycho's SNR
 <cit.>.
Their results imply
A≈1 on the scale l_Δρ∼ R_ SNR,
giving d_ RMI∼0.2 R_ SNR(l_Δρ/R_ SNR)(r_c/4)^-1.
Thus, the magnetic field amplification through
the turbulent dynamo induced by the RMI-driven turbulence
may explain the condition M_ A,turb≃1
at R∼0.9 R_ SNR from the shock front
by this orders-of-magnitude estimation,
which is consistent with our results for Tycho's SNR.
Our results for the inner circles (R0.9 R_ SNR),
which show somewhat flatter spectra than the outer (R0.9 R_ SNR) ones at relatively large scales λ/R0.2,
may be ascribed
to the interaction between the well-developed GS95 turbulence and the RTI driven turbulence.
To examine this interpretation, MHD simulations solving the interaction between the well-developed GS95 turbulence
and the RTI driven turbulence are required.
We will study this in forthcoming paper.
There are other possibilities to amplify the magnetic field at the vicinity of the shock
such as the Bell instability <cit.>,
which is usually expected as the mechanism responsible for the magnetic field amplification
at the upstream region leading to the acceleration of knee-energy CR protons.
This instability occurs resulting from the interaction between leaking CRs from the shock and the background plasma.
If the upstream field has been already amplified, the trans-Alfvénic condition can be satisfied in the region just behind the shock.
This situation may be consistent with our results for Tycho's SNR.
Moreover, the leaking CRs excite an acoustic instability <cit.>.
The upstream plasma affected by the leaking CRs, i.e. shock precursor,
interacts with the density fluctuations pre-existing in the ISM.
This interaction also
leads to the field amplification by a turbulent dynamo process
in the upstream region <cit.>.
Indeed, <cit.> pointed out that
the shock crossing time of the precursor length
is large enough to lead to full development
of the precursor dynamo in partially ionized ISM.
Note that the Balmer line emissions from Tycho's SNR
indicate the interaction between the shock and the partially ionized ISM
 <cit.>.
The above possible amplification mechanisms predict different evolution tracks
of the magnetic energy spectrum.
For the RMI inducing amplification on the downside
(i.e. turbulent dynamo in super-Alfvénic turbulence),
magnetic field disturbances on the scales larger than l_ A,
at which the turbulent velocity is equal to the Alfvén velocity,
grow with time <cit.>.
It indicates that the l_ A
evolves toward a larger scale with increasing a distance from the shock front.
On the other hand, if the field has significantly been amplified upstream, such evolution would not be seen.
Moreover,
<cit.> pointed out that the field amplified by the Bell instability damps downstream.
Therefore, we can assess the real amplification mechanism from the spatial variation
of the magnetic energy spectrum.
The correlation for the most outer circle with the radius R=1.00R_ SNR
roughly shows the single power-law with the Kolmogorov-like scaling,
which favors an well-amplified field just behind the shock, i.e. the upstream field amplification.
It would be confirmed once the evolution track of spectrum within the most outer circle is resolved.
Supposing that the strength of amplified magnetic field is ∼100 μ G <cit.>,
the scale length of the gyroradius of the knee-energy CRs,
r_g, knee∼0.01  pc(E/10^15.5  eV)(B/100 μ G)^-1,
is shorter than the spatial resolution of the present data.
The scale length of r_g, knee will be resolved and the mechanism of field amplification may be distinguished,
if we have data of higher spatial resolution (say, 0.001R_ SNR).
Once we obtain the magnetic energy spectrum at the length scale less than r_g, knee,
we can estimate the diffusion coefficient of the knee-energy CRs and
the possibilities of their acceleration in SNR.
We thus need a higher sensitivity with sub arcsecond resolution at GHz band;
this would be a science case of the Square Kilometre Array (SKA).
Finally, our method would be available not only for SNRs
but also other astrophysical objects with a spherical-shell structure,
such as radio relics in galaxy clusters.








§ ACKNOWLEDGEMENTS






We are grateful Dr. Brian J. Williams who kindly provided us with
the VLA image used in this paper.
We thank Prof. Jungyeon Cho for valuable comments to complete this work.
We also thank the anonymous referee for his/her comments
to further improve the paper.
This work is supported by Grant-in-aids
for JSPS Fellows (15J08894, JS) and JSPS KAKENHI Grants: 15H03639,
15K17614, 17H0110 (TA), 15K05039 (TI), and 15K05080 (YF).







mnras



























§ MEASURING METHOD OF MAGNETIC FIELD CORRELATION


Here we provide the measuring method of magnetic energy spectrum in SNRs
by applying the method developed for the ISM.
<cit.> provided a mathematical formalism describing
how second order correlation functions of synchrotron intensities are related with magnetic field disturbances.
An emissivity of a synchrotron emission per frequency ν depends on the strength of magnetic field
component perpendicular to the line of sight, |B_n|=B_n, as



    i_ν(r)
       =   Kν^-α
    B_n(r)^1+α, 
       =   Kν^1-γ
    B_n(r)^γ,



where K is a function depending on the density of relativistic electrons,
α=(s-1)/2, s is the power-law index of the CR electron energy spectrum
and we have defined γ≡1+α for simplicity.
Thus, the second-order correlation function of the synchrotron emissivity,



    C^(2)_i_ν,γ(l)
       =   ⟨ i_ν(r)i_ν(r+l)  ⟩_r,



is related with the magnetic field correlation as



    C^(2)_i_ν,γ(l)∝⟨ B_n(r)^γB_n(r+l)^γ⟩_r.



When γ=1, which is equivalent to α=0 and s=1,
and spatial distribution of CR electrons is uniform (i.e. K is constant),
C^(2)_i_ν,γ becomes identical to the magnetic field correlation function.
The case of γ=2 (α=1 and s=3) is also simple and can be representative
for the case of arbitrary γ.
Omitting the notations as
B_n( r)→ B_n and B_n(r+l)→ B_n',
we obtain



    ⟨ B_n^2 B_n'^2 ⟩_r 
    =⟨ B_n^4+B_n'^4⟩_r/2
    -1/2⟨( B_n+B_n' )^2
    ( B_n-B_n' )^2
    ⟩_r.



If we decompose the magnetic field into the mean component B̅_n=⟨ B_n ⟩_r
and the fluctuating component Δ B_n=B_n-B̅_n, the above equation can be written as



    ⟨ B_n^2 B_n'^2 ⟩_r   =   ⟨ B_n^4+B_n'^4⟩_r/2
       -   2B̅_n^4⟨( 1+Δ B_n+Δ B_n'/B̅_n)^2
    ( Δ B_n- Δ B_n'/B̅_n)^2
    ⟩_r.



For small standard deviation of the field
√(⟨ (B_n -B̅_n)^2⟩_r)/B̅_n∼
|(Δ B_n+Δ B_n')/B̅_n|<1,
the correlation function of B_n^2 becomes



    ⟨ B_n^2 B_n'^2 ⟩_r   ≈   ⟨ B_n^4+B_n'^4⟩_r/2
    -2B̅_n^4 ⟨( Δ B_n- Δ B_n'/B̅_n)^2
    ⟩_r
       =    4B̅_n^2⟨Δ B_n Δ B_n' ⟩_r+ const.,



where we have assumed the isotropic turbulent field as
⟨ B_n^4 ⟩_r=⟨ B_n'^4 ⟩_r and
⟨Δ B_n^2 ⟩_r=⟨Δ B_n'^2 ⟩_r.
Thus, ⟨ B_n^2 B_n'^2 ⟩_r
reproduces the second-order correlation function of the magnetic field disturbances.
Note that even if we consider a completely random field,
this approximation would be applicable for small scales.
This is because
the field disturbances on the larger scales act as a guide field for
the field disturbances on the smaller scales <cit.>. 
<cit.> showed that



    ⟨ B_n^2 B_n'^2 ⟩_r/⟨ B_n^4 ⟩_r
    - ⟨ B_n^2 ⟩_r^2   ≈   ⟨ B_n^γ B_n'^γ⟩_r/⟨ B_n^2γ⟩_r
    -⟨ B_n^γ⟩_r^2 ,



for several γ for a power-law correlation function of B_n.
In the range of 1.2≤γ≤3, they reported that the maximum difference of ⟨ B_n^γ B_n'^γ⟩_r
from ⟨ B_n^2 B_n'^2 ⟩_r
is only 3%.
This suggests that the correlation functions can be written as



    ⟨ B_n(r)^γ B_n(r+l)^γ⟩_r≈ P(γ)⟨ B_n(r)^2B_n(r+l)^2⟩_r,



where P is a function of γ.
This argument is numerically confirmed by <cit.>.
They performed synthetic observations of synchrotron emissions from simulated magnetic field
and derived the Fourier power spectrum from the correlation of the observed synchrotron polarization intensity
for the parameter range 1.5≤γ≤4.
They found that the power spectrum reproduced
the spectral index of the given magnetic field.
For SNRs, observations of the radio synchrotron intensity per frequency show
the power-law spectrum with the index α≈0.6
that indicates γ≈1.6 and s≈2.2
<cit.>.
Hence, C_i,γ^(2) can reproduce
the second-order correlation function of
magnetic field disturbance in the SNRs.
In order to measure the second-order correlation function of
magnetic field disturbance from
the synchrotron emissions, we must consider the projection effect.
We define the observed intensity of synchrotron emission per frequency
at the two-dimensional sky position X=(x,y) as



    I_ν(X)=∫_0^L(X) Kν^1-γ
    B_n(X,z)^γ
    dz,



where z represents the coordinate along the line of sight and
L(X) is the extent of the emission region.
Note that L is a function of X in general.
The second-order correlation function for I_ν is written as



    C^(2)_I_ν(λ)
       =   ∫  I_ν(X) I_ν(X') d^2X/∫ d^2X
       ≡   ⟨ I_ν(X)I_ν(X+λ)  ⟩_X,



where λ≡X'-X is the position vector of two separated positions in the sky X and
X' (see also Eq. (<ref>) of the main text).
The correlation function can be represented as



    C^(2)_I_ν(λ)=Kν^1-γ∫_0^L(X) dz ∫_0^L'(X') dz'
    
    ×⟨ B_n(X,z)^γB_n(X+λ,z')^γ⟩_X.



For the constant L(X)=L_0, <cit.> and <cit.> demonstrated
that C^(2)_I_ν reproduces the scaling relation of
a given magnetic field correlation.
However, if L varies spatially, C^(2)_I_ν is affected by the geometrical structure of
the emission region, which is usually unknown.
However, fortunately, the emission regions of some young SNRs are known to be a spherical shell <cit.>.
Therefore, if we select the points X and X' on the concentric circle of SNR image,
the condition of L(X)= constant is satisfied.







In Figure <ref>, we show a schematic of the SNR shell and projected image.










