Analytical solution of Kompaneets equation
Analytical solution of Kompaneets equation, describing the movement of shock front from the strong point explosion in the medium with the density changing as the hyperbolic tangent, was obtained. Solution allows to restore all shock front and investigate its evolution for arbitrary values of the den...
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Головна астрономічна обсерваторія НАН України
2012
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| Цитувати: | Analytical solution of Kompaneets equation / A.V. Karnaushenko // Advances in Astronomy and Space Physics. — 2012. — Т. 2., вип. 1. — С. 39-41. — Бібліогр.: 7 назв. — англ. |
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Karnaushenko, A.V. 2017-06-04T17:32:44Z 2017-06-04T17:32:44Z 2012 Analytical solution of Kompaneets equation / A.V. Karnaushenko // Advances in Astronomy and Space Physics. — 2012. — Т. 2., вип. 1. — С. 39-41. — Бібліогр.: 7 назв. — англ. 2227-1481 https://nasplib.isofts.kiev.ua/handle/123456789/119149 Analytical solution of Kompaneets equation, describing the movement of shock front from the strong point explosion in the medium with the density changing as the hyperbolic tangent, was obtained. Solution allows to restore all shock front and investigate its evolution for arbitrary values of the density change and position of the explosion point. The solution is applied for description of interaction of supernova remnants with molecular clouds. en Головна астрономічна обсерваторія НАН України Advances in Astronomy and Space Physics Analytical solution of Kompaneets equation Article published earlier |
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Analytical solution of Kompaneets equation |
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Analytical solution of Kompaneets equation Karnaushenko, A.V. |
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Analytical solution of Kompaneets equation |
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Analytical solution of Kompaneets equation |
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Analytical solution of Kompaneets equation |
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Analytical solution of Kompaneets equation |
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analytical solution of kompaneets equation |
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Karnaushenko, A.V. |
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Karnaushenko, A.V. |
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2012 |
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Advances in Astronomy and Space Physics |
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Головна астрономічна обсерваторія НАН України |
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Article |
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Analytical solution of Kompaneets equation, describing the movement of shock front from the strong point explosion in the medium with the density changing as the hyperbolic tangent, was obtained. Solution allows to restore all shock front and investigate its evolution for arbitrary values of the density change and position of the explosion point. The solution is applied for description of interaction of supernova remnants with molecular clouds.
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2227-1481 |
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https://nasplib.isofts.kiev.ua/handle/123456789/119149 |
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Analytical solution of Kompaneets equation / A.V. Karnaushenko // Advances in Astronomy and Space Physics. — 2012. — Т. 2., вип. 1. — С. 39-41. — Бібліогр.: 7 назв. — англ. |
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2025-11-25T23:28:32Z |
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1850581040427433984 |
| fulltext |
Analytical solution of Kompaneets equation
A.V. Karnaushenko∗
Advances in Astronomy and Space Physics, 2, 39-41 (2012)
© A.V. Karnaushenko, 2012
Radio Astronomy Institute, National Academy of Sciences of Ukraine, Chervonopraporna st., 4, 61002, Kharkiv, Ukraine
Analytical solution of Kompaneets equation, describing the movement of shock front from the strong point
explosion in the medium with the density changing as the hyperbolic tangent, was obtained. Solution allows to
restore all shock front and investigate its evolution for arbitrary values of the density change and position of the
explosion point. The solution is applied for description of interaction of supernova remnants with molecular clouds.
Key words: ISM: clouds, ISM: supernova remnants, shock waves
introduction
Interaction of the supernova remnant (SNR) with
inhomogeneities of the interstellar medium (ISM)
and especially with molecular clouds (MC) is of great
interest. Interaction in�uences on the shape of the
remnant and its evolution. Also in the region of in-
teraction between SNR and MC, due to collisional
pumping, the hydroxyl maser emission appears at
1720 MHz [3]. Observation on this frequency allows
to detect the interaction of the SNR with MC by the
regular way [6]. The gamma emission [1] from SNR
interacting with MC has also a great interest due to
ability to throw light on the acceleration of the nu-
clear component of cosmic rays on the shock waves
in SNR.
Among various approaches to the solution of the
problem the special place is occupied by the Kom-
paneets equation (KE) [4] which gives a good qual-
itative description of the phenomenon. Analytical
solution of KE allows to restore shock front and in-
vestigate its evolution. In the given short message
the continuation of our previous work [2] with ana-
lytical expressions for the shock front movement are
presented.
kompaneets equation for
interaction of supernova
remnant and cloud
For the description of interaction of a shock wave
with a molecular cloud we chose the following density
distribution:
ρ(z) = ρ0 · (a− b · tanh(z/z∗)), (1)
where z∗ is the scale of inhomogeneity of the medium
between MC and ISM. At z → +∞ ρ(z) tends to
small but �nite density of the interstellar medium
ρISM ≡ ρ(+∞) = ρ0 · (a − b). At z → −∞
ρ(z) tends to the density of the molecular cloud
ρMC ≡ ρ(−∞) = ρ0 · (a + b). Parameter γ2 =
a− b
a + b
describes the density changes.
Chosen density distribution allows to solve the
KE equation analytically and to investigate be-
haviour of the shock front (SF). On the other hand,
it allows adequate SNR penetration analysis.
KE for the SF in non-uniform medium with plane
strati�cation along z-axis is [4]:
(
∂r
∂y
)2
− 1
ϕ(z)
·
[(
∂r
∂z
)2
+ 1
]
= 0, (2)
here r = r(z, y) describes the SF form in the cylin-
drical coordinates as a function of z coordinate and
�Kompaneets time� y which equals:
y =
t∫
0
dt ·
√
E0λ(Γ2 − 1)
2ρ0V (t)
, (3)
where E0 is the energy of explosion at the moment
t = 0 in medium with density ρ0 in point of ex-
plosion, V (t) is the volume limited by the SF, Γ is
an adiabatic index, λ is the dimensionless factor of
an order of unit, considering proportionality of the
pressure behind the front to energy density of explo-
sion E0/V (t). Function ϕ ≡ ρ(z)/ρ0 describes the
medium density distribution. The basic assumption
based on (1) includes constant pressure behind the
SF at adiabatic stage of SNR evolution.
∗a.karnaushenko@gmail.com
39
Advances in Astronomy and Space Physics A.V. Karnaushenko
As it is known [4], the general integral of the equa-
tion (2) can be found by method of envelope con-
struction of the partial solutions of KE obtained by
the method of separation of variables. It looks like:
r = ±z∗
∫ z/z∗
z0/z∗
dx
√
ξ2 · ϕ(z)− 1 + ξy + µ, (4)
where ξ is a separation constant, z0 is the point of
explosion. The integration constant µ will be con-
sidered as a function of ξ: µ = µ(ξ). The condition
of envelope construction dr/dξ = 0 converts ξ into a
function ξ = ξ(z, y) which should be found from the
equations:
y = ±z∗
∫ z/z∗
z0/z∗
dx
ξϕ(x)√
ξ2 · ϕ(x)− 1
− µ′,
dµ
dξ
≡ µ′, (5)
with the account of initial conditions. Substituting
(5) into (4) we got:
r = ±z∗
∫ z/z∗
z0/z∗
dx√
ξ2 · ϕ(x)− 1
+ µ− ξµ′. (6)
As initial conditions we demanded that a lim-
iting case (y → 0, t → 0) the solution of (4) is
the Sedov solution for a homogeneous medium. In
this case µ(ξ) = 0, and SF represents a sphere
of radius R = λ
[
E0t
2/ρ0
]1/5, that corresponds to
r2 + (z − z0)
2 = y2.
analytical Solution
of Kompaneets equation
In the regions adjoining to the leading points of
SF, according to results in [7, 5], the function µ(ξ)
can be chosen equal to zero at any y. We passed to
dimensionless values: r → r/z∗, y → y/z∗, z → z/z∗.
Then the equations for the SF shape r(z, y) are re-
duced to
r(z, ξ) = ±
∫ z
z0
dx√
ξ2 · ϕ(x)− 1
,
y(z, ξ) = ±
∫ z
z0
dx
ξϕ(x)√
ξ2 · ϕ(x)− 1
. (7)
Integrals are calculated easily by using inverse hy-
perbolic functions [2], and the expressions for the
radius of the cross-section of the SF in the region
moving towards the MC (z > z0) are:
r(z, ξ) =
1
2C−
· ln C− + k(z)
C− − k(z)
· C− − k(z0)
C− + k(z0)
+
+
1
2C+
· ln C+ − k(z)
C+ + k(z)
· C+ + k(z0)
C+ − k(z0)
, (8)
y(z, ξ) =
ξ
2
a− b
C−
· ln C− + k(z)
C− − k(z)
· C− − k(z0)
C− + k(z0)
+
+
ξ
2
a + b
C+
· ln C+ − k(z)
C+ + k(z)
· C+ + k(z0)
C+ − k(z0)
, (9)
where C± and k(z) stand for:
C± =
√
(a± b)ξ2 − 1,
k(z) =
√
ξ2(a− b · tanh z)− 1. (10)
In the region moving to the periphery of the MC
(z < z0):
r(z, ξ) =
1
2C−
· ln C− − k(z)
C− + k(z)
· C− + k(z0)
C− − k(z0)
+
+
1
2C+
· ln C+ + k(z)
C+ − k(z)
· C+ − k(z0)
C+ + k(z0)
, (11)
y(z, ξ) =
ξ
2
a− b
C−
· ln C− − k(z)
C− + k(z)
· C− + k(z0)
C− − k(z0)
+
+
ξ
2
a + b
C+
· ln C+ + k(z)
C+ − k(z)
· C+ − k(z0)
C+ + k(z0)
. (12)
However this expressions describe only a part of
SF. In the intermediate region the surface of SF is
described by solutions (5-6) with µ(ξ) 6= 0 and can
be obtained from continuity condition of r and y at
z = zextr which corresponds to the plane where the
derivative ∂r
∂z
changes its sign and the wave radius
is maximum at the given time [7, 5]. It gives
µ(ξ) = − ξ(a− b)√
(a− b)ξ2 − 1
×
× ln
√
(a− b)ξ2 − 1−
√
(a− b tanh z0)ξ2 − 1√
(a− b)ξ2 − 1 +
√
(a− b tanh z0)ξ2 − 1
−
− ξ(a + b)√
(a + b)ξ2 − 1
×
× ln
√
(a + b)ξ2 − 1 +
√
(a− b tanh z0)ξ2 − 1√
(a + b)ξ2 − 1−
√
(a− b tanh z0)ξ2 − 1
.
(13)
40
Advances in Astronomy and Space Physics A.V. Karnaushenko
And �nally:
r(z, ξ) =
1
2C−
· ln C− − k(z)
C− + k(z)
· C− − k(z0)
C− + k(z0)
+
+
1
2C+
· ln C+ + k(z)
C+ − k(z)
· C+ + k(z0)
C+ − k(z0)
, (14)
y(z, ξ) =
ξ
2
a− b
C−
· ln C− − k(z)
C− + k(z)
· C− − k(z0)
C− + k(z0)
+
+
ξ
2
a + b
C+
· ln C+ + k(z)
C+ − k(z)
· C+ + k(z0)
C+ − k(z0)
. (15)
It can be seen, that the analytical solution of
Kompaneets equation for SF with given density dis-
tribution consist of three solutions using which we
can restore the form of the whole shock front and
investigate its evolution.
shock front evaluation
In Fig. 1 the cross-section of the SF is presented,
dashed lines correspond to the intermediate region
of the SF.
Fig. 1: Cross-section of the SF for the following param-
eters: z0 = 1,γ2 ∼ 10−3.
Similarly SF evolution can be investigated for ar-
bitrary values of the density changes and position of
the explosion point (see Fig. 2 and Fig. 3). It is ob-
vious, that the form of the SF signi�cantly changes,
depending on chosen values of parameters.
conclusions
The analytical solution of the Kompaneets equa-
tion describing the evolution of the SF in the non-
uniform medium for density distribution in a form
of the hyperbolic tangent from ISM to MC was ob-
tained. Solution allows to restore the form of the
whole shock front and investigate its evolution for
arbitrary values of parameters: density changes, po-
sition of the explosion point, time. This solution can
be applied for numerous problems such as: detection
and investigation of the interaction between SNRs
and MCs by observing the line of 1720 MHz hydroxyl
maser emission and other maser lines; gamma emis-
sion from SNR interacting with MC.
Fig. 2: Cross-section of the SF for the following parame-
ters: y = 1.5, γ2 ∼ 10−3; dashed lines correspond to the
intermediate region.
Fig. 3: Cross-section of the SF for the following param-
eters: y = 1.7, z0 = 1; dashed lines correspond to the
intermediate region.
references
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Ivashchenko G., 33-36
[3] Frail D.A., Goss W.M. & Slysh V. I. 1994, ApJ, 424, L111
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