数理解析研究所講究録 1425 巻 2005 年 94-97

94
エネルギー散逸を伴う遷音速流に対する二次元固有値問題
阪大レーザー研、

村上 匡且 (Masakatsu Murakami)

Institute of Laser Engineering, Osaka University
Self-similar solutions play a crucial role in many branches of physics, in particular, forsuch fields as hydrodynamic phenomena in astrophysics. For example, the Larson-Penston(LP) solution
facilitates qualitative analysis of complex hydrodynamic flows of gravitational
collapse of an isothermal gaseous sphere, 3,4 which is proposed to explain the qualitative
dynamics in the early stage of star formation. However, the effect of radiative heat conduction
is expected to play an important role in such a temporal domain that substantial dissociation andionization of molecules and atoms proceed with contraction of the system and that theisothermal assumption is not appropriate any more. Still, among a large variety of self-similarsolutions of the hydrodynamic equations, those which include heat conduction are relatively
Still more, to the best of our knowledge, there has been no other publications on the
self-similar solution, which simultaneously treats both the self-gravity and the non-linear heat
conductivity. A striking difference between the outputs of the LP model, for example, and thepresent model is found in the physical picture of the core formation. The former and the latter
respectively describe a decreasing and increasing core mass with time. Figure 1 highlights theoutput of the present work showing the temporal evolution of the density and velocity profiles
at sequential times. As can be seen in Fig. 1 the core shrinks with time, where
corresponds to the collapse time. The central density increases in proportion to t-2, that is theuniversal scaling regardless of the degree of the heat conductivity. At the same time, the coremass also increases due to mass accretion. One more important feature of the present model,
which is essentially different from the conventional ones obtained under the isothermal or the
adiabatic assumptions, is that all the scales of the physical quantities are uniquely determined asa function of time only.
The one-dimensional spherical gas-dynamical equations with both self-gravity and heat
conductivity are$\mathrm{f}\mathrm{e}\mathrm{w}.5\cdot 7$$\mathrm{t}=0$$(1)$$\frac{\partial \mathrm{p}}{\partial \mathrm{t}}+\frac{1\partial}{\mathrm{r}^{2}\partial \mathrm{r}}(\mathrm{r}^{\mathrm{z}_{\mathrm{P}^{\mathfrak{U})=0}}},$$\frac{\partial \mathrm{u}}{\partial \mathrm{t}}+\mathrm{u}\frac{\partial \mathrm{u}}{\partial \mathrm{r}}=-\frac{1}{\mathrm{p}}\frac{\partial \mathrm{p}}{\partial \mathrm{r}}-\frac{\partial\phi}{\partial \mathrm{r}}$

,

(2)

,

(3)

$\frac{1\partial}{\mathrm{r}^{2}\partial \mathrm{r}}(\mathrm{r}^{2}\frac{\partial\phi}{\partial \mathrm{r}})=4\pi \mathrm{G}\mathrm{p}$(4)$\mathrm{p}(\frac{\partial\epsilon}{\partial \mathrm{t}}+\mathrm{u}\frac{\partial\epsilon}{\partial \mathrm{r}})+\frac{\mathrm{p}}{\mathrm{r}^{2}}\frac{\partial}{\partial \mathrm{r}}(\mathrm{r}^{2}\mathrm{u})=\frac{1\partial}{\mathrm{r}^{2}\partial \mathrm{r}}(\mathrm{r}^{2}1\mathrm{C}\frac{\partial \mathrm{T}}{\partial \mathrm{r}})$where is the pressure, the mass density, the specific internal energy, the flow velocity,
the gravity potential, and the gravity constant. We assume the ideal gas equation of state(EOS) in the form,$\mathrm{e}$$\mathrm{r}$$\mathrm{p}$$\mathrm{u}$$\mathrm{G}$$\phi$$\mathrm{T}=\mathrm{p}/\mathrm{p}=(\gamma-1)\epsilon$,(5)

where is the specific heats ratio. Equation (4), described by the one-temperature model,
includes the non-linear heat conduction term on the right hand side, where we assume apower-law dependence for the conduction coefficient,$\gamma$$\theta 5$$\hat{\varpi_{\tilde{\Phi}}\mathrm{g}}$$\hat{\xi u.)\Phi 00>}$$\dot{\alpha}$$\mathrm{o}’ \mathrm{g}\mathrm{c}0\}n$$\frac{\frac{>}{\overline{8}}}{>\Phi}\backslash$$\Xi\alpha u\mathrm{s}\emptyset 1$$\xi$

Radius (cm)Fig. 2 The eigenstructure of the, and $\gamma=5/3$ .
,case,Fig. 1 Temporal evolution of the density andvelocity profiles at different sequential times.
$\kappa=\kappa_{0}\mathrm{T}^{\mathrm{n}}/\mathrm{p}^{\mathrm{m}}$$\mathrm{r}\mathrm{n}=2$$\mathrm{n}=13/2$(6),

are assumed.
andwith , , and being constants. For normal physical values,
parameters,
of
the
terms
in
generality
below,
the
keep
we
given
In the numerical calculations
set,andreferenceofthevaluesusing
the, , and , but also show specific forms
hydrogenionizedfullyina$13/2$ , which describing the opacity due to inverse bremsstrahlung
plasma together with $\gamma=5/3$ .
We introduce the following similarity ansatz,$\mathrm{n}>0$$\kappa_{0}$$\mathrm{m}$$\mathrm{m}>0$$\mathrm{n}$$\mathrm{m}=2$$\mathrm{m}$$\mathrm{n}$$\mathrm{n}=$$\gamma$$\xi=\mathrm{r}/\mathrm{R}(\mathrm{t})$,$\mathrm{R}(\mathrm{t})=\mathrm{A}|\mathrm{t}|^{1/\mathrm{a}}$$\mathrm{u}=\frac{\mathrm{A}}{\mathrm{a}}|\mathrm{t}|^{\mathrm{b}/\mathrm{a}}\mathrm{v}(\xi)$

,

(7)

,

$\mathrm{p}=\mathrm{B}|\mathrm{t}|^{-2}\mathrm{g}(\xi)$$\frac{\partial\phi}{\partial \mathrm{r}}=\frac{\mathrm{A}\mathrm{B}\mathrm{G}}{\xi^{2}}|\mathrm{t}|^{(\mathrm{e}-1)/\mathrm{a}}\Omega(\xi)$,,$\mathrm{T}=(\frac{\mathrm{A}}{\mathrm{a}})^{2}|\iota|^{\mathrm{c}/\mathrm{a}}\tau(\xi)$

,

(8)
(9)

,

$\Omega(\xi)=4\pi\int_{0}^{\xi}\xi^{2}\mathrm{g}(\xi)\mathrm{d}\xi$

is the temporal
, and
where
the scales ofdefining
constantscharacteristic scale length of the system; A and are positive(4)
are reduced(2),
andthe radius and the mass density, respectively. Then, Eqs. (1),respectively to the following ordinary differential equations,$1-\mathrm{a}=\mathrm{b}=\mathrm{c}/2=1+\mathrm{d}/2=\mathrm{e}/2=(1+2\mathrm{m})/(3+2\mathrm{m}-2\mathrm{n})$$\mathrm{R}(\mathrm{t})$$\mathrm{B}$$-(\pm\xi-\mathrm{v})\mathrm{g}’+(\pm \mathrm{d}+\mathrm{v}’+2\mathrm{v}/\xi)\mathrm{g}=0$

(10)

,

$\pm \mathrm{b}\mathrm{v}-(\pm\xi-\mathrm{v})\mathrm{v}’+(\mathrm{g}\tau)’/\mathrm{g}+\mathrm{K}_{1}\Omega/\xi^{2}=0$

(11)

,

$\frac{\pm \mathrm{c}\tau-(\pm\xi-\mathrm{v})\tau’}{\gamma-1}+(\mathrm{v}’+2\mathrm{v}/\xi)\tau=\mathrm{K}_{2}\frac{(\xi^{2}\mathrm{g}^{-\mathfrak{n}1}\tau^{\mathrm{n}}\tau’)’}{\mathrm{g}\xi^{2}}$

,

(12)

,
where the prime denotes the derivative with respect to , and concerning the double signs,, respectively. In the following analysis,
andthe upper and lower sign correspond to. Since Eq. (3) is automatically, and therefore
we focus on the time domain,
satisfied, its reduced form does not appear in the above set of equations. As can be seen in
Eqs. (11) and (12), the present system is characterized by the two dimensionless parameters,
and , defined by$\pm$$\mathrm{x}$$\mathrm{t}>0$$\mathrm{t}<0$$\mathrm{K}_{1}$$\mathrm{K}_{2}$$\mathrm{t}<0$$|\mathrm{t}|=-\mathrm{t}$88Normalized Core Mass
20
6
810

$\mathrm{M}_{\mathrm{c}\mathrm{o}oe}/\mathrm{M}$

4

.

30

40

Fig. 3 Temporal evolution of the core parameters,
andvs Tc, for the reference case,(solid line). Bothand the core mass are directly
related to time in the present model. The dashed line
labeled by Pidedenotes temperature .
density, on which the ideal EOS pressure, Pi&,. The
balances with the radiation pressure,denotes thedashed line labeled bytemperature vs density, on which the absorption
coefficient due to inverse bremsstrahlung, ,balances with the absorption coefficient due toThomson scattering, . The dotted curveshowing solar nebula formation process is justfor comparison with the present model, forwhich the time and mass axes do not apply.
$\mathrm{n}=13/2$$\mathrm{m}=2$$\mathrm{r}_{\mathrm{C}}$$\mathrm{P}\mathrm{c}$$\mathrm{v}\mathrm{s}$$=\mathrm{P}_{\mathrm{f}\mathrm{f}1]}$$\mathrm{P}_{\mathrm{I}\mathrm{f}\mathrm{f}\mathrm{i}}$$\mathrm{k}\mathrm{f}\mathrm{f}$$=\mathrm{k}_{\mathrm{e}}$$\mathrm{k}\mathrm{f}\mathrm{f}$$\mathrm{k}_{\mathrm{e}}$Central Density$\mathrm{K}_{\mathrm{I}}=\mathrm{a}^{2}\mathrm{G}\mathrm{B}$,$\mathrm{p}_{\mathrm{c}}\langle \mathrm{g}/\mathrm{c}\mathrm{m}^{3})$(13)$\mathrm{K}_{2}=\frac{\kappa_{0}}{\mathrm{a}\mathrm{B}^{\mathrm{m}+1}}(\frac{\mathrm{A}}{\mathrm{a}})^{2\mathrm{n}-2}$

and , we need two more physical conditions.
To detennine a unique set of parameters,prescriptionthat the right integration curve smoothly passes
The first one is quite an orthodox
through the singular point which is located somewhere at a finite distance from the center. Notethat the LP solution does not need energy equation (4) under the isothermal assumption, and, which is determined
the system is described in terms of only a single unknown constant,by this first condition. The second one is somewhat less obvious compared with the first one,but still seems enough natural, namely, that both the density and the temperature converge tozero simultaneously with increasing the radius. Thus, the system under consideration is
reduced to a two-dimensional eigenvalue problem.
, the, the density,
Figure 2 shows the eigenstructure for the temperature,, of the system under the eigenvalues thus
, and the heat flux,
velocity,
obtained, where the curves are assigned with labels corresponding to the original physicalquantities just for simplicity. Also two other dimensionless quantities of interest are shown in, which isFig. 2. The first one is the ratio of the plasma pressure to the gravity,
given in the form,$\mathrm{K}_{1}$$\mathrm{K}_{2}$$\mathrm{K}_{1}$$\tau$$\propto \mathrm{T}$$\mathrm{g}\propto \mathrm{p}$$\mathrm{q}=-\kappa\nabla \mathrm{T}$$\mathrm{v}\propto \mathrm{u}$$\Psi$

l$\Psi$$(\mathrm{n}-1)^{}$$/6\pi(\mathrm{n}-\mathrm{m} -3/2)^{2}\mathrm{K}_{1}$$\equiv|\nabla \mathrm{p}|/|\mathrm{p}\nabla\phi|$$(\xi<<1)$$\equiv\frac{|\nabla \mathrm{p}|}{|\mathrm{p}\nabla\phi|}=$(14)$\mathrm{K}_{\mathrm{I}}\xi^{-2(\mathrm{n}-1)(2+3\mathrm{m}-\mathrm{n})/\mathrm{n}(1+2\mathrm{m})}$$(\xi>>1)$

The second one is the Peclet number mentioned in the introduction, i.e., the ratio of the heating
work, which is given in the form,
to the mechanical compression$(\mathrm{p}\mathrm{d}\mathrm{V})$$\mathrm{P}\mathrm{e}\equiv|\frac{\mathrm{p}\nabla.\cdot \mathrm{u}}{\nabla \mathrm{q}}|=\{\begin{array}{l}4(\mathrm{n}-\mathrm{l})/(4\mathrm{n}-6\mathfrak{m}-7)(\xi<<1)\{1-(\mathrm{m}+2/3)/(\gamma-1)\mathrm{n}\}^{-1}(\xi>>\mathrm{l})\end{array}$

For the reference case,$\mathrm{m}=2$, $\mathrm{n}=13/2$ , and $\gamma=5/3$ , Eqs. (14) and (15) are reduced to

(15)

97

(16)

$\{\begin{array}{l}\mathrm{P}\mathrm{e}arrow 3.1,\Psiarrow 0.71(\xi<<1)\mathrm{P}\mathrm{e}arrow 2.6,\Psiarrow 0.62\xi^{-0.5\mathrm{l}}(\xi>>\mathrm{l})\end{array}$

0.6 $(\xi<<1)$ , which is close to that derived above. In the outer
For the LP solution,
region for $\xi>>1$ , however, for the LP solution remains constant 0.2, which contrasts
in Eq. (16). The constancy of Pe tells that the entropy is persistently emitted
withoutward in the entire space. For such a non-adiabatic implosions, the self-similar solution
predicts that the flow pattern should approach an asymptotical regime in which it ceases todepend on the initial entropy, in other words, the system will “forget” certain aspects of theinitial state. In that sense, the radiative heat conductivity is expected to substantially affect the
self-organization process of the core formation.
Figure 3 shows the temporal evolution of the core parameters, versus , as the solidline, which are both expressed as a function of time. Here it should be noted that the core massis also uniquely measured as a function of time, which appears as the uppermost axis in Fig. 3.
, hydrogen atoms are
At temperatures of few thousands and densities of about 10
expected to be enough ionized. Thus, we can roughly fix the applicable parameter domain of,
few 10 a few 10 , Pc
the present solution from Fig. 3:
core
mass.
The
solar
the
times
about
ten
of
few a few tens of years, and the core mass
formation
nebula
solar
the
with
example,evolves at rather high temperatures compared, forNote that, as is clear from the model description, the present analysis is not limited
to such a special combination of physical effects as the ideal EOS and the inverse
bremsstrahlung opacity, but more extensively applies to other physical situations, under which
one will obtain a different solution and scalings correspondingly.
In summary, taking both the gravity and the radiative heat conductivity into accountsimultaneously, a new class of self-similar solution for spherical implosions of gaseous
masses has been found. The solution has been investigated in detail in terms of the constants,
similarity
id , which characterize the radiative heat conductivity. Under the appropriate
novel
the
to
reducedissystemansatz and variable transformations, the hydrodynarnic
quantitative
uniqueathat
two-dimensional eigenvalue problem. The physical implication isrelations between the gravity and the heat conductivity indwells in the self-similar dynamics.
as inAlso, it has tuned out that the present system has no free parameter to control the system
LP
the
to
Furthermore,
contrastinthe cases under adiabatic or isothermal assumptions.
thefromaccretion,
resultswhich
solution, the core mass increases with time due to mass
persistent entropy emission via radiation.
$\Psi$––$\Psi$$\Psi$$arrow 0$$\mathrm{T}_{\mathrm{C}}$$\mathrm{P}\mathrm{c}$$\mathrm{g}/\mathrm{c}\mathrm{m}^{3}$$\mathrm{K}$-$\mathrm{T}=\mathrm{a}$$\mathrm{K}$$=10^{-11}- 10^{-9}\mathrm{g}/\mathrm{c}\mathrm{m}^{3}$-$\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{c}\mathrm{e}\mathrm{s}\mathrm{s}.8$$\mathrm{m}$$\mathrm{n}$REFE RENCES
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lLarson,$2\mathrm{P}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{o}\mathrm{n}$$3\mathrm{S}\mathrm{h}\mathrm{u}$$\mathrm{R}.\mathrm{B}.$,$\mathrm{M}.\mathrm{V}.$$\mathrm{F}.\mathrm{H}.$$4\mathrm{F}\mathrm{o}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{r}$$\mathrm{A}\mathrm{p}\mathrm{J}$$\mathrm{A}\mathrm{p}\mathrm{J}$$\mathrm{P}.\mathrm{N}.$$5\mathrm{P}\mathrm{a}\mathrm{k}\mathrm{u}\mathrm{l}\mathrm{a}$$\mathrm{S}.\mathrm{V}.$$6\mathrm{C}\mathrm{o}\mathrm{g}\mathrm{g}\mathrm{e}\mathrm{s}\mathrm{h}\mathrm{a}\mathrm{l}\mathrm{l}$$7\mathrm{M}\mathrm{u}\mathrm{r}\mathrm{a}\mathrm{k}\mathrm{a}\mathrm{m}\mathrm{i}$$8\mathrm{T}\mathrm{s}\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{n}\mathrm{u}\mathrm{t}\mathrm{e}\mathrm{r}$$\mathrm{W}.\mathrm{M}.$$|\mathrm{t}|=\mathrm{a}$