Рівняння стану металевого водню та атомарного гелію і внутрішня будова Юпітера та Сатурна
Запропоновано рiвняння стану сумiшi металевого водню та атомарного гелiю. Дослiджений iнтервал тискiв, температур i густин вiдповiдає умовам найбiльших планет сонячної системи – Юпiтера i Сатурна. Вважається, що речовина планети являє собою сумiш протонiв, атомiв гелiю та електронiв. Для знаходження...
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Швець, В.Т. Швець, Т.В. Рачинський, С.Є. 2010-11-05T14:34:19Z 2010-11-05T14:34:19Z 2010 Рівняння стану металевого водню та атомарного гелію і внутрішня будова Юпітера та Сатурна / В.Т. Швець, Т.В. Швець, С.Є. Рачинський // Укр. фіз. журн. — 2010. — Т. 55, № 2. — С. 252-258. — Бібліогр.: 29 назв. — укр. 2071-0194 PACS 61.25.Mv, 62.50.-p, 64.30.Ef, 96.15.Nd https://nasplib.isofts.kiev.ua/handle/123456789/13391 537.311.31 Запропоновано рiвняння стану сумiшi металевого водню та атомарного гелiю. Дослiджений iнтервал тискiв, температур i густин вiдповiдає умовам найбiльших планет сонячної системи – Юпiтера i Сатурна. Вважається, що речовина планети являє собою сумiш протонiв, атомiв гелiю та електронiв. Для знаходження тиску сумiшi використовується теорiя збурень за електрон-протонною та електрон-атомною взаємодiями. Електронна пiдсистема розглядається у наближеннi випадкових фаз. Взаємодiї протонiв, атомiв, атомiв i протонiв враховуються у наближеннi твердих сфер. Проаналiзовано застосовнiсть моделi полiтропного шару до моделювання внутрiшньої будови Юпiтера i Сатурна та запропоновано конкретне значення iндексу полiтропи. Знайдено густину, тиск i температуру на Юпiтерi та Сатурнi як функцiї вiдстанi до їх центра. Оцiнено можливi долi водню i гелiю у складi планет. Предложено уравнение состояния смеси металлического водорода и атомарного гелия. Исследованный интервал давлений, температур и плотностей соответствует условиям наибольших планет солнечной системы Юпитера и Сатурна. Считается, что вещество планеты представляет собой смесь протонов, атомов гелия и электронов. Для нахождения давления смеси используется теория возмущений по электрон-протонному и электрон-атомному взаимодействиям. Электронная подсистема рассматривается в приближении случайных фаз. Взаимодействие протонов, атомов, атомов и протонов учитывается в приближении твердых сфер. Проанализирована применимость модели политропного шара для моделирования внутренней структуры Юпитера и Сатурна и предложено конкретное значение индекса политропы. Найдены плотность, давление и температура Юпитера и Сатурна как функции расстояния до их центра. Оценена возможная доля водорода и гелия в составе планет. An equation of state for a mixture of metallic hydrogen and atomic helium has been proposed. The explored intervals of pressure, temperature, and density correspond to the conditions on the largest solar system planets, Jupiter and Saturn. The substance of a planet is modelled as a mixture of protons, helium atoms, and electrons. A theory, where the electron-proton and electron-atom interactions are considered as a perturbation, has been used to find the pressure in the mixture. The electron subsystem is analyzed in the random phase approximation, and the proton-proton, atom-atom, and proton-atom interactions in the hard-sphere approximation. The applicability of the polytropic sphere model for the simulation of Jupiter’s and Saturn’s internal structures has been analyzed, and a specific value for the polytropic index has been proposed. The density, pressure, and temperature on Jupiter and Saturn as functions of the distance from the planet center have been found. Possible fractions of hydrogen and helium in the planet composition have been estimated. uk Відділення фізики і астрономії НАН України Астрофізика і космологія Рівняння стану металевого водню та атомарного гелію і внутрішня будова Юпітера та Сатурна Уравнение состояния металлического водорода и атомарного гелия и внутреннее строение Юпитера и Сатурна Equation of State for Metallic Hydrogen and Atomic Helium and the Interior of Jupiter and Saturn Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
Рівняння стану металевого водню та атомарного гелію і внутрішня будова Юпітера та Сатурна |
| spellingShingle |
Рівняння стану металевого водню та атомарного гелію і внутрішня будова Юпітера та Сатурна Швець, В.Т. Швець, Т.В. Рачинський, С.Є. Астрофізика і космологія |
| title_short |
Рівняння стану металевого водню та атомарного гелію і внутрішня будова Юпітера та Сатурна |
| title_full |
Рівняння стану металевого водню та атомарного гелію і внутрішня будова Юпітера та Сатурна |
| title_fullStr |
Рівняння стану металевого водню та атомарного гелію і внутрішня будова Юпітера та Сатурна |
| title_full_unstemmed |
Рівняння стану металевого водню та атомарного гелію і внутрішня будова Юпітера та Сатурна |
| title_sort |
рівняння стану металевого водню та атомарного гелію і внутрішня будова юпітера та сатурна |
| author |
Швець, В.Т. Швець, Т.В. Рачинський, С.Є. |
| author_facet |
Швець, В.Т. Швець, Т.В. Рачинський, С.Є. |
| topic |
Астрофізика і космологія |
| topic_facet |
Астрофізика і космологія |
| publishDate |
2010 |
| language |
Ukrainian |
| publisher |
Відділення фізики і астрономії НАН України |
| format |
Article |
| title_alt |
Уравнение состояния металлического водорода и атомарного гелия и внутреннее строение Юпитера и Сатурна Equation of State for Metallic Hydrogen and Atomic Helium and the Interior of Jupiter and Saturn |
| description |
Запропоновано рiвняння стану сумiшi металевого водню та атомарного гелiю. Дослiджений iнтервал тискiв, температур i густин вiдповiдає умовам найбiльших планет сонячної системи – Юпiтера i Сатурна. Вважається, що речовина планети являє собою сумiш протонiв, атомiв гелiю та електронiв. Для знаходження тиску сумiшi використовується теорiя збурень за електрон-протонною та електрон-атомною взаємодiями. Електронна пiдсистема розглядається у наближеннi випадкових фаз. Взаємодiї протонiв, атомiв, атомiв i протонiв враховуються у наближеннi твердих сфер. Проаналiзовано застосовнiсть моделi полiтропного шару до моделювання внутрiшньої будови Юпiтера i Сатурна та запропоновано конкретне значення iндексу полiтропи. Знайдено густину, тиск i температуру на Юпiтерi та Сатурнi як функцiї вiдстанi до їх центра. Оцiнено можливi долi водню i гелiю у складi планет.
Предложено уравнение состояния смеси металлического водорода и атомарного гелия. Исследованный интервал давлений, температур и плотностей соответствует условиям наибольших планет солнечной системы Юпитера и Сатурна. Считается, что вещество планеты представляет собой смесь протонов, атомов гелия и электронов. Для нахождения давления смеси используется теория возмущений по электрон-протонному и электрон-атомному взаимодействиям. Электронная подсистема рассматривается в приближении случайных фаз. Взаимодействие протонов, атомов, атомов и протонов учитывается в приближении твердых сфер. Проанализирована применимость модели политропного шара для моделирования внутренней структуры Юпитера и Сатурна и предложено конкретное значение индекса политропы. Найдены плотность, давление и температура Юпитера и Сатурна как функции расстояния до их центра. Оценена возможная доля водорода и гелия в составе планет.
|
| issn |
2071-0194 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/13391 |
| citation_txt |
Рівняння стану металевого водню та атомарного гелію і внутрішня будова Юпітера та Сатурна / В.Т. Швець, Т.В. Швець, С.Є. Рачинський // Укр. фіз. журн. — 2010. — Т. 55, № 2. — С. 252-258. — Бібліогр.: 29 назв. — укр. |
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ASTROPHYSICS AND COSMOLOGY
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 2 251
EQUATION OF STATE FOR METALLIC HYDROGEN
AND ATOMIC HELIUM AND THE INTERIOR
OF JUPITER AND SATURN
V.T. SHVETS, T.V. SHVETS, S.YE. RACHYBSKIY
Odesa State Academy of Refrigeration
(1/3, Dvoryans’ka Str., Odesa 65026, Ukraine; e-mail: valtar@ paco. net )
PACS 61.25.Mv, 62.50.-p,
64.30.Ef, 96.15.Nd
c©2010
An equation of state for a mixture of metallic hydrogen and atomic
helium has been proposed. The explored intervals of pressure, tem-
perature, and density correspond to the conditions on the largest
solar system planets, Jupiter and Saturn. The substance of a
planet is modelled as a mixture of protons, helium atoms, and
electrons. A theory, where the electron-proton and electron-atom
interactions are considered as a perturbation, has been used to
find the pressure in the mixture. The electron subsystem is ana-
lyzed in the random phase approximation, and the proton-proton,
atom-atom, and proton-atom interactions in the hard-sphere ap-
proximation. The applicability of the polytropic sphere model for
the simulation of Jupiter’s and Saturn’s internal structures has
been analyzed, and a specific value for the polytropic index has
been proposed. The density, pressure, and temperature on Jupiter
and Saturn as functions of the distance from the planet center
have been found. Possible fractions of hydrogen and helium in the
planet composition have been estimated.
1. Introduction
After hydrogen in the metallic state under a pressure of
1.4 Mbar and a temperature of 3000 K has been dis-
covered [1, 2] and its properties under terrestrial condi-
tions have been experimentally studied in detail, there
appeared a real possibility to research its property un-
der such conditions that nowadays cannot be repro-
duced on the Earth. From this point of view, giant
planets can be regarded as natural laboratories. Ow-
ing to the available models for such planets [3–6], we
know a number of their thermodynamic characteristics
such as the density, pressure, and temperature, as well
as their dependences on the distance reckoned from
the planet center. All the models are based on the
mechanical equilibrium equation for a planet and the
polytrope equation. They are known for rather a long
time [7, 8]. However, a key characteristic of the sub-
stance, which plays a basic role when calculating the
density, pressure, and temperature, is the equation of
state. Note that, when obtaining metallic hydrogen
under terrestrial conditions, only one of three param-
eters – pressure, density, and temperature – namely,
pressure, was measured [1, 2]. Concerning giant plan-
ets, we may assert that none of those characteristics
can be measured in the central part of the planet. To-
day, the equilibrium properties of metallic hydrogen are
widely studied [9–11], including its equation of state
[12–14]. Every improving correction to the equation
of state allows the whole set of thermodynamic char-
acteristics of the planet to be calculated more precisely,
and the initial model of the planet, which is based on
the polytrope equation as well, to be made more spe-
cific.
This work aims at studying the equation of state
for a mixture of metallic hydrogen and atomic helium
in the density and temperature ranges that are char-
acteristic of giant planets. Note that the equation
has been earlier studied for a density of 0.6 g/cm3
and a temperature of 3000 K which are typical of
conditions needed for the production of metallic hy-
drogen under terrestrial conditions. On the basis of
those researches, some important thermodynamic char-
acteristics for such giant planets of the solar system
as Jupiter and Saturn have been specified. In par-
ticular, we proposed an algorithm for finding the he-
lium concentration in the central regions of those plan-
ets.
V.T. SHVETS, T.V. SHVETS, S.YE. RACHYBSKIY
Fig. 1. Dependences of the Jupiter density on the distance from
the planet center for various values of polytrope index
2. Equation of Mechanical Equilibrium of the
Planet
The equation of mechanical equilibrium of the planet
looks like [6–8]
1
r2
d
dr
[
r2
ρ(r)
dP (r)
dr
]
= −4πGρ(r). (2.1)
Here, P (r) and ρ(r) are the planet pressure and den-
sity, respectively, as the functions of the distance to the
planet center, and G is the gravitational constant. This
equation describes a planet with a spherically symmetric
substance distribution and without taking its rotation
into account. The relation between the pressure and
the density is conventionally described by the polytrope
equation
P (r) = cρ1+1/n(r), (2.2)
which follows from the assumption on the convective
mechanism of heat transfer between different planet lay-
ers. Here, c is an arbitrary constant, and n the polytrope
index.
The planet specificity reveals itself through the bound-
ary conditions at the center and on the surface of the
planet:
ρ(R) = 0, (2.3)
ρ(0) = ρ0, (2.4)
c(1 + 1/n) lim ρ1/n−1(r)
r→R
dρ(r)
dr
= −GM
R2
, (2.5)
dρ(0)
dr
= 0. (2.6)
Here, M is the mass of the planet, R its radius, and ρ0
the planet substance density at the planet center. The
mass and the radius of the planet are considered to be
known quantities, and the density at the center is an
additional parameter of the problem. Four boundary
conditions allow one to obtain the partial solution of the
equation and determine the constants ρ0 and c.
Making use of the substitutions u = ρ1/n, y = u/u0,
and x = λr, the equilibrium equation can be transformed
into a dimensionless form
1
x2
d
dx
(
x2 dy
dx
)
+ yn = 0, (2.7)
which is the Emden equation. The parameter u0 is de-
fined by the relation ρ(0) = un
0 . Now, both the density
and the pressure of the planet can be expressed in terms
of the solution y(x) of the Emden equation and the di-
mensionless radius of the planet x1, the latter being the
solution of the equation y(x1) = 0:
ρ(r) = − x1M
4πR3y′(x1)
yn
(x1
R
r
)
, (2.8)
P (r) =
GM2
4π(1 + n)R4 [y′(x1)]
2 y
1+n
(x1
R
r
)
. (2.9)
The only unknown parameter is the polytrope in-
dex n. In the case n = 1, the Emden equation is
linear and has an analytical solution. The latter is
adopted to correspond to a partially degenerate elec-
tron gas. This value of the polytrope index is the
most popular today [3, 5]. In Figs. 1 and 2, it corre-
sponds to a density of about 4 g/cm3 at the Jupiter
center and a pressure of about 40 Mbar. The hori-
zontal line corresponds to the density, at which hydro-
gen was obtained in the metallic state under terrestrial
conditions. The figures demonstrate that, if n = 1,
the favorable conditions for hydrogen to transform into
the metallic state arise already at a distance of 0.1 ra-
dius from the planet surface. If n = 3/2, these condi-
tions are satisfied at a distance of 0.2 radius from the
planet surface, and the density and the pressure in the
planet center are close to 7 g/cm3 and 60 Mbar, respec-
tively.
252 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 2
EQUATION OF STATE FOR METALLIC HYDROGEN
Fig. 2. Dependences of the Jupiter pressure on the distance from
the planet center for various values of the polytrope index
In this work, we adopt n = 3/2, which corresponds
to a completely degenerate electron gas. As is shown in
Fig. 3, the difference between the pressures in the central
region of Saturn in the indicated cases of the polytrope
index is rather substantial.
As is seen from Fig. 2, where the dependences
for the pressure on Jupiter are depicted, the favor-
able conditions for hydrogen to transform into the
metallic state also arise at a distance of 0.1 radius,
at n = 1, and 0.2 radius, at n = 3/2, of the
planet. Since two thermodynamic parameters simulta-
neously get necessary values at these points, the lat-
ter are to be considered as points that divide the
metallic and molecular phases of hydrogen. The third
thermodynamic parameter, the temperature, accord-
ing to the equation of state that was studied in
works [12–14], also has a necessary value of about
3000 K.
Since the temperature at the center of planets, accord-
ing to various estimations, amounts to approximately
10000–20000 K, which is several percent of the Fermi
energy, it is sufficient that the account of temperature
effects to the internal planet energy be confined to cor-
rections that are only linear in the temperature. Such a
correction is the kinetic energy of the proton and he-
lium subsystems. The temperature correction to the
electron gas energy is quadratic in the temperature, so
that it can be totally neglected, and the electron gas
can be considered as completely degenerate. It is easy
to show that, if n = 3/2, almost 80% of the Jupiter
Fig. 3. Dependences of the Saturn pressure on the distance from
the planet center for various values of the polytrope index
mass is located in the region of hydrogen metalliza-
tion.
3. Model of Ideal Electron, Proton, and Atomic
Gases
The polytrope index n = 3/2 corresponds to the models
of classical and degenerate ideal gases. We apply the
first model to describe the proton subsystem of metallic
hydrogen and helium atoms. In so doing, we consider the
planet substance to be chemically homogeneous and he-
lium atoms to be neutral, which completely corresponds
to the modern representation concerning conditions for
helium metallization [15, 16]. We confine the consider-
ation to the planet region, where helium metallization
occurs.
Let c be the concentration of the electron gas. Let us
define it as a ratio between the number of electrons and
the number of protons and helium atoms. In this case,
the pressure created by atoms and protons is
Pp+He(ρ, T, r) =
ρ(r)
MHc+MHe(1− c)
kBT (r), (3.1)
where MH and MHe are the masses of a proton and a
helium atom, respectively; and ρ(r) is the planet sub-
stance density which is determined from the equation of
mechanical equilibrium for the planet. Correspondingly,
for the density of electrons, we have
ne(c, r) = c
ρ(r)
MHc+MHe(1− c)
. (3.2)
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 2 253
V.T. SHVETS, T.V. SHVETS, S.YE. RACHYBSKIY
The second model – the model of degenerate ideal gas
– is applied to describe the electron gas which is formed
at the hydrogen metallization in the central region of the
planet. The corresponding equation of state reads
Pe(ρ, r) = 2ne(c, r)εF (c, r)/5, (3.3)
where Pe(ne, r) is the pressure in the degenerate gas cre-
ated by electrons at the distance r from the planet cen-
ter, and m is the electron mass. Hence, if interactions
are not taken into account, n = 3/2 for a mixture of hy-
drogen and helium, irrespective of whether they are in
the molecular or metallic state. The total pressure that
arises in the central region of the planet is
P (ρ, T, r) = Pp+He(ρ, T, r) + Pe(ρ, r). (3.4)
The numerical analysis of this formula shows that both
Jupiter and Saturn cannot consist of hydrogen only, be-
cause the pressure of electron gas is several times higher
in this case than the pressure calculated from the equa-
tion of planet mechanical equilibrium, provided that the
model of polytropic layer with n = 3/2 is used. If the
electron concentration, as well as the concentration of
hydrogen in the central region of the planet, amounts to
only 0.761, those pressures are practically coincide for
Jupiter. For Saturn, the limiting concentration of elec-
tron gas is 0.649. In our opinion, the main factor that
is responsible for such a concentration is the availability
of other elements – first of all, helium – in the planet
structure. This conclusion has to be specified, first of
all, by taking the proton subsystem into consideration,
as well as interactions in the system.
Since the temperature-induced contributions to the
thermodynamic potentials and the pressure are small
corrections, the results of calculations of the planet tem-
perature are sensitive to the electron gas concentration.
At a concentration of 0.761, the temperature of Jupiter
is zero. This means that it is the temperature positivity
that fixes the upper limit of the electron gas concen-
tration. At a concentration of 0.74, the temperature in
the planet center achieves 20000 K. This circumstance
explains the large divergence available in literary data
concerning the temperature.
The planet temperature can be found by equating the
pressures determined in the ideal gas and polytropic
layer models. In this case, we obtain the temperature
as a function of the distance to the planet center and
the electron gas concentration. Nowadays, no additional
information on the temperature can be obtained. To
determine the planet temperature, another equation is
needed, which would enable one to determine the elec-
tron gas concentration. As such, let us take the equa-
tion of state for interacting electron, proton, and helium
gases. In this case, the role of interactions in the forma-
tion of pressure magnitude on the planet can be eluci-
dated as well.
4. Internal and Free Energies
The Hamiltonian of the electron subsystem in metallic
hydrogen can be taken in the form similar to that used
for simple liquid metals [17]. The internal energy of the
system is obtained by averaging the Hamiltonian over
the Gibbs canonical ensemble
E = 〈H〉 = Ei + Ee + Eie. (4.1)
For the contribution of the proton subsystem to the en-
ergy, we have
Ei = 〈Hi〉 = N
3
2
kBT +N
1
2V
∑
q
′V (q)[Si(q)− 1]. (4.2)
Here, T is the absolute temperature of the systems. The
first term on the right-hand side is the kinetic energy of
protons. The second one is the Madelung energy which
makes allowance for the interaction between charged pro-
tons, nuclei, and ions. The neutral helium atoms give
no contribution to this energy. The quantity Si(q) is the
static structural factor of the proton subsystem.
The energy of the electron subsystem and the interac-
tion energy of electron and proton subsystems are conve-
nient to be examined together. Their sum – the ground
state energy of the electron gas in the proton-induced
field – can be expanded in a series in the electron-proton
interaction:
Ee = 〈He〉+ 〈Hie〉 =
∞∑
n=0
En. (4.3)
In turn, every electron-proton interaction term should
be expanded into a series in the electron-electron in-
teraction. For the zero-order term with respect to the
electron-proton interaction, it looks like
E0 = N
(
1.105
rs
− 0.458
rs
− 0.058 + 0.016 ln rs
)
. (4.4)
Here, rs is the Brueckner parameter of nonideality. The
first term corresponds to the kinetic energy of electrons,
the second to the Hartree–Fock energy [18]. The third
and fourth terms correspond to the correlation energy,
254 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 2
EQUATION OF STATE FOR METALLIC HYDROGEN
for which the Nozières–Pines interpolation formula [18,
19] was applied.
The term of the second order with respect to the
electron-proton interaction – the so-called band struc-
ture energy – looks like [20–24]
E2 = N
−1
4π2
∞∫
0
π(q)
ε(q)
V 2(q)S(q)q2dq. (4.5)
Here, π(q) is the polarization function, and ε(q) is the
dielectric permittivity of the electron gas in the random
phase approximation, which takes the exchange interac-
tion and electron correlations into account in the local
field approximation [25].
Helium atoms, providing their availability, give a con-
tribution to the band structure energy. This contribu-
tion can be included by making the substitution
V 2(q)S(q)→ c(1− c)[VH(q)− VHe(q)]2+
+c2V 2
H(q)SH,H(q) + c(1− c)VH(q)VHe(q)SH,He(q)+
+(1− c)2V 2
He(q)SHe,He(q), (4.6)
where VH(q) and VHe(q) are the formfactors for the
electron-proton and electron-atom interactions, respec-
tively; and SH,H(q), SH,He(q), and SHe,He(q) are the par-
tial proton, proton-atom and atomic, respectively, pair
structure factors [26, 27].
As was shown in works [13, 14], the third-order, with
respect to the electron-proton interaction potential, term
is essential only in the vicinity of the transition point of
hydrogen into the metallic state. Its absolute and rel-
ative values quickly fall down with the growth of the
density. Already at a density of the order of 1 g/cm3,
it amounts to a few percent of the second-order term.
At densities of 5–7 g/cm3 which are characteristic of
the central regions of giant planets, it can be totally
neglected. It is much more true for the third-order,
with respect to the electron-atomic interaction potential,
term, because it is much less than the electron-proton-
interaction term of the third order at every density.
According to the definition of free energy
F = E − TS, (4.7)
S is the entropy of the system. It can be taken in the
hard-sphere approximation [26, 27],
S = Sgas + S(η), (4.8)
where
Sgas =
5
2
+
3
2
ln
(
MC
HM
1−C
He kBT
2πn2/3
)
−
−c ln(c)− (1− c) ln(1− c) (4.9)
is the entropy of ideal proton and atomic gases, n is its
density,
S(η)/kB = −2 ln(1− η) + 6
(
1− 1
1− η
)
+
+15c(1− c)η(1− λ)1.7 (4.10)
is the interpolation formula for the contribution associ-
ated with the interaction between hard spheres, η is the
total packing density for protons and atoms, and λ is
the ratio between the diameters of proton- and atom-
simulating hard spheres.
The only system parameter in the hard-sphere models
is the packing densities for protons and atoms which
are directly expressed in terms of corresponding hard
sphere diameters. To find them, the idea of effective
proton-proton pair interaction is used [28]. Its important
property is that it contains no fitting parameters, but
depends only on the system density. The hard sphere
diameter, i.e. the minimal approach distance for protons
at a given temperature, is determined from the condition
of equality between the kinetic and potential energies
of two protons at their approach to each other. The
hard sphere diameter and the packing density for helium
atoms are determined in much the same way [29].
5. Discussion of Results
In Fig. 4, the distributions of the pressure on Jupiter
calculated in various approximations are depicted. The
planet substance was assumed to consist of two compo-
nents: hydrogen and helium. Hydrogen was assumed
to completely dissociate into protons and electrons, and
helium to remain in the non-ionized atomic state. This
assumption is true only for the central region of the
planet, provided that the distance to the center does
no exceed 0.8 times the planet radius. The planet sub-
stance is considered as chemically homogeneous. The
only unknown parameter of the system is the electron
gas concentration. In the framework of the microscopic
model, this parameter governs both the planet tempera-
ture and the pressure. For the further consideration, we
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 2 255
V.T. SHVETS, T.V. SHVETS, S.YE. RACHYBSKIY
Fig. 4. Jupiter pressure calculated in the second order of perturba-
tion theory with respect to the electron-proton and electron-atom
interactions
select the value of this parameter that provides an equal-
ity between the pressures obtained in the framework of
the polytropic model of the planet and calculated in the
framework of microscopic model for a planet, which takes
into account every interaction; both pressures are cal-
culated at the planet center, where, owing to a high
density of the electron gas, the role of interactions is
the least. The corresponding electron concentration is
equal to 0.747. In Fig. 5, the pressure distribution cal-
culated using this value and the equation of state for
the ideal degenerate electron gas is shown. It is evident
that the model of ideal degenerate electron gas and the
model of interacting electron-atom-proton liquid corre-
spond rather well to the equation of state obtained in the
framework of the polytropic model. At the same time,
whereas the equation of state for the ideal degenerate
electron gas better describes the pressure profile closer
to the planet surface, the microscopic equation of state
better describes the behavior of the equation of state,
which was obtained in the framework of the polytropic
model, closer to the planet center. In particular, the
temperature at the Jupiter center, which corresponds to
a concentration of 0.747, turns out to be 14000 K.
Note that the difference between pressures obtained in
the framework of either microscopic or polytropic model
of Jupiter testifies that the equation of state for an in-
teracting system is not a simple power-law dependence.
In the central region of Jupiter, where the electron gas
density is extremely high, this difference is almost un-
noticeable. When approaching the planet surface, the
Fig. 5. Jupiter pressure calculated in the second order of perturba-
tion theory with respect to the electron-proton and electron-atom
interactions
electron gas density quickly falls down, and this differ-
ence becomes substantial.
In Fig. 5, the solutions of the equation of state ob-
tained in the framework of the polytropic, microscopic
(taking interactions into account), and ideal-electron-gas
models for Saturn are depicted. The electron gas con-
centration on the planet was taken to be 0.64. This
concentration gives rise to a temperature of 4000 K at
the planet center. For Saturn, the radius of the region,
where the conditions are favorable for the hydrogen met-
allization, does not exceed 0.6 of the planet radius in
the polytropic model and 0.5 of the planet radius in the
microscopic model with interactions. Respectively, the
mass of the planet substance that is concentrated in the
metallization region amounts to 0.6 of the planet mass in
the first case and to 0.5 of the planet mass in the second
one. Hence, in the case of Saturn, the proposed micro-
scopic model describes a considerably smaller region of
the planet than that in the case of Jupiter. Accordingly,
the accuracy of such a description is lower as well.
6. Conclusions
The model of almost free electrons is extremely good for
the description of a substance under conditions that take
place in the central regions of giant planets. The pertur-
bation theory series of expansion in the electron-proton
and electron-atomic interactions quickly converges. The
contribution of interactions in a proton-helium system to
the equation of state falls within the interval of 15–20%.
256 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 2
EQUATION OF STATE FOR METALLIC HYDROGEN
The concentration of conduction electrons for Jupiter
amounts to 0.747, which testifies to the presence of many
other substances (approximately 25%) – first of all, he-
lium – in the Jupiter structure. The helium fraction
in the central region of the planet is much higher than
18%, which is typical of the Jupiter atmosphere. The
polytropic model is well substantiated for Jupiter.
The concentration of conduction electrons for Saturn
amounts to 0.64, which testifies to the presence of plenty
of other substances (about 35%) in the planet structure.
The helium fraction in the central part of the planet
has to be much higher than 11%, this value being typi-
cal of the Saturn atmosphere. The lower concentration
of helium in the Saturn atmosphere than that in the
Jupiter one gives rise to a lower concentration of he-
lium in the central part of the planet. Hence, it cannot
exceed 25%, the value characteristic of Jupiter. There-
fore, the fraction of other substances in the Saturn struc-
ture is rather substantial. This circumstance makes the
hydrogen-helium model used in this work rather approx-
imative for Saturn.
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Translated from Ukrainian by O.I. Voitenko
РIВНЯННЯ СТАНУ МЕТАЛЕВОГО ВОДНЮ
ТА АТОМАРНОГО ГЕЛIЮ I ВНУТРIШНЯ
БУДОВА ЮПIТЕРА ТА САТУРНА
В.Т. Швець, Т.В. Швець, С.Є. Рачинський
Р е з ю м е
Запропоновано рiвняння стану сумiшi металевого водню та
атомарного гелiю. Дослiджений iнтервал тискiв, температур
i густин вiдповiдає умовам найбiльших планет сонячної систе-
ми – Юпiтера i Сатурна. Вважається, що речовина планети
являє собою сумiш протонiв, атомiв гелiю та електронiв. Для
знаходження тиску сумiшi використовується теорiя збурень за
електрон-протонною та електрон-атомною взаємодiями. Еле-
ктронна пiдсистема розглядається у наближеннi випадкових
фаз. Взаємодiї протонiв, атомiв, атомiв i протонiв враховую-
ться у наближеннi твердих сфер. Проаналiзовано застосовнiсть
моделi полiтропного шару до моделювання внутрiшньої будо-
ви Юпiтера i Сатурна та запропоновано конкретне значення
iндексу полiтропи. Знайдено густину, тиск i температуру на
Юпiтерi та Сатурнi як функцiї вiдстанi до їх центра. Оцiнено
можливi долi водню i гелiю у складi планет.
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 2 257
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