Barometric Formula for Ultrarelativistic Degenerate Fermi-Gases
- PDF / 88,967 Bytes
- 5 Pages / 594 x 792 pts Page_size
- 15 Downloads / 243 Views
BRIEF COMMUNICATIONS
BAROMETRIC FORMULA FOR ULTRARELATIVISTIC DEGENERATE FERMI-GASES
A. E. Dubinov
An exact explicit barometric formula is derived for an ultrarelativistic degenerate Fermi-gas governed by the Chandrasekhar equation of state. Keywords: ultrarelativistic degenerate Fermi-gas: Chandrasekhar equation of state: barometric formula
1. Introduction
The barometric formula for an ideal gas relates the gas density n(x) in a force potential field U(x) to the height x for a specified temperature T and the concentration n0 at height x = 0. In a uniform gravitational field the barometric formula has the form [1,2]
(1) RFNC-VNIIEF, Sarov, Russia (2) SarFTI NRNU MEPhI, Sarov, Nizhnyi Novgorod Region, Russia, e-mail: [email protected]
Original article submitted July 2, 2020. Translated from Astrofizika, Vol. 63, No. 4, pp. 663-666 (November 2020) 580
0571-7256/19/6304-0580 ©2020 Springer Science+Business Media, LLC
§ mgx · n x n0 exp¨ ¸, © kT ¹
(1)
where m is the mass of the gas particles, g is the acceleration of gravity, and k is the Boltzmann constant. In practice it is often sufficient to know the dependence of the density on the potential energy of the particles in the force field. In this case the barometrric formula takes the form of the well known Boltzmann distribution,
§ U · n U n0 exp¨ ¸ © kT ¹
(2)
for a gas with the equation of state p = nkT at a fixed temperature (the isothermal case), where p is the gas pressure. Barometric formulas and distributions are known for other equations of state of gases. For example, an analogous formula for an adiabatic equation of state is used to describe processes in plasmas [3,4]; a barometric formula has been obtained and analyzed for a van-der-Waals gas [5]; a barometric formula has been derived [6] for a warm Fermi-gas, which has been applied in the form of a distribution in theories of ion acoustic waves in plasmas with quantum-degenerate electrons [7,8]. Chandrasekhar [9,10] obtained an equation of state for an ultrarelativistic degenerate gas, i.e., a gas with a Fermi energy substantially higher than the rest energy of the particles. This equation has been used repeatedly to describe processes in the interior of white dwarfs [11,12] and in theories of wave processes in ultrarelativistic degenerate plasmas [13-20]. Nevertheless, a barometric formula for a gas with the Chandrasekhar equation of state has not been reported anywhere. This article reports a simple derivation of the barometric formula for an ultrarelativistic degenerate gas obeying the Chandrasekhar equation of state,
P
in which A
A ªK 2K2 3 1 K2 3arcsinh K º , «¬ »¼
S m 4 c 5 3 h 3 is a coefficient and K
pF mc
3
3 h3n 8S m 3 c 3
3
(3)
n is the relativistic degeneracy parameter, nF
where P is the gas pressure, n is the density of the gas particles, m is their mass, pF is the relativistic Fermi momentum, nF is the density of the gas particles for which the Fermi momentum equals mc, c is the speed of light, and h is the Planck constant.
2. Derivation of the ba
Data Loading...
