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# VbvAstE-001

Book Boris V. Vasiliev Astrophysics

## where C Ω is a

where C Ω is a constant. Thus and E x = 2 C Ω x, E y = 2 C Ω y, E z = −4 C Ω z (8.10) and we obtain important equations: div E Ω = 0 (8.11) ρ Ω = 0; (8.12) γg Ω = ρE Ω. (8.13) Since centrifugal force must be contra-balanced by electric force and γ 2Ω 2 x = ρ 2C Ω x (8.14) C Ω = γ Ω2 ρ The potential of a positive uniform charged ball is ϕ(r) = Q ( ) 3 R 2 − r2 2R 2 = Ω2 √ G (8.15) (8.16) The negative charge on the surface of a sphere induces inside the sphere the potential ϕ(R) = − Q R (8.17) where according to Eq.(8.4) Q = √ GM, and M is the mass of the star. Thus the total potential inside the considered star is √ ) GM ϕ Σ = (1 − r2 + √ Ω2 r 2 (3cos 2 θ − 1) (8.18) 2R R 2 G Since the electric potential must be equal to zero on the surface of the star, at r = a and r = c ϕ Σ = 0 (8.19) and we obtain the equation which describes the equilibrium form of the core of a rotating star (at a2 −c 2 a 2 ≪ 1) a 2 − c 2 ≈ 9 Ω 2 a 2 2π Gγ . (8.20) 63

8.3 The angular velocity of the apsidal rotation Taking into account of Eq.(8.20) we have ω Ω ≈ 27 Ω 2 (8.21) 20π Gγ If both stars of a close pair induce a rotation of periastron, this equation transforms to ω Ω ≈ 27 20π ( Ω 2 1 + 1 ) , (8.22) G γ 1 γ 2 where γ 1 and γ 2 are densities of star cores. The equilibrium density of star cores is known (Eq.(2.18)): γ = 16 A Z 3 9π 2 Z mp . (8.23) If we introduce the period of ellipsoidal rotation P = 2π Ω rotation of periastron U = 2π , we obtain from Eq.(8.21) ω a 3 B and the period of the where and ( ) 2 P P ≈ U T 2∑ ξ i, (8.24) 1 √ 243 π 3 T = τ 0 ≈ 10τ 0, (8.25) 80 √ a 3 B τ 0 = ≈ 7.7 · 10 2 sec (8.26) G m p ξ i = Z i A i(Z i + 1) 3 . (8.27) 8.4 The comparison of the calculated angular velocity of the periastron rotation with observations Because the substance density (Eq.(8.23)) is depending approximately on the second power of the nuclear charge, the periastron movement of stars consisting of heavy elements will fall out from the observation as it is very slow. Practically the obtained equation (8.24) shows that it is possible to observe the periastron rotation of a star consisting of light elements only. The value ξ = Z/[AZ 3 ] is equal to 1/8 for hydrogen, 0.0625 for deuterium, 1.85 · 10 −2 for helium. The resulting value of the periastron rotation of double stars will be 64

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