6 Internal structure constant

Observations of binary systems allow us to determine the internal structure constant, k₂, which is related to the variation of density within the star and is an important parameter of stellar evolution models. It is best studied in binary systems with eccentric orbits that show apsidal motion. The period of rotation of the periastron in eccentric eclipsing binaries does not allow us to derive the individual internal stellar constant of the component stars.

 

 

Table 4: Triple eccentric eclipsing binaries with the AM and LTE.

System	Spectral	e	P	U	e3	P3	Source
	type		[days]	[years]		[years]
HS Her	B4+A4	0.021	1.637	72.4	0.7	74.1	Wolf (2001)
U Oph	B5+B5	0.003	1.677	21.2	0.22	38.7	Kämper (1986)
DR Vul	B0+B0.5	0.095	2.251	36.3	0.73	62.8	Wolf et al. (1999)
YY Sgr	B6+B5	0.159	2.628	290.4	0.44	44.3	Wolf (2000)
RU Mon	B7+B7	0.396	3.585	347	0.46	73.3	Wolf et al. (1999)
TV Cet	F2+F5	0.055	9.103	30000	0.25	28.5	this paper

The observational average value of $k_{2, \rm obs}$ is given by the relation

$$k_{2, \rm obs} = \frac{1}{c_{21} + c_{22}} \, \frac{P_{\rm a}}{U} = \frac{1}{c_{21} + c_{22}} \, \frac{\dot{\omega}}{360} ,$$ (3)

where c₂₁ and c₂₂ are functions of the orbital eccentricity, fractional radii, the masses of the components and the ratio between rotational velocity of the stars and Keplerian velocity (Kopal 1978).

Taking into account the value of the eccentricity and the masses of the components, one has to subtract from $\dot{\omega}$ a relativistic correction $\dot{\omega}_{\rm rel}$ (Levi-Civita 1937; Giménez 1985)

$$\dot{\omega}_{\rm rel} = \; 5.45 \times 10^{-4} \: \frac{1}{1-e^2} \: \Bigl( \frac{M_1 + M_2}{P} \Bigr) ^{2/3},$$ (4)

where M_i denotes the individual masses of the components in solar units, or

$$\dot{\omega}_{\rm rel} = \; 1.2 \times 10^{-8} \: \Bigl( \frac {K_1 + K_2} {\sin i} \Bigr) ^2,$$ (5)

if K_i is expressed in kms^-1.

The values of $\dot{\omega}_{\rm rel}$ and resulting mean internal structure constants $k_{2, \rm obs}$ are given in Table 3. Theoretical values $k_{2, \rm theo}$ according to available theoretical models for the internal stellar structure along the main sequence, computed by Claret & Giménez (1992) for a variety of masses and chemical compositions, are also given.