The second expression in (D.1.4) is a gray-body approximation. A correction for different-sized components made use of a theorem demonstrated by Russell (1948) that the light distribution as given by (D.1.3) is the same for a uniformly bright star of surface brightness J0(1 - 1 x) and having axes 1 + Na, 1 + Nb, and 1 + Nc, where n = (15 + x)(1 - 4K+2 y) ^ (15 + x)(1 + y) 1 5)

15 - 5x 15 - 5x and where the factor multiplying y is in the range -0.996 to -0.957 for the expected range of K (0.0018-0.018, respectively).

The effect of the ellipticity on the light curve is to provide an observed variation in system flux:

Icomp = (1 - 2 Nz cos2 0) (Imax - IW f - ^ ./2). (D.1.6)

where z = n2 sin2 i, and the observed flux is

The coefficient A2 is intrinsically negative, and Russell & Merrill (1952, p. 43)

express the peak light as lmax = A0 - A2, and Nz = -—. The rectified

light then becomes

Irect = (.A°, A2)lc0mi = lmax(1 - L1 f - L2fz), (D.1.8)

l1 l2

where L1 = -1— and L2 = -1—. A rectification in phase is also required because lmax lmax of the ellipticity of the components; the rectified phase is computed from sin2 0 = ' . (D.1.9)

sin2 0

This completes the rectification due to the deviation of the stars from sphericity. The effects of gravity brightening and reflection must be treated now. The effect of centrifugal forces is to widen the equatorial axes at the expense of the polar one and causes the star to brighten toward the rotation poles; the effect of tidal distortion (increasing the star's diameter along the line of centers) is to darken it at the extremes of the axis a. The net center-to-limb variation across an ellipsoid is given by (D.1.3). Express the coordinates in the directions of the ellipsoid axes as X, Y, and Z and take the gravity-darkening relative to the Y-axis, a = b(1 + u) , c = b(1 - v), so that (D.1.3) may be written as

As cos y is the direction cosine of the normal to the ellipsoid at the position X, cos y = rX/a2. Setting X = a cos ß, and neglecting second-order terms in Z and u, the relative surface brightness J in (D.1.10) becomes

J = Jo {1 - x + x cos p - u\x cos p + 4 y(1 - x )cos2 p + (4 y - 1)x cos3 p]}.

The theoretical basis for the treatment of the reflection effect is that of Milne (1926) who assumed a parallel beam of incoming radiation, with corrections from Sen (1948) for penumbral effects. The expected enhancement in light from the second star "reflection" is

where an approximation may be used for f (e)

Here e is the phase angle from full phase (i.e., from mid-secondary minimum for star 2). The net result of the reflection effect alone (Russell & Merrill 1952, p. 45) is

= (Lcr2h + Lhrc2)(0.30 - 0.10 cos2 i + 0.10 sin2 i cos 20)

Note that both ellipticity and reflection contribute cos 20 values, but with opposite sign, and the effect of reflection is to decrease the effect of ellipticity. The cos 0 contribution is due, however, to the reflection effect. Note that this is true for Russell's ellipsoid model, but not for an equipotential model.

Instead of removing the enhancement due to the reflection effect the prescription requires the addition of light at other phases. It is done this way to avoid iterative rectification. Given a situation in which the light curve outside the eclipse is well represented by a truncated Fourier series involving only terms A0-4, and star 2 is heated by star 1, the total radiation reemitted by star 2 shows up in the effects on coefficients A3 and A4 as

assuming r2 < 0.4 and

A4 = (0.002 - 0.050r22 - 0.006r?)L 1r22 < 0.0012L 1 (D.1.16)

(Russell & Merrill 1952, p. 52). For cases, where the distortion of the figures of the stars is large, Russell and Merrill cite Kopal (1946, Eqs. (210) and (220.1) on pp. 135 and 139):

A crude form of the mass-luminosity relation (m a L0 26) was then used to evaluate the magnitude of the expressions. As A3 <0.33r^ and thus 0.001 < A3 < 0.01 for 0.23 < r2 < 0.41 and A4 < 0.16r25, so that A4 < 0.001 for r2 < 0.36 .

For cases where both sine and cosine terms must be used to represent the maxima, the general prescription is to apply "empirical corrections" (Russell & Merrill 1952, pp. 53-54). They recommended dividing by the expression

if the light perturbations are thought to affect both stars in proportion to their brightnesses, but otherwise producing no shape or surface brightness asymmetry. If star 1 alone were affected, they recommended subtraction of the sine terms from all phases but those at which star 1 was eclipsed, and during the eclipse of star 1, the subtraction of (1 - f)(B1 sin 0 + B2 sin 20), where f is the fraction of that star's light which is obscured at any particular phase.

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